Documento de Trabajo - 2016/07 The Pruned State-Space System for Non-Linear DSGE Models: Theory and Empirical Applications

Martin M. Andreasen

(Aarhus University and CREATES)

Jesús Fernández-Villaverde

(University of Pennsylvania, NBER, and CEPR)

Juan F. Rubio-Ramírez

(Emory University, Federal Reserve Bank of Atlanta, and FEDEA)

fedea

Martin M. Andreasen

Jesús Fernández-Villaverde

Aarhus University and CREATES

University of Pennsylvania, NBER, and CEPR

Juan F. Rubio-Ramírez

Emory University, Federal Reserve Bank of Atlanta, and FEDEA

April 21, 2016

Abstract

This paper studies the pruned state-space system for higher-order perturbation approximations to DSGE models. We show the stability of the pruned approximation up to third order and provide closed-form expressions for first and second unconditional moments and impulse response functions. Our results introduce GMM estimation and impulse-response matching for DSGE models approximated up to third order and provide a foundation for indirect inference and SMM. As an application, we consider a New Keynesian model with Epstein-Zin-Weil preferences and two novel feedback effects from long-term bonds to the real economy, allowing us to match the level and variability of the 10-year term premium in the U.S. with a low relative risk aversion of 5.

Keywords: Epstein-Zin-Weil preferences, Feedback-effects from long-term bonds, Higher-order perturbation approximation, Yield curve. JEL: C15, C53, E30.

*We thank Mads Dang, Frank Diebold, Robert Kollmann, Dirk Krueger, Johannes Pfeifer, Eric Renault, Francisco Ruge-Murcia, Frank Schorfheide, Eric Swanson, the editor, and referees for useful comments. Remarks and suggestions from seminar participants at numerous institutions are much appreciated. Beyond the usual disclaimer, we must note that any views expressed herein are those of the authors and not necessarily those of the Federal Reserve Bank of Atlanta or the Federal Reserve System. We acknowledge access to computer facilities provided by the Danish Center for Scientific Computing (DCSC). We acknowledge support from CREATES - Center for Research in Econometric Analysis of Time Series (DNRF78), funded by the Danish National Research Foundation. Finally, we also thank the NSF for financial support.

Corresponding author: Fuglesangs Allé 4, 8210 Aarhus V, Denmark, email: mandreasen@econ.au.dk; telephone +0045 87165982.

1 Introduction

The perturbation method approximates the solution to dynamic stochastic general equilibrium (DSGE) models by higher-order Taylor series expansions around the steady state (see Judd and Guu (1997) and Schmitt-Grohé and Uribe (2004), among others). These approximations have grown in popularity, mainly because they allow researchers to quickly and accurately solve DSGE models with many state variables and inherent non-linearities to analyze uncertainty shocks or time-varying risk premia (see Fernández-Villaverde, Guerrón-Quintana, Rubio-Ramírez and Uribe (2011) and Rudebusch and Swanson (2012), among others).

Although higher-order approximations are intuitive and straightforward to compute, they often generate explosive sample paths even when the corresponding linearized solution is stable. As noted by Kim, Kim, Schaumburg and Sims (2008), these explosive sample paths arise because the higher-order terms generate unstable steady states in the approximated system. The presence of explosive behavior complicates any model evaluation because no unconditional moments exist in this approximation. It also means that any estimation method using unconditional moments, such as the generalized method of moments (GMM) or the simulated method of moments (SMM), is inapplicable because it relies on finite moments from stationary and ergodic probability distributions.

For second-order approximations, Kim, Kim, Schaumburg and Sims (2008) suggest eliminating explosive sample paths by applying a pruning method that omits terms of higher-order effects than the considered approximation order when the system is iterated forward in time. To illustrate the idea, suppose we have a solution for capital that depends on a quadratic function of , as present in many DSGE models solved to second order. If we iterate this equation one period forward and substitute for its own quadratic function of , we obtain an expression for that depends on , , , and . The pruning method omits the terms and capturing third- and fourth-order effects to obtain a second-order approximation of when expressed as a function of the current state variable .

This paper extends the pruning method to perturbation approximations of any order and shows how pruning greatly facilitates inference of DSGE models. Special attention is devoted to second- and third-order approximations, which are widely used. We first show that our pruning method ensures stable sample paths, provided the linearized solution is stable. Given this key result, we then provide closed-form solutions for first and second unconditional moments and impulse response functions (IRFs). We also derive conditions for the existence of third and fourth unconditional moments to compute skewness and kurtosis.

Ruge-Murcia (2013) reviews the use of GMM in the context of DSGE models. Non-explosive sample paths are also required for likelihood methods, for instance, when using the particle filter outlined in Fernández-Villaverde and Rubio-Ramírez (2007).

The econometric implications of these results are significant as most of the existing moment-based estimation methods for linearized DSGE models now carry over to non-linear approximations. For models solved up to third order, this includes GMM estimation based on first and second unconditional moments and matching model-implied IRFs to their empirical counterparts. Our results are also useful when estimating DSGE models using Bayesian methods, for instance, when conducting inference using a limited information likelihood function from unconditional moments, as suggested by Kim (2002), or when doing posterior model evaluations on unconditional moments, as in An and Schorfheide (2007). If simulations are needed to calculate higher-order unconditional moments such as skewness or kurtosis, then our results provide a foundation for SMM as in Duffie and Singleton (1993) and different types of indirect inference as in Smith (1993). Finally, our results are also relevant to researchers who prefer to calibrate their models as in Cooley and Prescott (1995), because the unconditional mean of a model solved with higher-order terms generally differs from its steady-state value. Given our results, researchers can now easily correct for these higher-order effects and non-linearly calibrate their models.

The suggested GMM estimation approach, its Bayesian equivalent, non-linear calibration, and IRF matching are promising because we can compute first and second unconditional moments or IRFs in a trivial amount of time for medium-size DSGE models solved up to third order. For the model described in Section 8 with seven state variables, it takes 0.75 second to find all first and second unconditional moments and only 0.08 second to compute the IRFs for 20 periods following a shock on an off-the-shelf laptop.

Matlab codes to implement our procedures are available on the authors' home pages; see, for instance, https://sites.google.com/site/mandreasendk/home-1. We also note that Dynare 4.4.0. has implemented our pruning method to simulate models approximated to third order.

Some papers in the literature have accounted for the difference between the steady state and the mean of the ergodic distribution by simulation; see, for instance, Fernández-Villaverde, Guerrón-Quintana, Rubio-Ramírez and Uribe (2011). These simulations are, however, computationally demanding, in particular, for very persistent processes, where a long sample path is required to accurately compute unconditional moments.

An application illustrates some of the new techniques that our paper makes available for DSGE models. We consider a rich New Keynesian economy with Calvo pricing, consumption habits, and Epstein-Zin-Weil preferences, which we estimate by GMM using first and second unconditional moments for the U.S. yield curve and five macro variables. Our New Keynesian model introduces two novel mechanisms that help us to improve our understanding of the interactions between financial markets, monetary policy, and the real economy. First, households deposit their savings in a financial intermediary. This financial intermediary invests in short- and long-term bonds and creates a wedge between the policy rate set by the monetary authority and the interest rate on deposits. Second, we augment the standard Taylor rule of the monetary authority to include the excess return in a longer-term bond, which is closely related to term premia. The first mechanism captures the frictions in the financial markets that induce differences between the policy rate and the interest rate faced by private agents. The second mechanism captures the observation that central banks also react to term premia, as seen during the recent financial crisis. Our two mechanisms depend on the degree of precautionary behavior and, therefore, are only operative when the model is solved using a third-order approximation. Thus, the methods derived in the present paper are essential for the quantitative analysis of the model.

Our model matches the mean and variability of the 10-year term premium with a reasonable risk aversion of 5, while simultaneously matching key moments for standard real macro variables. We illustrate the importance of a positive steady-state inflation in driving this result, as it amplifies the non-linearities in the price dispersion index related to Calvo pricing and produces the desired conditional heteroscedasticity in the stochastic discount factor. Notably, an unpruned third-order approximation to our model gives explosive sample paths and is, therefore, unable to “see” this novel channel for term premia volatility, which we uncover when using our pruning method. Thus, our model and our pruning method go a long way in resolving the bond risk premium puzzle described in Rudebusch and Swanson (2008) without postulating highly risk-averse households, as in much of the existing literature.

The rest of the paper is structured as follows. Section 2 introduces the problem. Section 3 presents the pruning method and the pruned state-space system for approximated DSGE models. Stability and unconditional moments of the pruned state-space system for second- and third-order approximations are derived in Section 4, with the closed-form expressions for the IRFs deferred to

Section 5. Section 6 studies the accuracy of the pruning method, and we discuss the econometric implications of the pruned state-space system in Section 7. Section 8 is devoted to our empirical application. Section 9 concludes. Detailed derivations and proofs are deferred to the Appendix and a longer Online Appendix available on the authors' home pages or on request.

2 The State-Space System

We consider the following class of DSGE models. Let be a vector of control variables, a vector of state variables, and an auxiliary perturbation parameter. To simplify the notation below, and are expressed in deviations from their steady state. The exact solution to the DSGE model is given by the state-space system

\[\mathbf {y} _ {t} = \mathbf {g} (\mathbf {x} _ {t}, \sigma),\tag{1}\]

\[\mathbf {x} _ {t + 1} = \mathbf {h} (\mathbf {x} _ {t}, \sigma) + \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1},\tag{2}\]

where contains the exogenous zero-mean innovations. We refer to (1) and (2) as the observation and state equations, respectively. Initially, we do not impose a distributional form for the innovations. In particular, the innovations may be non-Gaussian. We only assume that is independent and identically distributed with finite second moments, denoted by . Additional moment restrictions will be imposed later in the paper. The perturbation parameter scales the square root of the covariance matrix for the innovations having dimension .⁴

In general, DSGE models do not have a closed-form solution and the functions and cannot be found explicitly. The perturbation method is a popular way to obtain Taylor series expansions to these functions around the steady state. When the functions and are solved up to first order, the state-space system is approximated by and in (1) and (2), respectively. Here, is an matrix with first-order derivatives of with respect to and is an matrix with first-order derivatives of with respect to .⁵ Given our assumptions about , this system has finite first and second unconditional moments if all eigenvalues of have modulus less than one. Furthermore, the approximated state-space system fluctuates around the steady state, which also corresponds to the unconditional mean. It is, therefore, straightforward to calibrate the structural parameters in the DSGE model from unconditional first and second moments or carry out a formal estimation using existing econometric tools for Bayesian inference, maximum likelihood, GMM, SMM, etc. (see Ruge-Murcia (2007)).

The assumption that innovations enter linearly in (2) may appear restrictive but is without loss of generality. As shown in Appendix A.1, the state vector can be extended to deal with non-linearities between and .

The first-order derivatives and of and with respect to are known to be zero (see Schmitt-Grohé and Uribe (2004)).

When the functions and are approximated beyond linearization, we could, in principle, apply the same method to construct the approximated state-space system with their higher-order Taylor series expansions. However, the resulting approximated state-space system cannot, in general, be shown to have any finite unconditional moments and may even display explosive dynamics. This occurs even when we simulate simple versions of the New Keynesian model with few endogenous state variables. Hence, it is hard to use this approximated state-space system to calibrate or even estimate model parameters. Consequently, it is useful to construct another approximated state-space system that has well-defined statistical properties when analyzing DSGE models solved beyond linearization. We explain now how this can be done.

3 The Pruning Method

Kim, Kim, Schaumburg and Sims (2008) suggest using a pruning method to construct the approximated state-space system for DSGE models solved to second order. We will refer to this approach as the pruned state-space system. Section 3.1 reviews the pruning method and explains its logic for the second-order approximation. Section 3.2 extends the method to a third-order approximation. The general procedure for constructing the pruned state-space system for any approximation order is straightforward, but deferred to Appendix A.2 in the interest of space. We finally relate our approach to the existing literature in Section 3.3.

3.1 Second-Order Approximation

The first step when constructing the pruned state-space system for the second-order approximation is to decompose the state variables into first-order effects and second-order effects as follows.

We start from the second-order Taylor series expansion of the state equation

\[\mathbf {x} _ {t + 1} ^ {(2)} = \mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t} ^ {(2)} + \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t} ^ {(2)} \otimes \mathbf {x} _ {t} ^ {(2)}\right) + \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2} + \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1},\tag{3}\]

where is the unpruned second-order approximation to the state variables. Here, is an matrix with the derivatives of with respect to and is an matrix containing derivatives taken with respect to . Substituting with into the right-hand side of (3) gives

\[\mathbf {h} _ {\mathbf {x}} \left(\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s}\right) + \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \left(\left(\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s}\right) \otimes \left(\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s}\right)\right) + \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2} + \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1}.\tag{4}\]

A law of motion for is derived by preserving only first-order effects in (4). We keep the first-order effects from the previous period and the innovations to obtain

\[\mathbf {x} _ {t + 1} ^ {f} = \mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t} ^ {f} + \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1}.\tag{5}\]

This expression for is the standard first-order approximation to the state equation. Note that is a polynomial in that only includes first-order terms. The first-order approximation to the observation equation is also standard and given by

\[\mathbf {y} _ {t} ^ {f} = \mathbf {g} _ {\mathbf {x}} \mathbf {x} _ {t} ^ {f}.\tag{6}\]

Accordingly, the pruned state-space system for the first-order approximation is given by (5) and (6), meaning that the pruned and unpruned state-space systems are identical in this case.

A law of motion for is derived by preserving only second-order effects in (4). Here, we include the second-order effects from the previous period , the squared first-order effects in the previous period , and the correction . Hence,

\[\mathbf {x} _ {t + 1} ^ {s} = \mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t} ^ {s} + \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) + \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}.\tag{7}\]

Equation (3) adopts the standard assumption that the model has a unique stable first-order approximation, which implies that all second- and higher-order terms are also unique (see Judd and Guu (1997) and Lan and Meyer-Gohde (2014)).

We do not include terms with and because they reflect third- and fourth-order effects, respectively. Note that is a polynomial in that only includes second-order terms.

The final step in setting up the pruned state-space system is to derive the expression for the observation equation. Using the same approach as above, we start from the second-order Taylor series expansion of the observation equation

\[\mathbf {y} _ {t} ^ {(2)} = \mathbf {g} _ {\mathbf {x}} \mathbf {x} _ {t} ^ {(2)} + \frac {1}{2} \mathbf {G} _ {\mathbf {x x}} \left(\mathbf {x} _ {t} ^ {(2)} \otimes \mathbf {x} _ {t} ^ {(2)}\right) + \frac {1}{2} \mathbf {g} _ {\sigma \sigma} \sigma^ {2},\tag{8}\]

where denotes the unpruned second-order approximation to the control variables. Here, is an matrix with the corresponding derivatives of with respect to and is an matrix containing derivatives with respect to . We only want to preserve effects up to second order, meaning that the pruned approximation to the control variables is given by

\[\mathbf {y} _ {t} ^ {s} = \mathbf {g} _ {\mathbf {x}} \left(\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s}\right) + \frac {1}{2} \mathbf {G} _ {\mathbf {x x}} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) + \frac {1}{2} \mathbf {g} _ {\sigma \sigma} \sigma^ {2}.\tag{9}\]

Here, we leave out terms with and because they reflect third- and fourth-order effects, respectively. To simplify notation, we treat as the sum of the first- and second-order effects, while only contains the second-order effects. Hence, is a polynomial in that includes all first- and second-order terms.

Accordingly, the pruned state-space system for the second-order approximation is given by (5), (7), and (9). The state vector in this system is thus extended to as we separately track first- and second-order effects. For completeness, the unpruned state-space system for the second-order approximation is given by (3) and (8).

3.2 Third-Order Approximation

We now construct the pruned state-space system for the third-order approximation. Following the steps outlined above, we start by decomposing the state variables into first-order effects , second-order effects , and third-order effects . The laws of motion for and are the same as in the previous section, and only the recursion for remains to be derived. The third-order Taylor series expansion to the state equation is (see Ruge-Murcia (2012))

Lan and Meyer-Gohde (2013b) employ perturbation to derive a stable non-linear approximation of and in terms of past innovations. By using (5), (7), and (9), we can also express as an infinite moving average in terms of past innovations.

\[\begin{array}{r c l} \mathbf {x} _ {t + 1} ^ {(3)} & = & \mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t} ^ {(3)} + \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t} ^ {(3)} \otimes \mathbf {x} _ {t} ^ {(3)}\right) + \frac {1}{6} \mathbf {H} _ {\mathbf {x x x}} \left(\mathbf {x} _ {t} ^ {(3)} \otimes \mathbf {x} _ {t} ^ {(3)} \otimes \mathbf {x} _ {t} ^ {(3)}\right) \\ & & + \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2} + \frac {3}{6} \mathbf {h} _ {\sigma \sigma \mathbf {x}} \sigma^ {2} \mathbf {x} _ {t} ^ {(3)} + \frac {1}{6} \mathbf {h} _ {\sigma \sigma \sigma} \sigma^ {3} + \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1}, \end{array}\tag{10}\]

where represents the unpruned third-order approximation to the state variables. Here, denotes an matrix containing derivatives of with respect to , is an matrix including derivatives with respect to , and is an matrix containing derivatives related to . We adopt the same procedure as before and substitute into the right-hand side of (10) to obtain

\[\begin{array}{r l} & {\mathbf {h _ {x}} \left(\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d}\right) + \frac {1}{2} \mathbf {H _ {x x}} \left(\left(\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d}\right) \otimes \left(\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d}\right)\right)} \\ & {+ \frac {1}{6} \mathbf {H _ {x x x}} \left(\left(\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d}\right) \otimes \left(\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {\mathrm{rd}}\right) \otimes \left(\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {\mathrm{rd}}\right)\right)} \\ & {+ \frac {1}{2} \mathbf {h _ {\sigma \sigma}} \sigma^ {2} + \frac {3}{6} \mathbf {h _ {\sigma \sigma x}} \sigma^ {2} \left(\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {\mathrm{rd}}\right) + \frac {1}{6} \mathbf {h _ {\sigma \sigma \sigma}} \sigma^ {3} + \sigma \pmb {\eta} \pmb {\epsilon_ {t + 1}}.} \end{array}\tag{11}\]

A law of motion for the third-order effects is derived by preserving only third-order terms in (11):

\[\mathbf {x} _ {t + 1} ^ {r d} = \mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t} ^ {r d} + \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {s}\right) + \frac {1}{6} \mathbf {H} _ {\mathbf {x x x}} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} ^ {f}\right) + \frac {3}{6} \mathbf {h} _ {\sigma \sigma \mathbf {x}} \sigma^ {2} \mathbf {x} _ {t} ^ {f} + \frac {1}{6} \mathbf {h} _ {\sigma \sigma \sigma} \sigma^ {3}.\tag{12}\]

As in the derivation of the law of motion for in (7), is interpreted as a variable when constructing (12). This means that and represent fourth- and fifth-order effects, respectively, and are therefore omitted. Note that is a polynomial in that only includes third-order terms.

The final step is to set up the expression for the observation equation. Using results in Ruge-Murcia (2012), the third-order Taylor series expansion is given by

\[\begin{array}{r c l} \mathbf {y} _ {t} ^ {(3)} & = & \mathbf {g} _ {\mathbf {x}} \mathbf {x} _ {t} ^ {(3)} + \frac {1}{2} \mathbf {G} _ {\mathbf {x x}} \left(\mathbf {x} _ {t} ^ {(3)} \otimes \mathbf {x} _ {t} ^ {(3)}\right) + \frac {1}{6} \mathbf {G} _ {\mathbf {x x x}} \left(\mathbf {x} _ {t} ^ {(3)} \otimes \mathbf {x} _ {t} ^ {(3)} \otimes \mathbf {x} _ {t} ^ {(3)}\right) \\ & & + \frac {1}{2} \mathbf {g} _ {\sigma \sigma} \sigma^ {2} + \frac {3}{6} \mathbf {g} _ {\sigma \sigma \mathbf {x}} \sigma^ {2} \mathbf {x} _ {t} ^ {(3)} + \frac {1}{6} \mathbf {g} _ {\sigma \sigma \sigma} \sigma^ {3}, \end{array}\tag{13}\]

where represents the unpruned third-order approximation to the control variables. In (13), denotes an matrix containing derivatives of with respect to , is an matrix including derivatives with respect to , and is an matrix containing derivatives related to . To simplify notation, we treat as the sum of the first-, second-, and third-order effects, while is only the third-order effect. Hence, preserving effects up to third-order gives

\[\begin{array}{r c l} \mathbf {y} _ {t} ^ {r d} & = & \mathbf {g} _ {\mathbf {x}} \left(\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d}\right) + \frac {1}{2} \mathbf {G} _ {\mathbf {x x}} \left(\left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) + 2 \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {s}\right)\right) \\ & & + \frac {1}{6} \mathbf {G} _ {\mathbf {x x x}} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) + \frac {1}{2} \mathbf {g} _ {\sigma \sigma} \sigma^ {2} + \frac {3}{6} \mathbf {g} _ {\sigma \sigma \mathbf {x}} \sigma^ {2} \mathbf {x} _ {t} ^ {f} + \frac {1}{6} \mathbf {g} _ {\sigma \sigma \sigma} \sigma^ {3}, \end{array}\tag{14}\]

which is a polynomial in that include all first-, second-, and third-order terms.

Thus, the pruned state-space system for the third-order approximation is given by (5), (7), (12), and (14). The state vector in this system is further extended to , as we need to separately track first-, second-, and third-order effects. For completeness, the unpruned state-space system for the third-order approximation is given by (10) and (13).

3.3 Related Literature

Lombardo and Sutherland (2007) pioneered the idea of separately keeping track of first- and second-order effects to solve for a second-order perturbation of DSGE models. We extend their idea to approximations beyond second order. Since the first circulation of our paper, Lombardo and Uhlig (2014) have presented an alternative derivation of our pruned state-space system, but only for models without interaction between the innovations and the state variables. We find, nevertheless, that our approach allows us to easily derive many results and that the lack of interaction between the innovations and the state variables in Lombardo and Uhlig (2014) is too restrictive in many models of interest.

Our pruning approach is also analyzed in Haan and Wind (2012), who highlight two potential disadvantages of the method. First, pruning induces a larger vector of states than the unpruned approximation. Second, the pruned state-space system for the kth-order approximation cannot exactly fit the exact solution if it happens to be a kth-order polynomial. We do not consider the large state vector to be a problem because we find it informative to assess how important each of the second- and third-order effects is relative to the first-order effects. In addition, current computing power makes memory considerations less of a constraint. Indeed, Section 8 shows that the pruned state-space system for a third-order approximation to a medium-size DSGE model is easily obtained and stored. We also view the second disadvantage as minor because an exact fit can be obtained by raising the approximation beyond order k, as also acknowledged by Haan and Wind (2012).

Our pruning scheme differs from the alternative presented in Haan and Wind (2012) along two dimensions. First, for approximations beyond second order, these authors include terms with higher-order effects than the approximation order. For example, in the case of a third-order approximation, their first proposal for a pruning scheme includes some fourth-order effects, whereas their second proposal includes some fifth- and sixth-order effects. Second, their pruning scheme is expressed around what they refer to as the stochastic steady state, while our pruning scheme is expressed around the steady state. An advantage of our choices (i.e., omitting all higher-order effects than the approximation order and approximating around the steady state) is that they allow the derivation of unconditional moments in closed form. Furthermore, approximating around the steady state is consistent with our treatment of as a variable.

We conclude by stressing that, if a non-linear perturbation approximation does not preserve monotonicity and convexity of the exact policy function - as seen for extreme calibrations of DSGE models - then pruning will not restore these properties. For small DSGE models, Haan and Wind (2012) propose the perturbation-plus approximation and show that it may restore these properties of the policy function. However, the perturbation-plus algorithm is numerically demanding, even for small models, and does not allow the unconditional moments to be obtained in closed form.

4 Statistical Properties of the Pruned System

This section shows that the pruned state-space system has well-defined statistical properties and presents our closed-form expressions for first and second unconditional moments. We proceed as follows. Section 4.1 extends the analysis in Kim, Kim, Schaumburg and Sims (2008) for a second-order approximation, and Section 4.2 conducts a similar analysis for a third-order approximation. Applying the steps below to higher-order approximations is conceptually transparent.

After we circulated the first version of our paper, Francisco Ruge-Murcia directed our attention to his unpublished work on pruning at third order (Kim and Ruge-Murcia (2009) and Ruge-Murcia (2012)). Ruge-Murcia's approach is broadly similar to the one in Haan and Wind (2012), but with additional approximations imposed to compute unconditional moments.

Although not explicitly considered in this paper, it is straightforward to compute conditional moments for the state and control variables based on the expressions provided below.

4.1 Second-Order Approximation

In this section it is convenient to consider a more compact representation of the pruned state-space system than the one provided in Section 3.1. Therefore, we introduce the vector

\[\mathbf {z} _ {t} ^ {(2)} \equiv \left[ \begin{array}{c c c} {\left(\mathbf {x} _ {t} ^ {f}\right) ^ {\prime}} & {\left(\mathbf {x} _ {t} ^ {s}\right) ^ {\prime}} & {\left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime}} \end{array} \right] ^ {\prime},\]

where the superscript for denotes the approximation order. The first elements in are the first-order effects, while the remaining part of contains second-order effects. The laws of motion for and are stated above and the evolution for is easily derived from (5). This allows us to write the laws of motion for the first- and second-order effects in (5) and (7) by the linear law of motion in

\[\mathbf {z} _ {t + 1} ^ {(2)} = \mathbf {A} ^ {(2)} \mathbf {z} _ {t} ^ {(2)} + \mathbf {B} ^ {(2)} \pmb {\xi} _ {t + 1} ^ {(2)} + \mathbf {c} ^ {(2)},\tag{15}\]

and the law of motion in (9) as

\[\mathbf {y} _ {t} ^ {s} = \mathbf {C} ^ {(2)} \mathbf {z} _ {t} ^ {(2)} + \mathbf {d} ^ {(2)}.\tag{16}\]

The expressions for , , , , , and are provided in Appendix A.3. Standard properties for the Kronecker product and block matrices imply that the system in (15) is stable with all eigenvalues of having modulus less than one, provided the same holds for ; see Appendix A.4. This result might also be directly inferred from (5) and (7) because is stable by assumption, is constructed from a stable process, and the autoregressive part of is stable. The system has finite unconditional second moments if the same holds for , which is equivalent to having finite unconditional fourth moments; see Appendix A.5. This further implies that explosive sample paths do not appear in the pruned state-space system (almost surely).

These results also hold for models with deterministic and stochastic trends, provided trending variables are appropriately scaled (see King and Rebelo (1999)).

The next step is to find the expressions for the first and second unconditional moments. The innovations are a function of , , and , and we directly have that . Hence, the unconditional mean of is . To obtain some intuition for the determinants of the mean in the pruned state-space system, we explicitly compute some of the elements in . The mean of is easily seen to be zero from (5). Equation (7) implies that the mean of is

\[\mathbb {E} \left[ \mathbf {x} _ {t} ^ {s} \right] = \left(\mathbf {I} - \mathbf {h} _ {\mathbf {x}}\right) ^ {- 1} \left(\frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \mathbb {E} \left[ \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \right] + \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}\right).\tag{17}\]

Adding the mean for the first- and second-order effects, we then obtain the mean of the state variables in the pruned second-order approximation:

\[\mathbb {E} \left[ \mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} \right] = \mathbb {E} \left[ \mathbf {x} _ {t} ^ {f} \right] + \mathbb {E} \left[ \mathbf {x} _ {t} ^ {s} \right].\tag{18}\]

Equations (17) and (18) show that the second-order effects correct the mean of the first-order effects to adjust for uncertainty in the model. The adjustment comes from the second derivative of the perturbation parameter and the mean of . The latter can be computed from (5) and is given by .

Since and , the mean value of the variables in a first-order approximation is their steady state, while the mean of the pruned second-order approximation is corrected by the second moment of the innovations. In other words, the mean of implied by the pruned state-space system will, in most cases, differ from the steady state. This result is crucial because it shows that we cannot, in general, ignore the term and simply use the steady state of the model to calibrate or estimate model parameters.

Let us now consider the unconditional second moments. Standard properties of a VAR(1) system imply that the variance-covariance matrix for is given by

\[\mathbb {V} \left(\mathbf {z} _ {t} ^ {(2)}\right) = \mathbf {A} ^ {(2)} \mathbb {V} \left(\mathbf {z} _ {t} ^ {(2)}\right) \left(\mathbf {A} ^ {(2)}\right) ^ {\prime} + \mathbf {B} ^ {(2)} \mathbb {V} \left(\boldsymbol {\xi} _ {t} ^ {(2)}\right) \left(\mathbf {B} ^ {(2)}\right) ^ {\prime},\]

because and are uncorrelated as is independent across time. Appendix A.5 explains how to calculate . Once is known, we solve for by standard methods for

discrete Lyapunov equations.

Our procedure for computing differs slightly from the one in Kim, Kim, Schaumburg and Sims (2008). They suggest using a second-order approximation to by letting the last elements in be zero. This eliminates all third- and fourth-order terms related to and seems inconsistent with the fact that in contains third- and fourth-order terms. We prefer to compute without further approximations, implying that corresponds to the sample moment in a long simulation of the pruned state-space system.

The variance of the combined first- and second-order effects for the state variables is obtained by taking the variance of , i.e.

\[\mathbb {V} \left(\mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {f}\right) = \mathbb {V} \left(\mathbf {x} _ {t} ^ {f}\right) + \mathbb {V} \left(\mathbf {x} _ {t} ^ {s}\right) + C o v \left(\mathbf {x} _ {t} ^ {f}, \mathbf {x} _ {t} ^ {s}\right) + C o v \left(\mathbf {x} _ {t} ^ {s}, \mathbf {x} _ {t} ^ {f}\right).\]

The auto-covariances for are for because and are uncorrelated for , given that is independent across time.

The closed-form expressions for all corresponding unconditional moments related to follow directly from the linear relationship between and in (16). That is,

\[\mathbb {E} \left[ \mathbf {y} _ {t} ^ {s} \right] = \mathbf {C} ^ {(2)} \mathbb {E} \left[ \mathbf {z} _ {t} ^ {(2)} \right] + \mathbf {d} ^ {(2)}, \mathbb {V} \left[ \mathbf {y} _ {t} ^ {s} \right] = \mathbf {C} ^ {(2)} \mathbb {V} \left[ \mathbf {z} _ {t} \right] \left(\mathbf {C} ^ {(2)}\right) ^ {\prime}, \text { and }\]

\[C o v \left(\mathbf {y} _ {t + l} ^ {s}, \mathbf {y} _ {t} ^ {s}\right) = \mathbf {C} ^ {(2)} C o v \left(\mathbf {z} _ {t + l} ^ {(2)}, \mathbf {z} _ {t} ^ {(2)}\right) \left(\mathbf {C} ^ {(2)}\right) ^ {\prime} \text {for} l = 1, 2, 3, \dots\]

Finally, the representation in (15) and (16) makes it straightforward to derive additional statistical properties for the system. In particular, the pruned state-space system has finite unconditional third and fourth moments if the same holds for , which is equivalent to having finite unconditional sixth and eighth moments; see Appendix A.6.

4.2 Third-Order Approximation

As we did for the second-order approximation, we start by deriving a more compact representation for the pruned state-space system than the one in Section 3.2. This is done based on the vector

\[\mathbf {z} _ {t} ^ {(3)} \equiv \left[ \begin{array}{l l l l l l} (\mathbf {x} _ {t} ^ {f}) ^ {\prime} & (\mathbf {x} _ {t} ^ {s}) ^ {\prime} & (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} & (\mathbf {x} _ {t} ^ {r d}) ^ {\prime} & (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {s}) ^ {\prime} & (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} \end{array} \right] ^ {\prime},\]

where the first part reproduces and the last three components denote third-order effects. The law of motion for was derived in Section 3.2, and recursions for and follow from (5) and (7). Hence, the law of motion for , and in (5), (7), and (12), respectively, can be represented by the linear law of motion in

\[\mathbf {z} _ {t + 1} ^ {(3)} = \mathbf {A} ^ {(3)} \mathbf {z} _ {t} ^ {(3)} + \mathbf {B} ^ {(3)} \pmb {\xi} _ {t + 1} ^ {(3)} + \mathbf {c} ^ {(3)}.\tag{19}\]

We also have that the control variables are linear in as

\[\mathbf {y} _ {t} ^ {r d} = \mathbf {C} ^ {(3)} \mathbf {z} _ {t} ^ {(3)} + \mathbf {d} ^ {(3)}.\tag{20}\]

The expressions for , , , , , and are provided in Appendix A.7.

Appendix A.8 shows that the system in (19) is stable, with all eigenvalues of having modulus less than one, provided the same holds for . Building on the intuition from the second-order approximation, this result follows from the fact that the new component of the state vector is constructed from stable processes and its autoregressive component is also stable. The stability of relies on being treated as a variable in the pruned state-space system. If, instead, we had interpreted as a constant and included the term in the law of motion for , then would have the autoregressive matrix , which may imply eigenvalues with modulus greater than one even when is stable. Moreover, the system in (19) and (20) has finite unconditional second moments if the same holds for . The latter is equivalent to having finite unconditional sixth moments; see Appendix A.9.

The next step is to compute the first and second unconditional moments. The innovations in (19) are a function of , , , , and . Thus, and It is interesting to explore the value of as it may change the mean of the state variables. From (12), we immediately have

\[\mathbb {E} \left[ \mathbf {x} _ {t} ^ {r d} \right] = (\mathbf {I} _ {n _ {x}} - \mathbf {h} _ {\mathbf {x}}) ^ {- 1} \left(\mathbf {H} _ {\mathbf {x x}} \mathbb {E} \left[ \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {s} \right] + \frac {1}{6} \mathbf {H} _ {\mathbf {x x x}} \mathbb {E} \left[ \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \right] + \frac {1}{6} \mathbf {h} _ {\sigma \sigma \sigma} \sigma^ {3}\right),\]

and simple algebra gives and Adding the mean for the first-, second- and third-order effects, we obtain If we next consider the standard case where all innovations have symmetric probability distributions, then , which in turn implies and . Furthermore, based on the results in Andreasen (2012), and are also zero when all innovations have symmetric probability distributions. Thus, and the unconditional mean of the state vector is not further corrected by the third-order effects when all innovations have zero third moments. A similar property holds for the control variables because they are a linear function of , and . This result is useful when calibrating or estimating DSGE models with symmetric probability distributions. On the other hand, if one or several innovations have non-symmetric probability distributions, then and may be non-zero and , implying that the unconditional mean has an additional uncertainty correction compared to a second-order approximation.

Let us now consider the unconditional second moments. The expression for the variance-covariance matrix of is slightly more complicated than the one for because is correlated with . This correlation arises from terms of the form in which are correlated with elements in . Hence,

\[\begin{array}{r c l} \mathbb {V} \left(\mathbf {z} _ {t} ^ {(3)}\right) & = & \mathbf {A} ^ {(3)} \mathbb {V} \left(\mathbf {z} _ {t} ^ {(3)}\right) \left(\mathbf {A} ^ {(3)}\right) ^ {\prime} + \mathbf {B} ^ {(3)} \mathbb {V} \left(\boldsymbol {\xi} _ {t} ^ {(3)}\right) \left(\mathbf {B} ^ {(3)}\right) ^ {\prime} \\ & & + \mathbf {A} ^ {(3)} C o v \left(\mathbf {z} _ {t} ^ {(3)}, \boldsymbol {\xi} _ {t + 1} ^ {(3)}\right) \left(\mathbf {B} ^ {(3)}\right) ^ {\prime} + \mathbf {B} ^ {(3)} C o v \left(\boldsymbol {\xi} _ {t + 1} ^ {(3)}, \mathbf {z} _ {t} ^ {(3)}\right) \left(\mathbf {A} ^ {(3)}\right) ^ {\prime}. \end{array}\]

The expressions for and are provided in Appendix A.9. The variance of

the combined first-, second- and third-order effects for the state variables is given by

\[\begin{array}{r c l} \mathbb {V} \left(\mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {r d}\right) & = & \mathbb {V} \left(\mathbf {x} _ {t} ^ {f}\right) + \mathbb {V} \left(\mathbf {x} _ {t} ^ {s}\right) + \mathbb {V} \left(\mathbf {x} _ {t} ^ {r d}\right) + C o v \left(\mathbf {x} _ {t} ^ {f}, \mathbf {x} _ {t} ^ {s}\right) + C o v \left(\mathbf {x} _ {t} ^ {f}, \mathbf {x} _ {t} ^ {r d}\right) \\ & & + C o v \left(\mathbf {x} _ {t} ^ {s}, \mathbf {x} _ {t} ^ {f}\right) + C o v \left(\mathbf {x} _ {t} ^ {s}, \mathbf {x} _ {t} ^ {r d}\right) + C o v \left(\mathbf {x} _ {t} ^ {r d}, \mathbf {x} _ {t} ^ {f}\right) + C o v \left(\mathbf {x} _ {t} ^ {r d}, \mathbf {x} _ {t} ^ {s}\right). \end{array}\]

The auto-covariances for are

\[C o v \left(\mathbf {z} _ {t + s} ^ {(3)}, \mathbf {z} _ {t} ^ {(3)}\right) = \left(\mathbf {A} ^ {(3)}\right) ^ {s} \mathbb {V} \left[ \mathbf {z} _ {t} ^ {(3)} \right] + \sum_ {j = 0} ^ {s - 1} \left(\mathbf {A} ^ {(3)}\right) ^ {s - 1 - j} \mathbf {B} ^ {(3)} C o v \left(\boldsymbol {\xi} _ {t + 1 + j} ^ {(3)}, \mathbf {z} _ {t} ^ {(3)}\right)\]

for

The closed-form expressions for all corresponding unconditional moments related to follow directly from the linear relationship between and in (20) and are given by

\[C o v \left(\mathbf {y} _ {t + l} ^ {r d}, \mathbf {y} _ {t} ^ {r d}\right) = \mathbf {C} ^ {(3)} C o v \left(\mathbf {z} _ {t + l} ^ {(3)}, \mathbf {z} _ {t} ^ {(3)}\right) \left(\mathbf {C} ^ {(3)}\right) ^ {\prime} \text {for} l = 1, 2, 3,..\]

Finally, the representation in (19) and (20) of the pruned state-space system and the fact that is a function of allow us to derive additional properties for the system. For instance, the pruned state-space system has finite unconditional third and fourth moments if the same holds for , which is equivalent to having finite ninth and twelfth moments; see Appendix A.10.

5 Generalized Impulse Response Functions

Another fruitful way to study the properties of DSGE models is to look at their IRFs. For the first-order approximation, these functions have simple expressions where the effects of shocks are scalable, symmetric, and independent of the state of the economy. For higher-order approximations, no closed-form expressions currently exist for these functions and simulation is, therefore, required. This section shows that the pruned state-space system allows us to derive closed-form solutions for these functions and avoid the use of simulation.

We consider the generalized impulse response function (GIRF) proposed by Koop, Pesaran and

Potter (1996). The GIRF for any variable in the model var (either a state or control variable) in period following a disturbance to the ith shock of size in period is defined as

\[G I R F _ {\mathbf {v a r}} (l, \nu_ {i}, \mathbf {w} _ {t}) = \mathbb {E} [ \mathbf {v a r} _ {t + l} | \mathbf {w} _ {t}, \epsilon_ {i, t + 1} = \nu_ {i} ] - \mathbb {E} [ \mathbf {v a r} _ {t + l} | \mathbf {w} _ {t} ],\]

where denotes the required state variables in period t. As we will see below, the content of depends on the approximation order. Using this definition, the GIRFs for the first-order effects have the simple and well-known expressions

\[G I R F _ {\mathbf {x} ^ {f}} (l, \nu_ {i}) = \mathbb {E} \left[ \mathbf {x} _ {t + l} ^ {f} | \mathbf {x} _ {t} ^ {f}, \epsilon_ {i, t + 1} = \nu_ {i} \right] - \mathbb {E} \left[ \mathbf {x} _ {t + l} ^ {f} | \mathbf {x} _ {t} ^ {f} \right] = \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \pmb {\eta} \pmb {\nu}\tag{21}\]

and

\[G I R F _ {\mathbf {y} ^ {f}} (l, \nu_ {i}) = \mathbf {g} _ {\mathbf {x}} G I R F _ {\mathbf {x} ^ {f}} (l, \nu_ {i}),\]

where has dimension and contains the size of the disturbances in period . For (21), we have and for . Here, and are scalable, symmetric, and independent of the state of the economy because the state vector enters symmetrically in the two conditional expectations for computing each of these GIRFs. Momentarily, we will see how the GIRFs for second- and third-order effects will not be scalable, symmetric, and independent of the state of the economy.

5.1 Second-Order Approximation

For the second-order effects , we have from (7) that

\[\mathbf {x} _ {t + l} ^ {s} = \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {s} + \sum_ {j = 1} ^ {l - 1} \mathbf {h} _ {\mathbf {x}} ^ {l - 1 - j} \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t + j} ^ {f} \otimes \mathbf {x} _ {t + j} ^ {f}\right) + \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2} \sum_ {j = 0} ^ {l - 1} \mathbf {h} _ {\mathbf {x}} ^ {l - 1 - j}.\tag{22}\]

The GIRF for is derived in Appendix A.11, showing that

\[\begin{array}{r c l} G I R F _ {\mathbf {x} ^ {f} \otimes \mathbf {x} ^ {f}} \left(l, \nu_ {i}, \mathbf {x} _ {t} ^ {f}\right) & = & \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} + \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \\ & & + \left(\mathbf {h} _ {\mathbf {x}} ^ {l - 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1}\right) (\sigma \boldsymbol {\eta} \boldsymbol {\nu} \otimes \sigma \boldsymbol {\eta} \boldsymbol {\nu} + \boldsymbol {\Lambda}), \end{array}\tag{23}\]

The expressions we derive below for the GIRFs may also be used for studying the joint effects of more than one disturbance to the economy. Further details are provided in the Online Appendix.

where

\[\boldsymbol {\Lambda} \equiv \left(\left(\sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S}) \otimes \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S})\right) - \left(\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}\right)\right) v e c (\mathbf {I}).\tag{24}\]

Here, is an diagonal matrix with and for . Using this expression and (22), we get the GIRF for the second-order effects

\[G I R F _ {\mathbf {x} ^ {s}} \left(l, \nu_ {i}, \mathbf {x} _ {t} ^ {f}\right) = \sum_ {j = 1} ^ {l - 1} \mathbf {h} _ {\mathbf {x}} ^ {l - 1 - j} \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} G I R F _ {\mathbf {x} ^ {f} \otimes \mathbf {x} ^ {f}} \left(j, \nu_ {i}, \mathbf {x} _ {t} ^ {f}\right).\tag{25}\]

The expressions in (23) to (25) reveal three implications about the GIRF for the second-order effects. First, it is not scalable as for . Second, the term means that the GIRF is not symmetric in positive and negative shocks. Third, it depends on the first-order effects of the state variables. Adding the GIRFs for the first- and second-order effects, we obtain the pruned GIRF for the state variables in a second-order approximation.

Finally, the pruned GIRF for the control variables is easily derived from (8) and previous results:

\[\begin{array}{r c l} G I R F _ {\mathbf {y} ^ {s}} (l, \nu_ {i}, \mathbf {x} _ {t} ^ {f}) & = & \mathbf {g} _ {\mathbf {x}} (G I R F _ {\mathbf {x} ^ {f}} (l, \nu_ {i}, \mathbf {x} _ {t} ^ {f}) + G I R F _ {\mathbf {x} ^ {s}} (l, \nu_ {i}, \mathbf {x} _ {t} ^ {f})) \\ & & + \frac {1}{2} \mathbf {G} _ {\mathbf {x x}} G I R F _ {\mathbf {x} ^ {f} \otimes \mathbf {x} ^ {f}} (l, \nu_ {i}, \mathbf {x} _ {t} ^ {f}). \end{array}\]

Another interesting result from our analytical expressions relates to the IRFs in a linearized solution for a positive or negative one-standard-deviation shock computed at the steady state. As shown in Appendix A.12, these IRFs coincide with the GIRFs in a pruned second-order approximation because , implying that these IRFs in a linearized solution are actually second-order accurate.

5.2 Third-Order Approximation

Using (12), we first note that for the third-order effects

\[\begin{array}{r c l} \mathbf {x} _ {t + l} ^ {r d} & = & \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {r d} + \sum_ {j = 0} ^ {l - 1} \mathbf {h} _ {\mathbf {x}} ^ {l - 1 - j} \left[ \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t + j} ^ {f} \otimes \mathbf {x} _ {t + j} ^ {s}\right) + \frac {1}{6} \mathbf {H} _ {\mathbf {x x x}} \left(\mathbf {x} _ {t + j} ^ {f} \otimes \mathbf {x} _ {t + j} ^ {f} \otimes \mathbf {x} _ {t + j} ^ {f}\right) \right] \\ & & + \sum_ {j = 0} ^ {l - 1} \mathbf {h} _ {\mathbf {x}} ^ {l - 1 - j} \left[ \frac {3}{6} \mathbf {h} _ {\sigma \sigma \mathbf {x}} \sigma^ {2} \mathbf {x} _ {t + j} ^ {f} + \frac {1}{6} \mathbf {h} _ {\sigma \sigma \sigma} \sigma^ {3} \right]. \end{array}\]

Simple algebra implies

\[\begin{array}{r c l} G I R F _ {\mathbf {x} ^ {r d}} \left(l, \nu_ {i}, \left(\mathbf {x} _ {t} ^ {f}, \mathbf {x} _ {t} ^ {s}\right)\right) & = & \sum_ {j = 1} ^ {l - 1} \mathbf {h} _ {\mathbf {x}} ^ {l - 1 - j} \mathbf {H} _ {\mathbf {x x}} G I R F _ {\mathbf {x} ^ {f} \otimes \mathbf {x} ^ {s}} \left(j, \nu_ {i}, \left(\mathbf {x} _ {t} ^ {f}, \mathbf {x} _ {t} ^ {s}\right)\right) \\ & & + \sum_ {j = 1} ^ {l - 1} \mathbf {h} _ {\mathbf {x}} ^ {l - 1 - j} \frac {1}{6} \mathbf {H} _ {\mathbf {x x x}} G I R F _ {\mathbf {x} ^ {f} \otimes \mathbf {x} ^ {f} \otimes \mathbf {x} ^ {f}} \left(j, \nu_ {i}, \mathbf {x} _ {t} ^ {f}\right) \\ & & + \sum_ {j = 1} ^ {l - 1} \mathbf {h} _ {\mathbf {x}} ^ {l - 1 - j} \frac {3}{6} \mathbf {h} _ {\sigma \sigma \mathbf {x}} \sigma^ {2} G I R F _ {\mathbf {x} ^ {f}} \left(j, \nu_ {i}\right). \end{array}\]

All terms are known except for and , which are derived in Appendix A.13. As was the case for the second-order effect, the GIRF for the third-order effect is not scalable, not symmetric, and depends on the first-order effects of the state variables . In addition, the GIRF for the third-order effects also depends on . Adding the GIRF for the first-, second-, and third-order effects, we obtain the pruned GIRF for the state variables in a third-order approximation.

The pruned GIRF for the control variables in a third-order approximation is

\[\begin{array}{r c l} G I R F _ {\mathbf {y} ^ {r d}} \left(l, \nu_ {i}, \left(\mathbf {x} _ {t} ^ {f}, \mathbf {x} _ {t} ^ {s}\right)\right) & = & \mathbf {g} _ {\mathbf {x}} \left(G I R F _ {\mathbf {x} ^ {f}} \left(l, \nu_ {i}\right) + G I R F _ {\mathbf {x} ^ {s}} \left(l, \nu_ {i}, \mathbf {x} _ {t} ^ {f}\right)\right) + \mathbf {g} _ {\mathbf {x}} G I R F _ {\mathbf {x} ^ {r d}} \left(l, \nu_ {i}, \left(\mathbf {x} _ {t} ^ {f}, \mathbf {x} _ {t} ^ {s}\right)\right) \\ & & + \frac {1}{2} \mathbf {G} _ {\mathbf {x x}} \left(G I R F _ {\mathbf {x} ^ {f} \otimes \mathbf {x} ^ {f}} \left(l, \nu_ {i}, \mathbf {x} _ {t} ^ {f}\right) + 2 G I R F _ {\mathbf {x} ^ {f} \otimes \mathbf {x} ^ {s}} \left(l, \nu_ {i}, \left(\mathbf {x} _ {t} ^ {f}, \mathbf {x} _ {t} ^ {s}\right)\right)\right) \\ & & + \frac {1}{6} \mathbf {G} _ {\mathbf {x x x}} G I R F _ {\mathbf {x} ^ {f} \otimes \mathbf {x} ^ {f} \otimes \mathbf {x} ^ {f}} \left(l, \nu_ {i}, \mathbf {x} _ {t} ^ {f}\right) \\ & & + \frac {3}{6} \mathbf {g} _ {\sigma \sigma \mathbf {x}} \sigma^ {2} G I R F _ {\mathbf {x} ^ {f}} (l, \nu_ {i}), \end{array} \tag {26}\]

where all terms are known.

5.3 Conditional Impulse Response Functions

Given that the expressions for the GIRFs in Sections 5.1 and 5.2 depend on the values of the state variables, we can use them to analyze how the responses to shocks depend on the business cycle. For example, in a simple stochastic neoclassical growth model, the economy may respond differently to a positive technological shock when capital is high than when it is low. The challenge is that in most DSGE models, the values of the state variables are unobserved. Hence, it may be challenging to specify relevant state values, except for the obvious benchmark as given by the unconditional mean. We suggest overcoming this problem by conditioning the GIRFs on some set of observables, such as the economy being in a recession (i.e., negative output growth). More concretely, consider a conditional GIRF of the form

\[G I R F _ {\mathbf {v a r}} (l, \nu_ {i}, A) = \int 1 _ {A} (\mathbf {w} _ {t}) f (\mathbf {w} _ {t}) G I R F _ {\mathbf {v a r}} (l, \nu_ {i}, \mathbf {w} _ {t}) d \mathbf {w} _ {t}\tag{27}\]

where A is a set defined by the criteria in the observables, is an indicator function, and is the unconditional density of . The integral in (27) can be evaluated by Monte Carlo integration, where draws of the states are obtained from a long simulated sample path of the pruned state-space system. The great advantage of a conditional GIRF is that it is defined on an observed set A and is therefore directly observable, in contrast to the GIRFs provided in Sections 5.1 and 5.2.

An example illustrates how we address the challenge mentioned two paragraphs ago. Imagine we are working with a simple stochastic neoclassical growth model and we want to calculate the GIRF of conditional on output growth being above a given threshold, say, 2% annualized. Then A is the set of , which implies that output growth is above this annualized 2%.

6 Accuracy of Pruning

This section analyzes how pruning affects the accuracy of the approximated solution. We start with the following proposition, which studies the approximation errors when the perturbation parameter tends to zero.

Proposition 1 For , the errors in and are of third order, whereas the errors in and are of fourth order.

The proof of this proposition is provided in Appendix A.14. The consequence of this result is that, for a given approximation order and , the errors in the pruned and unpruned state-space systems are of the same order. When we account for uncertainty in the model by letting , the stability of the pruned state-space system ensures that approximation errors do not accumulate over time. A similar convenient property does not necessarily hold in the unpruned state-space system, which therefore may generate explosive sample paths.

To obtain further insights into the accuracy of pruning under uncertainty, one would have to consider a particular DSGE model and study the accuracy of a pruned and an unpruned state-space system. For small models, this can be done by comparing simulated sample paths to a highly accurate projection approximation as in Lan and Meyer-Gohde (2013a). For larger models where the projection method or other global approximation methods are computationally infeasible, accuracy may be explored based on Euler equation errors, as in an earlier version of this paper (see Andreasen, Fernández-Villaverde and Rubio-Ramírez (2013)). Given the analyzed models, the two aforementioned papers show that our pruning scheme does not worsen accuracy (and often it improves it) when compared to the unpruned state-space system. However, these findings are model-specific.

Nevertheless, we can offer some intuition of why we found in previous versions of this paper that the Euler equation errors of the pruned approximation along the simulations were smaller than those from the unpruned one. Unpruned approximations are subject to what we call microbursts of instability. Often, the simulations are hit by relatively large innovations. These innovations push the simulation toward an explosive path. At the same time, it is also often the case that after a few periods, a large innovation of opposite sign sends the simulation back into a stable path. During these periods of transitory explosive paths (our microbursts of instability), the Euler equation errors of the unpruned approximation are poor. In comparison, pruned approximations are not subject to these microbursts. We will often have small microbursts of instability that do not reach the threshold and are kept in the simulation while triggering poor accuracy. This behavior is documented in Appendix A.15, where we show that for a simple stochastic neoclassical growth model, the pruned solution does better when we are farther away from the steady state.

In general, regardless of whether pruning improves accuracy around the steady state or not, one can also adopt the view that pruning is a simple and transparent way of eliminating explosive sample paths. One may, therefore, argue that the cautious approach is to prune perturbation approximations. Unpruned perturbation approximations may also be useful for models where explosive sample paths rarely appear, but these unpruned approximations simply may be inapplicable to many models of interest that frequently generate explosive sample paths, including the New Keynesian model presented below in Section 8.

7 Econometric Implications of the Pruning Method

Our results in Sections 4 and 5 allow us to implement standard moment matching methods in non-linearly approximated DSGE models. For approximations up to third order, these methods include GMM estimation (Hansen (1982)) based on first and second unconditional moments. More generally, we could also estimate DSGE models by matching autocorrelation functions, the spectral density, or other functions of interest. Our results are also useful for a Bayesian researcher. The work by Kim (2002) shows how to build a limited information likelihood function from optimal GMM estimation. Equipped with priors, we may then carry out a Bayesian analysis, where the asymptotic distribution for the posterior equals the limiting distribution of GMM.

Another possibility is to match model-implied GIRFs to their empirical counterparts. This approach was popularized by Christiano, Eichenbaum and Evans (2005) for a linearized model. However, for non-linear approximations, we need to move beyond a VAR to document our empirical GIRFs because the linear structure of a VAR can only produce IRFs that are scalable, symmetric, and independent of the state of the economy. The methods proposed by Jorda (2005) and Matthes and Barnichon (2014) are, thus, more appropriate for DSGE models solved non-linearly. Given that the GIRFs for non-linear approximations depend on unobserved state variables, some care is needed to ensure that the empirical and model-implied GIRFs are comparable along this dimension, i.e., that they are evaluated at the same state values. A natural possibility is to compute model-implied GIRFs at the unconditional mean of the states and compare them to the empirical GIRFs at the sample mean. An alternative is to use the conditional GIRF in (27) to consider empirical and model-implied GIRFs on some criteria for the observables, such as i) recessions vs. expansions, ii) high vs. low inflation, iii) high vs. low conditional volatility of output, etc.

A third way to obtain the empirical GIRFs is to derive these functions from an estimated auxiliary model consistent with the pruned state-space system, as in Aruoba, Bocola and Schorfheide (2013) for the scalar case. Our work provides the theoretical foundation for constructing the auxiliary model consistent with the pruned state-space system in the multivariate case and beyond a second-order approximation.

In relation to GMM estimation and IRF matching, our results enable researchers to determine the stochastic specification of the structural innovations semi-parametrically, i.e., by only estimating the moments of without assuming a given probability distribution. The ability to identify moments of , and the DSGE model in general may be examined using the procedure from linearized models in Iskrev (2010) on the pruned state-space system, as in Mutschler (2015) for a second-order approximation.

If we want to use higher-order moments such as skewness and kurtosis in the estimation, then simulations are generally needed. Although it is possible to compute closed-form expressions for skewness and kurtosis in DSGE models when the pruning method is applied, the memory requirement for such computations is extremely onerous and only applicable to small models with a few state variables. But even if we use simulation, our analysis provides a foundation for SMM following Duffie and Singleton (1993) and indirect inference as considered in Smith (1993), Dridi, Guay and Renault (2007), and Creel and Kristensen (2011), among others. This is because pruning ensures that the model-implied processes are stationary (possibly following a transformation), as required for the limiting distribution of these simulation-based estimators.

8 An Application

We now present an empirical application to illustrate the GMM estimation methodology that our paper makes available. We also show results for the GIRFs and the conditional GIRFs shown in Section 5. We focus on a New Keynesian model with two novel features. First, we introduce a financial intermediary that trades short- and long-term government bonds. This intermediary generates a wedge between the policy rate set up by the monetary authority and the interest rate faced by private agents in the economy. This wedge arises in our model from time variation in the conditional second moments of the stochastic discount factor, whereas this premium in models with a banking sector is often motivated by steady-state frictions (see Bernanke, Gertler and Gilchrist (1999) or Gertler and Karadi (2011) among others). Thus, our model combines the macro-finance literature focusing on stochastic discount factors with the recent work on financial intermediation in DSGE models. Our second innovation is to consider a central bank that sets the policy rate not only based on the inflation and output gap, but also on a measure of term premia. This extension is motivated by the recent financial crisis, where several central banks engaged in large asset purchases to stimulate the economy by affecting term premia (see Gagnon, Raskin, Rernache and Sack (2011) and Joyce, Lasaosa, Stevens and Tong (2011)). In summary, our model displays two feedback effects from long-term bonds to the real economy: i) a wedge between the policy rate and the interest rate faced by the households and ii) a policy rate that depends on a measure of term premia. As we will show below, each of these feedback effects helps our model to overcome the counterintuitive result of Tallarini (2000) that real allocations are essentially unaffected by the amount of risk in the economy. Consequently, our model generates a much richer environment for monetary policy than the standard New Keynesian model. These features make our application of interest on its own beyond illustrating our new estimation methodology.

Building on our work, Mutschler (2015) provides closed-form expressions for skewness and kurtosis for second-order approximations to DSGE models.

See also Ruge-Murcia (2012) for a Monte Carlo study and application of SMM based on the neoclassical growth model solved up to third order.

See Peralta-Alva and Santos (2012) for a summary of the literature on the relation between estimation methods and numerical errors in simulation.

We proceed by outlining the model in Sections 8.1 to 8.5, describing our solution and estimation method in Sections 8.6 and 8.7, and presenting estimation results in Sections 8.8 to 8.12. We finish by showing, in Section 8.13, some of the GIRFs and conditional GIRFs of the model.

8.1 Households

We consider a representative household with recursive preferences as in Epstein and Zin (1989) and Weil (1990). Using the convenient formulation proposed by Rudebusch and Swanson (2012), the value function of the household can be written as

\[V _ {t} \equiv \left\{ \begin{array}{c l} u _ {t} + \beta \left(\mathbb {E} _ {t} \left[ V _ {t + 1} ^ {1 - \phi_ {3}} \right]\right) ^ {\frac {1}{1 - \phi_ {3}}} & \text {if} u _ {t} > 0 \text {for all} t \\ u _ {t} - \beta \left(\mathbb {E} _ {t} \left[ (- V _ {t + 1}) ^ {1 - \phi_ {3}} \right]\right) ^ {\frac {1}{1 - \phi_ {3}}} & \text {if} u _ {t} < 0 \text {for all} t \end{array} \right.,\tag{28}\]

where is the conditional expectation given information in period t and is the subjective discount factor. For higher values of , these preferences imply higher levels of risk aversion if the utility kernel is always positive, and vice versa for . The main benefit of the Epstein-Zin-Weil preferences is to disentangle risk aversion from the intertemporal elasticity of substitution (IES) when ; otherwise (28) simplifies to standard expected utility.

We let the utility kernel display separability between consumption and hours worked

\[u _ {t} \equiv \frac {d _ {t}}{1 - \phi_ {2}} \left(\left(\frac {c _ {t} - b c _ {t - 1}}{z _ {t} ^ {*}}\right) ^ {1 - \phi_ {2}} - 1\right) + \phi_ {0} \frac {(1 - h _ {t}) ^ {1 - \phi_ {1}}}{1 - \phi_ {1}},\tag{29}\]

where b controls the degree of internal habit formation. The variable with introduces preference shocks, where we omit the traditional AR(1) term to ensure that variations in long-term interest rates and term premia in our model are explained by consumption dynamics and not by persistent preference shocks. As in An and Schorfheide (2007), utility from habit-adjusted consumption in (29) is expressed relative to the deterministic trend in the economy to guarantee the existence of a balanced growth path. We simultaneously include habit formation and Epstein-Zin-Weil preferences because the literature has shown that the New Keynesian model needs both features to jointly match various macro and financial moments (see Hordahl, Tristani and Vestin (2008) and Binsbergen, Fernandez-Villaverde, Koijen and Rubio-Ramirez (2012)).

The budget constraint at time t reads:

\[c _ {t} + \frac {i _ {t}}{\Upsilon_ {t}} + b _ {t} + T _ {t} = w _ {t} h _ {t} + r _ {t} ^ {k} k _ {t} + \frac {b _ {t - 1} \exp \left\{r _ {t - 1} ^ {b} \right\}}{\pi_ {t}} + d i v _ {t} ^ {h}.\tag{30}\]

Resources are spent on consumption, investment , a one-period deposit in the financial intermediary at the net nominal risk-free deposit rate , and a lump-sum tax . The variable denotes a deterministic trend in the real relative price of investment: . Letting denote the real wage and the real price of capital , resources consist of real labor income , real income from capital services sold to firms , real returns from deposits in the previous period, and firm dividends to households . Here, is gross inflation.

The law of motion for is

\[k _ {t + 1} = (1 - \delta) k _ {t} + i _ {t} - \frac {\kappa}{2} \left(\frac {i _ {i}}{k _ {t}} - \psi\right) ^ {2} k _ {t},\tag{31}\]

where introduces capital adjustment costs as in Jermann (1998). The constant ensures

The constant ensures a stable level of and in the steady state when is close to one. As shown in Table 1, for all estimated models, the steady-state value of , , is substantially below zero.

that these adjustment costs are zero along the balanced growth path of the economy.

8.2 The Financial Intermediary

As mentioned above, the representative household makes one-period deposits in a perfectly competitive financial intermediary, which invests deposits in short- and long-term government bonds. The household may also overdraw this deposit, i.e., can be negative, in which case the financial intermediary shorts these bonds. The short-term bond, for simplicity, is assumed to be the one-period bond, whereas the maturity of the long-term bond is denoted by L > 1. In our implementation, we set L to reflect the 10-year interest rate, but other maturities may be considered.

The behavior of the financial intermediary is solely determined by the deposit rate . To state its expression, let the ex ante holding period return on the kth bond be

\[h r _ {t, k} \equiv \mathbb {E} _ {t} \left[ \log P _ {t + 1, k - 1} - \log P _ {t, k} \right],\tag{32}\]

where is the nominal price in period t of a zero-coupon bond maturing in period . The excess holding period return is then , where is the one-period nominal policy rate set by the central bank. We then assume that the deposit rate is equal to the ex ante holding period return on the invested bond portfolio, i.e.,

\[r _ {t} ^ {b} \equiv (1 - \omega) \times h r _ {t, 1} + \omega \times h r _ {t, L} = r _ {t} + \omega \times x h r _ {t, L}\tag{33}\]

because and . Here, denotes the fraction invested by the financial intermediary in the long-term government bond. The value of is determined by factors exogenous to the model. For example, financial regulation forces many mutual funds to keep large shares of their bonds in short maturities, regardless of their preferred investment strategies. Endogenizing this and other factors determining is well beyond the scope of this paper. Nevertheless, we will treat as a free parameter in our estimation procedure and infer the average portfolio weight from our model.

To clarify the behavior of this financial intermediary, suppose for a moment that . In this case, the financial intermediary only holds the one-period government bond and (33) simplifies to . Thus, our framework recovers the standard specification considered in most New Keynesian models where the deposit rate equals the one-period policy rate set by the central bank.

Another possibility is to assume that . This introduces a feedback effect from long-term government bonds to the real economy as the excess holding period return affects and the household's consumption decision. For instance, an increase in due to a higher term premium during a recession will tend to increase the deposit rate and encourage the household to postpone consumption. Given that is non-zero due to uncertainty, this feedback effect from long-term government bonds to the real economy operates through a precautionary saving channel. We will exploit this insight below to derive an efficient perturbation solution to our model.

A careful inspection of our framework reveals that it is related to the risk-premium shocks in Smets and Wouters (2007), where an exogenous shock drives a wedge between the policy rate and the interest rate faced by the households. When (33) is substituted into the consumption Euler equation, i.e., with , denoting the marginal utility of habit-adjusted consumption, we obtain a similar wedge, except that this wedge is endogenously generated within our model. Note also that, if we were to follow Smets and Wouters (2007) and use a standard log-linear approximation to our model, then for all t, implying that (33) would reduce to the standard specification where , even when .

Having outlined how the deposit rate is determined, we next describe how the financial intermediary prices government bonds. Given that the financial intermediary is owned by the households and, therefore, acts in their interest, we determine the price of these bonds by the stochastic discount factor of the representative household. That is,

\[P _ {t, k} = \mathbb {E} _ {t} \left[ \beta \frac {\lambda_ {t + 1}}{\lambda_ {t}} \frac {1}{\pi_ {t + 1}} P _ {t + 1, k - 1} \right],\tag{34}\]

for with . The nominal yield curve with continuous compounding is then given by for .

8.3 Firms

A perfectly competitive representative firm produces final output by aggregating a continuum of intermediate goods using the production function with . This generates the demand function , with aggregate price level . The intermediate good is produced by a monopolistic competitor using the production function . Here, is a deterministic trend that follows , and where . As in Altig, Christiano, Eichenbaum and Linde (2011), we define , which denotes the technological trend in the economy.

The intermediate firms maximize the net present value of real profit with respect to capital, labor, and prices given a nominal rigidity. We consider price-setting à la Calvo (1983), where contracts expire with probability in each period. Whenever a contract expires, firms set their optimal nominal prices, which otherwise are equal to past prices, i.e., .

8.4 Monetary and Fiscal Policy

A central bank sets the policy rate based on a desire to stabilize the inflation gap and the output gap , subject to smoothing changes in . Here, refers to steady-state inflation. As in Justiniano and Primiceri (2008) and Rudebusch and Swanson (2012), the output gap is measured in deviation from the deterministic trend in output, which equals times production in the normalized steady state . As we argued above, with a financial intermediary investing in long-term government bonds, the deposit rate offered to households is no longer fully determined by the central bank's policy rate due to changes in . Thus, the central bank may find it useful also to account for variability in when setting its policy rate. For instance, term premia typically increase during recessions and this generates upward pressure on and the deposit rate within our framework. Then, a central bank may consider a larger reduction in the policy rate than required with to offset the negative impact from higher term premia on economic activity. Also, the central bank may provide different policy responses to shocks that create the same inflation and output gaps on impact, but affect and term premia asymmetrically, for instance, because the shocks differ in their persistence. More concretely, we postulate that monetary policy follows an augmented Taylor rule of the form

\[\begin{array}{r c l} r _ {t} & = & (1 - \rho_ {r}) r _ {s s} + \rho_ {r} r _ {t - 1} + (1 - \rho_ {r}) \left(\beta_ {\pi} \log \left(\frac {\pi_ {t}}{\pi_ {s s}}\right) + \beta_ {y} \log \left(\frac {y _ {t}}{z _ {t} ^ {*} Y _ {s s}}\right)\right) \\ & & + (1 - \rho_ {r}) \beta_ {x h r} (x h r _ {t, L} - \mathbb {E} [ x h r _ {t, L} ]) \end{array}\tag{35}\]

where we omit monetary policy shocks as the literature has documented that they have a tiny effect on term premia.

When implementing (35), we approximate by with , as it has the convenient representation . In contrast, the steady-state value of equals zero and is a poor approximation of . Finally, note that if we were to solve our model by a log-linearization, then for all t and (35) would reduce to the standard Taylor rule even if .

Government consumption grows with the economy as in Rudebusch and Swanson (2012), where

\[\log \left(\frac {G _ {t + 1}}{G _ {s s}}\right) = \rho_ {G} \log \left(\frac {G _ {t}}{G _ {s s}}\right) + \sigma_ {G} \epsilon_ {G, t + 1}\]

and . Government consumption and the interest on government debt are paid with lump-sum taxes. Given that a version of Ricardian equivalence holds in our economy, we do not need to specify the timing of these taxes and simply write the resource constraint of the economy as .

8.5 Model Aggregation

The aggregated resource constraint in the goods market is , where is the price dispersion index. The dynamic of this endogenous state variable is

\[s _ {t + 1} = (1 - \alpha) \tilde {p} _ {t} ^ {- \eta} + \alpha \pi_ {t} ^ {\eta} s _ {t},\tag{36}\]

where and denotes the optimal nominal price in period . The relation between inflation and the newly optimized prices is

\[1 = (1 - \alpha) \tilde {p} _ {t} ^ {1 - \eta} + \alpha \pi_ {t} ^ {\eta - 1}.\tag{37}\]

Also, household deposits are zero in equilibrium (their net assets are in terms of capital investment), implying that net profit by the financial intermediary is also zero. Thus, the net worth of the financial intermediary does not change across time and may be ignored when solving the model.

This is illustrated in Rudebusch and Swanson (2012), who also show that it is the systematic part of monetary policy that has a large impact on term premia. We will find a similar result below.

There is an alternative, yet fully equivalent, representation of our model with complete markets, which is described in Appendix A.17. In that representation of the model, the households do not need to rely on the financial intermediary to invest their savings in government bonds, but the Taylor rule depends instead on and .

8.6 An Efficient Perturbation Approximation

To solve the model, we first induce stationarity by eliminating trending variables with appropriate transformations; see Appendix A.16. The desired policy functions that characterize the equilibrium dynamics of the model are then obtained by employing a third-order perturbation approximation. We require at least a third-order approximation to generate variation in and capture the feedback effects from long-term government bonds to the real economy. Our quarterly model with L set to reflect the 10-year interest rate has seven state variables and 54 control variables.

The standard approach in the literature to efficiently compute a higher-order perturbation of DSGE models with a yield curve exploits the fact that bond prices beyond the policy rate typically do not affect allocations and prices (i.e., consumption, inflation, etc.). Taking advantage of that property, these models are approximated by a two-step procedure, where the first step solves the model without bond prices exceeding one period, after which, in a second step, all remaining bond prices are computed recursively based on (34). This two-step procedure reduces the size of the simultaneous equation systems to be solved and, with it, the computation burden of the approximation (see Hordahl, Tristani and Vestin (2008), Binsbergen, Fernandez-Villaverde, Koijen and Rubio-Ramirez (2012), and Andreasen and Zabczyk (2015)).

We cannot apply this two-step procedure to our model when or because the long-term bond price affects the deposit rate and the policy rate through and, hence, all allocations and prices. Fortunately, the terms associated with the perfect foresight solution of our model -i.e., and - may be found with the standard two-step procedure even when or because is equal to zero under perfect foresight. Once we have computed these terms, we only need to find the derivatives involving the perturbation parameter using the full model. The whole three-step procedure is formally described in Appendix A.18 and constitutes a new numerical contribution to the literature. Accordingly, our three-step procedure allows us to compute a third-order solution to our model with feedback effects in just 3.7 seconds, whereas it takes 6.2 seconds when using the standard one-step perturbation algorithm of Binning (2013). This improvement in computational speed of more than 40% greatly facilitates the estimation, as the perturbation approximation must be computed for many different parameter values.

The relatively large number of control variables is needed to compute all bond prices within the 10-year maturity range.

8.7 Data and Moments for GMM

We employ the following quarterly time series to estimate our model: i) consumption growth , ii) investment growth , iii) inflation , iv) the 1-quarter nominal interest rate , v) the 10-year nominal interest rate , vi) the 10-year ex post excess holding period return , vii) the log ratio of government spending to GDP , and viii) the log of hours . The presence of a short- and long-term interest rate captures the slope of the yield curve, whereas the excess holding period return is included as a noisy proxy for the 10-year term premium. All series are stored in data with dimension . Our sample goes from 1961.Q3 to 2007.Q4. The end date is set to avoid the complications created by the zero lower bound of the nominal interest rate. See Appendix A.19 for a description of the data series.

We want to explore whether our model can match the mean, the variance, the contemporaneous covariances, and the persistence in the data. Hence, we let

\[\mathbf {q} _ {t} \equiv \left[ \begin{array}{c} \mathbf {d a t a} _ {t} \\ d i a g \left(\mathbf {d a t a} _ {t} \mathbf {d a t a} _ {t} ^ {\prime}\right) \\ v e c h \left(\widetilde {\mathbf {d a t a}} _ {t} \widetilde {\mathbf {d a t a}} _ {t} ^ {\prime}\right) \\ d i a g \left(\mathbf {d a t a} _ {t} \mathbf {d a t a} _ {t - 1} ^ {\prime}\right) \end{array} \right],\tag{38}\]

where denotes the diagonal elements of a matrix and refers to the first six elements of . We omit moments on the contemporaneous correlation relating to and due to the parsimonious specification of government spending and the labor market in our model.

These computations are done in Matlab 2014a on a Fujitsu laptop with an Intel(R) Core(TM) i5-4200M CPU @ 2.50 GHz.

Letting contain the structural parameters, our GMM estimator is given by

\[\hat {\boldsymbol {\theta}} _ {G M M} = \underset {\boldsymbol {\theta} \in \boldsymbol {\Theta}} {\arg \min} \left(\frac {1}{T} \sum_ {t = 1} ^ {T} \mathbf {q} _ {t} - \mathbb {E} \left[ \mathbf {q} _ {t} (\boldsymbol {\theta}) \right]\right) ^ {\prime} \mathbf {W} \left(\frac {1}{T} \sum_ {t = 1} ^ {T} \mathbf {q} _ {t} - \mathbb {E} \left[ \mathbf {q} _ {t} (\boldsymbol {\theta}) \right]\right).\]

Here, is a positive definite weighting matrix and contains the model-implied moments computed in closed form using the above formulas. We use the conventional two-step implementation of GMM by letting in a preliminary first step to obtain , where denotes the long-run variance of when re-centered around its sample mean. Our final estimates are obtained using the optimal weighting matrix , where denotes the long-run variance of our moments re-centered around . The long-run variances in both steps are estimated by the Newey-West estimator using 10 lags, but our results are robust to using more lags.

We estimate all structural parameters in our model except for a few poorly identified parameters. That is, we let and as typically considered for the U.S. economy. We also impose to get an average markup of 20%, and we let to obtain a Frisch labor supply elasticity in the neighborhood of 0.5.

8.8 Estimation Results I: The Benchmark Model

As a convenient benchmark, we first estimate our model without feedback effects from long-term bonds by imposing and . This version of our model is denoted . The estimated parameters in Table 1 are fairly standard with investment adjustment costs ( ), little curvature in the periodic utility of consumption ( ), and sizeable habits ( ). The latter implies a relatively low steady-state intertemporal elasticity of substitution ( ), which in the presence of internal habits is

\[I E S _ {s s} = \frac {1}{\phi_ {2}} \left[ \frac {\left(1 - \frac {b}{\mu_ {z ^ {*} , s s}}\right) (\mu_ {z ^ {*} , s s} - \beta b)}{\mu_ {z ^ {*} , s s} + b \beta + \beta b ^ {2} \mu_ {z ^ {*} , s s} ^ {- 1}} \right],\tag{39}\]

The Frisch labor supply in our model is and hence is affected by the steady-steady labor supply , which is close to 1/3 (see Table 1).

or simply with and . As in much of the existing macro-finance literature, we find extreme levels of relative risk aversion ( ), even when accounting for a variable labor supply as in Swanson (2012). Using the general formulas provided in Swanson (2013), our utility function in (29) implies

\[R R A = \frac {\phi_ {2}}{\frac {1 - b \mu_ {z ^ {*} , s s} ^ {- 1}}{1 - \beta b} + \frac {\phi_ {2}}{\phi_ {1}} \frac {W _ {s s} (1 - h _ {s s})}{C _ {s s}}} + \phi_ {3} \frac {1 - \phi_ {2}}{\frac {1 - b \mu_ {z ^ {*} , s s} ^ {- 1}}{1 - \beta b} - \frac {(1 - b \mu_ {z ^ {*} , s s} ^ {- 1}) ^ {\phi_ {2}}}{1 - \beta b} C _ {s s} ^ {\phi_ {2} - 1} + \frac {W _ {s s} (1 - h _ {s s})}{C _ {s s}} \frac {1 - \phi_ {2}}{1 - \phi_ {1}}},\tag{40}\]

where and are the real wage and consumption in the normalized steady state. Our estimated level of risk aversion is clearly too high to be consistent with the micro-evidence. For instance, Barsky, Juster, Kimball and Shapiro (1997) find a RRA between 3.8 and 15.7 in surveys, and Mehra and Prescott (1985) argue that a plausible level of relative risk aversion should not exceed 10. However, a key contribution of the present model is to demonstrate the sizeable reduction in risk aversion that follows when introducing feedback effects from long-term bonds. We also find a moderate degree of nominal frictions with prices being re-optimized roughly every fifth quarter ( ), and a central bank assigning more weight to stabilize inflation than output ( vs. ), subject to smoothing changes in the policy rate ( ).

Table 2 shows that our benchmark model reproduces all means, in particular the short- and long-term interest rates of and , respectively. We only match the mean inflation rate of due to a large precautionary saving correction that lowers the annual steady-state inflation rate of to obtain a model-implied inflation rate of . The model is also successful in matching the variability in the data, except for a too low standard deviation in the 10-year excess holding period return (12.93% vs. ). A satisfying performance is also seen for the first-order autocorrelations and the contemporaneous correlations (bottom of Table 2).

Of considerable interest is the implied term premia from our model. Following Rudebusch and Swanson (2012), we define term premia as , where is the yield-to-maturity on a zero-coupon bond under risk-neutral valuation by the financial intermediary, i.e., . Our benchmark model has a 10-year term premium with a mean of 145 basis points, which is close to the average slope of the yield curve (139 basis points) that serves as an observable proxy for the average term premium. We also find substantial variation in the 10-year term premium having a standard deviation of 116 basis points. This is in line with the variability in the 10-year term premium obtained in various Gaussian affine term structure models for our sample: i) the three- and four-factor models of Andreasen and Meldrum (2014) with bias-adjusted factor dynamics have a standard deviation of 105 and 115 basis points, respectively, and ii) the five-factor model of Adrian, Crump and Moench (2013) has a standard deviation of 121 basis points. Finally, our model is consistent with another noisy measure of term premia variability, namely the standard deviation of the slope for the 10-year yield curve, which equals 139 basis points.

Household wealth is measured by the present value of lifetime consumption in (40) as recommended by Swanson (2013). Given that and are unaffected by , we then use (40) to back out the value of the Epstein-Zin-Weil coefficient for a given value of the RRA during the estimation. Rudebusch and Swanson (2012) explain why the benchmark model requires high risk aversion to match post-war U.S. data.

8.9 Understanding the Volatility of the Term Premium

Although high risk aversion helps to increase the mean term premium, it does not necessarily generate a highly volatile term premium. To understand the main mechanism behind the variability in , recall that its volatility is directly related to the degree of heteroscedasticity in the stochastic discount factor , i.e., the variation of . The three shocks in our model are all homoscedastic. Thus, the model is endogenously generating heteroscedasticity in , as captured by our third-order perturbation. But, what is the source of this large heteroscedasticity? A possibility is to consider the effect of the price dispersion index , which is an endogenous state variable. Combining (36) and (37), its law of motion is then given by

\[s _ {t + 1} = (1 - \alpha) ^ {\frac {1}{1 - \eta}} \left[ 1 - \alpha \pi_ {t} ^ {\eta - 1} \right] ^ {\frac {\eta}{\eta - 1}} + \alpha \pi_ {t} ^ {\eta} s _ {t},\]

which is highly non-linear to ensure , as shown by Schmitt-Grohe and Uribe (2007). The first term in the expression for does not generate much heteroscedasticity with being well below one given our estimates. The second term , on the other hand, may generate extreme levels of heteroscedasticity because and we generally also have . Also, the degree of heteroscedasticity is increasing in the mean of both variables. A higher value of clearly increases , but also the steady state of , given that for . This effect is illustrated in Figure 1 by considering a sample path with positive steady-state inflation ( ) and one without . In the case of positive steady-state inflation (left panels), we see more extreme observations and hence more heteroscedasticity compared to the case of no steady-state inflation (right panels). Note also how the capital stock and the price dispersion with positive steady-state inflation attain very low and high values, respectively, just before observation 9,000, at which point the unpruned state-space system explodes. A similar divergence in sample path does not appear for , where the two approximations generate nearly identical time series. Accordingly, positive steady-state inflation serves as a channel to generate heteroscedasticity in and, hence, variation in , as required to produce the volatile 10-year term premium in our model.

The first column in Table 4 shows that this channel has a large effect, as the standard deviation of the 10-year term premium falls from 116 basis points with positive steady-state inflation to just 1.42 basis points when . To further decompose the effects of positive steady-state inflation, we adopt the standard decomposition of risk premia into the market price of risk times the quantity of risk. As in Cochrane (2001), we let , implying that the quantity of risk equals . Table 4 shows that omitting positive steady-state inflation lowers the standard deviation in the by a factor of 100, whereas the standard deviation of falls by a factor of . Hence, positive steady-state inflation mainly generates a volatile term premium in our model by increasing the variability in the quantity of risk. Table 4 further shows that positive steady-state inflation also affects the mean term premium, which falls from 145 basis points to just 32 basis points when , although RRA equals 615.7! This fall is due to a reduction in the mean of , which lowers the , whereas the level for the is nearly unaffected.

Thus, accounting for positive steady-state inflation serves as a key new channel to endogenously generate heteroscedasticity in the New Keynesian model and produce a 10-year term premium with the desired level and variability. Importantly, an unpruned third-order approximation to our model results in explosive sample paths and is unable to “detect” this novel channel, which we uncover by using our pruning scheme for a third-order perturbation.

Swanson (2015) also emphasizes the importance of the price dispersion index as a source of heteroscedasticity in the New Keynesian model but without noticing the importance of positive steady-state inflation for this channel.

8.10 Estimation Results II: The First Feedback Effect

Our next step is to introduce the first feedback effect from long-term bonds by allowing , while still maintaining that the central bank does not respond to the excess holding period return ( ). This version of our model is referred to as . Table 1 shows that the financial intermediary is estimated to allocate a large fraction of its investments to long-term bonds with . A standard t-test clearly rejects the null hypothesis of at conventional significant levels, which provide support for our first feedback channel from long-term bonds. Another important property of relates to the estimated degree of RRA, which is only 23, and thus substantially lower than in the benchmark model. For the remaining parameters, we find minor changes compared to our benchmark model, except for the policy rule and investment adjustment costs.

Table 2 shows that delivers a satisfying fit to the considered moments despite its lower risk aversion. To quantify the performance of compared to the benchmark model, Table 3 reports objective functions from our two-step GMM procedure. Only the objective functions from the first step use the same weighting matrix and are, therefore, comparable across models. They show that fits the data better than the benchmark model (14.546 vs. 16.929).

However, risk aversion in is estimated very imprecisely with a large standard error of 31, and it is likely that RRA can be lowered further with only a minor reduction in the goodness of fit. Consistent with the micro-evidence provided in Barsky, Juster, Kimball and Shapiro (1997), we restrict RRA to 5 and re-estimate our model with the first feedback effect. Table 2 verifies our conjecture as this restricted model with low risk aversion provides nearly the same fit as the unrestricted model. In particular, matches the slope of the yield curve, while simultaneously fitting key moments for the five macro variables. We also note from Table 3 that provides a better overall fit to the data than our benchmark model. Here, we report the P-value from the J-test for model misspecification, showing that we are unable to reject (and all the other models). That is, the observed differences between empirical and model-implied moments in Table 2 are not unusual given the sample variation in the empirical moments. However, this finding should be interpreted with caution as the J-test has low power due to our sample size (T = 186). Finally, generates a realistic 10-year term premium with a mean of 140 basis points and a standard deviation of 93 basis points, as seen from Table 4.

The estimates in step 1 are very similar to those reported for step 2 in Table 1, implying that the objective functions in step 1 serve as a good metric for model comparison.

Thus, allowing the financial intermediary to invest in long-term bonds goes a long way in resolving the bond risk premium puzzle described in Rudebusch and Swanson (2008) without postulating highly risk-averse households as in much of the existing literature (see Andreasen (2012), Binsbergen, Fernandez-Villaverde, Koijen and Rubio-Ramirez (2012), and Rudebusch and Swanson (2012), among others).

8.11 Understanding the First Feedback Effect

We now explore the mechanisms that enable and to generate a large and variable term premium without relying on high risk aversion. Suppose for and consider increasing to some positive value less than one. This increase in lowers as and, hence, the current bond price because

\[P _ {t, L} = \mathbb {E} _ {t} \left[ M _ {t, t + 1} \right] \mathbb {E} _ {t} \left[ P _ {t + 1, L - 1} \right] + C o v _ {t} \left(M _ {t, t + 1}, P _ {t + 1, L - 1}\right).\]

This fall in increases according to (32). But a higher induces a further fall in and the current bond price , which generates an even larger increase in . That is, generates a “feedback multiplication effect” that amplifies the level and variability in . An explicit way to see the implication of this feedback loop is to use a first-order approximation of the logarithmic and exponential function in (32) to obtain (see Appendix A.20):

\[x h r _ {t, L} \approx \frac {1}{1 - \omega \frac {\mathbb {E} _ {t} [ P _ {t + 1 , L - 1} ]}{P _ {s s , L - 1}}} \left[ \left(\frac {\mathbb {E} _ {t} [ P _ {t + 1 , L - 1} ]}{P _ {s s , L - 1}} - 1\right) (r _ {t} - r _ {s s}) - C o v _ {t} \left(\frac {M _ {t , t + 1}}{M _ {s s , s s + 1}}, \frac {P _ {t + 1 , L - 1}}{P _ {s s , L - 1}}\right) \right].\tag{41}\]

The first term in (41) is the risk-neutral component of , whereas is the required compensation by the risk averse household for carrying risk. The expression in (41) shows that both terms in are amplified by the factor when . Therefore, our model requires lower volatility in and, correspondingly, lower risk aversion, to match short- and long-term interest rates. Finally, extending the expression for the term premium in Rudebusch and Swanson (2012) to our model, it follows that

\[\begin{array}{r c l} {T P _ {t, k}} & \approx & {- \frac {1}{k P _ {s s , k}} \mathbb {E} _ {t} \left[ \sum_ {j = 0} ^ {k - 1} e ^ {\left\{- \sum_ {m = 0} ^ {j - 1} (r _ {t + m} + \omega \times x h r _ {t + m, L}) \right\}} C o v _ {t + j} (M _ {t + j, t + j + 1}, P _ {t + j + 1, k - j - 1}) \right]} \\ & & {- \frac {1}{k P _ {s s , k}} \mathbb {E} _ {t} \left[ e ^ {\left\{- \sum_ {m = 0} ^ {k - 1} \omega \times x h r _ {t + m, L} \right\}} - 1 \right].} \end{array}\tag{42}\]

This expression demonstrates how a higher level and variability in translates into a larger and more volatile term premium, which is to be expected given that both variables measure compensation for risk.

To illustrate the magnitude of the multiplication effect from long-term bonds on term premia, we momentarily set in , whereas all remaining parameters are as reported in Table 1. Omitting the first feedback channel from long-term bonds reduces the mean of the 10-year term premium from 140 to 9 basis points, and similarly for the standard deviation which falls from 93 to 4 basis points. Table 4 documents that the feedback channel from long-term bonds reduces the mean and standard deviation of by a factor of 100 in and compared to the benchmark model. This, in turn, leads to a similar reduction in the corresponding moments for the . Hence, and generate a high and volatile term premium by increasing the quantity of risk. As observed for the benchmark model, we see in Table 4 that positive steady-state inflation is essential for and to generate the desired level and variability of the term premium even with the first feedback from long-term bonds. Hence, it would be hard to discover this novel feedback effect without our pruning method, as an unpruned state-space system generates explosive sample paths when we assume positive steady-state inflation.

8.12 Estimation Results III: The First and Second Feedback Effects

We finally introduce our second feedback effect from long-term bonds by allowing the central bank to respond to variation in , which is closely related to term premia as shown in (41) and (42). That is, we let and refer to this model as . Table 1 shows that this second feedback effect from long-term bonds has a small effect in the model as , a point estimate not sufficiently far from zero to be statistically significant. However, the effect from our second feedback effect is somewhat larger when RRA is restricted to 5 in .

Here, and the response of the central bank to is now significant given a standard error of 0.094 for . That is, our model implies a reduction in the policy rate when term premia and increase, as the central bank tries to offset the rise in the deposit rate with a lower policy rate. Although we end our sample in 2007.Q4, this finding is consistent with monetary policy during the recent financial crisis, where the Federal Reserve undertook vigorous policy measures to stimulate economic activity in response to elevated levels of term premia.

Table 2 shows that matches most of the moments considered, in particular all mean values and the slope of the yield curve. Table 3 documents how outperforms both and the benchmark model in terms of overall goodness of fit, although with unrestricted risk aversion does somewhat better than . The term premium is also found to be consistent with empirical moments, as generates a 10-year term premium with a mean of 142 basis points and a standard deviation of 132 basis points (see Table 4). As before, positive steady-state inflation is essential for and to generate the desired level and variability in the term premium by “activating” the two novel feedback effects from long-term bonds to the real economy considered in this paper.

8.13 GIRFs and Conditional GIRFs

Our next exercise is to report the GIRFs following positive one-standard-deviation shocks in to technology, government spending, and preferences (Figures 2 to 4). These functions are computed for a log-linearized solution and a third-order approximation using (26) with the relevant state variables at their unconditional means. All the GIRFs have the expected pattern, and we therefore direct attention to the effects of higher-order terms, i.e., the differences between the marked and unmarked lines. Shocks to technology and government spending have substantial non-linear effects on consumption and investment, mainly because these shocks generate considerable variation in the ex ante excess holding period return and the term premium. This finding reveals that higher-order effects, and hence the amount of risk in the economy, affect real allocations in our model, which therefore overturns the result of Tallarini (2000) that risk does not matter for real allocations.

A key advantage of computing second- and third-order approximations is that we can analyze the effects of different shocks conditional on the state of the economy. This is illustrated in Figure 5, where we show how the response of consumption, investment, and to a positive one-standard-deviation shock to technology is larger when the economy is in a recession than when it is not. The intuition is that, when the economy is in a recession, consumption and capital tend to be low, and hence the marginal utility of extra consumption and the marginal return of additional investment are higher than usual. A similar exercise is done in Figure 6, except that now we compare the situation where we condition on high vs. low inflation. When inflation is high, consumption, investment, and interest rates respond more vigorously than when inflation is low. When inflation is high, nominal rigidities are particularly damaging, since firms that are not able to change their prices are far from the price they would set under flexible prices. A positive productivity shock translates into lower inflation through lower marginal costs and, hence, it alleviates these pernicious effects of nominal rigidities. When inflation is low, nominal rigidities are less of a constraint on firm behavior and a positive technology shock is less useful for firms. The asymmetries in responses to shocks documented by Figures 5 and 6 demonstrate how the methods we present in our paper allow researchers to probe deeper into the behavior of their models and uncover economic mechanisms that would otherwise remain hidden.

9 Conclusion

This paper extends the pruning method by Kim, Kim, Schaumburg and Sims (2008) to third- and higher-order approximations, with special attention devoted to models solved up to third order. Conditions for the existence of first and second unconditional moments are derived, and their values are provided in closed form. The existence of higher-order unconditional moments in the form of skewness and kurtosis is also established. We also analyze GIRFs and provide simple closed-form expressions for these functions.

The econometric implications of our findings are significant, as most of the existing moment-based estimation methods for linearized DSGE models now carry over to non-linear approximations. For approximations up to third order, this includes GMM estimation based on first and second unconditional moments and matching model-implied GIRFs to their empirical counterparts. When simulations are needed, our analysis also provides a foundation for different types of indirect inference and SMM. These results are not just relevant for classical inference, as the moment conditions in optimal GMM estimation may be used to build a limited information likelihood function, from which Bayesian inference may be carried out.

We define a recession as a quarter where there is (detrended) negative output in the current and the previous two periods. Otherwise, the economy is in expansion.

High inflation is defined as inflation larger than one standard deviation of inflation. Otherwise the economy is defined to be in a low inflation regime. The corresponding conditional GIRFs for government spending and preference shocks are omitted in the interest of space.

To illustrate one of the new estimation methods that our paper makes available, we revisit the term structure implications of the New Keynesian model. We first demonstrate a new channel to amplify the level and time variation in term premia by accounting for positive steady-state inflation. Given this more realistic term premium, we then introduce two feedback effects from long-term bonds to the real economy, and we show that they enable the New Keynesian model to generate a high and variable term premium with the same low risk aversion as found in the micro-evidence. We once again emphasize that our pruning scheme has greatly facilitated the discovery of these new channels and helped us to address the long-standing bond premium puzzle.

A Appendix

A.1 Non-linearities Between State Variables and Innovations

To illustrate how non-linearities between and can be addressed in our framework, let be an expanded state vector where the innovations now appear as state variables. The new state equation is, then, given by

\[\mathbf {v} _ {t + 1} = \left[ \begin{array}{c} \mathbf {h} (\mathbf {v} _ {t}, \sigma) \\ \mathbf {0} \end{array} \right] + \sigma \left[ \begin{array}{c} \mathbf {0} _ {n _ {x}} \\ \mathbf {u} _ {t + 1} \end{array} \right],\]

where is of dimension , and the new observation equation is

\[\mathbf {y} _ {t} = \mathbf {g} (\mathbf {v} _ {t}, \sigma).\]

Thus, any model with non-linearities between state variables and innovations may be rewritten into our notation with only linear innovations.

As an illustration, consider a neoclassical growth model with stochastic volatility. Using standard notation, the equilibrium conditions are given by:

\[\begin{array}{c} {c _ {t} ^ {- \gamma} = \beta \mathbb {E} _ {t} \left[ c _ {t + 1} ^ {- \gamma} \left(a _ {t + 1} \alpha k _ {t + 1} ^ {\alpha - 1} + 1 - \delta\right) \right]} \\ {c _ {t} + k _ {t + 1} = a _ {t} k _ {t} ^ {\alpha} + (1 - \delta) k _ {t}} \\ {\log a _ {t + 1} = \rho \log a _ {t} + \sigma_ {a, t + 1} \epsilon_ {a, t + 1}} \end{array}\]

and

\[\log \left(\frac {\sigma_ {a , t + 1}}{\sigma_ {a , s s}}\right) = \rho_ {\sigma} \log \left(\frac {\sigma_ {a , t}}{\sigma_ {a , s s}}\right) + \epsilon_ {\sigma , t + 1}.\]

We then rewrite these conditions as:

\[c _ {t} ^ {- \gamma} = \mathbb {E} _ {t} \left[ \beta c _ {t + 1} ^ {- \gamma} \left(\exp \left\{\rho \log a _ {t} + \sigma_ {a, s s} \exp^ {\left\{\rho_ {\sigma} \log \left(\frac {\sigma_ {a , t}}{\sigma_ {a , s s}}\right) + \epsilon_ {\sigma , t + 1} \right\}} \epsilon_ {a, t + 1} \right\} \alpha k _ {t + 1} ^ {\alpha - 1} + 1 - \delta\right) \right]\]

\[c _ {t} + k _ {t + 1} = a _ {t} k _ {t} ^ {\alpha} + (1 - \delta) k _ {t}\]

\[\log a _ {t} = \rho \log a _ {t - 1} + \sigma_ {a, t} \epsilon_ {a, t}\]

\[\log \left(\frac {\sigma_ {a , t}}{\sigma_ {a , s s}}\right) = \rho_ {\sigma} \log \left(\frac {\sigma_ {a , t - 1}}{\sigma_ {a , s s}}\right) + \epsilon_ {\sigma , t}\]

\[\epsilon_ {a, t + 1} = \sigma u _ {a, t + 1}\]

and

\[\epsilon_ {\sigma , t + 1} = \sigma u _ {\sigma , t + 1},\]

where the extended state vector is and is the perturbation parameter scaling the innovations and .

If, instead, the volatility process is specified as a GARCH(1,1) model, then the equilibrium

conditions can be expressed as:

\[\begin{array}{r} c _ {t} ^ {- \gamma} = \mathbb {E} _ {t} \left[ \beta c _ {t + 1} ^ {- \gamma} \left(\exp^ {\{\rho \log a _ {t} + \sigma_ {a, t + 1} \epsilon_ {a, t + 1} \}} \alpha k _ {t + 1} ^ {\alpha - 1} + 1 - \delta\right) \right] \\ c _ {t} + k _ {t + 1} = a _ {t} k _ {t} ^ {\alpha} + (1 - \delta) k _ {t} \\ \log a _ {t} = \rho \log a _ {t - 1} + \sigma_ {a, t} \epsilon_ {a, t} \\ \sigma_ {a, t + 1} ^ {2} = (1 - \rho_ {1}) \sigma_ {a, s s} ^ {2} + \rho_ {1} \sigma_ {a, t} ^ {2} + \rho_ {2} \sigma_ {a, t} ^ {2} \epsilon_ {a, t} ^ {2} \end{array}\]

and

\[\epsilon_ {a, t + 1} = \sigma u _ {t + 1},\]

where the extended state vector is and is the perturbation parameter scaling . As in Andreasen (2012), the constant term in the GARCH process is scaled by to ensure that in the steady state where .

A.2 Pruned State-Space Beyond Third Order

The pruned state-space system for the kth-order approximation based on the kth-order Taylor series expansions of and are obtained by: i) decomposing the state variables into first-, second-, ... , and kth-order effects, ii) setting up laws of motion for the state variables capturing only first-, second-, ... , and kth-order effects, and iii) constructing the expression for control variables by preserving only effects up to kth-order. In comparison, the unpruned state-space system for the kth-order approximation is given by the kth-order Taylor series expansions of and .

A.3 Coefficients for the Pruned State-Space System at Second Order

\[\mathbf {A} ^ {(2)} \equiv \left[ \begin{array}{c c c} \mathbf {h _ {x}} & \mathbf {0} & \mathbf {0} \\ \mathbf {0} & \mathbf {h _ {x}} & \frac {1}{2} \mathbf {H _ {x x}} \\ \mathbf {0} & \mathbf {0} & \mathbf {h _ {x}} \otimes \mathbf {h _ {x}} \end{array} \right],\]

\[\mathbf {B} ^ {(2)} \equiv \left[ \begin{array}{c c c c} \sigma \eta & \mathbf {0} & \mathbf {0} & \mathbf {0} \\ \mathbf {0} & \mathbf {0} & \mathbf {0} & \mathbf {0} \\ \mathbf {0} & \sigma \eta \otimes \sigma \eta & \sigma \eta \otimes \mathbf {h _ {x}} & \mathbf {h _ {x}} \otimes \sigma \eta \end{array} \right],\]

\[\pmb {\xi} _ {t + 1} ^ {(2)} \equiv \left[ \begin{array}{c} \pmb {\epsilon} _ {t + 1} \\ \pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1} - v e c (\mathbf {I} _ {n _ {e}}) \\ \pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \\ \mathbf {x} _ {t} ^ {f} \otimes \pmb {\epsilon} _ {t + 1} \end{array} \right],\]

\[\mathbf {c} ^ {(2)} \equiv \left[ \begin{array}{c} \mathbf {0} \\ \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2} \\ (\sigma \pmb {\eta} \otimes \sigma \pmb {\eta}) v e c (\mathbf {I} _ {n _ {e}}) \end{array} \right],\]

\[\mathbf {C} ^ {(2)} \equiv \left[ \begin{array}{l l l} \mathbf {g _ {x}} & \mathbf {g _ {x}} & \frac {1}{2} \mathbf {G _ {x x}} \end{array} \right],\]

and

\[\mathbf {d} ^ {(2)} \equiv \frac {1}{2} \mathbf {g} _ {\sigma \sigma} \sigma^ {2}.\]

A.4 Second Order: Stability

First, note that all eigenvalues of are strictly less than one. To see this, we work with

\[\begin{array}{r c l} p (\lambda) & = & \left| \mathbf {A} - \lambda \mathbf {I} _ {2 n _ {x} + n _ {x} ^ {2}} \right| \\ & = & \left| \left[ \begin{array}{c c c} \mathbf {h} _ {\mathbf {x}} - \lambda \mathbf {I} _ {n _ {x}} & \mathbf {0} _ {n _ {x} \times n _ {x}} & \mathbf {0} _ {n _ {x} \times n _ {x} ^ {2}} \\ \mathbf {0} _ {n _ {x} \times n _ {x}} & \mathbf {h} _ {\mathbf {x}} - \lambda \mathbf {I} _ {n _ {x}} & \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \\ \mathbf {0} _ {n _ {x} ^ {2} \times n _ {x}} & \mathbf {0} _ {n _ {x} ^ {2} \times n _ {x}} & \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} - \lambda \mathbf {I} _ {n _ {x} ^ {2}} \end{array} \right] \right| \\ & = & \left| \begin{array}{c c} \mathbf {B} _ {1 1} & \mathbf {B} _ {1 2} \\ \mathbf {B} _ {2 1} & \mathbf {B} _ {2 2} \end{array} \right| \\ & = & | \mathbf {B} _ {1 1} | | \mathbf {B} _ {2 2} |, \end{array}\]

where we let

\[\begin{array}{r c l} \mathbf {B} _ {1 1} & \equiv & \left[ \begin{array}{c c} \mathbf {h} _ {\mathbf {x}} - \lambda \mathbf {I} _ {n _ {x}} & \mathbf {0} _ {n _ {x} \times n _ {x}} \\ \mathbf {0} _ {n _ {x} \times n _ {x}} & \mathbf {h} _ {\mathbf {x}} - \lambda \mathbf {I} _ {n _ {x}} \end{array} \right], \\ \mathbf {B} _ {1 2} & \equiv & \left[ \begin{array}{c} \mathbf {0} _ {n _ {x} \times n _ {x} ^ {2}} \\ \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \end{array} \right], \\ \mathbf {B} _ {2 1} & \equiv & \left[ \begin{array}{c c} \mathbf {0} _ {n _ {x} ^ {2} \times n _ {x}} & \mathbf {0} _ {n _ {x} ^ {2} \times n _ {x}} \end{array} \right], \end{array}\]

and

\[\mathbf {B} _ {2 2} \equiv \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} - \lambda \mathbf {I} _ {n _ {x} ^ {2}}\]

and we use the fact that

\[\left| \begin{array}{c c} \mathbf {U} & \mathbf {C} \\ \mathbf {0} & \mathbf {Y} \end{array} \right| = | \mathbf {U} | | \mathbf {Y} |,\]

where is an matrix and is an matrix. Hence,

\[p \left(\lambda\right) = \left| \left[ \begin{array}{c c} \mathbf {h _ {x}} - \lambda \mathbf {I} _ {n _ {x}} & \mathbf {0} _ {n _ {x} \times n _ {x}} \\ \mathbf {0} _ {n _ {x} \times n _ {x}} & \mathbf {h _ {x}} - \lambda \mathbf {I} _ {n _ {x}} \end{array} \right] \right| \left| \mathbf {h _ {x}} \otimes \mathbf {h _ {x}} - \lambda \mathbf {I} _ {n _ {x} ^ {2}} \right| = \left| \mathbf {h _ {x}} - \lambda \mathbf {I} _ {n _ {x}} \right| \left| \mathbf {h _ {x}} - \lambda \mathbf {I} _ {n _ {x}} \right| \left| \mathbf {h _ {x}} \otimes \mathbf {h _ {x}} - \lambda \mathbf {I} _ {n _ {x} ^ {2}} \right|.\]

The eigenvalues are determined from or . The absolute values of all eigenvalues to the first problem are strictly less than one by assumption. That is, . This is also the case for the second problem because the eigenvalues to are for and .

A.5 Second Order: Unconditional Second Moments

For the variance, we have

\[\mathbb {V} \left(\mathbf {z} _ {t + 1} ^ {(2)}\right) = \mathbf {A} ^ {(2)} \mathbb {V} \left(\mathbf {z} _ {t} ^ {(2)}\right) \left(\mathbf {A} ^ {(2)}\right) ^ {\prime} + \mathbf {B} ^ {(2)} \mathbb {V} \left(\boldsymbol {\xi} _ {t + 1} ^ {(2)}\right) \left(\mathbf {B} ^ {(2)}\right) ^ {\prime}\]

as

\[\begin{array}{r l} & {\mathbb {E} \left[ \mathbf {z} _ {t} ^ {(2)} \left(\pmb {\xi} _ {t + 1} ^ {(2)}\right) ^ {\prime} \right] = \mathbb {E} \left[ \begin{array}{c c} \mathbf {x} _ {t} ^ {f} \pmb {\epsilon} _ {t + 1} ^ {\prime} & \mathbf {x} _ {t} ^ {f} \left(\pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1} - v e c \left(\mathbf {I} _ {n e}\right)\right) ^ {\prime} \\ \mathbf {x} _ {t} ^ {s} \pmb {\epsilon} _ {t + 1} ^ {\prime} & \mathbf {x} _ {t} ^ {s} \left(\pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1} - v e c \left(\mathbf {I} _ {n e}\right)\right) ^ {\prime} \\ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) \pmb {\epsilon} _ {t + 1} ^ {\prime} & \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1} - v e c \left(\mathbf {I} _ {n e}\right)\right) ^ {\prime} \\ \mathbf {x} _ {t} ^ {f} \left(\pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} & \mathbf {x} _ {t} ^ {f} \left(\mathbf {x} _ {t} ^ {f} \otimes \pmb {\epsilon} _ {t + 1}\right) ^ {\prime} \\ \mathbf {x} _ {t} ^ {s} \left(\pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} & \mathbf {x} _ {t} ^ {s} \left(\mathbf {x} _ {t} ^ {f} \otimes \pmb {\epsilon} _ {t + 1}\right) ^ {\prime} \\ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} & \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\mathbf {x} _ {t} ^ {f} \otimes \pmb {\epsilon} _ {t + 1}\right) ^ {\prime} \end{array} \right] = \mathbf {0}} \end{array}\]

Now, we only need to compute :

\[\begin{array}{r l r} \mathbb {V} (\pmb {\xi} _ {t + 1} ^ {(2)}) & = & \mathbb {E} \left[ \left[ \begin{array}{c} \pmb {\epsilon} _ {t + 1} \\ \pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1} - v e c (\mathbf {I} _ {n _ {e}}) \\ \pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \\ \mathbf {x} _ {t} ^ {f} \otimes \pmb {\epsilon} _ {t + 1} \end{array} \right] \left[ \begin{array}{c} \pmb {\epsilon} _ {t + 1} \\ \pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1} - v e c (\mathbf {I} _ {n _ {e}}) \\ \pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {\boldsymbol {f}} \\ \mathbf {x} _ {t} ^ {\boldsymbol {f}} \otimes \pmb {\epsilon} _ {t + 1} \end{array} \right] ^ {\prime} \right] \\ & = & \left[ \begin{array}{c c} \mathbf {I} _ {n _ {e}} & \mathbb {E} [ \pmb {\epsilon} _ {t + 1} (\pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1}) ^ {\prime} ] \\ \mathbb {E} [ (\pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1}) \pmb {\epsilon} _ {t + 1} ^ {\prime} ] & \mathbb {E} [ (\pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1} - v e c (\mathbf {I} _ {n _ {e}})) (\pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1} - v e c (\mathbf {I} _ {n _ {e}})) ^ {\prime} ] \\ \mathbf {0} & \mathbf {0} \\ \mathbf {0} & \mathbf {0} \end{array} \right] \\ & & \left[ \begin{array}{c c} \mathbf {0} & \mathbf {0} \\ \mathbf {0} & \mathbf {0} \end{array} \right]. \\ & & \mathbb {E} [ (\pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {\boldsymbol {f}}) (\pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {\boldsymbol {f}}) ^ {\prime} ] & \mathbb {E} [ (\pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {\boldsymbol {f}}) (\mathbf {x} _ {t} ^ {\boldsymbol {f}} \otimes \pmb {\epsilon} _ {t + 1}) ^ {\prime} ] \\ \mathbb {E} [ (\mathbf {x} _ {t} ^ {\boldsymbol {f}} \otimes \pmb {\epsilon} _ {t + 1}) (\pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {\boldsymbol {f}}) ^ {\prime} ] & \mathbb {E} [ (\mathbf {x} _ {t} ^ {\boldsymbol {f}} \otimes \pmb {\epsilon} _ {t + 1}) (\mathbf {x} _ {t} ^ {\boldsymbol {f}} \otimes \pmb {\epsilon} _ {t + 1}) ^ {\prime} ] \end{array}\right). \end{array}\]

This variance is finite when has a finite fourth moment. All elements in this matrix can be computed element-by-element.

A.6 Second Order: Unconditional Third and Fourth Moments

We consider the system , where A is stable and are mean-zero innovations. Thus, the pruned state-space representations of DSGE models belong to this class. For notational convenience, the system is expressed in deviation from its mean as . Therefore

\[\begin{array}{r} \mathbf {x} _ {t + 1} = (\mathbf {I} - \mathbf {A}) \mathbb {E} [ \mathbf {x} _ {t} ] + \mathbf {A x} _ {t} + \mathbf {v} _ {t + 1} \Rightarrow \\ \mathbf {x} _ {t + 1} - \mathbb {E} [ \mathbf {x} _ {t} ] = \mathbf {A} (\mathbf {x} _ {t} - E [ \mathbf {x} _ {t} ]) + \mathbf {v} _ {t + 1} \Rightarrow \\ \mathbf {z} _ {t + 1} = \mathbf {A z} _ {t} + \mathbf {v} _ {t + 1} \end{array}\]

We then have

\[\begin{array}{r c l} \mathbf {z} _ {t + 1} \otimes \mathbf {z} _ {t + 1} & = & (\mathbf {A z} _ {t} + \mathbf {v} _ {t + 1}) \otimes (\mathbf {A z} _ {t} + \mathbf {v} _ {t + 1}) \\ & = & \mathbf {A z} _ {t} \otimes \mathbf {A z} _ {t} + \mathbf {A z} _ {t} \otimes \mathbf {v} _ {t + 1} + \mathbf {v} _ {t + 1} \otimes \mathbf {A z} _ {t} + \mathbf {v} _ {t + 1} \otimes \mathbf {v} _ {t + 1}, \end{array}\]

\[\begin{array}{r c l} \mathbf {z} _ {t + 1} \otimes \mathbf {z} _ {t + 1} \otimes \mathbf {z} _ {t + 1} & = & \mathbf {A z} _ {t} \otimes \mathbf {A z} _ {t} \otimes \mathbf {A z} _ {t} + \mathbf {A z} _ {t} \otimes \mathbf {A z} _ {t} \otimes \mathbf {v} _ {t + 1} \\ & & + \mathbf {A z} _ {t} \otimes \mathbf {v} _ {t + 1} \otimes \mathbf {A z} _ {t} + \mathbf {A z} _ {t} \otimes \mathbf {v} _ {t + 1} \otimes \mathbf {v} _ {t + 1} \\ & & + \mathbf {v} _ {t + 1} \otimes \mathbf {A z} _ {t} \otimes \mathbf {A z} _ {t} + \mathbf {v} _ {t + 1} \otimes \mathbf {A z} _ {t} \otimes \mathbf {v} _ {t + 1} \\ & & + \mathbf {v} _ {t + 1} \otimes \mathbf {v} _ {t + 1} \otimes \mathbf {A z} _ {t} + \mathbf {v} _ {t + 1} \otimes \mathbf {v} _ {t + 1} \otimes \mathbf {v} _ {t + 1} \end{array}\]

and

\[\begin{array}{r c l} \mathbf {z} _ {t + 1} \otimes \mathbf {z} _ {t + 1} \otimes \mathbf {z} _ {t + 1} \otimes \mathbf {z} _ {t + 1} & = & \mathbf {A z} _ {t} \otimes \mathbf {A z} _ {t} \otimes \mathbf {A z} _ {t} \otimes \mathbf {A z} _ {t} + \mathbf {A z} _ {t} \otimes \mathbf {A z} _ {t} \otimes \mathbf {A z} _ {t} \otimes \mathbf {v} _ {t + 1} \\ & & + \mathbf {A z} _ {t} \otimes \mathbf {A z} _ {t} \otimes \mathbf {v} _ {t + 1} \otimes \mathbf {A z} _ {t} + \mathbf {A z} _ {t} \otimes \mathbf {A z} _ {t} \otimes \mathbf {v} _ {t + 1} \otimes \mathbf {v} _ {t + 1} \\ & & + \mathbf {A z} _ {t} \otimes \mathbf {v} _ {t + 1} \otimes \mathbf {A z} _ {t} \otimes \mathbf {A z} _ {t} + \mathbf {A z} _ {t} \otimes \mathbf {v} _ {t + 1} \otimes \mathbf {A z} _ {t} \otimes \mathbf {v} _ {t + 1} \\ & & + \mathbf {A z} _ {t} \otimes \mathbf {v} _ {t + 1} \otimes \mathbf {v} _ {t + 1} \otimes \mathbf {A z} _ {t} + \mathbf {A z} _ {t} \otimes \mathbf {v} _ {t + 1} \otimes \mathbf {v} _ {t + 1} \otimes \mathbf {v} _ {t + 1} \\ & & + \mathbf {v} _ {t + 1} \otimes \mathbf {A z} _ {t} \otimes \mathbf {A z} _ {t} \otimes \mathbf {A z} _ {t} + \mathbf {v} _ {t + 1} \otimes \mathbf {A z} _ {t} \otimes \mathbf {A z} _ {t} \otimes \mathbf {v} _ {t + 1} \\ & & + \mathbf {v} _ {t + 1} \otimes \mathbf {A z} _ {t} \otimes \mathbf {v} _ {t + 1} \otimes \mathbf {A z} _ {t} + \mathbf {v} _ {t + 1} \otimes \mathbf {A z} _ {t} \otimes \mathbf {v} _ {t + 1} \otimes \mathbf {v} _ {t + 1} \\ & & + \mathbf {v} _ {t + 1} \otimes \mathbf {v} _ {t + 1} \otimes \mathbf {A z} _ {t} \otimes \mathbf {A z} _ {t} + \mathbf {v} _ {t + 1} \otimes \mathbf {v} _ {t + 1} \otimes \mathbf {A z} _ {t} \otimes \mathbf {v} _ {t + 1} \\ & & + \mathbf {v} _ {t + 1} \otimes \mathbf {v} _ {t + 1} \otimes \mathbf {v} _ {t + 1} \otimes \mathbf {A z} _ {t} + \mathbf {v} _ {t + 1} \otimes \mathbf {v} _ {t + 1} \otimes \mathbf {v} _ {t + 1} \otimes \mathbf {v} _ {t + 1}. \end{array}\]

Thus, to solve for , the innovations need to have a finite third moment. At second order, depends on , meaning that must have a finite sixth moment. Similarly, to solve for , the innovations need to have finite fourth moments. At second order, depends on , meaning that must have a finite eighth moment.

A.7 Coefficients for the Pruned State-Space System at Third Order

\[\begin{array}{r l} & {\mathbf {A} ^ {(3)} \equiv \left[ \begin{array}{c c c c c c} \mathbf {h _ {x}} & 0 & 0 & 0 & 0 & 0 \\ 0 & \mathbf {h _ {x}} & \frac {1}{2} \mathbf {H _ {x x}} & 0 & 0 & 0 \\ 0 & 0 & \mathbf {h _ {x}} \otimes \mathbf {h _ {x}} & 0 & 0 & 0 \\ \frac {3}{6} \mathbf {h _ {\sigma \sigma x} \sigma^ {2}} & 0 & 0 & \mathbf {h _ {x}} & \mathbf {H _ {x x}} & \frac {1}{6} \mathbf {H _ {x x x}} \\ \mathbf {h _ {x}} \otimes \frac {1}{2} \mathbf {h _ {\sigma \sigma} \sigma^ {2}} & 0 & 0 & 0 & \mathbf {h _ {x}} \otimes \mathbf {h _ {x}} & \mathbf {h _ {x}} \otimes \frac {1}{2} \mathbf {H _ {x x}} \\ 0 & 0 & 0 & 0 & 0 & \mathbf {h _ {x}} \otimes \mathbf {h _ {x}} \otimes \mathbf {h _ {x}} \end{array} \right],} \\ & {\mathbf {B} ^ {(3)} \equiv \left[ \begin{array}{c c c c c c} \sigma \eta & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & \sigma \eta \otimes \sigma \eta & \sigma \eta \otimes \mathbf {h _ {x}} & \mathbf {h _ {x}} \otimes \sigma \eta \\ 0 & 0 & 0 & 0 \\ \sigma \eta \otimes \frac {1}{2} \mathbf {h _ {\sigma \sigma} \sigma^ {2}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \sigma \eta \otimes \mathbf {h _ {x}} & \sigma \eta \otimes \frac {1}{2} \mathbf {H _ {x x}} & 0 & 0 \\ 0 & \sigma \eta \otimes \mathbf {h _ {x}} \otimes \mathbf {h _ {x}} & \mathbf {h _ {x}} \otimes \mathbf {h _ {x}} \otimes \sigma \eta & \mathbf {h _ {x}} \otimes \sigma \eta \otimes \mathbf {h _ {x}} \\ \end{array} \right],} \end{array}\]

\[\begin{array} { r l } & { \left[ \begin{array} { c c c c } \mathbf { 0 } & \mathbf { 0 } & \mathbf { 0 } & \mathbf { 0 } \\ \mathbf { 0 } & \mathbf { 0 } & \mathbf { 0 } & \mathbf { 0 } \\ \mathbf { 0 } & \mathbf { 0 } & \mathbf { 0 } & \mathbf { 0 } \\ \mathbf { 0 } & \mathbf { 0 } & \mathbf { 0 } & \mathbf { 0 } \\ \mathbf {\delta _ { t + 1 } } & \sigma \eta \otimes \sigma \eta & \sigma \eta \otimes \mathbf { h _ { x } } \otimes \sigma \eta & \sigma \eta \otimes \sigma \eta \otimes \mathbf { h _ { x } } & \sigma \eta \otimes \sigma \eta \otimes \sigma \eta \\ \hline \end{array} \right] , } \\ & { \pmb { \xi _ { t + 1 } ^ { ( 3 ) } } \equiv \left[ \begin{array} { c } \pmb { \epsilon _ { t + 1 } } \\ \pmb { \epsilon _ { t + 1 } } \otimes \pmb { \epsilon _ { t + 1 } } - v e c ( \mathbf { I _ { n _ { e } } } ) \\ \pmb { \epsilon _ { t + 1 } } \otimes \mathbf { x _ { t } ^ { f } } \\ \mathbf { x _ { t } ^ { f } } \otimes \pmb { \epsilon _ { t + 1 } } \\ \pmb { \epsilon _ { t + 1 } } \otimes \mathbf { x _ { t } ^ { s } } \\ \pmb { \epsilon _ { t + 1 } } \otimes \mathbf { x _ { t } ^ { f } } \otimes \mathbf { x _ { t } ^ { f } } \\ \mathbf { x _ { t } ^ { f } } \otimes \mathbf { x _ { t } ^ { f } } \otimes \pmb { \epsilon _ { t + 1 } } \\ \mathbf { x _ { t } ^ { f } } \otimes \pmb { \epsilon _ { t + 1 } } \otimes \mathbf { x _ { t } ^ { f } } \\ \mathbf { x _ { t } ^ { f } } \otimes \pmb { \epsilon _ { t + 1 } } \otimes \pmb { \epsilon _ { t + 1 } } \\ \pmb { \epsilon _ { t + 1 } } \otimes \mathbf { x _ { t } ^ { f } } \otimes \pmb { \epsilon _ { t + 1 } } \\ \pmb { \epsilon _ { t + 1 } } \otimes \pmb { \epsilon _ { t + 1 } } \otimes \mathbf { x _ { t } ^ { f } } \\ ( \pmb { \epsilon _ { t + 1 } } \otimes \pmb { \epsilon _ { t + 1 } } \otimes \pmb { \epsilon _ { t + 1 } } ) - E [ ( \pmb { \epsilon _ { t + 1 } } \otimes \pmb { \epsilon _ { t + 1 } } \otimes \pmb { \epsilon _ { t + 1 } } ) ] \\ \end{array} \right] , } \\ & \textbf { c } ^ { ( 3 ) } = \left[ \begin{array} { c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c } & 0 _ { n _ { x } \times 1 } \\ & \frac { 1}{2} {\bf h} _ { \sigma \sigma } {\sigma ^ { 2 } } \\ & ( \sigma \eta \otimes \sigma \eta ) v e c ( {\bf I} _ { n _ { e } } ) \\ & \frac { 1}{6} {\bf h} _ { \sigma \sigma \sigma } {\sigma ^ { 3 } } \\ & {\bf 0} _ { n _ { x } ^ { 2 } \times 1 } \\ ( \sigma \eta \otimes \sigma \eta \otimes \sigma \eta ) E [ ( \pmb { \epsilon _ { t + 1 } } \otimes \pmb { \epsilon _ { t + 1 } } \otimes \pmb { \epsilon _ { t + 1} } ) ] \\ \end{array} \right] , \\ & C ^ { ( 3 ) } = [ ~ g _ { x } + ~ {\frac { 3}{ 6 }} g _ { \sigma \sigma x } σ ^ 2 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ $ [~ [~ g _ { x x} +~ {\frac { 3}{ 6}} g _ { o s o x} ] / [ g _ { x x} +~ {\frac { 3}{ 6}} g _ { o s o x} ] / [ g _ { x x} +~ {\frac { 3}{ 6}} g _ { o s o x} ] / [ g _ { x x} +~ {\frac { 3}{ 6}} g _ { o s o x} ] / [ g _ { x x} +~ {\frac { 3}{ 5}} g _ { o s o x} ] / [ g _ { x x} +~ {\frac { 3}{ 6}} g _ { o s o x} ] / [ g _ { x x} +~ {\frac { 3}{ 6}} g _ { o s o x} ] / [ g _ { x x} +~ {\frac { 3}{ 6}} g _ { o s o x} ] / [ g _ { y y} + [ g _ { y y} + {\frac { 3}{ 6}} g _ { y y} ] / [ g _ { y y} + [ g _ { y y} + {\frac { 3}{ 6}} g _ { y y} ] / [ g _ { y y} + [ g _ { y y} + [ g _ { y y} + [ g _ { y y} + [ g _ { y y} + [ g _ { y y} + [ g _ { y y} + [ g _ { y y} + [ g _ { y y} + [ g _ { y y} + [ g _ { y y} + [ g _ { y y} + ] / [ g _ { y y} + [ g _ { y y} + [ g _ { y y} + [ g _ { y y} + [ g _ { y y} + [ g _ { y y} + [ g _ { y y} + [ g _ { y y} + [ g _ { y y} + [ g _ { y y} + [ g _ { y y} - [ g _ { y y} - [ g _ { y y} - [ g _ { y y} - [ g _ { y y} - [ g _ { y y} - [ g _ { y y} - [ g _ { y y} - [ g _ { y y} - [ g _ { y y} - [ g _ { y y} - [ g _ { y y} - [ g _ { y y} - [ g _ {\mathrm{e}} - [ g _ {\mathrm{e}} - [ g _ {\mathrm{e}} - [ g _ {\mathrm{e}} - [ g _ {\mathrm{e}} - [ g _ {\mathrm{e}} - [ g _ {\mathrm{e}} - [ g _ {\mathrm{e}} - [ g _ {\mathrm{e}} - [ g _ {\mathrm{e}} - [ g _ {\mathrm{e}} - [ g _ {\mathrm{e}}- [ g _ {\mathrm{e}} - [ g _ {\mathrm{e}} - [ g _ {\mathrm{e}} - [ g _ {\mathrm{e}} - [ g _ {\mathrm{e}} - [ g _ {\mathrm{e}} - [ g _ {\mathrm{e}} - [ g _ {\mathrm{e}} - [ g _ {\mathrm{e}} - [ g _ {\mathrm{e}} - [ g _ {\mathrm{e}} - | p a r e d i n e r s i n e r d i n e r s i n e r d i n e r s i n e r d i n e r s i n e r d i n e r s i n e r d i n e r s i n e r d i n e r s i n e r d i n e r s i n e r d i n e r s i n e r d i n e r s i n e r d i n e r s i n e l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l | p a r e d i n e r s i n e r d i n e r s i n e r d i n e r s i n e r d i n e r s i n e r d i n e r s i n e r d i n e r s i n e r d i n e r s i n e r d i n e r s i n e r d i n e r s i n e r d i n er s i n e r d i n e r s i n e r d i n e r s i n e r d i n e r s i n e r d i n e r s i n e r d i n e r s i n e r d i n e r s i n e r d i n e r s i n e r d i n e r s i n e r d i n e r s i n e r d i n er s i n e t h a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a b a w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w u v e d ] , \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ . m a x = (m) ^ {- j}, m = (m) ^ {- k}, m = (k) ^ {- m}, m = (k) ^ {- n}, m = (n) ^ {- k}, m = (n) ^ {- m}, m = (n) ^ {- k}, m = (n) ^ {- m}, m = (n) ^ {- k}, m = (n) ^ {- m}, m = (n) ^ {- k}, m = (n) ^ {- m}, m = (n) ^ {- k}, m = (n) ^ {- m}, m = (n) ^ {- k}, m = (n) ^ {- m}, m = (n) ^ {- k}. \\ . m a x = (m) ^ {- j}, m = (m) ^ {- k}, m = (k) ^ {- m}, m = (k) ^ {- m}, m = (k) ^ {- k}, m = (k) ^ {- m}, m = (k) ^ {- k}, m = (k) ^ {- m}, m = (k) ^ {- k}, m = (k) ^ {- m}, m = (k) ^ {- k}, m = (k) ^ {- m}, m = (k) ^ {- k}. \\ . m a x = (m) ^ {- j}, m = (m) ^ {- k}, m = (k) ^ {- m}, m = (k) ^ {- m}, m = (k) ^ {- k}, m = (k) ^ {- m}, m = (k) ^ {- k}, m = (k) ^ {- m}, m = (k) ^ {- k}, m = (k) ^ {- m},m a x = (m) ^ {- j}, m = (m) ^ {- k}, m = (k) ^ {- m}, m = (k) ^ {- k}, m = (k) ^ {- k}, m = (k) ^ {- m}, m = (k) ^ {- k}, m = (k) ^ {- k}. \\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. ,\\ * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,$ M, M, M, M, M, M, M, M, M, M, M, M, M, M, M, M, M, M, M, M, M, M, M, M, M, M, M, M, M, M, M, M, M, M, M, M, M, M, M, M, M, M, M, M, M, M, M, M, M, M, M |< img src="c", "c">\\ ;M,a,b,c,d,e,f,g,h,i,j,k,l,m,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,u,w,x,y,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z,Z.Z,A,B,C,D,E,F,G,H,I,J,K,L,M,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N-N,A,B,C,D,E,F,G,H,I,J,K,L,M,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,NN,A,B,C,D,E,F,G,H,I,J,K,L,M,NN,A,B,C,D,E,F,G,H,I,J,K,L,M,NN,A,B,C,D,E,F,G,H,I,J,K,L,M,NN,A,B,C,D,E,F,G,H,I,J,K,L,M N,A,B,C,D,E,F,G,H,I,J,K,L,M N,A,B,C,D,E,F,G,H,I,J,K,L,M N,A,B,C,D,E,F,G,H,I,J,K,L,M N,A,B,C,D,E,F,G,H,I,J,K,L,M N,A,B,C,D,E,F,G,H,I,J,K,L,M N,A,B,C,D,E,F,G,H,I,J,K,L,M N,A,B,C,D,E,F,G,H,I,J,K,L,M N,A B,C,D,E,F,G,H,I,J,K,L,M N A,B,C,D,E,F,G,H,I,J,K,L,M N A,B,C,D,E,F,G,H,I,J,K,L,M N A,B,C,D,E,F,G,H,I,J,K,L,M N A,B,C,D,E,F,G,H,I,J,K,L,M N A,B,C,D,E,F,G,H,I,J,K,L,M N A,B,C,D,E,F,G,H,I,J,K,L,M N A,B,C,D,E,F,G,H,I,J,K,L,M N A,B,C D,E,F,G,H,I,J,K,L,M N A,B,C,D,E,F,G,H,I,J,K,L,M N A,B,C,D,E,F,G,H,I,J,K,L,M N A B,C,D,E,F,G,H,I,J,K,L,M N A B,C,D,E,F,G,H,I,J,K,L,M N A B,C,D,E,F,G,H,I,J,K,L,M N A B,C,D,E,F,G,H,I,J,K,L,M N A B,C,D,E,F,G,H,I,J,K,L,M N A B,C,D,E,F,G,H,I,J,K,L,M N A B,C,D,E,F,G,H,I,J,K,L,M N A B,C D,E,F,G,H,I,J,K,L,M N A B,C,D,E,F,G,H,I,J,K,L,M N A B,C,D,E,F,G,H,I,J,K,L,M N A B,C,D,E,F,G,H,I,J,K,L,M N A B,C,D,E,F,G,H,I,J,K,L,M N A B,C,D,E,F,G,H,I,J,K,L,M N A B,C,D,E,F,G,H,I,J,K,L,M N A B,C,D,E F,G,H,I,J,K,L,M N A B,C,D,E,F,G,H,I,J,K,L,M N A B,C,D,E,F,G,H,I,J,K,L,M N A B,C,D,E,F,G,H,I,J,K,L,M N A B,C,D,E,F,G,H,I,J,K,L,M N A B,C,D,E,F,G,H,I,J,K,L,M N A B,C,D,E,F,G,H,I,J,K,L,M N A B,C,D,E,F,G H,I,J,K,L,M N A B,C,D,E,F,G,H,I,J,K,L,M N A B,C,D,E,F,G,H,I,J,K,L,M N A B,C,D,E,F,G,H,I,J,K,L,M N A B,C,D,E,F,G,H,I,J,K,L,M N A B,C,D,E,F,G,H,I,J,K,L,M N A B,C,D,E,F,G,H,I,J,K,L,M N A B,C,D,E,F,G,H,I,j,K,L,M N A B,C,D,E,F,G,H,I,J,K,L,M N A B,C,D,E,F,G,H,I,J,K,L,M N A B,C,D,E,F,G,H,I,J,K,L,M N A B,C,D,E,F,G,H,I,J,K,L,M N A B,C,D,E,F,G,H,I,J,K,L,M N A B,C,D,E,F,G,H,I,J,K,L,M N A B,C,D,E,F,G,H,I,j,K,l,M N A B,C,D,E,F,G,H,I,j,K,l,M N A B,C,D,E,F,G,H,I,j,K,l,M N A B,C,D,E,F,G,H,I,j,K,l,M N A B,C,D,E,F,G,H,I,j,K,l,M N A B,C,D,E,F,G,H,I,j,K,l,M N A B,C,D,E,F,G,H,I,j,K,l,M N A B,C,D,E,F,G,H,I,j,K,l,M |< img src="c", "c">\\ ;M,a,b,c,d,e,f,g,h,i,j,k,l,m,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,p,q,r,s,t,v,w,x,y,k,l,m,n,p,q,r,s,t,v,w,x,y,k,l,m,n,p,q,r,s,t,v,w,x,y,k,l,m,n,p,q,r,s,t,v,w,x,y,k,l,m,n,p,q,r,s,t,v,w,x,y,k,l,m,n,p,q,r,s,t,v,w,x,y,k,l,m,n,p,q,r,s,t,v,w,x,y,k,l,m,n,p,q,r,s,t,v,w,x,y,k,l,m,n,p,q,r,s,t,v,w,X,y,k,l,m,n,p,q,r,s,t,v,w,x,y,k,l,m,n,p,q,r,s,t,v,w,x,y,k,l,m,n,p,q,r,s,t,v,w,x,y,k,l,m,n,p,q,r,s,t,v,w,x,y,k,l,m,n,p,q,r,s,t,v,w,x,y,k,l,m,n,p,q,r,s,t,v,w,x,y,k,l,m,n,p,q,r,s,t,v,w,x,y,k,l,m,n,p,q,r,S,t,v,w,x,y,k,l,m,n,p,q,r,s,t,v,w,x,y,k,l,m,n,p,q,r,s,t,v,w,x,y,k,l,m,n,p,q,r,s,t,v,w,x,y,k,l,m,n,p,q,r,s,t,v,w,x,y,k,l,m,n,p,q,r,s,t,v,w,x,y,k,l,m,n,p,q,r,s,t,v,w,x,y,k,l,m,n,p,q,r,s,t,v,w,x,y,k,l,m,,v,w,x,y,k,l,m,,v,w,x,y,k,l,m,,v,w,x,y,k,l,m,,v,w,x,y,k,l,m,,v,w,x,y,k,l,m,,v,w,x,y,k,l,m,,v,w,x,y,k,l,m,,v,w,x,y,k,l,m,,v,w,x,y,k,l,m,,v,w,x,y,k,l,m,,v,w,x,y,k,l,m,,v,w,x,y,k,l,m,,v,w,x,y-k,l,m,,v,w,x,y,k,l,m,,v,w,x,y,k,l,m,,v,w,x,y,k,l,m,,v,w,x,y,k,l,m,,v,w,x,y,k,l,m,,v,w,x,y,k,l,m,,v,w,x,y,k,l,m,,v,w,x,y,k,l,m,,v,w,x,y,k,l,m,,v,w,x,y,k,l,m,,v,w,x,y,k,l,m,,v,w,x,y,k,l,m,, v,w,x,y,k,l,m,,v,w,x,y,k,l,m,,v,w,x,y,k,l,m,,v,w,x,y,k,l,m,,v,w,x,y,k,l,m,,v,w,x,y,k,l,m,,v,w,x,y,k,l,m,,v,w,x,y,k,l,m,,v,w,x,y,k,l,m,,v,w,x,y,k,l,m,,v,w,x,y,k,l,m,,v,w,x,y,k,l,m,,v,w,x,y.k,l,m,,v,w,x,y.k,l,m,,v,w,x,y.k,l,m,,v,w,x,y.k,l,m,,v,w,x,y.k,l,m,,v,w,x,y.k,l,m,,v,w,x,y.k,l,m,,v,w,x,y.k,l,m,,v,w,x,y.k,l,m,,v,w,x,y.k,l,m,,v,w,x,y.k,l,m,,v,w,x,y.k,l,m,,v,w,x,y.k,l,m,, v,w,x,y.k,l,m,, v,w,x,y.k,l,m,, v,w,x,y.k,l,m,, v,w,x,y.k,l,m,, v,w,x,y.k,l,m,, v,w,x,y.k,l,m,, v,w,x,y.k,l,m,, v,w,x,y.k,l,m,, v,w,x,y.k,l,m,, v,w,x,y.k,l,m,, v,w,x,y.k,l,m,, v,w,x,y.k,l,m,, v,w,x,y. K S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S uS uS uS uS uS uS uS uS uS uS uS uS uS uS uS uS uS uS uS uS uS uS uS uS uS uS uS uS uS uS uS uS uS uS uS uS uS uS uS uS uS uS uS uS uS uS uS uS uS uS uTssuTssuTssuTssuTssuTssuTssuTssuTssuTssuTssuTssuTssuTssuTssuTssuTssuTssuTssuTssuTssuTssuTssuTssuTssuTssuTssuTssuTssuTssuTssuTssuTssuTssuUSSUSSUSSUSSUSSUSSUSSUSSUSSUSSUSSUSSUSSUSSUSSUSSUSSUSSUSSUSSUSSUSSUSSUSSUSSUSSUSSUSSUSSUSSUSSUSSUSSUSSUSSUSSUSSUSSUSSUSSUSSUSSUSSUSSUSSUSSUSSUSSUSSUSSUSTOUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUSUS US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US US< ecel>< nl>\]

A.8 Third Order: Stability

To prove stability:

\[\begin{array}{l} p (\lambda) = \left| \mathbf {A} ^ {(3)} - \lambda \mathbf {I} \right| \\ = \left| \left[ \begin{array}{c c c c c c} \mathbf {h} _ {\mathbf {x}} - \lambda \mathbf {I} & \mathbf {0} & \mathbf {0} & \mathbf {0} & \mathbf {0} & \mathbf {0} \\ \mathbf {0} & \mathbf {h} _ {\mathbf {x}} - \lambda \mathbf {I} & \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} & \mathbf {0} & \mathbf {0} & \mathbf {0} \\ \mathbf {0} & \mathbf {0} & \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} - \lambda \mathbf {I} & \mathbf {0} & \mathbf {0} & \mathbf {0} \\ \frac {3}{6} \mathbf {h} _ {\sigma \sigma \mathbf {x}} \sigma^ {2} & \mathbf {0} & \mathbf {0} & \mathbf {h} _ {\mathbf {x}} - \lambda \mathbf {I} & \mathbf {H} _ {\mathbf {x x}} & \frac {1}{6} \mathbf {H} _ {\mathbf {x x x}} \\ \mathbf {h} _ {\mathbf {x}} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2} & \mathbf {0} & \mathbf {0} & \mathbf {0} & \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} - \lambda \mathbf {I} & \mathbf {h} _ {\mathbf {x}} \otimes \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \\ \mathbf {0} & \mathbf {0} & \mathbf {0} & \mathbf {0} & \mathbf {0} & \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} - \lambda \mathbf {I} \\ \end{array} \right] \\ = \left| \left[ \begin{array}{c c} \mathbf {B} _ {1 1} & \mathbf {B} _ {1 2} \\ \mathbf {B} _ {2 1} & \mathbf {B} _ {2 2} \end{array} \right] \right|, \\ w h e r e \\ \mathbf {B} _ {1 1} \equiv \left[ \begin{array}{c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c} \\ [ 0 ] ^ {- 1 / 2}, \\ [ 0 ] ^ {- 1 / 2}, \\ [ 0 ] ^ {- 1 / 2}, \\ [ 0 ] ^ {- 1 / 2}, \\ [ 0 ] ^ {- 1 / 2}, \\ [ 0 ] ^ {- 1 / 2}, \\ [ 0 ] ^ {- 1 / 2}, \\ [ 0 ] ^ {- 1 / 2}, \\ [ 0 ] ^ {- 2 / 2}, \\ [ 0 ] ^ {- 2 / 2}, \\ [ 0 ] ^ {- 2 / 2}, \\ [ 0 ] ^ {- 2 / 2}, \\ [ 0 ] ^ {- 2 / 2}, \\ [ 0 ] ^ {- 2 / 2}, \\ [ 0 ] ^ {- 2 / 2}, \\ [ 0 ] ^ {- 2 / 2}, \\ [ 0 ]. \\ [ 0 ]. \\ [ 0 ]. \\ [ 0 ]. \\ [ 0 ]. \\ [ 0 ]. \\ [ 0 ]. \\ [ 0 ]. \\ [ 0 ]. \\ [ 0 ]. \\ [ 0 ]. \\ [ 0 ]. \\ [ 0 ]. \\ [ 0 ]. \\ [ 0 ]. \\ [ 0 ]. \\ [ 0 ]. \\ [ 0 ]. \\ [ 0 ]. \\ [ 0 ]. \\ [ 0 ]). \\ [ 0 ]. \\ [ 0 ]. \\ [ 0 ]. \\ [ 0 ]. \\ [ 0 ]. \\ [ 0 ]. \\ [ 0 ]. \\ [ 0 ]. \\ [ 0 ]. \\ [ 0 ]. \\ [ 0 ]. \\ [ 0 ]. \\ [ 0 ]. \\ [ 0 ]. \\ [ 0 ]. \\ [ 0 ]. \\ [ 0 ]. \\ [ 0 ]. \\ [ 0 ]. \\ [ 0]. \\ [ 0 ]. \\ [ 0 ]. \\ [ 0 ]. \\ [ 0 ]. \\ [ 0 ]. \\ [ 0 ]. \\ [ 0 ]. \\ [ 0 ]. \\ [ 0 ]. \\ [ 0 ]. \\ [ 0 ]. \\ [ 0 ]. \\ [ 0 ]. \\ [ 0 ]. \\ [ 0 ]. \\ [ 0 ]. \\ [ 0 ]. \\ [ 0 ]. \\ [ 0 ]. \\ [ 0. ] ^ {- 1 / 2}. \\ [ h _ \sigma x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y s u n g t a l l i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d e f o r e r e r e s t a t e r e s t a t e r e s t a t e r e s t a t e r e s t a t e r e s t a t e r e s t a t e r e s t a t e r e s t a t e r e s t a t e r e s t a t e r e s t a t e r e s t a t e r e s t a t e r e s t a t e t a t e r e s t a t e r e s t a t e r e s t a t e r e s t a t e r e s t a t e r e s t a t e r e s t a t e r e s t a t e r e s t a t e r e s t a t e r e s t a t e r e s t a t e r e s t a t e r e s t o f o r e r e s t o f o r e r e s t o f o r e r e s t o f o r e r e s t o f o r e r e s t o f o r e r e s t o f o r e r e s t o f o r e r e s t o f o r e r e s t o f o r e r e s t o f o r e r e s t o f o r u v a l l i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d j u a l l i n d j u a l l i n d j u a l l j u a l l j u a l l j u a l l j u a l l j u a l l j u a l l j u a l l j u a l l j u a l l j u a l l j u a l l j u a l l j u a l l j u a l l j u a l l j u a l l j u a l l j u a l l j u a l l j u a l | k o m p h o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b u m p h o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b u m p p h o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b u m p h o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b u m p h o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b o w b u m p h o w b o w b o w b o w b o w b o w b o w b o w b o w b u m p h o w b o w b o w b o w b o w b o w b ow b u m p h o w b o w b o w b o w b ow b u m p h o w b o w b ow b ow b u m p h o w b o w b ow b ow b u m p h o w b o w b ow b ow b u m p h o w b o w b ow b ow b u m p h o w b ow b ow b u m p h o w b ow b v a m p h h a m p h a m p h a m p h a m p h a m p h a m p h a m p h a m p h a m p h a m p h a m p h a m p h a m p h a m p h a m p h a m p h a m p h a m p h a m p h a m p h a m p h a m p h a m p h a m p h a m p h a m q u m p h a m p h a m p h a m p h a m p h a m p h a m p h a m p h a m p h a m p h a m p h a m p h a m p h a m p h a m p h a m p h a m p h a m p h a m p h a m p h a m q u m p h a m q u m p h a m q u m p h a m q u m p h a m q u m p h a m q u m p h a m q u m p h a m q u m p h a m q u m p h a m q u m p h a m q u m p h a m q u m p h q u m p h q u m p h q u m p h q u m p h q u m p h q u m p h q u m p h q u m p h q u m p h q u m p h q u m p h q u m p h q u m p h q u m p h q u m p h q u m p h q u m p h q u m p h q u m p h q u m p H q u m p h q u m p H q u m p H q u m p H q u m p H q u m p H q u m p H q u m p H q u m p H q u m p H q u m p H q u m p H q u m p H q u m p H q u m p H q u m p H q u m p H q u m p H q u m p H q u m p H q u m p H q u m p W U W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O NN O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N NO M I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | K S T A L E R S T A L E R S T A L E R S T A L E R S T A L E R S T A L E R S T A L E R S T A L E R S T A L E R S T A L E R S T A L E R S T A L E R S T A L E R S T A L E R S T A L E R S T A L E R S T A L E R S T A F A F A F A F A F A F A F A F A F A F A F A F A F A F A F A F A F A F A F A F A F A F A F A F A F A F A F A F A F A F A F A F A F A F A F A F A F A F A F A F A F A F A F A F A F A F A F A F A F A F A | K S T A L E R S T A L E R S T A L E R S T A L E R S T A L E R S T A L E R S T A L E R S T A L E R S T A L E R S T A L E R S T A L E R S T A L E R S T A L E R S T A L E R S T A L E R S T A L E R S | K S T A L E R S T A L E R S T A L E R S T A L E R S T A L E R S T A L E R S T A L E R S T A L E R S T A L E R S T A L E R S T A L E R S T A L E R S T A L E R S T A L E R S T A L E R S T A L E R S ] | K S T A L E R S T A L E R S T A L E R S T A L E R S T A L E R S T A L E R S T A L E R S T A L E R S T A L E R S T A L E R S T A L E R S T A L E R S T A L E R S T A L E R S T A L E R S T A L E R S (T) | K S T A L E R S T A L E R S T A L E R S T A L E R S T A L E R S T A L E R S T A L E R S T | K S T A L E R S T | K S T | K S T | K S T | K S T | K S T | K S T | K S T | K S T | K S T | K S T | K S T | K S T | K S T | K S T | K S T | K S T | K S T | K S T | K S T | K S T | K S T | K S T | K S T | K S T | K S T | k s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s.s.t.kvcs.kvcs.kvcs.kvcs.kvcs.kvcs.kvcs.kvcs.kvcs.kvcs.kvcs.kvcs.kvcs.kvcs.kvcs.kvcs.kvcs.kvcs.kvcs.kvcs.kvcs.kvcs.kvcs.kvcs.kvcs.kvcs.kvcs.kvcs.kvcs.kvcs.kvcs.kvcs.kvcs.kvcs kvcs.kvcs.kvcs.kvcs.kvcs.kvcs.kvcs.kvcs.kvcs.kvcs.kvcs.kvcs.kvcs.kvcs.kvcs.kvcs.kvcs.kvcs.kvcs.kvcs.kvcs.kvcs.kvcs.kvcs.kvcs.kvcs.kvcs.kvcs.kvcs.kvcs.kvcs.kvcs.kvcs.k vcs k vcs k vcs k vcs k vcs k vcs k vcs k vcs k vcs k vcs k vcs k vcs k vcs k vcs k vcs k vcs k vcs k vcs k vcs k vcs k vcs k vcs k vcs k vnsk vnsk vnsk vnsk vnsk vnsk vnsk vnsk vnsk vnsk vnsk vnsk vnsk vnsk vnsk vnsk vnsk vnsk vnsk vnsk vnsk vnsk vnsk vnsk vnsk vnsk vnsk vnsk vnsk vnsk vnsk vnsk vnsk vnsk. Vnsk: * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< nl>\]

\[\begin{array}{l} = | \mathbf {B} _ {1 1} | | \mathbf {B} _ {2 2} | \\ = | \mathbf {h} _ {\mathbf {x}} - \lambda \mathbf {I} | | \mathbf {h} _ {\mathbf {x}} - \lambda \mathbf {I} | | \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} - \lambda \mathbf {I} | | \mathbf {B} _ {2 2} | \\ (\text { using the result from the proof of proposition 1 }) \\ = | \mathbf {h} _ {\mathbf {x}} - \lambda \mathbf {I} | | \mathbf {h} _ {\mathbf {x}} - \lambda \mathbf {I} | | \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} - \lambda \mathbf {I} | | \mathbf {h} _ {\mathbf {x}} - \lambda \mathbf {I} | | \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} - \lambda \mathbf {I} | | \mathbf {h} _ {\boldsymbol {\texttt {x}}} \otimes \mathbf {h} _ {\boldsymbol {\texttt {x}}} \otimes \mathbf {h} _ {\boldsymbol {\texttt {x}}} - \lambda \mathbf {I} | \end{array}\]

(using the rule on block determinants repeatedly on ).

The eigenvalue solves , which implies:

\[\left| \mathbf {h} _ {\mathbf {x}} - \lambda \mathbf {I} \right| = 0 \text {or} \left| \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} - \lambda \mathbf {I} \right| = 0 \text {or} \left| \left(\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}\right) - \lambda \mathbf {I} \right| = 0\]

The absolute values of all eigenvalues to the first problem are strictly less than one by assumption. That is . This is also the case for the second problem, because the eigenvalues to are for and . The same argument ensures that the absolute values of all eigenvalues to the third problem are also less than one. This shows that all eigenvalues of have modulus less than one.

A.9 Third Order: Unconditional Second Moments

For the variance, we have

\[\begin{array}{r c l} \mathbb {V} \left[ \mathbf {z} _ {t + 1} ^ {(3)} \right] & = & \mathbf {A} ^ {(3)} \mathbb {V} \left[ \mathbf {z} _ {t} ^ {(3)} \right] \left(\mathbf {A} ^ {(3)}\right) ^ {\prime} + \mathbf {B} ^ {(3)} \mathbb {V} \left[ \boldsymbol {\xi} _ {t + 1} ^ {(3)} \right] \left(\mathbf {B} ^ {(3)}\right) ^ {\prime} \\ & & + \mathbf {A} ^ {(3)} C o v \left[ \mathbf {z} _ {t} ^ {(3)}, \boldsymbol {\xi} _ {t + 1} ^ {(3)} \right] \left(\mathbf {B} ^ {(3)}\right) ^ {\prime} + \mathbf {B} ^ {(3)} C o v \left[ \boldsymbol {\xi} _ {t + 1} ^ {(3)}, \mathbf {z} _ {t} ^ {(3)} \right] \left(\mathbf {A} ^ {(3)}\right) ^ {\prime} \end{array}\]

Contrary to a second-order approximation, . This is seen as follows:

\[\begin{array} { r l } & { \mathbb { E } \left[ \mathbf { z } _ { t } ^ { ( 3 ) } \left( \pmb { \xi } _ { t + 1 } ^ { ( 3 ) } \right) ^ { \prime } \right] = \mathbb { E } \left[ \left[ \begin{array} { c } \mathbf { x } _ { t } ^ { f } \\ \mathbf { x } _ { t } ^ { s } \\ \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \\ \mathbf { x } _ { t } ^ { r d } \\ \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { s } \\ \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \end{array} \right] \right. } \\ & \times \left[ \begin{array} { c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c } \boldsymbol { \epsilon } _ { t + 1 } ^ { \prime } & ( \pmb { \epsilon } _ { t + 1 } \otimes \pmb { \epsilon } _ { t + 1 } - v e c ( \mathbf { I } _ { n _ { e} } ) ) ^ { \prime } & ( \pmb { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } ) ^ { \prime } & ( \mathbf { x } _ { t } ^ { f } \otimes \pmb { \epsilon } _ { t + 1 } ) ^ { \prime } & ( \pmb { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { s } ) ^ { \prime } & ( \pmb { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) ^ { \prime } \\ ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \otimes \pmb { \epsilon } _ { t + 1 } ) ^ { \prime } & ( \mathbf { x } _ { t } ^ { f } \otimes \pmb { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } ) ^ { \prime } & ( \mathbf { x } _ { t } ^ { f } \otimes \pmb { \epsilon } _ { t + 1 } \otimes \pmb { \epsilon } _ { t + 1 } ) ^ { \prime } & ( \pmb { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \otimes \pmb { \epsilon } _ { t + 1 } ) ^ { \prime } \\ ( \pmb { \epsilon } _ { t + 1 } \otimes \pmb { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } ) ^ { \prime } & ( ( ( \pmb { \epsilon } _ { t + 1 } \otimes \pmb { \epsilon } _ { t + 1 } \otimes \pmb { \epsilon } _ { t + 1 } ) - E [ ( ( \pmb { \epsilon } _ { t + 1 } \otimes ( | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | & = \left[ \begin{array} l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l R _ { 1 , 1 } & R _ { 1 , 2 } & R _ { 1 , 3 } & 0 _ n _ { x } \times n _ { e } ^ { 3 } e m a n d o u s s u a d i o n g o r e d o u s s u a d i o n g o r e d o u s s u a d i o n g o r e d o u s s u a d i o n g o r e d o u s s u a d i o n g o r e d o u s s u a d i o n g o r e d o u s s u a d i o n g o r e d o u s s u a d i i n g o r e d o u s s u a d i i n g o r e d o u s s u a d i i n g o r e d o u s s u a d i i n g o r e d o u s s u a d i i n g o r e d o u s s u a d i i n g o r e d o u s s u a d i i n g o r e d o u ss u a d i i n g o r e d o u s s u a d i i n g o r e d o u s s u a d i i n g o r e d o u s s u a d i i n g o r e d o u s s u a d i i n g o r e d o u s s u a d i i n g o r e d o u s s u a d i i n g o r k e m a n d e d e d e d e d e d e d e d e d e d e d e d e d e d e d e d e d e d e d e d e d e d e d e d e d e d e d e d e d e d e d e d e d e d e d e d e d e d e d e d e d e d e d e d e d e d e d e d e d e d e d . \\ R _ { 2 , 1 } & R _ { 2 , 2 } & R _ { 2 , 3 } & 0 _ n _ { x } \times n _ { e } ^ { 3 } e m a n d o u s s u a d i o n g o r e d o u s s u a d i i n g o r e d o u s s u a d i i n g o r e d o u s s u a d i i n g o r e d o u s s u a d i i n g o r e d o u s s u a d i i n g o r e d o u s s u a d i j i n g o r e d o u s s u a d i j i n g o r e d o u s s u a d i j i n g o r e d o u s s u a d i j i n g o r e d o u s s u a d i j j i n g o r e d o u s s u a d i j j i n g o r e d o u s s u a d i j k i n g o r e d o u s s u a d i j k j i n g o r e d o u s s u a d i j k k j i n g o r e d o u s s u a d i j k k j i n g o r e d o u s s u a d i j k k j i n g o r e d o u s s u a d i j k k j i n g o r e d o u s s u a d i j k k j i n g o r e d o u s s u a d i j k k j i n q , \\ R _ { 3 , 1 } & R _ { 3 , 2 } & R _ { 3 , 3 } & 0 _ n _ { x } ^ { 2 } \times n _ { e } ^ { 3 } e m a n d o u s s u a d i o n g o r e d o u s s u a d i j k j i n g o r e d o u s s u a d i j k j i n g o r e d o u s s u a d i j k j i n g o r e d o u s s u a d i j k j i n g o r e d o u s s u a d i j k j i n g o r e d o u s s u a d i j k j i n g o r e d o u s , \\ R _ { 4 , 1 } & R _ { 4 , 2 } & R _ { 4 , 3 } & 0 _ n _ { x } \times n _ { e } ^ { 3 } e m a n d o u s s u a d i j k j i n g o r e d o u s s u a d i j k j i n g o r e d o u s s u a d i j k j i n g o r e d o u s s u a d i j k j i n g o r e d o u s , \\ R _ { 5 , 1 } & R _ { 5 , 2 } & R _ { 5 , 3 } & 0 _ n _ { x } ^ { 2 } \times n _ { e } ^ { 3 } e m a n d o u s s u a d i j k j i n g o r e d o u s s u a d i j k j i n g o r e d o u s , \\ R _ { 6 , 1 } & R _ { 6 , 2 } & R _ { 6 , 3 } & 0 _ n _ { x } ^ { 3 } \times n _ { e } ^ { 3 } e m a n d o u s s u a d i j k j i n g o r e d o u s , \\ & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & \\ & = & [ R _ { 1 , 1 } & R _ { 1 , 2 } & R _ { 1 , 3 } & 0 _ n _ { x } \times n _ { e } ^ { 3 } e m a n d o u s s u a d i j k j i n g o r e D O U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U I ] \\ & = & [ R _ {\mathrm{e}} ] [ R _ {\mathrm{e}} ] [ R _ {\mathrm{e}} ] [ R _ {\mathrm{e}} ] [ R _ {\mathrm{e}} ] [ R _ {\mathrm{e}} ] [ R _ {\mathrm{e}} ] [ R _ {\mathrm{e}} ] [ R _ {\mathrm{e}} ] [ R _ {\mathrm{e}} ] [ R _ {\mathrm{e}} ] [ R _ {\mathrm{e}} ] [ R _ {\mathrm{e}} ] [ R _ {\mathrm{e}} ] [ R _ {\mathrm{e}} ] [ R _ {\mathrm{e}} ] [ R _ {\mathrm{e}} ] [ R _ {\mathrm{e}} ] [ R _ {\mathrm{e}} ] [ R _ {\mathrm{e}} ] [ R _ {\mathrm{e}} ] [ R _ {\mathrm{e}} ] [ R _ {\mathrm{e}} ] [ K ] ] \\ & = & [ R _ {\mathrm{e}} ] [ R _ {\mathrm{e}} ] [ R _ {\mathrm{e}} ] [ R _ {\mathrm{e}} ] [ R _ {\mathrm{e}} ] [ R _ {\mathrm{e}} ] [ R _ {\mathrm{e}} ] [ R _ {\mathrm{e}} ] [ R _ {\mathrm{e}} ] [ R _ {\mathrm{e}} ] [ R _ \textit {\scriptsize p h} ] [ K ] ] \\ & = & [ R _ {\mathrm{e}} ] [ R _ {\mathrm{e}} ] [ R _ {\mathrm{e}} ] [ R _ {\mathrm{e}} ] [ R _ {\mathrm{e}} ] [ R _ {\mathrm{e}} ] [ R _ {\mathrm{e}} ] [ R _ {\mathrm{e}} ] [ R _ {\mathrm{e}} ] [ R _ {\mathrm{e}} ] [ K ] [ K ] [ K ] [ K ] [ K ] [ K ] [ K ] [ K ] [ K ] [ K ] [ K ] [ K ] [ K ] [ K ] [ K ] [ K ] [ K ] [ K ] [ K ] [ K ] [ K ] [ K ] [ K ] [ K ] [ K ] [ K ] [ K ] [ K ] [ K ] [ K ] [ K ] [ K ] [ K ] [ K ] [ M ] [ M ] \\ & = & [ R _ {\mathrm{e}} ] [ R _ {\mathrm{e}} ] [ R _ {\mathrm{e}} ] [ R _ {\mathrm{e}} ] [ R _ {\mathrm{e}} ] [ R _ {\mathrm{e}} ] [ R _ {\mathrm{e}} ] [ R _ {\mathrm{e}} ] [ R _ {\mathrm{e}} ] [ R _ {\mathrm{e}} ] [ R _ {\infty} ] [ M ] [ M ] [ M ] [ M ] [ M ] [ M ] [ M ] [ M ] [ M ] [ M ] [ M ] [ M ] [ M ] [ M ] [ M ] [ M ] [ M ] [ M ] [ M ] [ M ] [ M ] [ M ] [ M ] [ M ] [ M ] [ M ] [ M ] [ M ] [ M ] [ M ] [ M ] [ M ] [ M ] [ M ] [ M ] [M ] [ M ] [ M ] [ M ] [ M ] [ M ] [ M ] [ M ] [ M ] [ M ] [ M ] [ M ] [ M ] [ M ] [ M ] [ M ] [ M ] [ M ] [ M ] [ M ] [ M ] [ M ] [ M ] [ M ] [ M ] [ M ] [ M ] [ M ] [ M ] [ M ] [ M ] [ M ] [ M ] {[ M ],} m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m mm m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W WWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWW< fcel>R_{1,1}R_{1,2}R_{1,3}R_{1,4}R_{1,5}R_{1,6}R_{1,7}R_{1,8}R_{1,9}R_{1,10}R_{1,11}R_{1,12}R_{1,13}R_{1,14}R_{1,15}R_{1,16}R_{1,17}R_{1,18}R_{1,19}R_{1,20}R_{1,21}R_{1,22}R_{1,23}R_{1,24}R_{1,25}R_{1,26}R_{1,27}R_{1,28}R_{1,29}R_{1,30}R_{1,31}R_{1,32}R_{1,33}R_{1,34}R_{1,35}R_{1,36}R_{1,37}R_{1,38}R_{1,39}R_{1,40}R_{1,41}R_{1,42}R_{1,43}R_{1,44}R_{1,45}R_{1,46}R_{1,47}R_{1,48}R_{1,49}R_{1,50}R_{1,51}R_{1,52}R_{1,53}R_{1,54}R_{1,55}R_{1,56}R_{1,57}R_{1,58}R_{1,59}R_{1,60}R_{1,61}R_{1,62}R_{1,63}R_{1,64}R_{1,65}R_{1,66}R_{1,67}R_{1,68}R_{1,69}R_{1,70}R_{1,71}R_{1,72}R_{1,73}R_{1,74}R_{1,75}R_{1,76}R_{1,77}R_{1,78}R_{1,79}R_{1,80}R_{1,81}R_{1,82}R_{1,83}R_{1,84}R_{1,85}R_{1,86}R_{1,87}R_{1,88}R_{1,89}R_{1,90}R_{1,91}R_{1,92}R_{1,93}R_{1,94}R_{1,95}R_{1,96}R_{1,97}R_{1,98}R_{1,99}R_2 , p h p h p h p h p h p h p h p h p h p h p h p h p h p h p h p h p h p h p h p h p h p h p h p h p h p h p h p h p h p h p h p h p h p h p h p h p h p h p h p h p h p h p h p h p h p h p h p h p h p h p H P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P N T N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N NNININININININININININININININININININININININININININININININININININININININININININININININININININININININININININININININININININININININININININININININININININININININININININININININININININININ IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN In IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN IN< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< nl>\]

\[= \left[ \begin{array}{c c c} \mathbf {0} & \mathbf {R} & \mathbf {0} \end{array} \right].\]

The matrix can easily be computed element-by-element. To compute , we consider

\[\begin{array} { r l } & \mathbb { E } \left[ \pmb { \xi } _ { t + 1 } ^ { ( 3 ) } \left( \pmb { \xi } _ { t + 1 } ^ { ( 3 ) } \right) ^ { \prime } \right] = \mathbb { E } \left[ \left[ \begin{array} { c } \pmb { \epsilon } _ { t + 1 } \\ \pmb { \epsilon } _ { t + 1 } \otimes \pmb { \epsilon } _ { t + 1 } - v e c \left( \mathbf { I } _ { n _ { e } } \right) \\ \pmb { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \\ \mathbf { x } _ { t } ^ { f } \otimes \pmb { \epsilon } _ { t + 1 } \\ \pmb { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { s } \\ \pmb { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \\ \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \otimes \pmb { \epsilon } _ { t + 1 } \\ \mathbf { x } _ { t } ^ { f } \otimes \pmb { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \\ \mathbf { x } _ { t } ^ { f } \otimes \pmb { \epsilon } _ { t + 1 } \otimes \pmb { \epsilon } _ { t + 1 } \\ \pmb { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \otimes \pmb { \epsilon } _ { t + 1 } \\ \pmb { \epsilon } _ { t + 1 } \otimes \pmb { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \\ (\pmb { \epsilon } _ { t + 1 } \otimes \pmb { \epsilon } _ { t + 1 } \otimes \pmb { \epsilon } _ { t + 1 }) - \mathbb { E } [ ( \pmb { \epsilon } _ { t + 1 } \otimes \pmb { \epsilon } _ { t + 1 } \otimes \pmb { \epsilon } _ { t + 1 } ) ] \\ & \\ & \times \left[ \begin{array} { c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c } \\ \pmb { \epsilon } _ { t + 1 } ^ { \prime } & ( \pmb { \epsilon } _ { t + 1 } \otimes \pmb { \epsilon } _ { t + 1 } - v e c ( \mathbf { I } _ { n _ { e} } ) ) ^ { \prime } & ( \pmb { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } ) ^ { \prime } & ( \mathbf { x } _ { t } ^ { f } \otimes \pmb { \epsilon } _ { t + 1 } ) ^ { \prime } & ( \pmb { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { s } ) ^ { \prime } & ( \pmb { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } ) ^ { \prime } \\ ( | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | & \\ & ( \| x \| ^ \prime * p / 2 * q / 2 * r / 2 * w / 2 * y / 2 * z / 2 * w / 2 * x / 2 * y / 2 * z / 2 * w / 2 * y / 2 * z / 2 * w / 2 * y / 2 * z / 2 * w / 2 * y / 2 * z / 2 * w / 2 * y / 2 * z / 2 * w / 2 * y / 2 * z / 2 * w / 2 * y / 2 * z / 2 * w / 2 * y / 2 * y / 2 * z / 2 * w / 2 * y / 2 * z / 2 * w / 2 * y / 2 * z / 2 * w / 2 * y / 2 * z / 2 * w / 2 * y / 2 * z / 2 * w / 2 * y / 2 * z / 2 * w / 2 * y / 2 * w / 2 * y / 2 * z / 2 * w / 2 * y / 2 * z / 2 * w / 2 * y / 2 * z / 2 * w / 2 * y / 2 * z / 2 * w / 2 * y / 2 * z / 2 * w / 2 * y / 2 * z / 2 * w / 2 * z / 2 * w / 2 * y / 2 * z / 2 * w / 2 * y / 2 * z / 2 * w / 2 * y / 2 * z / 2 * w / 2 * y / 2 * z / 2 * w / 2 * y / 2 * z / 2 * w / 2 * y / 2 * z / 2 * y / 2 . \\ & \\ & ( \| x \| ^ {\prime *} q / q ^ {*} ( \| x \| ^ {\prime *} q ^ {*} ( \| x \| ^ {\prime *} q ^ {*} ( \| x \| ^ {\prime *} q ^ {*} ( \| x \| ^ {\prime *} q ^ {*} ( \| x \| ^ {\prime *} q ^ {*} ( \| x \| ^ {\prime *} q ^ {*} ( \| x \| ^ {\prime *} q ^ {*} ( \| x \| ^ {\prime *} q ^ {*} ( \| x \| ^ {\prime *} q ^ {*} ( \| x \| ^ {\prime *} k ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ] . \\ & \\ & ( \| x \| ^ {\prime *} q / q ^ {*} ( \| x \| ^ {\prime *} q ^ {*} ( \| x \| ^ {\prime *} q ^ {*} ( \| x \| ^ {\prime *} q ^ {*} ( \| x \| ^ {\prime *} q ^ {*} ( \| x \| ^ {\prime *} q ^ {*} ( \| x \| ^ {\prime *} q ^ {*} ( \| x \| ^ {\prime *} q ^ {*} ( \| x \| _ T a i n d o u s s i n d o u s s i n d o u s s i n d o u s s i n d o u s s i n d o u s s i n d o u s s i n d o u s s i n d o u s s i n d o u s s i n d o u s s i n d o u s s i n d o u s s i n d o u s s i n d o u s s i m a l l o g h i n e r e f f i l l o g h i n e r e f f i l l o g h i n e r e f f i l l o g h i n e r e f f i l l o g h i n e r e f f i l l o g h i n e r e f f i l l o g h i n e r e f f i l l o g h i n e r e f f i l l o g h i n e l o g h i n e r e f f i l l o g h i n e r e f f i l l o g h i n e r e f f i l l o g h i n e r e f f i l l o g h i n e r e f f i l l o g h i n e r e f f i l l o g h i n e r e f f i l l o g h i n e r o f h i n e r e f f i l l o g h i n e r e f f i l l o g h i n e r e f f i l l o g h i n e r e f f i l l o g h i n e r e f f i l l o g h i n e r e f f i l l o g h i n e r e f f i l l o g h i n e r e f f f i l l o g h i n e r e f f i l l o g h i n e r e f f i l l o g h i n e r e f f i l l o g h i n e r e f f i l l o g h i n e r e f f i l l o g h i n e r e f f i l l o g h i n e r e f f i l l o g h j a b b a d a d a b a d a b a d a b a d a b a d a b a d a b a d a b a d a b a d a b a d a b a d a b a d a b a d a b a d a b a d a b a d a b a d a b a d a b a d a b a d a b a d a b a d a b a d a b a d a b a d a b a d a d a b a d a b a d a b a d a b a d a b a d a b a d a b a d a b a d a b a d a b a d a b a d a b a d a b a d a b a d a b a d a b a d a b a d a b a d a b a d a b a d a b a d a b a d a b a d a b b u r r u m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r [ ( \| x \| ^ {\prime *} q / q ^ {*} ( \| x \| ^ {\prime *} q ^ {*} ( \| x \| ^ {\prime *} q ^ {*} ( \| x \| ^ {\prime *} q ^ {*} ( \| x \| ^ {\prime *} q ^ {*} ( \| x \| ^ {\prime *} q ^ {*} ( \| x \| ^ {\prime *} q ^ {*} ( \| x \| ^ {\prime *} q ^ {*} ( \| x \| ^ {\prime *} q^ {*} ( \| x \| ^ {\prime *} q^ {*} ( \| x \| ^ {\prime *} q^ {*} ( \| x \| ^ {\prime *} q^ {*} ( \| x \| ^ {\prime *} q^ {*} ( \| x \| ^ {\prime *} q^ {*} ( \| x \| ^ {\prime *} q^ {*} ( \| x \| ^ {\prime *} q^ {*} ( \| x \| ^ {\prime *} q^ {*} ( \| x \| ^ {\prime *} q^* ( \| x \| ^ {\prime *} q^* ( \| x \| ^ {\prime *} q^* ( \| x \| ^ {\prime *} q^* ( \| x \| ^ {\prime *} q^* ( \| x \| ^ {\prime *} q^* ( \| x \| ^ {\prime *} q^* ( \| x \| ^ {\prime *} q^* ( \| x \| ^ {\prime *} q^* ( \| x \| ^ {\prime *} q^* (\| x \| ^ {\prime *} q^* (\| x \| ^ {\prime *} q^* (\| x \| ^ {\prime *} q^* (\| x \| ^ {\prime *} q^* (\| x \| ^ {\prime *} q^* (\| x \| ^ {\prime *} q^* (\| x \| ^ {\prime *} q^* (\| x \| ^ {\prime *} q^* (\| x \| ^ {\prime *} q^* (\|x\| _ T A i n d o u s s i n d o u s s i n d o u s s i n d o u s s i n d o u s s i n d o u s s i n d o u s s i n d o u s s i n d o u s s i n d o u s s i n d o u s s i n d o u s s i n d o u s s i n d o u s s i u v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v vv w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W WWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBw BwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBw Bw Bw Bw Bw Bw Bw Bw Bw Bw Bw Bw Bw Bw Bw Bw Bw Bw Bw Bw Bw Bw Bw Bw Bw Bw Bw Bw Bn Bn Bn Bn Bn Bn Bn Bn Bn Bn Bn Bn Bn Bn Bn Bn Bn Bn Bn Bn Bn Bn Bn Bn Bn Bn Bn Bn Bn Bn Bn Bn Bn Bn Bn Bn Bn Bn Bn Bn Bn Bn Bn Bn Bn Bn Bn Bn Bn Bn Bm Bm Bm Bm Bm Bm Bm Bm Bm Bm Bm Bm Bm Bm Bm Bm Bm Bm Bm Bm Bm Bm Bm Bm Bm Bm Bm Bm Bm Bm Bm Bm Bm Bm Bm Bm Bm Bm Bm Bm Bm Bm Bm Bm Bm Bm Bm Bm Bm Bm BybMbMbMbMbMbMbMbMbMbMbMbMbMbMbMbMbMbMbMbMbMbMbMbMbMbMbMbMbMbMbMbMbMbMbMbMbMbMbMbMbMbMbMbMbMbMbMbMbMbMbMbMbMbMbMbMbMbMbMbMbMbMbMbMbMbMbMbMbMBm BybMbMbMbMbMbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbmbbm ABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABABB< fcel>\(\cdots\)< fcel>\(\cdots\)< fcel>\(\cdots\)< fcel>\(\cdots\)< fcel>\(\cdots\)< fcel>\(\cdots\)< fcel>\(\cdots\)< fcel>\(\cdots\)< fcel>\(\cdots\)< fcel>\(\cdots\)< fcel>\(\cdots\)< fcel>\(\cdots\)< fcel>\(\cdots\)< fcel>\(\cdots\)< fcel>\(\cdots\)< fcel>\(\cdots\)< fcel>\(\cdots\)< fcel>\(\ddots\)< fcel>\(\cdots\)< fcel>\(\cdots\)< fcel>\(\cdots\)< fcel>\(\cdots\)< fcel>\(\cdots\)< fcel>\(\cdots\)< fcel>\(\cdots\)< fcel>\(\cdots\)< fcel>\(\cdots\)< fcel>\(\cdots\)< fcel>\(\cdots\)< fcel>\(\cdots\)< fcel>\(\cdots\)< fcel>\(\cdots\)< fcel>\(\cdots\)< fcel>\(\cdots\)< fcel>\(D_{t}^{(t)}^{(t)}^{(t)}^{(t)}^{(t)}^{(t)}^{(t)}^{(t)}^{(t)}^{(t)}^{(t)}^{(t)}^{(t)}^{(t)}^{(t)}^{(t)}^{(t)}^{(t)}^{(t)}^{(t)}^{(t)}^{(t)}^{(t)}^{(t)}^{(t)}^{(t)},\)< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< nl>\]

Note that contains squared, meaning that must have a finite sixth moment for to be finite. Again, all elements in can be computed element-by-element. For further details, we refer to the paper's Online Appendix, which also discusses how can be computed in a more memory-efficient manner.

For the auto-covariance, we have

\[\begin{array}{r c l} C o v \left(\mathbf {z} _ {t + 1} ^ {(3)}, \mathbf {z} _ {t} ^ {(3)}\right) & = & C o v \left(\mathbf {c} ^ {(3)} + \mathbf {A} ^ {(3)} \mathbf {z} _ {t} ^ {(3)} + \mathbf {B} ^ {(3)} \boldsymbol {\xi} _ {t + 1} ^ {(3)}, \mathbf {z} _ {t} ^ {(3)}\right) \\ & = & \mathbf {A} ^ {(3)} C o v \left(\mathbf {z} _ {t} ^ {(3)}, \mathbf {z} _ {t} ^ {(3)}\right) + \mathbf {B} ^ {(3)} C o v \left(\boldsymbol {\xi} _ {t + 1} ^ {(3)}, \mathbf {z} _ {t} ^ {(3)}\right) \end{array}\]

and

\[\begin{array}{r c l} C o v \left(\mathbf {z} _ {t + 2} ^ {(3)}, \mathbf {z} _ {t} ^ {(3)}\right) & = & C o v \left(\mathbf {c} ^ {(3)} + \mathbf {A} ^ {(3)} \mathbf {z} _ {t + 1} ^ {(3)} + \mathbf {B} ^ {(3)} \pmb {\xi} _ {t + 2} ^ {(3)}, \mathbf {z} _ {t} ^ {(3)}\right) \\ & = & C o v \left(\mathbf {c} ^ {(3)} + \mathbf {A} ^ {(3)} \left(\mathbf {c} ^ {(3)} + \mathbf {A} ^ {(3)} \mathbf {z} _ {t} ^ {(3)} + \mathbf {B} ^ {(3)} \pmb {\xi} _ {t + 1} ^ {(3)}\right) + \mathbf {B} ^ {(3)} \pmb {\xi} _ {t + 2} ^ {(3)}, \mathbf {z} _ {t} ^ {(3)}\right) \\ & = & C o v \left(\mathbf {c} ^ {(3)} + \mathbf {A} ^ {(3)} \mathbf {c} ^ {(3)} + \left(\mathbf {A} ^ {(3)}\right) ^ {2} \mathbf {z} _ {t} ^ {(3)} + \mathbf {A} ^ {(3)} \mathbf {B} ^ {(3)} \pmb {\xi} _ {t + 1} ^ {(3)} + \mathbf {B} ^ {(3)} \pmb {\xi} _ {t + 2} ^ {(3)}, \mathbf {z} _ {t} ^ {(3)}\right) \\ & = & C o v \left(\left(\mathbf {A} ^ {(3)}\right) ^ {2} \mathbf {z} _ {t} ^ {(3)}, \mathbf {z} _ {t} ^ {(3)}\right) + C o v \left(\mathbf {A} ^ {(3)} \mathbf {B} ^ {(3)} \pmb {\xi} _ {t + 1} ^ {(3)}, \mathbf {z} _ {t} ^ {(3)}\right) + C o v \left(\mathbf {B} ^ {(3)} \pmb {\xi} _ {t + 2} ^ {(3)}, \mathbf {z} _ {t} ^ {(3)}\right) \\ & = & \left(\mathbf {A} ^ {(3)}\right) ^ {2} C o v \left(\mathbf {z} _ {t} ^ {(3)}, \mathbf {z} _ {t} ^ {(3)}\right) + \mathbf {A} ^ {(3)} \mathbf {B} ^ {(3)} C o v \left(\pmb {\xi} _ {t + 1} ^ {(3)}, \mathbf {z} _ {t} ^ {(3)}\right) + \mathbf {B} ^ {(3)} C o v \left(\pmb {\xi} _ {t + 2} ^ {(3)}, \mathbf {z} _ {t} ^ {(3)}\right). \end{array}\]

So, for

\[C o v \left(\mathbf {z} _ {t + s} ^ {(3)}, \mathbf {z} _ {t} ^ {(3)}\right) = \left(\mathbf {A} ^ {(3)}\right) ^ {s} \mathbb {V} \left[ \mathbf {z} _ {t} ^ {(3)} \right] + \sum_ {j = 0} ^ {s - 1} \left(\mathbf {A} ^ {(3)}\right) ^ {s - 1 - j} \mathbf {B} ^ {(3)} C o v \left(\boldsymbol {\xi} _ {t + 1 + j} ^ {(3)}, \mathbf {z} _ {t} ^ {(3)}\right)\]

and we therefore only need to compute :

\[\begin{array} { r l } & { \mathbb { E } \left[ \mathbf { z } _ { t } ^ { ( 3 ) } \left( \pmb { \xi } _ { t + 1 + j } ^ { ( 3 ) } \right) ^ { \prime } \right] = \mathbb { E } \left[ \left[ \begin{array} { c } \mathbf { x } _ { t } ^ { f } \\ \mathbf { x } _ { t } ^ { s } \\ \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \\ \mathbf { x } _ { t } ^ { r d } \\ \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { s } \\ \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \end{array} \right] \right. } \\ & \times \left[ \begin{array} { c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c } \boldsymbol { \epsilon } _ { t + 1 + j } ^ { \prime } & ( \pmb { \epsilon } _ { t + 1 + j } \otimes \pmb { \epsilon } _ { t + 1 + j } - v e c ( \mathbf { I } _ { n _ { e} } ) ) ^ { \prime } & ( \pmb { \epsilon } _ { t + 1 + j } \otimes \mathbf { x } _ { t + j } ^ { f } ) ^ { \prime } \\ ( \mathbf { x } _ { t + j } ^ { f } \otimes \pmb { \epsilon } _ { t + 1 + j } ) ^ { \prime } & ( \pmb { \epsilon } _ { t + 1 + j } \otimes \mathbf { x } _ { t + j } ^ { s } ) ^ { \prime } & ( \pmb { \epsilon } _ { t + 1 + j } \otimes \mathbf { x } _ { t + j } ^ { f } \otimes \mathbf { x } _ { t + j } ^ { f } ) ^ { \prime } \\ ( \mathbf { x } _ { t + j } ^ { f } \otimes \mathbf { x } _ { t + j } ^ { f } \otimes \pmb { \epsilon } _ { t + 1 + j } ) ^ { \prime } & ( \mathbf { x } _ { t + j } ^ { f } \otimes \pmb { \epsilon } _ { t + 1 + j } \otimes \mathbf { x } _ { t + j } ^ { f } ) ^ { \prime } \\ ( \mathbf { x } _ { t + j } ^ { f } \otimes \pmb { \epsilon } _ { t + 1 + j } \otimes \pmb { \epsilon } _ { t + 1 + j } ) ^ { \prime } & ( \pmb { \epsilon } _ { t + 1 + j } \otimes \mathbf { x } _ { t + j } ^ { f } \otimes \pmb { \epsilon } _ { t + 1 + j } ) ^ { \prime } \\ ( \pmb { \epsilon } _ { t + 1 + j } \otimes \pmb { \epsilon } _ { t + 1 + j } \otimes \mathbf { x } _ { t + j } ^ { f } ) ^ { \prime } & ( ( ( \pmb { \epsilon } _ { t + 1 + j } \otimes \pmb { \epsilon } _ { t + 1 + j } \otimes \pmb { \epsilon } _ { t + 1 + j } ) - E [ ( ( \pmb { \epsilon } _ { t + 1 } \otimes \pmb { \epsilon } _ { t + 1 } \otimes ( | | | | | ) ] ^ { \prime } ] \\ = & \left[ \begin{array} c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c a n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i i n d i n d i n d i i n d i n d i i n d i i n d i i n d i i n d i i n d i i n d i i n d i i n d i i n d i i n d i i n d i i n d i i n d i i n d i i n d i i n d i i n d i i n d i i n d i i n d i i n d i i n d i i n d i i n d i i n d e l o w e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e l o w e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e r e l o w e w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w u p p h a l l a m b a l l a m b a l l a m b a l l a m b a l l a m b a l l a m b a l l a m b a l l a m b a l l a m b a l l a m b a l l a m b a l l a m b a l l a m b a l l a m b a l l a m b a l l a m b a l l a m b a l l b u l l a m b u l l a m b u l l a m b u l l a m b u l l a m b u l l a m b u l l a m b u l l a m b u l l a m b u l l a m b u l l a m b u l l a m b u l l a m b u l l a m b u l l a m b u l l a m b u l l a m b u l | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | & = & R _ { 1 , 1 } ^ { j } R _ { 1 , 2 } ^ { j } R _ { 1 , 3 } ^ { j } R _ { 2 , 1 } ^ { j } R _ { 2 , 2 } ^ { j } R _ { 2 , 3 } ^ { j } R _ { 3 , 1 } ^ { j } R _ { 3 , 2 } ^ { j } R _ { 3 , 3 } ^ { j } R _ { 4 , 1 } ^ { j } R _ { 4 , 2 } ^ { j } R _ { 4 , 3 } ^ { j } R _ { 5 , 1 } ^ { j } R _ { 5 , 2 } ^ { j } R _ { 5 , 3 } ^ { j } R _ { 6 , 1 } ^ { j } R _ { 6 , 2 } ^ { j } R _ { 6 , 3 } ^ { j } R _ 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , . \\ & = & [ {\bf 0} {\bf R} ^ {j} {\bf R} ] . \\ & & = [ {\bf R} {\bf R} ] . \\ & = & [ {\bf R} {\bf R} ] . \\ & = & [ {\bf R} {\bf R} ] . \\ & = & [ {\bf R} {\bf R} ] . \\ & = & [ {\bf R} {\bf R} ] . \\ & = & [ {\bf R} {\bf R} ] . \\ & = & [ {\bf R} {\bf G} ] . \\ & = & [ {\bf G} {\bf G} ] . \\ & = & [ {\bf G} {\bf G} ] . \\ & = & [ {\bf G} {\bf G} ] . \\ & = & [ {\bf G} {\bf G} ] . \\ & = & [ {\bf G} {\bf G} ] . \\ & = & [ {\bf G} {\bf G} ] .\\ & = & [ {\bf G} {\bf G} ] . \\ & = & [ {\bf G} {\bf G} ] . \\ & = & [ {\bf G} {\bf G} ] . \\ & = & [ {\bf G} {\bf G} ] . \\ & = & [ {\bf G} {\bf G} ] . \\ & = & [ {\bf G} ] . \\ & = & [ {\bf G} ] . \\ & = & [ {\bf G} ] . \\ & = & [ {\bf G} ] . \\ & = & [ {\bf G} ] . \\ & = & [ {\bf G} ] . \\ & = & [ {\bf G} ] . \\ & = & [ {\bf G} ] . \\ & = & [ {{\bf G}} ] . \\ & = & [ {\bf G} ] . \\ & = & [ {\bf G} ] . \\ & = & [ {\bf G} ] . \\ & = & [ {\bf G} ] . \\ & = & [ {\bf G} ] . \\ & = & [ {\bf G} ] . \\ & = & [ {\bf G} ] . \\ & = & [ ] . \\ & = & [ {\bf G} ] . \\ & = & [ {\bf G} ] . \\ & = & [ {\bf G} ] . \\ & = & [ {\bf G} ] . \\ & = & [ {\bf G} ] . \\ & = & [ {\bf G} ] . \\ & = & [ {\bf G} ] . \\ & = & [ ({\bf G}) ] . \\ & = & [ {\bf G}) ] . \\ & = & [ ({\bf G}) ] . \\ & = & [ ({\bf G}) ] . \\ & = & [ ({\bf G}) ] . \\ & = & [ ({\bf G}) ] . \\ & = & [ ({\bf G}) ] . \\ & = & [ ({\bf G}) ] . \\ & = & [ ({\bf G}) ] . \\ & = & [ ({\bf F}) ] . \\ & = & [ ({\bf F}) ] . \\ & = & [ ({\bf F}) ] . \\ & = & [ ({\bf F}) ] . \\ & = & [ ({\bf F}) ] . \\ & = & [ ({\bf F}) ] . \\ & = & [ ({\bf F}) ] . \\ & = & [ ({\bf F}) ] . \\ & = & [ ({\bf F}) ] . \\ & = & [ ({\bf F}) ] . \\ & = & [ ({\bf F}) ] . \\ & = & [ ({\bf F}) ] . \\ & = & [ ({\bf F}) ] . \\ & = & [ ({\bf F}) ] . \\ & = & [ ({\bf F}) ] .. \\ & = & [ ({\bf F}) ] .. \\ & = & [ ({\bf F}) ] .. \\ & = & [ ({\bf F}) ] .. \\ & = & [ ({\bf F}) ] .. \\ & = & [ ({\bf F}) ] .. \\ & = & [ ({\bf F}) ] .. \\ & = & [ ({\bf F}) ] .. \\ * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * ]. \\ * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / * / ** ! ** ! ** ! ** ! ** ! ** ! ** ! ** ! ** ! ** ! ** ! ** ! ** ! ** ! ** ! ** ! ** ! ** ! ** ! ** ! ** ! ** ! ** ! ** ! ** ! ** ! ** ! ** ! ** ! ** ! ** ! ** ! ** ! ** ! ** ! ** ! ** ! ** ! ** ! ** ! ** ! ** ! ** ! ** ! ** ! ** ! ** ! ** ! ** ! ** ! **! ** ! ** ! ** ! ** ! ** ! ** ! ** ! ** ! ** ! ** ! ** ! ** ! ** ! ** ! ** ! ** ! ** ! ** ! ** ! ** ! ** ! ** ! **! **! **! **! **! **! **! **! **! **! **! **! **! **! **! **! **! **! **! **! **! **! **! **! **! **! **! **! **! **! **! **! **! **! **! **! **! **! **! **! **! **! **! **! **! **! **! **! **! **! **!\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\rbrack ; ].\end{array} \right]\]

The matrix can then be computed element-by-element. For further details, see the paper's Online Appendix.

A.10 Third Order: Unconditional Third and Fourth Moments

The proof proceeds as for a second-order approximation. At third order, the only difference is that also depends on . Hence, unconditional third moments exist if has a finite ninth moment, and the unconditional fourth moment exists if has a finite twelfth moment.

A.11 GIRFs: Second Order

We first note that

\[\begin{array}{l} \mathbf {x} _ {t + l} ^ {f} \otimes \mathbf {x} _ {t + l} ^ {f} = \left(\mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j}\right) \otimes \left(\mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j}\right) \\ = \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} \\ + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j}. \end{array}\]

Next, let

\[\begin{array}{l} \tilde {\mathbf {x}} _ {t + l} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + l} ^ {f} = \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\delta} _ {t + j} + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\delta} _ {t + j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \\ + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\delta} _ {t + j} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\delta} _ {t + j}, \end{array}\]

where we define such that and for . This means

\[\begin{array} { r l } & G I R F _ { \mathbf { x } ^ { f } \otimes \mathbf { x } ^ { f } } \left( l , \nu _ { i } , \mathbf { x } _ { t } ^ { f } \right) = \mathbb { E } \left[ \mathbf { x } _ { t + l } ^ { f } \otimes \mathbf { x } _ { t + l } ^ { f } | \mathbf { x } _ { t } ^ { f } , \epsilon _ { i , t + 1 } = \nu _ { i } \right] - \mathbb { E } \left[ \mathbf { x } _ { t + l } ^ { f } \otimes \mathbf { x } _ { t + l } ^ { f } | \mathbf { x } _ { t } ^ { f } \right] \\ & = \mathbb { E } \left[ \tilde { \mathbf { x } } _ { t + l } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + l } ^ { f } | \mathbf { x } _ { t } ^ { f } \right] - \mathbb { E } \left[ \mathbf { x } _ { t + l } ^ { f } \otimes \mathbf { x } _ { t + l } ^ { f } | \mathbf { x } _ { t } ^ { f } \right] \\ & { = \mathbb { E } [ \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \delta _ { t + 1 } + \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \delta _ { t + 1 } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } } \\ & { + \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta _ { t + j } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta _ { t + j } } \\ & { - \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \epsilon _ { t + j } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \epsilon _ { t + j } | \mathbf { x } _ { t } ^ { f } ] } \\ & { = \mathbb { E } [ \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } ( \boldsymbol { \nu } + ( \mathbf { I - S} ) \epsilon _ { t + 1} ) + \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta} ( \boldsymbol { \nu } + ( \mathbf { I - S} ) \epsilon _ { t + 1} ) \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f }} \\ & + ( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta} ( \boldsymbol { \nu } + ( \mathbf { I - S} ) \epsilon _ { t + 1} ) + \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta} \epsilon _ { t + j } ) \otimes ( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta} ( \boldsymbol { \nu } + ( \mathbf { I - S} ) \epsilon _ { t + 1} ) + \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j} \sigma \boldsymbol { \eta} \epsilon _ { t + j } ) \\ & - ( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta} \epsilon _ { t + 1 } + \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j} \sigma \boldsymbol { \eta} \epsilon _ { t + j } ) ) \otimes ( ( {\mathbf h} _ { {\mathbf x} }\]

With and we then obtain (23).

A.12 Second-Order Accuracy of Linear IRFs

Let and suppose and for . These assumptions imply

\[\begin{array}{l} \sigma \boldsymbol {\eta} \boldsymbol {\nu} \otimes \sigma \boldsymbol {\eta} \boldsymbol {\nu} + \boldsymbol {\Lambda} = \sigma \boldsymbol {\eta} \boldsymbol {\nu} \otimes \sigma \boldsymbol {\eta} \boldsymbol {\nu} + ((\sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S}) \otimes \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S})) - (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta})) v e c (\mathbf {I}) \\ = (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}) \left\{\mathbf {S} \otimes \mathbf {S} + ((\mathbf {I} - \mathbf {S}) \otimes (\mathbf {I} - \mathbf {S})) - \mathbf {I} \otimes \mathbf {I} \right\} v e c (\mathbf {I}) \\ = (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}) \left\{2 (\mathbf {S} \otimes \mathbf {S}) - \mathbf {I} \otimes \mathbf {S} - \mathbf {S} \otimes \mathbf {I} \right\} v e c (\mathbf {I}) \end{array}\]

because and , where has dimension . Next, let with all remaining elements of equal to zero. Hence, can be written as and . This implies

\[\begin{array}{l}\sigma \boldsymbol{\eta}\boldsymbol{\nu}\otimes \sigma \boldsymbol{\eta}\boldsymbol {\nu} + \boldsymbol {\Lambda} = (\sigma \boldsymbol {\eta}\otimes \sigma \boldsymbol {\eta})\left\{-\sum_{\substack{j = 1\\ i\neq j}}^{n_{\varepsilon}}\mathbf{D}_{j}\otimes \mathbf{D}_{i} - \sum_{\substack{j = 1\\ i\neq j}}^{n_{\varepsilon}}\mathbf{D}_{i}\otimes \mathbf{D}_{j}\right\} vec\left(\sum_{k = 1}^{n_{\varepsilon}}\mathbf{D}_{k}\right)\\ \\ = (\sigma \boldsymbol {\eta}\otimes \sigma \boldsymbol {\eta})\left\{-\sum_{\substack{j = 1\\ i\neq j}}^{n_{\varepsilon}}\sum_{k = 1}^{n_{\varepsilon}}\left(\mathbf{D}_{j}\otimes \mathbf{D}_{i}\right)vec\left(\mathbf{D}_{k}\right) - \sum_{\substack{j = 1\\ i\neq j}}^{n_{\varepsilon}}\sum_{k = 1}^{n_{\varepsilon}}\left(\mathbf{D}_{i}\otimes \mathbf{D}_{j}\right)vec\left(\mathbf{D}_{k}\right)\right\} \end{array}\]

\[\begin{array}{l} = (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}) \left\{- \sum_ {j = 1 \atop i \neq j} ^ {n _ {\varepsilon}} \sum_ {k = 1} ^ {n _ {\varepsilon}} v e c \left(\mathbf {D} _ {i} \mathbf {D} _ {k} \mathbf {D} _ {j}\right) - \sum_ {j = 1 \atop i \neq j} ^ {n _ {\varepsilon}} \sum_ {k = 1} ^ {n _ {\varepsilon}} v e c \left(\mathbf {D} _ {j} \mathbf {D} _ {k} \mathbf {D} _ {i}\right) \right\} \\ = \mathbf {0} \end{array}\]

because is only different from the zero matrix when i = k = j, but we have . Thus, and , which proves that GIRFs in a pruned second-order approximation reduces to the IRFs in a linearized solution.

A.13 GIRFs: Third Order

\[\begin{array}{l} \text {eriving} G I R F _ {\mathbf {x} ^ {f} \otimes \mathbf {x} ^ {f} \otimes \mathbf {x} ^ {f}} \left(j, \boldsymbol {\nu}, \mathbf {x} _ {t} ^ {f}\right) \quad \text {We first note that} \\ \mathbf {x} _ {t + l} ^ {f} \otimes \mathbf {x} _ {t + l} ^ {f} \otimes \mathbf {x} _ {t + l} ^ {f} \\ = \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \\ + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \\ + \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} + \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\alpha} _ {t + j} \\ + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldbm {\epsilon}. \\ H i n e r t h e f o r i n e s f o r i n e s f o r i n e s f o r i n e s f o r i n e s f o r i n e s f o r i n e s f o r i n e s f o r i n e s f o r i n e s f o r i n e s f o r i n e s f o r i n e s f o r i n e s f o r i n e s f a r d. \\ (2, 3) = (2, 4) = (2, 5) = (2, 6) = (2, 7) = (2, 8) = (2, 9) = (2, 1 0) = (2, 1 1) = (2, 1 2) = (2, 1 3) = (2, 1 4) = (2, 1 5) = (2, 1 6) = (2, 1 7) = (2, 1 8) = (2, 1 9) = (2, 2 0) = (2, 2 1) = (2, 2 2) = (2, 2 3) = (2, 2 4) = (2, 2 5) = (2, 2 6) = (2, 2 7) = (2, 2 8) = (2, 2 9) = (2, 3 0) = (2, 3 1) = (2, 3 2) = (2, 3 3) = (2, 3 4) = (2, 3 5) = (2, 3 6) = (2, 3 7) = (2, 3 8) = (2, 3 9) = (2, 4 0) = (2, 4 1) = (2, 4 2) = (2, 4 3) = (2, 4 4) = (2, 4 5) = (2, 4 6) = (2, 4 7) = (2, 4 8) = (2, 4 9) = (2, 5 0) = (2, 5 1) = (2, 5 2) = (2, 5 3) = (2, 5 4) = (2, 5 5) = (2, 5 6) = (2, 5 7) = (2, 5 8) = (2, 5 9) = (2, 6 0) = (2, 6 1) = (2, 6 2) = (2, 6 3) = (2, 6 4) = (2, 6 5) = (2, 6 6) = (2, 6 7) = (2, 6 8) = (2, 6 9) = (2, 7. A B C D E S T O M P O L I N D S T O M P O L I N D S T O M P O L I N D S T O M P O L I N D S T O M P O L I N D S T O M P O L I N D S T O M P O L I N D S T O M P O L I N D S T O M P O L I N D S T O M P O L I N D S T O M P O L I N C O M P O L I N D S T O M P O L I N D S T O M P O L I N D S T O M P O L I N D S T O M P O L I N D S T O M P O L I N D S T O M P O L I N D S T O M P O L I N D S T O M P O L I N D S T O M P O L I N D S S U P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P & . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . & . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ . \\ .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. & .. | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | \end{array}\]

Using the definition of from Appendix A.11, we have

\[\begin{array}{r l} & {\tilde {\mathbf {x}} _ {t + l} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + l} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + l} ^ {f} = \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\delta} _ {t + j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f}} \\ & {\quad + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\delta} _ {t + j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f}} \\ & {\quad + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\delta} _ {t + j} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\delta} _ {t + j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f}} \\ & {\quad + \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\delta} _ {t + j}} \\ & {\quad + \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\delta} _ {t + j} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\δ} _ {t + j}} \\ & {\quad + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\delta} _ {t + j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\delta} _ {t + j}} \\ & {\quad + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\delta} _ {t + j} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\η} \pmb {\delta} _ {t + j} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\η} \pmb {\delta} _ {t + j}.} \end{array}\]

Simple algebra gives

\[\begin{array}{r l} & G I R F _ {\mathbf {x} ^ {f} \otimes \mathbf {x} ^ {f} \otimes \mathbf {x} ^ {f}} (j, \nu_ {i}, \mathbf {x} _ {t} ^ {f}) = \mathbb {E} [ \tilde {\mathbf {x}} _ {t + l} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + l} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + l} ^ {f} | \mathbf {x} _ {t} ^ {f} ] - \mathbb {E} [ \mathbf {x} _ {t + l} ^ {f} \otimes \mathbf {x} _ {t + l} ^ {f} \otimes \mathbf {x} _ {t + l} ^ {f} | \mathbf {x} _ {t} ^ {f} ] \\ & = \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \\ & + \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} \otimes ((\mathbf {h} _ {\mathbf {x}} ^ {l} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l}) (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f})) \\ & + ((\mathbf {h} _ {\mathbf {x}} ^ {l} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l}) (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f})) \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} \\ & + (\mathbf {h} _ {\mathbf {x}} ^ {l - 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1}) [ (\sigma \boldsymbol {\eta} \boldsymbol {\nu} \otimes \sigma \boldsymbol {\eta} \boldsymbol {\nu}) + \boldsymbol {\Lambda} ] \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \\ & + \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes (\mathbf {h} _ {\mathbf {x}} ^ {l - 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1}) [ (\sigma \boldsymbol {\eta} \boldsymbol {\nu} \otimes \sigma \boldsymbol {\eta} \boldsymbol {\nu}) + \boldsymbol {\Lambda} ] \\ & + \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} \end{array}\]

\[\begin{array}{r l} & {+ \left(\mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S}) \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S}) - \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta}\right) v e c (\mathbf {I})} \\ & {+ \left(\mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S}) \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S})\right) v e c (\mathbf {I})} \\ & {+ \left(\mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S}) \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S}) \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu}\right) v e c (\mathbf {I})} \\ & {+ \left(\left(\mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S}) \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu}\right) \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S})\right) v e c (\mathbf {I})} \\ & {+ \left\{\mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S}) \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S}) \otimes \mathbf {h} _ {\mathbf {x i}} ^ {l - 1} \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S}) \right\} \mathbf {m} ^ {3} (\epsilon_ {t + 1}, \epsilon_ {t + 1}, \epsilon_ {t + 1})} \\ & {+ \sum_ {j = 2} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} \otimes (\mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta}) v e c (\mathbf {I})} \\ & {+ \sum_ {j = 2} ^ {l} (\mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta}) v e c (\mathbf {I}) \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu}} \\ & {+ \sum_ {j = 2} ^ {l} (\mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta}) v e c (\mathbf {I})} \\ & {- (\mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta}) \mathbf {m} ^ {3} (\epsilon_ {t + 1}, \epsilon_ {t + 1}, \epsilon_ {t + 1}),} \end{array}\]

where has dimension and contains all the third moments of

Deriving Using the law of motion for , we first note that

\[\begin{array}{l} \mathbf {x} _ {t + l} ^ {f} \otimes \mathbf {x} _ {t + l} ^ {s} = (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {l} (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {s}) + \sum_ {j = 0} ^ {l - 1} (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {l - 1 - j} (\mathbf {h} _ {\mathbf {x}} \otimes \frac {1}{2} \mathbf {H} _ {\mathbf {x x}}) (\mathbf {x} _ {t + j} ^ {f} \otimes \mathbf {x} _ {t + j} ^ {f} \otimes \mathbf {x} _ {t + j} ^ {f}) \\ \quad + \sum_ {j = 0} ^ {l - 1} (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {l - 1 - j} (\mathbf {h} _ {\mathbf {x}} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}) \mathbf {x} _ {t + j} ^ {f} \\ \quad + \sum_ {j = 0} ^ {l - 1} (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {l - 1 - j} (\sigma \boldsymbol {\eta} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}) \boldsymbol {\epsilon} _ {t + 1 + j} \\ \quad + \sum_ {j = 0} ^ {l - 1} (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {l - 1 - j} (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) (\boldsymbol {\epsilon} _ {t + 1 + j} \otimes \mathbf {x} _ {t + j} ^ {s}) \\ \quad + \sum_ {j = 0} ^ {l - 1} (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {l - 1 - j} (\sigma \boldsymbol {\eta} \otimes \frac {1}{2} \mathbf {H} _ {\mathbf {x x}}) (\boldsymbol {\epsilon} _ {t + 1 + j} \otimes \mathbf {x} _ {t + i} ^ {f} \otimes \mathbf {x} _ {t + j} ^ {f}) \end{array}\]

Using the definition of from Appendix A.11, we obtain

\[\begin{array}{l} \tilde {\mathbf {x}} _ {t + l} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + l} ^ {s} = (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {l} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {s}\right) + \sum_ {j = 0} ^ {l - 1} (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {l - 1 - j} \left(\mathbf {h} _ {\mathbf {x}} \otimes \frac {1}{2} \mathbf {H} _ {\mathbf {x x}}\right) \left(\tilde {\mathbf {x}} _ {t + j} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + j} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + j} ^ {f}\right) \\ \quad + \sum_ {j = 0} ^ {l - 1} (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {l - 1 - j} \left(\mathbf {h} _ {\mathbf {x}} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}\right) \tilde {\mathbf {x}} _ {t + j} ^ {f} \\ \quad + \sum_ {j = 0} ^ {l - 1} (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {l - 1 - j} \left(\sigma \boldsymbol {\eta} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}\right) \boldsymbol {\delta} _ {t + 1 + j} \\ \quad + \sum_ {j = 0} ^ {l - 1} (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {l - 1 - j} \left(\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}\right) \left(\boldsymbol {\delta} _ {t + 1 + j} \otimes \tilde {\mathbf {x}} _ {t + j} ^ {s}\right) \\ \quad + \sum_ {j = 0} ^ {l - 1} (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {l - 1 - j} \left(\sigma \boldsymbol {\eta} \otimes \frac {1}{2} \mathbf {H} _ {\mathbf {x x}}\right) \left(\boldsymbol {\delta} _ {t + 1 + j} \otimes \tilde {\mathbf {x}} _ {t + j} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + j} ^ {f}\right) \end{array}\]

Simple algebra then implies

\[\begin{array}{r l} & G I R F _ {\mathbf {x} ^ {f} \otimes \mathbf {x} ^ {s}} (j, \nu_ {i}, (\mathbf {x} _ {t} ^ {f}, \mathbf {x} _ {t} ^ {s})) = \sum_ {j = 1} ^ {l - 1} (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {l - 1 - j} (\mathbf {h} _ {\mathbf {x}} \otimes \frac {1}{2} \mathbf {H} _ {\mathbf {x x}}) G I R F _ {\mathbf {x} ^ {f} \otimes \mathbf {x} ^ {f} \otimes \mathbf {x} ^ {f}} (j, \nu_ {i}, \mathbf {x} _ {t} ^ {f}) \\ & + \sum_ {j = 1} ^ {l - 1} (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {l - 1 - j} (\mathbf {h} _ {\mathbf {x}} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}) G I R F _ {\mathbf {x} ^ {f}} (j, \nu_ {i}) \\ & + (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {l - 1} (\sigma \eta \pmb {\nu} \otimes (\mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t} ^ {s} + \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) + \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2})) \end{array}\]

A.14 Proof of Proposition 1

Second Order Let us first consider the state variables. Provided that the unpruned state-space system is stable, we know that

\[\mathbf {x} _ {t + 1} ^ {(2)} = \mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t} ^ {(2)} + \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t} ^ {(2)} \otimes \mathbf {x} _ {t} ^ {(2)}\right) + \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2} + \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1} + O \left(\sigma^ {3}\right),\]

i.e., the errors are of third order when . Comparing the pruned state-space system to this expression, we obtain

\[\mathbf {x} _ {t + 1} ^ {f} + \mathbf {x} _ {t + 1} ^ {s} - \mathbf {x} _ {t + 1} ^ {(2)} = \mathbf {h} _ {\mathbf {x}} \left(\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} - \mathbf {x} _ {t} ^ {(2)}\right) + \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} - \mathbf {x} _ {t} ^ {(2)} \otimes \mathbf {x} _ {t} ^ {(2)}\right) + O \left(\sigma^ {3}\right).\]

To show that , algebra gives

\[\begin{array}{r c l} \mathbf {x} _ {t + 1} ^ {f} \otimes \mathbf {x} _ {t + 1} ^ {f} - \mathbf {x} _ {t + 1} ^ {(2)} \otimes \mathbf {x} _ {t + 1} ^ {(2)} & = & (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} - \mathbf {x} _ {t} ^ {(2)} \otimes \mathbf {x} _ {t} ^ {(2)}) + (\mathbf {h} _ {\mathbf {x}} \otimes \boldsymbol {\eta}) ((\mathbf {x} _ {t} ^ {f} - \mathbf {x} _ {t} ^ {(2)}) \otimes \sigma \boldsymbol {\epsilon} _ {t + 1}) \\ & & + (\boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) (\sigma \boldsymbol {\epsilon} _ {t + 1} \otimes (\mathbf {x} _ {t} ^ {f} - \mathbf {x} _ {t} ^ {(2)})) + O (\sigma^ {3}) \end{array}\]

We know that and, therefore, . This shows that , given that all eigenvalues of have modulus less than one. This in turn shows that . For the controls we easily obtain

\[\mathbf {y} _ {t} ^ {s} - \mathbf {y} _ {t} ^ {(2)} = \mathbf {g} _ {\mathbf {x}} \left(\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} - \mathbf {x} _ {t} ^ {(2)}\right) + \frac {1}{2} \mathbf {G} _ {\mathbf {x x}} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} - \mathbf {x} _ {t} ^ {(2)} \otimes \mathbf {x} _ {t} ^ {(2)}\right) + O \left(\sigma^ {3}\right).\]

Given that and , we have as desired.

Third Order Let us first consider the state variables. Provided that the unpruned state-space system is stable, we know that

\[\begin{array}{r c l} \mathbf {x} _ {t + 1} ^ {(3)} & = & \left(\mathbf {h} _ {\mathbf {x}} + \frac {3}{6} \mathbf {h} _ {\sigma \sigma \mathbf {x}} \sigma^ {2}\right) \mathbf {x} _ {t} ^ {(3)} + \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t} ^ {(3)} \otimes \mathbf {x} _ {t} ^ {(3)}\right) + \frac {1}{6} \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t} ^ {(3)} \otimes \mathbf {x} _ {t} ^ {(3)} \otimes \mathbf {x} _ {t} ^ {(3)}\right) \\ & & + \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2} + \frac {1}{6} \mathbf {h} _ {\sigma \sigma \sigma} \sigma^ {3 \prime} + \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1} + O (\sigma^ {4}), \end{array}\]

that is, the errors are of fourth order when . Comparing the pruned state-space system to this expression, we have

\[\begin{array}{r l r} {\mathbf {x} _ {t + 1} ^ {f} + \mathbf {x} _ {t + 1} ^ {s} + \mathbf {x} _ {t + 1} ^ {r d} - \mathbf {x} _ {t + 1} ^ {(3)}} & = & {\mathbf {h} _ {\mathbf {x}} \left(\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d} - \mathbf {x} _ {t} ^ {(3)}\right)} \\ & & {+ \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \left(\left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) + \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {s}\right) + \left(\mathbf {x} _ {t} ^ {s} \otimes \mathbf {x} _ {t} ^ {f}\right) - \mathbf {x} _ {t} ^ {(3)} \otimes \mathbf {x} _ {t} ^ {(3)}\right)} \\ & & {+ \frac {1}{6} \mathbf {H} _ {\mathbf {x x x}} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} - \mathbf {x} _ {t} ^ {(3)} \otimes \mathbf {x} _ {t} ^ {(3)} \otimes \mathbf {x} _ {t} ^ {(3)}\right) + \frac {3}{6} \mathbf {h} _ {\sigma \sigma \mathbf {x}} \sigma^ {2} \left(\mathbf {x} _ {t} ^ {f} - \mathbf {x} _ {t} ^ {(3)}\right)} \end{array}\]

We know that , and therefore . We clearly also have . For the final term, some algebra implies

\[\begin{array}{r l} & {\mathbf {x} _ {t + 1} ^ {f} \otimes \mathbf {x} _ {t + 1} ^ {f} + \mathbf {x} _ {t + 1} ^ {f} \otimes \mathbf {x} _ {t + 1} ^ {s} + \mathbf {x} _ {t + 1} ^ {s} \otimes \mathbf {x} _ {t + 1} ^ {f} - \mathbf {x} _ {t + 1} ^ {(3)} \otimes \mathbf {x} _ {t + 1} ^ {(3)}} \\ {=} & {(\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {s} \otimes \mathbf {x} _ {t} ^ {f} - \mathbf {x} _ {t} ^ {(3)} \otimes \mathbf {x} _ {t} ^ {(3)})} \\ & {+ (\mathbf {h} _ {\mathbf {x}} \otimes \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} + \boldsymbol {\eta} \boldsymbol {\epsilon} _ {\mathbf {t + 1}} \otimes \mathbf {h} _ {\mathbf {x}}) ((\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} - \mathbf {x} _ {t} ^ {(3)}) \sigma)} \\ & {+ (\mathbf {h} _ {\mathbf {x}} \otimes \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} + \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \otimes \mathbf {h} _ {\mathbf {x}}) (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} - \mathbf {x} _ {t} ^ {(3)} \otimes \mathbf {x} _ {t} ^ {(3)} \otimes \mathbf {x} _ {t} ^ {(3)})} \\ & {+ (\mathbf {h} _ {\mathbf {x}} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} + \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \otimes \mathbf {h} _ {\mathbf {x}}) \sigma^ {2} (\mathbf {x} _ {t} ^ {f} - \mathbf {x} _ {t} ^ {(3)})} \\ & {+ (\boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} \otimes \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} + \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \otimes \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1}) (\sigma (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} - \mathbf {x} _ {t} ^ {(3)} \otimes \mathbf {x} _ {t} ^ {(3)}))} \end{array}\]

We know that , so . Similarly, , so . Finally, , and, thus, . Hence, and, therefore, , given that all eigenvalues of have modulus less than one. For the controls we easily obtain

\[\begin{array}{r c l} \mathbf {y} _ {t} ^ {r d} - \mathbf {y} _ {t} ^ {(3)} & = & \mathbf {g} _ {\mathbf {x}} \left(\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d} - \mathbf {x} _ {t} ^ {(3)}\right) + \frac {1}{2} \mathbf {G} _ {\mathbf {x x}} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {s} \otimes \mathbf {x} _ {t} ^ {f} - \mathbf {x} _ {t} ^ {(3)} \otimes \mathbf {x} _ {t} ^ {(3)}\right) \\ & & + \frac {1}{6} \mathbf {G} _ {\mathbf {x x x}} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} - \mathbf {x} _ {t} ^ {(3)} \otimes \mathbf {x} _ {t} ^ {(3)} \otimes \mathbf {x} _ {t} ^ {(3)}\right). \end{array}\]

Given that , , and , we clearly have as desired.

A.15 An Assessment of Accuracy

As a supplement to the accuracy studies mentioned in the main text, we briefly consider the performance of pruning on the stochastic neoclassical growth model, the workhorse of modern macroeconomics. Here, a representative household selects a sequence of consumption and investment to solve

\[\begin{array}{r l} & {\underset {\{c _ {t}, i _ {t} \} _ {t = 0} ^ {\infty}} {\max} \mathbb {E} _ {t} \sum_ {t = 0} ^ {\infty} \beta^ {t} \frac {c _ {t} ^ {1 - \gamma}}{1 - \gamma}} \\ & {\quad \mathrm{s.t.} c _ {t} + i _ {t} = a _ {t} k _ {t} ^ {\alpha}} \\ & {\quad k _ {t + 1} = (1 - \delta) k _ {t} + i _ {t}} \\ & {\log a _ {t + 1} = \rho_ {a} \log a _ {t} + \sigma_ {a} \epsilon_ {a, t + 1}, \epsilon_ {a, t + 1} \sim \mathcal {N I D} (0, 1).} \end{array}\]

The calibration is conventional: , , , , , . We consider three cases for : 2, 5, and 25. The first value, , is a standard calibration for risk aversion. The value is at the high end of estimated risk aversions. Finally, is an extreme calibration well beyond the values compatible with micro-evidence. We assess the accuracy of the pruned and unpruned state-space system based on a fourth-order projection approximation, which is sufficiently accurate to be used as a stand-in for the exact solution. In particular, we measure the root mean squared errors (RMSE) between each perturbation solution and the projection solution.

Table A.1 shows that the accuracy of the pruned and unpruned state-space systems at third order is roughly the same for (with a trivially small advantage for the pruned solution). The pruned state-space system is a bit less accurate than the unpruned approximation when . There is more deterioration of accuracy for the pruned solution when . The results are, however, biased against pruning in this case, since the unpruned solution explodes in 53 out of the 500 simulated sample paths. The reason why the pruned solution loses some accuracy as increases is that precautionary behavior becomes larger and the terms eliminated by pruning may carry information relevant to the solution. But, of course, these terms also generate explosive paths. Note also that the pruned state-space system clearly outperforms the standard first-order approximation for all the considered values of .

Table A.1: Stochastic Neoclassical Growth Model: Accuracy Test Approximation errors are computed based on an accurate fourth-order projection solution. Moments are computed from 500 sample paths of length 4,500 observations with a burn-in of 500 periods. The regression reads , where per refers to a perturbation approximation and proj to the projection solution. The circumflex denotes percentage deviation from steady state. denotes the number of explosive sample paths. The reported values are averages across non-explosive sample paths.

To obtain further insight into the accuracy of pruning, we regress the approximation errors on the distance of each state variable from the steady state and lagged pricing errors (needed to get a well-specified regression). For , the intercepts in these regressions are higher for the pruned than the unpruned state-space system, whereas the slope coefficients are smaller with pruning. Hence, for more non-linear models, the unpruned state-space system is more accurate around the steady state, but its performance deteriorates faster away from the steady state compared to the pruned system. Of course, we should emphasize once more that the results in Table A.1 depend on

We also checked the log case . Given how linear the model is when we have a log utility function, the pruned and unpruned solutions are nearly identical and they display the same level of accuracy.

the model considered.

A.16 Making the DSGE Model Stationary

We eliminate all trending variables in the model by adopting the transformation , , , , , , , and . Here is the Lagrangian multiplier for the law of motion for capital and for the value function in equation (28); see Rudebusch and Swanson (2012). Hence, , and the value of that eliminates capital adjustment costs in the steady state is therefore given by .

The transformed equilibrium conditions are summarized below:

From these equilibrium conditions, it is straightforward to derive a closed-form solution for the steady state of the model.

A.17 An Alternative Interpretation

The deposit rate only enters in equations 6, 12, and 15 of the model summary in Appendix A.16. But note that and when substituted into the Taylor rule we get

\[\begin{array}{r c l} {r _ {t} ^ {b}} & = & {(1 - \rho_ {r}) r _ {s s} + \rho_ {r} r _ {t - 1} ^ {b} + (1 - \rho_ {r}) \left(\beta_ {\pi} \log \left(\frac {\pi_ {t}}{\pi_ {s s}}\right) + \beta_ {y} \log \left(\frac {y _ {t}}{z _ {t} ^ {*} Y _ {s s}}\right)\right)} \\ & & {+ \omega \times x h r _ {t, L} - \rho_ {r} \omega \times x h r _ {t - 1, L} + (1 - \rho_ {r}) \beta_ {x h r} (x h r _ {t, L} - X _ {t, L}).} \end{array}\]

Given this substitution, only enters in equations 6 and 15 of the model summary in Appendix A.16. This implies that our model is equivalent to a standard New Keynesian model with market completeness, but with a Taylor rule for that depends on past and current values of excess holding period return on the long bond.

A.18 An Efficient Perturbation Approximation

To formally present our efficient perturbation approximation, consider the decomposition and similarly for all derivatives of . Here, refers to the control variables needed to solve the model without feedback effects from long-term bond prices to the real economy (when and ), whereas denotes the remaining variables related to pricing government bonds and computing excess holding period returns. Our three-step perturbation approximation is then:

Step 1: Solve for and by a standard perturbation algorithm using a version of our model without feedback effects from government bonds to the real economy. This version of our model has only 11 control variables and 18 equations and is solved using the Matlab codes of Binning (2013).

Step 2: Use the perturbation algorithm of Andreasen and Zabczyk (2015) to recursively solve for , given the derivatives obtained in Step 1.

Step 3: With the derivatives obtained in Steps 1 and 2, solve for and by the standard perturbation algorithm when using the full model with 54 control variables and 61 equations.

To maximize the efficiency of our perturbation algorithm, steps 2 and 3 are computed using a FORTRAN implementation accessible via MEX files in Matlab.

A.19 Data for the Application

We use data from the Federal Reserve Bank of St. Louis covering the period 1961.Q3 to 2007.Q4, giving a total of 186 observations. The annualized growth rate in consumption is calculated from real consumption expenditures (PCECC96). The series for real private fixed investment (FPIC96) is used to calculate the growth rate in investment. Both growth rates are expressed in per capita terms based on the total population in the US. The ratio of government spending to output is computed as government consumption expenditures and investments divided by gross domestic production. The annual inflation rate is for consumer prices. The 3-month nominal interest rate is measured by the rate in the secondary market (TB3MS), and the 10-year nominal rate is from Gürkaynak, Sack and Wright (2007). As in Rudebusch and Swanson (2012), observations for the 10-year interest rate from 1961.Q3 to 1971.Q3 are calculated by extrapolation of the estimated curves in Gürkaynak, Sack and Wright (2007). All moments related to interest rates are expressed in annualized terms. Finally, we use average weekly hours of production and non-supervisory employees in manufacturing (AWHMAN) as provided by the Bureau of Labor Statistics. The series is normalized by dividing it by five times 24 hours, giving a mean level of 0.34.

A.20 Approximate Expression for Excess Holding Period Return

\[x h r _ {t, L} = \mathbb {E} _ {t} \left[ \log \left(P _ {t + 1, L - 1}\right) - \log \left\{\mathbb {E} _ {t} \left[ M _ {t, t + 1} \right] \mathbb {E} _ {t} \left[ P _ {t + 1, L - 1} \right] + C o v _ {t} \left(M _ {t, t + 1}, P _ {t + 1, L - 1}\right) \right\} \right] - r _ {t}.\]

To first order,

\[\log (x _ {t} + y _ {t}) \approx \log (x _ {s s} + y _ {s s}) + \frac {1}{x _ {s s} + y _ {s s}} (x _ {t} - x _ {s s}) + \frac {1}{x _ {s s} + y _ {s s}} (y _ {t} - y _ {s s})\]

and let and , implying that . Hence,

\[\begin{array}{r l} & x h r _ {t, L} \approx \mathbb {E} _ {t} \left[ \log \left(P _ {t + 1, L - 1}\right) \right] - \log P _ {s s, L} - \frac {1}{P _ {s s , L}} \left(\mathbb {E} _ {t} \left[ M _ {t, t + 1} \right] \mathbb {E} _ {t} \left[ P _ {t + 1, L - 1} \right] - M _ {s s, s s + 1} P _ {s s, L - 1}\right) \\ & \quad - \frac {1}{P _ {s s , L}} C o v _ {t} \left(M _ {t, t + 1}, P _ {t + 1, L - 1}\right) - r _ {t} \\ & \quad = \mathbb {E} _ {t} \left[ \log \left(P _ {t + 1, L - 1}\right) \right] - \log P _ {s s, L} - \frac {\mathbb {E} _ {t} \left[ P _ {t + 1 , L - 1} \right]}{P _ {s s , L}} e ^ {- r _ {t} - \omega \times x h r _ {t, L}} + 1 - \frac {C o v _ {t} \left(M _ {t , t + 1} , P _ {t + 1 , L - 1}\right)}{P _ {s s , L}} - r _ {t} \\ & \quad \approx \mathbb {E} _ {t} \left[ \log \left(\frac {P _ {t + 1 , L - 1}}{P _ {s s , L}}\right) \right] - \frac {\mathbb {E} _ {t} \left[ P _ {t + 1 , L - 1} \right]}{P _ {s s , L}} \left(e ^ {- r _ {s s}} \left(1 - r _ {t} - \omega \times x h r _ {t, L} + r _ {s s}\right)\right) - \frac {C o v _ {t} \left(M _ {t , t + 1} , P _ {t + 1 , L - 1}\right)}{P _ {s s , L}}. \end{array}\]

\[+ 1 - r _ {t}\]

as

\[\mathbb {E} _ {t} \left[ M _ {t, t + 1} \right] = e ^ {- r _ {t} - \omega \times x h r _ {t, L}}, P _ {s s, L} = M _ {s s, s s + 1} P _ {s s, L - 1}\]

and

\[e ^ {- r _ {t} - \omega \times x h r _ {t, L}} \approx e ^ {- r _ {s s}} \left(1 - r _ {t} - \omega \times x h r _ {t, L} + r _ {s s}\right).\]

Finally, using , we obtain (41).

References

1. Adrian, T., Crump, R. K. and Moench, E. (2013), ‘Pricing the term structure with linear regressions’, Journal of Financial Economics 110, 110–138.

References

1. Altig, D., Christiano, L. J., Eichenbaum, M. and Linde, J. (2011), ‘Firm-specific capital, nominal rigidities and the business cycle’, Review of Economic Dynamics 14, 225–247.

References

1. An, S. and Schorfheide, F. (2007), ‘Bayesian analysis of DSGE models’, Econometric Review 26(2-4), 113–172.

References

1. Andreasen, M. M. (2012), ‘On the effects of rare disasters and uncertainty shocks for risk premia in non-linear DSGE models’, Review of Economic Dynamics 15, 295–316.

References

1. Andreasen, M. M., Fernández-Villaverde, J. and Rubio-Ramírez, J. (2013), The pruned state-space system for non-linear DSGE models: Theory and empirical applications, Working Paper 18983, National Bureau of Economic Research.

References

1. Andreasen, M. M. and Meldrum, A. (2014), Dynamic term structure models: the best way to enforce the zero lower bound, Working paper, Aarhus University.

References

1. Andreasen, M. M. and Zabczyk, P. (2015), ‘Efficient bond price approximations in non-linear equilibrium-based term structure models’, Studies in Nonlinear Dynamics and Econometrics 19(1), 1–34.

References

1. Aruoba, S. B., Bocola, L. and Schorfheide, F. (2013), A new class of nonlinear time series models for the evaluation of DSGE models, Working paper, University of Pennsylvania.

References

1. Barsky, R. B., Juster, F. T., Kimball, M. S. and Shapiro, M. D. (1997), ‘Preference parameters and behavioral heterogeneity: An experimental approach in the health and retirement study’, Quarterly Journal of Economics 112, 537–579.

References

1. Bernanke, B. S., Gertler, M. and Gilchrist, S. (1999), ‘The financial accelerator in a quantitative business cycle framework’, Handbook of Macroeconomics 1, 1341–1393.

References

1. Binning, A. (2013), Solving second and third-order approximations to DSGE models: A recursive Sylvester equation solution, Working paper, Norges Bank.

References

1. Binsbergen, J. H. V., Fernandez-Villaverde, J., Koijen, R. S. and Rubio-Ramirez, J. (2012), ‘The term structure of interest rates in a DSGE model with recursive preferences’, Journal of Monetary Economics 59, 634–648.

References

1. Calvo, G. A. (1983), ‘Staggered prices in a utility-maximizing framework’, Journal of Monetary Economics 12, 383–398.

References

1. Christiano, L. J., Eichenbaum, M. and Evans, C. L. (2005), ‘Nominal rigidities and the dynamic effects of a shock to monetary policy’, Journal of Political Economy 113, 1–45.

References

1. Cochrane, J. H. (2001), Asset Pricing, Princeton, NJ.

References

1. Cooley, T. F. and Prescott, E. (1995), Economic Growth and Business Cycles, Chapter 1 in Frontiers of Business Cycle Research, Ed. T. F. Cooley, Princeton University Press.

References

1. Creel, M. and Kristensen, D. (2011), Indirect likelihood inference, Working paper, Barcelona GSE.

References

1. Dridi, R., Guay, A. and Renault, E. (2007), ‘Indirect inference and calibration of dynamic stochastic general equilibrium models’, Journal of Econometrics 136, 397–430.

References

1. Duffie, D. and Singleton, K. J. (1993), ‘Simulated moments estimation of Markov models of asset prices’, Econometrica 61(4), 929–952.

References

1. Epstein, L. G. and Zin, S. E. (1989), ‘Substitution, risk aversion, and the temporal behavior of consumption and asset returns: A theoretical framework’, Econometrica 57(4), 937–969.

References

1. Fernández-Villaverde, J., Guerrón-Quintana, P., Rubio-Ramírez, J. F. and Uribe, M. (2011), ‘Risk matters: The real effects of volatility shocks’, American Economic Review 101, 2530–2561.

References

1. Fernández-Villaverde, J. and Rubio-Ramírez, J. F. (2007), ‘Estimating macroeconomic models: A likelihood approach’, Review of Economic Studies 74, 1–46.

References

1. Gagnon, J., Raskin, M., Rernache, J. and Sack, B. (2011), ‘Large-scale asset purchases by the Federal Reserve: Did they work?’, Federal Reserve Bank of New York, Research Paper Series - Economic Policy Review 17(1), 41–59.

References

1. Gertler, M. and Karadi, P. (2011), ‘A model of unconventional monetary policy’, Journal of Monetary economics 58, 17–34.

References

1. Gürkaynak, R., Sack, B. and Wright, J. (2007), ‘The U.S. Treasury yield curve: 1961 to the present’, Journal of Monetary Economics 54, 2291–2304.

References

1. Haan, W. J. D. and Wind, J. D. (2012), ‘Nonlinear and stable perturbation-based approximations’, Journal of Economic Dynamic and Control 36, 1477–1497.

References

1. Hansen, L. P. (1982), ‘Large sample properties of generalized method of moments estimators’, Econometrica 50(4), 1029–1054.

References

1. Hordahl, P., Tristani, O. and Vestin, D. (2008), ‘The yield curve and macroeconomic dynamics’, The Economic Journal 118, 1937–1970.

References

1. Iskrev, N. (2010), ‘Local identification in DSGE models’, Journal of Monetary Economics 57(2), 189–202.

References

1. Jermann, U. J. (1998), ‘Asset pricing in production economics’, Journal of Monetary Economics 41, 257–275.

References

1. Jorda, O. (2005), ‘Estimation and inference of impulse responses by local projections’, American Economic Review 95(1), 161–182.

References

1. Joyce, M. A. S., Lasaosa, A., Stevens, I. and Tong, M. (2011), ‘The financial market impact of quantitative easing in the United Kingdom’, International Journal of Central Banking 7(3), 113–161.

References

1. Judd, K. L. and Guu, S.-M. (1997), ‘Asymptotic methods for aggregate growth models’, Journal of Economic Dynamics and Control 21, 1025–1042.

References

1. Justiniano, A. and Primiceri, G. E. (2008), ‘The time-varying volatility of macroeconomic fluctuations’, American Economic Review 98(3), 604–641.

References

1. Kim, J., Kim, S., Schaumburg, E. and Sims, C. A. (2008), ‘Calculating and using second-order accurate solutions of discrete time dynamic equilibrium models’, Journal of Economic Dynamics and Control 32, 3397–3414.

References

1. Kim, J. and Ruge-Murcia, F. J. (2009), ‘How much inflation is necessary to grease the wheels?’, Journal of Monetary Economics 56, 365–377.

References

1. Kim, J.-Y. (2002), ‘Limited information likelihood and Bayesian analysis’, Journal of Econometrics 107, 175–193.

References

1. King, R. G. and Rebelo, S. T. (1999), ‘Resuscitating real business cycles’, Handbook of Macroeconomics 1, 927–1007.

References

1. Koop, G., Pesaran, M. H. and Potter, S. M. (1996), ‘Impulse response analysis in nonlinear multivariate models’, Journal of Econometrics 74, 119–147.

References

1. Lan, H. and Meyer-Gohde, A. (2013a), Pruning in perturbation DSGE models - Guidance from nonlinear moving average approximations, Discussion papers sfb 649, Humboldt University.

References

1. Lan, H. and Meyer-Gohde, A. (2013b), ‘Solving DSGE models with a nonlinear moving average’, Journal of Economic Dynamics and Control 37(12), 2643–2667.

References

1. Lan, H. and Meyer-Gohde, A. (2014), ‘Solvability of perturbation solutions in DSGE models’, Journal of Economic Dynamics and Control 45(C), 366–388.

References

1. Lombardo, G. and Sutherland, A. (2007), ‘Computing second-order accurate solutions for rational expectation models using linear solution methods’, Journal of Economic Dynamics and Control 31, 515–530.

References

1. Lombardo, G. and Uhlig, H. (2014), A theory of pruning, Working paper series number 1696, European Central Bank.

References

1. Matthes, C. and Barnichon, R. (2014), Measuring the non-linear effects of monetary policy, Working paper, CREi.

References

1. Mehra, R. and Prescott, E. C. (1985), ‘The equity premium: A puzzle’, Journal of Monetary Economics 15, 145–161.

References

1. Mutschler, W. (2015), ‘Identification of DSGE models - The effects of higher-order approximation and pruning’, Journal of Economic Dynamics and Control 56, 34–54.

References

1. Peralta-Alva, A. and Santos, M. S. (2012), Analysis of numerical errors, Working paper, Federal Reserve Bank of St. Louis.

References

1. Rudebusch, G. D. and Swanson, E. T. (2008), ‘Examining the bond premium puzzle with a DSGE model’, Journal of Monetary Economics 55, 111–126.

References

1. Rudebusch, G. D. and Swanson, E. T. (2012), ‘The bond premium in a DSGE model with long-run real and nominal risks’, American Economic Journal: Macroeconomics 4(1), 1–43.

References

1. Ruge-Murcia, F. (2012), ‘Estimating nonlinear DSGE models by the simulated method of moments: With an application to business cycles’, Journal of Economic Dynamics and Control 35, 914–938.

References

1. Ruge-Murcia, F. (2013), Generalized method of moments estimation of DSGE models, In Handbook of Research Methods and Applications in Empirical Macroeconomics, N. Hashimzade and M. Thornton (Eds.), Edward Elgar Publishing.

References

1. Ruge-Murcia, F. J. (2007), ‘Methods to estimate dynamic stochastic general equilibrium models’, Journal of Economic Dynamics and Control 31, 2599–2636.

References

1. Schmitt-Grohé, S. and Uribe, M. (2004), ‘Solving dynamic general equilibrium models using a second-order approximation to the policy function’, Journal of Economic Dynamics and Control 28, 755–775.

References

1. Schmitt-Grohe, S. and Uribe, M. (2007), Optimal inflation stabilization in a medium-scale macroeconomic model, in K. Schmidt-Hebbel and R. Mishkin, eds, 'Moneatry Policy Under Inflation Targeting', Central Bank of Chile, Santiago, Chile, pp. 125–186.

References

1. Smets, F. and Wouters, R. (2007), ‘Shocks and frictions in US business cycles: A Bayesian DSGE approach’, American Economic Review 97(3), 586–606.

References

1. Smith, J. A. A. (1993), ‘Estimating nonlinear time-series models using simulated vector autoregressions’, Journal of Applied Econometrics 8, Supplement: Special Issue on Econometric Inference Using Simulation Techniques, S63–S84.

References

1. Swanson, E. (2012), ‘Risk aversion and the labor margin in dynamic equilibrium models’, American Economic Review 102(4), 1663–1691.

References

1. Swanson, E. (2013), Risk aversion, risk premia, and the labor margin with generalized recursive preferences, Working paper series, Federal Reserve Bank of San Francisco.

References

1. Swanson, E. (2015), A Macroeconomic Mmodel of Equities and Real, Nominal, and Defaultable Bonds, Technical report, University of California, Irvine.

References

1. Tallarini, T. D. (2000), ‘Risk-sensitive real business cycles’, Journal of Monetary Economics 45, 507–532.

References

1. Weil, P. (1990), ‘Nonexpected utility in macroeconomics’, Quarterly Journal of Economics 105(1), 29–42.

Table 1: Estimation Results The reported estimates are from the second step in GMM using the optimal weighting matrix with 10 lags in the Newey-West estimator. For and , the value of the is restricted to 5 and not estimated. For these two models, preliminary results show that , which is also imposed for and .

Table 2: Model Fit All variables are expressed in annualized terms, except for and .

Newey-West estimator using 10 lags. The objective function in step 2, denoted , is computed using the optimal weighting matrix with 10 lags in the Newey-West estimator. The P-value is for the J-test for model misspecification based on the objective function in step 2. Table 2: Model Fit (continued)

Table 3: Model Specification Test

The objective function in step 1, denoted , is computed with the weighting matrix

\[\mathbf {W} _ {T} = \operatorname{diag} \left(\hat {\mathbf {S}} _ {\text { mean }} ^ {- 1}\right)\]

\[\hat {\mathbf {S}} _ {m e a n}\]

Table 4: Decomposing the 10-year term premium Moments for the 10-year term premium are reported in annualized basis points, whereas moments for the remaining variables are at a quarterly frequency and unscaled. Moments for the quantity of risk cannot be computed directly by the perturbation method (because and, hence, the market price of risk are zero in the steady state), and we therefore compute these moments from simulated sample paths of 1,000,000 observations for and the market price of risk.

Figure 1: Simulated sample path

The capital stock, the price dispersion index, and the inflation rate are expressed in deviation from the deterministic steady state, whereas consumption growth is de-meaned. Unless stated otherwise, all parameters are from . All variables are expressed at a quarterly level.

Figure 2: GIRFs: Technology shock

GIRFs following a positive one-standard-deviation shock to technology. The GIRFs are computed at the unconditional mean of the states using the estimated parameters for . All GIRFs are expressed in deviation from the steady state, except for excess holding period return and term premium, which are expressed in annualized basis points from their unconditional means.

Figure 3: GIRFs: Government shock

GIRFs following a positive one-standard-deviation shock to government spending. The GIRFs are computed at the unconditional mean of the states using the estimated parameters for . All GIRFs are expressed in deviation from the steady state, except for excess holding period return and term premium, which are expressed in annualized basis points from their unconditional means.

Figure 4: GIRFs: Preference shock

GIRFs following a positive one-standard-deviation shock to preferences. The GIRFs are computed at the unconditional mean of the states using the estimated parameters for . All GIRFs are expressed in deviation from the steady state, except for excess holding period return and term premium, which are expressed in annualized basis points from their unconditional means.

Figure 6: Conditional GIRFs: High vs. low inflation

Figure 5: Conditional GIRFs: Expansions vs. Recessions

GIRFs following a positive one-standard-deviation shock to technology using the estimated parameters for . The state values representing recessions are defined from episodes in a simulated sample path with detrended negative output in the current and the previous two periods; otherwise, the economy is defined to be in expansion. The GIRFs are computed as the average across 500 draws from expansions and recessions. All GIRFs are expressed in deviation from the steady state, except for excess holding period return and term premium, which are expressed in annualized basis points from their unconditional means.

GIRFs following a positive one-standard-deviation shock to technology using the estimated parameters for . The state values representing high inflation are defined from episodes with inflation larger than one standard deviation of inflation in a simulated sample path; otherwise, the economy is defined to be in a low inflation regime. The GIRFs are computed as the average across 500 draws from regimes of high and low inflation. All GIRFs are expressed in deviation from the steady state, except for excess holding period return and term premium, which are expressed in annualized basis points from their unconditional means.

April 18, 2016

Contents

1 The class of DSGE model 5 2 The pruning scheme: 5 2.1 Second order approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Third order approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 Summary: NO pruning up to third order. 11 2.4 Summary: pruning up to third order. 12 2.5 Increasing efficiency for the simulation in FORTRAN. 12 2.6 Increasing efficiency for the simulation in MATLAB. 15 3 Stastical properties: Second-order approximation 17 3.1 Covariance-stationary 17 3.2 Method 1: Formulas for the first and second moments 19 3.2.1 Computing the variance of the innovations 21 3.3 Method 2: Formulas for the first and second moments 25 3.3.1 For 26 3.3.2 For 27 3.4 Method 3: Simple formulas for first and second moments 32 3.4.1 First moments 33 3.4.2 Second moments 34 3.5 Decomposition: The total variance of the state variables 38 3.6 The auto-correlations 39 3.6.1 The innovations 39 3.6.2 The auto-covariances 40 4 Stastical properties: Third order approximation 41 4.1 Covariance-stationary 41 4.2 Method 1: Formulas for the first and second moments 45 4.2.1 Efficient computing of BCov 49 4.2.2 Computing 50 4.3 Method 2: Formulas for the first and second moments 53 4.3.1 For 57 4.3.2 For 57

4.3.3 For Var 59 4.3.4 For Var 63 4.3.5 For Var 74 4.3.6 For Var 77 4.3.7 For Var 87 4.4 Method 3: Simple formulas for first and second moments 111 4.4.1 First moments 112 4.4.2 Second moments 113 4.5 The auto-correlations 114 4.5.1 The innovations 114 4.5.2 The covariances 116 4.5.3 Computing Cov 117 5 The Dynare++ notation 128 6 Pruning scheme in Dynare++: 129 6.1 Second order approximation: 129 6.2 Second order approximation: a convenient representation 131 6.3 Third order approximation: 132 7 Dynare++ notation and statistical properties: second order 134 7.1 Co-variance stationarity 134 7.2 First and second moments 136 7.2.1 Computing Var 137 8 Equivalence between the SGU-notation and the Dynare notation 138 9 Existence of Skewness and Kurtosis 139 10 Impulse response functions - the definition by Andreasen 140 10.1 At first order 140 10.2 At second order 141 10.3 At third order 144 10.3.1 For 147 10.3.2 For 155 10.3.3 Summarizing 160 11 Impulse response functions - GIRF 161 11.1 At first order 161 11.2 At second order 162 11.3 At third order 166 11.3.1 For 168 11.3.2 For 176 11.3.3 Summarizing 181

12 Impulse response functions - GIRF version 2 182 12.1 The model for the conditional information . 182 12.2 At first order . 183 12.3 At second order . 184 12.4 Second order: at the steady state with shock size of unity . 188 12.5 At third order . 189 12.5.1 For . 191 12.5.2 For . 205 12.5.3 Summarizing . 210 13 Alternative notation with in the state vector 210 14 Accuracy of the pruned state-space system with 212 14.1 Some auxiliary expressions . 213 14.2 Proof for a second order approximation . 215 14.2.1 First order terms . 215 14.2.2 Second order terms . 216 14.3 Proof for a third order approximation . 217 14.3.1 First order terms . 217 14.3.2 Second order terms . 217 14.3.3 Third order terms . 218 15 The pruning schemes in Den Haan and De Wind (2012) 221 15.1 First proposal . 221 15.2 Second proposal . 222 16 A New Keynesian Model 223 16.1 Households . 223 16.2 Firms . 225 16.2.1 Final Good producers . 225 16.2.2 Intermediate Good Producer . 226 16.2.3 Marginal costs . 228 16.2.4 The recursive representation of the price relation . 229 16.3 Financial intermediary . 232 16.4 The central bank . 233 16.5 Aggregation . 234 16.5.1 The goods market: final good producer . 234 16.6 The goods market: The relation between the optimale price and the price index . 235 16.7 The resource constraint . 236 16.8 The periodic utility function of the representative household . 237 16.9 Summarizing . 238 16.10A transformation of the DSGE model . 239 16.11Market completeness . 248 16.12The intertemporal elasticity of substitution (IES) . 248 16.12.1 External habit formation . 249 16.12.2 Internal habit formation . 250 16.12.3 Comparing internal and external habits . 255 16.13The Frisch labor supply elasticity . 256 16.14Measures of relative risk aversion . 256 16.14.1 External Habits . 256 16.14.2 Internal habits . 259

16.15 The steady state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 16.16 The observables and their moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 16.16.1 Calculating the observables . . . . . . . . . . . . . . . . . . . . . . . . 266 16.16.2 Moments of growth rates and excess holding period return . . . . . . . . . . . . 267 16.17 Understanding the dynamics of the price dispersion index . . . . . . . . . . . . . . 271 16.18 Understanding term premium in the model 272 16.18.1 Term premium 272 16.18.2 The excess holding period return 274

This technical appendix explains in great detail the derivations carried out in relation to our paper. In addition to the material reported in the paper, this technical appendix also provides some additional results - for instance alternative ways of computing second moments (at second and third order) and how to directly implement pruning based on the Dynare notation.

1 The class of DSGE model

We consider the class of DSGE models where the set of equilibrium conditions can be written as

\[E _ {t} \left[ \mathbf {f} \left(\mathbf {y} _ {t + 1}, \mathbf {y} _ {t}, \mathbf {x} _ {t + 1}, \mathbf {x} _ {t}\right) \right] = \mathbf {0}.\tag{1}\]

Here, is the conditional expectation given information available at time t. The vector is the set of state variables (pre-determined variables) and has dimension . The vector contains the set of control variables (non pre-determined variables) and has dimension . We also let .

The state vector is partitioned as , where with dimension contains the set of endogenous state variables and with dimension contains the set of exogenous state variables. Note also that .

For the exogenous state variables we assume that

\[\mathbf {x} _ {2, t + 1} = \mathbf {h} (\mathbf {x} _ {2, t}, \sigma) + \sigma \tilde {\boldsymbol {\eta}} \boldsymbol {\epsilon} _ {t + 1},\tag{2}\]

where has dimension , and thus, has dimension . We assume throughout that , that is the innovations are identical and independent distributed with mean zero and covariance matrix I. Further moment requirements on will be imposed later.

The general solution to this class of DSGE model is given by

\[\mathbf {y} _ {t} = \mathbf {g} (\mathbf {x} _ {t}, \sigma)\tag{3}\]

\[\mathbf {x} _ {t + 1} = \mathbf {h} (\mathbf {x} _ {t}, \sigma) + \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1}\tag{4}\]

\[\boldsymbol {\eta} = \left[ \begin{array}{c} \mathbf {0} \\ \tilde {\boldsymbol {\eta}} \end{array} \right]\tag{5}\]

where the functions and are unknown. We will therefore approximate these functions up to any desired order. This is done around the non-stochastic steady state, i.e. and . Formally, the expression for non-stochastic steady state is given as the solution of to

\[\mathbf {f} \left(\mathbf {y} _ {s s}, \mathbf {y} _ {s s}, \mathbf {x} _ {s s}, \mathbf {x} _ {s s}\right) = \mathbf {0}.\tag{6}\]

Note also that and .

2 The pruning scheme:

2.1 Second order approximation

We start by partitioning the state vector using the approximated expression

\[\mathbf {x} _ {t} = \mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s},\]

where denotes the first order terms and denotes the second order terms.

A second-order approximation of the state equation reads (for )

\[\begin{array}{l} x _ {t + 1} (j, 1) = \mathbf {h} _ {\mathbf {x}} (j,:) \mathbf {x} _ {t} + \frac {1}{2} \mathbf {x} _ {t} ^ {\prime} \mathbf {h} _ {\mathbf {x x}} (j,:,:) \mathbf {x} _ {t} + \frac {1}{2} h _ {\sigma \sigma} (j, 1) \sigma^ {2} + \sigma \pmb {\eta} (j,:) \pmb {\epsilon} _ {t + 1} \\ \Updownarrow \end{array}\]

\[\begin{array}{c} x _ {t + 1} ^ {f} (j, 1) + x _ {t + 1} ^ {s} (j, 1) = \mathbf {h} _ {\mathbf {x}} (j,:) (\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s}) + \frac {1}{2} (\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s}) ^ {\prime} \mathbf {h} _ {\mathbf {x x}} (j,:,:) (\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s}) \\ + \frac {1}{2} h _ {\sigma \sigma} (j, 1) \sigma^ {2} + \sigma \boldsymbol {\eta} (j,:) \boldsymbol {\epsilon} _ {t + 1} \end{array}\]

\[\begin{array}{r} x _ {t + 1} ^ {f} (j, 1) + x _ {t + 1} ^ {s} (j, 1) = \mathbf {h} _ {\mathbf {x}} (j,:) \mathbf {x} _ {t} ^ {f} + \mathbf {h} _ {\mathbf {x}} (j,:) \mathbf {x} _ {t} ^ {s} + \frac {1}{2} \left(\left(\mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \mathbf {h} _ {\mathbf {x x}} (j,:,:) + (\mathbf {x} _ {t} ^ {s}) ^ {\prime} \mathbf {h} _ {\mathbf {x x}} (j,::)\right) (\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s}) \\ + \frac {1}{2} h _ {\sigma \sigma} (j, 1) \sigma^ {2} + \sigma \pmb {\eta} (j,:) \pmb {\epsilon} _ {t + 1} \end{array}\]

↓

\[\begin{array}{r l} & x _ {t + 1} ^ {f} (j, 1) + x _ {t + 1} ^ {s} (j, 1) = \mathbf {h} _ {\mathbf {x}} (j,:) \mathbf {x} _ {t} ^ {f} + \mathbf {h} _ {\mathbf {x}} (j,:) \mathbf {x} _ {t} ^ {s} \\ & \qquad + \frac {1}{2} \left(\left(\mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \mathbf {h} _ {\mathbf {x x}} (j,:,:) \mathbf {x} _ {t} ^ {f} + \left(\mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \mathbf {h} _ {\mathbf {x x}} (j,:,:) \mathbf {x} _ {t} ^ {s} + (\mathbf {x} _ {t} ^ {s}) ^ {\prime} \mathbf {h} _ {\mathbf {x x}} (j,:,:) \mathbf {x} _ {t} ^ {f} + (\mathbf {x} _ {t} ^ {s}) ^ {\prime} \mathbf {h} _ {\mathbf {x x}} (j,:,:) \mathbf {x} _ {t} ^ {s}\right) \\ & \qquad + \frac {1}{2} h _ {\sigma \sigma} (j, 1) \sigma^ {2} + \sigma \pmb {\eta} (j,:) \pmb {\epsilon} _ {t + 1} \end{array}\]

↓

\[\begin{array}{r l} & x _ {t + 1} ^ {f} (j, 1) + x _ {t + 1} ^ {s} (j, 1) = \mathbf {h} _ {\mathbf {x}} (j,:) \mathbf {x} _ {t} ^ {f} + \mathbf {h} _ {\mathbf {x}} (j,:) \mathbf {x} _ {t} ^ {s} \\ & \qquad + \frac {1}{2} \left(\left(\mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \mathbf {h} _ {\mathbf {x x}} (j,:,:) \mathbf {x} _ {t} ^ {f} + 2 (\mathbf {x} _ {t} ^ {f}) ^ {\prime} \mathbf {h} _ {\mathbf {x x}} (j,:,:) \mathbf {x} _ {t} ^ {s} + (\mathbf {x} _ {t} ^ {s}) ^ {\prime} \mathbf {h} _ {\mathbf {x x}} (j,:,:) \mathbf {x} _ {t} ^ {s}\right) \\ & \qquad + \frac {1}{2} h _ {\sigma \sigma} (j, 1) \sigma^ {2} + \sigma \pmb {\eta} (j,:) \pmb {\epsilon} _ {t + 1} \end{array}\]

due to the symmetry of .

A law of motion for the first order terms is thus

\[x _ {t + 1} ^ {f} (j, 1) = \mathbf {h} _ {\mathbf {x}} (j,:) \mathbf {x} _ {t} ^ {f} + \sigma \pmb {\eta} (j,:) \pmb {\epsilon} _ {t + 1}\]

A law of motion for the second order terms is thus

\[x _ {t + 1} ^ {s} (j, 1) = \mathbf {h} _ {\mathbf {x}} (j,:) \mathbf {x} _ {t} ^ {s} + \frac {1}{2} \left(\left(\mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \mathbf {h} _ {\mathbf {x x}} (j,:,:) \mathbf {x} _ {t} ^ {f}\right) + \frac {1}{2} h _ {\sigma \sigma} (j, 1) \sigma^ {2}\]

\[(\text { for } i = 1, 2,... n _ {y})\]

\[\begin{array}{l} \text {Inserting the decomposition of the state variables into the control variables we get} \\ y _ {t} ^ {s} (i, 1) = \mathbf {g} _ {\mathbf {x}} (i,:) \mathbf {x} _ {t} + \frac {1}{2} \mathbf {x} _ {t} ^ {\prime} \mathbf {g} _ {\mathbf {x x}} (i,:,: ) \mathbf {x} _ {t} + \frac {1}{2} g _ {\sigma \sigma} (i, 1) \sigma^ {2} \\ \Updownarrow \\ y _ {t} ^ {s} (i, 1) = \mathbf {g} _ {\mathbf {x}} (i,:) (\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s}) + \frac {1}{2} (\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s}) ^ {\prime} \mathbf {g} _ {\mathbf {x x}} (i,:,: ) (\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s}) + \frac {1}{2} g _ {\sigma \sigma} (i, 1) \sigma^ {2} \\ \Updownarrow \\ y _ {t} ^ {s} (i, 1) = \mathbf {g} _ {\mathbf {x}} (i,:) (\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s}) + \frac {1 / 2}{2} ((\mathbf {x} _ {t} ^ {f}) ^ {\prime} \mathbf {g} _ {\mathbf {x x}} (i,:,: ) + (\mathbf {x} _ {t} ^ {s}) ^ {\prime} \mathbf {g} _ {\mathbf {x x}} (i,:,:)) (\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s}) \\ \qquad + \frac {1}{2} g _ {\sigma \sigma} (i, 1) \sigma^ {2} \\ \Updownarrow \\ y _ {t} ^ {s} (i, 1) = \mathbf {g} _ {\mathbf {x}} (i,:) (\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s}) \\ \qquad + \frac {1}{2} ((\mathbf {x} _ {t} ^ {f}) ^ {\prime} \mathbf {g} _ {\mathbf {x x}} (i,:,: ) \mathbf {x} _ {t} ^ {f} + 2 (\mathbf {x} _ {t} ^ {f}) ^ {\prime} \mathbf {g} _ {\mathbf {x x}} (i,:,: ) \mathbf {x} _ {t} ^ {s} + (\mathbf {x} _ {t} ^ {s}) ^ {\prime} \mathbf {g} _ {\mathbf {x x}} (i,:,: ) \mathbf {x} _ {t} ^ {s}) \\ \qquad + \frac {1}{2} g _ {\sigma \sigma} (i, 1) \sigma^ {2} \end{array}\]

due to the symmetry of

We want to preserve terms up to second order, hence the pruned approximation is

\[y _ {t} ^ {s} (i, 1) = \mathbf {g} _ {\mathbf {x}} (i,:) (\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s}) + \frac {1}{2} (\mathbf {x} _ {t} ^ {f}) ^ {\prime} \mathbf {g} _ {\mathbf {x x}} (i,:,:) \mathbf {x} _ {t} ^ {f} + \frac {1}{2} g _ {\sigma \sigma} (i, 1) \sigma^ {2}\]

because is a third order term and is a fourth order term

2.2 Third order approximation

We decompose the state vector using the approximated expression

\[\mathbf {x} _ {t} = \mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d},\]

where the new term denotes the third order term.

A third order approximation of the state equation reads (for )

\[\begin{array}{r l} & {{x _ {t + 1} \left(j, 1\right) = \mathbf {h} _ {\mathbf {x}} \left(j,:\right) \mathbf {x} _ {t} + \sigma \pmb {\eta} \left(j,:\right) \pmb {\epsilon} _ {t + 1}}} \\ & {{\quad + \frac {1}{2} \mathbf {x} _ {t} ^ {\prime} \mathbf {h} _ {\mathbf {x x}} \left(j, :,:\right) \mathbf {x} _ {t} + \frac {1}{2} h _ {\sigma \sigma} \left(j, 1\right) \sigma^ {2}}} \\ & {{\quad + \frac {1}{6} \mathbf {x} _ {t} ^ {\prime} \left[ \begin{array}{c} {{\mathbf {x} _ {t} ^ {\prime} \mathbf {h} _ {\mathbf {x x x}} (j, 1, :,:) \mathbf {x} _ {t}}} \\ {{\dots}} \\ {{\mathbf {x} _ {t} ^ {\prime} \mathbf {h} _ {\mathbf {x x x}} (j, n _ {x},:,:) \mathbf {x} _ {t}}} \end{array} \right] + \frac {3}{6} \mathbf {h} _ {\sigma \sigma \mathbf {x}} \left(j,:\right) \sigma^ {2} \mathbf {x} _ {t} + \frac {1}{6} h _ {\sigma \sigma \sigma} \left(j, 1\right) \sigma^ {3}}} \\ {{\mathrm{个}}} \end{array}\]

\[\begin{array}{r l} & {{x _ {t + 1} ^ {f} \left(j, 1\right) + x _ {t + 1} ^ {s} \left(j, 1\right) + x _ {t + 1} ^ {r d} \left(j, 1\right) = \mathbf {h} _ {\mathbf {x}} \left(j,: \right\rangle \left(\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d}\right) + \sigma \pmb {\eta} \left(j,: \right\rangle \pmb {\epsilon} _ {t + 1}}} \\ & {{\quad + \frac {1}{2} \left(\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d}\right) ^ {\prime} \mathbf {h} _ {\mathbf {x x}} \left(j,:,:) \left(\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d}\right) + \frac {1}{2} h _ {\sigma \sigma} \left(j, 1\right) \sigma^ {2}}} \\ & \quad + \frac {1}{6} \left(\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d}\right) ^ {\prime} \left[ \begin{array}{c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c} & {{\left(\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d}\right) ^ {\prime} \mathbf {h} _ {\mathbf {x x x}} (j, 1,:,:) \left(\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d}\right)}} \\ & {{\qquad \qquad \qquad \qquad \qquad \qquad \qquad \dots}} \\ & {{\left(\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d}\right) ^ {\prime} \mathbf {h} _ {\mathbf {x x x}} (j, n _ {x},,:,:) \left(\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d}\right)}} \\ & {{\quad + \frac {3}{6} \mathbf {h} _ {\sigma \sigma \mathbf {x}} \left(j,:\right) \sigma^ {2} \left(\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d}\right) + \frac {1}{6} h _ {\sigma \sigma \sigma} \left(j, 1\right) \sigma^ {3}}} \\ {{\mathrm{个}}} & {{\quad ,}} \end{array}\]

\[\begin{array}{r l} & x _ {t + 1} ^ {f} (j, 1) + x _ {t + 1} ^ {s} (j, 1) + x _ {t + 1} ^ {r d} (j, 1) = \mathbf {h} _ {\mathbf {x}} (j,:) (\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d}) + \sigma \boldsymbol {\eta} (j,:) \boldsymbol {\epsilon} _ {t + 1} \\ & \quad + \frac {1}{2} ((\mathbf {x} _ {t} ^ {f}) ^ {\prime} \mathbf {h} _ {\mathbf {x x}} (j,:,:) + (\mathbf {x} _ {t} ^ {s}) ^ {\prime} \mathbf {h} _ {\mathbf {x x}} (j,:,:) + (\mathbf {x} _ {t} ^ {r d}) ^ {\prime} \mathbf {h} _ {\mathbf {x x}} (j,:,:))) (\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d}) + \frac {1}{2} h _ {\sigma \sigma} (j, 1) \sigma^ {2} \\ & \quad + \frac {1}{6} (\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d}) ^ {\prime} \times \\ & \qquad \left[ \begin{array}{c} (\left(\mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \mathbf {h} _ {\mathbf {x x x}} (j, 1,:,:) + (\mathbf {x} _ {t} ^ {s}) ^ {\prime} \mathbf {h} _ {\mathbf {x x x}} (j, 1,:,:) + (\mathbf {x} _ {t} ^ {r d}) ^ {\prime} \mathbf {h} _ {\mathbf {x x x}} (j, 1,:,:))) (\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d}) \\ \dots \\ (\left(\mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \mathbf {h} _ {\mathbf {x x x}} (j, n _ {x},:,:) + (\mathbf {x} _ {t} ^ {s}) ^ {\prime} \mathbf {h} _ {\mathbf {x x x}} (j, n _ {x},:,:) + (\mathbf {x} _ {t} ^ {r d}) ^ {\prime} \mathbf {h} _ {\mathbf {x x x}} (j, n _ {x},:,:))) (\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d}) \end{array} \right] \\ & \quad + \frac {3}{6} \mathbf {h} _ {\sigma \sigma \mathbf {x}} (j,:) \sigma^ {2} (\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d}) + \frac {1}{6} h _ {\sigma \sigma \sigma} (j, 1) \sigma^ {3} \\ & \hat {\boldsymbol {\alpha}}. \end{array}\]

↓

\[\begin{array}{r l} & x _ {t + 1} ^ {f} (j, 1) + x _ {t + 1} ^ {s} (j, 1) + x _ {t + 1} ^ {r d} (j, 1) = \mathbf {h} _ {\mathbf {x}} (j,:) (\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d}) + \sigma \pmb {\eta} (j,:) \pmb {\epsilon} _ {t + 1} \\ & \qquad + \frac {1}{2} \left((\mathbf {x} _ {t} ^ {f}) ^ {\prime} \mathbf {h} _ {\mathbf {x x}} (j,:,:) \mathbf {x} _ {t} ^ {f} + (\mathbf {x} _ {t} ^ {f}) ^ {\prime} \mathbf {h} _ {\mathbf {x x}} (j,:,:) \mathbf {x} _ {t} ^ {s} + (\mathbf {x} _ {t} ^ {f}) ^ {\prime} \mathbf {h} _ {\mathbf {x x}} (j,:,:) \mathbf {x} _ {t} ^ {r d}\right) \end{array}\]

\[\begin{array}{r l} & {+ \frac {1}{2} \left((\mathbf {x} _ {t} ^ {s}) ^ {\prime} \mathbf {h} _ {\mathbf {x x}} (j,:,:) \mathbf {x} _ {t} ^ {f} + (\mathbf {x} _ {t} ^ {s}) ^ {\prime} \mathbf {h} _ {\mathbf {x x}} (j,:,:) \mathbf {x} _ {t} ^ {s} + (\mathbf {x} _ {t} ^ {s}) ^ {\prime} \mathbf {h} _ {\mathbf {x x}} (j,:,:) \mathbf {x} _ {t} ^ {r d}\right)} \\ & {+ \frac {1}{2} \left((\mathbf {x} _ {t} ^ {r d}) ^ {\prime} \mathbf {h} _ {\mathbf {x x}} (j,:,:) \mathbf {x} _ {t} ^ {f} + (\mathbf {x} _ {t} ^ {r d}) ^ {\prime} \mathbf {h} _ {\mathbf {x x}} (j,:,:) \mathbf {x} _ {t} ^ {s} + (\mathbf {x} _ {t} ^ {r d}) ^ {\prime} \mathbf {h} _ {\mathbf {x x}} (j,:,:) \mathbf {x} _ {t} ^ {r d}\right)} \\ & {+ \frac {1}{2} h _ {\sigma \sigma} (j, 1) \sigma^ {2}} \\ & {+ \frac {1}{6} (\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d}) ^ {\prime} \times} \\ & {\left[ \begin{array}{c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c} & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & \\ & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & \\ \end{array} \right]} \\ & {\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \dots} \\ & {\qquad + \frac {3}{6} \mathbf {h} _ {\sigma \sigma \mathbf {x}} (j,:) \sigma^ {2} (\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d}) + \frac {1}{6} h _ {\sigma \sigma \sigma} (j, 1) \sigma^ {3}} \end{array}\]

\[\begin{array} { r l } & x _ { t + 1 } ^ { f } \left( j , 1 \right) + x _ { t + 1 } ^ { s } \left( j , 1 \right) + x _ { t + 1 } ^ { r d } \left( j , 1 \right) = \mathbf { h _ { x } } \left( j , : \right) \left( \mathbf { x } _ { t } ^ { f } + \mathbf { x } _ { t } ^ { s } + \mathbf { x } _ { t } ^ { r d } \right) + \sigma \pmb { \eta } \left( j , : \right) \pmb { \epsilon } _ { t + 1 } \\ & + \frac { 1 } { 2 } \left( \left( \mathbf { x } _ { t } ^ { f } \right) ^ { \prime } \mathbf { h _ { x x } } \left( j , : , : \right) \mathbf { x } _ { t } ^ { f } + 2 \left( \mathbf { x } _ { t } ^ { f } \right) ^ { \prime } \mathbf { h _ { x x } } \left( j , : , : \right) \mathbf { x } _ { t } ^ { s } + 2 \left( \mathbf { x } _ { t } ^ { f } \right) ^ { \prime } \mathbf { h _ { x x } } \left( j , : , : \right) \mathbf { x } _ { t } ^ { r d } \right) \\ & + \frac { 1 } { 2 } \left( ( \mathbf { x } _ { t } ^ { s } ) ^ { \prime } \mathbf { h _ { x x } } \left( j , : , : \right) \mathbf { x } _ { t } ^ { s } + 2 \left( \mathbf { x } _ { t } ^ { s } \right) ^ { \prime } \mathbf { h _ { x x } } \left( j , : , : \right) \mathbf { x } _ { t } ^ { r d } + ( \mathbf { x } _ { t } ^ { r d } ) ^ { \prime } \mathbf { h _ { x x } } \left( j , : , : \right) \mathbf { x } _ { t } ^ { r d } \right) \\ & + \frac { 1 } { 2 } h _ { \sigma \sigma } \left( j , 1 \right) \sigma ^ { 2 } \\ & + \frac { 1 } { 6 } \sum _ { \gamma = 1 } ^ { n _ { x } } \left( x _ { t } ^ { f } \left( \gamma , 1 \right) + x _ { t } ^ { s } ( \gamma , 1 ) + x _ { t } ^ { r d } ( \gamma , 1 ) \right) ( \mathbf { x } _ { t } ^ { f } ) ^ { \prime } \mathbf { h _ { x x x } } ( j , \gamma , : , : ) ( \mathbf { x } _ { t } ^ { f } + \mathbf { x } _ { t } ^ { s } + \mathbf { x } _ { t } ^ { r d } ) \\ & + \frac { 1 } { 6 } \sum _ { \gamma = 1 } ^ { n _ { x } } \left( x _ { t } ^ { f } ( \gamma , 1 ) + x _ { t } ^ { s } ( \gamma , 1 ) + x _ { t } ^ { r d } ( \gamma , 1 ) \right) ( \mathbf { x } _ { t } ^ { s } ) ^ { \prime } \mathbf { h _ { x x x } } ( j , \gamma , : , : ) ( \mathbf { x } _ { t } ^ { f } + \mathbf { x } _ { t } ^ { s } + \mathbf { x } _ { t } ^ { r d } ) \\ & + \frac { 1 } { 6 } \sum _ { \gamma = 1 }^{ n _ { x } } \left( x _ { t } ^ { f } ( \gamma , 1 ) + x _ { t } ^ { s } ( \gamma , 1 ) + x _ { t } ^ { r d} ( \gamma , 1 ) \right) ( \mathbf { x } _ { t } ^ { r d } ) ^ { \prime } \mathbf { h _ { x x x } } ( j , \gamma , : , : ) ( \mathbf { x } _ { t } ^ { f } + \mathbf { x } _ { t } ^ { s } + \mathbf { x } _ { t } ^ { r d } ) \\ & + \frac { 3 } { 6 } \mathbf { h _ { \sigma \sigma x } } ( j , : ) \sigma ^ { 2 } ( \mathbf { x } _ { t } ^ { f } + \mathbf { x } _ { t } ^ { s } + \mathbf { x } _ { t } ^ { r d} ) + \frac { 1}{ 6} h _ { \sigma \sigma \sigma }\left( j , 1 \right) \sigma ^ { 3 } . \end{array}\]

due to symmetry in

↓

\[\begin{array} { r l } & x _ { t + 1 } ^ { f } \left( j , 1 \right) + x _ { t + 1 } ^ { s } \left( j , 1 \right) + x _ { t + 1 } ^ { r d } \left( j , 1 \right) = \mathbf { h } _ { \mathbf { x } } \left( j , : \right) \left( \mathbf { x } _ { t } ^ { f } + \mathbf { x } _ { t } ^ { s } + \mathbf { x } _ { t } ^ { r d } \right) + \sigma \pmb { \eta } \left( j , : \right) \pmb { \epsilon } _ { t + 1 } \\ & \quad + \frac { 1 } { 2 } \left( \left( \mathbf { x } _ { t } ^ { f } \right) ^ { \prime } \mathbf { h } _ { \mathbf { x x } } \left( j , : , : \right) \mathbf { x } _ { t } ^ { f } + 2 \left( \mathbf { x } _ { t } ^ { f } \right) ^ { \prime } \mathbf { h } _ { \mathbf { x x } } \left( j , : , : \right) \mathbf { x } _ { t } ^ { s } + 2 \left( \mathbf { x } _ { t } ^ { f } \right) ^ { \prime } \mathbf { h } _ { \mathbf { x x } } \left( j , : , : \right) \mathbf { x } _ { t } ^ { r d } \right) \\ & \quad + \frac { 1 } { 2 } \left( ( \mathbf { x } _ { t } ^ { s } ) ^ { \prime } \mathbf { h } _ { \mathbf { x x } } \left( j , : , : \right) \mathbf { x } _ { t } ^ { s } + 2 \left( \mathbf { x } _ { t } ^ { s } \right) ^ { \prime } \mathbf { h } _ { \mathbf { x x } } \left( j , : , : \right) \mathbf { x } _ { t } ^ { r d } + ( \mathbf { x } _ { t } ^ { r d } ) ^ { \prime } \mathbf { h } _ { \mathbf { x x } } \left( j , : , : \right) \mathbf { x } _ { t } ^ { r d } \right) \\ & \quad + \frac { 1 } { 2 } h _ { \sigma \sigma } \left( j , 1 \right) \sigma ^ { 2 } \\ & \quad + \frac { 1 } { 6 } \sum _ { \gamma = 1 } ^ { n _ { x } } \left( x _ { t } ^ { f } \left( \gamma , 1 \right) + x _ { t } ^ { s } \left( \gamma , 1 \right) + x _ { t } ^ { r d } \left( \gamma , 1 \right) \right) \\ & \qquad \times \left( ( \mathbf { x } _ { t } ^ { f } ) ^ { \prime } \mathbf { h } _ { \mathbf { x x x } } ( j , \gamma , : , : ) \mathbf { x } _ { t } ^ { f } + ( \mathbf { x } _ { t } ^ { f } ) ^ { \prime } \mathbf { h } _ { \mathbf { x x x } } ( j , \gamma , : , : ) \mathbf { x } _ { t } ^ { s } + ( \mathbf { x } _ { t } ^ { f } ) ^ { \prime } \mathbf { h } _ { \mathbf { x x x } } ( j , \gamma , : , : ) \mathbf { x } _ { t } ^ { r d} \right) \\ & \quad + \frac { 1 } { 6 } \sum _ { \gamma = 1 } ^ { n _ { x } } \left( x _ { t } ^ { f } \left( \gamma , 1 \right) + x _ { t } ^ { s } \left( \gamma , 1 \right) + x _ { t } ^ { r d} \left( \gamma , 1 \right) \right) \\ & \qquad \times ( ( \mathbf { x } _ { t } ^ { s } ) ^ { \prime } \mathbf { h } _ { \mathbf { x x x } } ( j , \gamma , : , : ) {\mathbf { x }} _ { t } ^ { f } + ( {\mathbf { x }} _ { t } ^ { s } ) ^ { \prime } {\mathbf { h }} _ { \mathbf { x x x } } ( j , \gamma , : , : ) {\mathbf { x }} _ { t } ^ { s } + ( {\mathbf { x }} _ { t } ^ { s } ) ^ { \prime } {\mathbf { h}} _ { \mathbf { x x x } } ( j , \gamma , : , : ) {\mathbf { x }} _ { t } ^ { r d} ) \\ & \quad + \frac { 1 } { 6 } \sum _ { \gamma = 1 } ^ { n _ { x } } \left( x _ { t } ^ { f } \left( \gamma , 1 \right) + x _ { t } ^ { s } ( 7 , 1 ) + x _ { t } ^ { r d} ( 7 , 1 ) + x _ { t } ^ {( 7 ) r d} ( 7 , 1 ) - ( 7 ) ( 7 ) ( 7 ) ( 7 ) ( 7 ) ( 7 ) ( 7 ) ( 7 ) ( 7 ) ( 7 ) ( 7 ) ( 7 ) ( 7 ) ( 7 ) ( 7 ) ( 7 ) ( 7 ) ( 7 ) ( 7 ) ( 7 ) ( 7 ) ( 7 ) ( 7 ) ( 7 ) ( 7 ) ( 7 . ) ( 7 ) ( 7 ) ( 7 ) ( 7 ) ( 7 ) ( 7 ) ( 7 ) ( 7 ) ( 7 ) ( 7 ) ( 7 ) ( 7 ) ( 7 ) ( 7 ) ( 7 ) ( 7 ) ( 7 ) ( 7 ) ( 7 ) ( 7 ) ( 7 ) ( 7 ) ( 7 ) ( 7 ) ( 3 ) ( 3 ) ( 3 ) ( 3 ) ( 3 ) ( 3 ) ( 3 ) ( 3 ) ( 3 ) ( 3 ) ( 3 ) ( 3 ) ( 3 ) ( 3 ) ( 3 ) ( 3 ) ( 3 ) ( 3 ) ( 3 ) ( 3 ) ( 3 ) ( 3 ) ( 3 ) ( 3 ) ( 3 ) ( 4 ) ( 4 ) ( 4 ) ( 4 ) ( 4 ) ( 4 ) ( 4 ) ( 4 ) ( 4 ) ( 4 ) ( 4 ) ( 4 ) ( 4 ) ( 4 ) ( 4 ) ( 4 ) ( 4 ) ( 4 ) ( 4 ) ( 4 ) ( 4 ) ( 4 ) ( 4 ) ( 4 ) ( 4 ) ( 5 ) | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | & X = X, Y = Z, W = X, X = Y, Y = Z, Z, W = Z, X = Z, Y = Z, Z, W = Z, X = Z, Y = Z, Z, W = Z, X = Z, Y = Z, Z, W = Z, X = Z, Y = Z, Z, W = Z, X = Z, Y = Z, Z, W = Z, X = Z, Y = Z, Z, W = Z, X = Z, Y = Z, Z, W = Z, X = Z, T = T = T = T = T = T = T = T = T = T = T = T = T = T = T = T = T = T = T = T = T = T = T = T = T = T = T = T = T = T = T = T = T = T = T = T = T = T = T = T = T = T = T = T = T = T = T = T = T = T = T < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ; < ;< |content_end|>\]

\[\begin{array} { r l } & + \frac { 1 } { 2 } \left( \left( \mathbf { x } _ { t } ^ { f } \right) ^ { \prime } \mathbf { h } _ { \mathbf { x x } } ( j , : , : ) \mathbf { x } _ { t } ^ { f } + 2 \left( \mathbf { x } _ { t } ^ { f } \right) ^ { \prime } \mathbf { h } _ { \mathbf { x x } } ( j , : , : ) \mathbf { x } _ { t } ^ { s } + 2 \left( \mathbf { x } _ { t } ^ { f } \right) ^ { \prime } \mathbf { h } _ { \mathbf { x x } } ( j , : , : ) \mathbf { x } _ { t } ^ { r d } \right) \\ & + \frac { 1 } { 2 } \left( ( \mathbf { x } _ { t } ^ { s } ) ^ { \prime } \mathbf { h } _ { \mathbf { x x } } ( j , : , : ) \mathbf { x } _ { t } ^ { s } + 2 \left( \mathbf { x } _ { t } ^ { s } \right) ^ { \prime } \mathbf { h } _ { \mathbf { x x } } ( j , : , : ) \mathbf { x } _ { t } ^ { r d } + \left( \mathbf { x } _ { t } ^ { r d } \right) ^ { \prime } \mathbf { h } _ { \mathbf { x x } } ( j , : , : ) \mathbf { x } _ { t } ^ { r d } \right) \\ & + \frac { 1 } { 2 } h _ { \sigma \sigma } ( j , 1 ) \sigma ^ { 2 } \\ & + \frac { 1 } { 6 } \sum _ { \gamma = 1 } ^ { n _ { x } } \left( x _ { t } ^ { f } ( \gamma , 1 ) + x _ { t } ^ { s } ( \gamma , 1 ) + x _ { t } ^ { r d } ( \gamma , 1 ) \right) \\ & \qquad \times \left( \left( \mathbf { x } _ { t } ^ { f } \right) ^ { \prime } \mathbf { h } _ { \mathbf { x x x } } ( j , \gamma , : , : ) \mathbf { x } _ { t } ^ { f } + 2 \left( \mathbf { x } _ { t } ^ { f } \right) ^ { \prime } \mathbf { h } _ { \mathbf { x x x } } ( j , \gamma , : , : ) \mathbf { x } _ { t } ^ { s } + 2 \left( \mathbf { x } _ { t } ^ { f } \right) ^ { \prime } \mathbf { h } _ { \mathbf { x x x } } ( j , \gamma , : , : ) \mathbf { x } _ { t } ^ { r d } \right) \\ & + \frac { 1 } { 6 } \sum _ { \gamma = 1 } ^ { n _ { x } } \left( x _ { t } ^ { f } ( \gamma , 1 ) + x _ { t } ^ { s } ( \gamma , 1 ) + x _ { t } ^ { r d } ( \gamma , 1 ) \right) \\ & \qquad \times \left. ( ( \mathbf { x } _ { t } ^ { s } ) ^ { \prime } \mathbf { h } _ { \mathbf { x x x } } ( j , \gamma , : , : ) \mathbf { x } _ { t } ^ { s } + 2 ( \mathbf { x } _ { t } ^ { s } ) ^ { \prime } \mathbf { h } _ { \mathbf { x x x } } ( j , \gamma , : , : ) \mathbf { x } _ { t } ^ { r d } + ( \mathbf { x } _ { t } ^ { r d} ) ^ { \prime } \mathbf { h } _ { \mathbf { x x x } } ( j , \gamma , : , : ) \mathbf { x } _ { t } ^ { r d} \right) \\ & { + \frac { 3 } { 6 } \mathbf { h } _ { \sigma \sigma x } ( j , : ) \sigma ^ { 2 } ( \mathbf { x } _ { t } ^ { f } + \mathbf { x } _ { t } ^ { s } + \mathbf { x } _ { t } ^ { r d} ) + \frac { 1 } { 6 } h _ { \sigma \sigma \sigma } ( j , 1 ) \sigma ^ { 3 } . } \end{array}\]

due to symmetries in

A law of motion for is then (as before)

\[x _ {t + 1} ^ {f} (j, 1) = \mathbf {h} _ {\mathbf {x}} (j,:) \mathbf {x} _ {t} ^ {f} + \sigma \boldsymbol {\eta} (j,:) \boldsymbol {\epsilon} _ {t + 1}\]

because we only keep first order terms

A law of motion for is then (as before)

\[x _ {t + 1} ^ {s} (j, 1) = \mathbf {h} _ {\mathbf {x}} (j,:) \mathbf {x} _ {t} ^ {s} + \frac {1}{2} \left(\left(\mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \mathbf {h} _ {\mathbf {x x}} (j,:,:) \mathbf {x} _ {t} ^ {f}\right) + \frac {1}{2} h _ {\sigma \sigma} (j, 1) \sigma^ {2}\]

because we only keep second order terms.

A law of motion for is then

\[\begin{array}{r c l} x _ {t + 1} ^ {r d} (j, 1) & = & \mathbf {h} _ {\mathbf {x}} (j,:) \mathbf {x} _ {t} ^ {r d} + \frac {2}{2} (\mathbf {x} _ {t} ^ {f}) ^ {\prime} \mathbf {h} _ {\mathbf {x x}} (j,:,:) \mathbf {x} _ {t} ^ {s} + \frac {1}{6} \sum_ {\gamma = 1} ^ {n _ {x}} x _ {t} ^ {f} (\gamma , 1) (\mathbf {x} _ {t} ^ {f}) ^ {\prime} \mathbf {h} _ {\mathbf {x x x}} (j, \gamma ,:,:) \mathbf {x} _ {t} ^ {f} \\ & & + \frac {3}{6} \mathbf {h} _ {\sigma \sigma \mathbf {x}} (j,:) \sigma^ {2} \mathbf {x} _ {t} ^ {f} + \frac {1}{6} h _ {\sigma \sigma \sigma} (j, 1) \sigma^ {3} \end{array}\]

Note that is in perturbation a variable and is therefore a third order effect.

Inserting the decomposition of the state variables into the control variables we get (for )

\[\begin{array}{r l} & y _ {t} ^ {r d} (i, 1) = \mathbf {g} _ {\mathbf {x}} (i,:) \mathbf {x} _ {t} + \frac {1}{2} \mathbf {x} _ {t} ^ {\prime} \mathbf {g} _ {\mathbf {x x}} (i,:,:) \mathbf {x} _ {t} + \frac {1}{2} g _ {\sigma \sigma} (i, 1) \sigma^ {2} \\ & \qquad + \frac {1}{6} \mathbf {x} _ {t} ^ {\prime} \left[ \begin{array}{c} \mathbf {x} _ {t} ^ {\prime} \mathbf {g} _ {\mathbf {x x x}} (i, 1,:,:) \mathbf {x} _ {t} \\ \dots \\ \mathbf {x} _ {t} ^ {\prime} \mathbf {g} _ {\mathbf {x x x}} (i, n _ {x},:,:) \mathbf {x} _ {t} \end{array} \right] + \frac {3}{6} \mathbf {g} _ {\sigma \sigma \mathbf {x}} (i,:) \sigma^ {2} \mathbf {x} _ {t} + \frac {1}{6} g _ {\sigma \sigma \sigma} (i, 1) \sigma^ {3} \end{array}\]

\[\begin{array}{r l} & y _ {t} ^ {r d} (i, 1) = \mathbf {g} _ {\mathbf {x}} (i,:) (\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d}) + \frac {1}{2} (\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d}) ^ {\prime} \mathbf {g} _ {\mathbf {x x}} (i,:,:) (\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d}) + \frac {1}{2} g _ {\sigma \sigma} (i, 1) \sigma^ {2} \\ & \qquad + \frac {1}{6} \sum_ {\gamma = 1} ^ {n _ {x}} (x _ {t} ^ {f} (\gamma , 1) + x _ {t} ^ {s} (\gamma , 1) + x _ {t} ^ {r d} (\gamma , 1)) (\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d}) ^ {\prime} \mathbf {g} _ {\mathbf {x x x}} (i, \gamma ,:,:) (\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d}) \\ & \qquad + \frac {3}{6} \mathbf {g} _ {\sigma \sigma \mathbf {x}} (i,:) \sigma^ {2} (\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d}) + \frac {1}{6} g _ {\sigma \sigma \sigma} (i, 1) \sigma^ {3} \end{array}\]

\[\begin{array}{r l}&y _ {t} ^ {r d} (i, 1) = \mathbf {g} _ {\mathbf {x}} (i,:) (\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d}) + \frac {1}{2} \left(\left(\mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \mathbf {g} _ {\mathbf {x x}} (i,:,:) + (\mathbf {x} _ {t} ^ {s}) ^ {\prime} \mathbf {g} _ {\mathbf {x x}} (i,:,:) + (\mathbf {x} _ {t} ^ {r d}) ^ {\prime} \mathbf {g} _ {\mathbf {x x}} (i,:,:) \right.\left. \right.\\&\quad \left. + \frac {1}{2} g _ {\sigma \sigma} (i, 1) \sigma^ {2} \right.\end{array}\]

\[\begin{array}{r l} & + \frac {1}{6} \sum_ {\gamma = 1} ^ {n _ {x}} \left(x _ {t} ^ {f} (\gamma , 1) + x _ {t} ^ {s} (\gamma , 1) + x _ {t} ^ {r d} (\gamma , 1)\right) \\ & \qquad \times \left(\left(\mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \mathbf {g} _ {\mathbf {x x x}} (i, \gamma ,:,:) + (\mathbf {x} _ {t} ^ {s}) ^ {\prime} \mathbf {g} _ {\mathbf {x x x}} (i, \gamma ,:,:) + (\mathbf {x} _ {t} ^ {r d}) ^ {\prime} \mathbf {g} _ {\mathbf {x x x}} (i, \gamma ,:,:))\right) \\ & \qquad \times \left(\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d}\right) \\ & + \frac {3}{6} \mathbf {g} _ {\sigma \sigma \mathbf {x}} (i,:) \sigma^ {2} (\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d}) + \frac {1}{6} g _ {\sigma \sigma \sigma} (i, 1) \sigma^ {3} \end{array}\]

↓

\[\begin{array} { r l } y _ { t } ^ { r d } \left( i , 1 \right) & = \mathbf { g } _ { \mathbf { x } } \left( i , : \right) \left( \mathbf { x } _ { t } ^ { f } + \mathbf { x } _ { t } ^ { s } + \mathbf { x } _ { t } ^ { r d } \right) \\ & + \frac { 1 } { 2 } \left( \left( \mathbf { x } _ { t } ^ { f } \right) ^ { \prime } \mathbf { g } _ { \mathbf { x x } } \left( i , : , : \right) \mathbf { x } _ { t } ^ { f } + \left( \mathbf { x } _ { t } ^ { s } \right) ^ { \prime } \mathbf { g } _ { \mathbf { x x } } \left( i , : , : \right) \mathbf { x } _ { t } ^ { f } + \left( \mathbf { x } _ { t } ^ { r d } \right) ^ { \prime } \mathbf { g } _ { \mathbf { x x } } \left( i , : , : \right) \mathbf { x } _ { t } ^ { f } \right) \\ & + \frac { 1 } { 2 } \left( \left( \mathbf { x } _ { t } ^ { f } \right) ^ { \prime } \mathbf { g } _ { \mathbf { x x } } \left( i , : , : \right) \mathbf { x } _ { t } ^ { s } + \left( \mathbf { x } _ { t } ^ { s } \right) ^ { \prime } \mathbf { g } _ { \mathbf { x x } } \left( i , : , : \right) \mathbf { x } _ { t } ^ { s } + \left( \mathbf { x } _ { t } ^ { r d } \right) ^ { \prime } \mathbf { g } _ { \mathbf { x x } } \left( i , : , : \right) \mathbf { x } _ { t } ^ { s } \right) \\ & + \frac { 1 } { 2 } \left( \left( \mathbf { x } _ { t } ^ { f } \right) ^ { \prime } \mathbf { g } _ { \mathbf { x x } } \left( i , : , : \right) \mathbf { x } _ { t } ^ { r d } + \left( \mathbf { x } _ { t } ^ { s } \right) ^ { \prime } \mathbf { g } _ { \mathbf { x x } } \left( i , : , : \right) \mathbf { x } _ { t } ^ { r d } + \left( \mathbf { x } _ { t } ^ { r d } \right) ^ { \prime } \mathbf { g } _ { \mathbf { x x } } \left( i , : , : \right) \mathbf { x } _ { t } ^ { r d } \right) \\ & + \frac { 1 } { 2 } g _ { \sigma \sigma } ( i , 1 ) \sigma ^ { 2 } \\ & + \frac { 1 } { 6 } \sum _ { \gamma = 1 } ^ { n _ { x } } ( x _ { t } ^ { f } ( \gamma , 1 ) + x _ { t } ^ { s } ( \gamma , 1 ) + x _ { t } ^ { r d } ( \gamma , 1 ) ) \\ & × ( ( ( \mathbf { x } _ { t } ^ { f} ) ^ { \prime } | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | & × ( ( {\bf x} _ { t } ^ { f} + {\bf x} _ { t } ^ { s} + {\bf x} _ { t } ^ { r d} ) \\ & + \frac 3 6 g _ { \sigma \sigma x } ( i , : ) \sigma ^ { 2 } ( {\bf x} _ { t } ^ { f} + {\bf x} _ { t } ^ { s} + {\bf x} _ { t } ^ { r d}) + \frac 1 6 g _ { \sigma \sigma \sigma} ( i , 1 ) \sigma ^ { 3 } \\ & . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .\]

↓

\[\begin{array}{r l} & y _ {t} ^ {r d} (i, 1) = \mathbf {g} _ {\mathbf {x}} (i,:) (\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d}) \\ & \quad + \frac {1}{2} \left(\left(\mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \mathbf {g} _ {\mathbf {x x}} (i,:,: ) \mathbf {x} _ {t} ^ {f} + 2 (\mathbf {x} _ {t} ^ {s}) ^ {\prime} \mathbf {g} _ {\mathbf {x x}} (i,:,: ) \mathbf {x} _ {t} ^ {f} + 2 (\mathbf {x} _ {t} ^ {r d}) ^ {\prime} \mathbf {g} _ {\mathbf {x x}} (i,:,: ) \mathbf {x} _ {t} ^ {f}\right) \\ & \quad + \frac {1}{2} \left((\mathbf {x} _ {t} ^ {s}) ^ {\prime} \mathbf {g} _ {\mathbf {x x}} (i,:,: ) \mathbf {x} _ {t} ^ {s} + 2 (\mathbf {x} _ {t} ^ {r d}) ^ {\prime} \mathbf {g} _ {\mathbf {x x}} (i,:,: ) \mathbf {x} _ {t} ^ {s} + (\mathbf {x} _ {t} ^ {r d}) ^ {\prime} \mathbf {g} _ {\mathbf {x x}} (i,:,: ) \mathbf {x} _ {t} ^ {r d}\right) \\ & \quad + \frac {1}{2} g _ {\sigma \sigma} (i, 1) \sigma^ {2} \\ & \quad + \frac {1}{6} \sum_ {\gamma = 1} ^ {n _ {x}} \left(x _ {t} ^ {f} (\gamma , 1) + x _ {t} ^ {s} (\gamma , 1) + x _ {t} ^ {r d} (\gamma , 1)\right) \\ & \qquad \times \left(\left(\mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \mathbf {g} _ {\mathbf {x x x}} (i, \gamma ,,:,: ) + (\mathbf {x} _ {t} ^ {s}) ^ {\prime} \mathbf {g} _ {\mathbf {x x x}} (i, \gamma ,,:,: ) + (\mathbf {x} _ {t} ^ {r d}) ^ {\prime} \mathbf {g} _ {\mathbf {x x x}} (i, \gamma ,,:,:)\right) \\ & \qquad \times (\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d}) \\ & \quad + \frac {3}{6} \mathbf {g} _ {\sigma \sigma \mathbf {x}} (i,:) \sigma^ {2} (\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d}) + \frac {1}{6} g _ {\sigma \sigma \sigma} (i, 1) \sigma^ {3} \end{array}\]

We want to preserve terms up to third order, hence the pruned approximation is

\[\begin{array}{r c l} y _ {t} ^ {r d} (i, 1) & = & \mathbf {g} _ {\mathbf {x}} (i,:) (\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d}) \\ & & + \frac {1}{2} \left((\mathbf {x} _ {t} ^ {f}) ^ {\prime} \mathbf {g} _ {\mathbf {x x}} (i,:,:) \mathbf {x} _ {t} ^ {f} + 2 (\mathbf {x} _ {t} ^ {s}) ^ {\prime} \mathbf {g} _ {\mathbf {x x}} (i,:,:) \mathbf {x} _ {t} ^ {f})\right) \\ & & + \frac {1}{2} g _ {\sigma \sigma} (i, 1) \sigma^ {2} \\ & & + \frac {1}{6} \sum_ {\gamma = 1} ^ {n _ {x}} x _ {t} ^ {f} (\gamma , 1) \left((\mathbf {x} _ {t} ^ {f}) ^ {\prime} \mathbf {g} _ {\mathbf {x x x}} (i, \gamma ,:,:) \mathbf {x} _ {t} ^ {f}\right) \\ & & + \frac {3}{6} \mathbf {g} _ {\sigma \sigma \mathbf {x}} (i,:) \sigma^ {2} \mathbf {x} _ {t} ^ {f} \\ & & + \frac {1}{6} g _ {\sigma \sigma \sigma} (i, 1) \sigma^ {3} \end{array}\]

\[i = 1, 2, \dots n _ {y}\]

2.3 Summary: NO pruning up to third order

The approximation of the state variables is here

\[\begin{array}{r c l} x _ {t + 1} \left(j, 1\right) & = & \mathbf {h} _ {\mathbf {x}} \left(j,:) \mathbf {x} _ {t} + \frac {1}{2} \mathbf {x} _ {t} ^ {\prime} \mathbf {h} _ {\mathbf {x x}} \left(j,:,:) \mathbf {x} _ {t} \right. \right. \\ & & + \frac {1}{6} \mathbf {x} _ {t} ^ {\prime} \left[ \begin{array}{c} \mathbf {x} _ {t} ^ {\prime} \mathbf {h} _ {\mathbf {x x x}} (j, 1,:,:) \mathbf {x} _ {t} \\ ... \\ \mathbf {x} _ {t} ^ {\prime} \mathbf {h} _ {\mathbf {x x x}} (j, n _ {x},:,:) \mathbf {x} _ {t} \end{array} \right] \\ & & + \frac {3}{6} \mathbf {h} _ {\sigma \sigma \mathbf {x}} \left(j,:) \sigma^ {2} \mathbf {x} _ {t} + \frac {1}{2} h _ {\sigma \sigma} \left(j, 1\right) \sigma^ {2} \right. \\ & & + \frac {1}{6} h _ {\sigma \sigma \sigma} \left(j, 1\right) \sigma^ {3} + \sigma \boldsymbol {\eta} \left(j,:) \boldsymbol {\epsilon} _ {t + 1} \right. \end{array}\tag{7}\]

(8)

for .

The approximation of the control variables is

(9)

\[\begin{array}{r c l} y _ {t} \left(i, 1\right) & = & \mathbf {g} _ {\mathbf {x}} \left(i,:) \mathbf {x} _ {t} \right. \\ & & + \frac {1}{2} \mathbf {x} _ {t} ^ {\prime} \mathbf {g} _ {\mathbf {x x}} \left(i,:,:) \mathbf {x} _ {t} \right. \\ & & + \frac {1}{6} \mathbf {x} _ {t} ^ {\prime} \left[ \begin{array}{c} \mathbf {x} _ {t} ^ {\prime} \mathbf {g} _ {\mathbf {x x x}} (i, 1,:,:) \mathbf {x} _ {t} \\ ... \\ \mathbf {x} _ {t} ^ {\prime} \mathbf {g} _ {\mathbf {x x x}} (i, n _ {x},:,:) \mathbf {x} _ {t} \end{array} \right] \\ & & + \frac {3}{6} \mathbf {g} _ {\sigma \sigma \mathbf {x}} \left(i,:) \sigma^ {2} \mathbf {x} _ {t} + \frac {1}{2} g _ {\sigma \sigma} \left(i, 1\right) \sigma^ {2} \right. \\ & & + \frac {1}{6} g _ {\sigma \sigma \sigma} \left(i, 1\right) \sigma^ {3} \end{array}\tag{10}\]

\[i = 1, 2, \dots n _ {y}\]

2.4 Summary: pruning up to third order

The approximation of the state variables is

\[\mathbf {x} _ {t + 1} ^ {f} = \mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t} ^ {f} + \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1}\tag{11}\]

\[x _ {t + 1} ^ {s} (j, 1) = \mathbf {h} _ {\mathbf {x}} (j,:) \mathbf {x} _ {t} ^ {s} + \frac {1}{2} (\mathbf {x} _ {t} ^ {f}) ^ {\prime} \mathbf {h} _ {\mathbf {x x}} (j,:,:) (\mathbf {x} _ {t} ^ {f}) + \frac {1}{2} h _ {\sigma \sigma} (j, 1) \sigma^ {2}\tag{12}\]

\[\begin{array}{r c l} x _ {t + 1} ^ {r d} (j, 1) & = & \mathbf {h} _ {\mathbf {x}} (j,:) \mathbf {x} _ {t} ^ {r d} + \frac {2}{2} (\mathbf {x} _ {t} ^ {f}) ^ {\prime} \mathbf {h} _ {\mathbf {x x}} (j,:,:) (\mathbf {x} _ {t} ^ {s}) \\ & & + \frac {1}{6} (\mathbf {x} _ {t} ^ {f}) ^ {\prime} \left[ \begin{array}{c} (\mathbf {x} _ {t} ^ {f}) ^ {\prime} \mathbf {h} _ {\mathbf {x x x}} (j, 1,:,:) (\mathbf {x} _ {t} ^ {f}) \\ \dots \\ (\mathbf {x} _ {t} ^ {f}) ^ {\prime} \mathbf {h} _ {\mathbf {x x x}} (j, n _ {x},:,:) (\mathbf {x} _ {t} ^ {f}) \end{array} \right] \\ & & + \frac {3}{6} \mathbf {h} _ {\sigma \sigma \mathbf {x}} (j,:) \sigma^ {2} \mathbf {x} _ {t} ^ {f} + \frac {1}{6} h _ {\sigma \sigma \sigma} (j, 1) \sigma^ {3} \end{array}\tag{13}\]

\[\mathbf {x} _ {t + 1} = \mathbf {x} _ {t + 1} ^ {f} + \mathbf {x} _ {t + 1} ^ {s} + \mathbf {x} _ {t + 1} ^ {r d}\tag{14}\]

for .

The approximation of the control variables is

\[\begin{array}{r c l} y _ {t} ^ {r d} (i, 1) & = & \mathbf {g} _ {\mathbf {x}} (i,:) (\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d}) \\ & & + \frac {1}{2} (\mathbf {x} _ {t} ^ {f}) ^ {\prime} \mathbf {g} _ {\mathbf {x x}} (i,:,:) (\mathbf {x} _ {t} ^ {f} + 2 \mathbf {x} _ {t} ^ {s}) \\ & & + \frac {1}{6} (\mathbf {x} _ {t} ^ {f}) ^ {\prime} \left[ \begin{array}{c} (\mathbf {x} _ {t} ^ {f}) ^ {\prime} \mathbf {g} _ {\mathbf {x x x}} (i, 1,:,:) (\mathbf {x} _ {t} ^ {f}) \\ \dots \\ (\mathbf {x} _ {t} ^ {f}) ^ {\prime} \mathbf {g} _ {\mathbf {x x x}} (i, n _ {x},:,:) (\mathbf {x} _ {t} ^ {f}) \end{array} \right] \\ & & + \frac {1}{2} g _ {\sigma \sigma} (i, 1) \sigma^ {2} + \frac {3}{6} \mathbf {g} _ {\sigma \sigma \mathbf {x}} (i,:) \sigma^ {2} \mathbf {x} _ {t} ^ {f} \\ & & + \frac {1}{6} g _ {\sigma \sigma \sigma} (i, 1) \sigma^ {3} \end{array}\tag{15}\]

for

2.5 Increasing efficiency for the simulation in FORTRAN

When simulating the pruned state space system, the efficiency can be improved by re-expressing some of the sums in the matrices and by using some of the symmetry in the second and third order terms (due to Young's theorem). This is useful in FORTRAN because we can reduce the number of summations. However, in MATLAB, this trick does not work as it induces more loops.

First, to re-express some of the summations implied by the matrix notation, recall the following rules for the vec and kronecker operators:

1.

\[2. \mathbf {A} \otimes \mathbf {B} = \left[ \begin{array}{c c c c} a _ {1 1} \mathbf {B} & a _ {1 2} \mathbf {B} & \ldots & a _ {1 n _ {x}} \mathbf {B} \\ a _ {2 1} \mathbf {B} & a _ {2 2} \mathbf {B} & \ldots & a _ {2 n _ {x}} \mathbf {B} \\ \ldots & \ldots & \ldots & \ldots \\ a _ {n _ {x} 1} \mathbf {B} & a _ {n _ {x} 2} \mathbf {B} & \ldots & a _ {n _ {x} n _ {x}} \mathbf {B} \end{array} \right]\]

3. hence

4. and hence

5. if AC and BD are defined

\[6. (\mathbf {A} + \mathbf {B}) \otimes (\mathbf {C} + \mathbf {D}) = \mathbf {A} \otimes \mathbf {C} + \mathbf {A} \otimes \mathbf {D} + \mathbf {B} \otimes \mathbf {C} + \mathbf {B} \otimes \mathbf {D} \text { if } \mathbf {A} + \mathbf {B} \text { and } \mathbf {C} + \mathbf {D} \text { are defined }\]

\[7. \left[ \mathbf {x} _ {t} ^ {\prime} \otimes \mathbf {x} _ {t} ^ {\prime} \right] = v e c \left(\left[ \mathbf {x} _ {t} \mathbf {x} _ {t} ^ {\prime} \right]\right) ^ {\prime} \iff \left[ \mathbf {x} _ {t} \otimes \mathbf {x} _ {t} \right] = v e c \left(\left[ \mathbf {x} _ {t} \mathbf {x} _ {t} ^ {\prime} \right]\right)\]

where has dimension and , and have dimension . Hence, we may also write the terms of the form in the following way

\[\begin{array}{r l}&{\mathbf {x} _ {t} ^ {\prime} \mathbf {h} _ {\mathbf {x x}} \left(j,:,:) \right. \mathbf {x} _ {t} = \left(\mathbf {x} _ {t} ^ {\prime} \otimes \mathbf {x} _ {t} ^ {\prime}\right) v e c \left( \right.\mathbf {h} _ {\mathbf {x x}} \left(j,:,:)\right)}\\&{\qquad = v e c \left( \right.\mathbf {h} _ {\mathbf {x x}} \left(j,:,:)\right) ^ {\prime} \left(\mathbf {x} _ {t} \otimes \mathbf {x} _ {t}\right)}\\&{\qquad = v e c \left( \right.\mathbf {h} _ {\mathbf {x x}} \left(j,:,:)\right) ^ {\prime} v e c \left([ \mathbf {x} _ {t} \mathbf {x} _ {t} ^ {\prime} ]\right)}\end{array}\]

To exploit the symmetry in the second and third order terms, we use the vech-operator which stacks all elements of a

matrix on or below the diagonal. For instance, if then

It then holds that

\[v e c \left(\mathbf {h} _ {\mathbf {x x}} (j,:,:)\right) ^ {\prime} v e c \left([ \mathbf {x} _ {t} \mathbf {x} _ {t} ^ {\prime} ]\right) = v e c h \left(2 \mathbf {h} _ {\mathbf {x x}} (j,:,:) - d i a g \left(\mathbf {h} _ {\mathbf {x x}} (j,:,:)\right)\right) ^ {\prime} v e c h \left(\mathbf {x} _ {t} \mathbf {x} _ {t} ^ {\prime}\right)\tag{16}\]

Here, is an with zeros except at the diagonal where the matrix has the diagonal elements of for . To realize the validity of the expression in (16), consider

\[\begin{array} { r l } & ( \mathbf { h } _ { \mathbf { x } \mathbf { x } } ( j , : , : ) ) ^ { \prime } v e c ( [ \mathbf { x } _ { t } \mathbf { x } _ { t } ^ { \prime } ] ) \\ & = \mathbf { x } _ { t } ^ { \prime } \mathbf { h } _ { \mathbf { x } \mathbf { x } } ( j , : , : ) \mathbf { x } _ { t } \\ & = \sum _ { h = 1 } ^ { n _ { x } } \sum _ { k = 1 } ^ { n _ { x } } \mathbf { h } _ { \mathbf { x } \mathbf { x } } ( j , : , : ) x _ { t } ( h ) x _ { t } ( k ) \\ & = \sum _ { h = 1 } ^ { n _ { x } } \mathbf { h } _ { \mathbf { x } \mathbf { x } } ( j , h , h ) x _ { t } ( h ) ^ { 2 } + 2 \sum _ { h = 1 } ^ { n _ { x } } \sum _ { k = h + 1 } ^ { n _ { x } } \mathbf { h } _ { \mathbf { x } \mathbf { x } } ( j , : , : ) x _ { t } ( h ) x _ { t } ( k ) \\ & = \mathbf { x } _ { t } ^ { \prime } d i a g ( \mathbf { h } _ { \mathbf { x } \mathbf { x } } ( j , : , : ) ) \mathbf { x } _ { t } + 2 \sum _ { h = 1 } ^ { n _ { x } } \sum _ { k = h + 1 } ^ { n _ { x } } \mathbf { h } _ { \mathbf { x } \mathbf { x } } ( j , : , : ) x _ { t } ( h ) x _ { t } ( k ) \\ & = \mathbf { x } _ { t } ^ { ' } d i a g ( \mathbf { h } _ { \mathbf { x } \mathbf { x } } ( j , : , : ) ) \mathbf { x } _ { t } + 2 \left[ v e c h ( \mathbf { h } _ { \mathbf { x } \mathbf { x } } ( j , : , : ) ) ^ { \prime } v e c h ( \mathbf { x } _ { t } \mathbf { x } _ { t } ^ { \prime } ) - \mathbf { x } _ { t } ^ { \prime } d i a g ( \mathbf { h } _ { \mathbf { x } \mathbf { x } } ( j , : , : ) ) \mathbf { x } _ { t } \right] \\ & = 2 v e c h ( \mathbf { h } _ { \mathbf { x } \mathbf { x } } ( j , : , : ) ) ^ { \prime } v e c h ( \mathbf { x } _ { t } \mathbf { x } _ { t } ^ { \prime } ) - \mathbf { x } _ { t } ^ { \prime } d i a g ( \mathbf { h } _ { {\mathbf { x }} {\mathbf { x}} } ( j , : , : ) ) \mathbf { x } _ { t } \\ & = 2 v e c h ( \mathbf { h } _ { {\mathbf { x }} {\mathbf { x}} } ( j , : , : ) ) ^ { \prime } v e c h ( \mathbf { x } _ { t } \mathbf { x } _ { t } ^ { \prime } ) - v e c ( d i a g ( \mathbf { h } _ { {\mathbf { x }} {\mathbf { x}} } ( j , : , : ) ) ) ^ { \prime } v e c ( \mathbf { x } _ { t } \mathbf { x } _ { t } ^ { \prime } ) \\ & = 2 v e c h ( \mathbf { h } _ { {\mathbf { x }} {\mathbf { x}} } ( j , : , : ) ) ^ { \prime } v e c h ( \mathbf { x } _ { t } \mathbf { x } _ { t } ^ { \prime } ) - v e c h ( d i a g ( \mathbf { h } _ { {\mathbf { x }} {\mathbf { x}} } ( j , : , : ) ) ) ^ { \prime } v e c h ( \mathbf { x } _ { t } \mathbf { x } _ { t } ^ { \prime } ) \\ & = [ 2 v e c h ( \mathbf { h } _ { {\mathbf { x }} {\mathbf { x}} } ( j , : , : ) ) - v e c h ( d i a g ( \mathbf { h } _ { {\mathbf { x }} {\mathbf { x}} } ( j , : , : ) ) ) ] ^ { \prime } v e c h ( \mathbf { x } _ { t } \mathbf { x } _ { t } ^ { \prime } ) \end{array}\]

WITHOUT PRUNING:

For the state variables in (7), we have

\[\begin{array}{r c l} x _ {t + 1} (j, 1) & = & \mathbf {h} _ {\mathbf {x}} (j,:) \mathbf {x} _ {t} + \hat {\mathbf {H}} _ {\mathbf {x x}} (j,:) v e c h (\mathbf {x} _ {t} \mathbf {x} _ {t} ^ {\prime}) \\ & & + \mathbf {x} _ {t} ^ {\prime} \left[ \begin{array}{c} \hat {\mathbf {H}} _ {\mathbf {x x x}} (j, 1,:) v e c h (\mathbf {x} _ {t} \mathbf {x} _ {t} ^ {\prime}) \\ ... \\ \hat {\mathbf {H}} _ {\mathbf {x x x}} (j, n _ {x},:) v e c h (\mathbf {x} _ {t} \mathbf {x} _ {t} ^ {\prime}) \end{array} \right] \\ & & + \frac {3}{6} \mathbf {h} _ {\sigma \sigma \mathbf {x}} (j,:) \sigma^ {2} \mathbf {x} _ {t} + \frac {1}{2} h _ {\sigma \sigma} (j, 1) \sigma^ {2} \\ & & + \frac {1}{6} h _ {\sigma \sigma \sigma} (j, 1) \sigma^ {3} + \sigma \boldsymbol {\eta} (j,:) \boldsymbol {\epsilon} _ {t + 1} \end{array}\tag{17}\]

for where we define

\[\hat {\mathbf {H}} _ {\mathbf {x x}} \left(1: n _ {x}, 1: n _ {x} (n _ {x} + 1) / 2\right) = \frac {1}{2} \left[ \begin{array}{c} v e c h \left(2 \mathbf {h} _ {\mathbf {x x}} \left(1, \therefore ,:\right) - d i a g \left(\mathbf {h} _ {\mathbf {x x}} \left(1, \therefore ,:\right)\right)\right) ^ {\prime} \\ \ldots \\ v e c h \left(2 \mathbf {h} _ {\mathbf {x x}} \left(n _ {x}, \therefore ,:\right) - d i a g \left(\mathbf {h} _ {\mathbf {x x}} \left(n _ {x}, \therefore ,:\right)\right)\right) ^ {\prime} \end{array} \right]\tag{18}\]

\[\hat {\mathbf {H}} _ {\mathbf {x x x}} \left(j, 1: n _ {x}, 1: n _ {x} (n _ {x} + 1) / 2\right) = \frac {1}{6} \left[ \begin{array}{c} v e c h \left(2 \mathbf {h} _ {\mathbf {x x x}} \left(j, 1,:,:\right) - d i a g \left(\mathbf {h} _ {\mathbf {x x x}} \left(j, 1,:,:\right)\right)\right) ^ {\prime} \\ \ldots \\ v e c h \left(2 \mathbf {h} _ {\mathbf {x x x}} \left(j, n _ {x},,:,:\right) - d i a g \left(\mathbf {h} _ {\mathbf {x x x}} \left(j, n _ {x},,:,:\right)\right)\right) ^ {\prime} \end{array} \right]\tag{19}\]

for . The advantage of this formulation compared to one which use all the symmetry in the third order terms is simply that we only need to compute once.

For the control variables in (9), we have

\[\begin{array}{r c l} y _ {t} (i, 1) & = & \mathbf {g} _ {\mathbf {x}} (i,:) \mathbf {x} _ {t} \\ & & + \hat {\mathbf {G}} _ {\mathbf {x x}} (i,:) v e c h (\mathbf {x} _ {t} \mathbf {x} _ {t} ^ {\prime}) \\ & & + \mathbf {x} _ {t} ^ {\prime} \left[ \begin{array}{c} \hat {\mathbf {G}} _ {\mathbf {x x x}} (i, 1,:) v e c h (\mathbf {x} _ {t} \mathbf {x} _ {t} ^ {\prime}) \\ ... \\ \hat {\mathbf {G}} _ {\mathbf {x x x}} (i, n _ {x},:) v e c h (\mathbf {x} _ {t} \mathbf {x} _ {t} ^ {\prime}) \end{array} \right] \\ & & + \frac {1}{2} g _ {\sigma \sigma} (i, 1) \sigma^ {2} + \frac {3}{6} \mathbf {g} _ {\sigma \sigma \mathbf {x}} (i,:) \sigma^ {2} \mathbf {x} _ {t} \\ & & + \frac {1}{6} g _ {\sigma \sigma \sigma} (i, 1) \sigma^ {3} \end{array}\tag{20}\]

for where we define

(21)

\[\begin{array}{c} \hat {\mathbf {G}} _ {\mathbf {x x}} (1: n _ {y}, 1: n _ {x} (n _ {x} + 1) / 2) = \frac {1}{2} \left[ \begin{array}{c} v e c h (2 \mathbf {g} _ {\mathbf {x x}} (1,:,:) - d i a g (\mathbf {g} _ {\mathbf {x x}} (1,:,:))) ^ {\prime} \\ \ldots \\ v e c h (2 \mathbf {g} _ {\mathbf {x x}} (n _ {y},:,:) - d i a g (\mathbf {g} _ {\mathbf {x x}} (n _ {y},:,:))) ^ {\prime} \end{array} \right] \\ \hat {\mathbf {G}} _ {\mathbf {x x x}} (i, 1: n _ {x}, 1: n _ {x} (n _ {x} + 1) / 2) = \frac {1}{6} \left[ \begin{array}{c} v e c h (2 \mathbf {g} _ {\mathbf {x x x}} (i, 1,:,:) - d i a g (\mathbf {g} _ {\mathbf {x x x}} (i, 1,:,:))) ^ {\prime} \\ \ldots \\ v e c h (2 \mathbf {g} _ {\mathbf {x x x}} (i, n _ {x},:,:) - d i a g (\mathbf {g} _ {\mathbf {x x x}} (i, n _ {x},:,:))) ^ {\prime} \end{array} \right] \end{array}\tag{22}\]

for .

WITH PRUNING:

For the state variables in (12) and (13), we have

\[x _ {t + 1} ^ {s} (j, 1) = \mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t} ^ {s} + \hat {\mathbf {H}} _ {\mathbf {x x}} (j,:) v e c h \left(\left(\mathbf {x} _ {t} ^ {f}\right) \left(\mathbf {x} _ {t} ^ {f}\right) ^ {\prime}\right) + \frac {1}{2} h _ {\sigma \sigma} (j, 1) \sigma^ {2}\tag{23}\]

\[\begin{array}{r c l} x _ {t + 1} ^ {r d} (j, 1) & = & \mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t} ^ {r d} + \hat {\mathbf {H}} _ {\mathbf {x x}} (j,:) \left(v e c h \left(\left(\mathbf {x} _ {t} ^ {f}\right) (\mathbf {x} _ {t} ^ {s}) ^ {\prime}\right) + v e c h \left((\mathbf {x} _ {t} ^ {s}) (\mathbf {x} _ {t} ^ {f}) ^ {\prime}\right)\right) \\ & & + \left(\mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \left[ \begin{array}{c} \hat {\mathbf {H}} _ {\mathbf {x x x}} (j, 1,:) v e c h \left(\left(\mathbf {x} _ {t} ^ {f}\right) (\mathbf {x} _ {t} ^ {f}) ^ {\prime}\right) \\ \ldots \\ \hat {\mathbf {H}} _ {\mathbf {x x x}} (j, n _ {x},:) v e c h \left(\left(\mathbf {x} _ {t} ^ {f}\right) (\mathbf {x} _ {t} ^ {f}) ^ {\prime}\right) \end{array} \right] \\ & & + \frac {3}{6} \mathbf {h} _ {\sigma \sigma \mathbf {x}} (j,:) \sigma^ {2} \mathbf {x} _ {t} ^ {f} + \frac {1}{6} h _ {\sigma \sigma \sigma} (j, 1) \sigma^ {3} \end{array}\tag{24}\]

For the control variables in (15), we have

\[\begin{array}{r c l} y _ {t} ^ {r d} (i, 1) & = & \mathbf {g} _ {\mathbf {x}} (i,:) (\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d}) \\ & & + \hat {\mathbf {G}} _ {\mathbf {x x}} (i,:) [ v e c h ((\mathbf {x} _ {t} ^ {f}) (\mathbf {x} _ {t} ^ {f}) ^ {\prime}) + v e c h ((\mathbf {x} _ {t} ^ {f}) (\mathbf {x} _ {t} ^ {s}) ^ {\prime}) + v e c h ((\mathbf {x} _ {t} ^ {s}) (\mathbf {x} _ {t} ^ {f}) ^ {\prime}) ] \\ & & + (\mathbf {x} _ {t} ^ {f}) ^ {\prime} [ \begin{array}{c} \hat {\mathbf {G}} _ {\mathbf {x x x}} (i, 1,:) v e c h ((\mathbf {x} _ {t} ^ {f}) (\mathbf {x} _ {t} ^ {f}) ^ {\prime}) \\ \dots \\ \hat {\mathbf {G}} _ {\mathbf {x x x}} (i, n _ {x},:) v e c h ((\mathbf {x} _ {t} ^ {f}) (\mathbf {x} _ {t} ^ {f}) ^ {\prime}) \end{array} ] \\ & & + \frac {1}{2} g _ {\sigma \sigma} (i, 1) \sigma^ {2} + \frac {3}{6} \mathbf {g} _ {\sigma \sigma \mathbf {x}} (i,:) \sigma^ {2} \mathbf {x} _ {t} ^ {f} + \frac {1}{6} g _ {\sigma \sigma \sigma} (i, 1) \sigma^ {3} \end{array}\tag{25}\]

2.6 Increasing efficiency for the simulation in MATLAB

In MATLAB the most important thing is to avoid for-loops. We therefore provide a representation based on the kronecker product which does not require any loops. Even without using the symmetry in the non-linear terms, this greatly increases the execution speed in MATLAB. Note first that

\[\begin{array}{l} \mathbf {x} _ {t} ^ {\prime} \mathbf {h} _ {\mathbf {x x}} (j,:,:) \mathbf {x} _ {t} = r e s h a p e (\mathbf {h} _ {\mathbf {x x}}, n _ {x}, n _ {x} ^ {2}) \left(\mathbf {x} _ {t} \otimes \mathbf {x} _ {t}\right) \\ \text {where} \\ r e s h a p e (\mathbf {h} _ {\mathbf {x x}}, n _ {x}, n _ {x} ^ {2}) = \left[ \begin{array}{c c c c} \mathbf {h} _ {\mathbf {x x}} \left(1, 1: n _ {x}, 1\right) ^ {\prime} & \mathbf {h} _ {\mathbf {x x}} \left(1, 1: n _ {x}, 2\right) ^ {\prime} & ... & \mathbf {h} _ {\mathbf {x x}} \left(1, 1: n _ {x}, n _ {x}\right) ^ {\prime} \\ \mathbf {h} _ {\mathbf {x x}} \left(2, 1: n _ {x}, 1\right) ^ {\prime} & \mathbf {h} _ {\mathbf {x x}} \left(2, 1: n _ {x}, 2\right) ^ {\prime} & ... & \mathbf {h} _ {\mathbf {x x}} \left(1, 1: n _ {x}, n _ {x}\right) ^ {\prime} \\ ... & ... & ... & ... \\ \mathbf {h} _ {\mathbf {x x}} \left(n _ {x}, 1: n _ {x}, 1\right) ^ {\prime} & \mathbf {h} _ {\mathbf {x x}} \left(n _ {x}, 1: n _ {x}, 2\right) ^ {\prime} & ... & \mathbf {h} _ {\mathbf {x x}} \left(n _ {x}, 1: n _ {x}, n _ {x}\right) ^ {\prime} \end{array} \right] \end{array}\]

And for the third order terms:

\[\begin{array}{l} \text {And for the third order terms.} \\ \mathbf {x} _ {t} ^ {\prime} \left[ \begin{array}{c} \mathbf {x} _ {t} ^ {\prime} \mathbf {h} _ {\mathbf {x x x}} (j, 1,:,: \mathbf {x} _ {t} \\ ... \\ \mathbf {x} _ {t} ^ {\prime} \mathbf {h} _ {\mathbf {x x x}} (j, n _ {x},,:,: \mathbf {x} _ {t} \end{array} \right] = \sum_ {j _ {1} = 1} ^ {n _ {x}} x _ {t} \left(j _ {1}, 1\right) \mathbf {x} _ {t} ^ {\prime} \mathbf {h} _ {\mathbf {x x x}} (j, j _ {1},,:,: \mathbf {x} _ {t} \\ = \sum_ {j _ {1} = 1} ^ {n _ {x}} \sum_ {j _ {2} = 1} ^ {n _ {x}} \sum_ {j _ {3} = 1} ^ {n _ {x}} x _ {t} \left(j _ {1}, 1\right) x _ {t} \left(j _ {2}, 1\right) x _ {t} \left(j _ {3}, 1\right) \mathbf {h} _ {\mathbf {x x x}} (j, j _ {1}, j _ {2}, j _ {3}) \\ = r e s h a p e (\mathbf {h} _ {\mathbf {x x x}}, n _ {x}, n _ {x} ^ {3}) \left(\mathbf {x} _ {t} \otimes \mathbf {x} _ {t} \otimes \mathbf {x} _ {t}\right) \end{array}\]

WITHOUT PRUNING:

For the state variables in (7), we have

\[\begin{array}{r l r} {\mathbf {x} _ {t + 1}} & = & {\mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t} + \tilde {\mathbf {H}} _ {\mathbf {x x}} (\mathbf {x} _ {t} \otimes \mathbf {x} _ {t}) + \tilde {\mathbf {H}} _ {\mathbf {x x x}} (\mathbf {x} _ {t} \otimes \mathbf {x} _ {t} \otimes \mathbf {x} _ {t})} \\ & & {+ \frac {3}{6} \mathbf {h} _ {\sigma \sigma \mathbf {x}} \sigma^ {2} \mathbf {x} _ {t} + \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2} + \frac {1}{6} \mathbf {h} _ {\sigma \sigma \sigma} \sigma^ {3} + \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1}} \end{array}\tag{26}\]

where we define

\[\tilde {\mathbf {H}} _ {\mathbf {x x}} \equiv \frac {1}{2} r e s h a p e \left(\mathbf {h} _ {\mathbf {x x}}, n _ {x}, n _ {x} ^ {2}\right)\tag{27}\]

\[\tilde {\mathbf {H}} _ {\mathbf {x x x}} \equiv \frac {1}{6} r e s h a p e \left(\mathbf {h} _ {\mathbf {x x x}}, n _ {x}, n _ {x} ^ {3}\right)\tag{28}\]

For the control variables in (9), we have

\[{\mathbf {y} _ {t}} = {\mathbf {g} _ {\mathbf {x}} \mathbf {x} _ {t} + \tilde {\mathbf {G}} _ {\mathbf {x x}} (\mathbf {x} _ {t} \otimes \mathbf {x} _ {t}) + \tilde {\mathbf {G}} _ {\mathbf {x x x}} (\mathbf {x} _ {t} \otimes \mathbf {x} _ {t} \otimes \mathbf {x} _ {t})}\tag{29}\]

\[+ \frac {3}{6} {\bf g} _ {\sigma \sigma {\bf x}} \sigma^ {2} {\bf x} _ {t} + \frac {1}{2} {\bf g} _ {\sigma \sigma} \sigma^ {2} + \frac {1}{6} {\bf g} _ {\sigma \sigma \sigma} \sigma^ {3}\tag{30}\]

where we define

\[\tilde {\mathbf {G}} _ {\mathbf {x x}} \equiv \frac {1}{2} r e s h a p e \left(\mathbf {g} _ {\mathbf {x x}}, n _ {y}, n _ {x} ^ {2}\right)\tag{31}\]

\[\tilde {\mathbf {G}} _ {\mathbf {x x x}} \equiv \frac {1}{6} r e s h a p e \left(\mathbf {g} _ {\mathbf {x x x}}, n _ {y}, n _ {x} ^ {3}\right)\tag{32}\]

WITH PRUNING:

For the state variables in (12) and (13), we have

\[\mathbf {x} _ {t + 1} ^ {s} = \mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t} ^ {s} + \tilde {\mathbf {H}} _ {\mathbf {x x}} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) + \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}\tag{33}\]

\[\begin{array}{r c l} \mathbf {x} _ {t + 1} ^ {r d} & = & \mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t} ^ {r d} + 2 \tilde {\mathbf {H}} _ {\mathbf {x x}} (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {s}) + \tilde {\mathbf {H}} _ {\mathbf {x x x}} (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) \\ & & + \frac {3}{6} \mathbf {h} _ {\sigma \sigma \mathbf {x}} \sigma^ {2} \mathbf {x} _ {t} ^ {f} + \frac {1}{6} \mathbf {h} _ {\sigma \sigma \sigma} \sigma^ {3} \end{array}\tag{34}\]

For the control variables in (15), we have

\[\begin{array}{r c l} \mathbf {y} _ {t} ^ {r d} & = & \mathbf {g} _ {\mathbf {x}} \left(\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d}\right) + \tilde {\mathbf {G}} _ {\mathbf {x x}} \left(\left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) + 2 \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {s}\right)\right) + \tilde {\mathbf {G}} _ {\mathbf {x x x}} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) \\ & & + \frac {1}{2} \mathbf {g} _ {\sigma \sigma} \sigma^ {2} + \frac {3}{6} \mathbf {g} _ {\sigma \sigma \mathbf {x}} \sigma^ {2} \mathbf {x} _ {t} ^ {f} + \frac {1}{6} \mathbf {g} _ {\sigma \sigma \sigma} \sigma^ {3} \end{array}\tag{35}\]

3 Stastical properties: Second-order approximation

3.1 Covariance-stationary

Proposition 1:

The pruned second-order approximation for , and is covariance-stationary if

1. the DSGE model has a unique stable equilibrium, i.e. all eigenvalue of have modulus less than one

2. has finite fourth moment

\[\begin{array}{r l} & P r o o f \\ & \mathrm{Notefirstthat} \\ & x _ {t + 1} ^ {s} (j, 1) = \mathbf {h} _ {\mathbf {x}} (j,:) \mathbf {x} _ {t} ^ {s} + \frac {1}{2} (\mathbf {x} _ {t} ^ {f}) ^ {\prime} \mathbf {h} _ {\mathbf {x x}} (j,:,:) (\mathbf {x} _ {t} ^ {f}) + \frac {1}{2} h _ {\sigma \sigma} (j, 1) \sigma^ {2} \\ & \Updownarrow \\ & \mathbf {x} _ {t + 1} ^ {s} = \mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t} ^ {s} + \tilde {\mathbf {H}} _ {\mathbf {x x}} v e c (\left[ (\mathbf {x} _ {t} ^ {f}) (\mathbf {x} _ {t} ^ {f}) ^ {\prime} \right]) + \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2} \\ & \mathrm{where} \tilde {\mathbf {H}} _ {\mathbf {x x}} \equiv \frac {1}{2} r e s h a p e (\mathbf {h} _ {\mathbf {x x}}, n _ {x}, n _ {x} ^ {2}) \\ & \Updownarrow \\ & \mathbf {x} _ {t + 1} ^ {s} = \mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t} ^ {s} + \tilde {\mathbf {H}} _ {\mathbf {x x}} (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) + \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2} \\ & \mathrm{because} (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) = v e c (\left[ (\mathbf {x} _ {t} ^ {f}) (\mathbf {x} _ {t} ^ {f}) ^ {\prime} \right]) \end{array}\]

We now form the extended state vector

\[\mathbf {z} _ {t} \equiv \left[ \begin{array}{c} \mathbf {x} _ {t} ^ {f} \\ \mathbf {x} _ {t} ^ {s} \\ \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \end{array} \right]\]

We know the law of motion for and , so we only need to find the law of motion for . Hence consider

\[\begin{array}{r l} & = \mathbf {h _ {x}} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h _ {x}} \mathbf {x} _ {t} ^ {f} + \mathbf {h _ {x}} \mathbf {x} _ {t} ^ {f} \otimes \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} + \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {h _ {x}} \mathbf {x} _ {t} ^ {f} + \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} \otimes \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} \\ & \text {using (A + B) \otimes (C + D) = A\otimes C + A\otimes D + B\otimes C + B\otimes D} \\ & = (\mathbf {h _ {x}} \otimes \mathbf {h _ {x}}) (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) + (\mathbf {h _ {x}} \otimes \sigma \boldsymbol {\eta}) (\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}) \\ & + (\sigma \boldsymbol {\eta} \otimes \mathbf {h _ {x}}) (\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}) + (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}) (\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}) \\ & \text {using (A\otimes B) (C\otimes D) = A C\otimes B D} \\ & = (\mathbf {h _ {x}} \otimes \mathbf {h _ {x}}) (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) + (\mathbf {h _ {x}} \otimes \sigma \boldsymbol {\eta}) (\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {{\epsilon_ {t + 1}}}) \\ & + (\sigma \boldsymbol {\eta} \otimes \mathbf {h _ {x}}) (\boldsymbol {\epsilon_ {t + 1}} \otimes \mathbf {x} _ {t} ^ {f}) + (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}) ((\boldsymbol {\epsilon_ {t + 1}} \otimes \boldsymbol {\epsilon_ {t + 1}}) - v e c (\mathbf {I} _ {n _ {e}})) + (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}) v e c (\mathbf {I} _ {n _ {e}}) \\ & \text {Note that E[(\epsilon_ {t + 1}\otimes\epsilon_ {t + 1})] = v e c(\mathbf {I} _ {n_{e}}). Thus} \end{array}\]

\[\left[ \begin{array}{c} \mathbf {x} _ {t + 1} ^ {f} \\ \mathbf {x} _ {t + 1} ^ {s} \\ \mathbf {x} _ {t + 1} ^ {f} \otimes \mathbf {x} _ {t + 1} ^ {f} \end{array} \right] = \left[ \begin{array}{c c c} \mathbf {h} _ {\mathbf {x}} & \mathbf {0} _ {n _ {x} \times n _ {x}} & \mathbf {0} _ {n _ {x} \times n _ {x} ^ {2}} \\ \mathbf {0} _ {n _ {x} \times n _ {x}} & \mathbf {h} _ {\mathbf {x}} & \widetilde {\mathbf {H}} _ {\mathbf {x x}} \\ \mathbf {0} _ {n _ {x} ^ {2} \times n _ {x}} & \mathbf {0} _ {n _ {x} ^ {2} \times n _ {x}} & \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} \end{array} \right] \left[ \begin{array}{c} \mathbf {x} _ {t} ^ {f} \\ \mathbf {x} _ {t} ^ {s} \\ \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \end{array} \right] + \left[ \begin{array}{c} \mathbf {0} _ {n _ {x} \times 1} \\ \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2} \\ (\sigma \pmb {\eta} \otimes \sigma \pmb {\eta}) v e c (\mathbf {I} _ {n _ {e}}) \end{array} \right]\]

\[\begin{array}{r l} & + \left[ \begin{array}{c c c c} \sigma \boldsymbol {\eta} & \mathbf {0} & \mathbf {0} & \mathbf {0} \\ \mathbf {0} & \mathbf {0} & \mathbf {0} & \mathbf {0} \\ \mathbf {0} & (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}) & \sigma \boldsymbol {\eta} \otimes \mathbf {h _ {x}} & \mathbf {h _ {x}} \otimes \sigma \boldsymbol {\eta} \end{array} \right] \left[ \begin{array}{c} \boldsymbol {\epsilon} _ {t + 1} \\ \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} - v e c (\mathbf {I} _ {n _ {e}}) \\ \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \\ \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \end{array} \right] \\ & \Updownarrow \end{array}\]

\[\mathbf {z} _ {t + 1} = \mathbf {A} \mathbf {z} _ {t} + \mathbf {c} + \mathbf {B} \boldsymbol {\xi} _ {t + 1}\tag{36}\]

where for because is independent across time

The absolute value of the eigenvalues in are all strictly less than one by assumption. Accordingly, all eigenvalues of are also strictly less than one. To see this note first that

\[p (\lambda) = \left| \mathbf {A} - \lambda \mathbf {I} _ {2 n _ {x} + n _ {x} ^ {2}} \right|\]

\[\begin{array}{r l} & {= \left| \left[ \begin{array}{c c c} {\bf h} _ {x} - \lambda {\bf I} _ {n _ {x}} & {\bf 0} _ {n _ {x} \times n _ {x}} & {\bf 0} _ {n _ {x} \times n _ {x} ^ {2}} \\ {\bf 0} _ {n _ {x} \times n _ {x}} & {\bf h} _ {x} - \lambda {\bf I} _ {n _ {x}} & \tilde {\bf H} _ {x x} \\ {\bf 0} _ {n _ {x} ^ {2} \times n _ {x}} & {\bf 0} _ {n _ {x} ^ {2} \times n _ {x}} & {\bf h} _ {x} \otimes {\bf h} _ {x} - \lambda {\bf I} _ {n _ {x} ^ {2}} \end{array} \right] \right|} \\ & {= \left| \begin{array}{c c} {\bf B} _ {1 1} & {\bf B} _ {1 2} \\ {\bf B} _ {2 1} & {\bf B} _ {2 2} \end{array} \right|} \\ & {\mathrm{wherewelet}} \\ & {{\bf B} _ {1 1} \equiv \left[ \begin{array}{c c} {\bf h} _ {x} - \lambda {\bf I} _ {n _ {x}} & {\bf 0} _ {n _ {x} \times n _ {x}} \\ {\bf 0} _ {n _ {x} \times n _ {x}} & {\bf h} _ {x} - \lambda {\bf I} _ {n _ {x}} \end{array} \right] \mathrm{whichis} 2 n _ {x} \times 2 n _ {x}} \\ & {{\bf B} _ {1 2} \equiv \left[ \begin{array}{c c} {\bf 0} _ {n _ {x} \times n _ {x} ^ {2}} \\ \tilde {\bf H} _ {x x} \end{array} \right] \mathrm{whichis} 2 n _ {x} \times n _ {x} ^ {2}} \\ & {{\bf B} _ {2 1} \equiv \left[ \begin{array}{c c} {\bf 0} _ {n _ {x} ^ {2} \times n _ {x}} & {\bf 0} _ {n _ {x} ^ {2} \times n _ {x}} \end{array} \right] \mathrm{whichis} n _ {x} ^ {2} \times 2 n _ {x}} \\ & {{\bf B} _ {2 2} \equiv {\bf h} _ {x} \otimes {\bf h} _ {x} - \lambda {\bf I} _ {n _ {x} ^ {2}} \mathrm{whichis} n _ {x} ^ {2} \times n _ {x} ^ {2}} \\ & \\ & = | {\bf B} _ {1 1} | | {\bf B} _ {2 2} | \\ & {\mathrm{using}} \\ & = \left| \begin{array}{c c} {\bf U} & {\bf C} \\ {\bf 0} & {\bf Y} \end{array} \right| = | {\bf U} | | {\bf Y} | \mathrm{whereUism×mandYisn×n.5.5.6.7.8.9.10.11.12.13.14.15.16.17.18.19.20.21.22.23.24.25.26.27.28.29.30.31.32.33.34.35.36.37.38.39.40.41.42.43.44.45.46.47.48.49.50.51.52.53.54.55.56.57.58.59.60.61.62.63.64.65.66.67.68.69.70.71.72.73.74.75.76.77.78.79.80.81.82.83.84.85.86.87.88.89.90.91.92.93.94.95.96.97.98.99.10.11.12.13.14.15.16.17.18.19.20.21.22.23.24.25.26.27.28.29.30.31.32.33.34.35.36.37.38.39.40.41.43.44.45.46.47.48.49.50.51.52,53-54-55-56-57-58-59-60-61-62-63-64-65-66-67-68-69-70-71-72-73-74-75-76-77-78-79-80-81-82-83-84-85-86-87-88-89-90-91-92-93-94-95-96-97-98-99-100-101-102-103-104-105-106-107-108-109-110-111-112-113-114-115-116-117-118-119-120-121-122-123-124-125-126-127-128-129-130-131-132-133-134-135-136-137-138-139-140-141-142-143-144-145-146-147-148-149-150-151-152-153-154-155-156-157-158-159-160-161-162-163-164-165-166-167-168-169-170-171 - | {\bf h}, | {\bf h}, | {\bf h}, | {\bf h}, | {\bf h}, | {\bf h}, | {\bf h}, | {\bf h}, | {\bf h}, | {\bf h}, | {\bf h}, | {\bf h}, | {\bf h}, | {\bf h}, | {\bf h}, | {\bf h}, | {\bf h}, | {\bf h}, | {\bf h}, | {\bf h}, | {\bf h}.} \\ & \\ & = | {\bf h}, | {\bf h}, | {\bf h}, | {\bf h}, | {\bf h}, | {\bf h}, | {\bf h}, | {\bf h}, | {\bf h}, | {\bf h}, | {\bf h}, | {\bf h}, | {\bf h}, | {\bf h}, | {\bf h}, | {\bf h}, | {\bf h}, | {\bf h}. \end{array}\]

\[\begin{array}{l} \text {Hence, the eigenvalue \lambda solves the problem} \\ p (\lambda) = 0 \\ \Updownarrow \\ | \mathbf {h _ {x}} - \lambda \mathbf {I} _ {n _ {x}} | | \mathbf {h _ {x}} - \lambda \mathbf {I} _ {n _ {x}} | | \mathbf {h _ {x}} \otimes \mathbf {h _ {x}} - \lambda \mathbf {I} _ {n _ {x} ^ {2}} | = 0 \\ \Updownarrow \end{array}\]

\[\left| \mathbf {h} _ {\mathbf {x}} - \lambda \mathbf {I} _ {n _ {x}} \right| = 0 \text {or} \left| \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} - \lambda \mathbf {I} _ {n _ {x} ^ {2}} \right| = 0\]

The absolute value of all eigenvalues to the first problem are strictly less than one. That is . This is also the case for the second problem because the eigenvalues to are for and

Thus, the system in (36) is covariance stationary if has finite first and second moment. It follows directly that and has finite second moments if has a finite fourth moment. The latter holds by assumption.

For the control variables we have

Q.E.D.

3.2 Method 1: Formulas for the first and second moments

This section computes first and second moments using the representation of the second-order system stated above. This method is fairly direct but has the computational disadvantage of requiring a lot of memory because we work directly with the big B matrix.

The system

The mean values are

\[\begin{array}{r} \mathbf {z} _ {t + 1} = \mathbf {c} + \mathbf {A z} _ {t} + \mathbf {B} \pmb {\xi} _ {t + 1} \\ \mathbf {y} _ {t} ^ {s} = \mathbf {D z} _ {t} + \frac {1}{2} \mathbf {g} _ {\sigma \sigma} \sigma^ {2} \end{array}\]

\[E \left[ \mathbf {z} _ {t} \right] = \left(\mathbf {I} _ {2 n _ {x} + n _ {x} ^ {2}} - \mathbf {A}\right) ^ {- 1} \mathbf {c}.\]

\[E [ \mathbf {y} _ {t} ] = \mathbf {D} E [ \mathbf {z} _ {t} ] + \frac {1}{2} \mathbf {g} _ {\sigma \sigma} \sigma^ {2}\]

For the variances we first have that

\[\begin{array}{r l} & E \left[ \mathbf {z} _ {t + 1} \mathbf {z} _ {t + 1} ^ {\prime} \right] = E \left[ \left(\mathbf {c} + \mathbf {A z} _ {t} + \mathbf {B} \pmb {\xi} _ {t + 1}\right) \left(\mathbf {c} + \mathbf {A z} _ {t} + \mathbf {B} \pmb {\xi} _ {t + 1}\right) ^ {\prime} \right] \\ & \qquad = E \left[ \left(\mathbf {c} + \mathbf {A z} _ {t} + \mathbf {B} \pmb {\xi} _ {t + 1}\right) \left(\mathbf {c} ^ {\prime} + \mathbf {z} _ {t} ^ {\prime} \mathbf {A} ^ {\prime} + \pmb {\xi} _ {t + 1} ^ {\prime} \mathbf {B} ^ {\prime}\right) \right] \\ & \qquad = E \left[ \mathbf {c} \left(\mathbf {c} ^ {\prime} + \mathbf {z} _ {t} ^ {\prime} \mathbf {A} ^ {\prime} + \pmb {\xi} _ {t + 1} ^ {\prime} \mathbf {B} ^ {\prime}\right) \right] \\ & \qquad + E \left[ \mathbf {A z} _ {t} \left(\mathbf {c} ^ {\prime} + \mathbf {z} _ {t} ^ {\prime} \mathbf {A} ^ {\prime} + \pmb {\xi} _ {t + 1} ^ {\prime} \mathbf {B} ^ {\prime}\right) \right] \\ & \qquad + E \left[ \mathbf {B} \pmb {\xi} _ {t + 1} \left(\mathbf {c} ^ {\prime} + \mathbf {z} _ {t} ^ {\prime} \mathbf {A} ^ {\prime} + \pmb {\xi} _ {t + 1} ^ {\prime} \mathbf {B} ^ {\prime}\right) \right] \\ & \qquad = E \left[ \mathbf {c c} ^ {\prime} + \mathbf {c z} _ {t} ^ {\prime} \mathbf {A} ^ {\prime} + \mathbf {c} \pmb {\xi} _ {t + 1} ^ {\prime} \mathbf {B} ^ {\prime} \right] \\ & \qquad + E \left[ \mathbf {A z} _ {t} \mathbf {c} ^ {\prime} + \mathbf {A z} _ {t} \mathbf {z} _ {t} ^ {\prime} \mathbf {A} ^ {\prime} + \mathbf {A z} _ {t} \pmb {\xi} _ {t + 1} ^ {\prime} \mathbf {B} ^ {\prime} \right] \\ & \qquad + E \left[ \mathbf {B} \pmb {\xi} _ {t + 1} \mathbf {c} ^ {\prime} + \mathbf {B} \pmb {\xi} _ {t + 1} \mathbf {z} _ {t} ^ {\prime} \mathbf {A} ^ {\prime} + \mathbf {B} \pmb {\xi} _ {t + 1} \pmb {\xi} _ {t + 1} ^ {\prime} \mathbf {B} ^ {\prime} \right] \\ & \qquad = \mathbf {c c} ^ {\prime} + \mathbf {c E} \left[ \mathbf {z} _ {t} ^ {\prime} \right] \mathbf {A} ^ {\prime} \\ & \qquad + \mathbf {A E} \left[ \mathbf {z} _ {t} \right] \mathbf {c} ^ {\prime} + \mathbf {A E} \left[ \mathbf {z} _ {t} \mathbf {z} _ {t} ^ {\prime} \right] \mathbf {A} ^ {\prime} + \mathbf {A E} \left[ \mathbf {z} _ {t} \pmb {\xi} _ {t + 1} ^ {\prime} \right] \mathbf {B} ^ {\prime} \\ & \qquad + \mathbf {B E} \left[ \pmb {\xi} _ {t + 1} \pmb {\mathrm{z}} _ {t} ^ {\prime} \right] \mathbf {A} ^ {\prime} + \mathbf {B E} \left[ \pmb {\xi} _ {t + 1} \pmb {\xi} _ {t + 1} ^ {\prime} \right] \mathbf {B} ^ {\prime} \end{array}\]

We then note that

\[\begin{array}{r l} & {\mathrm{Wethennotethat}} \\ & {E \left[ \mathbf {z} _ {t} \pmb {\xi} _ {t + 1} ^ {\prime} \right] = E \left[ \left[ \begin{array}{c} \mathbf {x} _ {t} ^ {f} \\ \mathbf {x} _ {t} ^ {s} \\ \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \end{array} \right] \left[ \begin{array}{c c c} \pmb {\epsilon} _ {t + 1} ^ {\prime} & (\pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1} - v e c (\mathbf {I} _ {n e})) ^ {\prime} & (\pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} & (\mathbf {x} _ {t} ^ {f} \otimes \pmb {\epsilon} _ {t + 1}) ^ {\prime} \end{array} \right] \right]} \\ & {= E \left[ \begin{array}{c c c c} \mathbf {x} _ {t} ^ {f} \pmb {\epsilon} _ {t + 1} ^ {\prime} & \mathbf {x} _ {t} ^ {f} (\pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1} - v e c (\mathbf {I} _ {n e})) ^ {\prime} & \mathbf {x} _ {t} ^ {f} (\pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} & \mathbf {x} _ {t} ^ {f} (\mathbf {x} _ {t} ^ {f} \otimes \pmb {\epsilon} _ {t + 1}) ^ {\prime} \\ \mathbf {x} _ {t} ^ {s} \pmb {\epsilon} _ {t + 1} ^ {\prime} & \mathbf {x} _ {t} ^ {s} (\pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1} - v e c (\mathbf {I} _ {n e})) ^ {\prime} & \mathbf {x} _ {t} ^ {s} (\pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} & \mathbf {x} _ {t} ^ {s} (\mathbf {x} _ {t} ^ {f} \otimes \pmb {\epsilon} _ {t + 1}) ^ {\prime} \\ (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) \pmb {\epsilon} _ {t + 1} ^ {\prime} & (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) (\pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1} - v e c (\mathbf {I} _ {n e})) ^ {\prime} & (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) (\pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} & (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) (\mathbf {x} _ {t} ^ {f} \otimes \pmb {\epsilon_ {t + 1}}) ^ {\prime} \end{array} \right]} \end{array}\]

\[= \left[ \begin{array}{c c c c} \mathbf {0} & \mathbf {0} & \mathbf {0} & \mathbf {0} \\ \mathbf {0} & \mathbf {0} & \mathbf {0} & \mathbf {0} \\ \mathbf {0} & \mathbf {0} & \mathbf {0} & \mathbf {0} \end{array} \right]\]

\[\begin{array}{r l} & {\mathrm{Thus}} \\ & {E \left[ \mathbf {z} _ {t + 1} \mathbf {z} _ {t + 1} ^ {\prime} \right] = \mathbf {c c} ^ {\prime} + \mathbf {c} E \left[ \mathbf {z} _ {t} ^ {\prime} \right] \mathbf {A} ^ {\prime} + \mathbf {A} E \left[ \mathbf {z} _ {t} \right] \mathbf {c} ^ {\prime} + \mathbf {A} E \left[ \mathbf {z} _ {t} \mathbf {z} _ {t} ^ {\prime} \right] \mathbf {A} ^ {\prime} + \mathbf {B} E \left[ \boldsymbol {\xi} _ {t + 1} \boldsymbol {\xi} _ {t + 1} ^ {\prime} \right] \mathbf {B} ^ {\prime}} \\ & {\qquad = \mathbf {c} E \left[ \mathbf {z} _ {t} ^ {\prime} \right] \mathbf {A} ^ {\prime} + \left(\mathbf {c} + \mathbf {A} E \left[ \mathbf {z} _ {t} \right]\right) \mathbf {c} ^ {\prime} + \mathbf {A} E \left[ \mathbf {z} _ {t} \mathbf {z} _ {t} ^ {\prime} \right] \mathbf {A} ^ {\prime} + \mathbf {B} E \left[ \boldsymbol {\xi} _ {t + 1} \boldsymbol {\xi} _ {t + 1} ^ {\prime} \right] \mathbf {B} ^ {\prime}} \end{array}\]

Note also that

\[\begin{array}{r l} & E \left[ \mathbf {z} _ {t} \right] E \left[ \mathbf {z} _ {t} \right] ^ {\prime} = (\mathbf {c} + \mathbf {A} E \left[ \mathbf {z} _ {t} \right]) (\mathbf {c} + \mathbf {A} E \left[ \mathbf {z} _ {t} \right]) ^ {\prime} \\ & \qquad = (\mathbf {c} + \mathbf {A} E \left[ \mathbf {z} _ {t} \right]) \mathbf {c} ^ {\prime} + (\mathbf {c} + \mathbf {A} E \left[ \mathbf {z} _ {t} \right]) E \left[ \mathbf {z} _ {t} ^ {\prime} \right] \mathbf {A} ^ {\prime} \\ & \qquad = (\mathbf {c} + \mathbf {A} E \left[ \mathbf {z} _ {t} \right]) \mathbf {c} ^ {\prime} + \mathbf {c} E \left[ \mathbf {z} _ {t} ^ {\prime} \right] \mathbf {A} ^ {\prime} + \mathbf {A} E \left[ \mathbf {z} _ {t} \right] E \left[ \mathbf {z} _ {t} ^ {\prime} \right] \mathbf {A} ^ {\prime} \end{array}\]

\[\begin{array}{r l} & {\mathrm{So}} \\ & {E \left[ \mathbf {z} _ {t + 1} \mathbf {z} _ {t + 1} ^ {\prime} \right] - E \left[ \mathbf {z} _ {t} \right] E \left[ \mathbf {z} _ {t} \right] ^ {\prime} = \mathbf {c} E \left[ \mathbf {z} _ {t} ^ {\prime} \right] \mathbf {A} ^ {\prime} + \left(\mathbf {c} + \mathbf {A} E \left[ \mathbf {z} _ {t} \right]\right) \mathbf {c} ^ {\prime} + \mathbf {A} E \left[ \mathbf {z} _ {t} \mathbf {z} _ {t} ^ {\prime} \right] \mathbf {A} ^ {\prime} + \mathbf {B} E \left[ \boldsymbol {\xi} _ {t + 1} \boldsymbol {\xi} _ {t + 1} ^ {\prime} \right] \mathbf {B} ^ {\prime}} \\ & {\qquad - \left(\mathbf {c} + \mathbf {A} E \left[ \mathbf {z} _ {t} \right]\right) \mathbf {c} ^ {\prime} - \mathbf {c} E \left[ \mathbf {z} _ {t} ^ {\prime} \right] \mathbf {A} ^ {\prime} - \mathbf {A} E \left[ \mathbf {z} _ {t} \right] E \left[ \mathbf {z} _ {t} ^ {\prime} \right] \mathbf {A} ^ {\prime}} \\ & {\qquad = \mathbf {A} E \left[ \mathbf {z} _ {t} \mathbf {z} _ {t} ^ {\prime} \right] \mathbf {A} ^ {\prime} + \mathbf {B} E \left[ \boldsymbol {\xi} _ {t + 1} \boldsymbol {\xi} _ {t + 1} ^ {\prime} \right] \mathbf {B} ^ {\ast} - \mathbf {A} E \left[ \mathbf {z} _ {t} \right] E \left[ \mathbf {z} _ {t} ^ {\prime} \right] \mathbf {A} ^ {\ast}} \\ & {\qquad = \mathbf {A} \left(E \left[ \mathbf {z} _ {t} \mathbf {z} _ {t} ^ {\prime} \right] - E \left[ \mathbf {z} _ {t} \right] E \left[ \mathbf {z} _ {t} ^ {\prime} \right]\right) \mathbf {A} ^ {\ast} + \mathbf {B} E \left[ \boldsymbol {\xi} _ {t + 1} \boldsymbol {\xi} _ {t + 1} ^ {\prime} \right] \mathbf {B} ^ {\ast}} \\ & {\Updownarrow} \end{array}\]

\[\operatorname{vec} \left(\operatorname{Var} \left(\mathbf {z} _ {t + 1}\right)\right) = \left(\mathbf {I} _ {\left(2 n _ {x} + n _ {x} ^ {2}\right) ^ {2}} - (\mathbf {A} \otimes \mathbf {A})\right) ^ {- 1} \operatorname{vec} \left(\mathbf {B V a r} \left(\boldsymbol {\xi} _ {t + 1}\right) \mathbf {B} ^ {\prime}\right)\]

Hence we only need to compute .

\[\begin{array} { r l } V a r \left( \pmb { \xi } _ { t + 1 } \right) = E & { } \left[ \begin{array} { c } \pmb { \epsilon } _ { t + 1 } \\ \pmb { \epsilon } _ { t + 1 } \otimes \pmb { \epsilon } _ { t + 1 } - v e c \left( \mathbf { I } _ { n _ { e } } \right) \\ \pmb { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \\ \mathbf { x } _ { t } ^ { f } \otimes \pmb { \epsilon } _ { t + 1 } \end{array} \right] \left[ \begin{array} { c } \pmb { \epsilon } _ { t + 1 } \\ \pmb { \epsilon } _ { t + 1 } \otimes \pmb { \epsilon } _ { t + 1 } - v e c \left( \mathbf { I } _ { n _ { e } } \right) \\ \pmb { \epsilon } _ { t + 1 } \otimes x _ { t } ^ { f } \\ x _ { t } ^ { f } \otimes \pmb { \epsilon } _ { t + 1 } \end{array} \right] ^ { \prime } \\ & = E \left[ \begin{array} { c c c c } \pmb { \epsilon } _ { t + 1 } \\ \pmb { \epsilon } _ { t + 1 } \otimes \pmb { \epsilon } _ { t + 1 } - v e c \left( \mathbf { I } _ { n _ { e } } \right) \\ \pmb { \epsilon } _ { t + 1 } \otimes x _ { t } ^ { f } & \pmb { \epsilon } _ { t + 1 } ^ { \prime } & ( \pmb { \epsilon } _ { t + 1 } \otimes \pmb { \epsilon } _ { t + 1 } - v e c \left( \mathbf { I } _ { n _ { e } } \right) ) ^ { \prime } & ( \pmb { \epsilon } _ { t + 1 } \otimes x _ { t } ^ { f } ) ^ { \prime } & ( x _ { t } ^ { f } \otimes \pmb { \epsilon } _ { t + 1 } ) ^ { \prime } \\ x _ { t } ^ { f } \otimes \pmb { \epsilon } _ { t + 1 } & & & \\ & = E \left[ \begin{array} { c c c c } \pmb { \epsilon } _ { t + 1 } \pmb { \epsilon } _ { t + 1 } ^ { \prime } & ( \pmb { \epsilon } _ { t + 1 } \otimes \pmb { \epsilon } _ { t + 1 } - v e c ( I _ { n _ { e} } ) ) ^ { \prime } \\ ( \pmb { \epsilon } _ { t + 1 } \otimes \pmb { \epsilon } _ { t + 1 } - v e c ( I _ { n _ { e} } ) ) p e r m a l l o w h s h i g h y ( p e r m a l l o w h s h i g h y ( p e r m a l l o w h s h i g h y ( p e r m a l l o w h s h i g h y ( p e r m a l l o w h s h i g h y ( p e r m a l l o w h s h i g h y ( p e r m a l l o w h s h i g g ) ) ) ^ { j } \\ ( p e r m a l l o w h s h i g h y ( p e r m a l l o w h s h i g h y ( p e r m a l l o w h s h i g h y ( p e r m a l l o w h s h i g h y ( p e r m a l l o w h s h i g h y ( p e r m a l l o w h s h i f ) ) ) ^ { j } \\ ( p e r m a l l o w h s h i g h y ( p e r m a l l o w h s h i g h y ( p e r m a l l o w h s h i g h y ( p e r m a l l o w h s h i g h y ( p e r m a l l o w h s h i f ) ) ) ^ { j } \\ ( p e r m a l l u v e d , p e r m a l l u v e d , p e r m a l l u v e d , p e r m a l l u v e d , p e r m a l l u v e d , p e r m a l l u v e d , p e r m a l l u v e d , p e r m a l l u v e d , p e r m a l l u v e d , p e r n o n d , p e r n o n d , p e r n o n d , p e r n o n d , p e r n o n d , p e r n o n d , p e r n o n d , p e r n o n d , p e r n o n d , p e r n o n d , p e r n o n d , p e r n o n d , p e r n o n d , p q u a d , p q u a d , p q u a d , p q u a d , p q u a d , p q u a d , p q u a d , p q u a d , p q u a d , p q u a d , p q u a d , p q u a d , p q u a d , p q u a d , p q u a d , p q u a d , p q u a d , p p q u a d , p q u a d , p q u a d , p q u a d , p q u a d , p q u a d , p q u a d , p q u a d , p q u a d , p q u a d , p q u a d , p q u a d , p q u a d , p q u a d , p q u a d , p q u a d , p q u u a d , p q u a d , p q u a d , p q u a d , p q u a d , p q u a d , p q u a d , p q u a d , p q u a d , p q u a d , p q u a d , p q u a d , p q u a d , p q u a d , p q u a d , p q u a d , p q u a d . \\ & = E [ U ] ^ j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k jk j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j k j z y [ U ] ^ i n f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f | U | ^ i n f f | | U | ^ i n f | | U | ^ i n | | U | ^ i n | | U | ^ i n | | U | ^ i n | | U | ^ i n | | U | ^ i n | | U | ^ i n | | U | ^ i n | | U | ^ i n | | U | ^ i n | | U | ^ i n | | U | ^ i n | | U | ^ i n | | U \| ^ i n | | U \| ^ i n | | U \| ^ i n | | U \| ^ { i n | | U \| ^ { i n | | U \| ^ { i n | | U \| ^ { i n | | U \| ^ { i n | | U \| ^ { i n | | U \| ^ { i n | | U \| ^ { i n | | U \| ^ { i n | | U \| ^ { i in t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t t . \\ & = E [ U ] ^ { i n f | I _ { n _ { e} } - E [ E [ E [ E [ E [ E [ E [ E [ E [ E [ E [ E [ E [ E [ E [ E [ E [ E [ E [ E [ E [ E [ E [ E [ E [ E [ E [ E [ E [ E [ E [ E [ E [ E [ E [ E [ E [ E [ E [ E [ E [ E [ E [ E [ E [ E [ E [ E [ E [ E [ E ] - E [ E ] - E [ E ] - E [ E ] - E [ E ] - E [ E ] - E [ E ] - E [ E ] - E [ E ] - E [ E ] - E [ E ] - E [ E ] - E [ E ] - E [ E ] - E [ E ] - E [ E ] - E [ E ] - E [ E ] - E [ E ] - E [ E ] - E [ E ]- E [ E ] - E [ E ] - E [ E ] - E [ E ] - E [ E ] - E [ F ] - F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F F W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W WWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWW< fcel>E = C, P, Q, R, S, T, U, V, V, X, Y, Z, D, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, X, Y, Z, D, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, X, Y, Z, D, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, X, Y, Z, D, H, I, J, K, L, M,N, O, P, Q, R, S, T, U, V, X, Y, Z, D, H, I, J, K, L, M,N, O, P, Q, R, S, T, U, V, X, Y, Z, D, H, I, J, K, L, M,N, O, P, Q, R, S, T, U, V, X, Y, Z,D, H, I, J, K, L, M,N, O, P, Q, R, S, T, U, V, X, Y; & = C (P) ^{i} (P) ^{i} (P) ^{i} (P) ^{i} (P) ^{i} (P) ^{i} (P) ^{i} (P) ^{i} (P) ^{i} (P) ^{i} (P) ^{i} (P) ^{i} (P) ^{i} (P) ^{i} (P) ^{i} (S) ^{i} (S) ^{i} (S) ^{i} (S) ^{i} (S) ^{i} (S) ^{i} (S) ^{i} (S) ^{i} (S) ^{i} (S) ^{i} (S) ^{i} (S) ^{i} (S) ^{i} (S) ^{i} (S) ~ ; & = C (P) ^{i} (P) ^{i} (P) ^{i} (P) ^{i} (P) ^{i} (P) ^{i} (P) ^{i} (P) ^{i} (P) ^{i} (P) ^{i} (P) ^{i} (P) ^{i} (P) ^{i} (P) ^{j} (P) ^{j} (P) ^{j} (P) ^{j} (P) ^{j} (P) ^{j} (P) ^{j} (P) ^{j} (P) ^{j} (P) ^{j} (P) ^{j} (P) ^{j} (P) ^{j} (P) ^{j} (P) ^{j} (\textbf {{\scriptsize {\bf {\alpha}}} )^{j},\textbf{{\alpha}} )^{j},\textbf{{\alpha}} )^{j},\textbf{{\alpha}} )^{j},\textbf{{\alpha}} )^{j},\textbf{{\alpha}} )^{j},\textbf{{\alpha}} )^{j},\textbf{{\alpha}} )^{j},\textbf{{\alpha}} )^{j},\textbf{{\alpha}} )^{j},\textbf{{\alpha}} . \\ & = C (\textbf {{\alpha}} )^{\mathrm{i}} (\textbf {{\alpha}} )^{\mathrm{i}} (\textbf {{\alpha}} )^{\mathrm{i}} (\textbf {{\alpha}} )^{\mathrm{i}} (\textbf {{\alpha}} )^{\mathrm{i}} (\textbf {{\alpha}} )^{\mathrm{i}} (\textbf {{\alpha}} )^{\mathrm{i}} (\textbf {{\alpha}} )^{\mathrm{i}} (\textbf {{\alpha}} )^{j},\textbf {{\alpha}} )^{\mathrm{i}} (\textbf {{\alpha}} )^{\mathrm{i}} (\textbf {{\alpha}} )^{\mathrm{i}} (\textbf {{\alpha}} )^{\mathrm{i}} (\textbf {{\alpha}} )^{\mathrm{i}} (\textbf {{\alpha}} )^{\mathrm{i}} (\textbf {{\alpha}} )^{\mathrm{i}} (\textbf {{\alpha}} )^ {\mathrm{i}} (\textbf {{\alpha}} )^{\mathrm{i}} (\textbf {{\alpha}} )^{\mathrm{i}} (\textbf {{\alpha}} )^{\mathrm{i}} (\textbf {{\alpha}} )^{\mathrm{i}} (\textbf {{\alpha}} )^{\mathrm{i}} (\textbf {{\alpha}} )^{\mathrm{i}} (\textbf {{\alpha}} )^{\mathrm{i}} (\textbf {{\beta }} )^{\mathrm{i}} (\textbf {{\beta }} )^{\mathrm{i}} (\textbf {{\beta }} )^{\mathrm{i}} (\textbf {{\beta }} )^{\mathrm{i}} (\textbf {{\beta }} )^{\mathrm{i}} (\textbf {{\beta }} )^{\mathrm{i}} (\textbf {{\beta }} )^{\mathrm{i}} (\textbf {{\beta }} )^{\mathrm{i}} (\textsf {{\beta }} )^{\mathrm{i}} (\textsf {{\beta }} )^{\mathrm{i}} (\textsf {{\beta }} )^{\mathrm{i}} (\textsf {{\beta }} )^{\mathrm{i}} (\textsf {{\beta }} )^{\mathrm{i}} (\textsf {{\beta }} )^{\mathrm{i}} (\textsf {{\beta }} )^{\mathrm{i}} (\textsf {{\beta }} )^{\mathrm{j}} (\textsf {{\beta }} )^{\mathrm{j}} (\textsf {{\beta }} )^{\mathrm{j}} (\textsf {{\beta }} )^{\mathrm{j}} (\textsf {{\beta }} )^{\mathrm{j}} (\textsf {{\beta }} )^{\mathrm{j}} (\textsf {{\beta }} )^{\mathrm{j}} (\textsf {{\beta }} )^{\mathrm{j}} (\textsf {{\beta }} / :), \\ & = C (\textsf {{\beta }} / :), C (\textsf {{\beta }} / :), C (\textsf {{\beta }} / :), C (\textsf {{\beta }} / :), C (\textsf {{\beta }} / :), C (\textsf {{\beta }} / :), C (\textsf {{\beta }} / :), C (\textsf {{\beta }} / :), C (\textsf {{\beta }} / :), C (\textsf {{\beta }} / :), G = C (\textsf {{\beta }} / :), G = C (\textsf {{\beta }} / :), G = C (\textsf {{\beta }} / :), G = C (\textsf {{\beta }} / :), G = C (\textsf {{\beta }} / :), G = C (\textsf {{\beta }} / :), G = C (\textsf {{\beta }} / :), G = C (\textsf {{\beta }}/ :), G = C (\textsf {{\beta }} / :), G = C (\textsf {{\beta }} / :), G = C (\textsf {{\beta }} / :), G = C (\textsf {{\beta }} / :), G = C (\textsf {{\beta }} / :), G = C (\textsf {{\beta }} / :), G = C (\textsf {{\beta }} / :), G = C (\textsf{{.0}},:), G = C (\textsf{{.0}},:), G = C (\textsf{{.0}},:), G = C (\textsf{{.0}},:), G = C (\textsf{{.0}},:), G = C (\textsf{{.0}},:), G = C (\textsf{{.0}},:), G = C (\textsf{{.0}},:), G = C (\textsf{{.0}},:), G = D (\textsf{{.0}},:), G = D (\textsf{{.0}},:), G = D (\textsf{{.0}},:), G = D (\textsf{{.0}},:), G = D (\textsf{{.0}},:), G = D (\textsf{{.0}},:), G = D (\textsf{{.0}},:), G = D (\textsf{{.0}},:), G = D (\textsf{{{.0}}},:), G = D ({.0},:}), G = D ({.0},:}), G = D ({.0},:}), G = D ({.0},:}), G = D ({.0},:}), G = D ({.0},:}), G = D ({.0},:}), G = D ({.0},:}), G = D ({.0},:}), G = D ({.0},:}), G = D ({.0},:}), G = D ({.0},:}), A = B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A BA bA bA bA bA bA bA bA bA bA bA bA bA bA bA bA bA bA bA bA bA bA bA bA bA bA bA bA bA bA bA bA bA bA bA bA bA bA bA bA bA bA bA bA bA bA bA bA bA bA bA< nl>\]

All elements in this matrix can be computed (and coded) directly as shown below. The variance of the control variables is then given by

\[V a r [ \mathbf {y} _ {t} ^ {s} ] = \mathbf {D} V a r [ \mathbf {z} _ {t} ] \mathbf {D} ^ {\prime}\]

3.2.1 Computing the variance of the innovations

\[\begin{array}{r l} & {\mathrm{for} E \left[ \epsilon_ {t + 1} \left(\epsilon_ {t + 1} \otimes \epsilon_ {t + 1}\right) ^ {\prime} \right]} \\ & {E \left[ \epsilon_ {t + 1} \left(\epsilon_ {t + 1} \otimes \epsilon_ {t + 1}\right) ^ {\prime} \right] = E \left[ \{\epsilon_ {t + 1} (\phi_ {1}, 1) \} _ {\phi_ {1} = 1} ^ {n _ {e}} \left(\left\{\epsilon_ {t + 1} (\phi_ {2}, 1) \left\{\epsilon_ {t + 1} (\phi_ {3}, 1) \right\} _ {\phi_ {3} = 1} ^ {n _ {e}} \right\} _ {\phi_ {2} = 1} ^ {n _ {e}}\right) ^ {\prime} \right]} \\ & {\mathrm{HencethequasiMATLABcodesare:}} \\ & {E \_ e p s \_ e p s 2 = z e r o s (n e, (n e) ^ {2})} \\ & {\mathrm{forphi1=1:ne}} \\ & {\quad i n d e x 2 = 0} \\ & {\quad f o r p h i 2 = 1: n e} \end{array}\]

3.3 Method 2: Formulas for the first and second moments

This section computes first and second moments using a slightly different representation of the second-order system than stated above. (Basically, this was the first representation we considered for computing these moments). The advantage of this method is that it compared to Method 1 is less memory intensive because some of the matrix multiplications are done by hand.

We start by deriving an alternative representation of the pruned state space system (the old representation). Hence consider

\[\begin{array}{l} \mathbf {x} _ {t + 1} ^ {f} \otimes \mathbf {x} _ {t + 1} ^ {f} = \left(\mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t} ^ {f} + \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1}\right) \otimes \left(\mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t} ^ {f} + \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1}\right) \\ \qquad = \mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t} ^ {f} + \mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t} ^ {f} \otimes \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} + \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t} ^ {f} + \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} \otimes \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} \\ \text {using (A + B)\otimes(C + D) = A\otimes C + A\otimes D + B\otimes C + B\otimes D}\\ \\ \qquad = (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) + (\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}) (\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}) \\ \qquad + (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) (\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}) + (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}) (\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}) \\ \text {using (A\otimes B)(C\otimes D) = AC\otimes BD}\\ \\ \qquad = (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) + \mathbf {v} (t + 1) \\ \text {where}\\ \\ \mathbf {v} (t + 1) = (\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}) (\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}) + (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) (\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}) + (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}) (\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {{t + 1}}) \\ \end{array}\]

Note that because is independent across time and therefore also independent of . Moreover, and .

\[\begin{array}{r l} & {\mathrm{Thus}} \\ & {\left[ \begin{array}{c} \mathbf {x} _ {t + 1} ^ {f} \\ \mathbf {x} _ {t + 1} ^ {s} \\ \mathbf {x} _ {t + 1} ^ {f} \otimes \mathbf {x} _ {t + 1} ^ {f} \end{array} \right] = \left[ \begin{array}{c c c} \mathbf {h} _ {\mathbf {x}} & \mathbf {0} _ {n _ {x} \times n _ {x}} & \mathbf {0} _ {n _ {x} \times n _ {x} ^ {2}} \\ \mathbf {0} _ {n _ {x} \times n _ {x}} & \mathbf {h} _ {\mathbf {x}} & \tilde {\mathbf {H}} _ {\mathbf {x x}} \\ \mathbf {0} _ {n _ {x} ^ {2} \times n _ {x}} & \mathbf {0} _ {n _ {x} ^ {2} \times n _ {x}} & \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} \end{array} \right] \left[ \begin{array}{c} \mathbf {x} _ {t} ^ {f} \\ \mathbf {x} _ {t} ^ {s} \\ \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \end{array} \right]} \\ & {\qquad + \left[ \begin{array}{c} \mathbf {0} _ {n _ {x} \times 1} \\ \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2} \\ (\sigma \pmb {\eta} \otimes \sigma \pmb {\eta}) v e c (\mathbf {I} _ {n _ {e}}) \end{array} \right] + \left[ \begin{array}{c} \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1} \\ \mathbf {0} _ {n _ {x} \times 1} \\ \mathbf {v} (t + 1) - (\sigma \pmb {\eta} \otimes \pmb {\sigma} \pmb {\eta}) v e c (\mathbf {I} _ {n _ {e}}) \end{array} \right]} \\ & {\Updownarrow} \end{array}\]

\[\mathbf {z} _ {t + 1} = \mathbf {c} + \mathbf {A} \mathbf {z} _ {t} + \tilde {\boldsymbol {\xi}} _ {t + 1}\tag{37}\]

where for because is independent across time. The expression for the controls are as above, i.e.

\[\mathbf {y} _ {t} ^ {s} = \mathbf {D} \mathbf {z} _ {t} + \frac {1}{2} \mathbf {g} _ {\sigma \sigma} \sigma^ {2}\]

The mean values are

\[E [ \mathbf {z} _ {t} ] = (\mathbf {I} _ {2 n _ {x} + n _ {x} ^ {2}} - \mathbf {A}) ^ {- 1} \mathbf {c}\]

\[E [ \mathbf {y} _ {t} ] = \mathbf {D} E [ \mathbf {z} _ {t} ] + \frac {1}{2} \mathbf {g} _ {\sigma \sigma} \sigma^ {2}\]

and the covariance matrix is

\[\begin{array}{l} V a r \left(\mathbf {z} _ {t + 1}\right) = \mathbf {A} V a r \left(\mathbf {z} _ {t}\right) \mathbf {A} ^ {\prime} + V a r \left(\widetilde {\boldsymbol {\xi}} _ {t + 1}\right) \\ \Updownarrow \end{array}\]

\[\begin{array}{l} v e c \left(V a r \left(\mathbf {z} _ {t + 1}\right)\right) = v e c \left(\mathbf {A} V a r \left(\mathbf {z} _ {t}\right) \mathbf {A} ^ {\prime}\right) + v e c \left(V a r \left(\widetilde {\boldsymbol {\xi}} _ {t + 1}\right)\right) \\ \Updownarrow \\ v e c \left(V a r \left(\mathbf {z} _ {t + 1}\right)\right) = \left(\mathbf {A} \otimes \mathbf {A}\right) v e c \left(V a r \left(\mathbf {z} _ {t}\right)\right) + v e c \left(V a r \left(\widetilde {\boldsymbol {\xi}} _ {t + 1}\right)\right) \\ \Updownarrow \\ v e c \left(V a r \left(\mathbf {z} _ {t + 1}\right)\right) \left(\mathbf {I} _ {(2 n _ {x} + n _ {x} ^ {2}) ^ {2}} - \left(\mathbf {A} \otimes \mathbf {A}\right)\right) = v e c \left(V a r \left(\widetilde {\boldsymbol {\xi}} _ {t + 1}\right)\right) \\ \Updownarrow \end{array}\]

\[\operatorname{vec} \left(\operatorname{Var} \left(\mathbf {z} _ {t + 1}\right)\right) = \left(\mathbf {I} _ {\left(2 n _ {x} + n _ {x} ^ {2}\right) ^ {2}} - (\mathbf {A} \otimes \mathbf {A})\right) ^ {- 1} \operatorname{vec} \left(\operatorname{Var} \left(\tilde {\boldsymbol {\xi}} _ {t + 1}\right)\right)\]

The variance of the control variables is then given by

\[V a r [ \mathbf {y} _ {t} ^ {s} ] = \mathbf {D} V a r [ \mathbf {z} _ {t} ] \mathbf {D} ^ {\prime}\]

Hence we only need to compute .

\[\begin{array}{l} V a r \left(\tilde {\boldsymbol {\xi}} _ {t + 1}\right) = E \left(\left[ \begin{array}{c} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} \\ \mathbf {0} _ {n _ {x} \times 1} \\ \mathbf {v} (t + 1) - (\sigma \boldsymbol {\eta} \otimes \boldsymbol {\sigma} \boldsymbol {\eta}) v e c (\mathbf {I} _ {n _ {e}}) \end{array} \right] \left[ \begin{array}{c} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} \\ \mathbf {0} _ {n _ {x} \times 1} \\ \mathbf {v} (t + 1) - (\sigma \boldsymbol {\eta} \otimes \boldsymbol {\sigma} \boldsymbol {\eta}) v e c (\mathbf {I} _ {n _ {\epsilon}}) \end{array} \right] ^ {\prime}\right) \\ = E \left(\left[ \begin{array}{c} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} \\ \mathbf {0} _ {n _ {x} \times 1} \\ \mathbf {v} (t + 1) - (\sigma \boldsymbol {\eta} \otimes \boldsymbol {\sigma} \boldsymbol {\eta}) v e c (\mathbf {I _ {n _ {e}}}) \end{array} \right] \left[ \begin{array}{c c c} \sigma \boldsymbol {\epsilon} _ {t + 1} ^ {\prime} \boldsymbol {\eta} ^ {\prime} & \mathbf {0} _ {1 \times n _ {x}} & \mathbf {v} ^ {\prime} (t + 1) - v e c (\mathbf {I _ {n _ {e}}}) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \boldsymbol {\sigma} \boldsymbol {\eta}) ^ {\prime} ] \\ \end{array} \right)\right) \\ = E \left( \begin{array}{c c c} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} \sigma \boldsymbol {\epsilon} _ {t + 1} ^ {\prime} \boldsymbol {\eta} ^ {\prime} & \mathbf {0} _ {n _ {x} \times n _ {x}} & \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} (\mathbf {v} ^ {\prime} (t + 1) - v e c (\mathbf {I _ {n _ {e}}}) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \boldsymbol {\sigma} \boldsymbol {\eta}) ^ {\prime}) \\ \mathbf {0} _ {n _ {x} \times n _ {x}} & \mathbf {0} _ {n _ {x} \times n _ {x}} & \mathbf {0} _ {n _ {x} \times n _ {x} ^ {2}} \\ (\mathbf {v} (t + 1) - (\sigma \boldsymbol {\eta} \otimes \boldsymbol {\sigma} \boldsymbol {\eta}) v e c (\mathbf {I _ {n _ {e}}})) \sigma \boldsymbol {\epsilon} _ {t + 1} ^ {\prime} \boldsymbol {\eta} ^ {\prime} & \mathbf {0} _ {n _ {x} \times n _ {x}} & V a r [ \boldsymbol {\xi_ {t + 1}} ] _ {3 3} \\ \end{array} \right) \end{array}\]

where

Recall that

\[\begin{array}{r l} & {V a r \left[ \tilde {\pmb {\xi}} _ {t + 1} \right] _ {3 3} \equiv (\mathbf {v} (t + 1) - (\sigma \pmb {\eta} \otimes \pmb {\sigma} \pmb {\eta}) v e c (\mathbf {I} _ {n _ {e}})) (\mathbf {v} ^ {\prime} (t + 1) - v e c (\mathbf {I} _ {n _ {e}}) ^ {\prime} (\sigma \pmb {\eta} \otimes \pmb {\sigma} \pmb {\eta}) ^ {\prime})} \\ & {\mathbf {v} (t + 1) = (\mathbf {h} _ {\mathbf {x}} \otimes \sigma \pmb {\eta}) (\mathbf {x} _ {t} ^ {f} \otimes \pmb {\epsilon} _ {t + 1}) + (\sigma \pmb {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) (\pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}) + (\sigma \pmb {\eta} \otimes \pmb {\sigma} \pmb {\eta}) (\pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1})} \end{array}\]

\[\begin{array}{l} \textbf {3 . 3 . 1} \quad \textbf {F o r} V a r \left[ \tilde {\boldsymbol {\xi}} _ {t + 1} \right] _ {1 3} \\ V a r \left[ \tilde {\boldsymbol {\xi}} _ {t + 1} \right] _ {1 3} \equiv E [ \sigma \eta \epsilon_ {t + 1} \left(\mathbf {v} ^ {\prime} (t + 1) - v e c \left(\mathbf {I} _ {n _ {e}}\right) ^ {\prime} (\sigma \eta \otimes \sigma \eta) ^ {\prime}\right) ] \\ = E [ \sigma \eta \epsilon_ {t + 1} (\left((\mathbf {h} _ {\mathbf {x}} \otimes \sigma \eta) (\mathbf {x} _ {t} ^ {f} \otimes \epsilon_ {t + 1}) + (\sigma \eta \otimes \mathbf {h} _ {\mathbf {x}}) (\epsilon_ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}) + (\sigma \eta \otimes \sigma \eta) (\epsilon_ {t + 1} \otimes \epsilon_ {t + 1})\right) ^ {\prime} \\ - v e c (\mathbf {I} _ {n _ {e}}) ^ {\prime} (\sigma \eta \otimes \sigma \eta) ^ {\prime}) ] \\ = E [ \sigma \eta \epsilon_ {t + 1} (\left(\mathbf {x} _ {t} ^ {f} \otimes \epsilon_ {t + 1}\right) ^ {\prime} (\mathbf {h} _ {\mathbf {x}} \otimes \sigma \eta) ^ {\prime} + (\epsilon_ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} (\sigma \eta \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime} + (\epsilon_ {t + 1} \otimes \epsilon_ {t + 1}) ^ {\prime} (\sigma \eta \otimes \sigma \eta) ^ {\prime} \\ - v e c (\mathbf {I} _ {n _ {e}}) ^ {\prime} (\sigma \eta \otimes \sigma \eta) ^ {\prime})) ] \\ = E [ \sigma \eta \epsilon_ {t + 1} (\mathbf {x} _ {t} ^ {f} \otimes \epsilon_ {t + 1}) ^ {\prime} (\mathbf {h} _ {\mathbf {x}} \otimes \sigma \eta) ^ {\prime} + \sigma \eta \epsilon_ {t + 1} (\epsilon_ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} (\sigma \eta \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime} + \sigma \eta \epsilon_ {t + 1} (\epsilon_ {t + 1} \otimes \epsilon_ {t + 1}) ^ {\prime} (\sigma \eta \otimes \sigma \eta) ^ {\prime}. \end{array}\]

\[\begin{array}{c} - \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1} v e c (\mathbf {I} _ {n _ {e}}) ^ {\prime} (\sigma \pmb {\eta} \otimes \pmb {\sigma} \pmb {\eta}) ^ {\prime} ] \\ = E [ \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1} (\mathbf {x} _ {t} ^ {f} \otimes \pmb {\epsilon} _ {t + 1}) ^ {\prime} (\mathbf {h} _ {\mathbf {x}} \otimes \sigma \pmb {\eta}) ^ {\prime} + \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1} (\pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} (\sigma \pmb {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime} + \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1} (\pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1}) ^ {\prime} (\sigma \pmb {\eta} \otimes \pmb {\sigma} \pmb {\eta}) ^ {\prime} ] \\ \text {ause} E [ \pmb {\epsilon} _ {t + 1} ] = \mathbf {0} \end{array}\]

because is independent of and . Hence, for shocks with a symmetry distribution .

For the implementation, consider:

\[E \left[ \pmb {\epsilon} _ {t + 1} \left(\pmb {\epsilon} _ {t + 1} ^ {\prime} \otimes \pmb {\epsilon} _ {t + 1} ^ {\prime}\right) \right]\]

\[\begin{array}{l} = E \left[ \boldsymbol {\epsilon} _ {t + 1} \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) ^ {\prime} \right] \\ = E \left[ \left\{\epsilon_ {t + 1} \left(\phi_ {1}, 1\right) \right\} _ {\phi_ {1} = 1} ^ {n _ {e}} \left(\left\{\epsilon_ {t + 1} \left(\phi_ {2}, 1\right) \left\{\epsilon_ {t + 1} \left(\phi_ {3}, 1\right) \right\} _ {\phi_ {3} = 1} ^ {n _ {e}} \right\} _ {\phi_ {2} = 1} ^ {n _ {e}}\right) ^ {\prime} \right] \end{array}\]

Hence the quasi MATLAB codes are :

\[\begin{array} { r l } & \textbf { 3 . 3 . 2 } \quad \textbf { F o r } V a r \left[ \tilde { \boldsymbol { \xi } } _ { t + 1 } \right] _ { 3 3 } \\ & { V a r \left[ \tilde { \boldsymbol { \xi } } _ { t + 1 } \right] _ { 3 3 } \equiv E [ ( \mathbf { v } ( t + 1 ) - ( \sigma \boldsymbol { \eta } \otimes \sigma \boldsymbol { \eta } ) v e c ( \mathbf { I } _ { n _ { e } } ) ) ( \mathbf { v } ^ { \prime } ( t + 1 ) - v e c ( \mathbf { I } _ { n _ { e } } ) ^ { \prime } ( \sigma \boldsymbol { \eta } \otimes \sigma \boldsymbol { \eta } ) ^ { \prime } ) ] } \\ & { = E [ \left( ( \mathbf { h } _ { \mathbf { x } } \otimes \sigma \boldsymbol { \eta } ) ( \mathbf { x } _ { t } ^ { f } \otimes \boldsymbol { \epsilon } _ { t + 1 } ) + ( \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x } } ) ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } ) + ( \sigma \boldsymbol { \eta } \otimes \sigma \boldsymbol { \eta } ) ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \boldsymbol { \epsilon } _ { t + 1 } ) - ( \sigma \boldsymbol { \eta } \otimes \sigma \boldsymbol { \eta } ) v e c ( \mathbf { I } _ { n _ { e } } ) \right) } \\ & { \qquad \left( \left( ( \mathbf { h } _ { \mathbf { x } } \otimes \sigma \boldsymbol { \eta } ) ( \mathbf { x } _ { t } ^ { f } \otimes \boldsymbol { \epsilon } _ { t + 1 } ) + ( \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x } } ) ( \boldsymbol{ \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } ) + ( \sigma \boldsymbol { \eta } \otimes \sigma \boldsymbol { \eta } ) ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \boldsymbol { \epsilon } _ { t + 1 } ) \right) ^ { \prime } - v e c ( \mathbf { I } _ { n _ { e } } ) ^ { \prime } ( \sigma \boldsymbol { \eta } \otimes \sigma \boldsymbol { \eta } ) ^ { \prime } ) ] } \\ & = E [ ( ( \mathbf { h } _ { \mathbf { x } } \otimes \sigma \boldsymbol { \eta } ) ( \mathbf { x } _ { t } ^ { f } \otimes \boldsymbol { \epsilon } _ { t + 1 } ) + ( \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x } } ) ( \boldsymbol { \epsilon} _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } ) + ( \sigma \boldsymbol { \eta } \otimes \sigma \boldsymbol { \eta} ) ( ( \boldsymbol{ {\epsilon} _{ t + 1 }} \otimes \boldsymbol{ {\epsilon} _{ t + 1} ) - v e c ( | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | ] . } \\ & = E [ ( ( [ (\mathbf { h } _ { x } ) ⓕ [ (\mathbf { x } _ { t } ^ { f} ⓕ [ (\mathbf { x } _ { t } ^ { f} ⓕ [ (\mathbf { x } _ { t } ^ { f} ⓕ [ (\mathbf { x } _ { t } ^ { f} ⓕ [ (\mathbf { x } _ { t } ^ { f} ⓕ [ (\mathbf { x } _ { s } ^ { f} ⓕ [ (\mathbf { x } _ { s } ^ { f} ⓕ [ (\mathbf { x } _ { s } ^ { f} ⓕ [ (\mathbf { x } _ { s } ^ { f} ⓕ [ (\mathbf { x } _ { s } ^ { f} ⓕ [ (\mathbf {\Sigma} ^ {\prime }\]

\[\begin{array} { l } \left( \left( \mathbf { x } _ { t } ^ { f } \otimes \boldsymbol { \epsilon } _ { t + 1 } \right) ^ { \prime } ( \mathbf { h } _ { \mathbf { x } } \otimes \sigma \boldsymbol { \eta } ) ^ { \prime } + \left( \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \right) ^ { \prime } ( \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { \prime } + \left( ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \boldsymbol { \epsilon } _ { t + 1 } ) ^ { \prime } - v e c ( \mathbf { I } _ { n _ { e } } ) ^ { \prime } \right) ( \sigma \boldsymbol { \eta } \otimes \sigma \boldsymbol { \eta } ) ^ { \prime } \right) \\ + ( \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x } } ) ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } ) \times \\ \left( \left( \mathbf { x } _ { t } ^ { f } \otimes \boldsymbol { \epsilon } _ { t + 1 } \right) ^ { \prime } ( \mathbf { h } _ { \mathbf { x } } \otimes \sigma \boldsymbol { \eta } ) ^ { \prime } + \left( \boldsymbol { \epsilon } _ { t + 1 } \ottimes \mathbf { x } _ { t } ^ { f } \right) ^ { \prime } ( \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { \prime } + ( ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \boldsymbol { \epsilon } _ { t + 1 } ) ^ { \prime } - v e c ( \mathbf { I } _ { n _ { e } } ) ^ { \prime } ) ( \sigma \boldsymbol { \eta } \otimes \sigma \boldsymbol { \eta } ) ^ { \prime } \right) \\ + ( \sigma \boldsymbol { \eta } \otimes \sigma \boldsymbol { \eta } ) ( ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \boldsymbol { \epsilon } _ { t + 1 } ) - v e c ( \mathbf { I } _ { n _ { e } } ) ) \times \\ \left( \left( \mathbf { x } _ { t } ^ { f } \otimes \boldsymbol { \epsilon } _ { t + 1 } \right) ^ { \prime } ( \mathbf { h } _ { \mathbf { x } } \otimes \sigma \boldsymbol { \eta } ) ^ { \prime } + ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } ) ^ { \prime } ( \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { \prime } + ( ( ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \boldsymbol { \epsilon } _ { t + 1 } ) ^ { \prime } - v e c ( \mathbf { I } _ { n _ { e } } ) ^ { \prime } ) ( \sigma \boldsymbol { \eta } \otimes \sigma \boldsymbol { \eta } ) ^ { \prime } ) ] \\ = E [ ( {\bf h} _ { {\bf x}} ] \\ + ( {\bf h} _ { {\bf x}} ] + ( {\bf h} _ { {\bf x}} ] + ( {\bf h} _ { {\bf x}} ] + ( {\bf h} _ { {\bf x}} ] + ( (\sigma [ [ (\boldsymbol {\theta} ] ] ) ] + ( (\sigma [ [ (\boldsymbol {\theta} ] ] ) ] + ( (\sigma [ [ (\boldsymbol {\theta} ] ] ) ] + ( (\sigma [ [ (\boldsymbol {\theta} ] ] ) ] + ( (\sigma [ [ (\boldsymbol {\theta} ] ] ] ) ] + ( (\sigma [ [ (\boldsymbol {\theta} ] ] ] ) ] + ( (\sigma [ [ (\boldsymbol {\theta} ] ] ] ) ] + ( (\sigma [ [ (\boldsymbol {\theta} ] ] ] ) ] + ( (\sigma [ [ ] ] ) ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ]. \\ = E [ ( {\bf h} _ { {\bf x}} ] + ( (\boldsymbol {\theta} [ [ (\boldsymbol {\theta} ] ] ) ] + ( (\boldsymbol {\theta} [ [ (\boldsymbol {\theta} ] ] ) ] + ( (\boldsymbol {\theta} [ [ (\boldsymbol {\theta} ] ] ) ] + ( (\boldsymbol {\theta} [ [ (\boldsymbol {\theta} ] ] ) ].) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , |, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 |, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 |), \\ = E [ ( | {\bf h} _ { {\bf x}} | > | {\bf r} |) < ; | {\bf r} | > | {\bf r} | > | {\bf r} | > | {\bf r} | > | {\bf r} | > | {\bf r} | > | {\bf r} | > | {\bf r} | > | {\bf r} | > | {\bf r} | > | {\bf r} | > | {\bf r} | > | {\bf r} | > | {\bf r} | > | {\bf r} | > |{\bf r} | > | {\bf r} | > | {\bf r} | > | {\bf r} | > | {\bf r} | > | {\bf r} | > | {\bf r} | > | {\bf r} | > | {\bf r} | > | {\bf r} | > | {\bf r} | > | {\bf r} | > | {\bf r} | > | {\bf r} | > | {\bf s} | + ( {{\bf h} _ { {\bf x}}} | > | {{\bf r}} |) < ; | {{\bf r}} | > | {{\bf r}} | > | {{\bf r}} | > | {{\bf r}} | > | {{\bf r}} | > | {{\bf r}} | > | {{\bf r}} | > | {{\bf r}} | > | {{\bf r}} | > | {{\bf r}} | > | {{\bf r}} | > | {{\bf r}} | > | {{\bf r}} | > | {{\cal H}} | + ( {{\bf h} _ { {\cal X}}} | > | {{\bf r}} |) < ; | {{\bf r}} | > | {{\bf r}} | > | {{\bf r}} | > | {{\bf r}} | > | {{\bf r}} | > | {{\bf r}} | > | {{\bf r}} | > | {{\bf r}} | > | {{\cal H}} | + ( {{\cal H}} _ { {\cal X}} & = E [ ( \| {\cal H} _ { {\cal X}} \| > \| {\cal R} \| ) < ; \| {\cal R} \| > \| {\cal R} \| > \| {\cal R} \| > \| {\cal R} \| > \| {\cal R} \| > \| {\cal R} \| > \| {\cal R} \| > \| {\cal R} \| > \| {\cal R} \| > \| {\cal R} \| > \| {\cal R} \| > \| {\cal R} \| > \| {\cal R} \| > \| {\cal R} \| > \| {\cal R} \| > \|{\cal R} \| > \| {\cal R} \| > \| {\cal R} \| > \| {\cal R} \| > \|{\cal R} \| > \|{\cal R} \| > \|{\cal R} \| > \|{\cal R} \| > \|{\cal R} \| > \|{\cal R} \| > \|{\cal R} \| > \|{\cal R} \| > \|{\cal R} \| > \|{\cal R} \| > \|{\cal R} \| > \|{\cal R} \| > \|{\cal R} \| > \|{\cal R} \| > \|{\cal S}, \\ + ( {{\cal H}} _ { {\cal X}} & = E [ ( \| {\cal H} _ { {\cal X}} \| > \| {\cal R} \| ) < ; \| ({\cal H}_ {t}) ^ {- *} / ({\cal H}_ {t}) ^ {- *} / ({\cal H}_ {t}) ^ {- *} / ({\cal H}_ {t}) ^ {- *} / ({\cal H}_ {t}) ^ {- *} / ({\cal H}_ {t}) ^ {- *} / ({\cal H}_ {t}) ^ {- *} / ({\cal H}_ {t}) ^ {- *} / ({\cal H}_ {t}) ^ {- *} / ({\cal C}), \\ + ( {{\cal H}} _ { {\cal X}} & = E [ ( \| {\cal H}_ {t}) ^ {- *} / ({\cal H}_ {t}) ^ {- *} / ({\cal H}_ {t}) ^ {- *} / ({\cal H}_ {t}) ^ {- *} / ({\cal H}_ {t}) ^ {- *} / ({\cal H}_ {t}) ^ {- *} / ({\cal H}_ {t}) ^ {- *} / ({\cal H}_ {t}) ^ {- *} / ({\cal H}_ {(m)} ) ^ {- *} / ({\cal H}_ {(m)} ) ^ {- *} / ({\cal H}_ {(m)} ) ^ {- *} / ({\cal H}_ {(m)} ) ^ {- *} / ({\cal H}_ {(m)} ) ^ {- *} / ({\cal H}_ {(m)} ) ^ {- *} / ({\cal H}_ {(m)} ) ^ {- *} / ({\cal H}_ {(m)} ) ^ {- *} / ({\cal H}_{m}). \\ + ({{\cal H}} _{ {\cal X}} & = E [ ( \| {\cal H}_{t}) ^ {- *} / ({\cal H}_{t}) ^ {- *} / ({\cal H}_{t}) ^ {- *} / ({\cal H}_{t}) ^ {- *} / ({\cal H}_{t}) ^ {- *} / ({\cal H}_{t}) ^ {- *} / ({\cal H}_{t}) ^ {- *} / ({\cal H}_{t}) ^ {- *} / ({\cal H}_{t}) ^ {- *} / ({\cal H}_{(m)} ) ^ {- *} / ({\cal H}_{(m)} ) ^ {- *} / ({\cal H}_{(m)} ) ^ {- *} / ({\cal H}_{(m)} ) ^ {- *} / ({\cal H}_{(m)} ) ^ {- *} / ({\cal H}_{(m)} ) ^ {- *} / ({\cal H}_{(m)} ) ^ {- *} / ({\cal H}_{(m)} ) ^ {- *} / ({\cal H}_{m}). \\ + ({{\cal H}} _{ {\cal X}} & = E [ ( \| {\cal H}_{t}) ^ {- *} / ({\cal H}_{t}) ^ {- *} / ({\cal H}_{t}) ^ {- *} / ({\cal H}_{t}) ^ {- *} / ({\cal H}_{t}) ^ {- *} / ({\cal H}_{s}\cdot) ^ {- *} / ({\cal H}_{s}\cdot) ^ {- *} / ({\cal H}_{s}\cdot) ^ {- *} / ({\cal H}_{s}\cdot) ^ {- *} / ({\cal H}_{s}\cdot) ^ {- *} / ({\cal H}_{s}\cdot) ^ {- *} / ({\cal H}_{s}\cdot) ^ {- *} / ({\cal H}_{s}\cdot) ^ {- *} / ({\cal H}_{s}\cdot) ^ {- *} / ({\cal H}_{s}\cdot) ^ {- *} / ({\cal H}_{s}\cdot) ^ {- *} / ({\cal H}_{s}\cdot) ^ {- *} /({\cal H}_{s}\cdot) ^ {- * }, \\ + ({{\cal H}} _{ {\cal X}} & = E [ ( \| {\cal H}_{t}) ^ {- *} / ({\cal H}_{t}) ^ {- *} / ({\cal H}_{t}) ^ {- *} / ({\cal H}_{t}) ^ {- *} / ({\cal H}_{t}) ^ {- *} / ({\cal H}_{t}) ^ {- *} / ({\cal H}_{t}) ^ {- *} / ({\cal H}_{t}\cdot) ^ {- *} / ({\cal H}_{t}\cdot) ^ {- *} / ({\cal H}_{t}\cdot) ^ {- *} / ({\cal H}_{t}\cdot) ^ {- *} / ({\cal H}_{t}\cdot) ^ {-* }, \\ + ({{\cal H}} _{ {\cal X}} & = E [ ( \| {\cal H}_{t}) ^ {- *} / ({\cal H}_{t}) ^ {- *} / ({\cal H}_{t}) ^ {- *} / ({\cal H}_{t}) ^ {- *} / ({\cal H}_{t}\cdot) ^ {-* }, \\ + ({{\cal H}} _{ {\cal X}} & = E [ ( \| {\cal H}_{t}) ^ {- *} / ({\cal H}_{t}) ^ {- *} / ({\cal H}_{t}) ^ {-* }, \\ + ({{\cal H}} _{ {\cal X}} & = E [ ( \| {\delta_ {[ t + k] }}) ^ {- * }, \\ + ({{\delta_ {[ t + k] }}}) < ;< |content_end|>\]

\[\begin{array}{r l} & {+ (\sigma \pmb {\eta} \otimes \mathbf {h _ {x}}) (\pmb {\epsilon} _ {t + 1} \pmb {\epsilon} _ {t + 1} ^ {\prime} \otimes \mathbf {x} _ {t} ^ {f} (\mathbf {x} _ {t} ^ {f}) ^ {\prime}) (\sigma \pmb {\eta} \otimes \mathbf {h _ {x}}) ^ {\prime}} \\ & {+ (\sigma \pmb {\eta} \otimes \mathbf {h _ {x}}) ((\pmb {\epsilon} _ {t + 1} \pmb {\epsilon} _ {t + 1} ^ {\prime} \otimes \mathbf {x} _ {t} ^ {f} \pmb {\epsilon} _ {t + 1} ^ {\prime}) - (\pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}) v e c (\mathbf {I} _ {n _ {e}}) ^ {\prime}) (\sigma \pmb {\eta} \otimes \pmb {\sigma \eta}) ^ {\prime}} \\ & {+ (\sigma \pmb {\eta} \otimes \pmb {\sigma \eta}) ((\pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1}) ((\mathbf {x} _ {t} ^ {f}) ^ {\prime} \otimes \pmb {\epsilon} _ {t + 1} ^ {\prime}) - v e c (\mathbf {I} _ {n _ {e}}) ((\mathbf {x} _ {t} ^ {f}) ^ {\prime} \otimes \pmb {\epsilon} _ {t + 1} ^ {\prime})) (\mathbf {h _ {x}} \otimes \sigma \pmb {\eta}) ^ {\prime}} \\ & {+ (\sigma \pmb {\eta} \otimes \pmb {\sigma \eta}) ((\pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1}) (\pmb {\epsilon} _ {t + 1} ^ {\prime} \otimes (\mathbf {x} _ {t} ^ {f}) ^ {\prime}) - v e c (\mathbf {I} _ {n _ {e}}) ((\pmb {\epsilon} _ {t + 1} ^ {\prime} \otimes (\mathbf {x} _ {t} ^ {f}) ^ {\prime})) (\sigma \pmb {\eta} \otimes \mathbf {h _ {x}}) ^ {\prime}} \\ & {+ (\sigma \pmb {\eta} \otimes \pmb {\sigma \eta}) ((\pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1}) - v e c (\mathbf {I} _ {n _ {e}})) ((\pmb {\epsilon} _ {t + 1} ^ {\prime} \otimes \pmb {\epsilon} _ {t + 1} ^ {\prime}) - v e c (\mathbf {I} _ {n _ {e}}) ^ {\prime}) (\sigma \pmb {\eta} \otimes \pmb {\sigma \eta}) ^ {\prime} ]} \end{array}\]

\[\begin{array} { l } = ( \mathbf { h } _ { \mathbf { x } } \otimes \sigma \boldsymbol { \eta } ) \left( E \left[ \mathbf { x } _ { t } ^ { f } \left( \mathbf { x } _ { t } ^ { f } \right) ^ { \prime } \right] \otimes \mathbf { I } _ { n _ { e } } \right) ( \mathbf { h } _ { \mathbf { x } } \otimes \sigma \boldsymbol { \eta } ) ^ { \prime } \\ + ( \mathbf { h } _ { \mathbf { x } } \otimes \sigma \boldsymbol { \eta } ) E \left( \mathbf { x } _ { t } ^ { f } \boldsymbol { \epsilon } _ { t + 1 } ^ { \prime } \otimes \boldsymbol { \epsilon } _ { t + 1 } \left( \mathbf { x } _ { t } ^ { f } \right) ^ { \prime } \right) ( \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { \prime } \\ + ( \mathbf { h } _ { \mathbf { x } } \otimes \sigma \boldsymbol { \eta } ) E \left( \left( \mathbf { x } _ { t } ^ { f } \boldsymbol { \epsilon } _ { t + 1 } ^ { \prime } \otimes \boldsymbol { \epsilon } _ { t + 1 } \boldsymbol { \epsilon } _ { t + 1 } ^ { \prime } \right) - \left( \mathbf { x } _ { t } ^ { f } \otimes \boldsymbol { \epsilon } _ { t + 1 } \right) v e c ( \mathbf { I } _ { n _ { e } } ) ^ { \prime } \right) ( \sigma \boldsymbol { \eta } \otimes \sigma \boldsymbol { \eta } ) ^ { \prime } \\ + ( \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x } } ) E \left( \boldsymbol { \epsilon } _ { t + 1 } \left( \mathbf { x } _ { t } ^ { f } \right) ^ { \prime } \otimes \mathbf { x } _ { t } ^ { f } \boldsymbol { \epsilon } _ { t + 1 } ^ { \prime } \right) ( \mathbf { h } _ { \mathbf { x } } \otimes \sigma \boldsymbol { \eta } ) ^ { \prime } \\ + ( \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x } } ) \left( \mathbf { I } _ { n _ { e } } \otimes E \left[ \mathbf { x } _ { t } ^ { f } \left( \mathbf { x } _ { t } ^ { f } \right) ^ { \prime } \right] \right) ( \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { \prime } \\ + ( \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x } } ) E \left( ( [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] , \\ + ( \sigma 5 0 ) E ( ( 6 0 ) E ( 6 0 ) E ( 6 0 ) E ( 6 0 ) E ( 6 0 ) E ( 6 0 ) E ( 6 0 ) E ( 6 0 ) E ( 6 0 ) E ( 6 0 ) E ( 6 0 ) E ( 6 0 ) E ( 6 0 ) E ( 6 0 ) E ( 6 0 ) E ( E ) E ( E ) E ( E ) E ( E ) E ( E ) E ( E ) E ( E ) E ( E ) E ( E ) E ( E ) E ( E ) E ( E ) E ( E ) E ( E ) E ( E ) E ( E ) E ( E ) E ( E ) E ( E ) E ( E ) E ( E ) E ( E ) E ( E ) E ( E ) E ( E ) E (E ) E ( E ) E ( E ) E ( E ) E ( E ) E ( E ) E ( E ) E ( E ) E ( E ) E ( E ) E ( E ) E ( E ) E ( E ) E ( E ) E ( E ) E ( E ) E ( E ) E ( E ) E ( E ) E ( E ) E (E )E (E )E (E )E (E )E (E )E (E )E (E )E (E )E (E )E (E )E (E )E (E )E (E )E (E )E (E )E (E )E (E )E (E )E (E )E (E )E (E )E (E )E (E )E (E )E (E )E (E )e q u a d i n e r s e c t i o n e r s e c t i o n e r s e c t i o n e r s e c t i o n e r s e c t i o n e r s e c t i o n e r s e c t i o n e r s e c t i o n e r s e c t i o n e r s e c t i o n e r s e c t i o n e r s e c t i on e r s e c t i o n e r s e c t i o n e r s e c t i o n e r s e c t i o n e r s e c t i o n e r s e c t i o n e r s e c t i o n e r s e c t i o n e r s e c t i o n e r s e c t i o n e r s e c t i o n & + (\sigma 5 0) ^ {- 1} \\ + (\sigma 5 0) ^ {- 1} & + (\sigma 5 0) ^ {- 1} & + (\sigma 5 0) ^ {- 1} & + (\sigma 5 0) ^ {- 1} & + (\sigma 5 0) ^ {- 1} & + (\sigma 5 0) ^ {- 1} & + (\sigma 5 0) ^ {- 1} & + (\sigma 5 0) ^ {- 1} & + ((\sigma 5 0) ^ {- 1} & + (\sigma 5 0) ^ {- 1} & + (\sigma 5 0) ^ {- 1} & + (\sigma 5 0) ^ {- 1} & + (\sigma 5 0) ^ {- 1} & + (\sigma 5 0) ^ {- 1} & + (\sigma 5 0) ^ {- 1} & + (\sigma_ {\mathrm{eff}}) ^ {- 1} & + (\sigma_ {\mathrm{eff}}) ^ {- 1} & + (\sigma_ {\mathrm{eff}}) ^ {- 1} & + (\sigma_ {\mathrm{eff}}) ^ {- 1} & + (\sigma_ {\mathrm{eff}}) ^ {- 1} & + (\sigma_ {\mathrm{eff}}) ^ {- 1} & + (\sigma_ {\mathrm{eff}}) ^ {- - 1} & + (\sigma_ {\mathrm{eff}}) ^ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = == = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = == / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |\]

note that is independent of and is know and

\[\begin{array}{r l} & {= (\mathbf {h _ {x}} \otimes \sigma \pmb {\eta}) \left(E \left[ \mathbf {x} _ {t} ^ {f} (\mathbf {x} _ {t} ^ {f}) ^ {\prime} \right] \otimes \mathbf {I} _ {n _ {e}}\right) (\mathbf {h _ {x}} \otimes \sigma \pmb {\eta}) ^ {\prime}} \\ & {+ (\mathbf {h _ {x}} \otimes \sigma \pmb {\eta}) E (\mathbf {x} _ {t} ^ {f} \pmb {\epsilon} _ {t + 1} ^ {\prime} \otimes \pmb {\epsilon} _ {t + 1} (\mathbf {x} _ {t} ^ {f}) ^ {\prime}) (\sigma \pmb {\eta} \otimes \mathbf {h _ {x}}) ^ {\prime}} \\ & {+ (\sigma \pmb {\eta} \otimes \mathbf {h _ {x}}) E (\pmb {\epsilon} _ {t + 1} (\mathbf {x} _ {t} ^ {f}) ^ {\prime} \otimes \mathbf {x} _ {t} ^ {f} \pmb {\epsilon} _ {t + 1} ^ {\prime}) (\mathbf {h _ {x}} \otimes \sigma \pmb {\eta}) ^ {\prime}} \\ & {+ (\sigma \pmb {\eta} \otimes \mathbf {h _ {x}}) (\mathbf {I} _ {n _ {e}} \otimes E [ \mathbf {x} _ {t} ^ {f} (\mathbf {x} _ {t} ^ {f}) ^ {\prime} ]) (\sigma \pmb {\eta} \otimes \mathbf {h _ {x}}) ^ {\prime}} \\ & {+ (\sigma \pmb {\eta} \otimes \pmb {\sigma} \pmb {\eta}) E ((\pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1}) - v e c (\mathbf {I} _ {n _ {e}})) ((\pmb {\epsilon} _ {t + 1} ^ {\prime} \otimes \pmb {\epsilon} _ {t + 1} ^ {\prime}) - v e c (\mathbf {I} _ {n _ {e}}) ^ {\prime}) (\sigma \pmb {\eta} \otimes \pmb {\sigma} \pmb {\eta}) ^ {\prime}} \\ & {{\mathrm{termswiththreeorone}} \pmb {\epsilon} _ {t + 1} {\mathrm{arezerobecause}} \mathbf {x} _ {t} ^ {f} {\mathrm{isindependentof}} \pmb {\epsilon} _ {t + 1} {\mathrm{and}} E [ \mathbf {x} _ {t} ^ {f} ] = 0.} \end{array}\]

\[\begin{array}{l} = (\mathbf {h _ {x}} \otimes \sigma \boldsymbol {\eta}) \left(E \left[ \mathbf {x} _ {t} ^ {f} \left(\mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \right] \otimes \mathbf {I} _ {n _ {e}}\right) (\mathbf {h _ {x}} \otimes \sigma \boldsymbol {\eta}) ^ {\prime} \\ + (\mathbf {h _ {x}} \otimes \sigma \boldsymbol {\eta}) E \left(\mathbf {x} _ {t} ^ {f} \boldsymbol {\epsilon} _ {t + 1} ^ {\prime} \otimes \boldsymbol {\epsilon} _ {t + 1} \left(\mathbf {x} _ {t} ^ {f}\right) ^ {\prime}\right) (\sigma \boldsymbol {\eta} \otimes \mathbf {h _ {x}}) ^ {\prime} \\ + (\sigma \boldsymbol {\eta} \otimes \mathbf {h _ {x}}) E \left(\boldsymbol {\epsilon} _ {t + 1} \left(\mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \otimes \mathbf {x} _ {t} ^ {f} \boldsymbol {\epsilon} _ {t + 1} ^ {\prime}\right) (\mathbf {h _ {x}} \otimes \sigma \boldsymbol {\eta}) ^ {\prime} \\ + (\sigma \boldsymbol {\eta} \otimes \mathbf {h _ {x}}) \left(\mathbf {I} _ {n _ {e}} \otimes E \left[ \mathbf {x} _ {t} ^ {f} \left(\mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \right]\right) (\sigma \boldsymbol {\eta} \otimes \mathbf {h _ {x}}) ^ {\prime} \\ + (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}) \times \end{array}\]

\[\begin{array} { r l }&E \left( ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \boldsymbol { \epsilon } _ { t + 1 } ) \left( \boldsymbol { \epsilon } _ { t + 1 } ^ { \prime } \otimes \boldsymbol { \epsilon } _ { t + 1 } ^ { \prime } \right) - ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \boldsymbol { \epsilon } _ { t + 1 } ) v e c ( \mathbf { I } _ { n _ { e } } ) ^ { \prime } - v e c ( \mathbf { I } _ { n _ { e } } ) \left( \boldsymbol { \epsilon } _ { t + 1 } ^ { \prime } \otimes \boldsymbol { \epsilon } _ { t + 1 } ^ { \prime } \right) + v e c ( \mathbf { I } _ { n _ { e } } ) v e c ( \mathbf { I } _ { n _ { e } } ) ^ { \prime } \right) ( \sigma \boldsymbol { \eta } \otimes \sigma \boldsymbol { \eta } ) ^ { \prime }\\&= ( \mathbf { h } _ { \mathbf { x } } \otimes \sigma \boldsymbol { \eta } ) \left( E \left[ \mathbf { x } _ { t } ^ { f } \left( \mathbf { x } _ { t } ^ { f } \right) ^ { \prime } \right] \otimes \mathbf { I } _ { n _ { e } } \right) ( \mathbf { h } _ { \mathbf { x } } \otimes \sigma \boldsymbol { \eta } ) ^ { \prime }\\&+ ( \mathbf { h } _ { \mathbf { x } } \otimes \sigma \boldsymbol { \eta } ) E \left( \mathbf { x } _ { t } ^ { f } \boldsymbol { \epsilon } _ { t + 1 } ^ { \prime } \otimes \boldsymbol { \epsilon } _ { t + 1 } \left( \mathbf { x } _ { t } ^ { f } \right) ^ { \prime } \right) ( \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { \prime }\\&+ ( \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x } } ) E \left( \boldsymbol { \epsilon } _ { t + 1 } \left( \mathbf { x } _ { t } ^ { f } \right) ^ { \prime } \otimes \mathbf { x } _ { t } ^ { f } \boldsymbol { \epsilon } _ { t + 1 } ^ { \prime } \right) ( \mathbf { h } _ { \mathbf { x } } \otimes \sigma \boldsymbol { \eta } ) ^ { \prime }\\&+ ( \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x } } ) \left( \right. \mathbf { I } _ { n _ { e } } \otimes E [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ( 0 , 0 , 0 ) ( 0 , 0 , 0 ) ( 0 , 0 , 0 ) ( 0 , 0 , 0 ) ( 0 , 0 , 0 ) ( 0 , 0 , 0 ) ( 0 , 0 , 0 ) ( 0 , 0 , 0 ) ( 0 , 0 , 0 ) ( 0 , 0 , 0 ) ( 0 , 0 , 0 . )\\&+ ( \sigma \boldsymbol { \eta } \otimes \sigma \boldsymbol { \eta } ) ×\\&( E ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \boldsymbol { \epsilon } _ { t + 1 } ) ( \boldsymbol { \epsilon } _ { t + 1 } ^ { \prime } \otimes \boldsymbol { \epsilon } _ { t + 1 } ^ { \prime } ) - v e c ( \mathbf { I } _ { n _ { e } } ) v e c ( \mathbf { I } _ { n _ { e } } ) ^ { \prime } - v e c ( \mathbf { I } _ { n _ { e } } ) v e c ( \mathbf { I } _ { n _ { e } } ) ^ { \prime } + v e c ( \mathbf { I } _ { n _ { e } } ) v e c ( \mathbf { I } _ { n _ { e } } ) ^ { \prime }) ( \sigma \boldsymbol { \eta } \otimes \sigma \boldsymbol { \eta } ) ^ { \prime }\\&= ( {\mathbf h} _ { {\mathbf x} }\otimes\sigma{\boldsymbol{ {\eta}}})\left( \right.E\left[ \right. {\mathbf x} _ { t } ^ { f }\left. \right) ^ {\prime} [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ) ( 0 , 0 , 0 ) ( 0 , 0 , 0 ) ( 0 , 0 , 0 ) ( 0 , 0 , 0 ) ( 0 , 0 , 0 ) ( 0 , 0 , 0 ) ( 0 , 0 , 0 ) ( 0 , 0 , 0 ) ( 0 , 0 , 0 ) ( 0 , 0 , 1 ) ( 0 , 1 , 1 ) ( 0 , 1 , 1 ) ( 0 , 1 , 1 ) ( 0 , 1 , 1 ) ( 0 , 1 , 1 ) ( 0 , 1 , 1 ) ( 0 , 1 , 1 ) ( 0 , 1 , 1 ) ( 0 , 1 , 1 ) ( 0 , 1 , 1 ) ( 0 , 2 , 2 ) ( 0 , 2 , 2 ) ( 0 , 2 , 2 ) ( 0 , 2 , 2 ) ( 0 , 2 , 2 ) ( 0 , 2 , 2 ) ( 0 , 2 , 2 ) ( 0 , 2 , 2 ) ( 0 , 2 , 2 ) ( 0 , 2 , 2 ) ( 0 , 2 , 3 ) ( 0 , 2 , 3 ) ( 0 , 2 , 3 ) ( 0 , 2 , 3 ) ( 0 , 2 , 3 ) ( 0 , 2 , 3 ) ( 0 , 2 , 3 ) ( 0 , 2 , 3 ) ( 0 , 2 , 3 ) ( 0 , 2 , 3 ) ( 0 , 2 , m a x i n d i s i o n d i s i o n d i s i o n d i s i o n d i s i o n d i s i o n d i s i o n d i s i o n d i s i o n d i s i o n d i s i o n d i s i o n d i s i o n d i s i o n d i s i o n d i s i o n d i s i o n d i d i s i o n d i d i s i o n d i d i s i o n d i d i s i o n d i d i s i o n d i d i s i o n d i d i s i o n d i d i s i o n d i d i s i o n d i d i s i o n d i d i s i o n d i d i s i o n d i d i s j u a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |&+ (\sigma {\boldsymbol {\eta}}\otimes\sigma {\boldsymbol {\eta}}) ×\\&+ (\sigma {\boldsymbol {\eta}}\otimes\sigma {\boldsymbol {\eta}}) ×\\&E ( E ({\boldsymbol {\epsilon}} _ { t + 1 }\otimes E ({\boldsymbol {\epsilon}} _ { t + 1 }\otimes E ({\boldsymbol {\epsilon}} _ { t + 1 }\otimes E ({\boldsymbol {\epsilon}} _ { t + 1 }\otimes E ({\boldsymbol {\epsilon}} _ { t + 1 }\otimes E ({\boldsymbol {\epsilon}} _ { t + 1 }\otimes E ({\boldsymbol {\epsilon}} _ { t + 1}\otimes E ({\boldsymbol {\epsilon}} _ { t + 1 }\otimes E ({\boldsymbol {\epsilon}} _ { t + 1 }\otimes E ({\boldsymbol {\epsilon}} _ { t + 1 }\otimes E ({\boldsymbol {\epsilon}} _ { t + 1 }\otimes E ({\boldsymbol {\epsilon}} _ { t + 1 }\otimes E ({\boldsymbol {\epsilon}} _ t + m a x i n d i s i o n d i s i o n d i s i o n d i s i o n d i s i o n d i s i o n d i s i o n d i s i o n d i s i o n d i s i o n d i s i o n d i s i o n d i s i o n d i s i o n d i s i o n d i s i m a x {}^ *{}^ *{}^ *{}^ *{}^ *{}^ *{}^ *{}^ *{}^ *{}^ *{}^ *{}^ *{}^ *{}^ *{}^ *{}^ *{}^ *{}^ *{}^ *{}^ *{}^ *{}^ *{}^ *{}^ *{}^ *{}^ *{}^ *{}^ *{}^ *{}^ *{}^ *{}^ *{}^ *{}^ *{}^{*}\\&= ( {\mathbf h} _ {\mathbf x}\otimes\sigma{\boldsymbol{\eta}})E\left( \right.E\left[ \right. {\mathbf x} _ { t } ^ { f }\left. \right) ^ {\prime} {[} {\mathbf x} _ {{t} ^{f }} ^{ f} {\mathbf x} _ {{t} ^{f }} ^{ f} {\mathbf x} _ {{t} ^{f }} ^{ f} {\mathbf x} _ {{t} ^{f }} ^{ f} {\mathbf x} _ {{t} ^{f }} ^{ f} {\mathbf x} _ {{t} ^{f }} ^{ f} {\mathbf x} _ {{t} ^{f }} {\mathbf x} _ {{t} ^{f }} ^{ f} {\mathbf x} _ {{t} ^{f }} ^{ f} {\mathbf x} _ {{t} ^{f }} ^{ f} {\mathbf x} _ {{t} ^{f }} ^{ f} {\mathbf x} _ {{t} ^{f }} ^{ f} {\mathbf x} _ {{t} ^{f }} ^{*} {\mathbf x} _ {{t} ^{f}} ^{ f} {\mathbf x} _ {{t} ^{f}} ^{ f} {\mathbf x} _ {{t} ^{f}} ^{ f} {\mathbf x} _ {{t} ^{f}} ^{ f} {\mathbf x} _ {{t} ^{f}} ^{ f} {\mathbf x} _ {{t} ^{f}} ^{ f} {\mathbf x} _ {{t} ^{f}} {\mathbf x} _ {{t} ^{f}} ^{ f} {\mathbf x} _ {{t} ^{f}} {\mathbf x} _ {{t} ^{f}} {\mathbf x} _ {{t} ^{f}} {\mathbf x} _ {{t} ^{f}} {\mathbf x} _ {{t} ^{f}} {\mathbf x} _ {{t} ^{f}} {\mathbf x} _ {{t} ^{f}} {\mathbf x} _ {{t} ^{f}} {\mathbf x} _ {t}\texttt{I}\texttt{I}\texttt{I}\texttt{I}\texttt{I}\texttt{I}\texttt{I}\texttt{I}\texttt{I}\texttt{I}\texttt{I}\texttt{I}\texttt{I}\texttt{I}\texttt{I}\texttt{I}\texttt{I}\texttt{I}\texttt{I}\texttt{I}\texttt{II}\texttt{II}\texttt{II}\texttt{II}\texttt{II}\texttt{II}\texttt{II}\texttt{II}\texttt{II}\texttt{II}\texttt{II}\texttt{II}\texttt{II}\texttt{II}\texttt{II}\texttt{II}\texttt{II}\texttt{II}\texttt{II}\texttt{II}\texttt{III}\texttt{III}\texttt{III}\texttt{III}\texttt{III}\texttt{III}\texttt{III}\texttt{III}\texttt{III}\texttt{III}\texttt{III}\texttt{III}\texttt{III}\texttt{III}\texttt{III}\texttt{III}\texttt{III}\texttt{III}\texttt{III}\texttt{III}\texttt{II}\texttt{II}\texttt{II}\texttt{II}\texttt{II}\texttt{II}\texttt{II}\texttt{II}\texttt{II}\texttt{II}\texttt{II}\texttt{II}\texttt{II}\texttt{II}\texttt{II}\texttt{II}\texttt{(I)}\end{array}\]

end

Note also that

\[\begin{array} { r l } & { \mathrm{Finallyconsiderthematrix(withdimension} n _ { e } ^ { 2 } \times n _ { e } ^ { 2 } ) } \\ & { E \left[ \left( \boldsymbol { \epsilon } _ { t + 1 } \boldsymbol { \epsilon } _ { t + 1 } ^ { \prime } \otimes \boldsymbol { \epsilon } _ { t + 1 } \boldsymbol { \epsilon } _ { t + 1 } ^ { \prime } \right) \right] } \\ & { = E [ \left[ \begin{array} { c } \epsilon _ { t + 1 } ( 1 , 1 ) \\ \epsilon _ { t + 1 } ( 2 , 1 ) \\ \cdots \\ \epsilon _ { t + 1 } ( n _ { e } , 1 ) \end{array} \right] \left[ \begin{array} { c c c c } \epsilon _ { t + 1 } ^ { \prime } ( 1 , 1 ) & \epsilon _ { t + 1 } ^ { \prime } ( 1 , 2 ) & \cdots & \epsilon _ { t + 1 } ^ { \prime } ( 1 , n _ { e } ) \end{array} \right] } \\ & { \otimes [ \left[ \begin{array} { c } \epsilon _ { t + 1 } ( 1 , 1 ) \\ \epsilon _ { t + 1 } ( 2 , 1 ) \\ \cdots \\ \epsilon _ { t + 1 } ( n _ { e } , 1 ) \end{array} \right] \left[ \begin{array} { c c c c } \epsilon _ { t + 1 ] } ^ { \prime } ( 1 , 1 ) & \epsilon _ { t + 1 ] } ^ { \prime } ( 1 , 2 ) & \cdots & \epsilon _ { t + 1 ] } ^ { \prime } ( 1 , n _ { e } ) \end{array} \right] ] } \\ & { = E [ \left[ \begin{array} { c c c c } \epsilon _ { t + 1 } ( 1 , 1 ) \epsilon _ { t + 1 ] } ^ { \prime } ( 1 , 1 ) & \epsilon _ { t + 1 } ( 1 , 1 ) \epsilon _ { t + 1 ] } ^ { \prime } ( 1 , 2 ) & \cdots & \epsilon _ { t + 1 } ( 1 , 1 ) \epsilon _ { t + 1 ] } ^ { \prime } ( 1 , n _ { e } ) \\ \epsilon _ { t + 1 } ( 2 , 1 ) \epsilon _ { t + 1 ] } ^ { \prime } ( 1 , 1 ) & \epsilon _ { t + 1 } ( 2 , 1 ) \epsilon _ { t + 1 ] } ^ { \prime } ( 1 , 2 ) & \cdots & \epsilon _ { t + 1 } ( 2 , 1 ) \epsilon _ { t + 1 ] } ^ { \prime } ( 1 , n _ { e } ) \\ \cdots & \cdots & \cdots & \cdots \\ \epsilon _ { t + 1 } ( n _ { e } , 1 ) \epsilon _ { t + 1 ] } ( 1 , 1 ) & \epsilon _ { t + 1 } ( n _ { e } , 1 ) \epsilon _ { t + 1 ] } ( 1 , 2 ) & \cdots & \epsilon _ { t + 1 } ( n _ { e } , 1 ) \epsilon _ { t + 1 ] } ^ { \prime } ( 1 , n _ { e } ) \\ \end{array} \right] } \\ & { \otimes [ \left[ \begin{array} { c c c c } \epsilon _ { t + 1 } ( 1 , 1 ) \epsilon _ { t + 1 ] } ^ { \prime } ( 1 , 1 ) & \epsilon _ { t + 1 } ( 1 , 1 ) \epsilon _ { t + 1 ] } ^ { \prime } ( 1 , 2 ) & \cdots & e _ { t + 1 } ( 1 , 1 ) e _ { t + 1 ] } ^ { \prime } ( 1 , n _ { e } ) \\ e _ { t + 1 } ( 2 , 1 ) e _ { t + 1 ] } ^ { \prime } ( 1 , 1 ) & e _ { t + 1 } ( 2 , 1 ) e _ { t + 1 ] } ^ { \prime } ( 1 , 2 ) & \cdots & e _ { t + 1 } ( 2 , 1 ) e _ { t + 1 ] } ^ { \prime } ( 1 , n _ { e } ) \\ \cdots & \cdots & \cdots & \cdots \\ e _ { t + 1 } ( n _ { e } , 1 ) e _ { t + 1 ] } ( 1 , 1 ) & e _ { t + 1 } ( n _ { e } , 1 ) e _ { t + 1 ] } ( 1 , 2 ) & \cdots & e _ { t + 1 } ( n _ { e } , 1 ) e _ { t + 1 ] } ^ { \prime } ( 1 , n _ { e } ) \\ \end{array} \right] ] } \\ & { = E [ \left[ \begin{array} { c c c c } e _ { t + 1 } ( 1 , 1 ) e _ { t + 1 ] } ^ { \prime } ( 1 , 1 ) {\bf A} _ { \epsilon \epsilon } & e _ { t + 1 } ( 1 , 1 ) e _ { t + 1 ] } ^ { \prime } ( 1 , 2 ) {\bf A} _ { \epsilon \epsilon } & \cdots & e _ { t + 1 } ( 1 , 1 ) e _ { t + 1 ] } ^ { \prime } ( 1 , n _ { e } ) {\bf A} _ { \epsilon \epsilon } \\ e _ { t + 1 } ( 2 , 1 ) e _ { t + 1 ] } ( 1 , 1 ) {\bf A} _ { \epsilon \epsilon } & e _ { t + 1 } ( 2 , 1 ) e _ { t + 1 ] } ^ { \prime } ( 1 , 2 ) {\bf A} _ { \epsilon \epsilon } & \cdots & e _ { t + 1 } ( 2 , 1 ) e _ { t + 1 ] } ^ { \prime } ( 1 , n _ { e } ) {\bf A} _ { \epsilon \epsilon } \\ \cdots & \cdots & \cdots & \cdots \\ e _ { t + 1 } ( n _ { e } , 1 ) e _ { t + 1 ] } ( 1 , 1 ) {\bf A} _ { \epsilon \epsilon } & e _ { t + 1 } ( n _ { e } , 1 ) e _ { t + 1 ] } ^ { \prime } ( 1 , 2 ) {\bf A} _ { \epsilon \epsilon } & \cdots & e _ { t + 1 } ( n _ { e } , 1 ) e _ { t + 1 ] } ^ { \prime } ( 1 , n _ { e } ) {\bf A} _ { \epsilon \epsilon } \\ \end{array} \right] } \\ & \mathrm{where} {\bf A} _ { \epsilon \epsilon } = [ [ e f i o r a l y s u p p h i m i n g r a i v i s i o n y s u p p h i m i n g r a i v i s i o n y s u p p h i m i n g r a i v i s i o n y s u p p h i m i n g r a i v i s i o n y s u p p h i m i n g r a i v i s i o n y s u p p h i m i n g r a i v i s i on y s u p p h i m i n g r a i v i s i o n y s u p p h i m i n g r a i v i s i o n y s u p p h i m i n g r a i v i s i o n y s u p p h i m i n g r a i v i s i o n y s u p p h i m i n g r a i v i s i o d y s u p p h i m i n g r a i v i s i o n y s u p p h i m i n g r a i v i s i o n y s u p p h i m i n g r a i v i s i o n y s u p p h i m i n g r a i v i s i o n y s u p p h i m i n g r a i v i s i o l y s u p p h i m i n g r a i v i s i o n y s u p p h i m i n g r a i v i s i o n y s u p p h i m i n g r a i v i s i o n y s u p p h i m i n g r a i v i s i o n y s u p p h i m i n g r a i v i s i f f o r k j . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .\]

\[\begin{array} { r l } & = E [ \left[ \begin{array} { c c } \epsilon _ { t + 1 } \left( 1 , 1 \right) \epsilon _ { t + 1 } ^ { \prime } \left( 1 , 1 \right) \left\{ A _ { \epsilon \epsilon } \left( \phi _ { 1 } , \phi _ { 2 } \right) \right\} _ { \phi _ { 1 } = 1 , \phi _ { 2 } = 1 } ^ { n _ { e } , n _ { e } } & \epsilon _ { t + 1 } \left( 1 , 1 \right) \epsilon _ { t + 1 } ^ { \prime } \left( 1 , 2 \right) \left\{ A _ { \epsilon \epsilon } \left( \phi _ { 1 } , \phi _ { 2 } \right) \right\} _ { \phi _ { 1 } = 1 , \phi _ { 2 } = 1 } ^ { n _ { e } , n _ { e } } \\ \epsilon _ { t + 1 } \left( 2 , 1 \right) \epsilon _ { t + 1 } ^ { \prime } \left( 1 , 1 \right) \left\{ A _ { \epsilon \epsilon } \left( \phi _ { 1 } , \phi _ { 2 } \right) \right\} _ { \phi _ { 1 } = 1 , \phi _ { 2 } = 1 } ^ { n _ { e } n _ { e } } & \epsilon _ { t + 1 } \left( 2 , 1 \right) \epsilon _ { t + 1 } ^ { \prime } \left( 1 , 2 \right) \left\{ A _ { \epsilon \epsilon } \left( \phi _ { 1 } , \phi _ { 2 } \right) \right\} _ { \phi _ { 1 } = 1 , \phi _ { 2 } = 1 } ^ { n _ { e } n _ { e } } \\ ... & ... \\ \epsilon _ { t + 1 } \left( n _ { e } , 1 \right) \epsilon _ { t + 1 } ^ { \prime } \left( 1 , 1 \right) \left\{ A _ { \epsilon \epsilon } \left( \phi _ { 1 } , \phi _ { 2 } \right) \right\} _ { \phi _ { 1 } = 1 , \phi _ { 2 } = 1 } ^ { n _ { e } . n _ { e } } & \epsilon _ { t + 1 } \left( n _ { e } , 1 \right) \epsilon _ { t + 1 } ^ { \prime } \left( 1 , 2 \right) \left\{ A _ { \epsilon \epsilon } \left( \phi _ { 1 } , \phi _ { 2 } \right) \right\} _ { \phi _ { 1 } = 1 , \phi _ { 2 } = 1 } ^ { n _ { e } . n _ { e } } \\ [ ( n _ { e } ) ] ^ { n _ { e } . n _ { e } } & \\ ... & [ ( n _ { e } ) ] ^ { n _ { e } . n _ { e } } \\ ... & [ ( n _ { e } ) ] ^ { n _ { e } . n _ { e } } \\ ... & [ ( n _ { e } ) ] ^ { n _ { e } . n _ { e } } \\ ... & [ ( n _ { e } ) ] ^ { n _ { e } . n _ { e } } \\ ... & [ ( n _ { e } ) ] ^ n _ { e } . n_{ e} - n_{e} - n_{e} - n_{e} - n_{e} - n_{e} - n_{e} - n_{e} - n_{e} - n_{e} - n_{e} - n_{e} - n_{e} - n_{e} - n_{e} - n_{e} - n_{e} - n_{e} - n_{e} - n_{e} - n_{e} - n_{c} ] \\ & \\ = E [ [ ( n _ { e } ) ] ^ n _ { e } . n _ { e } - n_{e} - n_{e} - n_{e} - n_{e} - n_{e} - n_{e} - n_{e} - n_{e} - n_{e} - n_{e} - n_{e} - n_{e} - n_{e} - n_{e} - n_{e} - n_{e} - n_{e} - n_{e} - n_{e} - n_{e}, \\ & \\ ... \\ [ ( n _ { e } ) ] ^ n _ { e } . n _ { e } - n_{e} - n_{e} - n_{e} - n_{e} - n_{e} - n_{e} - n_{e} - n_{e} - n_{e} - n_{e} - n_{e} - n_{e} - n_{e} - n_{e} - n_{e}, \\ & ... \\ [ ( n _ { e } ) ] ^ n _{ e } . n_{e} - n_{e} - n_{e} - n_{e} - n_{e} - n_{e} - n_{e} - n_{e} - n_{e} - n_{e} - n_{e} - n_{e}, \\ & ... \\ [ ( n _{ e}) ] ^ n_{ e} . n_{e} - n_{e} - n_{e} - n_{e} - n_{e} - n_{e} - n_{e} - n_{e} - n_{e} - n_{e}, \\ & ... \\ [ ( n _{ e}) ] ^ n_{ e} . n_{e} - n_{e} - n_{e} - n_{e} - n_{e} - n_{e} - n_{{\bf {\Phi}}}\]

3.4 Method 3: Simple formulas for first and second moments

This section computes first and second moments at second order using a more direct approach. The advantage of this method is that we do not in a second-order approximation re-compute some of the moments already know from a first-order approximation. A direct implication is that we in Method 3 only need to invert smaller matrices than in Method 1 and 2.

3.4.1 First moments

Note first that

\[E [ \mathbf {x} _ {t} ] = E [ \mathbf {x} _ {t} ^ {f} ] + E [ \mathbf {x} _ {t} ^ {s} ]\]

For the first order effects, we have due to stationary of the linear model

\[E \left[ \mathbf {x} _ {t} ^ {f} \right] = \mathbf {h} _ {\mathbf {x}} E \left[ \mathbf {x} _ {t - 1} ^ {f} \right] + \sigma \pmb {\eta} E \left[ \pmb {\epsilon} _ {t} \right]\]

\[\begin{array}{l} \stackrel {\triangledown} {\mathbf {I}} E [ \mathbf {x} _ {t} ^ {f} ] - \mathbf {h} _ {\mathbf {x}} E [ \mathbf {x} _ {t - 1} ^ {f} ] = E [ \boldsymbol {\epsilon} _ {t} ] \\ \Updownarrow \end{array}\]

\[\begin{array}{l} \left(\mathbf {I} - \mathbf {h} _ {\mathbf {x}}\right) E \left[ \mathbf {x} _ {t} ^ {f} \right] = E \left[ \boldsymbol {\epsilon} _ {t} \right] \\ \Updownarrow \end{array}\]

\[\begin{array}{l} E \left[ \mathbf {x} _ {t} ^ {f} \right] = \left(\mathbf {I} - \mathbf {h} _ {\mathbf {x}}\right) ^ {- 1} E \left[ \boldsymbol {\epsilon} _ {t} \right] \\ \Updownarrow \end{array}\]

\[E \left[ \mathbf {x} _ {t} ^ {f} \right] = \mathbf {0}\]

since

For the second order effects we have

\[\begin{array}{l} E \left[ x _ {t + 1} ^ {s} (j, 1) \right] = \mathbf {h} _ {\mathbf {x}} (j,:) E \left[ \mathbf {x} _ {t} ^ {s} \right] + \frac {1}{2} E \left[ \left(\mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \mathbf {h} _ {\mathbf {x x}} (j,:,:) \left(\mathbf {x} _ {t} ^ {f}\right) \right] + \frac {1}{2} h _ {\sigma \sigma} (j, 1) \sigma^ {2} \\ \Updownarrow \end{array}\]

\[\left(\mathbf {I} - \mathbf {h} _ {\mathbf {x}}\right) E \left[ \mathbf {x} _ {t} ^ {s} \right] = \tilde {\mathbf {H}} _ {\mathbf {x x}} E \left[ v e c \left(\left[ \left(\mathbf {x} _ {t} ^ {f}\right) \left(\mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \right]\right) \right] + \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}\]

where

To compute , consider

\[\begin{array}{l} V a r \left(\mathbf {x} _ {t} ^ {f}\right) = \mathbf {h} _ {\mathbf {x}} V a r \left(\mathbf {x} _ {t} ^ {f}\right) \mathbf {h} _ {\mathbf {x}} ^ {\prime} + \sigma^ {2} \boldsymbol {\eta} \boldsymbol {\eta} ^ {\prime} \\ \Downarrow \end{array}\]

\[\begin{array}{l} v e c \left(V a r \left(\mathbf {x} _ {t} ^ {f}\right)\right) = v e c \left(\mathbf {h} _ {\mathbf {x}} V a r \left(\mathbf {x} _ {t} ^ {f}\right) \mathbf {h} _ {\mathbf {x}} ^ {\prime}\right) + v e c \left(\sigma^ {2} \boldsymbol {\eta} \boldsymbol {\eta} ^ {\prime}\right) \\ \Updownarrow \end{array}\]

\[\begin{array}{l} v e c \left(V a r \left(\mathbf {x} _ {t} ^ {f}\right)\right) = (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) v e c \left(V a r \left(\mathbf {x} _ {t} ^ {f}\right)\right) + v e c \left(\sigma^ {2} \boldsymbol {\eta} \boldsymbol {\eta} ^ {\prime}\right) \\ \Updownarrow \end{array}\]

\[\begin{array}{l} \left(\mathbf {I} _ {n _ {x} ^ {2}} - \left(\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}\right)\right) v e c \left(V a r \left(\mathbf {x} _ {t} ^ {f}\right)\right) = v e c \left(\sigma^ {2} \boldsymbol {\eta} \boldsymbol {\eta} ^ {\prime}\right) \\ \Updownarrow \end{array}\]

\[\operatorname{vec} \left(\operatorname{Var} \left(\mathbf {x} _ {t} ^ {f}\right)\right) = \left(\mathbf {I} _ {n _ {x} ^ {2}} - \left(\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}\right)\right) ^ {- 1} \operatorname{vec} \left(\sigma^ {2} \boldsymbol {\eta} \boldsymbol {\eta} ^ {\prime}\right)\]

Notice there that

\[V a r \left(\mathbf {x} _ {t} ^ {f}\right) = E \left[ \left(\mathbf {x} _ {t} ^ {f} - E \left[ \mathbf {x} _ {t} ^ {f} \right]\right) \left(\mathbf {x} _ {t} ^ {f} - E \left[ \mathbf {x} _ {t} ^ {f} \right]\right) ^ {\prime} \right] = E \left[ \mathbf {x} _ {t} ^ {f} \left(\mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \right]\]

since

\[\stackrel {\triangledown} {v e c} \left(E \left[ \mathbf {x} _ {t} ^ {f} \left(\mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \right]\right) = v e c \left(V a r \left(\mathbf {x} _ {t} ^ {f}\right)\right)\]

Hence,

\[E \left[ \mathbf {x} _ {t} ^ {s} \right] = \left(\mathbf {I} - \mathbf {h} _ {\mathbf {x}}\right) ^ {- 1} \left(\tilde {\mathbf {H}} _ {\mathbf {x x}} E \left[ v e c \left(\left[ \left(\mathbf {x} _ {t} ^ {f}\right) \left(\mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \right]\right) \right] + \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}\right)\]

The mean value of the control variables is given by

\[E \left[ \mathbf {y} _ {t} ^ {s} \right] = \mathbf {g} _ {\mathbf {x}} \left(E \left[ \mathbf {x} _ {t} ^ {f} \right] + E \left[ \mathbf {x} _ {t} ^ {s} \right]\right) + \tilde {\mathbf {G}} _ {\mathbf {x x}} E \left[ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) \right] + \frac {1}{2} \mathbf {g} _ {\sigma \sigma} \sigma^ {2}\]

\[E \left[ \mathbf {y} _ {t} ^ {s} \right] = \mathbf {g} _ {\mathbf {x}} E \left[ \mathbf {x} _ {t} ^ {s} \right] + \tilde {\mathbf {G}} _ {\mathbf {x x}} E \left[ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) \right] + \frac {1}{2} \mathbf {g} _ {\sigma \sigma} \sigma^ {2}\]

3.4.2 Second moments

We need to compute , , , and as

this will allow us to find . This is because

\[\begin{array}{r l} & {V a r \left(\mathbf {z} _ {t}\right) = E \left[ \left(\mathbf {z} _ {t} - E \left[ \mathbf {z} _ {t} \right]\right) \left(\mathbf {z} _ {t} - E \left[ \mathbf {z} _ {t} \right]\right) ^ {\prime} \right]} \\ & {\qquad = E \left[ \left(\mathbf {z} _ {t} - E \left[ \mathbf {z} _ {t} \right]\right) \left(\mathbf {z} _ {t} ^ {\prime} - E \left[ \mathbf {z} _ {t} ^ {\prime} \right]\right) \right]} \\ & {\qquad = E \left[ \mathbf {z} _ {t} \mathbf {z} _ {t} ^ {\prime} - \mathbf {z} _ {t} E \left[ \mathbf {z} _ {t} ^ {\prime} \right] - E \left[ \mathbf {z} _ {t} \right] \mathbf {z} _ {t} ^ {\prime} + E \left[ \mathbf {z} _ {t} \right] E \left[ \mathbf {z} _ {t} ^ {\prime} \right] \right]} \\ & {\qquad = E \left[ \mathbf {z} _ {t} \mathbf {z} _ {t} ^ {\prime} \right] - E \left[ \mathbf {z} _ {t} \right] E \left[ \mathbf {z} _ {t} ^ {\prime} \right]} \end{array}\]

and

\[\begin{array}{r l} & {E \left[ \mathbf {z} _ {t} \mathbf {z} _ {t} ^ {\prime} \right] = E \left[ \left[ \begin{array}{c} \mathbf {x} _ {t} ^ {f} \\ \mathbf {x} _ {t} ^ {s} \\ \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \end{array} \right] \left[ \begin{array}{c c c} (\mathbf {x} _ {t} ^ {f}) ^ {\prime} & (\mathbf {x} _ {t} ^ {s}) ^ {\prime} & (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} \end{array} \right] \right]} \\ & {\quad = E \left[ \begin{array}{c c c} \mathbf {x} _ {t} ^ {f} (\mathbf {x} _ {t} ^ {f}) ^ {\prime} & \mathbf {x} _ {t} ^ {f} (\mathbf {x} _ {t} ^ {s}) ^ {\prime} & \mathbf {x} _ {t} ^ {f} (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} \\ \mathbf {x} _ {t} ^ {s} \mathbf {x} _ {t} ^ {f} & \mathbf {x} _ {t} ^ {s} (\mathbf {x} _ {t} ^ {s}) ^ {\prime} & \mathbf {x} _ {t} ^ {s} (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} \\ (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) \mathbf {x} _ {t} ^ {f} & (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) (\mathbf {x} _ {t} ^ {s}) ^ {\prime} & (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} \end{array} \right]} \end{array}\]

\[\begin{array}{l} \textbf {F i n d i n g} E \left[ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \right] \\ \text {From above:} \\ \mathbf {x} _ {t + 1} ^ {f} \otimes \mathbf {x} _ {t + 1} ^ {f} = (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) + \mathbf {v} (t + 1) \\ \text {where} \\ \mathbf {v} (t + 1) = (\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}) \left(\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) + (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) + (\sigma \boldsymbol {\eta} \otimes \boldsymbol {\sigma} \boldsymbol {\eta}) (\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}) \\ \text {and} E [ \mathbf {v} (t + 1) ] = (\sigma \boldsymbol {\eta} \otimes \boldsymbol {\sigma} \boldsymbol {\eta}) v e c (\mathbf {I} _ {n _ {e}}). \text {Note that} (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) \text {and} \mathbf {v} (t + 1) ^ {\prime} \text {are uncorrelated}. \end{array}\]

So

\[\begin{array} { r l } & E \left( \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } \right) \left( \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } \right) ^ { \prime } \\ & \quad = E \{ ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) \left( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \right) + \mathbf { v } ( t + 1 ) \} \{ ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) \left( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \right) + \mathbf { v } ( t + 1 ) \} ^ { \prime } \\ & \quad = E \{ ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) \left( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \right) + \mathbf { v } ( t + 1 ) \}\{ ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) ^ { \prime } ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { \prime } + \mathbf { v } ( t + 1 ) ^ { \prime } \} \\ & = E [ ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) ( ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) ^ { \prime } ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { \prime } + \mathbf { v } ( t + 1 ) ^ { \prime } ) ] \\ & + E [ \mathbf { v } ( t + 1 ) ( ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) ^ { \prime } ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { \prime } + \mathbf { v } ( t + 1 ) ^ { \prime } ) \\ & = E [ ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) ( ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ {\mathrm{f}} ) ^ { \prime } ( \mathbf { h} _ { \mathbf { x} } \otimes \mathbf { h} _ { \mathbf { x} }) ^ { \prime } + ( \mathbf { h} _ { \mathbf { x} } \otimes \mathbf { h} _ { \mathbf { x} ) } ( ( \mathbf { x } _ {\mathrm{f}} ^ {\mathrm{f}} \otimes \mathbf { x} _ {\mathrm{f}} ^ {\mathrm{f}} ) ( v ( t + 1 ) ^ { \prime} ) \\ & + E [ ( v ( t + 1 ) ( ( v (\mathrm{h} ) ^ {\prime} ) + v ( t + 1 ) ) v ( t + 1 ) ^ { \prime} ) ] \\ & = ( v (\mathrm{h} _ {\mathrm{x}}) E [ ( v (\mathrm{h} _ {\mathrm{x}}) E [ ( v (\mathrm{h} _ {\mathrm{x}}) E [ ( v (\mathrm{h} _ {\mathrm{x}}) E [ ( v (\mathrm{h} _ {\mathrm{x}}) E [ ( v (\mathrm{h} _ {\mathrm{x}}) E [ ( v (\mathrm{h} _ {\mathrm{x}}) E [ ( v (\mathrm{h} _ {\textup{t+1}}) e [ v (\mathrm{h} _ {\textup{t+1}}) e [ v (\mathrm{h} _ {\textup{t+1}}) e [ v (\mathrm{h} _ {\textup{t+1}}) e [ v (\mathrm{h} _ {\textup{t+1}}) e [ v (\mathrm{h} _ {\textup{t+1}}) e [ v (\mathrm{h} _ {\textsuperscript - 1} e [ v (\mathrm{h} _ {\textup{t+1}}) e [ v (\mathrm{h} _ {\textup{t+1}}) e [ v (\mathrm{h} _ {\textup{t+1}}) e [ v (\mathrm{h} _ {\textup{t+1}}) e [ v (\mathrm{h} _ {\textup{t+1}}) e [ v (\mathrm{h} _{\textup{t+1}}) e [ v (\mathrm{h} _ {\textup{t+1}}) e [ v (\mathrm{h} _ {\textup{t+1}}) e [ v (\mathrm{h} _ {\textup{t+1}}) e [ v (\mathrm{h} _ {\textup{t+1}}) e [ v (\mathrm{h} _ {\textup{t+1}}) e[ v (\mathrm{h} _ {\textup{t+1}}) e [ v (\mathrm{h} _ {\textup{t+1}}) e [ v (\mathrm{h} _ {\textup{t+1}}) e [ v (\mathrm{h} _ {\textup{t+1}}) e [ v (\mathrm{h} _ {\textup{t+1}}) e [ v (\mathrm{h} _ {\textop{t+1}}) e [ v (\mathrm{h} _ {\textup{t+1}}) e [ v (\mathrm{h} _ {\textup{t+1}}) e [ v (\mathrm{h} _ {\textup{t+1}}) e [ v (\mathrm{h} _ {\textup{t+1}}) e [ v (\mathrm{h} _ {\textup{t+1}}) e [ v(\textup{t+1}) e [ v (\textup{t+1}) e [v (\textup{t+1}) e [v (\textup{t+1}) e [v (\textup{t+1}) e [v (\textup{t+1}) e [v (\textup{t+1}) e [v (\textup{t+1}) e [v (\textup{t+1}) e [v (\textup{t+1}) e [v (\textup{t+1}) e [v(\textup{t+1}) e [v (\textup{t+1}) e [v (\textup{t+1}) e [v (\textup{t+1}) e [v (\textup{t+1}) e [v (\textup{t+1}) e [v (\textup{t+1}) e [v (\textup{t+1}) e [v (\textup{t+1}) e [v (\mathrm{h} _ {\mathrm{x}} ) ^ { / 2 }\]

Finding

\[\begin{array} { r l } & E \left[ \mathbf { x } _ { t + 1 } ^ { s } \left( \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } \right) ^ { \prime } \right] = E \left[ \left( \mathbf { h } _ { \mathbf { x } } \mathbf { x } _ { t } ^ { s } + \tilde { \mathbf { H } } _ { \mathbf { x x } } \left( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \right) + \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } \right) \left( ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) \left( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \right) + \mathbf { v } ( t + 1 ) ^ { \prime } \right) ^ { \prime } \right] \\ & = E \left[ \left( \mathbf { h } _ { \mathbf { x } } \mathbf { x } _ { t } ^ { s } + \tilde { \mathbf { H } } _ { \mathbf { x x } } \left( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \right) + \frac 1 2 \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } \right) \left( \left( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \right) ^ { \prime } ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { \prime } + \mathbf { v } ( t + 1 ) ^ { \prime } \right) \right] \\ & = E \left[ \mathbf { h } _ { \mathbf { x } } \mathbf { x } _ { t } ^ { s } \left( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \right) ^ { \prime } ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { \prime } + \mathbf { h } _ { \mathbf { x } } \mathbf { x } _ { t } ^ { s } \mathbf { v } ( t + 1 ) ^ { \prime } \right] \\ & + E \left[ \tilde { \mathbf { H } } _ { \mathbf { x x } } \left( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \right) \left( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \right) ^ { \prime } ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { \prime } + \tilde { \mathbf { H } } _ { \mathbf { x x } } \left( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \right) \mathbf { v } ( t + 1 ) ^ { \prime } \right] \\ & + E [ {\frac 12} {\mathbf h} _ { \sigma \sigma } {\sigma^ 2} ( {\mathbf x} _ { t } ^ { f } {\otimes} {\mathbf x} _ { t } ^ { f} ) ^ { \prime } ( {\mathbf h} _ { \mathbf x} {\otimes} {\mathbf h} _ { \mathbf x} ) ^ { \prime } + {\frac 12} {\mathbf h} _ { \sigma \sigma } {\sigma^ 2} {\mathbf v} ( t + 1 ) ^ { \prime } ] \\ & = {\mathbf h} _ {\mathbf x} E [ {\mathbf x} _ { t } ^ { s } ( {\mathbf x} _ { t } ^ { f } {\otimes} {\mathbf x} _ { t } ^ { f} ) ^ { \prime } ] ( {\mathbf h} _ { \mathbf x} {\otimes} {\mathbf h} _ { \mathbf x} ) ^ { \prime } + {\mathbf h} _ {\mathbf x} E [ {\mathbf x} _ { t } ^ { s} ] E [ {\mathbf v} ( t + 1 ) ^ { \prime } ] \\ & + \\ & + E [ {\tilde {\mathbf H}} _ {\mathrm{xx}} E [ ( {\mathbf x} _ { t } ^ { f } {\otimes} {\mathbf x} _ { t } ^ { f} ) ( {\mathbf x} _ { t } ^ { f } {\otimes} {\mathbf x} _ { t } ^ { f} ) ^ { \prime } ] ( {\mathbf h} _ { \mathbf x} {\otimes} {\mathbf h} _ { \mathbf x} ) ^ { \prime } + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & - \\ & e n d i n g \\ & c = e n d i n g e [ {\mathbf c} ] E [ {\mathbf c} ] E [ {\mathbf v} ( t + 1 ) ^ { \prime } ] + \\ & - E [ {\tilde {\mathbf H}} _ {\mathrm{xx}} E [ ( {\mathbf c} ] E [ ( {\mathbf v} ( t + 1 ) ^ { \prime }\right) ] - E [ {\tilde {\mathbf H}} _ {\mathrm{xx}} E [ ( {\mathbf c} ] E [ ({\mathbf h} _ {\mathrm{x}} {\otimes} {\mathbf h} _ {\mathrm{x}} ) ^ { \prime }\right) - E [ ({\tilde {\mathbf H}} _ {\mathrm{xx}} E [ ({\mathbf c} ] E [ ({\mathbf v} ( t + 1 ) ^ { \prime }\right) ] \\ & + \\ & - E [ {\tilde {\mathbf H}} _ {\sigma s} E [ ( ({\mathbf c} ] E [ ({\mathbf h} _ {\mathrm{x}} {\otimes} {\mathbf c} _ {\mathrm{x}} ) ^{ ' }\right) - E [ ({\tilde {\mathbf H}} _ {\sigma s} E [ ({\mathbf v} ( t + 1 ) ^{ ' }\right) ] - \\ & - E [ ({\tilde {\mathbf H}} _ {\sigma s} E [ ({\tilde {\mathbf H}} _ {\sigma s}) ^{ ' }\right) - E [ ({\tilde {\mathbf H}} _ {\sigma s} E [ ({\tilde {\mathbf H}} _ {\sigma s}) ^{ ' }\right) - \\ & - E [ ({\tilde {\mathbf H}} _ {\sigma s} E [ ({\tilde {\mathbf H}} _ {\sigma s}) ^{ ' }\right) - \\ & - E [ ({\tilde {\mathbf H}} _ {\sigma s} E [ ({\tilde {\mathbf H}} _ {\sigma s}) ^{ ' }\right) - \\ & - E [ ({\tilde {\mathbf C}} _ {-}\right) - \\ & - E [ ({\tilde {\mathcal C}} _ {-}\right) - \\ & - E [ ({\tilde {\mathcal C}} _ {-}\right) - \\ & - E [ ({\tilde {\mathcal C}} _ {-}\right) - \\ & - E [ ({\tilde {\mathcal C}} _ {-}\right) - \\ & - E [ ({\tilde {\mathcal C}} _ {-}\right) - \\ & - E [ ({\tilde {\mathcal C}} _ {-}\right) - \\ & - E [ ({\tilde {\mathcal C}} _ {-}\right) - \\ & - E [ ({\tilde {\mathcal C}} _ {-}\right) - \\ & - E [ ({\tilde {\mathcal C}} _ {-}\right) - \\ & - E [ ({\tilde {\mathcal C}} _ {-}\right) - \\ & . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , e n d i n g e n d i n g e n d i n g e n d i n g e n d i n g e n d i n g e n d i n g e n d i n g e n d i n g e n d i n g e n d i n g e n d i n g e n d i n g e n d i n g e n d i n g e n d i n g e n d i n g e n d e n d i n g e n d i n g e n d i n g e n d i n g e n d i n g e n d i n g e n d i n g e n d i n g e n d i n g e n d i n g e n d i n g e n d i n g e n d i n g e n d i n g e n d i n g e n d i n g e n d j o w a l y, b o u l l o w a l y, b o w a l y, b o w a l y, b o w a l y, b o w a l y, b o w a l y, b o w a l y, b o w a l y, b o w a l y, b o w a l y, b o w a l y, b o w a l y, b o w a l y, b o w a l y, b o w a l y, b o o w a l y, b o w a l y, b o w a l y, b o w a l y, b o w a l y, b o w a l y, b o w a l y, b o w a l y, b o w a l y, b o w a l y, b o w a l y, b o w a l y, b o w a l y, b o w a l y, b o u w a l y, b o w a l y , b o w a l y , b o w a l y , b o w a l y , b o w a l y , b o w a l y , b o w a l y , b o w a l y , b o w a l y , b o w a l y , b o w a l y , b o w a l y , b o w a l y , b o w a l y , b o w a l y , b O W A N D I N G R S U N D I N G R S U N D I N G R S U N D I N G R S U N D I N G R S U N D I N G R S U N D I N G R S U N D I N G R S U N D I N G R S U N D I N G R S U N D I N G R S U N D I N G R S U N D I N G R S U N D IN G R S U N D I N G R S U N D I N G R S U N D I N G R S U N D I N G R S U N D I N G R S U N D I N G R S U N D I N G R S U N D I N G R S U N D I N G R S U N D I N G R S U N D I N G R S U N D I N G R S I N D I N G R S I N D I N G R S I N D I N G R S I N D I N G R S I N D I N G R S I N D I N G R S I N D I N G R S I N D I N G R S I N D I N G R S I N D I N G R S I N D I N G R S I N D I N G R S I N D IN G R S I N D I N G R S I N D I N G R S I N D I N G R S I N D I N G R S I N D I N G R S I N D I N G R S I N D I N G R S I N D I N G R S I N D I N G R S I N D I N G R S I N D I N G R S I N D I N G R SI N D I N G R SI N D I N G R SI N DI N G R SI N DI N G R SI N DI N G R SI N DI N G R SI N DI N G R SI N DI N G R SI N DI N G R SI N DI N G R SI N DI N G R SI N DI N G R SI N DI N G R SI N DI N G R SI N DI N G R s T A P L A P L A P L A P L A P L A P L A P L A P L A P L A P L A P L A P L A P L A P L A P L A P L A P L A P L A P L A P L A P L A P L A P L A P L A P L A P L A P L A P L A P L A P L A P L A P L A P L A P L B O W A W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W WWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWW< fcel>E = E = E = E = E = E = E = E = E = E = E = E = E = E = E = E = E = E = E = E = E = E = E = E = E = E = E = E = E = E = E = E = E = E = E = E = E = E = E = E = E = E = E = E = E = E = E = E = E = E = E =E = F / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V /V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V / V /V/< ecel>< nl>\]

\[\begin{array} { r l } & { = E \left[ \mathbf { h _ { x } } \mathbf { x _ { t } ^ { s } } \left( ( \mathbf { x _ { t } ^ { s } } ) ^ { \prime } \mathbf { h _ { x } ^ { \prime } } + ( \mathbf { x _ { t } ^ { f } } \otimes \mathbf { x _ { t } ^ { f } } ) ^ { \prime } \tilde { \mathbf { H _ { x x } ^ { \prime } } } + \frac { 1 } { 2 } \mathbf { h _ { \sigma \sigma } ^ { \prime } } \sigma ^ { 2 } \right) \right] } \\ & { + E \left[ \tilde { \mathbf { H _ { x x } } } ( \mathbf { x _ { t } ^ { f } } \otimes \mathbf { x _ { t } ^ { f } } ) ( ( \mathbf { x _ { t } ^ { s } } ) ^ { \prime } \mathbf { h _ { x } ^ { \prime } } + ( \mathbf { x _ { t } ^ { f } } \otimes \mathbf { x _ { t } ^ { f } } ) ^ { \prime } \tilde { \mathbf { H _ { x x } ^ { \prime } } } + \frac 1 2 \mathbf { h _ { \sigma \sigma } ^ { \prime } } \sigma ^ { 2 } \right) \right] } \\ & { + E \left[ \frac 1 2 \mathbf { h _ { \sigma \sigma } } \sigma ^ { 2 } ( ( \mathbf { x _ { t } ^ { s } } ) ^ { \prime } \mathbf { h _ { x } ^ { \prime } } + ( \mathbf { x _ { t } ^ { f } } \otimes \mathbf { x _ { t } ^ { f } } ) ^ { \prime } \tilde { \mathbf { H _ { x x } ^ { \prime } } } + \tfrac 1 2 \mathbf { h _ { \sigma \sigma } ^ { \prime } } \sigma ^ { 2 } \right) \right] } \\ & = E [ [ \mathbf h _ { x } x _ { t } ^ { s } ( x _ { t } ^ { s }) ^ {\prime} h _ { x } ^ { \prime } + h _ { x } x _ { t } ^ { s ( x _ { t } ^ { f } \otimes x _ { t } ^ { f )} ) ^ {\prime} \tilde { H _ { x x } ^ { \prime } + h _ { x } x _ { t } ^ { s} \frac 1 2 h _ { \sigma \sigma} ^ { \prime} \sigma ^ { 2} ]}} \\ & + E [ [ \tilde { \mathbf { H _ { x x } } } ( x _ { t } ^ { f } \otimes x _ { t } ^ f ) ( x _ { t } ^ s ) ^ {\prime} h _ { x } ^ { \prime } + \tilde { H _ { x x } } ( x _ { t } ^ { f } \otimes x _ { t } ^ f ) ( x _ { t } ^ { f } \otimes x _ { t } ^ { f )} ) ^ {\prime} \tilde { H _ { x x } ^ { \prime } + \tilde { H _ { x x } ( x _ { t } ^ { f } \otimes x _ { t } ^ { f )} ) \frac 1 2 h _ { \sigma \sigma} ^ { \prime} \sigma ^ { 2} ]}} \\ & + E [ [ \frac 1 2 h _ { \sigma \sigma } \sigma ^ 2 ( x _ { t } ^ s ) ^ {\prime} h _ { x } ^ { \prime } + \frac 1 2 h _ { \sigma \sigma } \sigma ^ { 2 ( x _ { t } ^ { f } \otimes x _ { t } ^ { f )} ) ^ {\prime} \tilde { H _ { x x } ^ { \prime } + \frac 1 2 h _ { \sigma \sigma } \sigma ^ { 2} \frac 1 2 h _ { \sigma \sigma} ^ { \prime} \sigma ^ { 2} ]}} \\ & \\ & = [ [ {\bf h _ { x }} E [ ({\bf x _ { t} ^ { s }} ( {\bf x _ { t} ^ { s }}) ^ {\prime} ] {\bf h _ { x} ^ {\prime}} + {\bf h _ { x }} E [ {\bf x _ { t} ^ { s }} ( {\bf x _ { t} ^ { f }} \otimes {\bf x _ { t} ^ { f }}) ^ {\prime} ] [ [ {\bf H _ { x x} ^ {\prime}} + {\bf h _ { x }} E [ {\bf x _ { t} ^ { s }} ] [ [ {\bf h _ {\sigma \sigma} ^ {\prime} }\]

Letting

\[\begin{array}{r l} & {\mathrm{Letting}} \\ & {\mathbf {c} \equiv \mathbf {h} _ {\mathbf {x}} E \left[ \mathbf {x} _ {t} ^ {s} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \right] \tilde {\mathbf {H}} _ {\mathbf {x x}} ^ {\prime} + \mathbf {h} _ {\mathbf {x}} E \left[ \mathbf {x} _ {t} ^ {s} \right] \frac {1}{2} \mathbf {h} _ {\sigma \sigma} ^ {\prime} \sigma^ {2}} \\ & {\qquad + \tilde {\mathbf {H}} _ {\mathbf {x x}} E \left[ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) (\mathbf {x} _ {t} ^ {s}) ^ {\prime} \right] \mathbf {h} _ {\mathbf {x}} ^ {\prime} + \tilde {\mathbf {H}} _ {\mathbf {x x}} E \left[ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \right] \tilde {\mathbf {H}} _ {\mathbf {x x}} ^ {\prime} + \tilde {\mathbf {H}} _ {\mathbf {x x}} E \left[ \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \right] \frac {1}{2} \mathbf {h} _ {\sigma \sigma} ^ {\prime} \sigma^ {2}} \\ & {\qquad + \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2} E \left[ (\mathbf {x} _ {t} ^ {s}) ^ {\prime} \right] \mathbf {h} _ {\mathbf {x}} ^ {\prime} + \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2} E \left[ (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} \right] \tilde {\mathbf {H}} _ {\mathbf {x x}} ^ {\prime} + \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2} \frac {1}{2} \mathbf {h} _ {\sigma \sigma} ^ {\prime} \sigma^ {2}} \\ & {\mathrm{wethereforehave(duesto stationarity)}} \\ & {E \left[ \mathbf {x} _ {t + 1} ^ {s} (\mathbf {x} _ {t + 1} ^ {s}) ^ {\prime} \right] = \mathbf {h} _ {\mathbf {x}} E [ \mathbf {x} _ {t} ^ {s} (\mathbf {x} _ {t} ^ {s}) ^ {\prime} ] \mathbf {h} _ {\mathbf {x}} ^ {\prime} + \mathbf {c}} \\ & {\Updownarrow} \\ & {E [ \mathbf {x} _ {t} ^ {s} (\mathbf {x} _ {t} ^ {s}) ^ {\prime} ] = v e c (c) (\mathbf {I} _ {n _ {x} ^ {2}} - (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}})) ^ {- 1}} \end{array}\]

\[\begin{array}{r l} & {\mathbf {F i n d i n g} E \left[ \mathbf {x} _ {t} ^ {f} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \right]} \\ & {E \left[ \mathbf {x} _ {t + 1} ^ {f} \left(\mathbf {x} _ {t + 1} ^ {f} \otimes \mathbf {x} _ {t + 1} ^ {f}\right) ^ {\prime} \right]} \\ & {\qquad = E \left[ \left(\mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t} ^ {f} + \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1}\right) \left((\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) + \mathbf {v} (t + 1)\right) ^ {\prime} \right]} \\ & {\qquad = E \left[ \left(\mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t} ^ {f} + \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1}\right) \left(\left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime} + \mathbf {v} (t + 1) ^ {\prime}\right) \right]} \\ & {\qquad = E \left[ \mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t} ^ {f} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime} + \mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t} ^ {f} \mathbf {v} (t + 1) ^ {\prime} \right]} \\ & {\qquad + E \left[ \left(\sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime} + \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1} \mathbf {v} (t + 1) ^ {\prime}\right) \right]} \\ & {\mathrm{Recallthat} \mathbf {v} (t + 1) = (\mathbf {h} _ {\mathbf {x}} \otimes \sigma \pmb {\eta}) (\mathbf {x} _ {t} ^ {f} \otimes \pmb {\epsilon} _ {t + 1}) + (\sigma \pmb {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) (\pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}) + (\sigma \pmb {\eta} \otimes \pmb {\sigma} \pmb {\eta}) (\pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1}) \mathrm{soweget}} \end{array}\]

\[\begin{array} { r l } & = \mathbf { h } _ { \mathbf { x } } E \left[ \mathbf { x } _ { t } ^ { f } \left( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \right) ^ { \prime } \right] ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { \prime } + \mathbf { 0 } + E \left[ \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + 1 } ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \boldsymbol { \epsilon } _ { t + 1 } ) ^ { \prime } ( \sigma \boldsymbol { \eta } \otimes \sigma \boldsymbol { \eta } ) ^ { \prime } \right] \\ & = \mathbf { h } _ { \mathbf { x } } E \left[ \mathbf { x } _ { t } ^ { f } \left( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \right) ^ { \prime } \right] ( \mathbf { h } _ { \mathbf { x } } \otimes h _ { \mathbf { x } } ) ^ { \prime } + \mathbf { 0 } + \sigma \boldsymbol { \eta } E [ \boldsymbol { \epsilon } _ { t + 1 } ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \boldsymbol { \epsilon } _ { t + 1 } ) ^ { \prime } ] ( \sigma \boldsymbol { \eta } \otimes \sigma \boldsymbol { \eta } ) ^ { \prime } \\ & \\ & v e c ( E [ \mathbf { x } _ { t + 1 } ^ { f } ( \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } ) ^ { \prime } ] ) = ( h _ { x } \otimes h _ { x } \otimes h _ { x } ) v e c ( E [ \mathbf { x } _ { t } ^ { f } ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) ^ { \prime } ] ) \\ & + v e c ( \sigma \boldsymbol { \eta } E [ \boldsymbol { \epsilon } _ { t + 1 } ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \boldsymbol { \epsilon } _ { t + 1 } ) ^ { \prime } ] ( \sigma \boldsymbol { \eta } \otimes \sigma \boldsymbol { \eta } ) ^ { \prime }) \\ & b e c a u s e v e c ( A B C ) = ( C ^ { \prime } \otimes A ) v e c ( B ) \\ & \\ & ( E [ \mathbf { x } _ { t + 1 } ^ { f } ( \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } ) ^ { \prime } ] ) = ( I _ { n _ { x } ^ { 3 } } - ( h _ { x } \otimes h _ { x } \otimes h _ { x } ) ) ^ { - 1 } v e c ( \sigma \boldsymbol { \eta } E [ \boldsymbol { \epsilon } _ { t + 1 } ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \boldsymbol { \epsilon } _ { t + 1 } ) ^ { \prime } ] ( \sigma \boldsymbol { \eta } \otimes \sigma \boldsymbol { \eta } ) ^ { \prime }) \\ & \\ & F i n d i n g E [ x _ { t } ^ { f } ( x _ { t } ^ { s } ) ^ { \prime } ] \\ & E [ x _ { t + 1 } ^ { f } ( x _ { t + 1 } ^ { s } ) ^ { \prime } ] = E [ ( h _ { x } x _ { t } ^ { f } + \sigma \boldsymbol { \eta }\epsilon _ { t + 1} ) ( h _ { x } x _ { t } ^ { s } + {\tilde {\mathbf H}} _ {\mathbf x} (x _ { t } ^ { f} \otimes x _ { t } ^ { f }) + {\frac 12} h _ {\sigma\sigma}\sigma^2 ) ^ { ' } ] \\ & = E [ h _ { x} x _ { t} ^ { f} ((x _ { t } ^ { s }) ^ {\prime} h _ { x} ^ {\prime} + (x _ { t } ^ { f} \otimes x _ { t} ^ { f}) ^ {\prime} {\tilde {\mathbf H}} _ {\mathbf x} ^ {\prime} + {\frac 12} h _ {\sigma\sigma} ^ {\prime}\sigma^2 ) ] \\ & = h _ { x} E [ x _ { t } ^ { f} ( x _ { t } ^ { s }) ^ {\prime} ] h _ { x} ^ {\prime} + h _ { x} E [ x _ { t } ^ { f} ( x _ { t } ^ { f} \otimes x _ { t} ^ { f}) ^ {\prime} ] {\tilde {\mathbf H}} _ {\mathbf x} ^ {\prime} \\ & \\ & v e c ( E [ x _ { t + 1 } ^ { f } ( x _ { t + 1 } ^ { s }) ^ {\prime} ]) = ( h _ { x} \otimes h _ { x} ) v e c ( E [ x _ { t } ^ { f } ( x _ { t } ^ { s }) ^ {\prime} ]) + v e c ( h _ { x} E [ x _ { t } ^ { f} ( x _ { t } ^ { f} \otimes x _ { t} ^ { f}) ^ {\prime} ] {\tilde {\mathbf H}} _ {\mathbf x} ^ {\prime} ) \\ & \\ & v e c ( E [ x _ { t } ^ { f} ( x _ { t } ^ { s }) ^ {\prime} ]) = ( I _ { n _ { x } ^ { 2 }} - ( h _ {\mathbf x} \otimes h _ {\mathbf x})) ^ {- 1} v e c ( h _ {\mathbf x} E [ x _ { t } ^ { f} ( x _ { t } ^ { f} \otimes x _ { t} ^ { f}) ^ {\prime} ] {\tilde {\mathbf H}} _ {\mathbf x} ^ {\prime} ) \\ & \\ & v e c ( E [ x _ {{t}} ^ {{f}} ( x {{t}} ^ {{s}}) ] ) = V e c ( H e q u a l o w i n e r e d i s i o n e d i s i o n e d i s i o n e d i s i o n e d i s i o n e d i s i o n e d i s i o n e d i s i o n e d i s i o n e d i s i o n e d i s i o n e d i s i o n e d i s i o n e d i s i o n e d i s i o n e n o u p l y , m a l l o w i n e r e d i s i o n e d i s i o n e d i s i o n e d i s i o n e d i s i o n e d i s i o n e d i s i o n e d i s i o n e d i s i o n e d i s i o n e d i s i o n e d i s i o n e d i s i o n e n o u p l y , M a l l o w i n e r e d i s i o n e d i s i o n e d i s i o n e d i s i o n e d i s i o n e d i s i o n e d i s i o n e d i s i o n e d i s i o n e d i s i o n e d i s i o n e d i s i o n e d i s i o l o w i n e r e d i s i o n e d i s i o n e d i s i o n e d i s i o n e d i s i o n e d i s i o n e d i s i o n e d i s i o n e d i s i o n e d i s i o n e d i s i o n e d i s i o n e d i s i o n e r e d i s i o n e d j u a l l o w i n e r e d j u a l l o w i n e r e d j u a l l o w i n e r e d j u a l l o w i n e r e d j u a l l o w i n e r e d j u a l l o w i n e r e d j u a l l o w i n e r e d j u a l l o w i n e r e d j u a l l o W a l l o w i n e r e d j u a l l o W a l l o w i n e r e d j u a l l o W a l l o w i n e r e d j u a l l o W a l l o w i n e r e d j u a l l o W a l l o w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w . \\ & \\ & v e c ( E [ X ] - V) = V. \\ & = V. \\ & = V. \\ & = V. \\ & = V. \\ & = V. \\ & = V. \\ & = V. \\ & = V. \\ & = V. \\ & = V. \\ & = V. \\ & = V. \\ & = V. \\ & = V. \\ & = V. \\ & = V. \\ & = V. \\ & = V. \\ & = V. \\ & = V. \\ * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * \\ & = V. \\ & = V. \\ & = V. \\ & = V. \\ & = V. \\ & = V. \\ & = V. \\ & = V. \\ & = V. \\ & = V. \\ & = V. \\ & = V. \\ & = V. \\ & = V. \\ & = V. \\ & = V. \\ & = V. \\ & = V. \\ & = V. \\ & = V . \\ & = V . \\ & = V . \\ & = V . \\ & = V . \\ & = V . \\ & = V . \\ & = V . \\ & = V . \\ & = V . \\ & = V . \\ & = V . \\ & = V . \\ & = V . \\ & = V . \\ & = V . \\ & = V . \\ & = V . \\ & = V . \\ & = V . \\ & = V .. \\ & = V . \\ & = V . \\ & = V . \\ & = V . \\ & = V . \\ & = V . \\ & = V . \\ & = V . \\ & = V . \\ & = V . \\ & = V . \\ & = V . \\ & = V . \\ & = V . \\ & = V . \\ & = V . \\ & = V . \\ & = V . \\ & = V . \\ & = V.. \\ & = V . \\ & = V . \\ & = V . \\ & = V . \\ & = V . \\ & = V . \\ & = V . \\ & = V . \\ & = V . \\ & = V . \\ & = V . \\ & = V . \\ & = V . \\ & = V . \\ & = V . \\ & = V . \\ & = V . \\ & = V . \\ & = V . \\ & = V . \end{array}\]

3.5 Decomposition: The total variance of the state variables

The previous subsection have computed and . We next discuss how the total variance of the state variables should be computed. Starting from our decomposition, we have

\[\mathbf {x} _ {t} = \mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s}\]

Hence, it is natural to compute moments of based on this decomposition. Hence, for the variance we get

\[\begin{array}{r c l} V a r (\mathbf {x} _ {t}) & = & V a r (\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s}) \\ & = & V a r (\mathbf {x} _ {t} ^ {f}) + V a r (\mathbf {x} _ {t} ^ {s}) + C o v (\mathbf {x} _ {t} ^ {f}, \mathbf {x} _ {t} ^ {s}) + C o v (\mathbf {x} _ {t} ^ {s}, \mathbf {x} _ {t} ^ {f}) \end{array}\]

Note that this procedure is fully consistent with the one adopted for the control variables. To realize that, let us consider the case where one element in simply reproduces one state variable. Then except if we want to have the k'th control variable to reproduce the j'th state variable, while all remaining derivatives of g are zero. Hence, we have

\[y (k) _ {t} = \mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s}.\]

Hence, .

One potential short-coming of this procedure for computing the moments of might be that we do not include all higher order terms. In the case of the variance, one could believe that we omit the fourth order term in our expression of . However, this is not the case. To realize this, recall

\[\mathbf {x} _ {t + 1} ^ {f} = \mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t} ^ {f} + \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1}\]

\[\mathbf {x} _ {t + 1} ^ {s} = \mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t} ^ {s} + \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) + \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}\]

The law of motion for the total state variable is

\[\begin{array}{r l} & {\mathbf {x} _ {t + 1} ^ {f} + \mathbf {x} _ {t + 1} ^ {s}} \\ {=} & {\mathbf {h} _ {\mathbf {x}} \left(\mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {f}\right) + \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) + \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2} + \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1}} \end{array}\]

We compute second moments of based on the first line in this expression. But, of course, we could equally well have computed the moments based on the second line in this expression. Adopting this alternative approach, we obtain

\[\begin{array}{r c l} V a r \left(\mathbf {x} _ {t + 1} ^ {f} + \mathbf {x} _ {t + 1} ^ {s}\right) & = & \mathbf {h} _ {\mathbf {x}} V a r \left(\mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {f}\right) \mathbf {h} _ {\mathbf {x}} + \frac {1}{4} \mathbf {H} _ {\mathbf {x x}} V a r \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) \mathbf {H} _ {\mathbf {x x}} ^ {\prime} + \sigma^ {2} \pmb {\eta} \pmb {\eta} ^ {\prime} \\ & & + \frac {1}{2} \mathbf {h} _ {\mathbf {x}} C o v \left(\mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {f}, \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) \mathbf {H} _ {\mathbf {x x}} ^ {\prime} \\ & & + \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} C o v \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}, \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {f}\right) \mathbf {h} _ {\mathbf {x}} ^ {\prime} \end{array}\]

It is evident that this expression for includes the term . Using this expression to solve directly for gives exactly the same expression for as the one we first suggested (which we obtain without solving more equations!)

3.6 The auto-correlations

This section derives the auto-correlations for the states and the control variables.

3.6.1 The innovations

We start by showing that and are uncorrelated for . To see this note that

\[E \left[ \left[ \begin{array}{c} \pmb {\epsilon} _ {t + 1} \\ \pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1} - v e c (\mathbf {I} _ {n _ {e}}) \\ \pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \\ \mathbf {x} _ {t} ^ {f} \otimes \pmb {\epsilon} _ {t + 1} \end{array} \right] \begin{array}{c c c c} \pmb {\epsilon} _ {t + 1 + s} ^ {\prime} & \left(\pmb {\epsilon} _ {t + 1 + s} \otimes \pmb {\epsilon} _ {t + 1 + s} - v e c (\mathbf {I} _ {n _ {e}})\right) ^ {\prime} & \left(\pmb {\epsilon} _ {t + 1 + s} \otimes \mathbf {x} _ {t + s} ^ {f}\right) ^ {\prime} & \left(\mathbf {x} _ {t + s} ^ {f} \otimes \pmb {\epsilon} _ {t + 1 + s}\right) ^ {\prime} \\ \end{array} \right]\]

\[= E \left[ \begin{array}{c c} \boldsymbol {\epsilon} _ {t + 1} \boldsymbol {\epsilon} _ {t + 1 + s} ^ {\prime} & \boldsymbol {\epsilon} _ {t + 1} \left(\boldsymbol {\epsilon} _ {t + 1 + s} \otimes \boldsymbol {\epsilon} _ {t + 1 + s} - v e c \left(\mathbf {I} _ {n _ {e}}\right)\right) ^ {\prime} \\ \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} - v e c \left(\mathbf {I} _ {n _ {e}}\right)\right) \boldsymbol {\epsilon} _ {t + 1 + s} ^ {\prime} & \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} - v e c \left(\mathbf {I} _ {n _ {e}}\right)\right) \left(\boldsymbol {\epsilon} _ {t + 1 + s} \otimes \boldsymbol {\epsilon} _ {t + 1 + s} - v e c \left(\mathbf {I} _ {n _ {e}}\right)\right) ^ {\prime} \\ \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) \boldsymbol {\epsilon} _ {t + 1 + s} ^ {\prime} & \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\boldsymbol {\epsilon} _ {t + 1 + s} \otimes \boldsymbol {\epsilon} _ {t + 1 + s} - v e c \left(\mathbf {I} _ {n _ {e}}\right)\right) ^ {\prime} \\ \left(\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) \boldsymbol {\epsilon} _ {t + 1 + s} ^ {\prime} & \left(\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) \left(\boldsymbol {\epsilon} _ {t + 1 + s} \otimes \boldsymbol {\epsilon} _ {t + 1 + s} - v e c \left(\mathbf {I} _ {n _ {e}}\right)\right) ^ {\prime} \end{array} \right.\]

\[\left. \begin{array}{c c} \pmb {\epsilon} _ {t + 1} \left(\pmb {\epsilon} _ {t + 1 + s} \otimes \mathbf {x} _ {t + s} ^ {f}\right) ^ {\prime} & \pmb {\epsilon} _ {t + 1} \left(\mathbf {x} _ {t + s} ^ {f} \otimes \pmb {\epsilon} _ {t + 1 + s}\right) ^ {\prime} \\ \left(\pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1} - v e c \left(\mathbf {I} _ {n _ {e}}\right)\right) \left(\pmb {\epsilon} _ {t + 1 + s} \otimes \mathbf {x} _ {t + s} ^ {f}\right) ^ {\prime} & \left(\pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1} - v e c \left(\mathbf {I} _ {n _ {e}}\right)\right) \left(\mathbf {x} _ {t + s} ^ {f} \otimes \pmb {\epsilon} _ {t + 1 + s}\right) ^ {\prime} \\ \left(\pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\pmb {\epsilon} _ {t + 1 + s} \otimes \mathbf {x} _ {t + s} ^ {f}\right) ^ {\prime} & \left(\pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\mathbf {x} _ {t + s} ^ {f} \otimes \pmb {\epsilon} _ {t + 1 + s}\right) ^ {\prime} \\ \left(\mathbf {x} _ {t} ^ {f} \otimes \pmb {\epsilon} _ {t + 1}\right) \left(\pmb {\epsilon} _ {t + 1 + s} \otimes \mathbf {x} _ {t + s} ^ {f}\right) ^ {\prime} & \left(\mathbf {x} _ {t} ^ {f} \otimes \pmb {\epsilon} _ {t + 1}\right) \left(\mathbf {x} _ {t + s} ^ {f} \otimes \pmb {\epsilon} _ {t + 1 + s}\right) ^ {\prime} \end{array} \right]\]

\[= \left[ \begin{array}{c c c c} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array} \right]\]

3.6.2 The auto-covariances

Recall that we have

\[\begin{array}{r l r} & & {\mathbf {z} _ {t} = \left[ \begin{array}{c} \mathbf {x} _ {t} ^ {f} \\ \mathbf {x} _ {t} ^ {s} \\ \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \end{array} \right]} \\ & & {\mathbf {z} _ {t + 1} = \mathbf {c} + \mathbf {A} \mathbf {z} _ {t} + \mathbf {B} \pmb {\xi} _ {t + 1}} \\ & & {\mathbf {y} _ {t} ^ {s} = \mathbf {D} \mathbf {z} _ {t} + \frac {1}{2} \mathbf {g} _ {\sigma \sigma} \sigma^ {2}} \end{array}\]

To find the one period auto-correlation, i.e. , we have

\[C o v \left(\mathbf {z} _ {t + 1}, \mathbf {z} _ {t}\right) = C o v \left(\mathbf {c} + \mathbf {A} \mathbf {z} _ {t} + \mathbf {B} \boldsymbol {\xi} _ {t + 1}, \mathbf {z} _ {t}\right) = \mathbf {A} C o v \left(\mathbf {z} _ {t}, \mathbf {z} _ {t}\right) = \mathbf {A} V a r \left(\mathbf {z} _ {t}\right)\]

because as shown above. And for two periods

\[\begin{array}{r l} & {C o v \left(\mathbf {z} _ {t + 2}, \mathbf {z} _ {t}\right) = C o v \left(\mathbf {c} + \mathbf {A} \mathbf {z} _ {t + 1} + \mathbf {B} \pmb {\xi} _ {t + 2}, \mathbf {z} _ {t}\right)} \\ & {\quad = C o v \left(\mathbf {A} \left(\mathbf {c} + \mathbf {A} \mathbf {z} _ {t} + \mathbf {B} \pmb {\xi} _ {t + 1}\right) + \mathbf {B} \pmb {\xi} _ {t + 2}, \mathbf {z} _ {t}\right)} \\ & {\quad = C o v \left(\mathbf {A} ^ {2} \mathbf {z} _ {t} + \mathbf {B} \pmb {\xi} _ {t + 2}, \mathbf {z} _ {t}\right)} \\ & {\quad = C o v \left(\mathbf {A} ^ {2} \mathbf {z} _ {t}, \mathbf {z} _ {t}\right)} \\ & {\quad = \mathbf {A} ^ {2} C o v \left(\mathbf {z} _ {t}, \mathbf {z} _ {t}\right)} \\ & {\quad = \mathbf {A} ^ {2} V a r \left(\mathbf {z} _ {t}\right)} \end{array}\]

Here, we use the fact that . This follows from the same arguments as above, that is consider

\[\begin{array}{r l} & E \left[ \mathbf {z} _ {t} \pmb {\xi} _ {t + 2} ^ {\prime} \right] = E \left[ \left[ \begin{array}{c} \mathbf {x} _ {t} ^ {f} \\ \mathbf {x} _ {t} ^ {s} \\ \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \end{array} \right] \left[ \begin{array}{c c c} \pmb {\epsilon} _ {t + 2} ^ {\prime} & (\pmb {\epsilon} _ {t + 2} \otimes \pmb {\epsilon} _ {t + 2} - v e c (\mathbf {I} _ {n e})) ^ {\prime} & (\pmb {\epsilon} _ {t + 2} \otimes \mathbf {x} _ {t + 1} ^ {f}) ^ {\prime} & (\mathbf {x} _ {t + 1} ^ {f} \otimes \pmb {\epsilon} _ {t + 2}) ^ {\prime} \end{array} \right] \right] \\ & = E \left[ \begin{array}{c c c c} \mathbf {x} _ {t} ^ {f} \pmb {\epsilon} _ {t + 2} ^ {\prime} & \mathbf {x} _ {t} ^ {f} (\pmb {\epsilon} _ {t + 2} \otimes \pmb {\epsilon} _ {t + 2} - v e c (\mathbf {I} _ {n e})) ^ {\prime} & \mathbf {x} _ {t} ^ {f} (\pmb {\epsilon} _ {t + 2} \otimes \mathbf {x} _ {t + 1} ^ {f}) ^ {\prime} & \mathbf {x} _ {t} ^ {f} (\mathbf {x} _ {t + 1} ^ {f} \otimes \pmb {\epsilon} _ {t + 2}) ^ {\prime} \\ \mathbf {x} _ {t} ^ {s} \pmb {\epsilon} _ {t + 2} ^ {\prime} & \mathbf {x} _ {t} ^ {s} (\pmb {\epsilon} _ {t + 2} \otimes \pmb {\epsilon} _ {t + 2} - v e c (\mathbf {I} _ {n e})) ^ {\prime} & \mathbf {x} _ {t} ^ {s} (\pmb {\epsilon} _ {t + 2} \otimes \mathbf {x} _ {t + 1} ^ {f}) ^ {\prime} & \mathbf {x} _ {t} ^ {s} (\mathbf {x} _ {t + 1} ^ {f} \otimes \pmb {\epsilon} _ {t + 2}) ^ {\prime} \\ (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) \pmb {\epsilon} _ {t + 1} ^ {\prime} & (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) (\pmb {\epsilon} _ {t + 2} \otimes \pmb {\epsilon} _ {t + 2} - v e c (\mathbf {I} _ {n e})) ^ {\prime} & (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) (\pmb {\epsilon} _ {t + 2} \otimes \mathbf {x} _ {t + 1} ^ {f}) ^ {\prime} & (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) (\mathbf {x} _ {t + 1} ^ {f} \otimes \pmb {\epsilon_ {t + 2}}) ^ {\prime} \end{array} \right] \end{array}\]

Hence, in the general case

For the control variables:

\[\begin{array}{r l} & C o v \left(\mathbf {y} _ {t + l} ^ {s}, \mathbf {y} _ {t} ^ {s}\right) = C o v \left(\mathbf {D z} _ {t + l} + \frac {1}{2} \mathbf {g} _ {\sigma \sigma} \sigma^ {2}, \mathbf {D z} _ {t} + \frac {1}{2} \mathbf {g} _ {\sigma \sigma} \sigma^ {2}\right) \\ & \qquad = C o v \left(\mathbf {D z} _ {t + l}, \mathbf {D z} _ {t}\right) \\ & \qquad = \mathbf {D C o v} \left(\mathbf {z} _ {t + l}, \mathbf {z} _ {t}\right) \mathbf {D ^ {\prime}} \\ & \qquad = \mathbf {D A ^ {l}} V a r \left(\mathbf {z} _ {t}\right) \mathbf {D ^ {\prime}} \end{array}\]

4 Stastical properties: Third order approximation

4.1 Covariance-stationary

Proposition 1:

The pruned third order approximation for , and is covariance-stationary if

1. the DSGE model has a unique stable equilibrium, i.e. all eigenvalue of have modulus less than 1

2. has finite sixth moment

\[\begin{array} { r l } & { \text {Proof} } \\ & { \text {Note first that} } \\ & { x _ { t + 1 } ^ { r d } ( j , 1 ) = \mathbf { h _ { x } } ( j , : ) \mathbf { x } _ { t } ^ { r d } + \frac { 2 } { 2 } \left( \mathbf { x } _ { t } ^ { f } \right) ^ { \prime } \mathbf { h _ { x x } } ( j , : , : ) \left( \mathbf { x } _ { t } ^ { s } \right) } \\ & { \qquad + \frac { 1 } { 6 } \left( \mathbf { x } _ { t } ^ { f } \right) ^ { \prime } \left[ \begin{array} { c } \left( \mathbf { x } _ { t } ^ { f } \right) ^ { \prime } \mathbf { h _ { x x x } } ( j , 1 , : , : ) \left( \mathbf { x } _ { t } ^ { f } \right) \\ \cdots \\ \left( \mathbf { x } _ { t } ^ { f } \right) ^ { \prime } \mathbf { h _ { x x x } } ( j , n _ { x } , : , : ) \left( \mathbf { x } _ { t } ^ { f } \right) \end{array} \right] + \frac { 3 } { 6 } \mathbf { h _ { \sigma \sigma x } } ( j , : ) \sigma ^ { 2 } \mathbf { x } _ { t } ^ { f } + \frac { 1 } { 6 } h _ { \sigma \sigma \sigma } ( j , 1 ) \sigma ^ { 3 } } \\ & { \Updownarrow } \\ & { \mathbf { x } _ { t + 1 } ^ { r d } = \mathbf { h _ { x } x _ { t } ^ { r d }} + 2 \tilde { \mathbf { H _ { x x } } } \left( \mathbf { x _ { t } ^ { f } \otimes x _ { t } ^ { s } } \right) + \tilde { \mathbf { H _ { x x x } } } \left( \mathbf { x _ { t } ^ { f } \otimes x _ { t } ^ { f } \otimes x _ { t } ^ { f } } \right) + \frac { 3 } { 6 } \mathbf { h _ { \sigma \sigma x } \sigma ^ { 2 } x _ { t } ^ { f } + \frac { 1 } { 6 } h _ { \sigma \sigma \sigma } \sigma ^ { 3 } }} \\ & \text {where} \tilde { \mathbf { H _ { x x } } } \equiv \frac { 1 } { 2 } r e s h a p e ( h _ { x x } , n _ { x } , n _ { x } ^ { 2 } ) \text {and} \tilde { \mathbf { H _ { x x x } } } \equiv \frac { 1 } { 6 } r e s h a p e ( h _ { x x x } , n _ { x } , n _ { x } ^ { 3 } ) . \text{So we need} \\ & { \left( \mathbf { x _ { t } ^ { f } \otimes x _ { t } ^ { s } } \right) \text {and} \left( \mathbf { x _ { t } ^ { f } \otimes x _ { t } ^ { f } \otimes x _ { t } ^ { f} }\right) . } \\ & \\ & {\mathrm{Hence,}} \\ & \left( \mathbf { x _ { t + 1 } ^ { f } \otimes x _ { t + 1 } ^ { s } } \right) = \left( h _ { x } x _ { t } ^ { f } + \sigma \eta \epsilon _ { t + 1 } \right) \otimes \left( h _ { x } x _ { t } ^ { s } + \tilde { H _ { x x } } ( x _ { t } ^ { f } \otimes x _ { t } ^ { f }) + \frac { 1 } { 2 } h _ { \sigma \sigma } \sigma ^ { 2 }\right) \\ & \\ & = h _ { x } x _ { t } ^ { f } \otimes h _ { x } x _ { t } ^ { s } + h _ { x } x _ { t } ^ { f } \otimes \tilde { H _ { x x } } ( x _ { t } ^ { f } \otimes x _ { t } ^ { f }) + h _ { x } x _ { t } ^ { f } \otimes \frac 1 2 h _ { \sigma \sigma } \sigma ^ { 2 } \\ & + \sigma \eta \epsilon _ { t + 1 } \otimes h _ { x } x _ { t } ^ { s } + \sigma \eta \epsilon _ { t + 1 } \otimes \tilde { H _ { x x } } ( x _ { t } ^ { f } \otimes x _ { t } ^ { f }) + \sigma \eta \epsilon _ { t + 1 } \otimes \frac 1 2 h _ { \sigma \sigma } \sigma ^ { 2 } \\ & \\ & {\mathrm{using} ( A + B ) \otimes ( C + D ) = A \otimes C + A \otimes D + B \otimes C + B \otimes D} \\ & \\ & = ( h _ { x } \otimes h _ { x ) } ) ( x _ { t } ^ { f } \otimes x _ { t } ^ { s }) + ( h _ { x } \otimes \tilde H _ { x x} ) ) ( x _ { t } ^ { f } \otimes x _ { t } ^ { f } \otimes x _ { t } ^ { f }) + ( h _ { x } \otimes \frac 1 2 h _ { \sigma \sigma} ) ( x _ { t } ^ { f } \otimes \sigma ^ { 2 }) \\ & + ( \sigma \eta \otimes h _ x ) ) ( e _ { t + 1 } \otimes x _ { t } ^ { s )} + ( \sigma \eta \otimes \tilde H _ { x x} ) ) ( e _ { t + 1} \otimes x _ { t } ^ { f } \otimes x _ { t } ^ { f )} + ( \sigma \eta \otimes {\frac 1 2} h _ { \sigma \sigma} ) ( e _ { t + 1} \otimes \sigma ^ { 2 }) \\ & {\mathrm{using} ( A ⓤ B ) ( C ⓤ D ) = A C ⓤ B D} \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & ; i n g e m a l o w i n e r e - i n g e m a l o w i n e r e - i n g e m a l o w i n e r e - i n g e m a l o w i n e r e - i n g e m a l o w i n e r e - i n g e m a l o w i n e r e - i n g e m a l o w i n e r e - i n g e m a l o w - i n g e m a l o w - i n g e m a l o w - i n g e m a l o w - i n g e m a l o w - i n g e m a l o w - i n g e m a l o w - i n g e m a l o w - i n g e m a l o w - i n g e m a l o w - i n g e m a l o W - i n g e m a l o W - i n g e m a l o W - i n g e m a l o W - i n g e m a l o W - i n g e m a l o W - i n g e m a l o W - i n g e m a l o W - i n g e m a l o W - i n g e m a l o W - i n g e m a l o M - i n g e m a l o M - i n g e m a l o M - i n g e m a l o M - i n g e m a l o M - i n g e m a l o M - i n g e m a l o M - i n g e m a l o M - i n g e m a l o M - i n g e m a l o M - i n g e m a l o N - i n g e m a l o N - i n g e m a l o N - i n g e m a l o N - i n g e m a l o N - i n g e m a l o N - i n g e m a l o N - i n g e m a l o N - i n g e m a l o N - i n g e m a l o N - i n g e m a l o M - i n g e m a l o M - i n g e m a l o M - i n g e m a l o M - i n g e m a l o M - i n g e m a l o M - i n g e m a l o M - i n g e m a l o M - i n g e m a l o M - i n g e m a l o K - i n g e m a l o K - i n g e m a l o K - i n g e m a l o K - i n g e m a l o K - i n g e m a l o K - i n g e m a l o K - i n g e m a l o K - i n g e m a l o K - i n g e m a l o K - i n g e m a l o M - i n g e m a l o M - i n g e m a l o M - i n g e m a l o M - i n g e m a l o M - i n g e m a l o M - i n g e m a l o M - i n g e m a l o M - i n g e m a l o M - i n g e m a l o L - i n g e m a l o L - i n g e m a l o L - i n g e m a l o L - i n g e m a l o L - i n g e m a l o L - i n g e m a l o L - i n g e m a l o L - i n g e m a l o L - i n g e m a l o L - i n g e m a l o M - i n g e m a l o M - i n g e m a l o M - i n g e m a l o M - i n g e m a l o M - i n g e m a l o M - i n g e m a l o M - i n g e m a l o M - i n g e m a l o M - i n g e m a l oM - i n g e m a l o M - i n g e m a l o M - i n g e m a l o M - i n g e m a l o M - i n g e m a l o M - i n g e m a l o M - i n g e m a l o M - i n g e m a l o M - i n g e m a l o M - i n g e m a l o S - i n g e m a l o S - i n g e m a l o S - i n g e m a l o S - i n g e m a l o S - i n g e m a l o S - i n g e m a l o S - i n g e m a l o S - i n g e m a l o S - i n g e m a l o S - i n g e s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s [ ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .\]

\[\begin{array}{l} = (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {s}) + (\mathbf {h} _ {\mathbf {x}} \otimes \tilde {\mathbf {H}} _ {\mathbf {x x}}) (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) + (\mathbf {h} _ {\mathbf {x}} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}) \mathbf {x} _ {t} ^ {f} \\ + (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) (\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {s}) + (\sigma \boldsymbol {\eta} \otimes \tilde {\mathbf {H}} _ {\mathbf {x x}}) (\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) + (\sigma \boldsymbol {\eta} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}) \boldsymbol {\epsilon} _ {t + 1} \end{array}\]

Recall from above

\[\begin{array} { r l } & { \mathrm{Recall~from~above} } \\ & { \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } = ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) + ( \mathbf { h } _ { \mathbf { x } } \otimes \sigma \boldsymbol { \eta } ) ( \mathbf { x } _ { t } ^ { f } \otimes \boldsymbol { \epsilon } _ { t + 1 } ) } \\ & { \qquad + ( \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x } } ) ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } ) + ( \sigma \boldsymbol { \eta } \otimes \sigma \boldsymbol { \eta } ) ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \boldsymbol { \epsilon } _ { t + 1 } ) } \\ & { \mathrm{So} } \\ & { \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } = ( \mathbf { h } _ { \mathbf { x } } \mathbf { x } _ { t } ^ { f } + \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + 1 } ) \otimes ( ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) + ( \mathbf { h } _ { \mathbf { x } } \otimes \sigma \boldsymbol { \eta } ) ( x _ { t } ^ { f } \otimes \epsilon _ { t + 1 } ) } \\ & { \qquad + ( \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x } } ) ( \boldsymbol { \epsilon } _ { t + 1 } \otimes x _ { t } ^ { f } ) + ( \sigma \boldsymbol { \eta } \otimes \sigma \boldsymbol { \eta} ) ( \epsilon _ { t + 1 } \otimes \epsilon _ { t + 1} ) ) } \\ & = [ h _ { x } x ] ^ { f } ( h _ { x } \otimes h _ { x }) ( x _ { t } ^ { f } \otimes x _ { t } ^ { f }) + h _ { x } x _ { t } ^ { f } ( h _ { x } \otimes c o n s) ( x _ { t } ^ { f } \otimes c o n s ) \\ & + [ h _ { x } x ] ^ { f } ( c o n s) ( c o n s) ( c o n s) ( c o n s) ( c o n s) ( c o n s) ( c o n s) ( c o n s) ( c o n s) ( c o n s) ( c o n s) ( c o n s) ( c o n s) ( c o n s) ( c o n s) ( c o n s) ( c o n s) ( c o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - [ c o n s ] ^ { f } , [ c o n s ] ^ { f }, [ c o n s ] ^ { f }, [ c o n s ] ^ { f }, [ c o n s ] ^ { f }, [ c o n s ] ^ { f }, [ c o n s ] ^ { f }, [ c o n s ] ^ { f }, [ c o n s ] ^ { f }, [ c o n s ] ^ { f }, [ c o n s ] ^ { f }, [ c o n s ] ^ { f }. , [ c o n s ] ^ { f }, [ c o n s ] ^ { f }, [ c o n s ] ^ { f }, [ c o n s ] ^ { f }, [ c o n s ] ^ { f }, [ c o n s ] ^ { f }, [ c o n s ] ^ { f }, [ c o n s ] ^ { f }, [ c o n s ] ^ { f }, [ c o n s ] ^ {f }, [ c o n s ] ^ { f }, [ c o n s ] ^ { f }, [ c o n s ] ^ { f }, [ c o n s ] ^ { f }, [ c o n s ] ^ { f }, [ c o n s ] ^ { f }, [ c o n s ] ^ { f }, [ c o n s ] ^ { f }, [ c o n s ] ^ { f }, [ c o n s ] ^ {f }. , [ c o n s ] ^ {f }. , [ c o n s ] ^ {f }. , [ c o n s ] ^ {f }. , [ c o n s ] ^ {f }. , [ c o n s ] ^ {f }. , [ c o n s ] ^ {f }. , [ c o n s ] ^ {f}. , [ c o n s ] ^ {f}. , [ c o n s ] ^ {f}. , [ c o n s ] ^ {f}. , [ c o n s ] ^ {f}. , [ c o n s ] ^ {f}. , [ c o n s ] ^ {f}. , [ c o n s ] ^ {f}. , [ c o n s ] ^ {f}. , [ c o n s ] ^ {f}. | , | , | | , | , | | , | , | | , | , | | , | , | | , | , | | , | , | | , | , | | , | , | | , | , | | , | , | | , | , | | , | , | | , | , | | , | , | | , | , | | , | , | | , | , | | , | , | | , | , | | , | , | | & {\cal O} = (\lambda_ {\mathrm{eff}}) (\lambda_ {\mathrm{eff}}) (\lambda_ {\mathrm{eff}}) (\lambda_ {\mathrm{eff}}) (\lambda_ {\mathrm{eff}}) (\lambda_ {\mathrm{eff}}) (\lambda_ {\mathrm{eff}}) (\lambda_ {\mathrm{eff}}) (\lambda_ {\mathrm{eff}}) (\lambda_ {\mathrm{eff}}) (\lambda_ {\mathrm{eff}}) (\lambda_ {\mathrm{eff}}) (l a m e g a l i v e r e d i a l i v e r e d i a l i v e r e d i a l i v e r e d i a l i v e r e d i a l i v e r e d i a l i v e r e d i a l i v e r e d i a l i v e r e d i a l i v e r e d i a l i v e r e d i a l i v e r e d i b j k u a l l i k u a l l i k u a l l i k u a l l i k u a l l i k u a l l i k u a l l i k u a l l i k u a l l i k u a l l i k u a l l i k u a l l i k u a l l i k u a l l i k u a l l i k u a l l i k u a l l i k u a / p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w & \\ & = (\lambda_ {\mathrm{eff}}) (\lambda_ {\mathrm{eff}}) (\lambda_ {\mathrm{eff}}) (\lambda_ {\mathrm{eff}}) (\lambda_ {\mathrm{eff}}) (\lambda_ {\mathrm{eff}}) (\lambda_ {\mathrm{eff}}) (\lambda_ {\mathrm{eff}}) (\lambda_ {\mathrm{eff}}) (\lambda_ {\mathrm{eff}}) (\lambda_ {\mathrm{eff}}), \\ & = (\lambda_ {\mathrm{eff}}) (\lambda_ {\mathrm{eff}}) (\lambda_ {\mathrm{eff}}) (\lambda_ {\mathrm{eff}}) (\lambda_ {\mathrm{eff}}) (\lambda_ {\mathrm{eff}}) (\lambda_ {\mathrm{eff}}) (\lambda_ {\mathrm{eff}}), \\ & = (\lambda_ {\mathrm{eff}}) (\lambda_ {\mathrm{eff}}) (\lambda_ {\mathrm{aff}}) (\lambda_ {\mathrm{aff}}) (\lambda_ {\mathrm{aff}}) (\lambda_ {\mathrm{aff}}), \\ & = (\lambda_ {\mathrm{eff}}) (\lambda_ {\mathrm{aff}}) (\lambda_ {\mathrm{aff}}) (\lambda_ {\mathrm{aff}}) (\lambda_ {\mathrm{aff}}), \\ & = (\lambda_ {\mathrm{eff}}) (\lambda_ {\mathrm{aff}}) (\lambda_ {\mathrm{aff}}) (\lambda_ {\mathrm{aff}}), \\ & = (\lambda_ {\mathrm{eff}}) (\lambda_ {\mathrm{aff}}) (\lambda_ {\mathrm{aff}}) (\lambda_ {\mathrm{aff}}), \\ & = (\lambda_ {\mathrm{eff}}) (\lambda_ {\mathrm{aff}}) (\lambda_ {\mathrm{aff}}), \\ & = (\lambda_ {\mathrm{eff}}) (\lambda_ {\mathrm{aff}}) (\lambda_ {\mathrm{aff}},). \\ & = (\lambda_ {\mathrm{eff}}) (\lambda_ {\mathrm{aff}},). \\ & = (\lambda_ {\mathrm{eff}}) (\lambda_ {\mathrm{aff}},). \\ & = (\lambda_ {\mathrm{eff}}) (\lambda_ {\mathrm{aff}},). \\ & = (\lambda_ {\mathrm{eff}}) (\lambda_ {\mathrm{aff}},). \\ & = (\lambda_ {\mathrm{eff}}) (\lambda_ {\mathrm{aff}},). \\ & = (\xi^ {- 1}) (\xi^ {- 1}), \\ & = (\xi^ {- 2}) (\xi^ {- 2}), \\ & = (\xi^ {- 3}) (\xi^ {- 3}), \\ & = (\xi^ {- 4}) (\xi^ {- 4}), \\ & = (\xi^ {- 5}) (\xi^ {- 5}), \\ & = (\xi^ {- 6}) (\xi^ {- 6}), \\ & = (\xi^ {- 7}) (\xi^ {- 7}), \\ & = (\xi^ {- 8}) (\xi^ {- 8}), \\ & = (\xi^ {- 9}) (\xi^ {- 9}), \\ & = (\xi^ {- 1 0}) (\xi^ {- 1 0}), \\ & = (\xi^ {- 1 1}) (\xi^ {- 1 1}), \\ & = (\xi^ {- 1 2}) (\xi^ {- 1 2}), \\ & = (\xi^ {- 1 3}) (\xi^ {- 1 3}), \\ & = (\xi^ {- 1 4}) (\xi^ {- 1 4}), \\ & = (\xi^ {- 1 5}) (\xi^ {- 1 5}), \\ & = (\xi^ {- 1 6}) (\xi^ {- 1 6}), \\ & = (\xi^ {- 1 7}) (\xi^ {- 1 7}), \\ & = (\xi^ {- 1 8}) (\xi^ {- 1 8}), \\ & = (\xi^ {- 1 9}) (\xi^ {- 1 9}), \\ & = (\xi^ {- 2 0}) (\xi^ {- 2 0}), \\ & = (\xi^ {- 2 1}) (\xi^ {- 2 1}), \\ & = (\xi^ {- 2 2}) (\xi^ {- 2 2}), \\ & = (\xi^ {- 2 3}) (\xi^ {- 2 3}), \\ & = (\xi^ {- 2 4}) (\xi^ {- 2 4}), \\ & = (\xi^ {- 2 5}) (\xi^ {- 2 5}), \\ & = (\xi^ {- 2 6}) (\xi^ {- 2 6}), \\ & = (\xi^ {- 2 7}) (\xi^ {- 2 7}), \\ & = (\xi^ {- 2 8}) (\xi^ {- 2 8}), \\ & = (\xi^ {- 2 9}) (\xi^ {- 2 9}), \\ & = (\xi^ {- 3 0}) (\xi^ {- 3 0}), \\ & = (\xi^ {- 3 1}) (\xi^ {- 3 1}), \\ & = (\xi^ {- 3 2}) (\xi^ {- 3 2}), \\ & = (\xi^ {- 3 3}) (\xi^ {- 3 3}), \\ & = (\xi^ {- 3 4}) (\xi^ {- 3 4}), \\ & = (\xi^ {- 3 5}) (\xi^ {- 3 5}), \\ & = (\xi^ {- 3 6}) (\xi^ {- 3 6}), \\ & = (\xi^ {- 3 7}) (\xi^ {- 3 7}), \\ & = (\xi^ {- 3 8}) (\xi^ {- 3 8}), \\ & = (\xi^ {- 3 9}) (\xi^ {- 3 9}), \\ & = (\xi^ {- 4} ) (x y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y, \\ & = (- z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z K S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S I N T I N T I N T I N T I N T I N T I N T I N T I N T I N T I N T I N T I N T I N T I N T I N T I N T I N T I N T I N T I N T I N T I N T I N T I N T I N T I N T I N T I N T I N T I N T I N T I N T I N T Y W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W< fcel>\end{array}\]

Thus we can construct the following extended system

\[\begin{array} { r l } & \left[ \begin{array} { c } \mathbf { x } _ { t + 1 } ^ { f } \\ \mathbf { x } _ { t + 1 } ^ { s } \\ \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } \\ \mathbf { x } _ { t + 1 } ^ { r d } \\ \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { s } \\ \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } \end{array} \right] = \left[ \begin{array} { c } \mathbf { 0 } _ { n _ { x } \times 1 } \\ \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } \\ ( \sigma \pmb { \eta } \otimes \sigma \pmb { \eta } ) v e c ( \mathbf { I } _ { n _ { e } } ) \\ \frac { 1 } { 6 } \mathbf { h } _ { \sigma \sigma \sigma } \sigma ^ { 3 } \\ \mathbf { 0 } _ { n _ { x } ^ { 2 } \times 1 } \\ ( \sigma \pmb { \eta } \otimes \pmb { \sigma } \pmb { \eta } \otimes \pmb { \sigma } \pmb { \eta } ) E [ ( \pmb { \epsilon } _ { t + 1 } \otimes \pmb { \epsilon } _ { t + 1 } \otimes \pmb { \epsilon } _ { t + 1 } ) ] \end{array} \right] \\ & + \left[ \begin{array} { c c c c c c } \mathbf { h } _ { \mathbf { x } } & \mathbf { 0 } _ { n _ { x } \times n _ { x } } & \mathbf { 0 } _ { n _ { x } \times n _ { x } ^ { 2 } } & \mathbf { 0 } _ { n _ { x } \times n _ { x } } & \mathbf { 0 } _ { n _ { x } \times n _ { x } ^ { 2 } } & \mathbf { 0 } _ { n _ { x } \times n _ { x } ^ { 3 } } \\ \mathbf { 0 } _ { n _ { x } \times n _ { x } } & \mathbf { h } _ { \mathbf { x } } & \tilde { H } _ { \mathbf { x x } } & \mathbf { 0 } _ { n _ { x } \times n _ { x } } & \mathbf { 0 } _ { n _ { x } \times n _ { x } ^ { 2 } } & \mathbf { 0 } _ { n _ { x } \times n _ { x } ^ { 3 } } \\ \mathbf { 0 } _ { n _ { x } ^ { 2 } \times n _ { x } } & \mathbf { 0 } _ { n _ { x } ^ { 2 } \times n _ { x } } & h _ { x } & 0. 2 h _ {\mathrm{xx}} & 0. 2 h _ {\mathrm{x}} & 0. 2 h _ {\mathrm{x}} \\ \frac { 3 } { 6 } h _ { \sigma \sigma x } \sigma ^ { 2 } & \mathbf { 0 } _ { n _ { x } \times n _ { x } } & \mathbf { 0 } _ { n _ { x } \times n _ { x } ^ { 2 } } & h _ { x } & 2 H _ { x x } & H _ { x x x } \\ ( h _ { x } \otimes \frac 1 2 h _ { \sigma \sigma } \sigma ^ { 2 } ) & 0 _ { n _ { x } ^ { 2 } \times n _ { x } } & 0 _ { n _ { x } ^ { 2 } \times n _ { x } ^ { 2 } } & 0 _ { n _ { x } ^ { 2 } \times n _ { x } } & ( h _ { x } \otimes h _ { x }) & ( h _ { x } \otimes H _ { x x }) \\ 0 _ { n _ { x } ^ { 3 } \times n _ { x } } & 0 _ { n _ { x } ^ { 3 } \times n _ { x } ^ { 2 } } & 0 _ { n _ { x } ^ { 3 } \times n _ { x } ^ { 2 } } & 0 _ { n _ { x } ^ { 3 } \times n _ { x } ^ { 2 } } & ( h _ { x} \otimes h _ { x} ) & h _ x x y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y . \\ & & & & & \\ & & & & & \\ & & & & & \\ & & & & & \\ & & & & & \\ & & & & & \\ & & & & & \\ & & & & & \\ & & & & & \\ & & & & & \\ & & & & & \\ & & & & & \\ & & & & & \\ & & & & & \\ & & & & & \\ & & & & & \\ & & & & & \\ & & - i j k a m a l o w i s e r e s s i g u e d i a l l i g u e d i a l l i g u e d i a l l i g u e d i a l l i g u e d i a l l i g u e d i a l l i g u e d i a l l i g u e d i a l l i g u e d i a l l i g u e d i a l l i g u e d i a l l i g u e d i a l l l i g u e d i a l l i g u e d i a l l i g u e d i a l l i g u e d i a l l i g u e d i a l l i g u e d i a l l i g u e d i a l l i g u e d i a l l i g u e d i a l l i g u e d i a l l i g u e d i a r e f o r e f o r e f o r e f o r e f o r e f o r e f o r e f o r e f o r e f o r e f o r e f o r e f o r e f o r e f o r e f o r e f o r e f o r e f o r e f o r e f o r e f o r e f o r e f o r e f o r e f f o r e f o r e f o r e f o r e f o r e f o r e f o r e f o r e f o r e f o r e f o r e f o r e f o r e f o r e f o r e f o r e f o r e f o r e f o r e f o r e f o r e f o r e f o r e f o r e f o r / f o r / f o r / f o r / f o r / f o r / f o r / f o r / f o r / f o r / f o r / f o r / f o r / f o r / f o r / f o r / f o r / f o r / f o r / f o r / f o r / f o r / f o r / f o r / f o r / f o r | , \\ - j k a m a l l i g u e d i a l l i g u e d i a l l i g u e d i a l l i g u e d i a l l i g u e d i a l l i g u e d i a l l i g u e d i a l l i g u e d i a l l i g u e d i a l l i g u e d i a l l i g u e d i | , \\ - j k a m a l l i g u e d i a l l i g u e d i a l l i g u e d i a l l i g u e d i a l l i g u e d i a l l i g u e d i | , \\ - j k a m a l l i g u e d i a l l i g u e d i a l l i g u e d i a l l i g u e d i | , \\ - j k a m a l l i g u e d i a l l i g u e d i a l l i g u e d i | , \\ - j k a m a l l i g u e d i a l l i g u e d i | , \\ - j k a m a l l i g u e d i | , \\ - j k a m a l l i g u e d i | , \\ - j k a m a l l i g u e d i | , \\ - j k a m a l l i g u e d i | , \\ - j k a m a l l i g u e d i | , \\ - j k a m a l l i g u e d i | , \\ - j j k a m a l l i g u e d j | , \\ - j j k a m a l l i g u e d j | , \\ - j j k a m a l l i g u e d j | , \\ - j j k a m a l l i g u e d j | , \\ - j j k a m a l l i g u e d j | , \\ - j j k a m a l l i g u e d j | , \\ - j j k a m b o w h s p h t h t h t h t h t h t h t h t h t h t h t h t h t h t h t h t h t h t h t h t h t h t h t h t h t h t h t h t h t h t h t h t h t h t h t h t h t h t h t h t h t h t h t h t h t h t h t h t h t h t h w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v vv v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v . \\ - j k q u s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s . \\ - j k q u c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c . \\ - j k q u m b o b b o b b o b b o b b o b b o b b o b b o b b o b b o b b o b b o b b o b b o b b o b b o b b o b b o b b o b b o b b o b b o b b o b b o b b o b b o b b o b b o b b o b b o b b o b b o b b o b b o b k z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z . \\ - j k q q u m b o b q u m b o b q u m b o b q u m b o b q u m b o b q u m b o b q u m b o b q u m b o b q u m b o b q u m b o b q u m b o b q u m b o b q u m b o b q u m b o b q u m b o b q u m b o b q u m b o b q u m B O U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W WWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWW< fcel>x, y, and z are the same as the same as the same as the same as the same as the same as the same as the same as the same as the same as the same as the same as the same as the same as the same as the same as the same as the same as the same as the same as the same as the same as the same as the same as the same as the same as the same as the same as the same as the same as the same as the same as the same as the same as it is in this view; for example, we can be able to see what we have an important role in how we want to do it. We can be able to see what we want to do it. We can be able to see what we want to do it. We can be able to see what we want to do it. We can be able to see what we want to do it. We can be able to see what we want to do it. We can be able to see what we want to do it. We can be able to see what we want to do it. We can be able to see what we want to do it. We can not be seen that we can be seen that we can be seen that we can be seen that we can be seen that we can be seen that we can be seen that we can be seen that we can be seen that we can be seen that we can be seen that we can be seen that we can be seen that we can be seen that we can be seen that we can be seen that we can be seen that we can be seen that we can be seen that we can be seen that we can be seen that we cannot be seen if we can be seen if we can be seen if we can be seen if we can be seen if we can be seen if we can be seen if we can be seen if we can be seen if we can be seen if we can be seen if we can be seen if we can be seen if we can be seen if we can be seen if we can be seen if we can be seen if we can be seen if we can be seen if we can be seen if we can be seen if we cannot be seen if we cannot be seen if we cannot be seen if we cannot be seen if we cannot be seen if we cannot be seen if we cannot be seen if we cannot be seen if we cannot be seen if we cannot be seen if we cannot be seen if we cannot be seen if we cannot be seen if we cannot be seen if we cannot be seen if we cannot be seen if we cannot be seen if we cannot be seen if we cannot be seen if we cannot be seen if we could not have an important role in which there is no significant change in which there is no significant change in which there is no significant change in which there is no significant change in which there is no significant change in which there is no significant change in which there is no significant change in which there is no significant change in which there is no significant change in which there is no significant change in which there is no significant change in which there is no significant change in which there is no significant change in which there is no significant change in which there is no significant change in )< fcel>x, y, and z are the same as the same as the same as the same as the same as the same as the same as the same as the same as the same as the same as the same as the same as the same as the same as the same as the same as the same as the same as the same as the same as the same as the same as the same as the same as the same as the same as the same as the same as the same as the different types of interest and interest are also given by us at all times.< nl>\]

\[+ \left[ \begin{array}{c c c c c c} \sigma \eta & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & (\sigma \eta \otimes \sigma \eta) & \sigma \eta \otimes \mathrm {h _ {x}} & \mathrm {h _ {x}} \otimes \sigma \eta & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ \sigma \eta \otimes \frac {1}{2} \mathrm {h _ {\sigma \sigma} \sigma^ {2}} & 0 & 0 & 0 & \sigma \eta \otimes \mathrm {h _ {x}} & \sigma \eta \otimes \tilde {\mathrm{H}} _ {\mathrm{xx}} \\ 0 & 0 & 0 & 0 & 0 & \sigma \eta \otimes \mathrm {h _ {x}} \otimes \mathrm {h _ {x}} \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ \mathrm {h _ {x}} \otimes \mathrm {h _ {x}} \otimes \sigma \eta & \mathrm {h _ {x}} \otimes \sigma \eta \otimes \mathrm {h _ {x}} & \mathrm {h _ {x}} \otimes \sigma \eta \otimes \sigma \eta & \sigma \eta \otimes \mathrm {h _ {x}} \otimes \sigma \eta & \sigma \eta \otimes \sigma \eta \otimes \mathrm {h _ {x}} & \sigma \eta \otimes \sigma \eta \otimes \sigma \eta \\ \end{array} \right]\]

\[\times \left[ \begin{array}{c} \pmb {\epsilon} _ {t + 1} \\ \pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1} - v e c (\mathbf {I} _ {n _ {e}}) \\ \pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \\ \mathbf {x} _ {t} ^ {f} \otimes \pmb {\epsilon} _ {t + 1} \\ \pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {s} \\ \pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \\ \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \pmb {\epsilon} _ {t + 1} \\ \mathbf {x} _ {t} ^ {f} \otimes \pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \\ \mathbf {x} _ {t} ^ {f} \otimes \pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1} \\ \pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \pmb {\epsilon} _ {t + 1} \\ \pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \\ (\pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1}) - E [ (\pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1}) ] \\ \end{array} \right]\]

↓

\[\mathbf {z} _ {t + 1} = \mathbf {c} + \mathbf {A} \mathbf {z} _ {t} + \mathbf {B} \boldsymbol {\xi} _ {t + 1}\tag{38}\]

The absolute value of the eigenvalues in are all strictly less than one by assumption. Accordingly, all eigenvalues of A are also strictly less than one. To see this note first that

\[\begin{array} { r l } & = \left| \left[ \begin{array} { c c c c c c } \mathbf { h } _ { \mathbf { x } } - \lambda \mathbf { I } & \mathbf { 0 } _ { n _ { x } \times n _ { x } } & \mathbf { 0 } _ { n _ { x } \times n _ { x } ^ { 2 } } & \mathbf { 0 } _ { n _ { x } \times n _ { x } } & \mathbf { 0 } _ { n _ { x } \times n _ { x } ^ { 2 } } & \mathbf { 0 } _ { n _ { x } \times n _ { x } ^ { 3 } } \\ \mathbf { 0 } _ { n _ { x } \times n _ { x } } & \mathbf { h } _ { \mathbf { x } } - \lambda \mathbf { I } & \tilde { \mathbf { H } } _ { \mathbf { x x } } & \mathbf { 0 } _ { n _ { x } \times n _ { x } } & \mathbf { 0 } _ { n _ { x } \times n _ { x } ^ { 2 } } & \mathbf { 0 } _ { n _ { x } \times n _ { x } ^ { 3 } } \\ \mathbf { 0 } _ { n _ { x } ^ { 2 } \times n _ { x } } & \mathbf { 0 } _ { n _ { x } ^ { 2 } \times n _ { x } } & \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } - \lambda \mathbf { I } & \mathbf { 0 } _ { n _ { x } ^ { 2 } \times n _ { x } } & \mathbf { 0 } _ { n _ { x } ^ { 2 } \times n _ { x } ^ { 2 } } & \mathbf { 0 } _ { n _ { x } ^ { 2 } \times n _ { x } ^ { 3 } } \\ \frac { 3 } { 6 } \mathbf { h } _ { \sigma \sigma \mathbf { x} } \sigma ^ { 2 } & \mathbf { 0 } _ { n _ { x } \times n _ { x } } & \mathbf { 0 } _ { n _ { x } \times n _ { x } ^ { 2 } } & \mathbf { h } _ { \mathbf { x } } - \lambda \mathbf { I } & 2 \mathbf { H } _ { \mathbf { x x } } & \tilde { \mathbf { H } } _ { \mathbf { x x x } } \\ \mathbf { h } _ { \mathbf { x} } \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } & \mathbf { 0 } _ { n _ { x } ^ { 2 } \times n _ { x } } & \mathbf { 0 } _ { n _ { x } ^ { 2 } \times n _ { x } ^ { 2 } } & \mathbf { 0 } _ { n _ { x } ^ { 2 } \times n _ { x } } & \mathbf { h } _ { \mathbf { x} } \otimes \mathbf { h } _ { \mathbf { x} } - \lambda \mathbf { I } & \mathbf { h } _ { \mathbf { x} } \otimes \tilde { \mathbf { H } } _ { \mathbf { x x} } \\ \mathbf { 0 } _ { n _ { x } ^ { 3 } \times n _ { x } } & \mathbf { 0 } _ { n _ { x } ^ { 3 } \times n _ { x } } & \mathbf { 0 } _ { n _ { x } ^ { 3 } \times n _ { x } ^ { 2 } } & \mathbf { 0 } _ { n _ { x } ^ { 3 } \times n _ { x } } & \mathbf { 0 } _ { n _ { x } ^ { 3 } \times n _ { x } ^ { 2 } } & \mathbf { h } _ { \mathbf { x} } \otimes \mathbf { h } _ { \mathbf { x} } \otimes \mathbf { h } _ { \mathbf { x} } - \lambda \mathbf { I} \\ \end{array} \right] \right| \\ & \quad = \left| \left[ \begin{array} { c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c a d e r e w e l e t} \\ & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & \\ & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & \\ & & & & & & & & & & & & & & & & & & & & \\ & & & & & & & & & & & \\ & & & & & \\ & b i v e r e w e l e t \\ B _ {\mathrm{11}} = [ b i v e r e w i t ] \\ B _ {\mathrm{12}} = [ b i v e r e w i t ] \\ B _ {\mathrm{21}} = [ b i v e r e w i t ] \\ B (x) = [ y , z ) = [ y , z ) = [ y , z ) = [ y , z ) = [ y , z ) = [ y , z ) = [ y , z ) = [ y , z ) = [ y , z ) = [ y , z ) = [ y , z ) = [ y , z ) = [ y , z ) = [ y , z ) = [ y , z ) = [ y , z ) = [ y , z ) = [ y, z ) = [ y, z ) = [ y, z ) = [ y, z ) = [ y, z ) = [ y, z ) = [ y, z ) = [ y, z ) = [ y, z ) = [ y, z ) = [ y, z ) = [ y, z ) = [ y, z ) = [ y, z ) = [ y, z ) = [ y, z ) = [ y, z ) = {[ y , z ]} \\ b i v e r e w i t ] \\ b i v e r e w i t ] \\ b i v e r e w i t ] \\ b i v e r e w i t ] \\ b i v e r e w i t ] \\ b i v e r e w i t ] \\ b i v e r e w i t ] \\ b i v e r e w i t ] \\ b i v e r e w i t ] \\ b i v e r e w i t ] / / \\ B (x) = [ y , z ) = [ y , z ) = [ y , z ) = [ y , z ) = [ y , z ) = [ y , z ) = [ y , z ) = [ y , z ) = [ y , z ) = [ y , z ) = [ y , z ) = [ y , z ) = [ y , z ) = [ y , z ) = [ y , z ) = [ y , s ] \\ b (y) = [ y , z ) = [ y , s ] \\ b (z) = [ y , s ] \\ b (w) = [ w , s ] \\ b (x) = [ w , s ] \\ b (y) = [ w , s ] \\ b (z) = [ w , s ] \\ b (w) = [ w , s ] \\ b (x) = [ w , s ] \\ b (y) = [ w , s ] \\ b (z) = [ w , s ] \\ b (w) = [ w , s ] \\ b (x) = [ w , s ] \\ b (y) = [ w , s ] \\ b (z) = [ w , s ] \\ b (w) = [ w s ] \\ b (x) = [ w s ] \\ b (y) = [ w s ] \\ b (z) = [ w s ] \\ b (w) = [ w s ] \\ b (x) = [ w s ] \\ b (y) = [ w s ] \\ b (z) = [ w s ] \\ b (w) = [ w s ] \\ b (x) = [ w s ] \\ b (y) = [ w s ] \\ b (z) = [ w s ] \\ b (w) = [ w m ] \\ b (x) = [ w m ] \\ b (y) = [ w m ] \\ b (z) = [ w m ] \\ b (w) = [ w m ] \\ b (x) = [ w m ] \\ b (y) = [ w m ] \\ b (z) = [ w m ] \\ b (w) = [ w m ] \\ b (x) = [ w m ] \\ b (y) = [ w m ] \\ b (z) = [ w m ] \\ b (w) = [ w m ], \\ b (x) = [ w m ], \\ b (y) = [ w m ], \\ b (z) = [ w m ], \\ b (w) = [ w m ], \\ b (x) = [ w m ], \\ b (y) = [ w m ], \\ b (z) = [ w m ], \\ b (w) = [ w m ], \\ b (x) = [ w m ], \\ b (y) = [ w m ], \\ b (z) = [ w m ], \\ b (w) = [ w m ], / / \\ b (x) = [ w m ], / / \\ b (y) = [ w m ], / / \\ b (z) = [ w m ], / / \\ b (w) = [ w m ], / / \\ b (x) = [ w m ], / / \\ b (y) = [ w m ], / / \\ b (z) = [ w m ], / / \\ b (w) = [ w m ], / / \\ b (x) = [ w m ], / / \\ b (y) = [ w m ], / / \\b (z) = [ w m ], / / \\b (w) = [ w m ], / / \\b (x) = [ w m ], / / \\b (y) = [ w m ], / / \\b (z) = [ w m ], / / \\b (w) = [ w m ], / / \\b (x) = [ w m ], / / \\b (y) = [ w m ], / / \\b (z) = [ w m ], / / \\b (w) = [ w m ], / / \\b (z) = [ w m ], / / \\b (w) = [ w m ], / / \\b (x) = [ w m ], / / \\b (y) = [ w m ], / / \\b (z) = [ w m ], / / \\b (w) = [ w m ], / / \\b (x) - q u a d o f f o r e a l l o g h a t e r e. A p p l o r e a l l o g h a t e r e. A p p l o r e a l l o g h a t e r e. A p p l o r e a l l o g h a t e r e. A p p l o r e a l l o g h a t e r e. A p p l o r e a l l o g h a t e r e. A p p l o r e a d l o g h a t e r e. A p p l o r e a d l o g h a t e r e. A p p l o r e a d l o g h a t e r e. A p p l o r e a d l o g h a t e r e. A p p l o r e a d l o g h a t e r e. A p p l o r e a d l o g h h a t e r e. A p p l o r e a d l o g h h a t e r e. A p p l o r e a d l o g h h a t e r e. A p p l o r e a d l o g h h a t e r e. A p p l o r e a d l o g h h a t e r e. A p p l o r e a d l o g g h a t e r e. A p p l o r e a d l o g g h a t e r e. A p p l o r e a d l o g g h a t e r e. A p p l o r e a d l o g g h a t e r e. A p p l o r e a d l o g g h a t e r e. A p p l o r e a d l o g h a t e r e. A p p l o r e a d l o g h a t e r e. A p p l o r e a d l o g h a t e r e. A p p l o r e a d l o g h a t e r e. A p p l o r e a d l o g h a t . E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .\]

\[\mathbf {B} _ {2 2} \equiv \left[ \begin{array}{c c c} \mathbf {h} _ {\mathbf {x}} - \lambda \mathbf {I} & 2 \tilde {\mathbf {H}} _ {\mathbf {x x}} & \tilde {\mathbf {H}} _ {\mathbf {x x x}} \\ \mathbf {0} _ {n _ {x} ^ {2} \times n _ {x}} & (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) - \lambda \mathbf {I} & (\mathbf {h} _ {\mathbf {x}} \otimes \tilde {\mathbf {H}} _ {\mathbf {x x}}) \\ \mathbf {0} _ {n _ {x} ^ {3} \times n _ {x}} & \mathbf {0} _ {n _ {x} ^ {3} \times n _ {x} ^ {2}} & (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) - \lambda \mathbf {I} \end{array} \right]\]

\[\begin{array}{c} = | \mathbf {B} _ {1 1} | | \mathbf {B} _ {2 2} | \\ \text {using} \left| \begin{array}{c c} \mathbf {U} & \mathbf {C} \\ \mathbf {0} & \mathbf {Y} \end{array} \right| = | \mathbf {U} | | \mathbf {Y} | \text {where} \mathbf {U} \text {is} m \times m \text {and} \mathbf {Y} \text {is} n \times n \end{array}\]

\[= \left| \mathbf {h} _ {\mathbf {x}} - \lambda \mathbf {I} \right| \left| \mathbf {h} _ {\mathbf {x}} - \lambda \mathbf {I} \right| \left| \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} - \lambda \mathbf {I} \right| \left| \mathbf {B} _ {2 2} \right|\]

using the results from the second order approximation

\[\begin{array}{r l r} & & {= | \mathbf {h _ {x}} - \lambda \mathbf {I} | | \mathbf {h _ {x}} - \lambda \mathbf {I} | | \mathbf {h _ {x}} \otimes \mathbf {h _ {x}} - \lambda \mathbf {I} | | \mathbf {h _ {x}} - \lambda \mathbf {I} | | (\mathbf {h _ {x}} \otimes \mathbf {h _ {x}}) - \lambda \mathbf {I} | | (\mathbf {h _ {x}} \otimes \mathbf {h _ {x}} \otimes \mathbf {h _ {x}}) - \lambda \mathbf {I} |} \\ & & {\mathrm{using~the~rule~on~block~determinants~repeatedly~on} \mathbf {B} _ {2 2}} \end{array}\]

Hence, the eigenvalue solves the problem

\[\begin{array}{l} p (\lambda) = 0 \\ \Updownarrow \\ | \mathbf {h} _ {\mathbf {x}} - \lambda \mathbf {I} | | \mathbf {h} _ {\mathbf {x}} - \lambda \mathbf {I} | | \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} - \lambda \mathbf {I} | | \mathbf {h} _ {\mathbf {x}} - \lambda \mathbf {I} | | (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) - \lambda \mathbf {I} | | (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) - \lambda \mathbf {I} | = 0 \\ \Updownarrow \end{array}\]

\[\left| \mathbf {h} _ {\mathbf {x}} - \lambda \mathbf {I} \right| = 0 \text {or} \left| \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} - \lambda \mathbf {I} \right| = 0 \text {or} \left| \left(\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}\right) - \lambda \mathbf {I} \right| = 0\]

The absolute value of all eigenvalues to the first problem are strictly less than one. That is i = 1, 2, ..., . This is also the case for the second problem because the eigenvalues to are for and . The same argument ensures that this is also the case for the third problem.

Thus, the system in (38) is covariance stationary if has finite first and second moment. It follows directly that and has finite second moments if has a sixth moment. The latter holds by assumption.

For the control variables we have

\[\begin{array}{r l} & y _ {t} ^ {r d} (i, 1) = \mathbf {g} _ {\mathbf {x}} (i,:) (\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d}) + \frac {1}{2} (\mathbf {x} _ {t} ^ {f}) ^ {\prime} \mathbf {g} _ {\mathbf {x x}} (i,:,: (\mathbf {x} _ {t} ^ {f} + 2 \mathbf {x} _ {t} ^ {s}) \\ & \qquad + \frac {1}{6} (\mathbf {x} _ {t} ^ {f}) ^ {\prime} \left[ \begin{array}{c} (\mathbf {x} _ {t} ^ {f}) ^ {\prime} \mathbf {g} _ {\mathbf {x x x}} (i, 1,:,: (\mathbf {x} _ {t} ^ {f}) \\ \dots \\ (\mathbf {x} _ {t} ^ {f}) ^ {\prime} \mathbf {g} _ {\mathbf {x x x}} (i, n _ {x},,:,: (\mathbf {x} _ {t} ^ {f})) \end{array} \right] + \frac {1}{2} g _ {\sigma \sigma} (i, 1) \sigma^ {2} + \frac {3}{6} \mathbf {g} _ {\sigma \sigma \mathbf {x}} (i,:) \sigma^ {2} \mathbf {x} _ {t} ^ {f} + \frac {1}{6} g _ {\sigma \sigma \sigma} (i, 1) \sigma^ {3} \\ & \end{array}\]

\[\begin{array}{r l} & {\mathbf {y} _ {t} ^ {r d} = \mathbf {g} _ {\mathbf {x}} \left(\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d}\right) + \tilde {\mathbf {G}} _ {\mathbf {x x}} \left(\left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) + 2 \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {s}\right)\right) + \tilde {\mathbf {G}} _ {\mathbf {x x x}} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right)} \\ & {\qquad + \frac {1}{2} \mathbf {g} _ {\sigma \sigma} \sigma^ {2} + \frac {3}{6} \mathbf {g} _ {\sigma \sigma \mathbf {x}} \sigma^ {2} \mathbf {x} _ {t} ^ {f} + \frac {1}{6} \mathbf {g} _ {\sigma \sigma \sigma} \sigma^ {3}} \\ & {\mathrm{where} \tilde {\mathbf {G}} _ {\mathbf {x x}} \equiv \frac {1}{2} r e s h a p e (\mathbf {g} _ {\mathbf {x x}}, n _ {y}, n _ {x} ^ {2}) \mathrm{and} \tilde {\mathbf {G}} _ {\mathbf {x x x}} \equiv \frac {1}{6} r e s h a p e (\mathbf {g} _ {\mathbf {x x x}}, n _ {y}, n _ {x} ^ {3})} \\ & {\Updownarrow} \end{array}\]

\[\mathbf {y} _ {t} ^ {r d} = \left[ \begin{array}{c c c c c} \mathbf {g} _ {\mathbf {x}} + \frac {3}{6} \mathbf {g} _ {\sigma \sigma \mathbf {x}} \sigma^ {2} & \mathbf {g} _ {\mathbf {x}} & \tilde {\mathbf {G}} _ {\mathbf {x x}} & \mathbf {g} _ {\mathbf {x}} & 2 \tilde {\mathbf {G}} _ {\mathbf {x x}} \\ \end{array} \right] \left[ \begin{array}{c} \mathbf {x} _ {t} ^ {f} \\ \mathbf {x} _ {t} ^ {s} \\ \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \\ \mathbf {x} _ {t} ^ {r d} \\ \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {s} \\ \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \end{array} \right] + \frac {1}{2} \mathbf {g} _ {\sigma \sigma} \sigma^ {2} + \frac {1}{6} \mathbf {g} _ {\sigma \sigma \sigma} \sigma^ {3}\]

\[= \mathbf {D} \mathbf {z} _ {t} + \frac {1}{2} \mathbf {g} _ {\sigma \sigma} \sigma^ {2} + \frac {1}{6} \mathbf {g} _ {\sigma \sigma \sigma} \sigma^ {3}\]

That is is linear function of and is therefore also covariance-stationary.

Q.E.D.

4.2 Method 1: Formulas for the first and second moments

This section computes first and second moments using the representation of the second-order system stated above. This method is fairly direct but has the computational disadvantage of requiring a lot of memory because we work directly with the big B matrix.

The system

The mean values are

\[\begin{array}{c} \mathbf {z} _ {t + 1} = \mathbf {c} + \mathbf {A} \mathbf {z} _ {t} + \mathbf {B} \boldsymbol {\xi} _ {t + 1} \\ \mathbf {y} _ {t} ^ {r d} = \mathbf {D} \mathbf {z} _ {t} + \frac {1}{2} \mathbf {g} _ {\sigma \sigma} \sigma^ {2} + \frac {1}{6} \mathbf {g} _ {\sigma \sigma \sigma} \sigma^ {3} \\ E [ \mathbf {z} _ {t} ] = (\mathbf {I} _ {3 n _ {x} + 2 n _ {x} ^ {2} + n _ {x} ^ {3}} - \mathbf {A}) ^ {- 1} \mathbf {c}. \\ E [ \mathbf {y} _ {t} ^ {r d} ] = \mathbf {D} E [ \mathbf {z} _ {t} ] + \frac {1}{2} \mathbf {g} _ {\sigma \sigma} \sigma^ {2} + \frac {1}{6} \mathbf {g} _ {\sigma \sigma \sigma} \sigma^ {3} \end{array}\]

For the variances we first have as above that

\[\begin{array}{r l} & E \left[ \mathbf {z} _ {t + 1} \mathbf {z} _ {t + 1} ^ {\prime} \right] = E \left[ \left(\mathbf {c} + \mathbf {A z} _ {t} + \mathbf {B} \boldsymbol {\xi} _ {t + 1}\right) \left(\mathbf {c} + \mathbf {A z} _ {t} + \mathbf {B} \boldsymbol {\xi} _ {t + 1}\right) ^ {\prime} \right] \\ & \qquad = E \left[ \mathbf {c c} ^ {\prime} + \mathbf {c z} _ {t} ^ {\prime} \mathbf {A} ^ {\prime} + \mathbf {c} \boldsymbol {\xi} _ {t + 1} ^ {\prime} \mathbf {B} ^ {\prime} \right] \\ & \qquad + E \left[ \mathbf {A z} _ {t} \mathbf {c} ^ {\prime} + \mathbf {A z} _ {t} \mathbf {z} _ {t} ^ {\prime} \mathbf {A} ^ {\prime} + \mathbf {A z} _ {t} \boldsymbol {\xi} _ {t + 1} ^ {\prime} \mathbf {B} ^ {\prime} \right] \\ & \qquad + E \left[ \mathbf {B} \boldsymbol {\xi} _ {t + 1} \mathbf {c} ^ {\prime} + \mathbf {B} \boldsymbol {\xi} _ {t + 1} \mathbf {z} _ {t} ^ {\prime} \mathbf {A} ^ {\prime} + \mathbf {B} \boldsymbol {\xi} _ {t + 1} \boldsymbol {\xi} _ {t + 1} ^ {\prime} \mathbf {B} ^ {\prime} \right] \\ & \qquad = \mathbf {c c} ^ {\prime} + \mathbf {c E} \left[ \mathbf {z} _ {t} ^ {\prime} \right] \mathbf {A} ^ {\prime} \\ & \qquad + \mathbf {A E} \left[ \mathbf {z} _ {t} \right] \mathbf {c} ^ {\prime} + \mathbf {A E} \left[ \mathbf {z} _ {t} \mathbf {z} _ {t} ^ {\prime} \right] \mathbf {A} ^ {\prime} + \mathbf {A E} \left[ \mathbf {z} _ {t} \boldsymbol {\xi} _ {t + 1} ^ {\prime} \right] \mathbf {B} ^ {\prime} \\ & \qquad + \mathbf {B E} \left[ \boldsymbol {\xi} _ {t + 1} \mathbf {z} _ {t} ^ {\prime} \right] \mathbf {A} ^ {\prime} + \mathbf {B E} \left[ \boldsymbol {\xi} _ {t + 1} \boldsymbol {\xi} _ {t + 1} ^ {\prime} \right] \mathbf {B} ^ {\prime} \end{array}\]

and

\[\begin{array}{l} \text {and} \\ E \left[ \mathbf {z} _ {t} \right] E \left[ \mathbf {z} _ {t} \right] ^ {\prime} = \left(\mathbf {c} + \mathbf {A} E \left[ \mathbf {z} _ {t} \right]\right) \left(\mathbf {c} + \mathbf {A} E \left[ \mathbf {z} _ {t} \right]\right) ^ {\prime} \\ = \left(\mathbf {c} + \mathbf {A} E \left[ \mathbf {z} _ {t} \right]\right) \mathbf {c} ^ {\prime} + \mathbf {c} E \left[ \mathbf {z} _ {t} ^ {\prime} \right] \mathbf {A} ^ {\prime} + \mathbf {A} E \left[ \mathbf {z} _ {t} \right] E \left[ \mathbf {z} _ {t} ^ {\prime} \right] \mathbf {A} ^ {\prime} \end{array}\]

Hence,

\[\begin{array}{r l} & E \left[ \mathbf {z} _ {t + 1} \mathbf {z} _ {t + 1} ^ {\prime} \right] - E \left[ \mathbf {z} _ {t} \right] E \left[ \mathbf {z} _ {t} \right] ^ {\prime} = \mathbf {c c} ^ {\prime} + \mathbf {c} E \left[ \mathbf {z} _ {t} ^ {\prime} \right] \mathbf {A} ^ {\prime} \\ & \quad + \mathbf {A} E \left[ \mathbf {z} _ {t} \right] \mathbf {c} ^ {\prime} + \mathbf {A} E \left[ \mathbf {z} _ {t} \mathbf {z} _ {t} ^ {\prime} \right] \mathbf {A} ^ {\prime} + \mathbf {A} E \left[ \mathbf {z} _ {t} \pmb {\xi} _ {t + 1} ^ {\prime} \right] \mathbf {B} ^ {\prime} \\ & \quad + \mathbf {B} E \left[ \pmb {\xi} _ {t + 1} \mathbf {z} _ {t} ^ {\prime} \right] \mathbf {A} ^ {\prime} + \mathbf {B} E \left[ \pmb {\xi} _ {t + 1} \pmb {\xi} _ {t + 1} ^ {\prime} \right] \mathbf {B} ^ {\prime} \\ & \quad - (\mathbf {c} + \mathbf {A} E [ \mathbf {z} _ {t} ]) \mathbf {c} ^ {\prime} - \mathbf {c} E [ \mathbf {z} _ {t} ^ {\prime} ] \mathbf {A} ^ {\prime} - \mathbf {A} E [ \mathbf {z} _ {t} ] E [ \mathbf {z} _ {t} ^ {\prime} ] \mathbf {A} \\ & = \mathbf {A} (E [ \mathbf {z} _ {t} \pmb {\xi} _ {t + 1} ^ {\prime} ] - E [ \mathbf {z} _ {t} ] E [ \mathbf {z} _ {t} ^ {\prime} ]) \mathbf {A} ^ {\prime} \\ & \quad + \mathbf {A} E [ \mathbf {z} _ {t} \pmb {\xi} _ {t + 1} ^ {\prime} ] \mathbf {B} ^ {\prime} + \mathbf {B} E [ \pmb {\xi} _ {t + 1} \pmb {\xi} _ {t + 1} ^ {\prime} ] \mathbf {A} ^ {\prime} \\ & \quad + \mathbf {B} E [ \pmb {\xi} _ {t + 1} \pmb {\xi} _ {t + 1} ^ {\prime} ] \mathbf {B} ^ {\prime} \\ & \Updownarrow \\ & V a r [ \mathbf {z} _ {t + 1} ] = A V a r [ \mathbf {z _ {t}} ] \mathbf {A} ^ {\prime} \\ & \quad + \mathbf {A} (E [ \mathbf {z} _ {t} \pmb {\xi} _ {t + 1} ^ {\prime} ] - E [ \mathbf {z} _ {t} ] E [ \pmb {\xi} _ {t + 1} ^ {\prime} ]) \mathbf {B} ^ {\prime} + \mathbf {B} (E [ \pmb {\xi} _ {t + 1} \pmb {\mathrm{z}} _ {t} ^ {\prime} ] - E [ \pmb {\xi} _ {t + 1} ] E [ \pmb {\mathrm{z}} _ {t} ^ {\prime} ]) \mathbf {A} ^ {\prime} \\ & \quad + \mathbf {B} E [ \pmb {\xi} _ {t + 1} \pmb {\xi} _ {t + 1} ^ {\prime} ] \mathbf {B} ^ {\prime} \end{array}\]

Notice that because

Moreover,

\[\begin{array}{c} V a r \left[ \mathbf {z} _ {t + 1} \right] = \mathbf {A} V a r \left[ \mathbf {z} _ {\mathbf {t}} \right] \mathbf {A} ^ {\prime} + \mathbf {B} V a r \left[ \boldsymbol {\xi} _ {t + 1} \right] \mathbf {B} ^ {\prime} + \mathbf {A} C o v \left[ \mathbf {z} _ {t}, \boldsymbol {\xi} _ {t + 1} \right] \mathbf {B} ^ {\prime} + \mathbf {B} C o v \left[ \boldsymbol {\xi} _ {t + 1}, \mathbf {z} _ {t} \right] \mathbf {A} ^ {\prime} \\ V a r \left[ \mathbf {y} _ {t} ^ {r d} \right] = \mathbf {D} V a r \left[ \mathbf {z} _ {t} \right] \mathbf {D} ^ {\prime} \end{array}\]

Contrary to a second-order approximation, we have that . This is seen as follows

\[\begin{array} { r l } & E \left[ \mathbf { z } _ { t } \pmb { \xi } _ { t + 1 } ^ { \prime } \right] = E \left[ \left[ \begin{array} { c } \mathbf { x } _ { t } ^ { f } \\ \mathbf { x } _ { t } ^ { s } \\ \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \\ \mathbf { x } _ { t } ^ { r d } \\ \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { s } \\ \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \end{array} \right] \right. \\ & \quad \times \left[ \begin{array} { c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c } \boldsymbol { \epsilon } _ { t + 1 } ^ { \prime } & ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \boldsymbol { \epsilon } _ { t + 1 } - v e c ( \mathbf { I } _ { n _ { e} } ) ) ^ { \prime } & ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } ) ^ { \prime } & ( \mathbf { x } _ { t } ^ { f } \otimes \boldsymbol { \epsilon } _ { t + 1 } ) ^ { \prime } & ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { s } ) ^ { \prime } & ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) ^ { \prime } \\ ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \otimes \boldsymbol { \epsilon } _ { t + 1 } ) ^ { \prime } & ( \mathbf { x } _ { t } ^ { f } \otimes \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } ) ^ { \prime } & ( \mathbf { x } _ { t } ^ { f } \otimes \boldsymbol { \epsilon } _ { t + 1 } \otimes \boldsymbol { \epsilon } _ { t + 1 } ) ^ { \prime } & ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \otimes \boldsymbol { \epsilon } _ { t + 1 } ) ^ { \prime } \\ ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } ) ^ { \prime } & ( ( ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \boldsymbol { \epsilon } _ { t + 1 } \otimes \boldsymbol { \epsilon } _ { t + 1 } ) - E [ ( ( \boldsymbol { \epsilon } _ { t + 1 } \otimes ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (\mathrm{次}) o n g h i n g h i n g h i n g h i n g h i n g h i n g h i n g h i n g h i n g h i n g h i n g h i n g h i n g h i n g h i n g h i n g h i n g h i n g h i n g h i n g h i n g h i n g h i n g h i n g h i n g h j e r m a l l o w e r m a l l o w e r m a l l o w e r m a l l o w e r m a l l o w e r m a l l o w e r m a l l o w e r m a l l o w e r m a l l o w e r m a l l o w e r m a l l o w e r m a l l o w e r m a l l o w e r m o l l o w e r m a l l o w e r m a l l o w e r m a l l o w e r m a l l o w e r m a l l o w e r m a l l o w e r m a l l o w e r m a l l o w e r m a l l o w e r m a l l o w e r m a l l o w e r m a l l o , \\ 0 _ { n _ { x } \times n _ { e } } & 0 _ { n _ { x } \times n _ { e } ^ { 2 } } & 0 _ { n _ { x } \times n _ { e } n _ { x } } & 0 _ { n _ { x } \times n _ { x } n _ { e } } & 0 _ { n _ { x } \times n _ { x } n _ { e } } & 0 _ { n _ { x } \times n _ { e } n _ { x } ^ { 2 } } & 0 _ { n _ { x } \times n _ { x } ^ { 2 } n _ { e } } & 0 _ { n _ { x } \times n _ { x } ^ { 2 } n _ { e } } & 0 _ { n _ { x } \times n _ { x } ^ { 2 } n _ { e } ^ { 2 }} & 0 _ { n _ { x } \times n _ { x } ^ { 2 } n _ { e } ^ { 2 }} & 0 _ { n _ { x } \times n _ { x } ^ { 2 } n _ { e } ^ { 2 }} & 0 _ { n _ { x }} \\ 0 _ { n _ { x } \times n _ { e } } & 0 _ { n _ { x } \times n _ { e } ^ { 2 } } & 0 _ { n _ { x } \times n _ { e } n _ { x } } & 0 _ { n _ { x } \times n _ { x } n _ { e } } & 0 _ n _ { x } \times N _ {\infty} ^ {\infty} N _ {\infty} ^ {\infty} N _ {\infty} ^ {\infty} N _ {\infty} ^ {\infty} N _ {\infty} ^ {\infty} N _ {\infty} ^ {\infty} N _ {\infty} ^ {\infty} N _ {\infty} ^ {\infty} N _ {\infty} ^ {\infty} N _ {\infty} ^ {\infty}N _ {\infty} ^ {\infty} N _ {\infty} ^ {\infty} N _ {\infty} ^ {\infty} N _ {\infty} ^ {\infty} N _ {\infty} ^ {\infty} N _ {\infty} ^ {\infty} N _ {\infty} ^ {\infty} N _ {\infty} ^ {\infty} N _ {\infty} ^ {\infty} N_{\infty} ^ {\infty} N_{\infty} ^ {\infty} N_{\infty} ^ {\infty} N_{\infty} ^ {\infty} N_{\infty} ^ {\infty} N_{\infty} ^ {\infty} N_{\infty} ^ {\infty} N_{\infty} ^ {\infty} N_{\infty} ^ {\infty} N_{\mathrm{的}} \\ 0 _ { n _ { x } ^ { 2 }\times n _ { e } } & 0 _ { n _ { x } ^ { 2 }\times n _ { e } ^ { 2 } } & 0 _ { n _ { x } ^ { 2 }\times n _ { e }\times n _ { x }} & 0 _ { n _ { x } ^ { 2 }\times n _ { x }\times n _ { e }} & 0 _ { n _ { x } ^ { 2 }\times n _ { x }\times n _ { e }} & 0 _ n _ { x }\timesn_ { x }\timesn_ { x }\timesn_ { x }\timesn_ { x }\timesn_ { x }\timesn_ { x }\timesn_ { x }\timesn_ { x }\timesn_ { x }\timesn_ { x }\timesn_ { x }\timesn_ { x }\timesn_ { x }\timesn_ { x }\timesn_ { x }\timesn_ { x }\timesn_ { x }\mathrm{e}\\ 0 _ { n _ { x }\times n _ { e }} & 0 _ n _ { x }\times n _ { e }\mathrm{e}^{ 2} ) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ (~~ )~ . \\ 0 _ n_{ x }\mathrm{e}\times n_{ e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{a}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{c}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{u}\\ 0 _ n_{ x }\mathrm{e}\times n_{ e }\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{e}\mathrm{u}\\ 0 _ n_{ x }\mathrm{\ell}_{x }\times n_{ y }\times n_{ y }\times n_{ y }\times n_{ y }\times n_{ y }\times n_{ y }\times n_{ y }\times n_{ y }\times n_{ y }\times n_{ y }\times n_{ y }\times n_{ y }\times n_{ y }\times n_{ y }\times n_{ y }\times n_{ y }\times n_{ y }\times n_{ y }\times n_{ y }\times n_{ y }\times n_{ u}\\ 0 _ n_{ x }\mathrm{\ell}_{x }\times n_{ y }\times n_{ y }\times n_{ y }\times n_{ y }\times n_{ y }\times n_{ y }\times n_{ y }\times n_{ y }\times n_{ y }\times n_{ y }\times n_{ y }\times n_{ y }\times n_{ y }\times n_{ y }\times n_{ y }\times n_{ y }\times n_{ u}\\ 0 _ n_{ x }\mathrm{\ell}_{x}\times n_{ y }\times n_{ y }\times n_{ y }\times n_{ y }\times n_{ y }\times n_{ y }\times n_{ y }\times n_{ y }\times n_{ y }\times n_{ y }\times n_{ y }\times n_{ u}\\ 0 _ n_{ x }\mathrm{\ell}_{x}\times n_{ y }\times n_{ y }\times n_{ y }\times n_{ y }\times n_{ y }\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot\cdot/\\ = [ 0 . R . R . ] & = [ R . R . R . ] \\ & = [ R . R . R . ] \\ & = [ R . R . R . ] \\ & = [ R . R . R . ] \\ & = [ R . R . R . ] \\ & = [ R . R . R . ] \\ & = [ R . R . R . ] \\ & = [ R . R . R . ] \\ & = [ R . R . R . ] \\ & = [ R . R . R . ] \\ = [ R . R . R . ] \\ & = [ R . R . R . ] \\ & = [ R . R . R . ] \\ & = [ R . R . R . ] \\ & = [ R . R . R . ] \\ & = [ R . R . R . ] \\ & = [ R . R . R . ] \\ & = [ R . R . R . ] \\ & = [ R . R . R . ] \\ & < [ R . R . R . ] \\ & = [ R . R . R . ] \\ & = [ R . R . R . ] \\ & = [ R . R . R . ] \\ & = [ R . R . R . ] \\ & = [ R . R . R . ] \\ & = [ R . R . R . ] \\ & = [ R . R . R . ] \\ & = [ R . R . R . ] \\ & = [ M S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U SU S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u , \\ & = [ M S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U S U.S u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u.s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u , \\ & = [ M A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C d A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B CD A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C D A B C K L K L K L K L K L K L K L K L K L K L K L K L K L K L K L K L K L K L K L K L K L K L K L K L K L K L K L K L K L K L K L K L K L K L K L K L K L K L K L K L K L K L K L K L K L K L K L K L K L K L KL K L K L K L K L K L K L K L K L K L K L K L K L K L K L K L K L K L K L K L K L K L K L K LK L K LK LK LK LK LK LK LK LK LK LK LK LK LK LK LK LK LK LK LK LK LK LK LK LK LK LK LK LK LK LK LK LK LK LK LK LK LK LK LK LK LK LK LK LK LK LK LK LK LK LK LkL kL kL kL kL kL kL kL kL kL kL kL kL kL kL kL kL kL kL kL kL kL kL kL kL kL kL kL kL kL kL kL kL kL kL kL kL kL kL kL kL kL kL kL kL kL kL kL kL kL kL skk kk kk skk skk skk skk skk skk skk skk skk skk skk skk skk skk skk skk skk skk skk skk skk skk skk skk skk skk skk skk skk skk skk skk skk skk skk skk skk skk skk skk skk skk skk skk skk skk skk skk skk skk< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< nl>\]

We now compute the non-zero elements in this matrix

end

\[\begin{array}{l} \text {2) The value of r_{1,10}} \\ r _ {1, 1 0} = E \left[ \mathbf {x} _ {t} ^ {f} \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) ^ {\prime} \right] \\ = E \left[ \left\{x _ {t} ^ {f} (\gamma_ {1}, 1) \right\} _ {\gamma_ {1} = 1} ^ {n _ {x}} \left\{\epsilon_ {t + 1} (\phi_ {1}, 1) \left\{x _ {t} ^ {f} (\gamma_ {2}, 1) \left\{\epsilon_ {t + 1} (\phi_ {2}, 1) \right\} _ {\phi_ {2} = 1} ^ {n _ {e}} \right\} _ {\gamma_ {2} = 1} ^ {n _ {x}} \right\} _ {\phi_ {1} = 1} ^ {n _ {e}} \right] \end{array}\]

3) The value of

\[\begin{array}{r l} & {r _ {1, 1 1} = E \left[ \mathbf {x} _ {t} ^ {f} \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \right]} \\ & {\qquad = E \left[ \left\{x _ {t} ^ {f} (\gamma_ {1}, 1) \right\} _ {\gamma_ {1} = 1} ^ {n _ {x}} \left\{\epsilon_ {t + 1} (\phi_ {1}, 1) \left\{\epsilon_ {t + 1} (\phi_ {2}, 1) \left\{x _ {t} ^ {f} (\gamma_ {2}, 1) \right\} _ {\gamma_ {2}} ^ {n _ {x}} \right\} _ {\phi_ {2} = 1} ^ {n _ {e}} \right\} _ {\phi_ {1} = 1} ^ {n _ {e}} \right]} \end{array}\]

\[\begin{array}{l} 4) \text {The value of} r _ {2, 9} \\ r _ {2, 9} = E \left[ \mathbf {x} _ {t} ^ {s} \left(\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) ^ {\prime} \right] \\ = E \left[ \left\{x _ {t} ^ {s} (\gamma_ {1}, 1) \right\} _ {\gamma_ {1} = 1} ^ {n _ {x}} \left\{x _ {t} ^ {f} (\gamma_ {2}, 1) \left\{\epsilon_ {t + 1} (\phi_ {1}, 1) \left\{\epsilon_ {t + 1} (\phi_ {2}, 1) \right\} _ {\phi_ {2} = 1} ^ {n _ {e}} \right\} _ {\phi_ {1} = 1} ^ {n _ {e}} \right\} _ {\gamma_ {2} = 1} ^ {n _ {x}} \right] \end{array}\]

5) The value of

\[\begin{array}{r l} & {r _ {2, 1 0} = E \left[ \mathbf {x} _ {t} ^ {s} \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) ^ {\prime} \right]} \\ & {\qquad = E \left[ \left\{x _ {t} ^ {s} (\gamma_ {1}, 1) \right\} _ {\gamma_ {1} = 1} ^ {n _ {x}} \left\{\epsilon_ {t + 1} (\phi_ {1}, 1) \left\{x _ {t} ^ {f} (\gamma_ {2}, 1) \left\{\epsilon_ {t + 1} (\phi_ {2}, 1) \right\} _ {\phi_ {2} = 1} ^ {n _ {e}} \right\} _ {\gamma_ {2} = 1} ^ {n _ {x}} \right\} _ {\phi_ {1} = 1} ^ {n _ {e}} \right]} \end{array}\]

6) The value of

\[\begin{array}{r l} & {r _ {2, 1 1} = E \left[ \mathbf {x} _ {t} ^ {s} \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \right]} \\ & {\qquad = E \left[ \left\{x _ {t} ^ {s} (\gamma_ {1}, 1) \right\} _ {\gamma_ {1} = 1} ^ {n _ {x}} \left\{\epsilon_ {t + 1} (\phi_ {1}, 1) \left\{\epsilon_ {t + 1} (\phi_ {2}, 1) \left\{x _ {t} ^ {f} (\gamma_ {2}, 1) \right\} _ {\gamma_ {2}} ^ {n _ {x}} \right\} _ {\phi_ {2} = 1} ^ {n _ {e}} \right\} _ {\phi_ {1} = 1} ^ {n _ {e}} \right]} \end{array}\]

\[\begin{array}{r l} & {\mathrm{7)Thevalueof} r _ {3, 9}} \\ & {r _ {3, 9} = E \left[ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\mathbf {x} _ {t} ^ {f} \otimes \pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1}\right) ^ {\prime} \right]} \\ & {\qquad = E \left[ \left\{x _ {t} ^ {f} (\gamma_ {1}, 1) \left\{x _ {t} ^ {f} (\gamma_ {2}, 1) \right\} _ {\gamma_ {2} = 1} ^ {n _ {x}} \right\} _ {\gamma_ {1} = 1} ^ {n _ {x}} \left\{x _ {t} ^ {f} (\gamma_ {3}, 1) \left\{\epsilon_ {t + 1} (\phi_ {1}, 1) \left\{\epsilon_ {t + 1} (\phi_ {2}, 1) \right\} _ {\phi_ {2} = 1} ^ {n _ {e}} \right\} _ {\phi_ {1} = 1} ^ {n _ {e}} \right\} _ {\gamma_ {3} = 1} ^ {n _ {x}} \right]} \end{array}\]

\[\begin{array}{r l} & {\mathrm{8)Thevalueof} r _ {3, 1 0}} \\ & {r _ {3, 1 0} = E \left[ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) ^ {\prime} \right]} \\ & {\qquad = E \left[ \left\{x _ {t} ^ {f} (\gamma_ {1}, 1) \left\{x _ {t} ^ {f} (\gamma_ {2}, 1) \right\} _ {\gamma_ {2} = 1} ^ {n _ {x}} \right\} _ {\gamma_ {1} = 1} ^ {n _ {x}} \left\{\epsilon_ {t + 1} (\phi_ {1}, 1) \left\{x _ {t} ^ {f} (\gamma_ {3}, 1) \left\{\epsilon_ {t + 1} (\phi_ {2}, 1) \right\} _ {\phi_ {2} = 1} ^ {n _ {e}} \right\} _ {\gamma_ {3} = 1} ^ {n _ {x}} \right\} _ {\phi_ {1} = 1} ^ {n _ {e}} \right]} \end{array}\]

9) The value of

\[\begin{array}{l} r _ {3, 1 1} = E \left[ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \right] \\ = E \left[ \left\{x _ {t} ^ {f} (\gamma_ {1}, 1) \left\{x _ {t} ^ {f} (\gamma_ {2}, 1) \right\} _ {\gamma_ {2} = 1} ^ {n _ {x}} \right\} _ {\gamma_ {1} = 1} ^ {n _ {x}} \left\{\epsilon_ {t + 1} (\phi_ {1}, 1) \left\{\epsilon_ {t + 1} (\phi_ {2}, 1) \left\{x _ {t} ^ {f} (\gamma_ {3}, 1) \right\} _ {\gamma_ {3} = 1} ^ {n _ {x}} \right\} _ {\phi_ {2} = 1} ^ {n _ {e}} \right\} _ {\phi_ {1} = 1} ^ {n _ {e}} \right] \end{array}\]

10) The value of

\[\begin{array}{r l} & {r _ {4, 9} = E \left[ \mathbf {x} _ {t} ^ {r d} \left(\mathbf {x} _ {t} ^ {f} \otimes \pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1}\right) ^ {\prime} \right]} \\ & {\qquad = E \left[ \left\{x _ {t} ^ {r d} (\gamma_ {1}, 1) \right\} _ {\gamma_ {1} = 1} ^ {n _ {x}} \left\{x _ {t} ^ {f} (\gamma_ {2}, 1) \left\{\epsilon_ {t + 1} (\phi_ {1}, 1) \left\{\epsilon_ {t + 1} (\phi_ {2}, 1) \right\} _ {\phi_ {2} = 1} ^ {n _ {e}} \right\} _ {\phi_ {1} = 1} ^ {n _ {e}} \right\} _ {\gamma_ {2} = 1} ^ {n _ {x}} \right]} \end{array}\]

11) The value of

\[\begin{array}{r l} & {r _ {4, 1 0} = E \left[ \mathbf {x} _ {t} ^ {r d} \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) ^ {\prime} \right]} \\ & {\qquad = E \left[ \left\{x _ {t} ^ {r d} (\gamma_ {1}, 1) \right\} _ {\gamma_ {1} = 1} ^ {n _ {x}} \left\{\epsilon_ {t + 1} (\phi_ {1}, 1) \left\{x _ {t} ^ {f} (\gamma_ {2}, 1) \left\{\epsilon_ {t + 1} (\phi_ {2}, 1) \right\} _ {\phi_ {2} = 1} ^ {n _ {e}} \right\} _ {\gamma_ {2} = 1} ^ {n _ {x}} \right\} _ {\phi_ {1} = 1} ^ {n _ {e}} \right]} \end{array}\]

\[\begin{array}{l} \text {12) The value of r_{4,11}} \\ r _ {4, 1 1} = E \left[ \mathbf {x} _ {t} ^ {r d} \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \right] \\ = E \left[ \left\{x _ {t} ^ {r d} (\gamma_ {1}, 1) \right\} _ {\gamma_ {1} = 1} ^ {n _ {x}} \left\{\epsilon_ {t + 1} (\phi_ {1}, 1) \left\{\epsilon_ {t + 1} (\phi_ {2}, 1) \left\{x _ {t} ^ {f} (\gamma_ {2}, 1) \right\} _ {\gamma_ {2}} ^ {n _ {x}} \right\} _ {\phi_ {2} = 1} ^ {n _ {e}} \right\} _ {\phi_ {1} = 1} ^ {n _ {e}} \right] \end{array}\]

\[\begin{array}{r l} & {\mathrm{13)Thevalueof} r _ {5, 9}} \\ & {r _ {5, 9} = E \left[ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {s}\right) \left(\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) ^ {\prime} \right]} \\ & {\qquad = E \left[ \left\{x _ {t} ^ {f} (\gamma_ {1}, 1) \{x _ {t} ^ {s} (\gamma_ {2}, 1) \} _ {\gamma_ {2} = 1} ^ {n _ {x}} \right\} _ {\gamma_ {1} = 1} ^ {n _ {x}} \left\{x _ {t} ^ {f} (\gamma_ {3}, 1) \left\{\epsilon_ {t + 1} (\phi_ {1}, 1) \{\epsilon_ {t + 1} (\phi_ {2}, 1) \} _ {\phi_ {2} = 1} ^ {n _ {e}} \right\} _ {\phi_ {1} = 1} ^ {n _ {e}} \right\} _ {\gamma_ {3} = 1} ^ {n _ {x}} \right]} \end{array}\]

14) The value of

\[r _ {5, 1 0} = E \left[ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {s}\right) \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) ^ {\prime} \right]\]

\[= E \left[ \left\{x _ {t} ^ {f} (\gamma_ {1}, 1) \left\{x _ {t} ^ {s} (\gamma_ {2}, 1) \right\} _ {\gamma_ {2} = 1} ^ {n _ {x}} \right\} _ {\gamma_ {1} = 1} ^ {n _ {x}} \left\{\epsilon_ {t + 1} (\phi_ {1}, 1) \left\{x _ {t} ^ {f} (\gamma_ {3}, 1) \left\{\epsilon_ {t + 1} (\phi_ {2}, 1) \right\} _ {\phi_ {2} = 1} ^ {n _ {e}} \right\} _ {\gamma_ {3} = 1} ^ {n _ {x}} \right\} _ {\phi_ {1} = 1} ^ {n _ {e}} \right]\]

15) The value of

\[\begin{array}{r l} & {r _ {5, 1 1} = E \left[ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {s}\right) \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \right]} \\ & {\qquad = E \left[ \left\{x _ {t} ^ {f} (\gamma_ {1}, 1) \left\{x _ {t} ^ {s} (\gamma_ {2}, 1) \right\} _ {\gamma_ {2} = 1} ^ {n _ {x}} \right\} _ {\gamma_ {1} = 1} ^ {n _ {x}} \left\{\epsilon_ {t + 1} (\phi_ {1}, 1) \left\{\epsilon_ {t + 1} (\phi_ {2}, 1) \left\{x _ {t} ^ {f} (\gamma_ {3}, 1) \right\} _ {\gamma_ {3} = 1} ^ {n _ {x}} \right\} _ {\phi_ {2} = 1} ^ {n _ {e}} \right\} _ {\phi_ {1} = 1} ^ {n _ {e}} \right]} \end{array}\]

\[\begin{array}{r l} & {\mathrm{16)Thevalueof} r _ {6, 9}} \\ & {r _ {6, 9} = E \left[ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) ^ {\prime} \right]} \\ & {\qquad = E \left[ \left\{x _ {t} ^ {f} (\gamma_ {1}, 1) \left\{x _ {t} ^ {f} (\gamma_ {2}, 1) \left\{x _ {t} ^ {f} (\gamma_ {3}, 1) \right\} _ {\gamma_ {3} = 1} ^ {n _ {x}} \right\} _ {\gamma_ {2} = 1} ^ {n _ {x}} \right\} _ {\gamma_ {1} = 1} ^ {n _ {x}} \left\{x _ {t} ^ {f} (\gamma_ {4}, 1) \left\{\epsilon_ {t + 1} (\phi_ {1}, 1) \left\{\epsilon_ {t + 1} (\phi_ {2}, 1) \right\} _ {\phi_ {2} = 1} ^ {n _ {e}} \right\} _ {\phi_ {1} = 1} ^ {n _ {e}} \right\} _ {\gamma_ {4} = 1} ^ {n _ {x}} \right]} \end{array}\]

17) The value of

\[\begin{array}{r l} & {r _ {6, 1 0} = E \left[ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \pmb {\epsilon} _ {t + 1}\right) ^ {\prime} \right]} \\ & {\qquad = E \left[ \left\{x _ {t} ^ {f} (\gamma_ {1}, 1) \left\{x _ {t} ^ {f} (\gamma_ {2}, 1) \left\{x _ {t} ^ {f} (\gamma_ {3}, 1) \right\} _ {\gamma_ {3} = 1} ^ {n _ {x}} \right\} _ {\gamma_ {2} = 1} ^ {n _ {x}} \right\} _ {\gamma_ {1} = 1} ^ {n _ {x}} \left\{\epsilon_ {t + 1} (\phi_ {1}, 1) \left\{x _ {t} ^ {f} (\gamma_ {4}, 1) \left\{\epsilon_ {t + 1} (\phi_ {2}, 1) \right\} _ {\phi_ {2} = 1} ^ {n _ {e}} \right\} _ {\gamma_ {4} = 1} ^ {n _ {x}} \right\} _ {\phi_ {1} = 1} ^ {n _ {e}} \right]} \end{array}\]

18) The value of

\[\begin{array}{r l} & {r _ {6, 1 1} = E \left[ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \right]} \\ & {\qquad = E \left[ \left\{x _ {t} ^ {f} (\gamma_ {1}, 1) \left\{x _ {t} ^ {s} (\gamma_ {2}, 1) \left\{x _ {t} ^ {f} (\gamma_ {3}, 1) \right\} _ {\gamma_ {3} = 1} ^ {n _ {x}} \right\} _ {\gamma_ {2} = 1} ^ {n _ {x}} \right\} _ {\gamma_ {1} = 1} ^ {n _ {x}} \left\{\epsilon_ {t + 1} (\phi_ {1}, 1) \left\{\epsilon_ {t + 1} (\phi_ {2}, 1) \left\{x _ {t} ^ {f} (\gamma_ {4}, 1) \right\} _ {\gamma_ {4} = 1} ^ {n _ {x}} \right\} _ {\phi_ {2} = 1} ^ {n _ {e}} \right\} _ {\phi_ {1} = 1} ^ {n _ {e}} \right]} \end{array}\]

Notice that all the required moments needed to compute these 18 terms are available from the covariance matrix at second order. Hence we only need to compute . This is done below.

4.2.1 Efficient computing of

The matrix is very big and we therefore by hand try to simplify the summations . Note that such a simplified expression is also useful when computing auto-correlations. We first note that

\[\begin{array}{l} = \mathbf {B} \left[ \begin{array}{c} 0 \\ \mathbf {R} ^ {\prime} \\ 0 \end{array} \right] \\ \text { because } E \left[ \mathbf {z} _ {t} \boldsymbol {\xi} _ {t + 1} ^ {\prime} \right] = \left[ \begin{array}{c c c} 0 & \mathbf {R} & 0 \end{array} \right] \end{array}\]

\[\left[ \begin{array}{c c c c c c} \sigma \eta & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & (\sigma \eta \otimes \sigma \eta) & \sigma \eta \otimes \mathrm {h _ {x}} & \mathrm {h _ {x}} \otimes \sigma \eta & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ \sigma \eta \otimes \frac {1}{2} \mathrm {h _ {\sigma\sigma} \sigma^ {2}} & 0 & 0 & 0 & \sigma \eta \otimes \mathrm {h _ {x}} & \sigma \eta \otimes \tilde {\mathrm{H}} _ {\mathrm{xx}} \\ 0 & 0 & 0 & 0 & 0 & \sigma \eta \otimes \mathrm {h _ {x}} \otimes \mathrm {h _ {x}} \end{array} \right]\]

\[\left. \begin{array}{c c c c c c} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array} \right]\]

\[\times \left[ \begin{array}{c} 0 \\ \mathbf {R} ^ {\prime} \\ 0 \end{array} \right]\]

\[= \left[ \begin{array}{c c c} & 0 & \\ & 0 & \\ & 0 & \\ & 0 & \\ & 0 & \\ {\left[ \begin{array}{l l l} \mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta} & \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta} & \sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \end{array} \right] \mathbf {R} ^ {\prime}} \end{array} \right]\]

\[= \left[ \begin{array}{l l} & 0 _ {n _ {x} \times (3 n _ {x} + 2 n _ {x} ^ {2} + n _ {x} ^ {3})} \\ & 0 _ {n _ {x} \times (3 n _ {x} + 2 n _ {x} ^ {2} + n _ {x} ^ {3})} \\ & 0 _ {n _ {x} ^ {2} \times (3 n _ {x} + 2 n _ {x} ^ {2} + n _ {x} ^ {3})} \\ & 0 _ {n _ {x} \times (3 n _ {x} + 2 n _ {x} ^ {2} + n _ {x} ^ {3})} \\ & 0 _ n _ {x} ^ {2} \times (3 n _ {x + 2 n _ {x} ^ {2} + n _ {x} ^ {3})} \\ \left[ \begin{array}{l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l} \mathbf {h _ {x}} \otimes \sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta} & \sigma \boldsymbol {\eta} \otimes \mathbf {h _ {x}} \otimes \sigma \boldsymbol {\eta} & \sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta} \otimes \mathbf {h _ {x}} & \end{array} \right] \mathbf {R ^ {\prime}} \end{array} \right]\]

We see that has dimensions and has dimensions . Thus

has dimensions

\[\begin{array}{r l} & {\mathrm{Hence,}} \\ & {C o v \left[ \mathbf {z} _ {t}, \pmb {\xi} _ {t + 1} \right] \mathbf {B} ^ {\prime}} \\ & {\qquad = \left(\mathbf {B} C o v \left[ \pmb {\xi} _ {t + 1}, \mathbf {z} _ {t} \right]\right) ^ {\prime}} \\ & {\qquad = \left[ \begin{array}{l l l l l l} 0 _ {n n \times n _ {x}} & 0 _ {n n \times n _ {x}} & 0 _ {n n \times n _ {x} ^ {2}} & 0 _ {n n \times n _ {x}} & 0 _ {n n \times n _ {x} ^ {2}} & \textbf {R} \left[ \begin{array}{l l l l} \textbf {h} _ {\mathbf {x}} \otimes \sigma \pmb {\eta} \otimes \sigma \pmb {\eta} & \sigma \pmb {\eta} \otimes \textbf {h} _ {\mathbf {x}} \otimes \sigma \pmb {\eta} & \sigma \pmb {\eta} \otimes \sigma \pmb {\eta} \otimes \textbf {h} _ {\mathbf {x}} \end{array} \right] ^ {\prime} \end{array} \right]} \\ & {\mathrm{where} n n = (3 n _ {x} + 2 n _ {x} ^ {2} + n _ {x} ^ {3})} \end{array}\]

4.2.2 Computing Var

We start by noticing that

\[\begin{array} { r l } E \left[ \pmb { \xi } _ { t + 1 } \pmb { \xi } _ { t + 1 } ^ { \prime } \right] = E & { } \left[ \left[ \begin{array} { c } \pmb { \epsilon } _ { t + 1 } \\ \pmb { \epsilon } _ { t + 1 } \otimes \pmb { \epsilon } _ { t + 1 } - v e c \left( \mathbf { I } _ { n _ { e } } \right) \\ \pmb { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \\ \mathbf { x } _ { t } ^ { f } \otimes \pmb { \epsilon } _ { t + 1 } \\ \pmb { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { s } \\ \pmb { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \\ \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \otimes \pmb { \epsilon } _ { t + 1 } \\ \mathbf { x } _ { t } ^ { f } \otimes \pmb { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \\ \mathbf { x } _ { t } ^ { f } \otimes \pmb { \epsilon } _ { t + 1 } \otimes \pmb { \epsilon } _ { t + 1 } \\ \pmb { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \otimes \pmb { \epsilon } _ { t + 1 } \\ \pmb { \epsilon } _ { t + 1 } \otimes \pmb { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \\ ( \pmb { \epsilon } _ { t + 1 } \otimes \pmb { \epsilon } _ { t + 1 } \otimes \pmb { \epsilon } _ { t + 1 } ) - E [ ( \pmb { \epsilon } _ { t + 1 } \otimes \pmb { \epsilon } _ { t + 1 } \otimes \pmb { \epsilon } _ { t + 1 } ) ] & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & & \\ & \\ & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & & \\ & = \left[ \begin{array} l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l | p | | p | | p | | p | | p | | p | | p | | p | | p | | p | | p | | p | | p | | p | | p | | p | | p | | p | | p | | p | | p | | p | | p | | p | | p | | p | | p | | p | | p | | p | | p | | p | | p | | p | p | | p | | p | | p | | p | | p | | p | | p | | p | | p | | p | | p | | p | | p | | p | | p | | p | | p | | p | | p | | p | | p | | p | | p | | p | | p | | p | | p | | p | | p _ { 1 , 1 } & p _ { 1 , 2 } & 0 & 0 & p _ { 1 , 5 } & p _ { 1 , 6 } & p _ { 1 , 7 } & p _ { 1 , 8 } & 0 & 0 & 0. 5 5 5. 5 5 5. 5 5 5. 5 5 5. 5 5 5. 5 5 5. 5 5 5. 5 5 5 . 5 5 5 . 5 5 5 . 5 5 5 . 5 5 5 . 5 5 5 . 5 5 5 . 5 5 5 . 5 5 5 . 5 5 5 . 5 5 5 . 5 5 5 . 5 5 5 . 5 5 5 . 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p _ { 3 , 3 } & p _ { 3 , 4 } & p _ { 3 , 5 } & p _ { 3 , 6 } & p _ { 3 , 7 } & p _ { 3 , 8 } & p _ { 3 , 9 } & p _ { 3 , 1 0 } & p _ { 3 , 1 1 } & p _ { 3 , 1 2 } \\ p _ { 4 , 4 } & p _ { 4 , 5 } & p _ { 4 , 6 } & p _ { 4 , 7 } & p _ { 4 , 8 } & p _ { 4 , 9 } & p _ { 4 , 1 0 } & p _ { 4 , 1 1 } & p _ { 4 , 1 2 } \\ p _ { 5 , 5 } & p _ { 5 , 6 } & p _ { 5 , 7 } & p _ { 5 , 8 } & p _ { 5 , 9 } & p _ { 5 , 1 0 } & p _ { 5 , 1 1 } & p _ { 5 , 1 2 } \\ p _ { 6 , 6 } & p _ { 6 , 7 } & p _ { 6 , 8 } & p _ { 6 , 9 } & p _ { 6 , 1 0 } & p _ { 6 , 1 1 } & p _ { 6 , 1 2 } \\ p _ { 7 , 7 } & p _ { 7 , 8 } & p _ { 7 , 9 } & p _ { 7 , 1 0 } & p _ { 7 , 1 1 } & p _ { 7 , 1 2 } \\ p _ y , y , y , y , y , y , y , y , y , y , y , y , y , y , y , y , y , y , y , y , y , y , y , y , y , y , y , y , y , y , y , y , y , y , y , y , y , y , y , y , y , y , y , y , y , y , y , y , y , y , y [ ( a [ b [ c [ d [ e [ f [ g [ h [ i [ j [ k [ m [ n ] [ o [ n ] [ o [ n ] [ o [ n ] [ o [ n ] [ o [ n ] [ o [ n ] [ o [ n ] [ o [ n ] [ o [ n ] [ o [ n ] [ o [ n ] [ o [ n ] [ o [ n ] [ o [ n ] [ o [ n ] [ o [ n ] [ o [ n ] [ o [ n ] [ o [ n ] [ o [ n ] [ u ] ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ), and the total sum of all elements is calculated by the sum of all elements in each element, with the total sum of all elements in each element, with the total sum of all elements in each element, with the total sum of all elements in each element, with the total sum of all elements in each element, with the total sum of all elements in each element, with the total sum of all elements in each element, with the total sum of all elements in each element, with the total sum of all elements in each element, with the total sum of all elements in each element, with the total sums of all elements in each element, with the total sum of all elements in each element, with the total sum of all elements in each element, with the total sum of all elements in each element, with the total sum of all elements in each element, with the total sum of all elements in each element, with the total sum of all elements in each element, with the total sum of all elements in each element, with the total sum of all elements in each element, with the total sum (a b c d) and b c d (b c d) and b c d (b c d) and b c d (b c d) and b c d (b c d) and b c d (b c d) and b c d (b c d) and b c d (b c d) and b c d (b c d) and b c d (b c d) and b c d (b c d) and b c d (b c d) and b c d (b e q u a d) a b c d (a b c d) a b c d (a b c d) a b c d (a b c d) a b c d (a b c d) a b c d (a b c d) a b c d (a b c d) a b c d (a b c d) a b c d (a b c d) a b c d (a b c d) a b c d (a b c d) a b c d (a b e q u a d) a b c d (a b e q u a d) a b c d (a b e q u a d) a b c d (a b e q u a d) a b c d (a b e q u a d) a b c d (a b e q u a d) a b c d (a b e q u a d) a b c d (a b e q u a d) a b c d (a e q u a d) a b c d (a e q u a d) a b c d (a e q u a d) a b c d (a e q u a d) a b c d (a e q u a d) a b c d (a e q u a d) a b c d (a e q u a d) a b c d (a e q u a d) a b c d (a e q u u a d) a b c d (a e q u u a d) a b c d (a e q u u a d) a b c d (a e q u u a d) a b c d (a e q u u a d) a b c d (a e q u u a d) a b c d (a e q u u a d) a b c d (a e q u u a d) a b c d (\texttt {\textit{ex}} ;\texttt{\textit{ex}} ;\texttt{\textit{ex}} ;\texttt{\textit{ex}} ;\texttt{\textit{ex}} ;\texttt{\textit{ex}} ;\texttt{\textit{ex}} ;\texttt{\textit{ex}} ;\texttt{\textit{ex}} ;\texttt{\textit{ex}} ;\texttt{\textit{ex}} ;\texttt{ex} ;\texttt{\textit{ex}} ;\texttt{\textit{ex}} ;\texttt{\textit{ex}} ;\texttt{\textit{ex}} ;\texttt{\textit{ex}} ;\texttt{\textit{ex}} ;\texttt{\textit{ex}} ;\texttt{\textit{ex}} ;\texttt{\textit{ex}} ;\texttt{\textit{ex}},\texttt{\textit{ex}} ;\texttt{\textit{ex}} ;\texttt{\textit{ex}} ;\texttt{\textit{ex}} ;\texttt{\textit{ex}} ;\texttt{\textit{ex}} ;\texttt{\textit{ex}} ;\texttt{\textit{ex}} ;\texttt{\textit{ex}} ;\texttt{\textit{ex}};,\texttt{\textit{ex}} ;,\texttt{\textit{ex}} ;,\texttt{\textit{ex}} ;,\texttt{\textit{ex}} ;,\texttt{\textit{ex}} ;,\texttt{\textit{ex}} ;,\texttt{\textit{ex}} ;,\texttt{\textit{ex}} ;,\texttt{\textit{ex}} ;,\texttt{\textit{ex}} ;,\texttt{\textit{ex}};,\texttt{\textit{ex}} ;,\texttt{\textit{ex}} ;,\texttt{\textit{ex}} ;,\texttt{\textit{ex}} ;,\texttt{\textit{ex}} ;,\texttt{\textit{ex}} ;,\texttt{\textit{ex}} ;,\texttt{\textit{ex}} ;,\texttt{\textit{ex}} ;,,\texttt{\textit{x}} ;,\texttt{\textit{x}} ;,\texttt{\textit{x}} ;,\texttt{\textit{x}} ;,\texttt{\textit{x}} ;,\texttt{\textit{x}} ;,\texttt{\textit{x}} ;,\texttt{\textit{x}} ;,\texttt{\textit{x}} ;,\texttt{\textit{x}} ;,\texttt{\textit{x}} ;,\texttt{\textit{x}} ;,,\texttt{\textit{x}} ;,\texttt{\textit{x}} ;,\texttt{\textit{x}} ;,\texttt{\textit{x}} ;,\texttt{\textit{x}} ;,\texttt{\textit{x}} ;,\texttt{\textit{x}} ;,\texttt{\textit{x}} ;,\texttt{\textit{x}} ;,\texttt{\textit{x}} ;,\texttt{\textit{x}} ;,\texttt{\textit{x}} ;,\texttt{\textit{x}} ;,\texttt{\textit{x}} ;,\texttt{\textit{x}} ;,\texttt{\textit{x}} ;,\texttt{\textit{x}} ;,\texttt{\textit{x}} ;,\texttt{\textit{x}} ;,\mathrm {\alpha }\mathrm {\beta }\mathrm {\eta}\mathrm {\mu}\mathrm {\eta}\mathrm {\rho}\mathrm {\eta}\mathrm {\sigma}\mathrm {\eta}\mathrm {\mu}\mathrm {\eta}\mathrm {\sigma}\mathrm {\eta}\mathrm {\sigma}\mathrm {\eta}\mathrm {\sigma}\mathrm {\eta}\mathrm {\sigma}\mathrm {\eta}\mathrm {\sigma}\mathrm {\eta}\mathrm {\sigma}\mathrm {\eta}\mathrm {\sigma}\mathrm {\eta}\mathrm {\sigma}\mathrm {\eta}\mathrm {\sigma}\mathrm {\eta}\mathrm {\sigma}\mathrm {\eta}\mathrm {\sigma}\mathrm {\eta}\mathrm {\sigma}\mathrm {\eta}\mathrm {\sigma}\mathrm {\tau}\mathrm {\tau}\mathrm {\tau}\mathrm {\tau}\mathrm {\tau}\mathrm {\tau}\mathrm {\tau}\mathrm {\tau}\mathrm {\tau}\mathrm {\tau}\mathrm {\tau}\mathrm {\tau}\mathrm {\tau}\mathrm {\tau}\mathrm {\tau}\mathrm {\tau}\mathrm {\tau}\mathrm {\tau}\mathrm {\tau}\mathrm {\tau}\mathrm {\tau}\mathrm {\tau}\mathrm {\tau}\mathrm {\tau}\mathrm {\tau}\mathrm {\gamma }\mathrm {\gamma }\mathrm {\gamma }\mathrm {\gamma }\mathrm {\gamma }\mathrm {\gamma }\mathrm {\gamma }\mathrm {\gamma }\mathrm {\gamma }\mathrm {\gamma }\mathrm {\gamma }\mathrm {\gamma }\mathrm {\gamma }\mathrm {\gamma }\mathrm {\gamma }\mathrm {\gamma }\mathrm {\gamma }\mathrm {\gamma }\mathrm {\gamma }\mathrm {\gamma }\mathrm {\gamma }\mathrm {\gamma }\mathrm {\gamma }\mathrm {\gamma }\mathrm {\gamma }\mathrm {\gamma}\end{array} \right] = P_{i}P_{i}P_{i}P_{i}P_{i}P_{i}P_{i}P_{i}P_{i}P_{i}P_{i}P_{i}P_{i}P_{i}P_{i}P_{i}P_{i}P_{i}P_{i}P_{i}P_{i}P_{i}P_{i}P_{i}P_{i}P_{i}S_{i}S_{i}S_{i}S_{i}S_{i}S_{i}S_{i}S_{i}S_{i}S_{i}S_{i}S_{i}S_{i}S_{i}S_{i}S_{i}S_{i}S_{i}S_{i}S_{i}S_{i}S_{i}S_{i}S_{i}S_{i} S_{i}S_{i}S_{i}S_{i}S_{i}S_{i}S_{i}S_{i}S_{i}S_{i}S_{i}S_{i}S_{i}S_{i}S_{i}S_{i}S_{i}S_{i}S_{i}S_{i}S_{i}S_{i}S_{i},\]

Only stating the elements on and above the diagonal. We first notice that can be computed as:

We next compute all the elements in this matrix. The method is illustrated below

\[\begin{array}{l} \text {2) for p_{1,2}} \\ E \left[ \boldsymbol {\epsilon} _ {t + 1} \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} - v e c \left(\mathbf {I} _ {n _ {e}}\right)\right) ^ {\prime} \right] = E \left[ \boldsymbol {\epsilon} _ {t + 1} \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) ^ {\prime} \right] \\ = E \left[ \{\epsilon_ {t + 1} (\phi_ {1}, 1) \} _ {\phi_ {1} = 1} ^ {n _ {e}} \left(\left\{\epsilon_ {t + 1} (\phi_ {2}, 1) \{\epsilon_ {t + 1} (\phi_ {3}, 1) \} _ {\phi_ {3} = 1} ^ {n _ {e}} \right\} _ {\phi_ {2} = 1} ^ {n _ {e}}\right) ^ {\prime} \right] \end{array}\]

Hence the quasi MATLAB codes are :

end

\[\begin{array}{r l} & {\mathrm{3)for} p _ {1, 5}} \\ & {E \left[ \pmb {\epsilon} _ {t + 1} \left(\pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {s}\right) ^ {\prime} \right] = E \left[ \pmb {\epsilon} _ {t + 1} \left(\pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {s}\right) ^ {\prime} \right]} \\ & {\qquad = E \left[ \left(\pmb {\epsilon} _ {t + 1} \otimes 1\right) \left(\pmb {\epsilon} _ {t + 1} ^ {\prime} \otimes \left(\mathbf {x} _ {t} ^ {s}\right) ^ {\prime}\right) \right]} \\ & {\qquad = E \left[ \pmb {\epsilon} _ {t + 1} \pmb {\epsilon} _ {t + 1} ^ {\prime} \otimes \left(\mathbf {x} _ {t} ^ {s}\right) ^ {\prime} \right]} \\ & {\qquad = \mathbf {I} \otimes E \left[ \left(\mathbf {x} _ {t} ^ {s}\right) ^ {\prime} \right]} \end{array}\]

\[\begin{array}{l} 4) \text {for} p _ {1, 6} \\ E \left[ \boldsymbol {\epsilon} _ {t + 1} \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \right] = E \left[ (\boldsymbol {\epsilon} _ {t + 1} \otimes 1) \left(\boldsymbol {\epsilon} _ {t + 1} ^ {\prime} \otimes \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime}\right) \right] \\ = E \left[ \boldsymbol {\epsilon} _ {t + 1} \boldsymbol {\epsilon} _ {t + 1} ^ {\prime} \otimes \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \right] \\ = \mathbf {I} \otimes E \left[ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \right] \end{array}\]

\[\begin{array}{r l} & {\mathrm{5)for} p _ {1, 7}} \\ & {E \left[ \pmb {\epsilon} _ {t + 1} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \pmb {\epsilon} _ {t + 1}\right) ^ {\prime} \right] = E \left[ 1 \otimes \pmb {\epsilon} _ {t + 1} \left(\left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \otimes \pmb {\epsilon} _ {t + 1} ^ {\prime}\right) \right]} \\ & {\qquad = E \left[ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \otimes \pmb {\epsilon} _ {t + 1} \pmb {\epsilon} _ {t + 1} ^ {\prime} \right]} \\ & {\qquad = E \left[ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \right] \otimes \mathbf {I}} \end{array}\]

4.3 Method 2: Formulas for the first and second moments

This section computes first and second moments using a slightly different representation of the third-order system than stated above. (Basically, this was the first representation we considered for computing these moments). The advantage of this method is that it compared to Method 1 is less memory intensive because some of the matrix multiplications are done by hand.

\[\begin{array}{r l} & {\mathrm{Wefirstrecallthat}} \\ & {\left(\mathbf {x} _ {t + 1} ^ {f} \otimes \mathbf {x} _ {t + 1} ^ {s}\right) = \left(\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}\right) \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {s}\right) + \left(\mathbf {h} _ {\mathbf {x}} \otimes \tilde {\mathbf {H}} _ {\mathbf {x x}}\right) \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) + \left(\mathbf {h} _ {\mathbf {x}} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}\right) \mathbf {x} _ {t} ^ {f}} \\ & {\qquad + (\sigma \pmb {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) (\pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {s}) + (\sigma \pmb {\eta} \otimes \tilde {\mathbf {H}} _ {\mathbf {x x}}) (\pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) + (\sigma \pmb {\eta} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}) \pmb {\epsilon} _ {t + 1}} \end{array}\]

\[\begin{array} { r l } & { \mathrm{We~also~know~that} } \\ & { \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } = ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) + ( \mathbf { h } _ { \mathbf { x } } \otimes \sigma \boldsymbol { \eta } ) ( \mathbf { x } _ { t } ^ { f } \otimes \boldsymbol { \epsilon } _ { t + 1 } ) } \\ & { \qquad + ( \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x } } ) ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } ) + ( \sigma \boldsymbol { \eta } \otimes \sigma \boldsymbol { \eta } ) ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \boldsymbol { \epsilon } _ { t + 1 } ) } \\ & { \mathrm{so} } \\ & { \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } = ( \mathbf { h } _ { \mathbf { x } } \mathbf { x } _ { t } ^ { f } + \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + 1 } ) \otimes ( ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) + ( \mathbf { h } _ { \mathbf { x } } \otimes \sigma \boldsymbol { \eta} ) ( \mathbf { x } _ { t } ^ { f } \otimes \boldsymbol { \epsilon } _ { t + 1 } ) } \\ & { \qquad + ( \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x } } ) ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } ) + ( \sigma \boldsymbol { \eta } \otimes \sigma \boldsymbol { \eta} ) ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \boldsymbol { \epsilon } _ { t + 1} ) ) } \\ & { = \mathbf { h } _ { \mathbf { x } } \mathbf { x } _ { t } ^ { f } \otimes ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) + \mathbf { h } _ { \mathbf { x } } \mathbf { x } _ { t } ^ { f } \otimes ( \mathbf { h } _ { \mathbf { x } } \otimes \sigma \boldsymbol { \eta} ) ( \mathbf { x } _ { t } ^ { f } \otimes \boldsymbol { \epsilon } _ { t + 1 } ) } \\ & + [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ ] / i n d ] / i n d ] / i n d ] / i n d ] / i n d ] / i n d ] / i n d ] / i n d ] / i n d ] / i n d ] / i n d ] / i n d ] / i n d ] / i n d ] / i n d ] / i n d ] / i n d ] / i n d , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ ] / i n d ] / i n d ] / i n d ] / i n d ] / i n d ] / i n d ] / i n d ] / i n d ] / i n d ] / i n d ] / i n d ] / i n d ] / i n d ] / i n d ] / i n d ] / i n d ] / i n d ] / i n d ] / i n d , . . . . . . . . , . . . , . , . , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | I = I ^ {\prime} I ^ {\prime} I ^ {\prime} I ^ {\prime} I ^ {\prime} I ^ {\prime} I ^ {\prime} I ^ {\prime} I ^ {\prime} I ^ {\prime} I ^ {\prime} I ^ {\prime} I ^ {\prime} I ^ {\prime} I ^ {\prime} I ^ {\prime} I ^ {\prime} I ^ {\prime} I ^ {\prime} I ^ {\prime} I ^ {\prime}, I ^ {\prime} I ^ {\prime} I ^ {\prime} I ^ {\prime} I ^ {\prime} I ^ {\prime} I ^ {\prime} I ^ {\prime} I ^ {\prime} I ^ {\prime} I ^ {\prime} I ^ {\prime} I ^ {\prime} I ^ {\prime} I ^ {\prime} I ^ {\prime} I ^ {\prime} I ^ {\prime} I ^ {\prime} I ^ {\prime}\]

\[\begin{array}{l} + (\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) (\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}) + (\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}) (\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}) \\ + (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) (\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) + (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}) (\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}) \\ + (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) (\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}) + (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}) (\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}) \\ = (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) + \mathbf {u} _ {t + 1} \\ \text {where} \mathbf {u} _ {t + 1} \equiv (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}) (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}) \\ + (\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) (\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}) + (\mathbf {h} _ {\mathcal {X}} \otimes \sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}) (\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}) \\ + (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \dot {\mathbf {h}} _ {\mathbf {x}}) (\boldsymbol {\epsilon_ {t + 1}} \otimes \dot {\mathbf {x}} _ {t} ^ {f} \otimes \dot {\mathbf {x}} _ {t} ^ {f}) + (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}) (\boldsymbol {\epsilon_ {t + 1}} \otimes \dot {\mathbf {x}} _ {t} ^ {f} \otimes \boldsymbol {\epsilon_ {t + 1}}) \\ + (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta} \otimes \dot {\mathbf {h}} _ {\mathbf {x}}) (\boldsymbol {\epsilon_ {t + 1}} \otimes \boldsymbol {\epsilon_ {t + 1}} \otimes \dot {\mathbf {x}} _ {t} ^ {f}) + (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}) (\boldsymbol {\epsilon_ {t + 1}} \otimes \boldsymbol {\epsilon_ {t + 1}} \otimes \boldsymbol {\epsilon_ {t + 1}}) \end{array}\]

Thus we can construct the following extended system

\[\begin{array} { r l } & \left[ \begin{array} { c } \mathbf { x } _ { t + 1 } ^ { f } \\ \mathbf { x } _ { t + 1 } ^ { s } \\ \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } \\ \mathbf { x } _ { t + 1 } ^ { r d } \\ \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { s } \\ \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } \end{array} \right] = \left[ \begin{array} { c c c c c c } \mathbf { h } _ { \mathbf { x } } & \mathbf { 0 } _ { n _ { x } \times n _ { x } } & \mathbf { 0 } _ { n _ { x } \times n _ { x } ^ { 2 } } & \mathbf { 0 } _ { n _ { x } \times n _ { x } } & \mathbf { 0 } _ { n _ { x } \times n _ { x } ^ { 2 } } & \mathbf { 0 } _ { n _ { x } \times n _ { x } ^ { 3 } } \\ \mathbf { 0 } _ { n _ { x } \times n _ { x } } & \mathbf { h } _ { \mathbf { x } } & \tilde { \mathbf { H } } _ { \mathbf { x x } } & \mathbf { 0 } _ { n _ { x } \times n _ { x } } & \mathbf { 0 } _ { n _ { x } \times n _ { x } ^ { 2 } } & \mathbf { 0 } _ { n _ { x } \times n _ { x } ^ { 3 } } \\ \mathbf { 0 } _ { n _ { x } ^ { 2 } \times n _ { x } } & \mathbf { 0 } _ { n _ { x } ^ { 2 } \times n _ { x } } & \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } & \mathbf { 0 } _ { n _ { x } ^ { 2 } \times n _ { x } } & \mathbf { 0 } _ { n _ { x } ^ { 2 } \times n _ { x } ^ { 2 } } & \mathbf { 0 } _ { n _ { x } ^ { 2 } \times n _ { x } ^ { 3 } } \\ \frac { 3 } { 6 } \mathbf { h } _ { \sigma \sigma \mathbf { x} } \sigma ^ { 2 } & \mathbf { 0 } _ { n _ { x } \times n _ { x } } & \mathbf { 0 } _ { n _ { x } \times n _ { x } ^ { 2 } } & \mathbf { h } _ { \mathbf { x } } & 2 \tilde {\mathbf { H }} _ { \mathbf { x x } } & \tilde {\mathbf { H }} _ { \mathbf { x x x } } \\ ( \mathbf { h } _ { \mathbf { x} } \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } ) & \mathbf { 0 } _ { n _ { x } ^ { 2 } \times n _ { x } } & \mathbf { 0 } _ { n _ { x } ^ { 2 } \times n _ { x } ^ { 2 } } & \mathbf { 0 } _ { n _ { x } ^ { 2 } \times n _ { x } } & ( \mathbf { h } _ { \mathbf { x} } \otimes \mathbf { h } _ { \mathbf { x} ) } & ( \mathbf { h } _ { \mathbf { x} } \otimes \tilde {\mathbf { H }} _ { \mathbf { x x} ) } \\ \mathbf { 0 } _ { n _ { x } ^ { 3 } \times n _ { x } } & \mathbf { 0 } _ { n _ { x } ^ { 3 } \times n _ { x } } & \mathbf { 0 } _ { n _ { x } ^ { 3 } \times n _ { x } ^ { 2 } } & \mathbf { 0 } _ { n _ { x } ^ { 3 } \times n _ { x } } & \mathbf { 0 } _ { n _ { x } ^ { 3 } \times n _ { x } ^ { 2 } } & ( \mathbf { h } _ { \mathbf { x} } \otimes \mathbf { h} _ { \mathbf { x} ) }\otimes \mathbf {\hat {} h} _ {\mathbf {\hat {} x} )} \\ \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ + & \\ & \\ & + \\ & \\ & + \\ & \\ & + \\ & \\ & + \\ & \\ & + \\ & \\ & + \\ & \\ & + \\ & \\ & + \\ & \\ & + \\ & \\ & + \\ & \\ & + \\ & \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & +; - E [ e ] = - E [ e ] = - E [ e ] = - E [ e ] = - E [ e ] = - E [ e ] = - E [ e ] = - E [ e ] = - E [ e ] = - E [ e ] = - E [ e ] = - E [ e ] = - E [ e ] = - E [ e ] = - E [ e ] = - E [ e ] = - E [ e ] = - E [ E ] = - E [ E ] = - E [ E ] = - E [ E ] = - E [ E ] = - E [ E ] = - E [ E ] = - E [ E ] = - E [ E ] = - E [ E ] = - E [ E ] = - E [ E ] = - E [ E ] = - E [ E ] = - E [ E ] = - E [ E ] = - E [ E ] = -E [ E ] = -E [ E ] = -E [ E ] = -E [ E ] = -E [ E ] = -E [ E ] = -E [E ] = -E [E ] = -E [E ] = -E [E ] = -E [E ] = -E [E ] = -E [E ] = -E [E ] = -E [E ] = -E [E ] = -E [E ] = -E [E ] = -E [E ] = -E [E ] = -E [E ] = -E [E ] = -E [E ] = -F [E ] = -F [E ] = -F [E ] = -F [E ] = -F [E ] = -F [E ] = -F [E ] = -F [E ] = -F [E ] = -F [E ] = -F [E ] = -F [E ] = -F [E ] = -F [E ] = -F [E ] = -F [E ] = -F [E ] ,\\ ( e ) ^ - s / a t a r i g u a r i g u a r i g u a r i g u a r i g u a r i g u a r i g u a r i g u a r i g u a r i g u a r i g u a r i g u a r i g u a r i g u a r i g u a r i g u a r i g u a r i g u a r i g u a r i g u a r i g U A R S O U T I N T I N T I N T I N T I N T I N T I N T I N T I N T I N T I N T I N T I N T I N T I N T I N T I N T I N T I N T I N T I N T I N T I N T I N T I N T I N T I N T I N T I N T I N T I N T I N T I N T I N t o m p o w e r e s s o m p o w e r e s s o m p o w e r e s s o m p o w e r e s s o m p o w e r e s s o m p o w e r e s s o m p o w e r e s s o m p o w e r e s s o m p o w e r e s s o m p o w e r e s s o m p o w e r e s t h e t h e t h e t h e t h e t h e t h e t h e t h e t h e t h e t h e t h e t h e t h e t h e t h e t h e t h e t h e t h e t h e t h e t h e t h e t h e t h e t h e t h e t h e t h e t h e t h e t h e l o d i v i d i v i d i v i d i v i d i v i d i v i d i v i d i v i d i v i d i v i d i v i d i v i d i v i d i v i d i v i d i v i d i v i d i v i d i v i d i v i d i v i d i v i d i v i d i v i d i v i d i v j o l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y b o w k o w k o w k o w k o w k o w k o w k o w k o w k o w k o w k o w k o w k o w k o w k o w k o w k o w k o w k o w k o w k o w k o w k o w k o w k o w k o w k o w k o w k o w k o w k o w k o w k o w k c o m p o w e r e s s o m p o w e r e s s o m p o w e r e s s o m p o w e r e s s o m p o w e r e s s o m p o w e r e s s o m p o w e r e s s o m p o w e r e s s o m p o w e r e s s o m p o w e r e s , c o m p o w e r e s s o m p o w e r e s s o m p o w e r e s s o m p o w e r e s s o m p o w e r e s s o m p o w e r e s s o m p o w e r e s s o m p o w e r e s s , c o m p o w e r e s s , c o m p o w e r e s s , c o m p o w e r e s s , c o m p o w e r e s s , c o m p o w e r e s s , c o m p o w e r e s s , c o m p o w e r e s s , c o m p o w e r e s s , c o m p . c o m p . c o m p . c o m p . c o m p . c o m p . c o m p . c o m p . c o m p . c o m p . c o m p . c o m p . c o m p . c o m p . c o m p . c o m p . c o m p . c o m p . c o m p . c o m p . c o m p. c o m p . c o m p . c o m p . c o m p . c o m p . c o m p . c o m p . c o m p . c o m p . c o m p . c o m p . c o m p . c o m p . c o m p . c o m p . c o m p . c o m p . c o m p . c o m p . c o m p ~ ; ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ $ ! $ : $ ! $ : $ ! $ : $ ! $ : $ ! $ : $ ! $ : $ ! $ : $ ! $ : $ ! $ : $ ! $ : $ ! $ : $ ! $ : $ ! $ : $ ! $ : $ ! $ : $ ! $ : $ ! $ : $ ! $ : $ ! $ : $ ! $ : $ ! $ : $ ! $ : $ ! $ : $ ! $ : $ ! $ : $ ! $: $ ! $ : $ ! $ : $ ! $ : $ ! $ : $ ! $ : $ ! $ : $ ! $ : $ ! $ : $ !$ : $ !$ : $ !$ : $ !$ : $ !$ : $ !$ : $ !$ : $ !$ : $ !$ : $ !$ : $ !$ : $ !$ : $ !$ : $ !$ : $ !$ : $ !$ : $ !$ : $ !$ : $ !$ : $ !$ : $ !$ : $ !$ : $ !$ : $ !$ : $ !$ : $ !|,\\ (e) ^ - s / a t a r i g u a r i g u a r i g u a r i g u a r i g u a r i g u a r i g u a r i g u a r i g u a r i g u a r i g u a r i g u a r i g u a r i g u a r i g u a r i g u a r i g u a r i g u a r i f , (e) ^ - s / a t a r i g u a r | , (e) ^ - s / a t a r | , (e) ^ - s / a t a r | , (e) ^ - s / a t a r | , (e) ^ - s / a t a r | , (e) ^ - s / a t a r | , (e) ^ - s / a t a r | , (e) ^ - s / a t a r | , (e) ^ - s / a t a r | , (e) ^-s / a t / a t | , (e) ^-s / a t / a t | , (e) ^-s / a t / a t | , (e) ^-s / a t / a t | , (e) ^-s / a t / a t | , (e) ^-s / a t / a t | , (e) ^-s / a t / a t | , (e) ^-s / a t / a t | > C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D C D CD A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A b A b A b A b A b A b A b A b A b A b A b A b A b A b A b A b A b A b A b A b A b A b A b A b A b A b A b A b A b A b A b A b A b A b A b A b A b A b A b A b A b A b A b A b A b A b A b A b A b A b A f ,a ,b ,c ,d ,e ,f ,g ,h ,i ,j ,k ,l ,m ,n ,o ,p ,q ,r ,w ,v ,w< fcel>+ F and G on the left of each other in the right of each other in the left of each other in the left of each other in the left of each other in the left of each other in the left of each other in the left of each other in the left of each other in the left of each other in the left of each other in the left of each other in the left of each other in the left of each other in the left of each other in the left of each other in the left of each other in the left of each other in the left of each other, and then, we can be used to use the plot as an example. We can be used to use the plot as an example. We can be used to use the plot as an example. We can be used to use the plot as an example. We can be used to use the plot as an example. We can be used to use the plot as an example. We can be used to use the plot as an example. We can be used to use the plot as an example. We can be used to use the plot as an example. We can be used for all or all possible points, and we can be used to use the plot as an example. We can be used to use the plot as an example. We can be used to use the plot as an example. We can be used to use the plot as an example. We can be used to use the plot as an example. We can be used to use the plot as an example. We can be used to use the plot as an example. We can be used to use the plot as an example. We can be using the plot as an example. We can be used to use the plot as an example. We can be used to use the plot as an example. We can be used to use the plot as an example. We can be used to use the plot as an example. We can be used to use the plot as an example. We can be used to use the plot as an example. We can be used to use the plot as an example. We can be used to use the plot as another example. We can be used to use the plot as another example. We can be used to use the plot as another example. We can be used to use the plot as another example. We can be used to use the plot as another example. We can be used to use the plot as another example. We can be used to use the plot as another example. We can be used to use the plot as another example. We can be used to use the plot as another example. We cannot be used to use the plot as another example. We cannot be used to use the plot as another example. We cannot be used to use the plot as another example. We cannot be used to use the plot as another example. We cannot be used to use the plot as another example. We cannot be used to use the plot as another example. We cannot be used to use the plot as another example. We cannot be used to use the plot as another example. We cannot be used to include our own data from our own data source, and we are not included in this image without any such text. The image is not included in this image without any such text. The image is not included in this image without any such text. The image is not included in this image without any such text. The image is not included in this image without any such text. The image is not included in this image without any such text. The image is not included in this image without any such text. The image is not included in this image without any such text. The image is not included in this image without any such time, time, and duration, time, and duration, time, and duration, time, and duration, time, and duration, time, and duration, time, and duration, time, and duration, time, and duration, time, and duration, time, and duration, time, and duration, time, and duration, time, and duration, time, and duration, time, and duration, time, and duration, time, and duration, time, and duration, time, and duration, time, and time, and duration, time, and duration, time, and duration, time, and duration, time, and duration, time, and duration, time, and duration, time, and duration, time, and duration, time, and duration, time, and duration, time, and duration, time, and duration, time, and duration, time, and duration, time, and duration, time, and duration, time, and duration, time, and duration, time, and duration,. For all times it is given by us at point z=15:15:15:15:15:15:15:15:15:15:15:15:15:15:15:15:15:15:15:15:15:15:15:15:15:15:15:15:15:15:15:15:15:15:48:48:48:48:48:48:48:48:48:48:48:48:48:48:48:48:48:48:48:48:48:48:48:48:48:48:48:48:48:48:48:48:48:48 :-79999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999A WtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWt WtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWtWt W t W t W t W t W t W t W t W t W t W t W t W t W t W t W t W t W t W t W t W t W t W t W t W t W t W t W t W t W t W t W t W t W t W t W t W t W t W t W t W t W t W t W t W t W t W t W t W t W t W t W_t W_t W_t W_t W_t W_t W_t W_t W_t W_t W_t W_t W_t W_t W_t W_t W_t W_t W_t W_t W_t W_t W_t W_t W_t W_t W_t W_t W_t W_t W_t W_t W_t W_t W_t W_t W_t W_t W_t W_t W_t W_t W_t W_t W_t W_t W_t W_t W_t W_t W_i^* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *< lcel>< nl>\]

\[\mathbf {z} _ {t + 1} = \mathbf {c} + \mathbf {A} \mathbf {z} _ {t} + \tilde {\boldsymbol {\xi}} _ {t + 1}\]

Hence, . The expression for the controls are as before, i.e.

\[\mathbf {y} _ {t} = \mathbf {D} \mathbf {z} _ {t} + \frac {1}{2} \mathbf {g} _ {\sigma \sigma} \sigma^ {2} + \frac {1}{6} \mathbf {g} _ {\sigma \sigma \sigma} \sigma^ {3}\]

The mean values are

\[E [ \mathbf {z} _ {t} ] = (\mathbf {I} _ {3 n _ {x} + 2 n _ {x} ^ {2} + n _ {x} ^ {3}} - \mathbf {A}) ^ {- 1} \mathbf {c}.\]

\[E [ \mathbf {y} _ {t} ] = \mathbf {D} E [ \mathbf {z} _ {t} ] + \frac {1}{2} \mathbf {g} _ {\sigma \sigma} \sigma^ {2} + \frac {1}{6} \mathbf {g} _ {\sigma \sigma \sigma} \sigma^ {3}\]

We showed above that

\[V a r \left[ \mathbf {z} _ {t + 1} \right] = \mathbf {A} V a r [ \mathbf {z} _ {\mathbf {t}} ] \mathbf {A} ^ {\prime} + \mathbf {B} V a r \left[ \boldsymbol {\xi} _ {t + 1} \right] \mathbf {B} ^ {\prime} + \mathbf {A} C o v \left[ \mathbf {z} _ {t}, \boldsymbol {\xi} _ {t + 1} \right] \mathbf {B} ^ {\prime} + \mathbf {B} C o v \left[ \boldsymbol {\xi} _ {t + 1}, \mathbf {z} _ {t} \right] \mathbf {A} ^ {\prime}\]

which is equivalent to

\[V a r \left[ \mathbf {z} _ {t + 1} \right] = \mathbf {A} V a r [ \mathbf {z} _ {\mathbf {t}} ] \mathbf {A} ^ {\prime} + V a r \left[ \tilde {\boldsymbol {\xi}} _ {t + 1} \right] + \mathbf {A} C o v \left[ \mathbf {z} _ {t}, \boldsymbol {\xi} _ {t + 1} \right] \mathbf {B} ^ {\prime} + \mathbf {B} C o v \left[ \boldsymbol {\xi} _ {t + 1}, \mathbf {z} _ {t} \right] \mathbf {A} ^ {\prime}\]

We have already known how to compute the and . Hence we only need to compute . Recall from above that

\[\tilde {\boldsymbol {\xi}} _ {t + 1} \equiv \left[ \begin{array}{c} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} \\ \mathbf {0} _ {n _ {x} \times 1} \\ \mathbf {v} (t + 1) - (\sigma \boldsymbol {\eta} \otimes \boldsymbol {\sigma} \boldsymbol {\eta}) v e c (\mathbf {I} _ {n _ {e}}) \\ \mathbf {0} _ {n _ {x} \times 1} \\ (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) (\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {s}) + (\sigma \boldsymbol {\eta} \otimes \tilde {\mathbf {H}} _ {\mathbf {x x}}) (\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) + (\sigma \boldsymbol {\eta} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}) \boldsymbol {\epsilon} _ {t + 1} \\ \mathbf {u} _ {t + 1} - E [ \mathbf {u} _ {t + 1} ] \end{array} \right]\]

where

\[\mathbf {v} (t + 1) \equiv (\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}) (\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}) + (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) (\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}) + (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}) (\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1})\]

and

\[\begin{array}{r c l} \mathbf {u} _ {t + 1} & \equiv & (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}) (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}) \\ & & + (\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) (\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}) + (\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}) (\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}) \\ & & + (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) (\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) + (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}) (\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}) \\ & & + (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) (\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}) + (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}) (\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}) \end{array}\]

Note that

\[E \left[ \mathbf {u} _ {t + 1} \right] = (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}) E \left[ \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) \right]\]

because is iid, , and . Note also that can be coded directly as:

\[\begin{array}{r l} & {= E [ \left[ \begin{array}{c} \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1} \\ \mathbf {0} _ {n _ {x} \times 1} \\ \mathbf {v} (t + 1) - (\sigma \pmb {\eta} \otimes \pmb {\sigma} \pmb {\eta}) v e c (\mathbf {I} _ {n _ {e}}) \\ \mathbf {0} _ {n _ {x} \times 1} \\ (\sigma \pmb {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) (\pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {s}) + (\sigma \pmb {\eta} \otimes \tilde {\mathbf {H}} _ {\mathbf {x x}}) (\pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) + (\sigma \pmb {\eta} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}) \pmb {\epsilon} _ {t + 1} \\ \mathbf {u} _ {t + 1} - E [ \mathbf {u} _ {t + 1} ] \end{array} \right]} \\ & \quad \times [ \begin{array}{c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c} \sigma \pmb {\epsilon} _ {t + 1} ^ {\prime} \pmb {\eta} ^ {\prime} & \mathbf {0} _ {1 \times n _ {x}} & \mathbf {v} (t + 1) ^ {\prime} - v e c (\mathbf {I} _ {n _ {e}}) ^ {\prime} (\sigma \pmb {\eta} \otimes \pmb {\sigma} \pmb {\eta}) ^ {\prime} & \mathbf {0} _ {1 \times n _ {x}} \\ (\pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {s}) ^ {\prime} (\sigma \pmb {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime} + (\pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} (\sigma \pmb {\eta} \otimes \tilde {\mathbf {H}} _ {\mathbf {x x}}) ^ {\prime} + \pmb {\epsilon} _ {t + 1} ^ {\prime} (\sigma \pmb {\eta} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}) ^ {\prime} & \mathbf {u} _ {t + 1} ^ {\prime} - E [ \mathbf {u} _ {t + 1} ^ {\prime} ] & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & \\ \end{array}\]

\[V a r \left(\widetilde {\boldsymbol {\xi}} _ {t + 1}\right) = E [ \left[ \begin{array}{c c c c c c} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} \sigma \boldsymbol {\epsilon} _ {t + 1} ^ {\prime} \boldsymbol {\eta} ^ {\prime} & \mathbf {0} _ {n _ {x} \times n _ {x}} & V a r \left[ \widetilde {\boldsymbol {\xi}} _ {t + 1} \right] _ {1 3} & \mathbf {0} _ {n _ {x} \times n _ {x}} & V a r \left[ \widetilde {\boldsymbol {\xi}} _ {t + 1} \right] _ {1 5} & V a r \left[ \widetilde {\boldsymbol {\xi}} _ {t + 1} \right] _ {1 6} \\ \mathbf {0} _ {n _ {x} \times n _ {x}} & \mathbf {0} _ {n _ {x} \times n _ {x}} & \mathbf {0} _ {n _ {x} \times n _ {x} ^ {2}} & \mathbf {0} _ {n _ {x} \times n _ {x}} & \mathbf {0} _ {n _ {x} \times n _ {x} ^ {2}} & \mathbf {0} _ {n _ {x} \times n _ {x} ^ {3}} \\ V a r \left[ \widetilde {\boldsymbol {\xi}} _ {t + 1} \right] _ {3 1} & \mathbf {0} _ {n _ {x} ^ {2} \times n _ {x}} & V a r \left[ \widetilde {\boldsymbol {\xi}} _ {t + 1} \right] _ {3 3} & \mathbf {0} _ {n _ {x} \times n _ {x}} & V a r \left[ \widetilde {\boldsymbol {\xi}} _ {t + 1} \right] _ {3 5} & V a r \left[ \widetilde {\boldsymbol {\xi}} _ {t + 1} \right] _ {3 6} \\ \mathbf {0} _ {n _ {x} \times n _ {x}} & \mathbf {0} _ {n _ {x} \times n _ {x}} & \mathbf {0} _ {n _ {x} \times n _ {x} ^ {2}} & \mathbf {0} _ {n _ {x} \times n _ {x}} & \mathbf {0} _ {{n _ {x}} \times n _ {{x}} ^ {2}} & \mathbf {0} _ {{n _ {x}} \times n _ {{x}} ^ {3}} \\ V a r \left[ \widetilde {\boldsymbol {\xi}} _ {t + 1} \right] _ {5 1} & \mathbf {0} _ {{n _ {{x}} ^ {2}} \times n _ {{x}} ^ {} x} & V a r \left[ \widetilde {\boldsymbol {\xi}} _ {{t + 1}} \right] _ {5 3} & \mathbf {0} _ {{n _ {{x}}} x n _ {{x}} ^ {} x} & V a r \left[ \widetilde {\boldsymbol {\xi}} _ {{t + 1}} \right] _ {5 5} & V a r \left[ \widetilde {\boldsymbol {\xi}} _ {{t + 1}} \right] _ {5 6} \\ V a r \left[ \widetilde {\boldsymbol {\xi}} _ {{t + 1}} \right] _ {6 1} & \mathbf {0} _ {{n _ {{x}} ^ {3}} x n _ {{x}} ^ {} x} & V a r \left[ \widetilde {\boldsymbol {\xi}} _ {{t + 1}} \right] _ {6 3} & \mathbf {0} _ {{n _ {{x}}} x n _ {{x}} ^ {} x} & V a r \left[ \widetilde {\boldsymbol {\xi}} _ {{t + 1}} \right] _ {6 5} & V a r \left[ \widetilde {\boldsymbol {\xi}} _ {{t + 1}} \right] _ {6 6} \end{array} \right]\]

where we have defined:

\[V a r \left[ \tilde {\pmb {\xi}} _ {t + 1} \right] _ {1 3} \equiv E [ \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1} \left(\mathbf {v} (t + 1) ^ {\prime} - v e c (\mathbf {I} _ {n _ {e}}) ^ {\prime} (\sigma \pmb {\eta} \otimes \pmb {\sigma} \pmb {\eta}) ^ {\prime}\right) ]\]

\[V a r \left[ \widetilde {\pmb {\xi}} _ {t + 1} \right] _ {3 3} \equiv E [ (\mathbf {v} (t + 1) - (\sigma \pmb {\eta} \otimes \pmb {\sigma} \pmb {\eta}) v e c (\mathbf {I} _ {n _ {e}})) (\mathbf {v} (t + 1) ^ {\prime} - v e c (\mathbf {I} _ {n _ {e}}) ^ {\prime} (\sigma \pmb {\eta} \otimes \pmb {\sigma} \pmb {\eta}) ^ {\prime}) ]\]

\[V a r \left[ \widetilde {\pmb {\xi}} _ {t + 1} \right] _ {1 5} \equiv E [ \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1} \left((\pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {s}) ^ {\prime} (\sigma \pmb {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime} + (\pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} (\sigma \pmb {\eta} \otimes \tilde {\mathbf {H}} _ {\mathbf {x x}}) ^ {\prime} + \pmb {\epsilon} _ {t + 1} ^ {\prime} (\sigma \pmb {\eta} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}) ^ {\prime}\right) ]\]

\[V a r \left[ \tilde {\pmb {\xi}} _ {t + 1} \right] _ {1 6} \equiv E [ \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1} \left(\mathbf {u} _ {t + 1} ^ {\prime} - E \left[ \mathbf {u} _ {t + 1} ^ {\prime} \right]\right) ]\]

\[\begin{array}{r l} & {V a r \left[ \tilde {\pmb {\xi}} _ {t + 1} \right] _ {3 5} \equiv E [ (\mathbf {v} (t + 1) - (\sigma \pmb {\eta} \otimes \pmb {\sigma} \pmb {\eta}) v e c (\mathbf {I} _ {n _ {e}}))} \\ & {\qquad \times \left((\pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {s}) ^ {\prime} (\sigma \pmb {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime} + (\pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} (\sigma \pmb {\eta} \otimes \tilde {\mathbf {H}} _ {\mathbf {x x}}) ^ {\prime} + \pmb {\epsilon} _ {t + 1} ^ {\prime} (\sigma \pmb {\eta} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}) ^ {\prime}\right) ]} \end{array}\]

\[V a r \left[ \tilde {\pmb {\xi}} _ {t + 1} \right] _ {3 6} \equiv E [ (\mathbf {v} (t + 1) - (\sigma \pmb {\eta} \otimes \pmb {\sigma} \pmb {\eta}) v e c (\mathbf {I} _ {n _ {e}})) (\mathbf {u} _ {t + 1} ^ {\prime} - E [ \mathbf {u} _ {t + 1} ^ {\prime} ]) ]\]

\[\begin{array}{r l} & V a r \left[ \tilde {\boldsymbol {\xi}} _ {t + 1} \right] _ {5 5} \equiv E [ \Big ((\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) (\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {s}) + (\sigma \boldsymbol {\eta} \otimes \tilde {\mathbf {H}} _ {\mathbf {x x}}) (\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) + (\sigma \boldsymbol {\eta} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}) \boldsymbol {\epsilon} _ {t + 1} \Big) \\ & \qquad \times \left((\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {s}) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime} + (\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \tilde {\mathbf {H}} _ {\mathbf {x x}}) ^ {\prime} + \boldsymbol {\epsilon} _ {t + 1} ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}) ^ {\prime}\right) ] \end{array}\]

\[V a r \left[ \tilde {\pmb {\xi}} _ {t + 1} \right] _ {5 6} \equiv E [ \left((\sigma \pmb {\eta} \otimes \mathbf {h _ {x}}) (\pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {s}) + (\sigma \pmb {\eta} \otimes \tilde {\mathbf {H}} _ {\mathbf {x x}}) (\pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) + (\sigma \pmb {\eta} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}) \pmb {\epsilon} _ {t + 1}\right) (\mathbf {u} _ {t + 1} ^ {\prime} - E [ \mathbf {u} _ {t + 1} ^ {\prime} ]) ]\]

\[V a r \left[ \tilde {\pmb {\xi}} _ {t + 1} \right] _ {6 6} \equiv E [ (\mathbf {u} _ {t + 1} - E [ \mathbf {u} _ {t + 1} ]) (\mathbf {u} _ {t + 1} ^ {\prime} - E [ \mathbf {u} _ {t + 1} ^ {\prime} ]) ]\]

We have already derived the expressions for and , and we will now compute the remaining terms.

4.3.1 For

\[\begin{array} { r l } & \mathrm{Note~ first~ that~} V a r \left[ \tilde { \pmb { \xi } } _ { t + 1 } \right] _ { 1 5 } \mathrm{~ has~ dimensions~} n _ { x } \times n _ { x } ^ { 2 } . \\ & V a r \left[ \tilde { \pmb { \xi } } _ { t + 1 } \right] _ { 1 5 } = E [ \sigma \pmb { \eta } \pmb { \epsilon } _ { t + 1 } \left( ( \pmb { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { s } ) ^ { \prime } ( \sigma \pmb { \eta } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { \prime } + ( \pmb { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) ^ { \prime } ( \sigma \pmb { \eta } \otimes \tilde { \mathbf { H } } _ { \mathbf { x x } } ) ^ { \prime } + \pmb { \epsilon } _ { t + 1 } ^ { \prime } ( \sigma \pmb { \eta } \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } ) ^ { \prime } ) ] \\ & = E [ \sigma \pmb { \eta } \pmb { \epsilon } _ { t + 1 } ( \pmb { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { s } ) ^ { \prime } ( \sigma \pmb { \eta } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { \prime } + \sigma \pmb { \eta } \pmb { \epsilon } _ { t + 1 } ( \pmb { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) ^ { \prime } ( \sigma \pmb { \eta } \otimes \tilde { \mathbf { H } } _ { \mathbf { x x } } ) ^ { \prime } + \sigma \pmb { \eta } \pmb { \epsilon } _ { t + 1 } \pmb { \epsilon } _ { t + 1 } ^ { \prime } ( \sigma \pmb { \eta } \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } ) ^ { \prime } ] \\ & = E [ \sigma \pmb { \eta } \pmb { \epsilon } _ { t + 1 } ( \pmb { \epsilon } _ { t + 1 } ^ { \prime } \otimes ( \mathbf { x } _ { t } ^ { s } ) ^ { \prime } ) ( \sigma \pmb { \eta } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { \prime } + \sigma \pmb { \eta } \pmb { \epsilon } _ { t + 1 } ( ( \pmb { \epsilon } _ { t + 1 } ^ { \prime } \otimes ( ( \mathbf { x } _ { t } ^ { f } ) ^ { \prime } ) ( ( x _ { t } ^ { f } ) ^ { ' } ) ( ( x _ { t } ^ { f } ) ^ { ' } ) ( ( x _ { t } ^ { f } ) ^ ' + 1 / 2 - 1 / 2 - 1 / 2 - 1 / 2 - 1 / 2 - 1 / 2 - 1 / 2 - 1 / 2 - 1 / 2 - 1 / 2 - 1 / 2 - 1 / 2 - 1 / 2 - 1 / 2 - 1 / 2 - 1 / 2 - 1 / 2 - 1 / n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n ] \\ & = E [ \sigma \pmb { \eta } ( ( e _ { t + 1 } ⓤ ) ( e _ { t + 1 } ^ { ' } ⓤ ) ( e _ { t + 1 } ⓤ ) ( e _ { t + 1 } ⓤ ) ( e _ { t + 1 } ⓤ ) ( e _ { t + 1 } ⓤ ) ( e _ { t + 1 } ⓤ ) ( e _ { t + 1 } ⓤ ) ( e _ { t + 1 } ⓤ ) ( e _ { t + 1 } ⓤ ) ( e _ {\mathrm{的}} ⓤ ) ( e _ {\mathrm{的}} ⓤ ) ( e _ {\mathrm{的}} ⓤ ) ( e _ {\mathrm{的}} ⓤ ) ( e _ {\mathrm{的}} ⓤ ) ( e _ {\mathrm{的}} ⓤ ) ( e _ {\mathrm{的}} ⓤ ) ( e _ {\mathrm{的}} ⓤ ) ( e _ {\mathrm{的}} ⓤ ). \\ & = E [ e [ o m i g u a l l i c k e ] o m i g u a l l i c k e ] o m i g u a l l i c k e ] o m i g u a l l i c k e ] o m i g u a l l i c k e ] o m i g u a l l i c k e ] o m i g u a l l i c k e ] o m i g u a l l i c k e ] o m i g u a l l l i c k e ] o m i g u a l l i c k e ] o m i g u a l l i c k e ] o m i g u a l l i c k e ] o m i g u a l l i c k e ] o m i g u a l l i c k e ] o m i g u a l l i c k e ] o m i g u a l l i c k e ] o o p o w i d j e c t i o n \\ & {\quad =} \\ & \quad 1) = o o p o w i d j e c t i o n (\textbf {\mu} (\textbf {\mu} _ {\textbf {\mu}}) (\textbf {\mu} _ {\textbf {\mu}}) (\textbf {\mu} _ {\textbf {\mu}}) (\textbf {\mu} _ {\textbf {\mu}}) (\textbf {\mu} _ {\textbf {\mu}}) (\textbf {\mu} _ {\textbf {\mu}}) (\textbf {\mu} _ {\textbf {\mu}}) (\textbf {\mu} _ {\textbf {\mu}}) (\boldsymbol {\sigma} ; | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | \quad & \quad + o o p o w i d j e c t i o n (\textbf {\mu} (\textbf {\mu} _ {\textbf {\mu}}) (E [ (\textbf {\mu} _ {\textbf {\mu}}) (\textbf {\mu} _ {\textbf {\mu}}) (\textbf {\mu} _ {\textbf {\mu}}) (\textbf {\mu} _ {\textbf {\mu}}) (\textbf {\mu} _ {\textbf {\mu}}) (\textbf {\mu} _ {\textbf {\mu}}) (\textbf {\mu} _ {\textbf {\mu}}) (\varepsilon ; |\varepsilon| ) (\varepsilon ; |\varepsilon| ) (\varepsilon ; |\varepsilon| ) (\varepsilon ; |\varepsilon| ) (\varepsilon ; |\varepsilon| ) (\varepsilon ; |\varepsilon| ) (\varepsilon ; |\varepsilon| ) (\varepsilon ; |\varepsilon| ) (\varepsilon ; |\varepsilon| ) (\varepsilon ; |\varepsilon| ) (e q : q : q ; q : q ; q : q ; q : q ; q : q ; q : q ; q : q ; q : q ; q : q ; q : q ; q : q ; q : q ; q : q ; q : q ; q : q ; q : q ; q : q ; q : q ; q : q ; q : q ; q : q ; q : q ; q : q ; q : q ; q : q ; q : q . \\ & \quad + o o p o w i d j e c t i o n (\sigma [ O ] (o o p o w i d j e c t i o n (\sigma [ O ] (o o p o w i d j e c t i o n (\sigma [ O ] (o o p o w i d j e c t i o n (\sigma [ O ] (o o p o w i d j e c t i o n (\sigma [ O ] (o o p o w i d j e c t i o n (\sigma [ O ] (i o f f o r s s y s y s y s y s y s y s y s y s y s y s y s y s y s y s y s y s y s y s y s y s y s y s y s y s y s y s y s y s y s y s y s y s y s y s y s y s y s y s y s y s y s y s y s y s y s y s y s y s y s y s a [ O ] ] \\ & \quad + o o p o w i d j e c t i o n (\sigma [ O ] (o o p o w i d j e c t i o n (\sigma [ O ] (o o p o w i d j e c t i o n (\sigma [ O ] (o o p o w i d j e c t i o n (\sigma [ O ] (o o p o w i d j e c t i o n (s u r b a l l u r b a l l u r b a l l u r b a l l u r b a l l u r b a l l u r b a l l u r b a l l u r b a l l u r b a l l u r b a l l u r b a l l u r b a l l u r b a l l u r b a l l u r b a l l u r b a l l u r b b a l l u r b a l l u r b a l l u r b a l l u r b a l l u r b a l l u r b a l l u r b a l l u r b a l l u r b a l l u r b a l l u r b a l l u r b a l l u r b a l l u r b a l l u r b a l l u r b a l I ] [ O ] [ O ] [ O ] [ O ] [ O ] [ O ] [ O ] [ O ] [ O ] [ O ] [ O ] [ O ] [ O ] [ O ] [ O ] [ O ] [ O ] [ O ] [ O ] [ O ] [ O ] [ O ] [ O ] [ O ] [ O ] [ O ] [ O ] [ O ] [ O ] [ O ] [ O ] [ O ] [ O ] [ O ], \\ & \quad + o o p o w i d j e c t i o n (\sigma [ O ] (o o p o w i d j e c t i o n (\sigma [ O ] (o o p o w i d j e c t i o n (\sigma [ O ] (o o p o w i d j e c t i o n (\sigma [ O ] (o o p o w i d j e c t i o n ({\cal S}) [ N ] [ N ] [ N ] [ N ] [ N ] [ N ] [ N ] [ N ] [ N ] [ N ] [ N ] [ N ] [ N ] [ N ] [ N ] [ N ] [ N ] [ N ] [ N ] [ N ] [ N ] [ N ] [ N ] [ N ] [ N ] [ N ] [ N ] [ N ] [ N ] [ N ] [ N ] [ N ] [ N ] [ N ] [ M S P P S P S P S P S P S P S P S P S P S P S P S P S P S P S P S P S P S P S P S P S P S P S P S P S P S P S P S P S P S P S P S P S P S P S P S P S P S P S P S P S P S P S P S P S P S P S P S P S P S P S P I T I T I T I T I T I T I T I T I T I T I T I T I T I T I T I T I T I T I T I T I T I T I T I T I T I T I T I T I T I T I T I T I T I T I T I T I T I T I T I T I T I T I T I T I T I T I T I T I T I T I I U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W WWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBwBw B w B w B w B w B w B w B w B w B w B w B w B w B w B w B w B w B w B w B w B w B w B w B w B w B w B w B w B w B w B w B w B w B w B w B w B w B w B w B w B w B w B w B w B w B w B w B w B w B w B w B we ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? \\ & \quad + o o p o w i d j e c t i o n (\sigma_ {[ O ]} (\sigma_ {[ O ]} (\sigma_ {[ O ]} (\sigma_ {[ O ]} (\sigma_ {[ O ]} (\sigma_ {[ O ]} (\sigma_ {[ O ]} (\sigma_ {[ O ]} (\sigma_ {[ O ]} (\sigma_ {[ O ]} (\sigma_ {[ O ]} (\sigma_ {[ O ]} (\sigma_ {[ O ]} (\sigma_ {[ O ]} (\sigma_ {[ O ]} (\rho_ {[ O ]} (\rho_ {[ O ]} (\rho_ {[ O ]} (\rho_ {[ O ]} (\rho_ {[ O ]} (\rho_ {[ O ]} (\rho_ {[ O ]} (\rho_ {[ O ]} (\rho_ {[ O ]} (\rho_ {[ O ]} (\rho_ {[ O ]} (\rho_ {[ O ]} (\rho_ {[ O ]} (\rho_ {[ O ]} (\rho_[[ O ]) (\rho_ [ O ]) ([ O ]) ([ O ]) ([ O ]) ([ O ]) ([ O ]) ([ O ]) ([ O ]) ([ O ]) ([ O ]) ([ O ]) ([ O ]) ([ O ]) ([ O ]) ([ O ]) ([ O ]) ([ O ]) ([ O ]) ([ O ]) ([ O ]) ([ O ]) ([ O ]) ([ O ]) ([ O ]) ([ O ]) ([ O ]) ([ O ]) ([ O ]) ([ O ]) ([ O ]) ([ O ]) ([ O ]) ([ O ]) ([ O ]) ([O ]) ([O ]) ([O ]) ([O ]) ([O ]) ([O ]) ([O ]) ([O ]) ([O ]) ([O ]) ([O ]) ([O ]) ([O ]) ([O ]) ([O ]) ([O ]) ([O ]) ([O ]) ([O ]) ([O ]) ([O ]) ([O ]) ([O ]) ([O ]) ([O ]) ([O ]) ([O ]) ([O ]) ([O ]) ([O ]) ([O ]) ([O ]) ([O ]) ([O)) ([O]) ([O]) ([O]) ([O]) ([O]) ([O]) ([O]) ([O]) ([O]) ([O]) ([O]) ([O]) ([O]) ([O]) ([O]) ([O]) ([O]) ([O]) ([O]) ([O]) ([O]) ([O]) ([O]) ([O]) ([O]) ([O]) ([O]) ([O]) ([O]) ([O]) ([O]) ([O]) ([O]) ([O]) ((I) ((I) ((I) ((I) ((I) ((I) ((I) ((I) ((I) ((I) ((I) ((I) ((I) ((I) ((I) ((I) ((I) ((I) ((I) ((I) ((I) ((I) ((I) ((I) ((I) ((I) ((I) ((I) ((I) ((I) ((I) ((I) ((I) ((I)((I) ((I) ((I) ((I) ((I) ((I) ((I) ((I) ((I) ((I) ((I) ((I) ((I) ((I) ((I) ((I) ((I) ((I) ((I) ((I) ((I) ((I) ((I) ((I) ((I) ((I) ((I) ((I) ((I) ((I) ((I) ((I) ((I) ((II) ((II) ((II) ((II) ((II) ((II) ((II) ((II) ((II) ((II) ((II) ((II) ((II) ((II) ((II) ((II) ((II) ((II) ((II) ((II) ((II) ((II) ((II) ((II) ((II) ((II) ((II) ((II) ((II) ((II) ((II) ((II) ((II) ((II< fcel>[ I C E F A R E F A R E F A R E F A R E F A R E F A R E F A R E F A R E F A R E F A R E F A R E F A R E F A R E F A R E F A R E F A R E F A R E F A R E F A R E F A R E F A R E F A R E F A R E F A R E F A R E F A, & < 0.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000< |content_end|>\]

4.3.2 For

Note first that has dimensions .

\[\begin{array}{l} V a r \left[ \tilde {\boldsymbol {\xi}} _ {t + 1} \right] _ {1 6} = E [ \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} \left(\mathbf {u} _ {t + 1} ^ {\prime} - E \left[ \mathbf {u} _ {t + 1} ^ {\prime} \right]\right) ] \\ = E [ \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} \mathbf {u} _ {t + 1} ^ {\prime} ] \\ = E [ \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} ((\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}) ^ {\prime} (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}) ^ {\prime} \\ \quad + (\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} (\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime} + (\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}) ^ {\prime} (\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}) ^ {\prime} \\ \quad + (\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime} + (\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}) ^ {\prime} \\ \quad + (\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime} + (\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}) ^ {\prime}) ] \\ = \sigma \boldsymbol {\eta} E (\boldsymbol {\epsilon} _ {t + 1} (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}) ^ {\prime}) (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}) ^ {\prime} \\ + \sigma \boldsymbol {\eta} E [ \boldsymbol {\epsilon} _ {t + 1} (\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} ] (\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime} + \sigma \boldsymbol {\eta} E [ \boldsymbol {\epsilon} _ {t + 1} (\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}) ^ {\prime} ] (\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}) ^ {\prime}. \end{array}\]

\[\begin{array} { r l } & + \sigma \eta E \left[ \epsilon _ { t + 1 } \left( \epsilon _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \right) ^ { \prime } \right] ( \sigma \eta \otimes \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { \prime } + \sigma \eta E \left[ \epsilon _ { t + 1 } \left( \epsilon _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \otimes \epsilon _ { t + 1 } \right) ^ { \prime } \right] ( \sigma \eta \otimes \mathbf { h } _ { \mathbf { x } } \otimes \sigma \eta ) ^ { \prime } \\ & + \sigma \eta E \left[ \epsilon _ { t + 1 } \left( \epsilon _ { t + 1 } \otimes \epsilon _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \right) ^ { \prime } \right] ( \sigma \eta \otimes \sigma \eta \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { \prime } + \sigma \eta E \left[ \epsilon _ { t + 1 } \left( \epsilon _ { t + 1 } \otimes \epsilon _ { t + 1 } \otimes \epsilon _ { t + 1 } \right) ^ { \prime } \right] ( \sigma \eta \otimes \sigma \eta \otimes \sigma \eta ) ^ { \prime } ] \\ & { = \sigma \eta E \left( ( 1 \otimes \epsilon _ { t + 1 } ) ( ( ( x _ { t } ^ { f } \otimes x _ { t } ^ { f } ) ^ { \prime } \otimes \epsilon _ { t + 1 } ^ { \prime } ) ) ( h _ { x } \otimes h _ { x } \otimes \sigma \eta ) ^ { \prime } + 0 - 2 E [ ( x _ { t } ^ { f } , x _ { t } ^ { f } , x _ { t } ^ { f } , x _ { t } ^ { f } , x _ { t } ^ { f } , x _ { t } ^ { f } , x _ { t } ^ { f } , x _ { t } ^ { f } , x _ { t } ^ { f } , x _ { t } ^ { f } , x _ { t } ^ { f } ] ] , } \\ & + \sigma \eta E [ ( e _ { t + 1 } , ( e _ { t + 1 } , ( e _ { t + 1 } , ( e _ { t + 1 } , ( e _ { t + 1 } , ( e _ { t + 1 } , ( e _ { t + 1 } , ( e _ { t + 1 } , ( e _ { t + 1 } , ( e _ { t + 1 } , ( e _ { t + 1 } ) , ( e _ { t + 1 } , ( e _ { t + 1 } , ( e _ { t + 1 } , ( e _ { t + 1 } , ( e _ { t + 1 } , ( e _ { t + 1 } , ( e _ { t + 1 } , ( e _ { t + 1 } , ( e _ { t + 1 } , ( e _ { t + 1 }\ , ( e _ { t + 1 } , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1}\ , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1 } , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1}\ , ( e _ { t + 1 }\ , ( e _ { t + 1}\ , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1}\ , ( e _ { t + 1 }\ , ( e _ { t + 1 } , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1 } , ( e _ { t + 1 }\ , ( e _ { t + 1}\ , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1 } , ( e _ { t + 1 }\ , ( e _ { t + 1 } , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1}\ , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1}\ , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1}\ , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1 } , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1}\ , ( e _ { t + 1}\ , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ { t + 1 }\ , ( e _ t - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - m a x i s c o r r i d i o n g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. ~ \\ & {\quad = \sigma \eta E [ (\mathbf {\nabla} x _ {\mathbf {\Phi} ^ {\prime}} ^ {\prime} (\mathbf {\Phi} ^ {\prime}) ^ {\prime} (\mathbf {\Phi} ^ {\prime}) (\mathbf {\Phi} ^ {\prime}) (\mathbf {\Phi} ^ {\prime}) (\mathbf {\Phi} ^ {\prime}) (\mathbf {\Phi} ^ {\prime}) (\mathbf {\Phi} ^ {\prime}) (\mathbf {\Phi} ^ {\prime}) (\mathbf {\Phi} ^ {\prime}) (\mathbf {\Phi} ^ {\prime}) (\mathbf {\Phi} ^ {\prime}) (\mathbf {\Phi} ^ {\prime} ) (\mathbf {\Phi} ^ {\prime}) (\mathbf {\Phi} ^ {\prime}) (\mathbf {\Phi} ^ {\prime}) (\mathbf {\Phi} ^ {\prime}) (\mathbf {\Phi} ^ {\prime}) (\mathbf {\Phi} ^ {\prime}) (\mathbf {\Phi} ^ {\prime}) (\mathbf {\Phi} ^ {\prime}) (\mathbf {\Phi} ^ {\prime}) (\mathbf {\Phi} ^ {\prime}),} \\ & \quad = [ (\alpha_ {- k} E [ (\alpha_ {- k} E [ (\alpha_ {- k} E [ (\alpha_ {- k} E [ (\alpha_ {- k} E [ (\alpha_ {- k} E [ (\alpha_ {- k} E [ (\alpha_ {- k} E [ (\alpha_ {- k} E [ (\alpha_ {- k} E [ (\alpha_ {- k} E [ (\alpha_ {- k} E [ (\alpha_ {- k} E [ (\alpha. v a l l o w i s c o r r o r s i o n d i s s c o r r o r s i o n d i s s c o r r o r s i o n d i s s c o r r o r s i o n d i s s c o r r o r s i o n d i s s c o r r o r s i o n d i s s c o r r o r s i o n d i s s c o r r o r s i o n d i s s s c o r r o r s i o n d i s s s c o r r o r s i o n d i s s s c o r r o r s i o n d i s s s c o r r o r s i o n d i s s s c o r r o r s i o n d i s s s c o r r o r s i o n d i s s s c o r r o r s j m a x i s c o r r o r s j m a x i s c o r r o r s j m a x i s c o r r o r s j m a x i s c o r r o r s j m a x i s c o r r o r s j m a x i s c o r r o r s j m a x i s c o r r o r s j m a x i s c o r r o r s l m a x i s c o r r o r s l m a x i s c o r r o r s l m a x i s c o r r o r s l m a x i s c o r r o r s l m a x i s c o r r o r s l m a x i s c o r r o r s l m a x i s c o r r o r s l m a x i s c o r l m a x i s c o r l m a x i s c o r l m a x i s c o r l m a x i s c o r l m a x i s c o r l m a x i s c o r l m a x i s c o r l m a x i s c o r l m a x i s c o r l m a x i s c o r l m a x i s c o r | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | & = [ S p I E [ (\alpha_ {- k} E [ (\alpha_ {- k} E [ (\alpha_ {- k} E [ (\alpha_ {- k} E [ (\alpha_ {- k} E [ (\alpha_ {- k} E [ (\alpha_ {- k} E [ (\alpha_ {- k} E [ (\alpha_ {- k} E [ (\alpha_ {- k} E [ (\alpha_ {- k} E [ (\alpha_ {- k} E [ (\beta u v a l l o w i s c o r r o r s i o n d i s s c o r r o r s i o n d i s s c o r r o r s i o n d i s s c o r r o r s i o n d i s s c o r r o r s i o n d i s s c o r l u v a l l o w i v a l l o w i v a l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / ] \\ & = [ S p I E [ (\alpha_ {- k} E [ (\alpha_ {- k} E [ (\alpha_ {- k} E [ (\alpha_ {- k} E [ (\alpha_ {- k} E [ (\alpha_ {- k} E [ (\alpha_ {- k} E [ (\alpha_ {- k} E [ (\alpha_ {- k} E [ (\alpha_ {- k} E [ (\alpha_ {- k} E [ (\alpha_ {{\bf z}} ; y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y Y ] \\ & = [ S p I E [ (\alpha_ {- k} E [ (\alpha_ {- k} E [ (\alpha_ {- k} E [ (\alpha_ {- k} E [ (\alpha_ {- k} E [ (\alpha_ {- k} E [ (\alpha_ {- k} E [ (\alpha_ {- k} E [ (\alpha_ {- k} E [ (\alpha_ {- k} E [ (\alpha_ {- k} E [ (\beta u v a l ] ) ) ) ) ) ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ][ S p I E [ (\alpha_ {- k} E[ (\alpha_ {- k} E[ (\alpha_ {- k} E[ (\alpha_ {- k} E[ (\alpha_ {- k} E[ (\alpha_ {- k} E[ (\alpha_ {- k} E[ (\alpha_ {- k} E[ (\alpha_ {- k} E[ (\alpha_ {- k} E[ (\alpha_ {- k} E[ (\alpha_ {- k} E[ (\alpha_ {- k} E[ (\alpha_ {{\bf z}} ; y v a v a v a v a v a v a v a v a v a v a v a v a v a v a v a v a v a v a v a v a v a v a v a v a v a v a v a v a v a v a v a v a v a v a v a v a v a v a v a v a v a v a v a v a v a v a v a v a v a v a v b u v a v b u v b u v b u v b u v b u v b u v b u v b u v b u v b u v b u v b u v b u v b u v b u v b u v b u v b u v b u v b u v b u v b u v b u v b u v b u v b u v b u v b u v b u v b u v b u v b u v b u v b u w ) ]. \\ & = [ S p I E [ (\alpha_ {- k} E[ (\alpha_ {- k} E[ (\alpha_ {- k} E[ (\alpha_ {- k} E[ (\alpha_ {- k} E[ (\alpha_ {- k} E[ (\alpha_ {- k} E[ (\alpha_ {- k} E[ (\alpha_ {- k} E[ (\alpha_ {- k} E[ (\alpha_ {- k} E[ (\alpha_ {- k} K U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U W ) ]. \\ & = [ S p I E [ (\alpha_ {- k} E[ (\alpha_ {- k} E[ (\alpha_ {- k} E[ (\alpha_ {- k} E[ (\alpha_ {- k} E[ (\alpha_ {- k} E[ (\alpha_ {- k} E[ (\alpha_ {- k} E[ (\alpha_ {- k} E[ (\alpha_ {- k} E[ (\alpha_ {- k} E[ ({\bf z})^{\prime})^{\prime})^{\prime})^{\prime})^{\prime})^{\prime})^{\prime})^{\prime})^{\prime})^{\prime})^{\prime})^{\prime})^{\prime})^{\prime})^{\prime})^{\prime})^{\prime})^{\prime})^{\prime})^{\prime})^{\prime})^{\prime})^{\prime})^{\prime})^{\prime})^{\ast}] : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :: A B C D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D B C D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D B CD A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D A D B CD B CD B CD B CD B CD B CD B CD B CD B CD B CD B CD B CD B CD B CD B CD B CD B CD B CD B CD B CD B CD B CD B CD B CD B CD B CD B CD B CD B CD B CD B CD B CD B CD B CD N M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M L N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N NNININININININININININININININININININININININININININININININININININININININININININININININININININININININININININININININININININININININININININININININININININININININININININININININININININININInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInInIn InIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIainIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIIaIIaIIaIIaIIaIIaIIaIIaIIaIIaIIaIIaIIaIIaIIaIIaIIaIIaIIaIIaIIaIIaIIaIIaIIaIIaIIaIIaIIaIIaIIaIIaIIaIIaIIaIIaIIaIIaIIaIIaIIaIIaIIaIIaIIaIIaIIaIIaIIaIIaIIbIIIbIIIbIIIbIIIbIIIbIIIbIIIbIIIbIIIbIIIbIIIbIIIbIIIbIIIbIIIbIIIbIIIbIIIbIIIbIIIbIIIbIIIbIIIbIIIbIIIbIIIbIIIbIIIbIIIbIIIbIIIbIIIbIIIbIIIbIIIbIIIbIIIbIIIbIIIbIIIbIIIbIIIbIIIbIIIbIIIbIIIbIIIbIIIbIIIbIIIb IIIBIIIIBHIVVVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVIVI VI V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V VV I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIII VIIIV VII VII VII VII VII VII\]

Hence we only need to compute directly the terms and .

We first note that

\[E \left[ \pmb {\epsilon} _ {t + 1} \left(\mathbf {x} _ {t} ^ {f} \otimes \pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \right]\]

\[\begin{array}{r l} & {\mathrm{And}} \\ & {E \left[ \epsilon_ {t + 1} \left(\epsilon_ {t + 1} \otimes \epsilon_ {t + 1} \otimes \epsilon_ {t + 1}\right) ^ {\prime} \right] = E \left[ \epsilon_ {t + 1} \left(\epsilon_ {t + 1} ^ {\prime} \otimes \epsilon_ {t + 1} ^ {\prime} \otimes \epsilon_ {t + 1} ^ {\prime}\right) \right]} \\ & {\qquad = E \left[ \epsilon_ {t + 1} \left(\epsilon_ {t + 1} ^ {\prime} \otimes \left\{\left\{\epsilon_ {t + 1} \left(1, \phi_ {3}\right) \right\} _ {\phi_ {3} = 1} ^ {n _ {e}} \epsilon_ {t + 1} \left(1, \phi_ {4}\right) \right\} _ {\phi_ {4} = 1} ^ {n _ {e}}\right) \right]} \\ & {\qquad = E \left[ \begin{array}{l} \epsilon_ {t + 1} \left(\phi_ {1}, 1\right) \\ \epsilon_ {t + 1} \left(\phi_ {2}, 1\right) \\ ... \\ \epsilon_ {t + 1} \left(\phi_ {4}, 1\right) \end{array} \left(\left\{\epsilon_ {t + 1} \left(1, \phi_ {2}\right) \left\{\left\{\epsilon_ {t + 1} \left(1, \phi_ {3}\right) \right\} _ {\phi_ {3} = 1} ^ {n _ {e}} \epsilon_ {t + 1} \left(1, \phi_ {4}\right) \right\} _ {\phi_ {4} = 1} ^ {n _ {e}} \right\} _ {\phi_ {2} = 1} ^ {n _ {e}}\right) \right]} \end{array}\]

Thus, the quasi Matlab codes are

4.3.3 For

\[\begin{array} { r l } & { \mathrm{otefirstthatVar} \left[ \tilde { \pmb { \xi } } _ { t + 1 } \right] _ { 3 5 } \mathrm{hasdimensions} n _ { x } ^ { 2 } \times n _ { x } ^ { 2 } . } \\ & { \quad V a r \left[ \tilde { \pmb { \xi } } _ { t + 1 } \right] _ { 3 5 } } \\ & { \equiv E [ ( \mathbf { v } ( t + 1 ) - ( \sigma \pmb { \eta } \otimes \pmb { \sigma } \pmb { \eta } ) v e c ( \mathbf { I } _ { n _ { e } } ) ) \left( ( \pmb { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { s } ) ^ { \prime } ( \sigma \pmb { \eta } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { \prime } + ( \pmb { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) ^ { \prime } ( \sigma \pmb { \eta } \otimes \tilde { \mathbf { H } } _ { \mathbf { x x } } ) ^ { \prime } + \pmb { \epsilon } _ { t + 1 } ^ { \prime } ( \sigma \pmb { \eta } \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } ) ^ { \prime } ) ] } \\ & { = E [ ( ( \mathbf { h } _ { \mathbf { x } } \otimes \sigma \pmb { \eta } ) ( \mathbf { x } _ { t } ^ { f } \otimes \pmb { \epsilon } _ { t + 1 } ) + ( \sigma \pmb { \eta } \otimes \mathbf { h } _ { \mathbf { x } } ) ( \pmb { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } ) + ( \sigma \pmb { \eta } \otimes \pmb { \sigma } \pmb { \eta} ) ( \pmb { \epsilon } _ { t + 1 } \otimes \pmb { \epsilon } _ { t + 1 } ) - ( \sigma \pmb { \eta } \otimes \pmb { \sigma } \pmb { \eta} ) v e c ( \mathbf { I } _ { n _ { e } } ) ) } \\ & { \qquad \times ( ( ( \pmb { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { s } ) ^ { \prime } ( \sigma \pmb { \eta } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { \prime } + ( \pmb { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ {t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) ^ { \prime } ( \sigma \pmb { \eta } \otimes \tilde { \mathbf { H } } _ { \mathbf { x x } } ) ^ { \prime } + \pmb { \epsilon } _ { t + 1 } ^ { \prime } ( \sigma \pmb { \eta } \odotslantslantslantslantslantslantslantslantslantslantslantslantslantslantslantslantslantslantslantslantslantslantslantslantslantslantslantslantslantslantslantslantslantslantslantslantslantslantslantslantslantslantslantslantslantslantslantslantslantslantslanomethesameral , ) ] } \\ & { = E [ ( {\bf h _ x} \otimes {\sigma} {\pmb {\eta} }) ( {\bf x} _ { t } ^ { f } \otimes {\pmb {\epsilon}} _ { t + 1 } ) ( {\pmb {\epsilon}} _ { t + 1 } \otimes {\bf x} _ { t } ^ { s } ) ^ { \prime } ( {\sigma} {\pmb {\eta}} \otimes {\bf h _ x} ) ^ { \prime }} \\ & { = ( {\bf h _ x} \otimes {\sigma} {\pmb {\eta} }) ( {\bf x} _ { t } ^ { f } \otimes {\pmb {\epsilon}} _ { t + 1 } ) ( [ {\pmb {\epsilon}} _ { t + 1 } \otimes {\bf x} _ { t } ^ { f } ] ( [ {\pmb {\epsilon}} _ { t + 1 } ] ( [ {\pmb {\chi}} _ { t + 1 } ] ) ^ { ' } ( [ {\pmb {\sigma} {\pmb {\eta}}} ] ( [ {\pmb {\chi}} _ { t + 1 } ] ) ^ { ' }} \\ & { = ( {\bf h _ x} \otimes {\sigma} {\pmb {\eta} }) ( {\bf x} _ { t } ^ { f } \otimes {\pmb {\epsilon}} _ { t + 1 } ) (\pmb {\epsilon} _ { t + 1 } ^ { ' } ( [ {\pmb {\chi}} _ { t + 1 } ] ) ^ { ' } ( [ {\pmb {\sigma} {\pmb {\eta}}} ] ( [ {\pmb {\chi}} _ { t + 1 } ] ) ^ { ' }} \\ & { = ( ( {\sigma} {\pmb {\eta}} \otimes {\bf h _ x} ) ( [ {\pmb {\epsilon}} _ { t + 1 } ] ( [ {\pmb {\chi}} _ { t + 1 } ] ) ^ { ' } ( [ {\sigma} {\pmb {\eta}} ] ( [ {\pmb {\chi}} _ { t + 1 } ] ) ^ { ' }} \\ & { = ( ( {\sigma} {\pmb {\eta}} ] ( [ {\pmb {\epsilon}} _ { t + 1 } ] ( [ {\pmb {\chi}} _ { t + 1 } ] ) ^ { ' } ( [ {\pmb {\epsilon}} _ { t + 1 } ] ( [ {\pmb {\chi}} _ { t + 1 } ] ) ^ { ' }} \\ & { = ( ( {\sigma} {\pmb {\eta}} ] ( [ {\pmb {\epsilon}} _ { t + 1 } ] ( [ {\pmb {\chi}} _ { t + 1 } ] ) ^ { ' } ( [ {\pmb {\sigma} {\pmb {\eta}}} ] ( [ {\pmb {\chi}} _ { t + 1 } ] ) ^ { ' }} \\ & = ( (\sigma {{\boldsymbol{\eta}}} ] ( [ {\pmb {\sigma} {\boldsymbol{\eta}}} ] ) ( [ {\pmb {\epsilon}} _ { t + 1 } ] ( [ {\pmb {\epsilon}} _ { t + 1 } ] ) ^ { ' }( [ {\pmb {\sigma} {{\boldsymbol{\eta}}} ] ( [ {\pmb {\chi}} _ { t + 1 } ] ) ^{ ' }} ) ^ - 1 / 2 - i j k q , j , k , m , n , n , m , n , m , n , m , n , m , n , m , n , m , n , m , n , m , n , m , n , m , n , m , n , m , n , m , n , m , n , m , n , m , n , m , n , m , n , m , n , m , n , m , n , m , n , m , n , m , n , m , n , n , m , n , m , n , m , n , m , n , m , n , m , n , m , n , m , n , m , n , m , n , m , n , m , n , m , n , m , n , m , n , m , n , m , n , m , n , m , n , m , n , m , n , m , n , m , n , n , m , m , n , m , n , m , n , m , n , m , n , m , n , m , n , m , n , m , n , m , n , m , n , m , n , m , n , m , n , m , n , m , n , m , n , m , n , m , n , m ] \\ & - ( | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | & - ( |\nabla_{e}^{\prime}\nabla_{e}\nabla_{e}\nabla_{e}\nabla_{e}\nabla_{e}\nabla_{e}\nabla_{e}\nabla_{e}\nabla_{e}\nabla_{e}\nabla_{e}\nabla_{e}\nabla_{e}\nabla_{e}\nabla_{e}\nabla_{e}\nabla_{e}\nablasq_{i,j}^{\prime }\nabla_{e}\nabla_{e}\nablasq_{j,j}^{\prime }\nabla_{e}\nablasq_{k,j}^{\prime }\nabla_{e}\nablasq_{l,j}^{\prime }\nabla_{e}\nablasq_{m,j}^{\prime }\nabla_{e}\nablasq_{n,j}^{\prime }\nabla_{e}\nablasq_{m,j}^{\prime }\nabla_{e}\nablasq_{m,j}^{\prime }\nabla_{e}\nablasq_{m,j}^{\prime }\nabla_{e}\nablasq_{m,j}^{\prime }\nabla_{e}\nablasq_{m,j}^{\prime }\nabla_{e}\nablasq_{m,j}^{\prime }\nabla_{{\bf d}_{i,j}}^{\prime }\nabla_{{\bf d}_{i,j}}^{\prime }\nabla_{{\bf d}_{i,j}}^{\prime }\nabla_{{\bf d}_{i,j}}^{\prime }\nabla_{{\bf d}_{i,j}}^{\prime }\nabla_{{\bf d}_{i,j}}^{\prime }\nabla_{{\bf d}_{i,j}}^{\prime },\quad i < j < k < l < j < k < l < j < k < l < j < k < l < j < k < l < j < k < l < j < k < l < j < k < l < j < k < l < j < k < l < j < k < l < j < k < l< |content_end|>\]

\[\begin{array} { r l } & ( \mathbf { h } _ { \mathbf { x } } \otimes \sigma \boldsymbol { \eta } ) \left( \mathbf { x } _ { t } ^ { f } \otimes \boldsymbol { \epsilon } _ { t + 1 } \right) \left( \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { s } \right) ^ { \prime } \left( \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x } } \right) ^ { \prime } \\ & \quad + ( \mathbf { h } _ { \mathbf { x } } \otimes \sigma \boldsymbol { \eta } ) \left( \mathbf { x } _ { t } ^ { f } \otimes \boldsymbol { \epsilon } _ { t + 1 } \right) \left( \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \right) ^ { \prime } \left( \sigma \boldsymbol { \eta } \otimes \tilde { \mathbf { H } } _ { \mathbf { x x } } \right) ^ { \prime } \\ & \quad + ( \mathbf { h } _ { \mathbf { x } } \otimes \sigma \boldsymbol { \eta } ) \left( \mathbf { x } _ { t } ^ { f } \otimes \boldsymbol { \epsilon } _ { t + 1 } \boldsymbol { \epsilon } _ { t + 1 } ^ { \prime } \right) ( \sigma \boldsymbol { \eta } \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } ) ^ { \prime } \\ & + ( \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x } } ) \left( \boldsymbol { \epsilon } _ { t + 1 } \boldsymbol { \epsilon } _ { t + 1 } ^ { \prime } \otimes \mathbf { x } _ { t } ^ { f } ( \mathbf { x } _ { t } ^ { s } ) ^ { \prime } \right) ( \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { \prime } \\ & \quad + ( \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x } } ) \left( \boldsymbol { \epsilon } _ { t + 1 } \boldsymbol { \epsilon } _ { t + 1 } ^ { \prime } \otimes \mathbf { x } _ { t } ^ { f } ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) ^ { \prime } \right) ( \sigma \boldsymbol { \eta } \otimes \tilde { \mathbf { H } } _ { \mathbf { x x } } ) ^ { \prime } \\ & + ( \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x } } ) ( \boldsymbol { \epsilon } _ { t + 1 } \boldsymbol { \epsilon } _ { t + 1 } ^ { \prime } \otimes \mathbf { x } _ { t } ^ { f } ) ( \sigma \boldsymbol { \eta } \otimes {\frac 1 2} {\mathbf h} _ { {\sigma} {\sigma} {\sigma} ^ { 2 }} ) ^ { \prime } \\ & + ( \sigma \boldsymbol { \eta } \otimes {\sigma} {\boldsymbol {\eta}} ) ( \boldsymbol { \epsilon } _ { t + 1 } \otimes {\boldsymbol {\epsilon}} _ { t + 1 ) } ( [ ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (\mathrm{大}) ) ) - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 ) ] \\ & = E [ \\ & ( ) \\ & ( ) \\ & ( ) \\ & + ( ) \\ & + ( ) \\ & + ( ) \\ & + ( ) \\ & + ( ) \\ & + ( ) \\ & + ( ) \\ & + ( ) \\ & + ( ) \\ & + ( ) \\ & + ( ) \\ & + ( ) \\ & + ( ) \\ & + ( ) \\ & + ( ) \\ & + ( ) \\ & + ( ) | \\ & + ( ) \\ & + ( ) | \\ & + ( ) | \\ & + ( ) | \\ & + ( ) | \\ & + ( ) | \\ & + ( ) | \\ & + ( ) | \\ & + ( ) | \\ & + ( ) | \\ & + ( ) | \\ & + ( ) | \\ & + ( ) | \\ & + ( ) | \\ & + ( ) | \\ & + ( ) | \\ * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * . \\ & = E [ \\ & ( ) \\ & (\mathbf { h _ { x }}\otimes\sigma\eta) (\mathbf { x _ { t }} ^ { f }\otimes\epsilon_ { t + 1 }) (\epsilon_ { t + 1 }\otimes\mathbf { x _ { t }} ^ { s }) ^ { ' }\left( {{\sigma\eta}\otimes\mathbf{h_{x}}}\right) ^ { ' }\end{array}\]

\[\begin{array}{r l} & {\mathrm{Notethat}} \\ & {(\sigma \pmb {\eta} \otimes \pmb {\sigma} \pmb {\eta}) (\pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1}) (\pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} (\sigma \pmb {\eta} \otimes \tilde {\mathbf {H}} _ {\mathbf {x x}}) ^ {\prime}} \\ & {= (\sigma \pmb {\eta} \otimes \pmb {\sigma} \pmb {\eta}) ((\pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1}) \otimes 1) (\pmb {\epsilon} _ {t + 1} ^ {\prime} \otimes (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime}) (\sigma \pmb {\eta} \otimes \tilde {\mathbf {H}} _ {\mathbf {x x}}) ^ {\prime}} \end{array}\]

So simply doing (for a 2 by 2 matrix)

\[r e s h a p e (E \left[ \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {s} \right], n x, n x) = E \left[ \begin{array}{c c} x ^ {f} (1, 1) x ^ {s} (1, 1) & x ^ {f} (2, 1) x ^ {s} (1, 1) \\ x ^ {f} (1, 1) x ^ {s} (2, 1) & x ^ {f} (2, 1) x ^ {s} (2, 1) \end{array} \right]\]

and we therefore need to transpose in the expression above.

\[\begin{array}{l} \text {And} \\ E \left[ \left(\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \right] \\ = E \left[ \left\{x _ {t} ^ {f} (\gamma_ {1}, 1) \left\{\epsilon_ {t + 1} (\phi_ {1}, 1) \right\} _ {\phi_ {1} = 1} ^ {n _ {e}} \right\} _ {\gamma_ {1} = 1} ^ {n _ {x}} \left(\left\{\epsilon_ {t + 1} (\phi_ {2}, 1) \left\{\left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) (\gamma_ {2}, 1) \right\} _ {\gamma_ {2} = 1} ^ {n _ {x} ^ {2}} \right\} _ {\phi_ {2} = 1} ^ {n _ {e}}\right) ^ {\prime} \right] \\ = E \left[ \left\{x _ {t} ^ {f} (\gamma_ {1}, 1) \left\{\epsilon_ {t + 1} (\phi_ {1}, 1) \right\} _ {\phi_ {1} = 1} ^ {n _ {e}} \right\} _ {\gamma_ {1} = 1} ^ {n _ {\mathrm{x}}} \left(\left\{\epsilon_ {t + 1} (\phi_ {2}, 1) \left\{x _ {t} ^ {f} (\gamma_ {2}, 1) \left\{x _ {t} ^ {f} (\gamma_ {3}, 1) \right\} _ {\gamma_ {3} = 1} ^ {n _ {\mathrm{x}}} \right\} _ {\gamma_ {2} = 1} ^ {n _ {\mathrm{x}}} \right\} _ {\phi_ {2} = 1} ^ {n _ {\mathrm{e}}}\right) ^ {\prime} \right] \end{array}\]

\[\begin{array} { l } = E [ \left( ( \mathbf { h } _ { \mathbf { x } } \otimes \sigma \boldsymbol { \eta } ) \left( \mathbf { x } _ { t } ^ { f } \otimes \boldsymbol { \epsilon } _ { t + 1 } \right) + ( \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x } } ) \left( \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \right) + ( \sigma \boldsymbol { \eta } \otimes \sigma \boldsymbol { \eta } ) \left( \boldsymbol { \epsilon } _ { t + 1 } \otimes \boldsymbol { \epsilon } _ { t + 1 } \right) - ( \sigma \boldsymbol { \eta } \otimes \sigma \boldsymbol { \eta } ) v e c ( \mathbf { I } _ { n _ { e } } ) \right) \\ \times ( \mathbf { u } _ { t + 1 } ^ { \prime } - E [ \mathbf { u } _ { t + 1 } ^ { \prime } ] ) ] \\ = E [ \left( ( \mathbf { h } _ { \mathbf { x } } \otimes \sigma \boldsymbol { \eta } ) \left( \mathbf { x } _ { t } ^ { f } \otimes \boldsymbol { \epsilon } _ { t + 1 } \right) + ( \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x} } ) \left( \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \right) + ( \sigma \boldsymbol { \eta } \otimes \sigma \boldsymbol { \eta } ) ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \boldsymbol { \epsilon } _ { t + 1 } ) - ( \sigma \boldsymbol { \eta } \otimes \sigma \boldsymbol { \eta } ) v e c ( \mathbf { I } _ { n _ { e } } ) \right) \mathbf { u } _ { t + 1 } ^ { \prime } ] \\ - E [ ( ( \mathbf { h } _ { \mathbf { x } } \otimes \sigma \boldsymbol { \eta } ) \left( \mathbf { x } _ { t } ^ { f } \otimes \boldsymbol { \epsilon } _ { t + 1 } \right) + ( \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x} } ) \left( (\boldsymbol { \epsilon } _ { t + 1 } \otimes (\boldsymbol { x } _ { t } ^ { f} ) + ( \sigma \boldsymbol { \eta } \otimes \sigma \boldsymbol { \eta} ) ( (\boldsymbol { \epsilon } _ { t + 1 } \otimes (\boldsymbol { \epsilon } _ { t + 1} ) - ( \sigma \boldsymbol { \eta } \otimes (\sigma \boldsymbol { \eta} ) v e c ( | {\bf I} _ { n _ { e} }\rangle ) ) ) E [ | {\bf u} _ { t + 1} ^ { \prime } ] \\ = E [ ( ( | {\bf h} _ { \mathbf { x } } \otimes (\sigma \boldsymbol { \eta} ) | ( | {\bf x} _ { t } ^ { f } | o t a l y s i n g r a b l y s i n g r a b l y s i n g r a b l y s i n g r a b l y s i n g r a b l y s i n g r a b l y s i n g r a b l y s i n g r a b l y s i n g r a b l y s i n g r a b l y s i n g r a b l y s i n g r a b d ] \\ - E [ ( 0 + 0 + 0 ) E [ | {\bf u} _ { t + 1} ^ { \prime } ] ] \\ = E [ ( | {\bf h} _ { \mathbf { x } } | o t a l y s i n g r a b l y s i n g r a b l y s i n g r a b l y s i n g r a b l y s i n g r a b l y s i n g r a b l y s i n g r a b l y s i n g r a b l y s i n g r a b l y s i n g r a b l y s i m p o w h e r e d i s e , \\ + ( o s o n g h a t h e r m a x i n g r a b l y s i n g r a b l y s i n g r a b l y s i n g r a b l y s i n g r a b l y s i n g r a b l y s i n g r a b l y s i n g r a b l y s i n g r a b l y s i n g r a b l y s i n g r a b l y s is o n g r a b l y s i n g r a b l y s i n g r a b l y s i n g r a b l y s i n g r a b l y s i n g r a b l y s i m p o w h e r e d i s e , \\ + ( o s o n g h a t h e r m a x i n g r a b l y s i n g r a bl y s i n g r a b l y s i n g r a b l y s i n g r a b l y s i n g r a b l y s i n g r a b l y s i n g r a b l y s i n g r a b l y s i n g r a b l y s i n g r a b l y s i m p o w h e r e d i s e , \\ - ( o s o n g h a t h e r m a x i n g r a b l y s i n g r a b l y s i n g r a b l y s i n g r a b l y s i n g r a b l y s i n g r a b l y s i n g r a b l y s i n g r a b l y s i n g r a b l y s i m p o w h e r e d i s e , \\ = E [ ( | {\bf h} _ { x } | o t a l y s i n g r a b l y s i n g r a b l y s i n g r a b l y s i n g r a b l y s i n g r a b l y s i n g r a b l y s i n g r a b l y s i n g r a b l y s i n g r a b l y s i m p o w h e r e d i s e , \\ - E [ ( | {\bf h} _ { x } | o t a l y s i n g r a b l y s i n g r a b l y s i n g r a b l y s i n g r a b l y s i n g r a b l y s i n g r a b l y s i m p o w h e r e d i s e , \\ - E [ ( | {\bf h} _ { x } | o t a l y s i n g r a b l | {\bf x} _ { t } ^ { f } | o t a l y s i n g r a b l | {\bf x} _ { t } ^ { f } | o t a l y s i n g r a b l | {\bf x} _ { t } ^ { f } | o t a l y s i n g r a b l | {\bf x} _ { t } ^ { f } | o t a l y s i n g r a b l ] \\ = E [ ( | {\bf h} _ { x } | o t a l y s i n g r a b l | {\bf x} _ { t } ^ { f } | o t a l y s i n g r a b l | {\bf x} _ { t } ^ { f } | o t a l y s i n g r a b l | {\bf x} _ { t } ^ { f } | o t A | {\bf x} _ { t } ^ { f } | o t A | {\bf x} _ { t } ^ { f } | o t A | {\bf x} _ { t } ^ { f } | o t A | {\bf x} _ { t } ^ { f } | o t A | {\bf x} _ { t } ^ { f } | o t A | {\bf x} _ { t }\]

\[\begin{array} { l } + \left( \mathbf { x } _ { t } ^ { f } \otimes \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \right) ^ { \prime } ( \mathbf { h } _ { \mathbf { x } } \otimes \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { \prime } + \left( \mathbf { x } _ { t } ^ { f } \otimes \boldsymbol { \epsilon } _ { t + 1 } \otimes \boldsymbol { \epsilon } _ { t + 1 } \right) ^ { \prime } ( \mathbf { h } _ { \mathbf { x } } \otimes \sigma \boldsymbol { \eta } \otimes \boldsymbol { \sigma } \boldsymbol { \eta } ) ^ { \prime } \\ + \left( \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \right) ^ { \prime } ( \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { \prime } + \left( \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \otimes \boldsymbol { \epsilon } _ { t + 1 } \right) ^ { \prime } ( \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x } } \otimes \boldsymbol { \sigma } \boldsymbol { \eta } ) ^ { \prime } \\ + \left( \boldsymbol { \epsilon } _ { t + 1 } \otimes \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \right) ^ { \prime } ( \sigma \boldsymbol { \eta } \otimes \boldsymbol { \sigma } \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { \prime } + 0 ) \\ ( \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x } } ) ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } ) ( ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \otimes \boldsymbol { \epsilon } _ { t + 1 } ) ^ { \prime } ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } \otimes \sigma \boldsymbol { \eta } ) ^ { \prime } \\ + ( \mathbf { x } _ { t } ^ { f } \otimes \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } ) ^ { \prime } ( \mathbf { h } _ { \mathbf { x } } \otimes \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { \prime } + ( \mathbf { x } _ { t } ^ { f } \otimes \boldsymbol { \epsilon } _ { t + 1 } \otimes \boldsymbol { \epsilon } _ { t + 1 } ) ^ { \prime } ( \mathbf { h } _ { \mathbf { x } } \otimes \sigma \boldsymbol { \eta } \otimes \boldsymbol { \sigma } \boldsymbol { \eta } ) ^ { \prime } \\ + ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) ^ { \prime } ( \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { \prime } + ( [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ], \\ - ( | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | = E [ ( {\bf h} _ {\bf x} {\otimes} {\boldsymbol {\sigma}} {\boldsymbol {\eta}}) ( {\bf x} _ {\mathrm{t}} ^ {\mathrm{f}} {\otimes} {\boldsymbol {\epsilon}} _ {\mathrm{t+1}}) ( ( {\bf x} _ {\mathrm{t}} ^ {\mathrm{f}} {\otimes} {\bf x} _ {\mathrm{t}} ^ {\mathrm{f}} {\otimes} {\boldsymbol{\epsilon}} _ {\mathrm{t+1}}) ^ { ' } ( {\bf h} _ {\bf x} {\otimes} {\bf h} _ {\bf x} {\otimes} {\boldsymbol{\sigma}} {\boldsymbol{\eta}} ) ^ { ' }\nRightarrow (\nRightarrow (\nRightarrow (\nRightarrow (\nRightarrow (\nRightarrow (\nRightarrow (\nRightarrow (\nRightarrow (\nRightarrow (\nRightarrow (\nRightarrow (\nRightarrow (\nRightarrow (\nRightarrow (\nRightarrow (\nRightarrow (\nRightarrow (\nRightarrow (\nRightarrow (\nRightarrow (\nRightarrow (\nRightarrow (\nRightarrow (\nRightarrow (\nRightarrow (\nRightarrow (\nRightarrow (\nRightarrow (\nRightarrow (\nRightarrow (\nRightarrow (\nRightarrow (\nRightarrow (\mathrm{e}) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ), & (a) , b) , c) , d) , e) , f) , g) , h) , i) , j) , k) , l) , m) , n) , o) , p) , q) , r) , s) , t) , u) , v) , w) , x) , y) , z) , w) , x) , y) , z) , w) , y) , z) , w) , z) , y) , z) , w) , z) , y) , z) , w) , z) , y) , z) , w) , z) , y) , z) , w) , z) , y) , z) , w) , z) , y) , z) , w) , z) , y) , z) , w) , z) , y) , z) , w) , z) , y), & (a, b; c; d; e; f; g; h; i; j; k; l; m; n; o; p; q; r; s; y; z; w; x; y; z; w; y; z; z; y; z; w; z); \\ (a, b; c; d; e; f; g; h; i; j; k; l; m; n; o; p; q; r; s; y; z; w; z; y; z; z; w; z); & (a, b; c; d; e; f; g; h; i; j; k; l; m; n; o; p; q; r; s; y; z; w; z); \\ (a, b; c; d; e; f; g; h; i; j; k; l; m; n; o; p; q; r; s; y; z; w; z); & (a, b; c; d; e; f; g; h ; i ; j ; k ; l ; m ; n ; o ; p ; q ; r ; s ; y ; z ; w ; z ; y ; z ; w ; z ); \\ (a, b; c; d; e; f ; g ; h ; i ; j ; k ; l ; m ; n ; o ; p ; q ; r ; s ; y ; z ; w ; z ; y ; z ; w ; z ); & (a, b ; c ; d ; e ; f ; g ; h ; i ; j ; k ; l ; m ; n ; o ; p ; q ; r ; s ; y ; z ; w ; z ); \\ (a, b ; c ; d ; e ; f ; g ; h ; i ; j ; k ; l ; m ; n ; o ; p ; q ; r ; s ; y ; z ; w ; z ); & (a, b ; c ; d ; e ; f ; g ; h ; i ; j ; k ; l ; m ; n ; o ; p ; q ; r ; s ; y ; z ; w ; z ); \\ (a, b ; c ; d ; e ; f ; g ; h ; i ; j ; k ; l ; m ; n ; o ; p ; q ; r ; s ; y ;z ; w ; z ); & (a, b ; c ; d ; e ; f ; g ; h ; i ; j ; k ; l ; m ; n ; o ; p ; q ; r ; s ; y ;z ; w ; z ); \\ (a, b ; c ; d ; e ; f ; g ; h . i j a ]) & (a, b : c : c : c : c : c : c : c : c : c : c : c : c : c : c : c : c : c : c : c : c : c : c : c : c : c : c : c : c : c : c : c : c : c : c : c : c : c : c : c : c : c : c : c : c : c : c : c : c : c : c :c :c :c :c :c :c :c :c :c :c :c :c :c :c :c :c :c :c :c :c :c :c :c :c :c :c :c :c :c :c :c :c :c :c :c :c :c :c :c :c :c :c :c :c :c :c :c :c :c :c : c: c: c: c: c: c: c: c: c: c: c: c: c: c: c: c: c: c: c: c: c: c: c: c: c: c: c: c: c: c: c: c: c: c: c: c: c: c: c: c: c: c: c: c: c: c: c: c: c: c: c :c: c: c: c: c: c: c: c: c: c: c: c: c: c: c: c: c: c: c: c: c: c: c: c: c: c: c: c: c: c: c: . \\ + (a, b): & (a, b): \\ + (a, b): & (a, b): \\ + (a, b): & (a, b): \\ + (a, b): & (a, b): \\ + (a, b): & (a, b): \\ + (a, b): & (a, b): \\ + (a, b): & (a, b): \\ + (a, b): & (a, b): \\ + (a, b): & (\textit \texttt \texttt \texttt \texttt \texttt \texttt \texttt \texttt \texttt \texttt \texttt \texttt \texttt \texttt \texttt \texttt \texttt \texttt \texttt \texttt \texttt \texttt \texttt \texttt \texttt \texttt \texttt \texttt \texttt \texttt \texttt \texttt \texttt \texttt\texttt\texttt\texttt\texttt\texttt\texttt\texttt\texttt\texttt\texttt\texttt\texttt\texttt\texttt\texttt\texttt\texttt\texttt\texttt\texttt\texttt\texttt\texttt\texttt\texttt\texttt\texttt\texttt\texttt\texttt\texttt\texttt\texttt{\xintop}\int_\int_\int_\int_\int_\int_\int_\int_\int_\int_\int_\int_\int_\int_\int_\int_\int_\int_\int_\int_\int_\int_\int_\int_\int_\int_\int_\int_\int_\int_\int_{\int_{\int_{\int_{|} \|}} \|}\overline \int_|\overline \int_|\overline \int_|\overline \int_|\overline \int_|\overline \int_|\overline \int_|\overline \int_|\overline \int_|\overline \int_|\overline \int_|\overline \int_|\overline \int_|\overline \int_|\overline \int_|\overbrace \int_|\]

\[\begin{array} { l } + ( \sigma \eta \otimes \sigma \eta ) ( \epsilon _ { t + 1 } \otimes \epsilon _ { t + 1 } ) ( \epsilon _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) ^ { \prime } ( \sigma \eta \otimes \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { \prime } \\ + ( \sigma \eta \otimes \sigma \eta ) ( \epsilon _ { t + 1 } \otimes \epsilon _ { t + 1 } ) ( \epsilon _ { t + 1 } \otimes \epsilon _ { t + 1 } \otimes \epsilon _ { t + 1 } ) ^ { \prime } ( \sigma \eta \otimes \sigma \eta \otimes \sigma \eta ) ^ { \prime } ] \\ - ( \sigma \eta \otimes \sigma \eta ) v e c ( \mathbf { I } _ { n _ { e } } ) E [ \mathbf { u } _ { t + 1 } ^ { \prime } ] \\ = E [ \\ ( \mathbf { h } _ { \mathbf { x } } \otimes \sigma \eta ) ( \mathbf { x } _ { t } ^ { f } ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) ^ { \prime } \otimes \epsilon _ { t + 1 } ^ { \prime } \epsilon _ { t + 1 } ) ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } \otimes \sigma \eta ) ^ { \prime } \\ + ( \mathbf { h } _ { \mathbf { x } } \otimes \sigma \eta ) ( \mathbf { x } _ { t } ^ { f } \otimes \epsilon _ { t + 1 } ) ( \mathbf { x } _ { t } ^ { f } \otimes \epsilon _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } ) ^ { \prime } ( \mathbf { h } _ { \mathbf { x } } \otimes \sigma \eta \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { \prime } \\ + ( \mathbf { h } _ { \mathbf { x } } \otimes \sigma \eta ) ( \mathbf { x } _ { t } ^ { f } ( \mathbf { x } _ { t } ^ { f } ) ^ { \prime } \otimes \epsilon _ { t + 1 } ( \epsilon _ { t + 1 } \otimes \epsilon _ { t + 1 } ) ^ { \prime } ) ( \mathbf { h } _ { \mathbf { x } } \otimes \sigma \eta \otimes \sigma \eta ) ^ { \prime } \\ + ( \mathbf { h } _ { \mathbf { x } } \otimes \sigma \eta ) ( \mathbf { x } _ { t } ^ { f } \otimes \epsilon _ { t + 1 } ) ( \epsilon _ { t + 1 } ^ { \prime } \otimes ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) ^ { \prime } ) ( \sigma \eta \otimes h _ { x } \otimes h _ { x } ) ^ { \prime } \\ + ( h _ { x } \otimes \sigma \eta ) ( x _ { t } ^ { f } \otimes e _ { t + 1 } ) ( e _ { t + 1 } ^ { f } & ( e _ { t + 1 } ^ { f } & ( e _ { t + 1 } ^ { f } & ( e _ { t + 1 } ^ { f } & ( e _ { t + 1 } ^ { f } & ( e _ { t + 1 } ^ { f } & ( e _ { t + 1 } ^ { f } & ( e _ { t + 1 } ^ { f } & ( a _ { t + 1 } ^ { f } & ( a _ { t + 1 } ^ { f } & ( a _ { t + 1 } ^ { f } & ( a _ { t + 1 } ^ { f } & ( a _ { t + 1 } ^ { f } & ( a _ { t + 1 } ^ { f } & ( a _ { t + 1 } ^ { f } & ( a _ {\mathrm{的}} ^ { f } & ( a _ {\mathrm{的}} ^ { f } & ( a _ {\mathrm{的}} ^ { f } & ( a _ {\mathrm{的}} ^ { f } & ( a _ {\mathrm{的}} ^ { f } & ( a _ {\mathrm{的}} ^ { f } & ( a _ {\mathrm{的}} ^ { f } & ( a _ {\mathrm{的}} ^ { f } & ( a _ {-} ^ {\prime} & ( a _ {\mathrm{的}} ^ {\prime} & ( a _ {-} ^ {\prime} & ( a _ {-} ^ {\prime} & ( a _ {-} ^ {\prime} & ( a _ {-} ^ {\prime} & ( a _ {-} ^ {\prime} & ( a _ {-} ^ {\prime} & ( a _ {-} ^ {\prime} & ( a _ {-} ^ {\prime} & ( a _ {-} ^ {\prime} & ( a _ {-} ^ {\prime} & ( a _ {- -} ^ {\prime} & ( a _ {-} ^ {\prime} & ( a _ {-} ^ {\prime} & ( a _ {-} ^ {\prime} & ( a _ {-} ^ {\prime} & ( a _ {-} ^ {\prime} & ( a _ {-} ^ {\prime} & ( a _ {-} ^ {\prime} & ( a _ {-} ^ {\prime} & ( a _ {-} ^ {\prime} & ( a _ {\mathrm{一}} ^ {\prime} & ( a _ {-} ^ {\prime} & ( a _ {-} ^ {\prime} & ( a _ {-} ^ {\prime} & ( a _ {-} ^ {\prime} & ( a _ {-} ^ {\prime} & ( a _ {-} ^ {\prime} & ( a _ {-} ^ {\prime} & ( a _ {-} ^ {\prime} & ( a _ {-} ^ {\prime} & ( a _ {\infty}) \\ + (\sigma , q) = (\sigma , q) = (\sigma , q) = (\sigma , q) = (\sigma , q) = (\sigma , q) = (\sigma , q) = (\sigma , q) = (\sigma , q) = (\sigma , q) = (\sigma , q) = (\sigma , q) = (\sigma , q) = (\sigma , q) = (\sigma , q) = (\sigma , q) = (\sigma , q) = (\sigma , p) = (\sigma , p) = (\sigma , p) = (\sigma , p) = (\sigma , p) = (\sigma , p) = (\sigma , p) = (\sigma , p) = (\sigma , p) = (\sigma , p) = (\sigma , p) = (\sigma , p) = (\sigma , p) = (\sigma , p) = (\sigma , p) = (\sigma , p) = (\sigma , p) = (\bar {\sigma}, p) = (\bar {\sigma}, p) = (\bar {\sigma}, p) = (\bar {\sigma}, p) = (\bar {\sigma}, p) = (\bar {\sigma}, p) = (\bar {\sigma}, p) = (\bar {\sigma}, p) = (\bar {\sigma}, p) = (\bar {\sigma}, p) = (\bar {\sigma}, p) = (\bar {\sigma}, p) = (\bar {\sigma}, p) = ((\bar {\sigma}, p) - (\bar {\sigma}, p)) \\ + (\sigma , q) = (\sigma , q) = (\sigma , q) = (\sigma , q) = (\sigma , q) = (\sigma , q) = (\sigma , q) = (\sigma , q) = (\bar {\sigma}, p) = (\bar {\sigma}, p) = (\bar {\sigma}, p) = (\bar {\sigma}, p) = (\bar {\sigma}, p) = (\bar {\sigma}, p) = (\bar {\sigma}, p) = (\bar {\sigma}, p) = (\bar {\sigma}, p) = (\bar {\sigma}, p) = (\bar {\sigma}, p) = (\bar {\sigma}, p) = (\sqrt [ n ]{p}) \\ + (\sigma , q) = (\sigma , q) = (\sigma , q) = (\sigma , q) = (\bar {\sigma}, p) = (\bar {\sigma}, p) = (\bar {\sigma}, p) = (\bar {\sigma}, p) = (\bar {\sigma}, p) = (\bar {\sigma}, p) = (\bar {\sigma}, p) = (\bar {\sigma}, p) = (\bar {\sigma}, p) = (\bar {\pi}, q) \\ + (\sigma , q) = (\sigma , q) = (\bar {\pi}, q) = (\bar {\pi}, q) = (\bar {\pi}, q) = (\bar {\pi}, q) = (\bar {\pi}, q) = (\bar {\pi}, q) = (\bar {\pi}, q) \\ + (\sigma , q) = (\sigma , q) = ((\bar {\pi}, q) - (\bar {\pi}, q)) ((\bar {\pi}, q - (\bar {\pi}, q)) ^ {'}) ((\bar {\pi}, q - (\bar {\pi}, q)) ^ {'}) ((\bar {\pi}, q - (\bar {\pi}, q)) ^ {'}) ((\bar {\pi}, q - (\bar {\pi}, q)) ^ {'}) ((\bar {\pi}, q - (\bar {\pi}, q)) ^ {'}) ((\bar {\pi}, q - (\bar {\pi}, q)) ^ {'}) ((\bar {\pi}, q - (\overline {{q}}), q)) \\ + (\sigma , q) = ((\bar {\pi}, q - (\overline {{q}}, q)) ((\overline {{q}}, q - (\overline {{q}}, q)) ((\overline {{q}}, q - (\overline {{q}}, q)) ((\overline {{q}}, q - (\overline {{q}}, q)) ((\overline {{q}}, q - (\overline {{q}}, q)) ((\overline {{q}}, q - (\overline {{q}}, q)) ((\overline {{q}}, q - (\overline {{q}}, q)) ((\overline {{{q}}}) \\ + (\overline {{q}}, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q. \\ - ((\overline {{q}}, q; s i n g e r e d i n g e r e d i n g e r e d i n g e r e d i n g e r e d i n g e r e d i n g e r e d i n g e r e d i n g e r e d i n g e r e d i n g e r e d i n g e r e d i n g e r e d i n g e r e d i n g e r e d i l o w i n g e r e d i n g e r e d i n g e r e d i n g e r e d i n g e r e d i n g e r e d i n g e r e d i n g e r e d i n g e r e d i n g e r e d i n g e r e d i n g e r e d i n g e r e d i n g e r e d i m b y c o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n s o n . \\ - ((\overline {{q}}, q; s i n g e r e d i n g e r e d i l o w i n g e r e d i l o w i n g e r e d i l o w i n g e r e d i l o w i n g e r e d i l o w i n g e r e d i l o w i n g e r e d i l o w i n g e r e d i l o w i n g e r e d i l o w i n g e r e d i l o w j y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y . \\ - ((\overline {{q}}, q; s i n g jy z) \\ - ((\overline {{q}}, q; s i n g jy z)) ((\overline {{q}}, q; s i n g jy z)) ((\overline {{q}}, q; s i n g jy z)) ((\overline {{q}}, q; s i n g jy z)) ((\overline {{q}}, q; s i n g jy z)) ((\overline {{q}}, q; s i n g jy z)) ((\overline {{q}}, q; s i n g jz); \\ - ((\overline {{q}}, q; s i n g jy z)) ((\overline {{q}}, q; s i n g jy z)) ((\overline {{q}}, q; s i n g jy z)) ((\overline {{q}}, q; s i n g jy z)) ((\overline {{q}}, q; s i n g jy z)) ((\overline {{q}}, q; s i ng jy z); \\ - ((\overline {{q}}, q; s i n g jy z)) ((\overline {{q}}, q; s i n g jy z)) ((\overline {{q}}, q; s i ng jy z)) ((\overline {{q}}, q; s i ng jy z)) ((\overline {{q}}, q; s i ng jy z)) ((\overline {{q}}, q; s i ng jy z); \\ - ((\overline {{q}}, q; s i n g jy z)) ((\overline {{q}}, q; s i ng jy z)) ((\overline {{q}}, q; s i ng jy z)) ((\overline {{q}}, q; s i ng jy z)) ((\overline {{q}}, q; s i ng jy z)) ((\overline {{q}}, q; s i ng jy z); \\ - ((\overline {{q}}, q; s i n g jy z)) ((\overline {{q}}, q ; s i ng jy z)) ((\overline {{q}}, q ; s i ng jy z)) ((\overline {{q}}, q ; s i ng jy z)) ((\overline {{q}}, q ; s i ng jy z)) ((\overline {{q}}, q ; s i ng jy z)) ((\overline {{q}}, q ; s i ng jy z); \\ - ((\overline {{q}}, q; s i n g jy z)) ((\overline {{q}}, q ; s i ng jy z)) ((\overline {{q}}, q ; s i ng jy z)) ((\overline {{q}}, q ; s i ng jy z)) ((\overline {{q}}, q ; s i ng jy z)) ((\overline{{q}} ,s i c m ) / 2 ) / 2 ) / 2 ) / 2 ) / 2 ) / 2 ) / 2 ) / 2 ) / 2 ) / 2 ) / 2 ) / 2 ) / 2 ) / 2 ) / 2 ) / 2 ) / 2 ) / 2 ) / 2 ) / 2 ) / 2 ) / 2 ) / 2 ) / 2 ) / 2 ) / 3 . \\ - ((\overline {{q}} ,s i c m ) / 2 ) / 2 ) / 2 ) / 2 ) / 2 ) / 2 ) / 2 ) / 2 ) / 2 ) / 2 ) / 2 ) / 2 ) / 2 ) / 2 ) / 2 ) / 2 ) / 2 ) / 2 ) / 2 ) / 2 ) / 2 ) / 2 ) / 2 ) - ((a b c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c . \\ - ((a b c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c . \\ - ((a b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b< fcel>+((a,b,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c,c;c,a,b,d,e,f,f,g,h,i,j,k,l,m,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,u,w,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,y,Y< nl>\tag{1}\]

2)

3)

4)

5)

6)

\[+ \left(\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}\right) E \left[ \left(\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) \left(\left(\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) ^ {\prime} \otimes \left(\mathbf {x} _ {t} ^ {f}\right) ^ {\prime}\right) \right] (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime}\tag{7}\]

\[+ \left(\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}\right) E \left[ \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) ^ {\prime} \right] \left(\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}\right) ^ {\prime}\tag{8}\]

\[+ \left(\sigma \pmb {\eta} \otimes \mathbf {h} _ {\mathbf {x}}\right) E \left| \left(\pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\mathbf {x} _ {t} ^ {f} \otimes \pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \right| \left(\mathbf {h} _ {\mathbf {x}} \otimes \sigma \pmb {\eta} \otimes \mathbf {h} _ {\mathbf {x}}\right) ^ {\prime}\tag{9}\]

\[+ (\sigma \pmb {\eta} \otimes \mathbf {h _ {x}}) E \left[ (\pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}) (\mathbf {x} _ {t} ^ {f} \otimes \pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1}) ^ {\prime} \right] (\mathbf {h _ {x}} \otimes \sigma \pmb {\eta} \otimes \pmb {\sigma} \pmb {\eta}) ^ {\prime}\tag{10}\]

\[+ \left(\sigma \pmb {\eta} \otimes \mathbf {h} _ {\mathbf {x}}\right) \left(\mathbf {I} _ {n _ {e}} \otimes E \left[ \mathbf {x} _ {t} ^ {f} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \right]\right) \left(\sigma \pmb {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}\right) ^ {\prime}\tag{11}\]

\[+ \left(\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}\right) \left(E \left[ \boldsymbol {\epsilon} _ {t + 1} \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) ^ {\prime} \right] \otimes E \left[ \mathbf {x} _ {t} ^ {f} \left(\mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \right]\right) \left(\sigma \boldsymbol {\eta} \otimes \boldsymbol {\sigma} \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}\right) ^ {\prime}\tag{12}\]

\[\begin{array}{r l} & + (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}) \left(E \left[ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \right] \otimes E \left[ (\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}) \boldsymbol {\epsilon} _ {t + 1} ^ {\prime} \right]\right) (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}) ^ {\prime} \\ & + (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}) E \left[ (\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}) \left(\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \right] (\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime} \\ & + (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}) \left(E \left[ (\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}) \boldsymbol {\epsilon} _ {t + 1} ^ {\prime} \right] \otimes E \left[ (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} \right]\right) (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime} \\ & + (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}) E \left[ (\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}) (\boldsymbol {\epsilon} _ {t + 1} ^ {\prime} \otimes (\boldsymbol {\epsilon} _ {t + 1} ^ {\prime} \otimes \boldsymbol {\epsilon} _ {t + 1} ^ {\prime})) \right] (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}) ^ {\prime} \end{array}\]

\[- (\sigma \boldsymbol {\eta} \otimes \boldsymbol {\sigma} \boldsymbol {\eta}) v e c (\mathbf {I} _ {n _ {e}}) E [ \mathbf {u} _ {t + 1} ^ {\prime} ] \tag {17}\]

Hence, we need to compute the remaining matrices directly. This is done below where the number relates to the row in the expression for

1) None

17) none

\[\begin{array} { r l } & { \textbf { 1 . 3 . 5 } \quad \textbf { F o r } V a r \left[ \tilde { \boldsymbol { \xi } } _ { t + 1 } \right] _ { 5 5 } } \\ & { \textbf { N o t e f i r s t t h a t } V a r \left[ \tilde { \boldsymbol { \xi } } _ { t + 1 } \right] _ { 5 5 } \text { has dimensions } n _ { x } ^ { 2 } \times n _ { x } ^ { 2 } . } \\ & { \quad V a r \left[ \tilde { \boldsymbol { \xi } } _ { t + 1 } \right] _ { 5 5 } \equiv E [ ( ( \sigma \eta \otimes \mathbf { h _ { x } } ) ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x _ { t } ^ { s } } ) + ( \sigma \eta \otimes \tilde { \mathbf { H } } _ { \mathbf { x x } } ) ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x _ { t } ^ { f } } \otimes \mathbf { x _ { t } ^ { f } } ) + ( \sigma \eta \otimes \frac { 1 } { 2 } \mathbf { h _ { \sigma \sigma } } \sigma ^ { 2 } ) \boldsymbol { \epsilon } _ { t + 1 } ) } \\ & { \qquad \quad \times ( ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x _ { t } ^ { s } } ) ^ { \prime } ( \sigma \eta \otimes \mathbf { h _ { x } } ) ^ { \prime } + ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x _ { t } ^ { f } } \otimes \mathbf { x _ { t } ^ { f } } ) ^ { \prime } ( \sigma \eta \otimes \tilde { \mathbf { H } } _ { \mathbf { x x } } ) ^ { \prime } + \boldsymbol { \epsilon } _ { t + 1 } ^ { \prime } ( \sigma \eta \otimes \frac { 1 } { 2 } \mathbf { h _ { \sigma \sigma } } \sigma ^ { 2 } ) ^ { \prime } ) ] } \\ & = E [ ( \sigma \eta \otimes \mathbf { h _ { x } } ) ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x _ { t } ^ { s } } ) ( ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x _ { t } ^ { s } } ) ^ { \prime } ( \sigma \eta \otimes \mathbf { h _ { x } } ) ^ { \prime } + ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x _ { t } ^ { f } } \otimes x _ { t } ^ { f } ) ^ { \prime } ( \sigma \eta \otimes \tilde { \mathbf { H } } _ { \mathbf { x x } } ) ^ { \prime } + \boldsymbol { \epsilon _ { t + 1 } ^ { \prime } ( \sigma \eta \otimes \frac { 1 } { 2 } h _ { \sigma \sigma } \sigma ^ { 2 } ) ^ { \prime } ) } \\ & + ( ( \sigma \eta \otimes \tilde { H } _ { x x } ) ( \boldsymbol { \epsilon _ { t + 1 } }\otimes\otimes\otimes\otimes\otimes\otimes\otimes\otimes\otimes\otimes\otimes\otimes\otimes\otimes\otimes\otimes\otimes\otimes\otimes\otimes\otimes\otimes\otimes\otimes\otimes\otimes\otimes\otimes\otimes\otimes\otimes\otimes\otimes\otimes\entimes\otimes\otimes\otimes\otimes\otimes\otimes\otimes\otimes\otimes\otimes\otimes\otimes\otimes\otimes\otimes\otimes\otimes\otimes\otimes\otimes\otimes\otimes\otimes\otimes\otimes\otimes\otimes\otimes\otimes\otimes\otimes\otimes\ottimes) ] , \\ & \qquad + ( ( \sigma \eta \otimes \frac 1 2 h _ { s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r c e d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n c e d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d e f f o r ,} \\ & {\qquad = E [ ( ( \sigma \eta \otimes h _ { x } ) ( \boldsymbol { \epsilon _ { t + 1 }} (\boldsymbol { x _ { t }} ^ { s }) ( ( \boldsymbol { \epsilon _ { t + 1 }} (\boldsymbol { x _ { t }} ^ { s }) ^ { \prime } ( ( \sigma \eta (\boldsymbol {\theta_ {\mathrm{X}}}) ^ { ' } - ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (\ v a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l u e m e g e , v e r e c e , v e r e c e , v e r e c e , v e r e c e , v e r e c e , v e r e c e , v e r e c e , v e r e c e , v e r e c e , v e r e c e , v e r e c e , v e r e c e , v e r e c e , v e r e c e , v e r e c e ) , v e r e c e , v e r e c e , v e r e c e , v e r e c e , v e r e c e , v e r e c e , v e r e c e , v e r e c e , v e r e c e , v e r e c e , v e r e c e , v e r e c e , v e r e c e , v e r e c e , w ) ,} \\ & \qquad + (\sigma / 2 h _ { x x}) (\boldsymbol { (\varepsilon _ { t + 1 }} (\varepsilon _ { t + 1 } (\varepsilon _ { t + 1 } (\varepsilon _ { t + 1 } (\varepsilon _ t + 1 (\varepsilon _ t + 1 (\varepsilon _ t + 1 (\varepsilon _ t + 1 (\varepsilon _ t + 1 (\varepsilon _ t + 1 (\varepsilon _ t + 1 (\varepsilon _ t + 1 (\varepsilon _ t + 1 (\varepsilon _ t + 1 (\varepsilon _ t + 1 (\varepsilon _ t + 1 (\varepsilon _ t + 1 (\varepsilon _ t + 1 (\varepsilon _ t + 1 (\varepsilon _ t + 1 (\varepsilon _ t + 1 (\varepsilon _ { t + 1 (\varepsilon _ { t + 1 ({v a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l | ) , v ) ) ) ) ) ) .} \\ & {\qquad + (\sigma / 2 h _ { x x}) (\boldsymbol{ (\varepsilon _ { t + 1 }} (\boldsymbol{ (\varepsilon _ { t + 1 }} (\boldsymbol{ (\varepsilon _ { t + 1 }} (\boldsymbol{ (\varepsilon _ { t + 1 }} (\boldsymbol{ (\varepsilon _ { t + 1 }} (\boldsymbol{ (\varepsilon _ { t + 1 }} (\boldsymbol{ (\varepsilon _ { t + 1 }} (\boldsymbol{\varepsilon_{t+ 1}} (\boldsymbol{ (\varepsilon_{t+ 1}} (\boldsymbol{ (\varepsilon_{t+ 1}} (\boldsymbol{ (\varepsilon_{t+ 1}} (\boldsymbol{ (\varepsilon_{t+ 1}} (\boldsymbol{ (\varepsilon_{t+ 1}} (\boldsymbol{ (\varepsilon_{t+ 1}} (\boldsymbol{ (\varepsilon_{t+ 1}}) ) ) ) ) ) ) .} \\ & {\qquad + (\sigma / 2 h _ { x x}) (\boldsymbol{ (\varepsilon_{t+ 1}} (\boldsymbol{ (\varepsilon_{t+ 1}} (\boldsymbol{ (\varepsilon_{t+ 1}} (\boldsymbol{ (\varepsilon_{t+ 1}} (\boldsymbol{ (\varepsilon_{t+ 1}} (\boldsymbol{ (\varepsilon_{t+ 1}}) ) ) ) ) ) .} \\ & {\qquad + (\sigma / 2 h _ { x x}) (\boldsymbol{ (\varepsilon_{t+ 1}} (\boldsymbol{ (\varepsilon_{t+ 1}} (\boldsymbol{ (\varepsilon_{t+ 1}} (\boldsymbol{ (\varepsilon_{t+ 1}}) ) ) ) ) .} \\ & {\qquad + (\sigma / 2 h _ { x x}) (\boldsymbol{ (\varepsilon_{t+ 1}} (\boldsymbol{ (\varepsilon_{t+ 1}} (\boldsymbol{ (\varepsilon_{t+ 1}}) ) ) ) ) .} \\ & {\qquad + (\sigma / 2 h _ { x x}) (\boldsymbol{ (\varepsilon_{t+ 1}} (\boldsymbol{ (\varepsilon_{t+ 1}} (\nabla^f^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^r)^k,} \\ & {\qquad + (\sigma / 2 h _ { x x}) (\boldsymbol{(\varepsilon_{t+ 1}} (\boldsymbol{(\varepsilon_{t+ 1}} (\nabla^f^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^s^c)^k,} \\ & {\qquad + (\sigma / 2 h _ { x x}) (\boldsymbol{(\varepsilon_{t+ 1}} (\nabla^{f}x) ^{ '}(x) ^{ '}(x) ^{ '}(x) ^{ '}(x) ^{ '}(x) ^{ '}(x) ^{ '}(x) ^{ '}(x) ^{ '}(x) ^{ '}(x) ^{ '}(x) ^{ '}(x) ^{ '}(x) ^{ '}(x) ^{ '}(x) ^{ '}(x) ^{ '}(x) ^{\prime},} \\ & {\qquad + (\sigma / 2 h _ { x x}) (\nabla^{f}x) ^{ '}(x) ^{ '}(x) ^{ '}(x) ^{ '}(x) ^{ '}(x) ^{ '}(x) ^{ '}(x) ^{ '}(x) ^{ '}(x) ^{ '}(x) ^{ '}(x) ^{ '}(x) ^{ '}(x) ^{ '}(x) ^{ '}(x) ;} \\ & {\qquad + (\sigma / 2 h _ { x x}) (\nabla^{f}x) ^{ '}(x) ^{ '}(x) ^{ '}(x) ^{ '}(x) ^{ '}(x) ^{ '}(x) ^{ '}(x) ^{ '}(x) ^{ '}(x) ^{ '}(x) ^{ '}(x) ^{ '}(x) ;} \\ & {\qquad + (\sigma / 2 h _ { x x}) (\nabla^{f}s}^{''}(\nabla^{f}s^{''}(\nabla^{f}s^{''}(\nabla^{f}s^{''}(\nabla^{f}s^{''}(\nabla^{f}s^{''}(\nabla^{f}s^{''}(\nabla^{f}s^{''}(\nabla^{f}s^{''}(\nabla^{f}s^{''}(\nabla^{f}s^{''}(\nabla^{f} s^{''}(\nabla^{f}s^{''}(\nabla^{f}s^{''}(\nabla^{f}s^{''}(\nabla^{f}s^{''}(\nabla^{f}s^{''}(\nabla^{f}s^{''}(\nabla^{f}s^{''}(\nabla^{f}s^{''}(\nabla^{f}s^{''}(\nabla^{f}\texttt{s}) ) ) ) .} \\ & {\qquad + (\sigma / 2 h _ { x x}) (\nabla^{f}s}^{''}(\nabla^{f}s^{''}(\nabla^{f}s^{''}(\nabla^{f}s^{''}(\nabla^{f}s^{''}(\nabla^{f}s^{''}(\nabla^{f}s^{''}(\nabla^{f}s^{''}({\nu}_{a}{}^{i}{}^{j}{}^{k}{}^{l}{}^{m}{}^{j}{}^{k}{}^{l}{}^{m}{}^{k}{}^{l}{}^{m}{}^{k}{}^{l}{}^{m}{}^{k}{}^{l}{}^{m}{}^{k}{}^{l}{}^{m}{}^{k}{}^{l}{}^{m}{}^{k}{}^{l}{}^{m}{}^{k}{}^{l}{}^{m}{}^{k}{}^{l}{}^{m}{}^{m}{}^{k}{}^{l}{}^{m}{}^{k}{}^{l}{}^{m}{}^{k}{}^{l}{}^{m}{}^{k}{}^{l}{}^{m}{}^{k}{}^{l}{}^{m}{}^{k}{}^{l}{}^{m}{}^{k}{}^{l}{}^{m}{}^{k}{}^{l}{}^{k}{}^{l}{}^{m}{}^{k}{}^{l}{}^{m}{}^{k}{}^{l}{}^{m}{}^{k}{}^{l}{}^{m}{}^{k}{}^{l}{}^{m}{}^{k}{}^{l}{}^{m}{}^{k}{}^{l}{}^{m}{}^{k}{}^{l}{}^{m}{}^{ k}-h;}\end{array}\]

\[\begin{array} { r l } & { ] } \\ & { = E [ } \\ & { \left( \sigma \eta \otimes \mathbf { h _ { x } } \right) \left( \epsilon _ { t + 1 } \otimes \mathbf { x _ { t } ^ { s } } \right) \left( \epsilon _ { t + 1 } ^ { \prime } \otimes \left( \mathbf { x _ { t } ^ { s } } \right) ^ { \prime } \right) \left( \sigma \eta \otimes \mathbf { h _ { x } } \right) ^ { \prime } } \\ & { + \left( \sigma \eta \otimes \mathbf { h _ { x } } \right) \left( \epsilon _ { t + 1 } \otimes \mathbf { x _ { t } ^ { s } } \right) \left( \epsilon _ { t + 1 } ^ { \prime } \otimes \left( \mathbf { x _ { t } ^ { f } } \otimes \mathbf { x _ { t } ^ { f } } \right) ^ { \prime } \right) \left( \sigma \eta \otimes \tilde { \mathbf { H _ { x x } } } \right) ^ { \prime } } \\ & { + \left( \sigma \eta \otimes \mathbf { h _ { x } } \right) \left( \epsilon _ { t + 1 } \otimes \mathbf { x _ { t } ^ { s } } \right) \left( \epsilon _ { t + 1 } ^ { \prime } \otimes 1 \right) \left( \sigma \eta \otimes \frac 12 \mathbf { h _ { s o } } \sigma ^ { 2 } \right) ^ { \prime } } \\ & { + \left( \sigma \eta \otimes \tilde { \mathbf { H _ { x x } } } \right) \left( \epsilon _ { t + 1 } \otimes \left( \mathbf { x _ { t } ^ { f } } \otimes \mathbf { x _ { t } ^ { f } } \right) \right) \left( \epsilon _ { t + 1 } ^ { \prime } \otimes ( \mathbf { x _ { t } ^ { s } } ) ^ { \prime } \right) ( \sigma \eta \otimes \mathbf { h _ { x } } ) ^ { \prime } } \\ & { + \left( \sigma \eta \otimes \tilde { \mathbf { H _ { x x } } } \right) \left( \epsilon _ { t + 1 } \otimes \mathbf { x _ { t } ^ { f } } \otimes \mathbf { x _ { t } ^ { f } } \right) \left( \epsilon _ { t + 1 } ^ { \prime } \otimes ( \mathbf { x _ { t } ^ { f } } \otimes \mathbf { x _ { t } ^ { f } } ) ^ { \prime } \right) ( \sigma \eta \otimes \tilde { \mathbf { H _ { x x } } } ) } \\ & { + \left( \sigma \eta \otimes \tilde { \mathbf { H _ { x x } } } \right) \left( \epsilon _ { t + 1 } \otimes \mathbf { x _ { t } ^ { f } } \otimes \mathbf { x _ { t } ^ { f } } \right) ( \epsilon _ { t + 1 } ^ { \prime } \otimes 1 ) ( \sigma \eta \otimes \frac 12 {\mathbf { h _ { s o }} } {\sigma ^ { 2 } ) ^ { ' } } } \\ & \\ & + ( \sigma \eta \otimes \frac 12 {\mathbf { h _ { s o }} } {\sigma ^ { 2 } ) ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (\mathrm{H})}) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ; \\ & \\ & = E [ \\ & (\sigma \eta \otimes {\mathbf { h _ { x } }}) ( [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ ] , i n d e - 1 ] , i n d e - 1 ] , i n d e - 1 ] , i n d e - 1 ] , i n d e - 1 ] , i n d e - 1 ] , i n d e - 1 ] , i n d e - 1 ] , i n d e - 1 ] , i n d e - 1 ] , i n d e - 1 ] , i n d e - 1] , i n d e - 1 ] , i n d e - 1 ] , i n d e - 1 ] , i n d e - 1 ] , i n d e - 1 ] , i n d e - 1 ] , i n d e - 1 ] , i n d e - 1 ] , i n d e - 1 ] , i n d e - 1 ] , i n d e - 1 ] . \\ & \\ & + ( [ (\sigma / * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * . \\ & \\ & + ( [ (\sigma / * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * . \\ & \\ & + ( [ (\sigma / * * * * . \\ & \\ & + ( [ (\sigma / * . \\ & \\ & + ( [ (\sigma / . \\ & \\ & + ( [ (\sigma / . \\ & \\ & + ( [ (\sigma / . \\ & \\ & + ( [ (\sigma / . \\ & \\ & + ( [ (\sigma / . \\ & \\ & + ( [ (\sigma / . \\ & \\ & + ( [ (\sigma / . \\ & \\ & + ( (\sigma / . \\ & \\ & + ( . \\ & \\ & + ( . \\ & \\ & + ( . \\ & \\ & + ( . \\ & \\ & + ( . \\ & \\ & + ( . \\ & \\ & + ( . \\ & \\ & + ( . \\ & \\ & + ( . \\ & \\ & + ( . \\ & \\ & + ( . \\ & \\ & + ( . \\ & \\ & + ( . \\ | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , |, | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | , | | \\ & + (\sigma / . \\ & + (\sigma / . \\ & + (\sigma / . \\ & + (\sigma / . \\ & + (\sigma / . \\ & + (\sigma / . \\ & + (\sigma / . \\ & + (\sigma / . \\ & + (\sigma / . \\ & + (\sigma / . \\ & + (\sigma / . \\ & + (\sigma / . ; \\ & + (\sigma / . ; \\ & + (\sigma / . ; \\ & + (\sigma / . ; \\ & + (\sigma / . ; \\ & + (\sigma / . ; \\ & + (\sigma / . ; \\ & + (\sigma / . ; \\ & + (\sigma / . ; \\ & + (\sigma / . ; \\ & + (\sigma / .; \\ & + (\sigma / . ; \\ & + (\sigma / . ; \\ & + (\sigma / . ; \\ & + (\sigma / . ; \\ & + (\sigma / . ; \\ & + (\sigma / . ; \\ & + (\sigma / . ; \\ & + (\sigma / . ; \\ & + (\sigma / . ; \\ & + (\sigma / . : \\ & + (\sigma / . : \\ & + (\sigma / . : \\ & + (\sigma / . : \\ & + (\sigma / . : \\ & + (\sigma / . : \\ & + (\sigma / . : \\ & + (\sigma / . : \\ & + (\sigma / . : \\ & + (\sigma / . : \\ & + (\sigma / .: \\ & + (\sigma / . : \\ & + (\sigma / . : \\ & + (\sigma / . : \\ & + (\sigma / . : \\ & + (\sigma / . : \\ & + (\sigma / . : \\ & + (\sigma / . : \\ & + (\sigma / . : \\ & + (\sigma / . : \\ & + (\sigma / .：;\\ & + (\sigma / . :;\\ & + (\sigma / . :;\\ & + (\sigma / . :;\\ & + (\sigma / . :;\\ & + (\sigma / . :;\\ & + (\sigma / . :;\\ & + (\sigma / . :;\\ & + (\sigma / . :;\\ & + (\sigma / . :;\\ & + (\sigma / . :);\\ & + (\sigma / . :;\\ & + (\sigma / . :;\\ & + (\sigma / . :;\\ & + (\sigma / . :;\\ & + (\sigma / . :;\\ & + (\sigma / . :;\\ & + (\sigma / . :;\\ & + (\sigma / . :;\\ & + (\sigma / . :;;\\ & + (\sigma / . :;\\ & + (\sigma / . :;\\ & + (\sigma / . :;\\ & + (\sigma / . :;\\ & + (\sigma / . :;\\ & + (\sigma / . :;\\ & + (\sigma / . :;\\ & + (\sigma / . :;\\ & + (\sigma / . :|;\\ & + (\sigma / . :|;\\ & + (\sigma / . :|;\\ & + (\sigma / . :|;\\ & + (\sigma / . :|;\\ & + (\sigma / . :|;\\ & + (\sigma / . :|;\\ & + (\sigma / . :|;\\ & + (\sigma / . :|;\\ &+ (i n d e - 1) ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! # a b c a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y b c a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a k u r p r o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r t h e m e g h e m e g h e m e g h e m e g h e m e g h e m e g h e m e g h e m e g h e m e g h e m e g h e m e g h e m e g h e m e g h e m e g h e m e g h e m e g h e m e g h e m e g h e m e g h e m e g h e m e k u r p r o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r s o r t h e m e g h h e m e g h e m e g h e m e g h e m e g h e m e g h e m e g h e m e g h e m e g h e m e g h e m e g h e m e g h e m e g h e m e g h e m e g h e m e g h e m e g h e m e g h e m e g h e m e g h e m e q u r p r o r s o r t h e m e g h h e m e g h e m e g h e m e g h e m e g h e m e g h e m e g h e m e g h e m e g h e m e g h e m e g h e m e g h e m e g h e m e g h e m e g h e m e g h e m e g h e m e g h e n u v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v j w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W WWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIw IwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIwIIeIIeIIeIIeIIeIIeIIeIIeIIeIIeIIeIIeIIeIIeIIeIIeIIeIIeIIeIIeIIeIIeIIeIIeIIeIIeIIeIIeIIeIIeIIeIIeIIeIIeIIeIIeIIeIIeIIeIIeIIeIIeIIeIIeIIeIIeIIeIIeIIeIIeIIiEaIIIbIXcXIXcXIXcXIXcXIXcXIXcXIXcXIXcXIXcXIXcXIXcXIXcXIXcXIXcXIXcXIXcXIXcXIXcXIXcXIXcXIXcXIXcXIXcXIXcXIXcXIXcXIXcXIXcXIXcXIXcXIXcXIXcXIXcXIXcXIAIIIbIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXIXXXC X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X Ix I x I x I x I x I x I x I x I x I x I x I x I x I x I x I x I x I x I x I x I x I x I x I x I x I x I x I x I x I x I x I x I x I x I x I x I x I x I x I x I x I x I x I x I x I x I x I x I x I x I x Ix Ix Ix Ix Ix Ix Ix Ix Ix Ix Ix Ix Ix Ix Ix Ix Ix Ix Ix Ix Ix Ix Ix Ix Ix Ix Ix Ix Ix Ix Ix Ix Ix Ix Ix Ix Ix Ix Ix Ix Ix Ix Ix Ix Ix Ix Ix Ix Ix Ix Ixi IIex IIex IIex IIex IIex IIex IIex IIex IIex IIex IIex IIex IIex IIex IIex IIex IIex IIex IIex IIex IIex IIex IIex IIex IIex IIex IIex IIex IIex IIex IIex IIex IIex IIex IIex IIex IIex IIex IIex IIex IIex IIex IIex IIex IIex IIex IIex IIex IIex IIex IIeq IIIeq IIIeq IIIeq IIIeq IIIeq IIIeq IIIeq IIIeq IIIeq IIIeq IIIeq IIIeq IIIeq IIIeq IIIeq IIIeq IIIeq IIIeq IIIeq IIIeq IIIeq IIIeq IIIeq IIIeq IIIeq IIIeq IIIeq IIIeq IIIeq IIIeq IIIeq IIIeq IIIeq IIIeq IIIeq IIIeq IIIeq IIIeq IIIeq IIIeq IIIeq IIIeq IIIeq IIIeq IIIeq IIIeq IIIeq IIIeq IIIeq IIIeq IVIdg IVIdg IVIdg IVIdg IVIdg IVIdg IVIdg IVIdg IVIdg IVIdg IVIdg IVIdg IVIdg IVIdg IVIdg IVIdg IVIdg IVIdg IVIdg IVIdg IVIdg IVIdg IVIdg IVIdg IVIdg IVIdg IVIdg IVIdg IVIdg IVIdg IVIdg IVIdg IVIdg IVIdgIVIdgIVIdgIVIdgIVIdgIVIdgIVIdgIVIdgIVIdgIVIdgIVIdgIVIdgIVIdgIVIdgIVIdgIVIdgIVIdgIVIdgIVIdgIVIdgIVIdgIVIdgIVIdgIVIdgIVIdgIVIdgIVIdgIVIdgIVIdgIVIdgIVIdgIVIdgIVIdgIVIdgIVAtIsItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItItIt< fcel>E = E[σ,9,0]< nl>\]

\[\begin{array}{r l} & {+ \left(\sigma \pmb {\eta} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}\right) \left(\mathbf {I} _ {n _ {e}} \otimes E \left[ (\mathbf {x} _ {t} ^ {s}) ^ {\prime} \right]\right) \left(\sigma \pmb {\eta} \otimes \mathbf {h} _ {\mathbf {x}}\right) ^ {\prime}} \\ & {\quad + \left(\sigma \pmb {\eta} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}\right) \left(\mathbf {I} _ {n _ {e}} \otimes E \left[ (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} \right]\right) \left(\sigma \pmb {\eta} \otimes \tilde {\mathbf {H}} _ {\mathbf {x x}}\right) ^ {\prime}} \\ & {\quad + \left(\sigma \pmb {\eta} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}\right) \left(\sigma \pmb {\eta} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}\right) ^ {\prime}} \end{array}\]

\[1) \quad (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) \left(\mathbf {I} _ {n _ {e}} \otimes E \left[ \mathbf {x} _ {t} ^ {s} \left(\mathbf {x} _ {t} ^ {s}\right) ^ {\prime} \right]\right) (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime}\]

\[{ 2 ) } { + ( \sigma \pmb { \eta } \otimes \mathbf { h _ { x } } ) \left( \mathbf { I } _ { n _ { e } } \otimes E \left[ \mathbf { x } _ { t } ^ { s } \left( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \right) ^ { \prime } \right] \right) \left( \sigma \pmb { \eta } \otimes \tilde { \mathbf { H } } _ { \mathbf { x x } } \right) ^ { \prime } }\]

\[+ \left(\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}\right) \left(\mathbf {I} _ {n _ {e}} \otimes E \left[ \mathbf {x} _ {t} ^ {s} \right]\right) \left(\sigma \boldsymbol {\eta} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}\right) ^ {\prime}) \tag {3}\]

\[4) \qquad + \left(\sigma \pmb {\eta} \otimes \tilde {\mathbf {H}} _ {\mathbf {x x}}\right) \left(\mathbf {I} _ {n _ {e}} \otimes E \left[ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) (\mathbf {x} _ {t} ^ {s}) ^ {\prime} \right]\right) (\sigma \pmb {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime}\]

\[5) \qquad + \left(\sigma \boldsymbol {\eta} \otimes \tilde {\mathbf {H}} _ {\mathbf {x x}}\right) \left(\mathbf {I} _ {n _ {e}} \otimes E \left[ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \right]\right) \left(\sigma \boldsymbol {\eta} \otimes \tilde {\mathbf {H}} _ {\mathbf {x x}}\right) ^ {\prime}\]

\[+ \left(\sigma \boldsymbol {\eta} \otimes \tilde {\mathbf {H}} _ {\mathbf {x x}}\right) \left(\mathbf {I} _ {n _ {e}} \otimes E \left[ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) \right]\right) \left(\sigma \boldsymbol {\eta} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}\right) ^ {\prime} \tag {6}\]

\[{ 7 ) } { + \left( \sigma \pmb { \eta } \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } \right) \left( \mathbf { I } _ { n _ { e } } \otimes E \left[ ( \mathbf { x } _ { t } ^ { s } ) ^ { \prime } \right] \right) ( \sigma \pmb { \eta } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { \prime } }\]

\[{ 8 ) } { + \left( \sigma \pmb { \eta } \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } \right) \left( \mathbf { I } _ { n _ { e } } \otimes E \left[ \left( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \right) ^ { \prime } \right] \right) \left( \sigma \pmb { \eta } \otimes \tilde { \mathbf { H } } _ { \mathbf { x x } } \right) ^ { \prime } }\]

\[9) \quad + \left(\sigma \boldsymbol {\eta} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}\right) \left(\sigma \boldsymbol {\eta} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}\right) ^ {\prime}\]

Note here that we already know , and from the variance of the states using a second order approximation. This is because

\[\begin{array}{r l} & {V a r \left[ \mathbf {x} _ {t} ^ {s} \right] = E \left[ \left(\mathbf {x} _ {t} ^ {s} - E \left[ \mathbf {x} _ {t} ^ {s} \right]\right) \left(\mathbf {x} _ {t} ^ {s} - E \left[ \mathbf {x} _ {t} ^ {s} \right]\right) ^ {\prime} \right]} \\ & {\qquad = E \left[ \left(\mathbf {x} _ {t} ^ {s} - E \left[ \mathbf {x} _ {t} ^ {s} \right]\right) \left(\left(\mathbf {x} _ {t} ^ {s}\right) ^ {\prime} - E \left[ \mathbf {x} _ {t} ^ {s} \right] ^ {\prime}\right) \right]} \\ & {\qquad = E \left[ \mathbf {x} _ {t} ^ {s} \left(\mathbf {x} _ {t} ^ {s}\right) ^ {\prime} - \mathbf {x} _ {t} ^ {s} E \left[ \mathbf {x} _ {t} ^ {s} \right] ^ {\prime} - E \left[ \mathbf {x} _ {t} ^ {s} \right] \left(\mathbf {x} _ {t} ^ {s}\right) ^ {\prime} + E \left[ \mathbf {x} _ {t} ^ {s} \right] E \left[ \mathbf {x} _ {t} ^ {s} \right] ^ {\prime} \right]} \\ & {\qquad = E \left[ \mathbf {x} _ {t} ^ {s} \left(\mathbf {x} _ {t} ^ {s}\right) ^ {\prime} \right] - E \left[ \mathbf {x} _ {t} ^ {s} \right] E \left[ \mathbf {x} _ {t} ^ {s} \right] ^ {\prime}} \\ & {\Updownarrow} \\ & {V a r \left[ \mathbf {x} _ {t} ^ {s} \right] + E \left[ \mathbf {x} _ {t} ^ {s} \right] E \left[ \mathbf {x} _ {t} ^ {s} \right] ^ {\prime} = E \left[ \mathbf {x} _ {t} ^ {s} \left(\mathbf {x} _ {t} ^ {s}\right) ^ {\prime} \right]} \end{array}\]

\[\begin{array}{r l} & {\mathrm{and}} \\ & {V a r \left(\mathbf {x} _ {t} ^ {s}, \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right)\right) = E \left[ (\mathbf {x} _ {t} ^ {s} - E [ \mathbf {x} _ {t} ^ {s} ]) \left((\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) - E \left[ (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) \right]\right) ^ {\prime} \right]} \\ & {\quad = E \left[ (\mathbf {x} _ {t} ^ {s} - E [ \mathbf {x} _ {t} ^ {s} ]) \left((\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} - E \left[ (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} \right]\right) \right]} \\ & {\quad = E [ \mathbf {x} _ {t} ^ {s} (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} - \mathbf {x} _ {t} ^ {s} E [ (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} ] - E [ \mathbf {x} _ {t} ^ {s} ] (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} + E [ \mathbf {x} _ {t} ^ {s} ] E [ (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} ] ]} \end{array}\]

\[\begin{array}{r l} & = E \left[ \mathbf {x} _ {t} ^ {s} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \right] - E \left[ \mathbf {x} _ {t} ^ {s} \right] E \left[ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \right] \\ & \Updownarrow \\ & V a r \left(\mathbf {x} _ {t} ^ {s}, \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right)\right) + E \left[ \mathbf {x} _ {t} ^ {s} \right] E \left[ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \right] = E \left[ \mathbf {x} _ {t} ^ {s} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \right] \end{array}\]

and

\[\begin{array} { r l }& V a r \left[\left( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \right)\right] = E \left[ \right.\left(\left( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \right) - E \left[\left( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \right)\right]\right)\left(\left( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \right) - E \left[\left( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \right)\right]\right) ^ { \prime }\\& = E \left[ \right.\left(\left( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \right) - E \left[\left( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \right)\right]\right)\left( \right.\left( \right. . 2 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . 2 . ,\\& = E [ ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) ^ { \prime } - ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) E [ ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) ^ { \prime } ]\\& - E [ ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) ] ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) ^ { \prime } + E [ ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) ] E [ ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) ^ { \prime } ] ]\\& = E [ ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) ^ { \prime } ] - E [ ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) ] E [ ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) ^ { \prime } ]\\& - E [ ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) ] E [ ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) ^ { \prime } ] + E [ ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) ] E [ ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) ^ { \prime } ]\\& = E [ ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) ^ { \prime } ] - E [ ( \mathbf { x } _ { t } ^ { f } \ottimes \mathbf { x } _ { t } ^ { f } ) ] E [ ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) ^ { \prime } ]\\&\Updownarrow\\&V a r [ ( \mathbf { x} _ { t } ^ { f } \otimes \mathbf { x} _ { t } ^ { f} ) ] + E [ ( \mathbf { x} _ { t } ^ { f } \otimes \mathbf { x} _ { t } ^ { f} ) ] E [ ( \mathbf { x} _ { t } ^ { f } \otimes \mathbf { x} _ { t } ^ { f} ) ^ { \prime } ] = E [ ( \mathbf { x} _ { t } ^ { f } \otimes \mathbf { x} _ { t } ^ { f} ) ( \mathbf { x} _ { t } ^ { f } \otimes \mathbf { x} _ { t } ^ { f} ) ^ { ' ] }\end{array}\]

4.3.6 For

Note first that has dimensions .

\[\begin{array} { r l } & { V a r \left[ \tilde { \pmb { \xi } } _ { t + 1 } \right] _ { 5 6 } } \\ & { = E [ ( ( \sigma \pmb { \eta } \otimes \mathbf { h } _ { \mathbf { x } } ) ( \pmb { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { s } ) + ( \sigma \pmb { \eta } \otimes \tilde { \mathbf { H } } _ { \mathbf { x x } } ) ( \pmb { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) + ( \sigma \pmb { \eta } \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } ) ( \pmb { \epsilon } _ { t + 1 } ) ( \mathbf { u } _ { t + 1 } ^ { \prime } - E [ \mathbf { u } _ { t + 1 } ^ { \prime } ] ) ] } \\ & { = E [ ( ( \sigma \pmb { \eta } \otimes \mathbf { h } _ { \mathbf { x } } ) ( \pmb { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { s } ) + ( \sigma \pmb { \eta } \otimes \tilde { \mathbf { H } } _ { \mathbb { x x } } ) ( \pmb { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) + ( \sigma \pmb { \eta } \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2} ) ( \pmb { \epsilon } _ { t + 1 } ) ( \mathbf { u } _ { t + 1 } ^ { \prime } - E [ \mathbf { u } _ { t + 1 } ^ { \prime } ] ) ] } \\ & { - ( ( \sigma \pmb { \eta } \otimes \mathbf { h } _ { \mathbf { x } } ) ( \pmb { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { s } ) + ( \sigma \pmb { \eta } \otimes \tilde { \mathbf { H } } _ { \mathbb { x x } } ) (\pmb { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) + ( \sigma \pmb { \eta } \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2} ) ( \pmb { \varepsilon } _ { t + 1 } ) E [ ( \mathbf { u } _ { t + 1 } ^ { \prime } ] ] } \\ & = E [ ( \sigma \pmb { \eta } \otimes \mathbf { h } _ { \mathbf { x } } ) ( \pmb { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { s} ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , e , i , j , k , l , m , n , o , p , q , r , s , t , u , v , w , x , y , z , w , x , y , z , w , y , z , w , z , w , y , z , w , z , w , y , z , w , z , w , y , z , w , y , z , w , y , z , w , y , z , w , y , z , w , y , z , w , y , z , w , y , z , w , y , z , w , y , z , w , y , z , w , y , z , w , y , z , w , y , z , w , y , z , w , y , z , w , y , z , y , z , w , y , z , w , y , z , w , y , z , w , y , z , w , y , z , w , y , z , w , y , z , w , y , z , w , y , z , w , y , z , w , y , z , w , y , z , w , y , z , w , y , z , w , y , z , w , y , y , z , w , y , z , w , y , z , w , y , z , w , y , z , w , y , z , w , y , z , w ] e \\ & \quad E = E [ (\sigma ; t ; t ; t ; t ; t ; t ; t ; t ; t ; t ; t ; t ; t ; t ; t ; t ; t ; t ; t ; t ; t ; t ; t ; t ; t ; t ; t ; t ; t ; t ; t ; t ; t ; t ; t ; t ; t ; t ; t ; t ; t ; t ; t ; t ; t ; t ; t ; t ; t ; t ; s i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d e f f e r e r e r g e r e f f e r e r g e r e f f e r e r g e r e f f e r e r g e r e f f e r e r g e r e f f e r e r g e r e f f e r e r g e r e f f e r e r g e r e f f e r e r g e r e f f e r e r g e r e f f e r e r g e r e f l e r e r g e r e f l e r e r g e r e f l e r e r g e r e f l e r e r g e r e f l e r e r g e r e f l e r e r g e r e f l e r e r g e r e f l e r e r g e r e f l e r e r g e r e f l e r e r g e r e f f e r e r g e r e f l e r e r g e r e f l e r e r g e r e f l e r e r g e r e f l e r e r g e r e f l e r e r g e r e f l e r e r g e r e f l e r e r g e r e f l e r e r g e r e f f o l o c a l o c a l o c a l o c a l o c a l o c a l o c a l o c a l o c a l o c a l o c a l o c a l o c a l o c a l o c a l o c a l o c a l o c a l o c a l o c a l o c a l o c a l o c a l o c a l o c a l o c o u p p h a b s i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n c o u p p h a b s i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d j o u p p h a b s i n d j o u p p h a b s i n d j o u p p h a b s j o u p p h a b s j o u p p h a b s j o u p p h a b s j o u p p h a b s j o u p p h a b s j o u p p h a b s j o u p p h a b s j o u p p h a b s j o u p p h a b s j o u p p h a b s j o u p p h a b s | j o u p p h a b s j o u p p h a b s j o u p p h a b s j o u p p h a b s j o u p p h a b s j o u p p h a b s j o u p p h a b s j o u p p h a b s j o u p p h a b s j o u p p h a b s j o u p p h a b s| j o u p p h a b s j o u p p h a b s j o u p p h a b s j o u p p h a b s j o u p p h a b s j o u p p h a b s j o u p p h a b s j o u p p h a b s | j o u p p h a b s j o u p p h a b s j o u p p h a bs | j o u p p h a b s j o u p p h a b s j o u p p h a b s | j o u p p h a b s j o u p p h a b s | j o u p p h a b s j o u p p h a b s | j o u p p h a b s j o u p p h a b s | j o u p p h a b s j o u p p h a b s | j o u p p h a b s j o u p p h a b s | j o u p q k : k : k : k : k : k : k : k : k : k : k : k : k : k : k : k : k : k : k : k : k : k : k : k : k : k : k : k : k : k : k : k : k : k : k : k : k : k : k : k : k : k : k : k : k : k : k : k : k : k : k ： ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ， ，， 。 ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” ” “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ " “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ “ " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " " * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! /! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ! / ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ?# ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? # ? @ >0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0) \\ & \quad E = E [ (\sigma; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T;T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; T; S O U P I N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D AN D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N< fcel>E = E [ (\sigma ;T;p) (\sigma ;T;p) (\sigma ;T;p) (\sigma ;T;p) (\sigma ;T;p) (\sigma ;T;p) (\sigma ;T;p) (\sigma ;T;p) (\sigma ;T;p) (\sigma ;T;p) (\sigma ;T;p) (\sigma ;T;p) (\sigma ;T;p) (\sigma ;T;p) (\sigma ;T;p) (\sigma ;T;p) (\sigma ;T;p) (\sigma ;W) (\sigma ;W) (\sigma ;W) (\sigma ;W) (\sigma ;W) (\sigma ;W) (\sigma ;W) (\sigma ;W) (\sigma ;W) (\sigma ;W) (\sigma ;W) (\sigma ;W) (\sigma ;W) (\sigma ;W) (\sigma ;W) (\sigma ;W) (\sigma ;W) (\sigma ;W) (\sigma ;W) (\sigma ;W) (\sigma ; W) (\sigma ;W) (\sigma ;W) (\sigma ;W) (\sigma ;W) (\sigma ;W) (\sigma ;W) (\sigma ;W) (\sigma ;W) (\sigma ;W) (\sigma ;W) (\sigma ;W) (\sigma ;W) (\sigma ;W) (\sigma ;W) (\sigma ;W) (\sigma ;W) (\sigma ;W) (\sigma ;W) (\sigma ;W) (\sigma ;S O U P I N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N DA N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A N D A M O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R OR O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O R O< ecel>< nl>\]

\[\begin{array} { r l } & + \left( \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \right) ^ { \prime } ( \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { \prime } + \left( \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \otimes \boldsymbol { \epsilon } _ { t + 1 } \right) ^ { \prime } ( \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x } } \otimes \boldsymbol { \sigma } \boldsymbol { \eta } ) ^ { \prime } \\ & + \left( \boldsymbol { \epsilon } _ { t + 1 } \otimes \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \right) ^ { \prime } ( \sigma \boldsymbol { \eta } \otimes \boldsymbol { \sigma } \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { \prime } + ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \boldsymbol { \epsilon } _ { t + 1 } \otimes \boldsymbol { \epsilon } _ { t + 1 } ) ^ { \prime } ( \sigma \boldsymbol { \eta } \otimes \boldsymbol { \sigma } \boldsymbol { \eta } \otimes \boldsymbol { \sigma } \boldsymbol { \eta } ) ^ { \prime } ) \\ & \\ & { + ( \sigma \boldsymbol { \eta } \otimes \tilde { \mathbf { H } } _ { \mathbf { x x } } ) ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) ( ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \otimes \boldsymbol { \epsilon } _ { t + 1 } ) ^ { \prime } ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } \otimes \sigma \boldsymbol { \eta } ) ^ { \prime }} \\ & \\ & { + ( \mathbf { x } _ { t } ^ { f } \otimes \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } ) ^ { \prime } ( \mathbf { h } _ { \mathbf { x } } \otimes \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { \prime } + ( \mathbf { x } _ { t } ^ { f } \otimes \boldsymbol { \epsilon } _ { t + 1 } \otimes \boldsymbol { \epsilon } _ { t + 1 } ) ^ { \prime } ( \mathbf { h } _ { \mathbf { x } } \otimes \sigma \boldsymbol { \eta } \otimes \boldsymbol { \sigma } \boldsymbol { \eta } ) ^ { \prime }} \\ & \\ & { + ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) ^ { \prime } ( \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { \prime } + ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \otimes \boldsymbol { \epsilon } _ { t + 1 } ) ^ { \prime } ( \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x } } \otimes \sigma \boldsymbol { \eta } ) ^ { \prime }} \\ & \\ & { + ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } ) ^ { \prime } ( \sigma \boldsymbol { \eta } \otimes \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x} } ) ^ { \prime } + ( ( \boldsymbol { \epsilon } _ { t + 1 } \otimes ( [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ) - ( | | | ) ^ {\prime} ) ^ {\prime }} \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ = ( 2 2 ) ^ {- 3 / 2} . 5 , 6 , 7 , 8 , 9 , 1 0 , 1 1 , 1 2 , 1 3 , 1 4 , 1 5 , 1 6 , 1 7 , 1 8 , 1 9 , 2 0 , 2 1 , 2 2 , 2 3 , 2 4 , 2 5 , 2 6 , 2 7 , 2 8 , 2 9 , 3 0 , 3 1 , 3 2 , 3 3 , 3 4 , 3 5 , 3 6 , 3 7 , 3 8 , 3 9 , 4 0 , 4 1 , 4 2 , 4 3 , 4 4 , 4 5 , 4 6 , 4 7 , 4 8 , 4 9 , 5 0 , 5 1 , 5 2 , 5 3 , 5 4 , 5 5 , 5 6 , 5 7 , 5 8 , 5. 6 , 5. 7 , 5. 8 , 5. 9 , 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .\]

using the definition of where

\[\begin{array} { r l } & { \mathbf { u } _ { t + 1 } ^ { \prime } \equiv \left( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \otimes \boldsymbol { \epsilon } _ { t + 1 } \right) ^ { \prime } ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } \otimes \sigma \boldsymbol { \eta } ) ^ { \prime } } \\ & { + \left( \mathbf { x } _ { t } ^ { f } \otimes \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \right) ^ { \prime } ( \mathbf { h } _ { \mathbf { x } } \otimes \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { \prime } + \left( \mathbf { x } _ { t } ^ { f } \otimes \boldsymbol { \epsilon } _ { t + 1 } \otimes \boldsymbol { \epsilon } _ { t + 1 } \right) ^ { \prime } ( \mathbf { h } _ { \mathbf { x } } \otimes \sigma \boldsymbol { \eta } \otimes \sigma \boldsymbol { \eta } ) ^ { \prime } } \\ & { + \left( \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \right) ^ { \prime } ( \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { \prime } + \left( \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \otimes \boldsymbol { \epsilon } _ { t + 1 } \right) ^ { \prime } ( \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x } } \otimes \sigma \boldsymbol { \eta } ) ^ { \prime } } \\ & { + \left( \boldsymbol { \epsilon } _ { t + 1 } \otimes \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \right) ^ { \prime } ( \sigma \boldsymbol { \eta } \otimes \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { \prime } + ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \boldsymbol { \epsilon } _ { t + 1 } \otimes \boldsymbol { \epsilon } _ { t + 1 } ) ^ { \prime } ( \sigma \boldsymbol { \eta } \otimes \sigma \boldsymbol { \eta } \otimes \sigma \boldsymbol { \eta } ) ^ { \prime }} \\ & \\ & = E [ ( \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x} } ) ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { s} ) ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \otimes \boldsymbol { \epsilon } _ { t + 1 } ) ^ { \prime } ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } \otimes \sigma \boldsymbol { \eta } ) ^ { \prime } \\ & \\ & + ( \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x} } ) ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { s} ) ( \mathbf { x } _ { t } ^ { f } \otimes \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } ) ^ { ' } ( \mathbf { h } _ { \mathbf { x } } \otimes \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x} } ) ^ { ' } \\ & \\ & + ( \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x} } ) ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { s} ) ( [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ ] o n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i m e d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n c l e m a l l e , \\ & \\ & + ( | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | \\ & \\ & + ( | σ η Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Κ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ ζ η Σ ζ η Σ ζ η Σ ζ η Σ ζ η Σ ζ η Σ ζ η Σ ζ η Σ ζ η Σ ζ η Σ; \\ & \\ & + ( σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ α r e s s o r p o r e r e s s o r p o r e r e s s o r p o r e r e s s o r p o r e r e s s o r p o r e r e s s o r p o r e r e s s o r p o r e r e s s o r p o r e r e s s o r p o r e r e s s o r p o r e r e s s o r p o r a l l e s o r p o r a l l e s o r p o r a l l e s o r p o r a l l e s o r p o r a l l e s o r p o r a l l e s o r p o r a l l e s o r p o r a l l e s o r p o r a l l e s o r p o r a l l e s o r p o r a l l l e s o r p o r a l l l e s o r p o r a l l l e s o r p o r a l l l e s o r p o r a l l l e s o r p o r a l l l e s o r p o r a l l l e s o r p o r a l l l e s o r p o r a l l l e s o r p o r a l k e l e s o r p o r a l k e l e s o r p o r a l k e l e s o r p o r a l k e l e s o r p o r a l k e l e s o r p o r a l k e l e s o r p o r a l k e l e s o r p o r a l k e l e s o r p o r a l k e l e s s o r p o r a l k e l e s s o r p o r a l k e l e s s o r p o r a l k e l e s s o r p o r a l k e l e s s o r p o r a l k e l e s s o r p o r a l k e l e s s o r p o r a l k e l e s s o r p o r a l l k e l e s s o r p o r a l k e l e s s o r p o r a l k e l e s s o r p o r a l k e l e s s o r p o r a l k e l e s s o r p o r a l k e l e s s o r p o r a l k e l e s s o r p o r a l k e l e s s o w h y w h y w h y w h y w h y w h y w h y w h y w h y w h y w h y w h y w h y w h y w h y w h y w h y w h y w h y w h y w h y w h y w h y w h y w h y w h y w h y w h y w h y w h y w h y w h y w h y w h y v w h y v w h y v w h y v w h y v w h y v w h y v w h y v w h y v w h y v w h y v w h y v w h y v w h y v w h y v w h y v w h y v w h y v w h y v w h y v w h y v w h y v w h y v w h y v w h y v w h y w w h y v w h y v w h y v w h y v w h y v w h y v w h y v w h y v w h y v w h y v w h y v w h y v w h y v w h y v w h y v w h y v w h y v w h y v w h y v w h y v w h y v w h y v w h y v w h y v w h yv w w h y v w w h y v w h y v w h y v w w h y v w w h y v w w h y v w w h y v w w h y v w w h y v w w h y v w w h y v w w h y v w w h y v w w h y v w w h y v w w h y v w w h y v w w h y v w w h y v w w h y v w w h y v w w h y v w w h y vw / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / : \\ & \\ & + (σση Σση Σση Σση Σση Σση Σση Σση Σση Σση Σση Σση Σση Σση Σση Σση Σση Σση Σση Σση Σση Σση Σση Σση Σση Σση Ρ u c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c . \\ & \\ & + (σση Σση β u b u b u b u b u b u b u b u b u b u b u b u b u b u b u b u b u b u b u b u b u b u b u b u b u b u b u b u b u b u b u b u b u b u b u b u b u b u b u b u b u b u b u b u b u b u b u b u b u b u b u f ) ^ ' ( [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ * ] * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ) ^ ' ( ) ^ ' ( ) ^ ' ( ) ^ ' ( ) ^ ' ( ) ^ ' ( ) ^ ' ( ) ^ ' ( ) ^ ' ( ) ^ ' ( ) ^ ' ( ) ^ ' ( ) ^ ' ( ) ^ ' ( ) ^ ' ( ) ^ ' ( ) ^ ' ( ) ^ ' ( ) ^ ' ( ) ^ ' ( ) ^ ' ( ) ^ ' ( ) ^ ' ( ) ^ ' ( ) ^ ' ( ) ^ ' (\texttt "o" (\texttt "o" (\texttt "o" (\texttt "o" (\texttt "o" (\texttt "o" (\texttt "o" (\texttt "o" (\texttt "o" (\texttt "o" (\texttt "o" (\texttt "o" (\texttt "o" (\texttt "o" (\texttt "o" (\texttt "o" (\texttt "o" (\texttt "\texttt "o" (\texttt "o" (\texttt "o" (\texttt "o" (\texttt "o" (\texttt "o" (\texttt "o" (\texttt "o" (\texttt "o" (\texttt "o" (\texttt "o" (\texttt "o" (\texttt "o" (\texttt "o" (\texttt "o" (\texttt "o" (\texttt "i" (\texttt "i" (\texttt "i" (\texttt "i" (\texttt "i" (\texttt "i" (\texttt "i" (\texttt "i" (\texttt "i" (\texttt "i" (\texttt "i" (\texttt "i" (\texttt "i" (\texttt "i" (\texttt "i" (\texttt "i" (\texttt "i" (\textit {\mu} (\textit {\mu} (\textit {\mu} (\textit {\mu} (\textit {\mu} (\textit {\mu} (\textit {\mu} (\textit {\mu} (\textit {\mu} (\textit {\mu} (\textit {\mu} (\textit {\mu} (\textit {\mu} (\textit {\mu} (\textit {\mu} (\textit {\mu} (\textit {\mu} (\textit{\mu} (\textit {\mu} (\textit {\mu} (\textit {\mu} (\textit {\mu} (\textit {\mu} (\textit {\mu} (\textit {\mu} (\textit {\mu} (\textit {\mu} (\textit {\mu} (\textit {\mu} (\textit {\mu} (\textit {\mu} (\textit {\mu} (\textit {\mu} (\textit {\mu} (\mathrm{e}) ^ {(s - 1)} ;^ {(s - 1)} ;^ {(s - 1)} ;^ {(s - 1)} ;^ {(s - 1)} ;^ {(s - 1)} ;^ {(s - 1)} ;^ {(s - 1)} ;^ {(s - 1)} ;^ {(s - 1)} ;^ {(s - 1)} ;^ {(s - 1)} ;^ {(s - 1)} ;^ (s -\mathrm{e}) ;^ {(s - 1)} ;^ {(s - 1)} ;^ {(s - 1)} ;^ {(s - 1)} ;^ {(s - 1)} ;^ {(s - 1)} ;^ {(s - 1)} ;^ {(s - 1)} ;^ {(s - 1)} ;^ {(s - 1)} ;^ {(s - 1)} ;^ {(s - 1)} ;^ {(t - 1)} ;^ {(t - 1)} ;^ {(t - 1)} ;^ {(t - 1)} ;^ {(t - 1)} ;^ {(t - 1)} ;^ {(t - 1)} ;^ {(t - 1)} ;^ {(t - 1)} ;^ {(t - 1)} ;^ {(t - 1)} ;^ {(t - 1)} ;^ {(t - 1)},^ {(t - 1)},^ {(t - 1)},^ {(t - 1)},^ {(t - 1)},^ {(t - 1)},^ {(t - 1)},^ {(t - 1)},^ {(t - 1)},^ {(t - 1)},^ {(t - 1)},^ {(t - 1)},^ {(t - 1)},^ {(t - 1)},^ {(t - 1)},^{(t - 1)},^ {(t - 1)},^ {(t - 1)},^ {(t - 1)},^ {(t - 1)},^ {(t - 1)},^ {(t - 1)},^ {(t - 1)},^ {(t - 1)},^ {(t - 1)},^ {(t - 1)},^ {(t - 1)},^ {(t - 1)},^ {(t - 1)},^ {(t-2)},^ {(t-2)},^ {(t-2)},^ {(t-2)},^ {(t-2)},^ {(t-2)},^ {(t-2)},^ {(t-2)},^ {(t-2)},^ {(t-2)},^ {(t-2)},^ {(t-2)},^ {(t-2)},^ {(t-2)},^ {(t-2)},^ {(t-2)},^ {(t-2)},^ {(T-2)},^ {(T-2)},^ {(T-2)},^ {(T-2)},^ {(T-2)},^ {(T-2)},^ {(T-2)},^ {(T-2)},^ {(T-2)},^ {(T-2)},^ {(T-2)},^ {(T-2)},^ {(T-2)},^ {(T-2)},^ {(T-2)},^ {(T-2)},^ {(T-2)}, ^{(T-2)}, ^{(T-2)}, ^{(T-2)}, ^{(T-2)}, ^{(T-2)}, ^{(T-2)}, ^{(T-2)}, ^{(T-2)}, ^{(T-2)}, ^{(T-2)}, ^{(T-2)}, ^{(T-2)}, ^{(T-2)}, ^{(T-2)}, ^{(T-2)}, ^{(T-2)}, ^{(T-2),*}, \\ & \\ & + (σση β u b u b) (α ω γ ν) (α ω γ ν) (α ω γ ν) (α ω γ ν) (α ω γ ν) (α ω γ ν) (α ω γ ν) (α ω γ ν) (α ω γ ν) (α ω γ ν) (α ω γ ν) (α ω γ ν) (α ω γ ν) (α ω γ ν) (α ω γ ν)< fcel>+ (σση β u b u b) (α ω γ ν) (α ω γ ν) (α ω γ ν) (α ω γ ν) (α ω γ ν) (α ω γ ν) (α ω γ ν) (α ω γ ν) (α ω γ ν) (α ω γ ν) (α ω γ ν) (α ω γ ν) (α ω γν)< fcel>+ (σση β u b u b) (α ω γ ν) (α ω γ ν) (α ω γ ν) (α ω γ ν) (α ω γ ν) (α ω γ ν) (α ω γ ν) (α ω γ ν) (α ω γ ν)< fcel>+ (σση β u b u b) (α ω γ ν) (α ω γ ν) (α ω γ ν) (αωγ ν) (α ω γ ν)< fcel>+ (σση β u b u b) (α ω γ ν) (α ω γ ν) (α ω γ ν) (α ω γ ν)< fcel>+ (σση β u b u b) (α ω γ ν) (α ω γ ν) (α ω γ ν)< fcel>+ (σση β u b u b) (α ω γ ν) (α ω γ ν) (α ω γ ν)< fcel>+ (σση β u b u b) (α ω γ ν) (α ω γ ν) (α ω γ ν)< fcel>+ (σση β u b u b))< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< nl>\]

\[\begin{array} { l } + \left( \sigma \eta \otimes \tilde { \mathbf { H } } _ { \mathbf { x } \mathbf { x } } \right) \left( \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \right) \left( \boldsymbol { \epsilon } _ { t + 1 } \otimes \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \right) ^ { \prime } ( \sigma \eta \otimes \sigma \eta \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { \prime } \\ + \left( \sigma \eta \otimes \tilde { \mathbf { H } } _ { \mathbf { x } \mathbf { x } } \right) \left( \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \right) ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \boldsymbol { \epsilon } _ { t + 1 } \otimes \boldsymbol { \epsilon } _ { t + 1 } ) ^ { \prime } ( \sigma \eta \otimes \sigma \eta \otimes \sigma \eta ) ^ { \prime } \\ + ( \sigma \eta \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } ) \boldsymbol { \epsilon } _ { t + 1 } ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \otimes \boldsymbol { \epsilon } _ { t + 1 } ) ^ { \prime } ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } \otimes \sigma \eta ) ^ { \prime } \\ + ( \sigma \eta \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } ) \boldsymbol { \epsilon } _ { t + 1 } ( \mathbf { x } _ { t } ^ { f } \otimes \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } ) ^ { \prime } ( \mathbf { h } _ { \mathbf { x } } \otimes \sigma \eta \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { \prime } \\ + ( \sigma \eta \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } ) \boldsymbol { \epsilon } _ { t + 1 } ( \mathbf { x } _ { t } ^ { f } \otimes \boldsymbol { \epsilon } _ { t + 1 } \otimes \boldsymbol { \epsilon } _ { t + 1 } ) ^ { \prime } ( \mathbf { h } _ { \mathbf { x } } \otimes \sigma \eta \otimes \sigma \eta ) ^ { \prime } \\ + ( \sigma \eta \otimes \frac { 1 } { 2 } {\mathbf h} _ { {\sigma} {\sigma}} {\sigma} ^ { 2 } ) {\boldsymbol { \epsilon }} _ { t + 1 } ( {\boldsymbol { {\epsilon}} _ { t + 1 }} {\otimes} {\boldsymbol { {\epsilon}} _ { t + 1 }} {\otimes} {\boldsymbol { {\epsilon}} _ { t + 1 }} ) ^ { \prime } ( {\boldsymbol { {\sigma}} {\eta}} {\otimes} {\boldsymbol { {\eta}}} {\otimes} {\boldsymbol { {\eta}}} {\otimes} {\boldsymbol { {\eta}}} ) ^ { \prime } \\ + ( ( 0 , 0 ) ( 0 , 0 ) ( 0 , 0 ) ( 0 , 0 ) ( 0 , 0 ) ( 0 , 0 ) ( 0 , 0 ) ( 0 , 0 ) ( 0 , 0 ) ( 0 , 0 ) ( 0 , 0 ) ( 0 , 0 ) ( 0 , 0 ) ( 0 , 0 ) ( 0 , 0 ) ( n , n ) ( n , n ) ( n , n ) ( n , n ) ( n , n ) ( n , n ) ( n , n ) ( n , n ) ( n , n ) ( n , n ) ( n , n ) ( n , n ) ( n , n ) ( n , n ) ( n , n ) ( n , n ) ( n , n ) ( n , n ) ( n , n ) ( n , n ) ( n ,n ) ) \\ = E [ ( 0 , 0 ) ( 0 , 0 ) ( 0 , 0 ) ( 0 , 0 ) ( 0 , 0 ) ( 0 , 0 ) ( 0 , 0 ) ( 0 , 0 ) ( 0 , 0 ) ( 0 , 0 ) ( 0 , 0 ) ( 0 , 0 ) ( 0 , 0 ) ( 0 , 0 ) ( q , q ; q ; q ; q ; q ; q ; q ; q ; q ; q ; q ; q ; q ; q ; q ; q ; q ; q ; q ; q ; q ; q ; q ; q ; q ; q ; q ; q ; q ; q ; q ; q ; q ; q ; q ; q ; q ; q ; q ; q ; q ; q ; q ; q ; q ; q ; q ; q ; q ; q ; q . \\ + ( 0 , 0 ) ( 0 , 0 ) ( 0 , 0 ) ( 0 , 0 ) ( 0 , 0 ) ( 0 , 0 ) ( 0 , 0 ) ( 0 , 0 ) ( 0 , 0 ) ( 0 , 0 ) ( 0 , 0 ) ( 0 , 0 ) ( 0 , 0 ) ( n , n ) ( n , m ) ( n , m ) ( n , m ) ( n , m ) ( n , m ) ( n , m ) ( n , m ) ( n , m ) ( n , m ) ( n , m ) ( n , m ) ( n , m ) ( n , m ) ( n , m ) ( n , m ) ( n , m ) ( n , m ) ( n , m ) ( n , m ) ( n , m ) ( n ,m ) \\ + ( (\sigma , p - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - t a r d i g e d i g e d i g e d i g e d i g e d i g e d i g e d i g e d i g e d i g e d i g e d i g e d i g e d i g e d i g e d i g e d i g e d i g e d i g e d i g e d i g e d i g e d i g e d i g e d i g e d i g i o u d i g e d i g e d i g e d i g e d i g e d i g e d i g e d i g e d i g e d i g e d i g e d i g e d i g e d i g e d i g e d i g e d i g e d i g e d i g e d i g e d i g e d i g e d i g e d i g e d i g h a c k a l l o w h a c k a l l o w h a c k a l l o w h a c k a l l o w h a c k a l l o w h a c k a l l o w h a c k a l l o w h a c k a l l o w h a c k a l l o w h a c k a l l o w h a c k a l l o w h a c k a l l o wh a c k a l l o w h a c k a l l o w h a c k a l l o w h a c k a l l o w h a c k a l l o w h a c k a l l o w h a c k a l l o w h a c k a l l o w h a c k a l l o w h a c k a l l o w h a c k a l l o w h h a c k a l l o w h h a c k a l l o w h h a c k a l l o w h h a c k a l l o w h h a c k a l l o w h h a c k a l l o w h h a c k a l l o w h h a c k a l l o w h h a c k a l l o w h h a c k a l l o w y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y & \\ + (\sigma , p - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s -s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - t a r d i g e d i g e d i g e d i g e d i g e d i g e d i g e d i g e d i g e d i g e d i g e d j u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b v a r b v a r b v a r b v a r b v a r b v a r b v a r b v a r b v a r b v a r b v a r b v a r b v a r b v a r b v a r b v v a r b v v a r b v v a r b v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v vvvi) \\ + (\sigma , p - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s - s -s- t a r d i g e d i g e d j u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u rb u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b u r b z ] \\ + (\sigma , p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p | \\ + (\sigma , p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p | \\ + (\sigma , p / p / p / p / p / p / p / p / p / p / p / p / p / p /p / p / p / p / p / p / p / p / p / p | \\ + (\sigma , p / p / p / p / p / p / p / p / p / p / p / p / p / p | \\ + (\sigma , p / p / p / p / p / p / p / p / p | \\ + (\sigma , p / p / p / p / p / p | \\ + (\sigma , p / p / p / p | \\ + (\sigma , p / p | \\ + (\sigma , p | \\ + (\sigma , p | \\ + (\sigma , p | \\ + (\sigma , p | \\ + (\sigma , p | \\ + (\sigma , p | \\ + (\sigma , p | \\ + (\sigma , p | \\ + (\sigma , p | \\ + (\sigma , p | \\ + (\sigma , p | \\ + (\sigma , p | \\ + (\sigma , p | \\ + (\sigma , p | \\ + (\sigma , p | \\ + (\sigma , p | \\ + (\sigma , p | \\ + (\sigma , p | \\ + (\sigma , p | \\ + (\sigma , p | \\ + (\sigma , p | \\ + (\sigma , p | \\ + (\sigma , p | \\ + (\sigma , p | \\ + (\sigma , p | \\ + (\sigma , p | \\ + (\sigma , p | \\ + (\sigma , p | \\ + (\sigma , p | \\+ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ]] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ]]\]

\[\begin{array} { r l } & { } \quad + 0 \\ & { } \quad + 0 \\ & { } \quad + \left( \sigma \eta \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } \right) \boldsymbol { \epsilon } _ { t + 1 } \left( \boldsymbol { \epsilon } _ { t + 1 } \otimes \boldsymbol { \epsilon } _ { t + 1 } \otimes \boldsymbol { \epsilon } _ { t + 1 } \right) ^ { \prime } \left( \sigma \eta \otimes \sigma \eta \otimes \sigma \eta \right) ^ { \prime } \\ & { } ] \\ & = E [ ( \sigma \eta \otimes \mathbf { h } _ { x } ) ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { s } ) ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \otimes \boldsymbol { \epsilon } _ { t + 1 } ) ^ { \prime } ( \mathbf { h } _ { x } \otimes \mathbf { h } _ { x } \otimes \sigma \eta ) ^ { \prime } \\ & { } \quad + ( \sigma \eta \otimes \mathbf { h } _ { x } ) ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { s } ) ( \mathbf { x } _ { t } ^ { f } \otimes \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } ) ^ { \prime } ( \mathbf { h } _ { x } \otimes \sigma \eta \otimes \mathbf { h } _ { x } ) ^ { \prime } \\ & { } \quad + ( \sigma \eta \otimes \mathbf { h } _ { x } ) ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { s } ) ( \mathbf { x } _ { t } ^ { f } \otimes \boldsymbol { \epsilon } _ { t + 1 } \ottimes \boldsymbol { \epsilon } _ { t + 1 } ) ^ { \prime } ( \mathbf { h } _ { x } \otimes \sigma \eta \otimes \sigma \eta ) ^ { \prime } \\ & { } \quad + ( \sigma \eta \otimes \mathbf { h } _ { x } ) ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { s } ) ( \boldsymbol { \epsilon ^ { \prime } _ { t + 1 } } \otimes ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) ^ { \prime } ) ( \sigma \eta \otimes \mathbf { h } _ { x } \otimes \mathbf { h } _ { x } ) ^ { \prime } \\ & { } \quad + ( \sigma \eta \otimes \mathbf { h } _ { x } ) ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { s } ) ( \boldsymbol { \epsilon _ { t + 1 } } \otimes ( \mathbf { x } _ { t } ^ { f } ) ^ { f } ) ( \sigma \eta \otimes h _ { x } \otimes \sigma \eta ) ^ { \prime } \\ & { } + ( \sigma \eta \otimes h _ { x } ) ( \boldsymbol { \epsilon _ { t + 1 } } \otimes x _ { t } ^ { s } ) ( ( e _ { t + 1 } ⓤ e _ { t + 1} ) ^ { f } ⓤ ( e _ { t + 1 } ⓤ e _ { t + 1} ⓤ e _ { t + 1} ) ^ { f } ⓤ ( e _ { t + 1 } ⓤ e _ { t + 1} ⓤ e _ { t + 1} ⓤ e _ { t + 1} ⓤ e _ { t + 1} ⓤ e _ { t + 1} ⓤ e _ { t + 1} ⓤ e _ { t + 1} ⓤ e _ { t + 1} ⓤ e _ { t + 1} ⓤ e _ { t + 1} C ) ( e _ { t + 1 } ⓤ e _ { t + 1} ⓤ e _ { t + 1} ⓤ e _ { t + 1} ⓤ e _ { t + 1} ⓤ e _ { t + 1} ⓤ e _ { t + 1} ⓤ e _ { t + 1} ⓤ e _ { t + 1} Ⓟ e _ { t + 1} Ⓟ e _ { t + 1} Ⓟ e _ { t + 1} Ⓟ e _ { t + 1} Ⓟ e _ { t + 1} Ⓟ e _ { t + 1} Ⓟ e _ { t + 1} Ⓟ e _ { t + 1} Ⓟ e _ { t + 1} Ⓟ c ) ( e _ { t + 1 } Ⓟ c ) ( e _ { t + 1 } Ⓟ c ) ( e _ { t + 1 } Ⓟ c ) ( e _ { t + 1 } Ⓟ c ) ( e _ { t + 1 } Ⓟ c ) ( e _ { t + 1 } Ⓟ c ) ( e _ { t + 1 } Ⓟ c ) ( e _ { t + 1 } Ⅲ c ) ( e _ { t + 1 } Ⅲ c ) ( e _ { t + 1 } Ⅲ c ) ( e _ { t + 1 } Ⅲ c ) ( e _ { t + 1 } Ⅲ c ) ( e _ { t + 1 } Ⅲ c ) ( e _ { t + 1 } Ⅲ c ) ( e _ { t + 1 } Ⅲ c ). \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ . \\ & \\ = E [ ( c o n g a [ (\sigma n g a [ (\sigma n g a [ (\sigma n g a [ (\sigma n g a [ (\sigma n g a [ (\sigma n g a [ (\sigma n g a [ (\sigma n g a [ (\sigma n g a [ (\sigma n g a [ (\sigma n g a [ (\sigma n g a [ (\sigma n g a [ (\sigma n g a [ (\sigma n g a [ (\sigma n g a [ (\sigma n g a ] [ (\sigma n g a [ (\sigma n g a [ (\sigma n g a [ (\sigma n g a [ (\sigma n g a [ (\sigma n g a [ (\sigma n g a [ (\sigma n g a [ (\sigma n g a [ (\sigma n g a [ (\sigma n g a [ (\sigma n g a [ (\sigma n g a [ (\sigma n g a [ (\sigma n g a [ (\sigma n g a [ (\sigma m i n j i k i o n j i k i o n j i k i o n j i k i o n j i k i o n j i k i o n j i k i o n j i k i o n j i k i o n j i k i o n j i k i o n j i k i o n j i k i o n j i k i o n j i k i o n j i k i o n j i k i o n j i , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .\tag{1}\]

2)

3)

4)

5)

6)

7)

\[+ \left(\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}\right) \left(\boldsymbol {\epsilon} _ {t + 1} \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) ^ {\prime} \otimes \mathbf {x} _ {t} ^ {s}\right) \left(\sigma \boldsymbol {\eta} \otimes \boldsymbol {\sigma} \boldsymbol {\eta} \otimes \boldsymbol {\sigma} \boldsymbol {\eta}\right) ^ {\prime}\tag{8}\]

\[+ \left(\sigma \boldsymbol {\eta} \otimes \tilde {\mathbf {H}} _ {\mathbf {x x}}\right) \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) ^ {\prime} \left(\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}\right) ^ {\prime}\tag{9}\]

\[+ \left(\sigma \boldsymbol {\eta} \otimes \tilde {\mathbf {H}} _ {\mathbf {x x}}\right) \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} (\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime}\tag{10}\]

\[+ \left(\sigma \boldsymbol {\eta} \otimes \tilde {\mathbf {H}} _ {\mathbf {x x}}\right) \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) ^ {\prime} (\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}) ^ {\prime}\tag{11}\]

\[+ \left(\sigma \pmb {\eta} \otimes \tilde {\mathbf {H}} _ {\mathbf {x x}}\right) \left(\mathbf {I} _ {n _ {e}} \otimes \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime}\right) (\sigma \pmb {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime}\tag{12}\]

13)

\[+ \left(\sigma \boldsymbol {\eta} \otimes \tilde {\mathbf {H}} _ {\mathbf {x x}}\right) \left(\boldsymbol {\epsilon} _ {t + 1} \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) ^ {\prime} \otimes \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\mathbf {x} _ {t} ^ {f}\right) ^ {\prime}\right) (\sigma \boldsymbol {\eta} \otimes \boldsymbol {\sigma} \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime}\tag{14}\]

\[+ \left(\sigma \boldsymbol {\eta} \otimes \tilde {\mathbf {H}} _ {\mathbf {x x}}\right) \left(\boldsymbol {\epsilon} _ {t + 1} \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) ^ {\prime} \otimes \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right)\right) (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta})\tag{15}\]

\[+ \left(\sigma \boldsymbol {\eta} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}\right) \left(E \left[ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \right] \otimes \mathbf {I} _ {n _ {e}}\right) (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}) ^ {\prime}\tag{16}\]

\[+ \left(\sigma \pmb {\eta} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}\right) \pmb {\epsilon} _ {t + 1} \left(\mathbf {x} _ {t} ^ {f} \otimes \pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \left(\mathbf {h} _ {\mathbf {x}} \otimes \sigma \pmb {\eta} \otimes \mathbf {h} _ {\mathbf {x}}\right) ^ {\prime}\tag{17}\]

\[+ \left(\sigma \boldsymbol {\eta} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}\right) \left(\mathbf {I} _ {n _ {e}} \otimes E \left[ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \right]\right) \left(\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}\right) ^ {\prime}\tag{18}\]

We thus need to explain how to compute each of these terms

4.3.7 For Var

Note first that has dimensions .

\[\begin{array}{l} V a r \left[ \tilde {\boldsymbol {\xi}} _ {t + 1} \right] _ {6 6} = E [ (\mathbf {u} _ {t + 1} - E [ \mathbf {u} _ {t + 1} ]) (\mathbf {u} _ {t + 1} ^ {\prime} - E [ \mathbf {u} _ {t + 1} ^ {\prime} ]) ] \\ \qquad = E [ \mathbf {u} _ {t + 1} \mathbf {u} _ {t + 1} ^ {\prime} - \mathbf {u} _ {t + 1} E [ \mathbf {u} _ {t + 1} ^ {\prime} ] - E [ \mathbf {u} _ {t + 1} ] \mathbf {u} _ {t + 1} ^ {\prime} + E [ \mathbf {u} _ {t + 1} ] E [ \mathbf {u} _ {t + 1} ^ {\prime} ] ] \\ \qquad = E [ \mathbf {u} _ {t + 1} \mathbf {u} _ {t + 1} ^ {\prime} ] - E [ \mathbf {u} _ {t + 1} ] E [ \mathbf {u} _ {t + 1} ^ {\prime} ] \end{array}\]

We already know so we only need to compute the first term. Hence

\[E \left[ \mathbf {u} _ {t + 1} \mathbf {u} _ {t + 1} ^ {\prime} \right] = E [\]

\[\begin{array}{r l} & {\left((\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}) (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}\right)} \\ & {+ (\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) (\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f})} \\ & {+ (\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}) (\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1})} \\ & {+ (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) (\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f})} \\ & {+ (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}) (\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1})} \\ & {+ (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) (\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f})} \\ & {+ (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}) (\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}))} \\ & {\left(\left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) ^ {\prime} (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}) ^ {\prime}} \\ & {+ (\left(\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} (\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime}} \\ & {+ (\left(\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) ^ {\prime} (\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}) ^ {\prime}} \\ & {+ (\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime}} \\ & {+ (\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}) ^ {\prime}} \\ & {+ (\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime}} \\ & {+ (\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}) ^ {\prime}) ]} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} ] \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} ] \\ & {} \\ & {{.}} \\ & {{.}} \\ & {{.}} \\ & {{.}} \\ & {{.}} \\ & {{.}} \\ & {{.}} \\ & {{.}} \\ & {{.}} \\ & {{.}} \\ & {{.}} \\ & {{.}} \\ & {{.}} \\ & {{.}} \\ & {{.}} \\ & {{.}} \\ & {{.}} \\ & {{.}} \\ & {{.}} \\ & {{.}} \\ & {{. .}} \\ & {{.}} \\ & {{.}} \\ & {{.}} \\ & {{.}} \\ & {{.}} \\ & {{.}} \\ & {{.}} \\ & {{.}} \\ & {{.}} \\ & {{.}} \\ & {{.}} \\ & {{.}} \\ & {{.}} \\ & {{.}} \\ & {{.}} \\ & {{.}} \\ & {{.}} \\ & {{.}} \\ & {{.}} \\ & {{[}. ]}. |,< |content_end|>\]

\[\begin{array}{l} = E [ \\ \left(\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}\right) \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) \\ \left(\left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) ^ {\prime} \left(\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}\right) ^ {\prime} \right. \\ + \left(\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \left(\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}\right) ^ {\prime} \\ + \left(\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) ^ {\prime} \left(\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}\right) ^ {\prime} \\ + \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime} \\ + \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}) ^ {\prime} \\ + \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime} \end{array}\]

\[\begin{array}{r l} & + (\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}) ^ {\prime}) \\ & + (\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) (\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}) \\ & (\left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) ^ {\prime} (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}) ^ {\prime} \\ & + (\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} (\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime} \\ & + (\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}) ^ {\prime} (\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}) ^ {\prime} \\ & + (\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime} \\ & + (\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}) ^ {\prime} \\ & + (\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime} \\ & + (\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}) ^ {\prime}) \end{array}\]

\[\begin{array}{r l} & {+ (\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}) (\mathbf {x} _ {t} ^ {f} \otimes \epsilon_ {t + 1} \otimes \epsilon_ {t + 1})} \\ & {\quad (\left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \epsilon_ {t + 1}\right) ^ {\prime} (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}) ^ {\prime}} \\ & {\quad + (\mathbf {x} _ {t} ^ {f} \otimes \epsilon_ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} (\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime}} \\ & {\quad + (\mathbf {x} _ {t} ^ {f} \otimes \epsilon_ {t + 1} \otimes \epsilon_ {t + 1}) ^ {\prime} (\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}) ^ {\prime}} \\ & {\quad + (\epsilon_ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime}} \\ & {\quad + (\epsilon_ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \epsilon_ {t + 1}) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}) ^ {\prime}} \\ & {\quad + (\epsilon_ {t + 1} \otimes \epsilon_ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime}} \\ & {\quad + (\epsilon_ {t + 1} \otimes \epsilon_ {t + 1} \otimes \epsilon_ {t + 1}) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}) ^ {\prime})} \end{array}\]

\[\begin{array}{r l} & {+ (\sigma \pmb {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) (\pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f})} \\ & {\quad ((\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \pmb {\epsilon} _ {t + 1}) ^ {\prime} (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \sigma \pmb {\eta}) ^ {\prime}} \\ & {\quad + (\mathbf {x} _ {t} ^ {f} \otimes \pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} (\mathbf {h} _ {\mathbf {x}} \otimes \sigma \pmb {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime}} \\ & {\quad + (\mathbf {x} _ {t} ^ {f} \otimes \pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1}) ^ {\prime} (\mathbf {h} _ {\mathbf {x}} \otimes \sigma \pmb {\eta} \otimes \sigma \pmb {\eta}) ^ {\prime}} \\ & {\quad + (\pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} (\sigma \pmb {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime}} \\ & {\quad + (\pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \pmb {\epsilon} _ {t + 1}) ^ {\prime} (\sigma \pmb {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \sigma \pmb {\eta}) ^ {\prime}} \\ & {\quad + (\pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} (\sigma \pmb {\eta} \otimes \sigma \pmb {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime}} \\ & {\quad + (\pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1}) ^ {\prime} (\sigma \pmb {\eta} \otimes \sigma \pmb {\eta} \otimes \sigma \pmb {\eta}) ^ {\prime})} \end{array}\]

\[\begin{array}{r l} & {+ (\sigma \pmb {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \pmb {\sigma} \pmb {\eta}) (\pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \pmb {\epsilon} _ {t + 1})} \\ & {\quad ((\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \pmb {\epsilon} _ {t + 1}) ^ {\prime} (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \sigma \pmb {\eta}) ^ {\prime}} \\ & {\quad + (\mathbf {x} _ {t} ^ {f} \otimes \pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} (\mathbf {h} _ {\mathbf {x}} \otimes \sigma \pmb {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime}} \\ & {\quad + (\mathbf {x} _ {t} ^ {f} \otimes \pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1}) ^ {\prime} (\mathbf {h} _ {\mathbf {x}} \otimes \sigma \pmb {\eta} \otimes \sigma \pmb {\eta}) ^ {\prime}} \end{array}\]

\[\begin{array}{l} + \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime} \\ + \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}) ^ {\prime} \\ + \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime} \\ + \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}) ^ {\prime}) \end{array}\]

\[\begin{array}{r l} & {+ (\sigma \pmb {\eta} \otimes \pmb {\sigma} \pmb {\eta} \otimes \mathbf {h _ {x}}) (\pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f})} \\ & {\quad ((\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \pmb {\epsilon} _ {t + 1}) ^ {\prime} (\mathbf {h _ {x}} \otimes \mathbf {h _ {x}} \otimes \sigma \pmb {\eta}) ^ {\prime}} \\ & {\quad + (\mathbf {x} _ {t} ^ {f} \otimes \pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} (\mathbf {h _ {x}} \otimes \sigma \pmb {\eta} \otimes \mathbf {h _ {x}}) ^ {\prime}} \\ & {\quad + (\mathbf {x} _ {t} ^ {f} \otimes \pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1}) ^ {\prime} (\mathbf {h _ {x}} \otimes \sigma \pmb {\eta} \otimes \pmb {\sigma} \pmb {\eta}) ^ {\prime}} \\ & {\quad + (\pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} (\sigma \pmb {\eta} \otimes \mathbf {h _ {x}} \otimes \mathbf {h _ {x}}) ^ {\prime}} \\ & {\quad + (\pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \pmb {\epsilon} _ {t + 1}) ^ {\prime} (\sigma \pmb {\eta} \otimes \mathbf {h _ {x}} \otimes \pmb {\sigma} \pmb {\eta}) ^ {\prime}} \\ & {\quad + (\pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} (\sigma \pmb {\eta} \otimes \pmb {\sigma} \pmb {\eta} \otimes \mathbf {h _ {x}}) ^ {\prime}} \\ & {\quad + (\pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1}) ^ {\prime} (\sigma \pmb {\eta} \otimes \pmb {\sigma} \pmb {\eta} \otimes \pmb {\sigma} \pmb {\eta}) ^ {\prime})} \end{array}\]

\[\begin{array} { r l } & + ( \sigma \pmb { \eta } \otimes \pmb { \sigma } \pmb { \eta } \otimes \pmb { \sigma } \pmb { \eta } ) ( \pmb { \epsilon } _ { t + 1 } \otimes \pmb { \epsilon } _ { t + 1 } \otimes \pmb { \epsilon } _ { t + 1 } ) \\ & { \quad ( ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \otimes \pmb { \epsilon } _ { t + 1 } ) ^ { \prime } ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } \otimes \sigma \pmb { \eta } ) ^ { \prime } } \\ & { + ( \mathbf { x } _ { t } ^ { f } \otimes \pmb { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } ) ^ { \prime } ( \mathbf { h } _ { \mathbf { x } } \otimes \sigma \pmb { \eta } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { \prime } } \\ & { + ( \mathbf { x } _ { t } ^ { f } \otimes \pmb { \epsilon } _ { t + 1 } \otimes \pmb { \epsilon } _ { t + 1 } ) ^ { \prime } ( \mathbf { h } _ { \mathbf { x } } \otimes \sigma \pmb { \eta } \otimes \sigma \pmb { \eta } ) ^ { \prime } } \\ & { + ( \pmb { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) ^ { \prime } ( \sigma \pmb { \eta } \otimes \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { \prime } } \\ & { + ( \pmb { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \otimes \pmb { \epsilon } _ { t + 1 } ) ^ { \prime } ( \sigma \pmb { \eta } \otimes \mathbf { h } _ { \mathbf { x } } \otimes \sigma \pmb { \eta } ) ^ { \prime } } \\ & { + ( \pmb { \epsilon } _ { t + 1 } \otimes \pmb { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } ) ^ { \prime } ( \sigma \pmb { \eta } \otimes \sigma \pmb { \eta } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { \prime } } \\ & + ( ( \pmb { \epsilon } _ { t + 1 } \otimes \pmb { \epsilon } _ { t + 1 } \otimes \pmb { \epsilon } _ { t + 1 } ) ^ { \prime } ( \sigma \pmb { \eta } \otimes \sigma \pmb { \eta } \otimes \sigma \pmb { \eta } ) ^ { \prime } ) \\ & {\quad ]} \\ & = E [ \\ & ( [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] . \\ & ( [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] . \\ & ( ( [ [ [ [ [ [ [ [ [ [ [ [ [ [ ] ] ] ] ] ] ] ] ] ] ] ) ^ { ' } ( ( | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | \\ & + ( ( \| x \| ) ^ {- 1 / 2} , \| x \| ) ^ {- 1 / 2} , \| y \| ) ^ {- 1 / 2} , \| z \| ) ^ {- 1 / 2} , \| w \| ) ^ {- 1 / 2} , \| x \| ) ^ {- 1 / 2} , \| y \| ) ^ {- 1 / 2} , \| w \| ) ^ {- 1 / 2} , \| z \| ) ^ {- 1 / 2} , \| w \| ) ^ {- 1 / 2} , \| z \| ) ^ {- 1 / 2} , \| w \| ) ^ {- 1 / 2} , \| y \| ) ^ {- 1 / 2} , \| z \| ) ^ {- 1 / 2} , \| w \| ) ^ {- 1 / 2} , \| z \| ) ^ {- 1 / 2} , \| w \| ) ^ {- 1 / 2} , \| y \| ) ^ {- 1 / 2} , \| z \| ) ^ {- 1 / 2} , \| w \| ) ^ {- 1 / 3} , \| y \| ) ^ {- 1 / 3} , \| z \| ) ^ {- 1 / 3} , \| w \| ) ^ {- 1 / 3} , \| z \| ) ^ {- 1 / 3} , \| y \| ) ^ {- 1 / 3} , \| z \| ) ^ {- 1 / 3} , \| w \| ) ^ {- 1 / 3} , \| y \| ) ^ {- 1 / 3} , \| z \| ) ^ {- 1 / 3} , \|w \| ) ^ {- 1 / 3} , \|y| ) ^ {- 1 / 3} , \|z| ) ^ {- 1 / 3} , \|w| ) ^ {- 1 / 3} , \|y| ) ^ {- 1 / 3} , \|z| ) ^ {- 1 / 3} , \|w| ) ^ {- 1 / 3} , \|y| ) ^ {- 1 / 3} , \|z| ) ^ {- 1 / 3} , \|w| ) ^ {- 1 / 3} , \|y| ) ^ {- 1 / 3}, \\ & + ( ( \| x \| ) ^ {- 1 / 2} , \| x \| ) ^ {- 1 / 2} , \| y \| ) ^ {- 1 / 2} , \| z \| ) ^ {- 1 / 2} , \| w \| ) ^ {- 1 / 2} , \| y| ) ^ {- 1 / 2} , \| z| ) ^ {- 1 / 2} , \|w| ) ^ {- 1 / 2} , \|y| ) ^ {- 1 / 2} , \|z| ) ^ {- 1 / 2} , \|w| ) ^ {- 1 / 2} , \|y| ) ^ {- 1 / 2} , ||x|| > - i n d i n e r e f f i c a l e n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i m e d o u s s e r s e r s e r s e r s e r s e r s e r s e r s e r s e r s e r s e r s e r s e r s e r s e r s e r s e r s e r s e r s e r s e r s e r s e r s e r s e r s e r s e r s e r s e r s e r s e r s e r s e r t h a v e r s e r s e r s e r s e r s e r s e r s e r s e r s e r s e r s e r s e r s e r s e r s e r s e r s e r s e r s e r s e r s e r s e r s e r s e r s e r s e r s e r s e r s e r s e r s e r s e r s c o l l o p a l l o p a l l o p a l l o p a l l o p a l l o p a l l o p a l l o p a l l o p a l l o p a l l o p a l l o p a l l o p a l l o p a l l o p a l l o p a l l o p a l l o p a l l o p a l l o p a l l o p u l l o p a l l o p a l l o p a l l o p a l l o p a l l o p a l l o p a l l o p a l l o p a l l o p a l l o p a l l o p a l l o p a l l o p a l l o p a l l o p a l l o p a l l o p a l l o p a l l o p g u m b u t h a v e c o n t h a v e c o n t h a v e c o n t h a v e c o n t h a v e c o n t h a v e c o n t h a v e c o n t h a v e c o n t h a v e c o n t h a v e c o n t h a v e c o n t h a v e c o n t h a v e c o n t h a w i k h u u t h a v e c o n t h a v e c o n t h a v e c o n t h a v e c o n t h a v e c o n t h a v e c o n t h a v e c o n t h a v e c o n t h a v e c o n t h a v e c o n t h a v e c o n t h a v e c o n t h A u u t h a v e c o n t h A u u t h a v e c o n t h A u u t h a v e c o n t h A u u t h a v e c o n t h A u u t h a v e c o n t h A u u t h a v e c o n t h A u u t h A v e c o n t h A u u t h A v e c o n t h A u u t h A v e c o n t h A u u t h A v e c o n t h A u u t h A v e c o n t h A u u t h A v e c o n t h A u u t h A v e c o n t h A u u t h A v e c o n t h A U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I N D S O W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T HW T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H WT H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W T H W THWTTHWTTHWTTHWTTHWTTHWTTHWTTHWTTHWTTHWTTHWTTHWTTHWTTHWTTHWTTHWTTHWTTHWTTHWTTHWTTHWTTHWTTHWTTHWTTHWTTHWTTHWTTHWTTHWTTHWTTHWTTHWTTHWTTHWTTTHTHWTTTHTHWTTTHTHWTTTHTHWTTTHTHWTTTHTHWTTTHTHWTTTHTHWTTTHTHWTTTHTHWTTTHTHWTTTHTHWTTTHTHWTTTHTHWTTTHTHWTTTHTHWTTTHTHWTTTHTHWTTTHTHWTTTHTHWTTTMTHTHWTTTMTHTHWTTTMTHTHWTTTMTHTHWTTTMTHTHWTTTMTHTHWTTTMTHTHWTTTMTHTHWTTTMTHTHWTTTMTHTHWTTTMTHTHWTTTMTHTHWTTTMTHTHWTTTMTHTHWTTTMTHTHWTTTMTHTHWTTTMTHTHW TTHTHWTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHTHT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HTRT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT RT HT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVA VTVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT TVAT VTVAT VTVAT VTVAT VTVAT VTVAT VTVAT VTVAT VTVAT VTVAT VTVAT VTVAT VTVAT VTVAT VTVAT VTVAT VTVAT VTVAT VTVAT VTVAT VTVAT VTVAT VTVAT VTVAT VTVAT VTVAT VTVAT VTVAT VTVAT VTVAT VTVAT VTVAT VTVAT VTV AT VTV AT VTV AT VTV AT VTV AT VTV AT VTV AT VTV AT VTV AT VTV AT VTV AT VTV AT VTV AT VTV AT VTV AT VTV AT VTV AT VTV AT VTV AT VTV AT VTV AT VTV AT VTV AT VTV AT VTV AT VTV AT VTV AT VTV AT VTV AT VTV AT VTV AT VTV AT VTV AT VTV ATVTVATTVTVATTVTVATTVTVATTVTVATTVTVATTVTVATTVTVATTVTVATTVTVATTVTVATTVTVATTVTVATTVTVATTVTVATTVTVATTVTVATTVTVATTVTVATTVTVATTVTVATTVTVATTVTVATTVTVATTVTVATTVTVATTVTVATTVTVATTVTVATTVTVATTVTVATTVTVATTVTVATTVTVATTGVVITTGVVITTGVVITTGVVITTGVVITTGVVITTGVVITTGVVITTGVVITTGVVITTGVVITTGVVITTGVVITTGVVITTGVVITTGVVITTGVVITTGVVITTGVVITTGVVITTGVVITTGVVITTGVVITTGVVITTGVVITTGVVITTGVVITTGVVITTGVVITTGVVITTGVVITTGVVITTGVVITTGVVITT GVVITTGVVITTGVVITTGVVITTGVVITTGVVITTGVVITTGVVITTGVVITTGVVITTGVVITTGVVITTGVVITTGVVITTGVVITTGVVITTGVVITTGVVITTGVVITTGVVITTGVVITTGVVITTGVVITTGVVITTGVVITTGVVITTGVVITTGVVITTGVVITTGVVITTGVVITTGVVITTGVVITTGWWITTGWWITTGWWITTGWWITTGWWITTGWWITTGWWITTGWWITTGWWITTGWWITTGWWITTGWWITTGWWITTGWWITTGWWITTGWWITTGWWITTGWWITTGWWITTGWWITTGWWITTGWWITTGWWITTGWWITTGWWITTGWWITTGWWITTGWWITTGWWITTGWWITTGWWITTGWWITTGWWITTGWWITGGVVITTGGVVITTGGVVITTGGVVITTGGVVITTGGVVITTGGVVITTGGVVITTGGVVITTGGVVITTGGVVITTGGVVITTGGVVITTGGVVITTGGVVITTGGVVITTGGVVITTGGVVITTGGVVITTGGVVITTGGVVITTGGVVITTGGVVITTGGVVITTGGVVITTGGVVITTGGVVITTGGVVITTGGVVITTGGVVITTGGVVITTGGVVITTGGVVITTGGVVITT GGVVITTGGVVITTGGVVITTGGVVITTGGVVITTGGVVITTGGVVITTGGVVITTGGVVITTGGVVITTGGVVITTGGVVITTGGVVITTGGVVITTGGVVITTGGVVITTGGVVITTGGVVITTGGVVITTGGVVITTGGVVITTGGVVittTGVAa b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b b< fcel>+ (ση)⊗ ση⊗ ση⊗ ση⊗ ση⊗ ση⊗ ση⊗ ση⊗ ση⊗ ση⊗ ση⊗ ση⊗ ση⊗ ση⊗ ση⊗ ση⊗ ση⊗ ση⊗ ση⊗ ση⊗ ση⊗ ση⊗ ση⊗ ση⊗ ση⊗ ση⊗ ση⊗ ση⊗ ση⊗ ση⊗ ση⊗ ση⊗ ση⊗ ση⊗ ση& (ση)⊗ ση⊗ ση⊗ ση⊗ ση⊗ ση⊗ ση⊗ ση⊗ ση⊗ ση⊗ ση⊗ ση⊗ ση⊗ ση⊗ ση⊗ ση⊗ ση⊗ ση⊗ ση⊗ ση⊗ ση⊗ ση⊗ ση⊗ ση⊗ ση⊗ ση⊗ ση⊗ ση⊗ ση⊗ ση⊗ ση⊗ ση⊗ σιοντακευτής τακευτής τακευτής τακευτής τακευτής τακευτής τακευτής τακευτής τακευτής τακευτής τακευτής τακευτής τακευτής τακεύντακεύντακεύντακεύντακεύντακεύντακεύντακεύντακεύντακεύντακεύντακεύντακεύντακεύντακεύντακεύντακεύντακεόντακεόντακεόντακεόντακεόντακεόντακεόντακεόντακεόντακεόντακεόντακεόντακεόντακεόντακεόντακεόντακεόνταkεόντακεόντακεόντακεόντακεόντακεόντακεόντακεόντακεόντακεόντακεόντακεόντακεόντακεόντακεόντακεόντακεόνtακεόνtακεόνtακεόνtακεόνtακεόνtακεόνtακεόνtακεόνtακεόνtακεόνtακεόνtακεόνtακεόνtακe κολούποποποποποποποποποποποποποποποποποποποποποποποποποποποποποποποποποποποποποποποποποποποποποποποποποποπΟ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Φ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ Χ ΥΠΤΗΤΗΤΗΤΗΤΗΤΗΤΗΤΗΤΗΤΗΤΗΤΗΤΗΤΗΤΗΤΗΤΗΤΗΤΗΤΗΤΗΤΗΤΗΤΗΤΗΤΗΤΗΤΗΤΗΤΗΤΗΤΗΤΗΤΗΤΗΤΗΤΗΤΗΤΗΤΗΤΗΤΗΤΗΤΗΤΗΤΗΤΗΤΗΤΗΤΗΤΑΚΕΚΕΚΕΚΕΚΕΚΕΚΕΚΕΚΕΚΕΚΕΚΕΚΕΚΕΚΕΚΕΚΕΚΕΚΕΚΕΚΕΚΕΚΕΚΕΚΕΚΕΚΕΚΕΚΕΚΕΚΕΚΕΚΕΚΕΚΕΚΕΚΕΚΕΚΕΚΕΚΕΚΕΚΕΚΕΚΕΚΕΚΕΚΕΚΕΚΕΚΑΙΝΙΝΙΝΙΝΙΝΙΝΙΝΙΝΙΝΙΝΙΝΙΝΙΝΙΝΙΝΙΝΙΝΙΝΙΝΙΝΙΝΙΝΙΝΙΝΙΝΙΝΙΝΙΝΙΝΙΝΙΝΙΝΙΝΙΝΙΝΙΝΙΝΙΝΙΝΙΝΙΝΙΝΙΝΙΝΙΝΙΝΙΝΙΝΙΝΙΝΙNIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIlllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc\]

\[\begin{array}{l} \left(\left(\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) ^ {\prime} (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}) ^ {\prime} \right. \\ + \left(\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} (\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime} \\ + \left(\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) ^ {\prime} (\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}) ^ {\prime} \\ + \left(\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime} \\ + \left(\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}) ^ {\prime} \\ + \left(\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime} \\ + \left(\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) (\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}) ^ {\prime}) \end{array}\]

\[\begin{array}{l} + (\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}) \\ \quad ((\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}) (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}) ^ {\prime} (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}) ^ {\prime} \\ \quad + (\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}) (\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} (\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime} \\ \quad + (\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}) (\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}) ^ {\prime} (\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}) ^ {\prime} \\ \quad + (\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}) (\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime} \\ \quad + (\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}) (\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}) ^ {\prime} \\ \quad + (\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}) (\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime} \\ + 0) \end{array}\]

\[\begin{array}{l} + (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) \\ \quad ((\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}) ^ {\prime} (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}) ^ {\prime} \\ \quad + (\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) (\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} (\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime} \\ \quad + (\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) (\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}) ^ {\prime} (\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}) ^ {\prime} \\ \quad + (\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) (\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime} \\ \quad + (\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) (\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}) ^ {\prime} \\ \quad + (\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) (\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime} \\ \quad + (\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) (\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}) ^ {\prime}) \end{array}\]

\[\begin{array}{l} + (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}) \\ \quad ((\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}) (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}) ^ {\prime} (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}) ^ {\prime} \\ \quad + (\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}) (\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} (\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime} \\ \quad + (\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}) (\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}) ^ {\prime} (\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}) ^ {\prime} \\ \quad + (\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}) (\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime} \\ \quad + (\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}) (\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}) ^ {\prime} \\ \quad + (\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}) (\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime} \end{array}\]

\[\begin{array}{l} + (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta} \otimes \mathbf {h _ {x}}) \\ \quad ((\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}) (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}) ^ {\prime} (\mathbf {h _ {x}} \otimes \mathbf {h _ {x}} \otimes \sigma \boldsymbol {\eta}) ^ {\prime} \\ \quad + (\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}) (\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} (\mathbf {h _ {x}} \otimes \sigma \boldsymbol {\eta} \otimes \mathbf {h _ {x}}) ^ {\prime} \\ \quad + (\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}) (\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}) ^ {\prime} (\mathbf {h _ {x}} \otimes \sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}) ^ {\prime} \\ \quad + (\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}) (\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \mathbf {h _ {x}} \otimes \mathbf {h _ {x}}) ^ {\prime} \\ \quad + (\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}) (\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \mathbf {h _ {x}} \otimes \sigma \boldsymbol {\eta}) ^ {\prime} \\ \quad + (\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}) (\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta} \otimes \mathbf {h _ {x}}) ^ {\prime} \\ + 0) \end{array}\]

\[\begin{array}{r l} & {+ (\sigma \pmb {\eta} \otimes \pmb {\sigma} \pmb {\eta} \otimes \pmb {\sigma} \pmb {\eta})} \\ & {\quad ((\pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1}) (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \pmb {\epsilon} _ {t + 1}) ^ {\prime} (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \sigma \pmb {\eta}) ^ {\prime}} \\ & {\quad + (\pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1}) (\mathbf {x} _ {t} ^ {f} \otimes \pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} (\mathbf {h} _ {\mathbf {x}} \otimes \sigma \pmb {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime}} \\ & {\quad + 0} \\ & {\quad + (\pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1}) (\pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} (\sigma \pmb {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime}} \\ & {\quad + 0} \\ & {\quad + 0} \\ & {\quad + (\pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1}) (\pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1}) ^ {\prime} (\sigma \pmb {\eta} \otimes \sigma \pmb {\eta} \otimes \sigma \pmb {\eta}) ^ {\prime}) ]} \end{array}\]

\[\left(\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}\right)\]

\[\left(\left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) ^ {\prime} \left(\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}\right) ^ {\prime} \right. \tag {1}\tag{2}\]

\[+ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) \left(\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \left(\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}\right) ^ {\prime}\tag{3}\]

\[+ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) \left(\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) ^ {\prime} \left(\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta} \otimes \boldsymbol {\sigma} \boldsymbol {\eta}\right) ^ {\prime}\tag{4}\]

\[+ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime}\]

\[+ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \boldsymbol {\sigma} \boldsymbol {\eta}) ^ {\prime}\tag{6}\]

\[+ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime}\tag{7}\]

\[+ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) ^ {\prime} \left(\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}\right) ^ {\prime})\]

\[+ \left(\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}\right)\tag{8}\]

\[\left(\left(\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) ^ {\prime} \left(\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}\right) ^ {\prime} \right.\tag{9}\]

\[+ \left(\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \left(\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}\right) ^ {\prime}\tag{10}\]

\[+ \left(\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) ^ {\prime} \left(\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}\right) ^ {\prime}\tag{11}\]

\[+ \left(\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime}\]

12)

\[+ \left(\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}) ^ {\prime}\tag{13}\]

\[+ \left(\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime}\tag{14}\]

\[+ \left(\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) ^ {\prime} \left(\sigma \boldsymbol {\eta} \otimes \boldsymbol {\sigma} \boldsymbol {\eta} \otimes \boldsymbol {\sigma} \boldsymbol {\eta}\right) ^ {\prime})\]

\[+ \left(\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}\right)\tag{15}\]

\[\left(\left(\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) ^ {\prime} \left(\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}\right) ^ {\prime} \right.\tag{16}\]

\[+ \left(\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) \left(\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \left(\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}\right) ^ {\prime}\tag{17}\]

\[+ \left(\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) \left(\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) ^ {\prime} (\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}) ^ {\prime}\tag{18}\]

\[+ \left(\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime}\tag{19}\]

\[+ \left(\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}) ^ {\prime}\tag{20}\]

\[+ \left(\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime})\]

\[+ \left(\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}\right)\tag{21}\]

\[\left(\left(\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) ^ {\prime} \left(\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}\right) ^ {\prime} \right.\tag{22}\]

\[+ \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \left(\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}\right) ^ {\prime}\tag{23}\]

\[+ \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) ^ {\prime} \left(\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta} \otimes \boldsymbol {\sigma} \boldsymbol {\eta}\right) ^ {\prime}\tag{24}\]

\[+ \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime}\tag{25}\]

\[+ \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \boldsymbol {\sigma} \boldsymbol {\eta}) ^ {\prime}\tag{26}\]

\[+ \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime}\tag{27}\]

\[+ \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) ^ {\prime} \left(\sigma \boldsymbol {\eta} \otimes \boldsymbol {\sigma} \boldsymbol {\eta} \otimes \boldsymbol {\sigma} \boldsymbol {\eta}\right) ^ {\prime})\]

\[+ \left(\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}\right)\tag{28}\]

\[\left(\left(\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) ^ {\prime} \left(\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}\right) ^ {\prime} \right.\tag{29}\]

\[+ \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) \left(\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \left(\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}\right) ^ {\prime}\tag{30}\]

\[+ \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) \left(\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) ^ {\prime} \left(\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}\right) ^ {\prime}\tag{31}\]

\[+ \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime}\tag{32}\]

\[+ \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \boldsymbol {\sigma} \boldsymbol {\eta}) ^ {\prime}\tag{33}\]

\[+ \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime})\]

\[+ \left(\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}\right)\tag{34}\]

\[\left(\left(\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) ^ {\prime} \left(\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}\right) ^ {\prime} \right.\tag{35}\]

\[+ \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} (\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime}\tag{36}\]

\[+ \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) ^ {\prime} \left(\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta} \otimes \boldsymbol {\sigma} \boldsymbol {\eta}\right) ^ {\prime}\]

37)

\[+ \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} (\sigma \pmb {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime}\tag{38}\]

\[+ \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}) ^ {\prime}\tag{39}\]

\[+ \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {\prime})\tag{40}\]

\[\left(\left(\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) ^ {\prime} \left(\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}\right) ^ {\prime} \right.\tag{41}\]

\[+ \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) \left(\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \left(\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}\right) ^ {\prime}\tag{42}\]

\[+ \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \left(\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}\right) ^ {\prime}\tag{43}\]

We next derive how to compute the moments in these terms.

end

end

\[\begin{array}{r l} & {4 0)} \\ & {E \left[ (\pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1}) (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \pmb {\epsilon} _ {t + 1}) ^ {\prime} \right] = E \left[ (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \pmb {\epsilon} _ {t + 1}) (\pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1}) ^ {\prime} \right] ^ {\prime}} \\ & {\mathrm{wherewealreadyknow} E \left[ (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \pmb {\epsilon} _ {t + 1}) (\pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1}) ^ {\prime \prime} \right] \mathrm{from7}).} \end{array}\]

\[\begin{array}{r l} & {\mathrm{41)}} \\ & {E \left[ (\pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1}) (\mathbf {x} _ {t} ^ {f} \otimes \pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} \right] = E \left[ (\mathbf {x} _ {t} ^ {f} \otimes \pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}) (\pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1}) ^ {\prime} \right] ^ {\prime}} \\ & {\mathrm{wherewealreadyknow} E \left[ (\mathbf {x} _ {t} ^ {f} \otimes \pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}) (\pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1}) ^ {\prime \prime} \right] \mathrm{from14}).} \end{array}\]

\[\begin{array}{r l} & {\mathrm{42)}} \\ & {E \left[ (\pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1}) (\pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) ^ {\prime} \right] = E \left[ (\pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) (\pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1}) ^ {\prime} \right] ^ {\prime}} \\ & {\mathrm{wherewealreadyknow} E \left[ (\pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) (\pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1}) ^ {\prime \prime} \right] \mathrm{from27).}} \end{array}\]

\[\begin{array}{l} 4 3) \\ E \left[ (\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}) (\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}) ^ {\prime} \right] \\ = E [ \left(\left\{\epsilon_ {t + 1} (\phi_ {1}, 1) \left\{\epsilon_ {t + 1} (\phi_ {2}, 1) \left\{\epsilon_ {t + 1} (\phi_ {3}, 1) \right\} _ {\phi_ {3} = 1} ^ {n _ {e}} \right\} _ {\phi_ {2} = 1} ^ {n _ {e}} \right\} _ {\phi_ {1} = 1} ^ {n _ {e}}\right) \\ \times \left(\left\{\epsilon_ {t + 1} (\phi_ {4}, 1) \left\{\epsilon_ {t + 1} (\phi_ {5}, 1) \left\{\epsilon_ {t + 1} (\phi_ {6}, 1) \right\} _ {\phi_ {6} = 1} ^ {n _ {e}} \right\} _ {\phi_ {5} = 1} ^ {n _ {e}} \right\} _ {\phi_ {4} = 1} ^ {n _ {e}}\right) ^ {\prime} ] \end{array}\]

The codes are given in the matlab file. (too big for displaying)

4.4 Method 3: Simple formulas for first and second moments

This section shows how to compute mean values up to third order in a very direct manner. As in the case of the second-order approximation, the advantage of Method 3 is that we do not recompute terms which are already known at a lower approximation order. As a result, the matrices which must be inverted are here smaller than in Method 1 and 2.

4.4.1 First moments

This section derives the unconditional mean value of and . Recall

\[\mathbf {x} _ {t} = \mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d}\]

Thus, we only need to find . Here

\[\begin{array}{r l} & {E \left[ \mathbf {x} _ {t + 1} ^ {r d} \right] = \mathbf {h} _ {\mathbf {x}} E \left[ \mathbf {x} _ {t} ^ {r d} \right] + 2 \tilde {\mathbf {H}} _ {\mathbf {x x}} E \left[ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {s}\right) \right] + \tilde {\mathbf {H}} _ {\mathbf {x x x}} E \left[ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) \right] + \frac {3}{6} \mathbf {h} _ {\sigma \sigma \mathbf {x}} \sigma^ {2} E \left[ \mathbf {x} _ {t} ^ {f} \right] + \frac {1}{6} \mathbf {h} _ {\sigma \sigma \sigma} \sigma^ {3}} \\ & {\Updownarrow} \end{array}\]

\[\left(\mathbf {I} _ {n _ {x}} - \mathbf {h} _ {\mathbf {x}}\right) E \left[ \mathbf {x} _ {t} ^ {r d} \right] = 2 \tilde {\mathbf {H}} _ {\mathbf {x x}} E \left[ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {s}\right) \right] + \tilde {\mathbf {H}} _ {\mathbf {x x x}} E \left[ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) \right] + \frac {1}{6} \mathbf {h} _ {\sigma \sigma \sigma} \sigma^ {3}\]

because is stationary and

\[E \left[ \mathbf {x} _ {t} ^ {r d} \right] = (\mathbf {I} _ {n _ {x}} - \mathbf {h} _ {\mathbf {x}}) ^ {- 1} \left(2 \tilde {\mathbf {H}} _ {\mathbf {x x}} E \left[ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {s}\right) \right] + \tilde {\mathbf {H}} _ {\mathbf {x x x}} E \left[ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) \right] + \frac {1}{6} \mathbf {h} _ {\sigma \sigma \sigma} \sigma^ {3}\right)\]

To compute recall that

\[\begin{array}{l} \mathbf {x} _ {t + 1} ^ {f} \otimes \mathbf {x} _ {t + 1} ^ {f} \otimes \mathbf {x} _ {t + 1} ^ {f} = (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) + (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}) (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}) \\ \quad + (\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) (\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}) + (\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}) (\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}) \\ \quad + (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) (\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) + (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}) (\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}) \\ \quad + (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) (\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}) + (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}) (\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}) \\ H e n c e \\ E [ \mathbf {x} _ {t + 1} ^ {f} \otimes \mathbf {x} _ {t + 1} ^ {f} \otimes \mathbf {x} _ {t + 1} ^ {f} ] = (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) E [ (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) ] + (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}) E [ (\boldsymbol {\epsilon_ {t + 1}} \otimes \boldsymbol {\epsilon_ {t + 1}} \otimes \boldsymbol {\epsilon_ {t + 1}}) ] \\ ₽ ₽ & . ₽ ₽ ₽ ₽ ₽ ₽ ₽ ₽ ₽ ₽ ₽ ₽ ₽ ₽ ₽ ₽ ₽ ₽ ₽ ₽ ₽ ₽ ₽ ₽ ₽ ₽ ₽ ₽ ₽ ₽ ₽ ₽ ₽ ₽ & . ₽ ₽ & . ₽ ₽ & . ₽ & . ₽ & . ₽ & . ₽ & . ₽ & . ₽ & . ₽ & . ₽ & . ₽ & . ₽ & . ₽ & . ₽ & . ₽ & . ₽ & . ₽ & . ₽ & . ₽ & . ₽ & . ₽ & . ₽ & .. & . ₽ & . ₽ & . ₽ & . ₽ & . ₽ & . ₽ & . ₽ & . ₽ & . ₽ & . ₽ & . ₽ & . ₽ & . ₽ & . ₽ & . ₽ & . ₽ & . ₽ & . ₽ & . ₽ & . ₉ & . ₉ & . ₉ & . ₉ & . ₉ & . ₉ & . ₉ & . ₉ & . ₉ & . ₉ & . ₉ & . ₉ & . ₉ & . ₉ & . ₉ & . ₉ & . ₉ & . ₉ & . ₉ & . ₉ & . ₎ & . ₎ & . ₎ & . ₎ & . ₎ & . ₎ & . ₎ & . ₎ & . ₎ & . ₎ & . ₎ & . ₎ & . ₎ & . ₎ & . ₎ & . ₎ & . ₎ & . ₎ & . ₎ & . ₎ & . ₅ & . ₅ & . ₅ & . ₅ & . ₅ & . ₅ & . ₅ & . ₅ & . ₅ & . ₅ & . ₅ & . ₅ & . ₅ & . ₅ & . ₅ & . ₅ & . ₅ & . ₅ & . ₅ & . ₅ & . ₎ & . ₎ & . ₎ & . ₎ & . ₎ & . ₎ & . ₎ & . ₎ & . ₎ & . ₎ & . ₎ & . ₎ & . ₎ & . ₎ & . ₎ & . ₎ & . ₎ & . ₎ & . ₎ & . ₮ & . ⓞ & . ∞ | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |\]

\[\left(\mathbf {I} _ {n _ {x} ^ {3}} - \left(\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}\right)\right) E \left[ \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \right] = (\sigma \pmb {\eta} \otimes \pmb {\sigma} \pmb {\eta} \otimes \pmb {\sigma} \pmb {\eta}) E [ (\pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1} \otimes \pmb {\epsilon} _ {t + 1}) ]\]

because is stationary

\[E \left[ \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \right] = \left(\mathbf {I} _ {n _ {x} ^ {3}} - \left(\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}\right)\right) ^ {- 1} (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}) E [ (\epsilon_ {t + 1} \otimes \epsilon_ {t + 1} \otimes \epsilon_ {t + 1}) ]\]

Note that this term is zero if all third moments of are zero.

\[\begin{array}{r l} & {\mathrm{Tocompute} E \left[ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {s}\right) \right] \mathrm{recallthat}} \\ & {\left(\mathbf {x} _ {t + 1} ^ {f} \otimes \mathbf {x} _ {t + 1} ^ {s}\right) = (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {s}\right) + \left(\mathbf {h} _ {\mathbf {x}} \otimes \tilde {\mathbf {H}} _ {\mathbf {x x}}\right) \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) + \left(\mathbf {h} _ {\mathbf {x}} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}\right) \mathbf {x} _ {t} ^ {f}} \\ & {\qquad + (\sigma \pmb {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) (\pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {s}) + (\sigma \pmb {\eta} \otimes \tilde {\mathbf {H}} _ {\mathbf {x x}}) (\pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) + (\sigma \pmb {\eta} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}) \pmb {\epsilon} _ {t + 1}} \end{array}\]

\[\begin{array}{l} \text {So} \\ E \left[ \left(\mathbf {x} _ {t + 1} ^ {f} \otimes \mathbf {x} _ {t + 1} ^ {s}\right) \right] = (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) E \left[ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {s}\right) \right] + \left(\mathbf {h} _ {\mathbf {x}} \otimes \tilde {\mathbf {H}} _ {\mathbf {x x}}\right) E \left[ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) \right] \\ \Updownarrow \end{array}\]

\[\left(\mathbf {I} _ {n _ {x} ^ {2}} - \left(\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}\right)\right) E \left[ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {s}\right) \right] = \left(\mathbf {h} _ {\mathbf {x}} \otimes \tilde {\mathbf {H}} _ {\mathbf {x x}}\right) E \left[ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) \right]\]

because is stationary

\[E \left[ \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {s} \right] = \left(\mathbf {I} _ {n _ {x} ^ {2}} - (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}})\right) ^ {- 1} \left(\mathbf {h} _ {\mathbf {x}} \otimes \tilde {\mathbf {H}} _ {\mathbf {x x}}\right) E \left[ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) \right]\]

Note that this term is zero if , which is the case if all third moments of are zero.

For the control variables, we have

\[\begin{array}{r l} & E \left[ \mathbf {y} _ {t} ^ {r d} \right] = \mathbf {g} _ {\mathbf {x}} \left(E \left[ \mathbf {x} _ {t} ^ {f} \right] + E \left[ \mathbf {x} _ {t} ^ {s} \right] + E \left[ \mathbf {x} _ {t} ^ {r d} \right]\right) + \tilde {\mathbf {G}} _ {\mathbf {x x}} \left(E \left[ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) \right] + 2 E \left[ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {s}\right) \right]\right) + \tilde {\mathbf {G}} _ {\mathbf {x x x}} E \left[ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) \right] \\ & + \frac {1}{2} \mathbf {g} _ {\sigma \sigma} \sigma^ {2} + \frac {3}{6} \mathbf {g} _ {\sigma \sigma \mathbf {x}} \sigma^ {2} E \left[ \mathbf {x} _ {t} ^ {f} \right] + \frac {1}{6} \mathbf {g} _ {\sigma \sigma \sigma} \sigma^ {3} \\ & = \mathbf {g} _ {\mathbf {x}} \left(E \left[ \mathbf {x} _ {t} ^ {s} \right] + E \left[ \mathbf {x} _ {t} ^ {r d} \right]\right) + \tilde {\mathbf {G}} _ {\mathbf {x x}} \left(E \left[ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {\mathrm{f}}\right) \right] + 2 E \left[ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {\mathrm{s}}\right) \right]\right) \\ & + \tilde {\mathbf {G}} _ {\mathbf {x x x}} E \left[ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {\mathrm{f}} \otimes \mathbf {x} _ {t} ^ {\mathrm{f}}\right) \right] + \frac {1}{2} \mathbf {g} _ {\sigma \sigma} \sigma^ {2} + \frac {1}{6} \mathbf {g} _ {\sigma \sigma \sigma} \sigma^ {3} \end{array}\]

Now recall, if all third moments of are zero, then and . Hence, we have the following

Corollary 1 The mean value in a third order approximation is identical to the mean value in a second order approximation if all third moments of are zero.

4.4.2 Second moments

We start by noticing that

\[\begin{array} { r l } & { \textit { V a r } \left( \mathbf { z } _ { t } \right) = E \left[ \left( \mathbf { z } _ { t } - E \left[ \mathbf { z } _ { t } \right] \right) \left( \mathbf { z } _ { t } - E \left[ \mathbf { z } _ { t } \right] \right) ^ { \prime } \right] } \\ & { \quad = E \left[ \left( \mathbf { z } _ { t } - E \left[ \mathbf { z } _ { t } \right] \right) \left( \mathbf { z } _ { t } ^ { \prime } - E \left[ \mathbf { z } _ { t } ^ { \prime } \right] \right) \right] } \\ & { \quad = E \left[ \mathbf { z } _ { t } \mathbf { z } _ { t } ^ { \prime } - \mathbf { z } _ { t } E \left[ \mathbf { z } _ { t } ^ { \prime } \right] - E \left[ \mathbf { z } _ { t } \right] \mathbf { z } _ { t } ^ { \prime } + E \left[ \mathbf { z } _ { t } \right] E \left[ \mathbf { z } _ { t } ^ { \prime } \right] \right] } \\ & { \quad = E \left[ \mathbf { z } _ { t } \mathbf { z } _ { t } ^ { \prime } \right] - E \left[ \mathbf { z } _ { t } \right] E \left[ \mathbf { z } _ { t } ^ { \prime } \right] } \\ & { \textnormal { a n d } } \\ & E \left[ \mathbf { z } _ { t } \mathbf { z } _ { t } ^ { \prime } \right] = E \left[ \left[ \begin{array} { c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c } & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & 1 \\ & & & & & & & & & & & & & & & & & & & & & 3 \\ & & & & 6 & 2 & 4 & 5 & 6 & 7 & 8 & 9 & 1 0 & 1 1 & 1 2 & 1 3 & 1 4 & 1 5 & 1 6 & 1 7 & 1 8 & 1 9 \\ 3 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 \\ 3 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 \\ 3 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 \\ 3 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 & 6 \\ e f f ( x , y ) = ( x , y ) ^ ( x , y ) / ( x , y ) / ( x , y ) ) / ( x , y ) ) ^ ( x , y ) / ( x , y ) / ( x , y ) ) / ( x , y ) ) / ( x , y ) ) ^ ( x , y ) / ( x , y ) / ( x , y ) ) / ( x , y ) ) / ( x , y ) ) ^ ( x , y ) / ( x , y ) / ( x , y ) / ( x , y ) ) / ( x , y ) ) / ( x , y ) ) ^ ( x , y ) / ( x , y ) / ( x , y ) / ( x , y ) / ( x , y ) / ( x , y ) / ( x , y ) / ( x , y ) / ( x , y ) / ( x , y ) / ( x , y ) / ( x , y ) / ( x , y ) / ( x , y ) / ( x , y ) / ( x , y ) / ( x , y ) / ( x , x ) / ( x , x ) / ( x , x ) / ( x , x ) / ( x , x ) / ( x , x ) / ( x , x ) / ( x , x ) / ( x , x ) / ( x , x ) / ( x , x ) / ( x , x ) / ( x , x ) / ( x , x ) / ( x , x ) / ( x , x ) / ( x , x ) / ( u , u ) ^ ( u , u ) / ( u , u ) / ( u , u ) / ( u , u ) / ( u , u ) / ( u , u ) / ( u , u ) / ( u , u ) / ( u , u ) / ( u , u ) / ( u , u ) / ( u , u ) / ( u , u ) / ( u , u ) / ( u , u ) / ( u , u ) / ( u , u ) / ( u , v ) ^ ( u , v ) / ( u , v ) / ( u , v ) / ( u , v ) / ( u , v ) / ( u , v ) / ( u , v ) / ( u , v ) / ( u , v ) / ( u , v ) / ( u , v ) / ( u , v ) / ( u , v ) ^ ( u , v ) / ( u , v ) / ( u , v ) / ( u , v ) / ( u , v ) / ( u , v ) / ( u , v ) / ( u , v ) ^ ( u , v ) / ( u , v ) / ( u , v ) / ( u , v ) / ( u , v ) ^ ( u , v ) / ( u , v ) / ( u , v ) / ( u , v ) ^ ( u , v ) / ( u , v ) / ( u , v ) ^ ( u , v ) / ( u , v ) / ( u , v ) ^ ( u , v ) / ( u , v ) ^ ( u , v ) / ( u , v ) ^ ( u , v ) / ( u , v ) ^ ( u , v ) / ( u , v ) ^ ( u , v ) / ( u , v ) ^ ( u , v ) / ( u , v ) ^ ( u , v ) / ( u , v ) ^ ( u , v ) / ( u . v . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s .s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . s . t h r m a l o n d i n g e r e f f i n g e r e f f i n g e r e f f i n g e r e f f i n g e r e f f i n g e r e f f i n g e r e f f i n g e r e f f i n g e r e f f i n g e r e f f i n g e r e f f i n g e r e f f i n g e r e f f i n g f i n g e r e f f i n g e r e f f i n g e r e f f i n g e r e f f i n g e r e f f i n g e r e f f i n g e r e f f i n g e r e f f i n g e r e f f i n g e r e f f i n g e r e f f i n g e r e f f i ng e r e f f i n g e r e f f i n g e r e f f i n g e r e f f i n g e r e f f i n g e r e f f i n g e r e f f i n g e r e f f i n g e r e f f i n g e r e f f i n g e r e f f i n g e r e f f i n g e r e l o w i n g i n g i n g i n g i n g i n g i n g i n g i n g i n g i n g i n g i n g i n g i n g i n g i n g i n g i n g i n g i n g i n g i n g i n g i n g i n g i n g i n g i n g i n g i n g i n g i n g i n g j o l l o w i n g j o l l o w i n g j o l l o w i n g j o l l o w i n g j o l l o w i n g j o l l o w i n g j o l l o w i n g j o l l o w i n g j o l l o w i n g j o l l o w i n g j o l l o w i n g j o l l o w j o l l o w j o l l o w j o l l o w j o l l o w j o l l o w j o l l o w j o l l o w j o l l o w j o l l o w j o l l o w j o l l o w j o l l o w j o l l o w j o l l o w j o l l o w j o l l o w j o l | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | I = I = I = I = I = I = I = I = I = I = I = I = I = I = I = I = I = I = I = I = I = I = I = I = I = I = I = I = I = I = I = I = I = I = I = I = I = I = I = I = I = I = I = I = I = I = I = I = I = I = I * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U *U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * U * \\ [ b ] = [ b ] + [ b ] + [ b ] + [ b ] + [ b ] + [ b ] + [ b ] + [ b ] + [ b ] + [ b ] + [ b ] + [ b ] + [ b ] + [ b ] + [ b ] + [ b ] + [ b ] + [ b ] + [ b ] + [ b ] + [ b ] + [ b ] + [ b ] + [ b ] + [ b ] + [ b ] - [ b ] + [ b ] + [ b ] + [ b ] + [ b ] + [ b ] + [ b ] + [ b ] + [ b ] + [ b ] + [ b ] + [ b ] + [ b ] + [ b ] + [ b ] + [ b ] + [ b ] + [ b ] - [ b ] + [ b ] + [ b ] + [ b ] + [ b ] + [ b ] + [ b ] - [ b ] + [ b ] + [ b ] + [ b ] + [ b ] + [ b ] + [ b ] - [ b ] + [ b ] + [ b ] + [ b ] + [ b ] + [ b ] + [ b ] - [ b ] + [ b ] + [ b ] + [ b ] - [ b ] + [ b ] + [ b ] - [ b ] + [ b ] + [ b ] - [ b ] + [ b ] - [ b ] + [ b ] - [ b ] - [ b ] - [ b ] - [ b ] - [ b ] - [ b ] - [ b ] - [ b ] - [ b ] - [ b ] - [ b ] - [ b ] - [ b ] - [ b ] - [ b ] - [ b ] - [ b ] - [ b ] - [ b ] - [ b ] - [ b ] - [ b ] - [ b ] - [ b ] - [ b ]- [ k _ { p } ^ { p } ; k _ { q } ^ { p } ; k _ { r } ^ { p } ; k _ { t } ^ { p } ; k _ { q } ^ { p } ; k _ { r } ^ { p } ; k _ { t } ^ { p } ; k _ { q } ^ { p } ; k _ { t } ^ { p } ; k _ { q } ^ { p } ; k _ { t } ^ { p } ; k _ { q } ^ { p } ; k _ { t } ^ { p } ; k _ { q } ^ { p } ; k _ { t } ^ { p } ; k _ { q } ^ { p } ; k _ { t } ^ { p } ; k _ { t } ^ { p } ; k _ { q } ^ { p } ; k _ { t } ^ { p } ; k _ { q } ^ { p } ; k _ { t } ^ { p } ; k _ { q } ^ { p } ; k _ { t } ^ { p } ; k _ { q } ^ { p } ; k _ { t } ^ { p } ; k _ Q R S T a m a t h a m a t h a m a t h a m a t h a m a t h a m a t h a m a t h a m a t h a m a t h a m a t h a m a t h a m a t h a m a t h a m a t h a m a t h a m a t h a m a t h a m a t h a m a t h a m a t h a m a t t h a m a t t h a m a t t h a m a t t h a m a t t h a m a t t h a m a t t h a m a t t h a m a t t h a m a t t h a m a t t h a m a t t h a m a t t h a m a t t h a m a t t h a m a t t h a m a t t h am a t t h a m a t t h a m a t t h a m a t t h a m a t t h a m a t t h a m a t t h a m a t t h a m a t t h a m a t t h a m a t t h a m a t t h a m a t t h a m a t t h a m a t t h a m a t t h a m a t t H A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W AW A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A W A N M O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N NO N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O N N O NN O NN O NN ONINN ONINN ONINN ONINN ONINN ONINN ONINN ONINN ONINN ONINN ONINN ONINN ONINN ONINN ONINN ONINN ONINN ONINN ONINN ONINN ONINN ONINN ONINN ONINN ONINN ONISINN ONISINN ONISINN ONISINN ONISINN ONISINN ONISINN ONISINN ONISINN ONISINN ONISINN ONISINN ONISINN ONISINN ONISINN ONISINN ONISINN ONISINN ONISINN ONISINN ON ISINN S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S F S< fcel>I have an example of the complex number representation in the complex number representation in the complex number representation in the complex number representation in the complex number representation in the complex number representation in the complex number representation in the complex number representation in the complex number representation in the complex number representation in the complex number representation in the complex number representation in the complex number representation in the complex number representation in the complex number representation in the complex number representation in the complex number representation in the complex number representation in the complex number representation in the complex number representation in the complex number representation as follows:< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< lcel>< nl>\]

\[\left. \begin{array}{c c} {\bf x} _ {t} ^ {f} \left({\bf x} _ {t} ^ {f} \otimes {\bf x} _ {t} ^ {s}\right) ^ {\prime} & {\bf x} _ {t} ^ {f} \left({\bf x} _ {t} ^ {f} \otimes {\bf x} _ {t} ^ {f} \otimes {\bf x} _ {t} ^ {f}\right) ^ {\prime} \\ {\bf x} _ {t} ^ {s} \left({\bf x} _ {t} ^ {f} \otimes {\bf x} _ {t} ^ {s}\right) ^ {\prime} & {\bf x} _ {t} ^ {s} \left({\bf x} _ {t} ^ {f} \otimes {\bf x} _ {t} ^ {f} \otimes {\bf x} _ {t} ^ {f}\right) ^ {\prime} \\ \left({\bf x} _ {t} ^ {f} \otimes {\bf x} _ {t} ^ {f}\right) \left({\bf x} _ {t} ^ {f} \otimes {\bf x} _ {t} ^ {s}\right) ^ {\prime} & \left({\bf x} _ {t} ^ {f} \otimes {\bf x} _ {t} ^ {f}\right) \left({\bf x} _ {t} ^ {f} \otimes {\bf x} _ {t} ^ {f} \otimes {\bf x} _ {t} ^ {f}\right) ^ {\prime} \\ {\bf x} _ {t} ^ {r d} \left({\bf x} _ {t} ^ {f} \otimes {\bf x} _ {t} ^ {s}\right) ^ {\prime} & {\bf x} _ {t} ^ {r d} \left({\bf x} _ {t} ^ {f} \otimes {\bf x} _ {t} ^ {f} \otimes {\bf x} _ {t} ^ {f}\right) ^ {\prime} \\ \left({\bf x} _ {t} ^ {f} \otimes {\bf x} _ {t} ^ {s}\right) \left({\bf x} _ {t} ^ {f} \otimes {\bf x} _ {t} ^ {s}\right) ^ {\prime} & \left({\bf x} _ {t} ^ {f} \otimes {\bf x} _ {t} ^ {s}\right) \left({\bf x} _ {t} ^ {f} \otimes {\bf x} _ {t} ^ {f} \otimes {\bf x} _ {t} ^ {f}\right) ^ {\prime} \\ \left({\bf x} _ {t} ^ {f} \otimes {\bf x} _ {t} ^ {f} \otimes {\bf x} _ {t} ^ {f}\right) \left({\bf x} _ {t} ^ {f} \otimes {\bf x} _ {t} ^ {s}\right) ^ {\prime} & \left({\bf x} _ {t} ^ {f} \otimes {\bf x} _ {t} ^ {f} \otimes {\bf x} _ {t} ^ {f}\right) \left({\bf x} _ {t} ^ {f} \otimes {\bf x} _ {t} ^ {f} \otimes {\bf x} _ {t} ^ {f}\right) ^ {\prime} \end{array} \right]\]

Hence, we need to find the following terms:

\[\begin{array}{r l} & {\mathrm{Hence,we} \mathbf {x} _ {t} ^ {f} \left(\mathbf {x} _ {t} ^ {r d}\right) ^ {\prime}, \mathbf {x} _ {t} ^ {s} \left(\mathbf {x} _ {t} ^ {r d}\right) ^ {\prime}, \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\mathbf {x} _ {t} ^ {r d}\right) ^ {\prime}, \mathbf {x} _ {t} ^ {r d} \left(\mathbf {x} _ {t} ^ {r d}\right) ^ {\prime}, \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {s}\right) \left(\mathbf {x} _ {t} ^ {r d}\right) ^ {\prime}, \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\mathbf {x} _ {t} ^ {r d}\right) ^ {\prime}} \\ & {- \mathbf {x} _ {t} ^ {f} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {s}\right) ^ {\prime}, \mathbf {x} _ {t} ^ {s} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {s}\right) ^ {\prime}, \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {s}\right) ^ {\prime}, \mathbf {x} _ {t} ^ {r d} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {s}\right) ^ {\prime}, \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {s}\right) \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {s}\right) ^ {\prime}, \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {s}\right) ^ {\prime}} \\ & {- \mathbf {x} _ {t} ^ {f} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime}, \mathbf {x} _ {t} ^ {s} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime}, \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime}, \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right)} \end{array}\]

All these terms are easy to compute using the procedure outlined above and previous results.

4.5 The auto-correlations

This section derives the auto-correlations for the states and the control variables.

4.5.1 The innovations

We first show that for To see this recall that

\[\begin{array} { r l } E \left[ \pmb { \xi } _ { t + 1 } \pmb { \xi } _ { t + 1 + s } ^ { \prime } \right] = E & { } \left[ \left[ \begin{array} { c } \pmb { \epsilon } _ { t + 1 } \\ \pmb { \epsilon } _ { t + 1 } \otimes \pmb { \epsilon } _ { t + 1 } - v e c \left( \mathbf { I } _ { n _ { e } } \right) \\ \pmb { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \\ \mathbf { x } _ { t } ^ { f } \otimes \pmb { \epsilon } _ { t + 1 } \\ \pmb { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { s } \\ \pmb { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \\ \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \otimes \pmb { \epsilon } _ { t + 1 } \\ \mathbf { x } _ { t } ^ { f } \otimes \pmb { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \\ \mathbf { x } _ { t } ^ { f } \otimes \pmb { \epsilon } _ { t + 1 } \otimes \pmb { \epsilon } _ { t + 1 } \\ \pmb { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \otimes \pmb { \epsilon } _ { t + 1 } \\ \pmb { \epsilon } _ { t + 1 } \otimes \pmb { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \\ ( \pmb { \epsilon } _ { t + 1 } \otimes \pmb { \epsilon } _ { t + 1 } \otimes \pmb { \epsilon } _ { t + 1 } ) - E [ ( \pmb { \epsilon } _ { t + 1 } \otimes \pmb { \epsilon } _ { t + 1 } \otimes \pmb { \epsilon } _ { t + 1 } ) ] \\ \end{array} \right] \\ & \times \left[ \begin{array} { c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c } \\ \pmb { \epsilon } _ { t + 1 + s } ^ { \prime } & ( \pmb { \epsilon } _ { t + 1 + s } \otimes \pmb { \epsilon } _ { t + 1 + s } - v e c ( \mathbf { I } _ { n _ { e} } ) ) ^ { \prime } & ( \pmb { \epsilon } _ { t + 1 + s } \otimes \mathbf { x } _ { t + s } ^ { f } ) ^ { \prime } \\ ( \mathbf { x } _ { t + s } ^ { f } \otimes \pmb { \epsilon } _ { t + 1 + s } ) ^ { \prime } & ( \pmb { \epsilon } _ { t + 1 + s } \otimes \mathbf { x } _ { t + s } ^ { s } ) ^ { \prime } & ( \pmb { \epsilon } _ { t + 1 + s } \otimes \mathbf { x } _ { t + s } ^ { f } \otimes \mathbf { x } _ { t + s } ^ { f } ) ^ { \prime } \\ ( \mathbf { x } _ { t + s } ^ { f } \otimes \mathbf { x } _ { t + s } ^ { f } \otimes \pmb { \epsilon } _ { t + 1 + s } ) ^ { \prime } & ( \mathbf { x } _ { t + s } ^ { f } \otimes \pmb { \epsilon } _ { t + 1 + s } \otimes \mathbf { x } _ { t + s } ^ { f } ) ^ { \prime } & ( \mathbf { x } _ { t + s } ^ { f } \otimes \pmb { \epsilon } _ { t + 1 + s } \otimes \pmb { \epsilon } _ { t + 1 + s } ) ^ { \prime } & ( \pmb { \epsilon } _ { t + 1 + s } \otimes \mathbf { x } _ { t + s } ^ { f } \otimes \pmb { \epsilon } _ { t + 1 + s } ) ^ { \prime } \\ ( \pmb { \epsilon } _ { t + 1 + s } \otimes \pmb { \epsilon } _ { t + 1 + s } \otimes \mathbf { x } _ { t + s } ^ { f } ) ^ { \prime } & ( ( ( \pmb { \epsilon } _ { t + 1 + s } \otimes \pmb { \epsilon } _ { t + 1 + s } \otimes \pmb { \epsilon } _ { t + 1 + s} ) - E [ ( ( \pmb { \epsilon } _ { t + 1 } \otimes \pmb { \epsilon } _ { t + 1 } \otimes ( | | | | ) ) ^ { ' } ] & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & \\ & . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .< fcel>. e m a d i o n g e n d i o n g e n d i o n g e n d i o n g e n d i o n g e n d i o n g e n d i o n g e n d i o n g e n d i o n g e n d i o n g e n d i o n g e n d i o n g e n d i o n g e n d i o n g e n d i o n g e n d e q u a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | C o r r o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o wh o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h o w h z e r r e r r e r r e r r e r r e r r e r r e r r e r r e r r e r r e r r e r r e r r e r r e r r e r r e r r e r r e r r e r r e r r e r r e r r e r r e r r e r r e r r e r r e r r e r r e r r e r r e r r u p p u a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l k , k , m , n , m , n , m , n , m , m , m , m , m , m , m , m , m , m , m , m , m , m , m , m , m , m , m , m , m , m , m , m , m , m , m , m , m , m , m , m , m , m , m , m , m , m , m , m , m , m , m , m , m , m , m , m , m , m ,m , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n ,n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n ) = - 2 5 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8, \\ = - 2, b i j k u i d i v e d i v e d i v e d i v e d i v e d i v e d i v e d i v e d i v e d i v e d i v e d i v e d i v e d i v e d i v e d i v e d i v e d i v e d i v e d i v e d i v e d i v e d i v e d i v e d i v e d i v i d u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u s u t a p p u a l k u i d i v e d i v e d i v e d i v e d i v e d i v e d i v e d i v e d i v e d i v e d i v e d i v e d i v e d i v e d i v e d i v e d i v e d i v e d i v e d i v e d i v e d i v e d i v e d i v e f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / q / q / q / q / q / q / q / q / q / q / q / q / q / q / q / q / q / q / q / q / q / q / q / q / q / q / q / q / q / q / q / q / q / q / q / q / q / q / q / q / q / q / q / q / q / q / q / q / q / q / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / p / q : \\ = - N M A B S A B S A B S A B S A B S A B S A B S A B S A B S A B S A B S A B S A B S A B S A B S A B S A B S A B S A B S A B S A B S A B S A B S A B S A B S A B S A B S A B S A B S A B S A B S A B S A B S A B S S A B S A B S A B S A B S A B S A B S A B S A B S A B S A B S A B S A B S A B S A B S A B S A B S A B S A B S A B S A B S A B S A B S A B S A B S A B S A B S A B S A B S A B S A B S A B S A B S A B T U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U R U U R U R U U R U U R U U R U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I II II III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< ecel>< nl>\]

We now inspect each of the rows in turn. Here, we need the following result that

\[\mathbf {x} _ {t + 1} ^ {f} = \mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t} ^ {f} + \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1}\]

\[\begin{array}{r l} & {\mathbf {x} _ {t + 2} ^ {f} = \mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t + 1} ^ {f} + \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 2}} \\ & {\qquad = \mathbf {h} _ {\mathbf {x}} \left(\mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t} ^ {f} + \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1}\right) + \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 2}} \\ & {\qquad = \mathbf {h} _ {\mathbf {x}} ^ {2} \mathbf {x} _ {t} ^ {f} + \mathbf {h} _ {\mathbf {x}} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} + \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 2}} \\ & {\dots} \\ & {\mathbf {x} _ {t + s} ^ {f} = \mathbf {h} _ {\mathbf {x}} ^ {s} \mathbf {x} _ {t} ^ {f} + \sum_ {i = 1} ^ {s} \mathbf {h} _ {\mathbf {x}} ^ {s - i} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + i}} \end{array}\]

1) Row with

Consider the sub-matrix

\[\begin{array}{l} E \{\boldsymbol {\epsilon} _ {t + 1} \left[ \begin{array}{c c} \boldsymbol {\epsilon} _ {t + 1 + s} ^ {\prime} & (\boldsymbol {\epsilon} _ {t + 1 + s} \otimes \boldsymbol {\epsilon} _ {t + 1 + s} - v e c (\mathbf {I} _ {n _ {e}})) ^ {\prime} \\ & (\boldsymbol {\epsilon} _ {t + 1 + s} \otimes \mathbf {x} _ {t + s} ^ {f}) ^ {\prime} \end{array} \right. \\ \left(\mathbf {x} _ {t + s} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1 + s}\right) ^ {\prime} \quad \left(\boldsymbol {\epsilon} _ {t + 1 + s} \otimes \mathbf {x} _ {t + s} ^ {s}\right) ^ {\prime} \quad \left(\boldsymbol {\epsilon} _ {t + 1 + s} \otimes \mathbf {x} _ {t + s} ^ {f} \otimes \mathbf {x} _ {t + s} ^ {f}\right) ^ {\prime} \\ \left(\mathbf {x} _ {t + s} ^ {f} \otimes \mathbf {x} _ {t + s} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1 + s}\right) ^ {\prime} \quad \left(\mathbf {x} _ {t + s} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1 + s} \otimes \mathbf {x} _ {t + s} ^ {f}\right) ^ {\prime} \quad \left(\mathbf {x} _ {t + s} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1 + s} \otimes \boldsymbol {\epsilon} _ {t + 1 + s}\right) ^ {\prime} \\ \left(\boldsymbol {\epsilon} _ {t + 1 + s} \otimes \boldsymbol {\epsilon} _ {t + 1 + s} \otimes \mathbf {x} _ {t + s} ^ {f}\right) ^ {\prime} \quad ((\boldsymbol {\epsilon} _ {t + 1 + s} \otimes \boldsymbol {\epsilon} _ {t + 1 + s} \otimes \boldsymbol {\epsilon} _ {t + 1 + s}) - E [ (\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}) ]) ^ {\prime} \\ ] \rbrace \\ = E \{\boldsymbol {\epsilon} _ {t + 1} \left[ \begin{array}{c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c} 0 & 0 & 0 & 0 & 0 & 0 & 0 ^ {\prime} \\ 0 & 0 & (\mathbf {x} _ {t + s} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1 + s} \otimes \boldsymbol {\epsilon} _ {t + 1 + s}) ^ {\prime} & (\boldsymbol {\epsilon} _ {t + 1 + s} \otimes \mathbf {x} _ {t + s} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1 + s}) ^ {\prime} \\ (\boldsymbol {\epsilon} _ {t + 1 + s} \otimes \boldsymbol {\epsilon} _ {t + 1 + s} \otimes \mathbf {x} _ {t + s} ^ {f}) ^ {\prime} & 0 & ] \rbrace \end{array} \right. \end{array}\]

Hence, we only need to study the term of the form

\[\begin{array}{r l} & E \left[ \boldsymbol {\epsilon} _ {t + 1} \left(\mathbf {x} _ {t + s} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1 + s} \otimes \boldsymbol {\epsilon} _ {t + 1 + s}\right) ^ {\prime} \right] \\ & \quad = E \left[ \boldsymbol {\epsilon} _ {t + 1} \left(\mathbf {h} _ {\mathbf {x}} ^ {s} \mathbf {x} _ {t} ^ {f} + \sum_ {i = 1} ^ {s} \mathbf {h} _ {\mathbf {x}} ^ {s - i} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + i} \otimes \boldsymbol {\epsilon} _ {t + 1 + s} \otimes \boldsymbol {\epsilon} _ {t + 1 + s}\right) ^ {\prime} \right] \\ & \quad = E \left[ \boldsymbol {\epsilon} _ {t + 1} \left(\mathbf {h} _ {\mathbf {x}} ^ {s} \mathbf {x} _ {t} ^ {d} \otimes \boldsymbol {\epsilon} _ {t + 1 + s} \otimes \boldsymbol {\epsilon} _ {t + 1 + s} + \sum_ {i = 1} ^ {s} \mathbf {h} _ {\mathbf {x}} ^ {s - i} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + i} \otimes \boldsymbol {\epsilon} _ {t + 1 + s} \otimes \boldsymbol {\epsilon} _ {t + 1 + s}\right) ^ {\prime} \right] = \\ & \quad = E \left[ \boldsymbol {\epsilon} _ {t + 1} \left(\mathbf {h} _ {\mathbf {x}} ^ {s} \mathbf {x} _ {t} ^ {d} \otimes \boldsymbol {\epsilon} _ {t + 1 + s} \otimes \boldsymbol {\epsilon} _ {t + 1 + s}\right) ^ {\prime} \right] \\ & \quad + E \left[ \boldsymbol {\epsilon} _ {t + 1} \left(\sum_ {i = 1} ^ {s} \mathbf {h} _ {\mathbf {x}} ^ {s - i} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + i} \otimes \boldsymbol {\epsilon} _ {t + 1 + s} \otimes \boldsymbol {\epsilon} _ {t + 1 + s}\right) ^ {\prime} \right] \\ & \qquad = 0 + E \left[ \boldsymbol {\epsilon} _ {t + 1} \left(\sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1 + s} \otimes \boldsymbol {\epsilon} _ {t + 1 + s}\right) ^ {\prime} \right] \\ & \qquad = E \left[ (\boldsymbol {\epsilon} _ {t + 1} \otimes 1) (\boldsymbol {\epsilon^ {\prime}} _ {t + 1} \boldsymbol {\eta^ {\prime}} \sigma \otimes (\boldsymbol {\epsilon_ {t + 1 + s}} \otimes \boldsymbol {\epsilon_ {t + 1 + s}}) ^ {\prime}) \right] \\ & \qquad = E [ (\boldsymbol {\epsilon_ {t + 1}} \otimes 1) (\boldsymbol {\epsilon^ {\prime}} _ {t + 1} \boldsymbol {\eta^ {\prime}} \sigma \otimes (\boldsymbol {\epsilon_ {t + 1 + s}} \otimes \boldsymbol {\epsilon_ {t + 1 + s}}) ^ {\prime}) ] \\ & = E [ (\boldsymbol {\epsilon_ {t + 1}} \boldsymbol {\epsilon^ {\prime}} _ {t + 1} \boldsymbol {\eta^ {\prime}} \sigma \otimes (\boldsymbol {\epsilon_ {t + 1 + s}} \otimes \boldsymbol {\epsilon_ {t + 1 + s}}) ^ {\prime}) ] \\ & = I _ {n _ {e}} \boldsymbol {\eta^ {\prime}} \sigma \otimes v e c (I _ {n _ {e}}) ^ {\prime} \\ & = E [ (\boldsymbol {\epsilon_ {t + 1}}) (\boldsymbol {\eta^ {\prime}} _ {t + 1}) ^ {- 1}, (\boldsymbol {\eta^ {\prime}} _ {t + 1}) (\boldsymbol {\eta^ {\prime}} _ {t + 1}) ] = E [ (\boldsymbol {\eta^ {\prime}} _ {t + 1}) (\boldsymbol {\eta^ {\prime}} _ {t + 1}) ] = E [ (\boldsymbol {\eta^ {\prime}} _ {t + 1}) (\boldsymbol {\eta^ {\prime}} _ {t + 1}) ] = E [ (\boldsymbol {\eta^ {\prime}} _ {t + 1}) (\boldsymbol {\eta^ {\prime}} _ {T}) ] = E [ (\boldsymbol {\eta^ {\prime}} _ {t + 1}) (\boldsymbol {\eta^ {\prime}} _ {T}) ] = E [ (\boldsymbol {\eta^ {\prime}} _ {T}) (\boldsymbol {\eta^ {\prime}} _ {T}) ] = E [ (\boldsymbol {\eta^ {\prime}} _ {T}) (\boldsymbol {\eta^ {\prime}} _ {T}) ] = E [ (\boldsymbol {\eta^ {\prime}} _ {T}) (\boldsymbol {\eta^ {\prime}} _ {T}) ] = E [ (\boldsymbol {\eta^ {\prime}} _ {T}) (\boldsymbol {\eta^ {\prime}}, t) ] = E [ (\boldsymbol {\eta^ {\prime}} _ {T}) (\boldsymbol {\eta^ {\prime}}, t) ] = E [ (\boldsymbol {\eta^ {\prime}} _ {T}) (\boldsymbol {\eta^ {\prime}}, t) ] = E [ (\boldsymbol {\eta^ {\prime}} _ {T}) (\boldsymbol {\eta^ {\prime}}, t) ] = E [ (\boldsymbol {\eta^ {\prime}} _ {S}) (\boldsymbol {\eta^ {\prime}}, t) ] = E [ (\boldsymbol {\eta^ {\prime}} _ {S}) (\boldsymbol {\eta^ {\prime}}, t) ] = E [ (\boldsymbol {\eta^ {\prime}} _ {S}) (\boldsymbol {\eta^ {\prime}}, t) ] = E [ (\boldsymbol {\eta^ {\prime}} _ {S}) (\boldsymbol {\eta^ {\prime}}, t) ] = E < T, \\ & = I _ {n _ {e}} I _ {n _ {e}} I _ {n _ {e}} I _ {n _ {e}} I _ {n _ {e}} I _ {n _ {e}} I _ {n _ {e}} I _ {n _ {e}} I _ {n _ {e}} I _ {n _ {e}} I _ {n _ {e}} I _ {n _ {e}} I _ {n _ {e}} I _ {N}. \\ & = I _ {(2)} I _ {(2)} I _ {(2)} I _ {(2)} I _ {(2)} I _ {(2)} I _ {(2)} I _ {(2)} I _ {(2)} I _ {(2)} I _ {(2)} I _ {(2)} I _ {(2)} I _ {(2)} I _ {(2)} I _ {(2)} I _ {(2)} I _ {(2)} I _ {(2)} I _ {(2)} I _ {(2)}, \\ & = I (I, N), I (N), I (N), I (N), I (N), I (N), I (N), I (N), I (N), I (N), I (N), I (N), I (N), I (N), I (N), I (N), I (N), I (N), I (N), I (N). \\ & = I (I, N), I (N), I (N), I (N), I (N), I (N), I (N), I (N), I (N), I (N), I (N), I (N), I (N), I (N), I (N). \\ & = I (I, N), I (N), I (N), I (N), I (N), I (N), I (N), I (N), I (N), I(3). \\ & = I (I, N), I (N), I (N), I (N), I (N), I (N), I (N), I (N), I (N), I (N), I (N), I (N). \\ & = I (I, N), I (N), I (N), I (N), I (N), I (N), I (N), I (N), I (N). \\ & = I (I, N), I (N), I (N), I (N), I (N), I (N), I (N), I (N). \\ & = I (I, N), I (N), I (N), I (N), I (N), I (N), I (N). \\ & = I (I, N), I (N), I (N), I (N), I (N), I (N). \\ & = I (I, N), I (N), I (N), I (N), I (N), I (N). \\ & = I (I, N), I (N), I (N), I (N), I (N). \\ & = I (I, N), I (N), I (N), I (N), I (N). \\ & = I (I, N), I (N), I (N), I (N). \\ & = I (I, N), I (N), I (N), I (N). \\ & = I (I, N), I (N), I (N), I (N). \\ & = I (I, N), I (N), I (N), I (N). \\ & = I (I, N), I (N), I (N), I (N). \\ & . \\ & = E [ (\alpha_ {- 1}), A ], \\ & . \\ & = E [ (\alpha_ {- 1}), B ], \\ & . \\ & = E [ (\alpha_ {- 1}), C ], \\ & . \\ & = E [ (\alpha_ {- 1}), D ], \\ & . \\ & = E [ (\alpha_ {- 1}), E ], \\ & . \\ & = E [ (\alpha_ {- 1}), F ], \\ & . \\ & = E [ (\alpha_ {- 1}), G ], \\ & . \\ & = E [ (\alpha_ {- 1}), H ], \\ & . \\ & = E [ (\alpha_ {- 1}), J ], \\ & . \\ & = E [ (\alpha_ {- 1}), K ], \\ & . \\ & = E [ (\alpha_ {- 1}), L ], \\ & . \\ & = E [ (\alpha_ {- 1}), M ], \\ & . \\ & = E [ (\alpha_ {- 1}), N ], \\ & . \\ & = E [ (\alpha_ {- 1}), O ], \\ & . \\ & = E [ (\alpha_ {- 1}), P ], \\ & . \\ & = E [ (\alpha_ {- 1}), Q ], \\ & . \\ & = E [ (\alpha_ {- 1}), R ], \\ & . \\ & = E [ (\alpha_ {- 1}), S ], \\ & . \\ & = E [ (\alpha_ {- 1}), T ], \\ & . \\ & = E [ (\alpha_ {- 1}), U ], \\ & . \\ & = E [ (\alpha_ {- 1}), V ], \\ & . \\ & = E [ (\alpha_ {- 1}), W ], \\ & . \\ & = E [ (\alpha_ {- 1}), X ], \\ & . \\ & = E [ (\alpha_ {- 1}), Y ], \\ & . \\ & = E [ (\alpha_ {- 1}), Z ], \\ & . \\ & = E [ (\alpha_ {- 1}), Z ], \\ & . \\ & = E [ (\alpha_ {- 1}), Z ], \\ & . \\ & = E [ (\alpha_ {- 1}), Z ], \\ & . \\ & = E [ (\alpha_ {- 1}), Z ], \\ & . \\ & = E [ (\alpha_ {- 1}), Z ], \\ & . \\ & = E [ (\alpha_ {- 2}), A ], \\ & . \\ & = E [ (\alpha_ {- 2}), A ], \\ & . \\ & = E [ (\alpha_ {- 2}), A ], \\ & . \\ & = E [ (\alpha_ {- 2}), A ], \\ & . \\ & = E [ (\alpha_ {- 2}), A ], \\ & . \\ & = E [ (\alpha_ {- 2}), A ], \\ & . \\ & = E [ (\alpha_ {- 3}), A ], \\ & . \\ & = E [ (\alpha_ {- 3}), A ], \\ & . \\ & = E [ (\alpha_ {- 3}), A ], \\ & . \\ & = E [ (\alpha_ {- 3}), A ], \\ & . \\ & = E [ (\alpha_ {- 3}), A ], \\ & . \\ & = E [ (\alpha_ {- 3}), A ], \\ & . \\ & = E [ (\alpha_ {{\mathrm{的}}})), A ]. \\ & . \\ & = E [ (\alpha_ {{\mathrm{的}}})), A ]. \\ & .; |, |. | > |. | > |. | > |. | > |. | > |. | > |. | > |. | > |. | > |. | > |. | > |. | > |. | > |. | > |. | > |. | > |. | > |. | > |. | > |. | > |. | > |. | > |. | > |. | > |. | > |. | < |>\]

2)

To be completed

4.5.2 The covariances

Recall that we have

\[\begin{array}{r l} & {\mathbf {z} _ {t} = \left[ \begin{array}{c} \mathbf {x} _ {t} ^ {f} \\ \mathbf {x} _ {t} ^ {s} \\ \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \\ \mathbf {x} _ {t} ^ {r d} \\ \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {s} \\ \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \end{array} \right]} \\ & {\mathbf {z} _ {t + 1} = \mathbf {c} + \mathbf {A z} _ {t} + \mathbf {B} \pmb {\xi} _ {t + 1}} \\ & {_ t ^ {r d} = \mathbf {D z} _ {t} + \frac {1}{2} \mathbf {g} _ {\sigma \sigma} \sigma^ {2} + \frac {1}{6} \mathbf {g} _ {\sigma \sigma \sigma} \sigma^ {3}} \end{array}\]

To find the one period auto-covariances, i.e. , we have

\[\begin{array}{l} C o v \left(\mathbf {z} _ {t + 1}, \mathbf {z} _ {t}\right) \\ = C o v \left(\mathbf {c} + \mathbf {A} \mathbf {z} _ {t} + \mathbf {B} \boldsymbol {\xi} _ {t + 1}, \mathbf {z} _ {t}\right) \\ = \mathbf {A} C o v \left(\mathbf {z} _ {t}, \mathbf {z} _ {t}\right) + \mathbf {B} C o v \left(\boldsymbol {\xi} _ {t + 1}, \mathbf {z} _ {t}\right) \end{array}\]

And for two periods

\[\begin{array}{r l} & {C o v \left(\mathbf {z} _ {t + 2}, \mathbf {z} _ {t}\right) = C o v \left(\mathbf {c} + \mathbf {A} \mathbf {z} _ {t + 1} + \mathbf {B} \pmb {\xi} _ {t + 2}, \mathbf {z} _ {t}\right)} \\ & {\quad = C o v \left(\mathbf {c} + \mathbf {A} \left(\mathbf {c} + \mathbf {A} \mathbf {z} _ {t} + \mathbf {B} \pmb {\xi} _ {t + 1}\right) + \mathbf {B} \pmb {\xi} _ {t + 2}, \mathbf {z} _ {t}\right)} \\ & {\quad = C o v \left(\mathbf {c} + \mathbf {A} \mathbf {c} + \mathbf {A} ^ {2} \mathbf {z} _ {t} + \mathbf {A B} \pmb {\xi} _ {t + 1} + \mathbf {B} \pmb {\xi} _ {t + 2}, \mathbf {z} _ {t}\right)} \\ & {\quad = C o v \left(\mathbf {A} ^ {2} \mathbf {z} _ {t}, \mathbf {z} _ {t}\right) + C o v \left(\mathbf {A B} \pmb {\xi} _ {t + 1}, \mathbf {z} _ {t}\right) + C o v \left(\mathbf {B} \pmb {\xi} _ {t + 2}, \mathbf {z} _ {t}\right)} \\ & {\quad = \mathbf {A} ^ {2} C o v \left(\mathbf {z} _ {t}, \mathbf {z} _ {t}\right) + \mathbf {A B} C o v \left(\pmb {\xi} _ {t + 1}, \mathbf {z} _ {t}\right) + \mathbf {B} C o v \left(\pmb {\xi} _ {t + 2}, \mathbf {z} _ {t}\right)} \end{array}\]

\[\begin{array}{r l} & C o v \left(\mathbf {z} _ {t + 2}, \mathbf {z} _ {t}\right) = \mathbf {A} C o v \left(\mathbf {z} _ {t + 1}, \mathbf {z} _ {t}\right) + \mathbf {B} C o v \left(\boldsymbol {\xi} _ {t + 2}, \mathbf {z} _ {t}\right) \\ & A n d f o r t h r e e p e r i o d s \\ & C o v \left(\mathbf {z} _ {t + 3}, \mathbf {z} _ {t}\right) = C o v \left(\mathbf {c} + \mathbf {A} \mathbf {z} _ {t + 2} + \mathbf {B} \boldsymbol {\xi} _ {t + 3}, \mathbf {z} _ {t}\right) \\ & \quad = C o v \left(\mathbf {c} + \mathbf {A} \left(\mathbf {c} + \mathbf {A c} + \mathbf {A} ^ {2} \mathbf {z} _ {t} + \mathbf {A B} \boldsymbol {\xi} _ {t + 1} + \mathbf {B} \boldsymbol {\xi} _ {t + 2}\right) + \mathbf {B} \boldsymbol {\xi} _ {t + 3}, \mathbf {z} _ {t}\right) \\ & \quad = C o v \left(\mathbf {c} + \mathbf {A c} + \mathbf {A} ^ {2} \mathbf {c} + \mathbf {A} ^ {3} \mathbf {z} _ {t} + \mathbf {A} ^ {2} \mathbf {B} \boldsymbol {\xi} _ {t + 1} + \mathbf {A B} \boldsymbol {\xi} _ {t + 2} + \mathbf {B} \boldsymbol {\xi} _ {t + 3}, \mathbf {z} _ {t}\right) \\ & \quad = C o v \left(\mathbf {A} ^ {3} \mathbf {z} _ {t}, \mathbf {z} _ {t}\right) + C o v \left(\mathbf {A} ^ {2} \mathbf {B} \boldsymbol {\xi} _ {t + 1}, \mathbf {z} _ {t}\right) + C o v \left(\mathbf {A B} \boldsymbol {\xi} _ {t + 2}, \mathbf {z} _ {t}\right) + C o v \left(\mathbf {B} \boldsymbol {\xi} _ {t + 3}, \mathbf {z} _ {t}\right) \\ & \quad = \mathbf {A} ^ {3} V a r (\mathbf {z} _ {t}) + \mathbf {A} ^ {2} B C o v (\boldsymbol {\xi} _ {t + 1}, \mathbf {z} _ {t}) + A B C o v (\boldsymbol {\xi} _ {t + 2}, \mathbf {z} _ {t}) + B C o v (\boldsymbol {\xi} _ {t + 3}, \mathbf {z} _ {t}) \\ & \quad = \mathbf {A} ^ {3} V a r (\mathbf {z} _ {t}) + \sum_ {i = 1} ^ {3} \mathbf {A} ^ {3 - i} B C o v (\boldsymbol {\xi} _ {t + i}, \mathbf {z} _ {t}) \end{array}\]

\[C o v \left(\mathbf {z} _ {t + 3}, \mathbf {z} _ {t}\right) = \mathbf {A} C o v \left(\mathbf {z} _ {t + 2}, \mathbf {z} _ {t}\right) + \mathbf {B} C o v \left(\boldsymbol {\xi} _ {t + 3}, \mathbf {z} _ {t}\right)\]

\[\begin{array}{r l} & {\mathrm{Henceingeneral}} \\ & {C o v \left(\mathbf {z} _ {t + s}, \mathbf {z} _ {t}\right) = \mathbf {A} ^ {s} V a r \left(\mathbf {z} _ {t}\right) + \sum_ {i = 1} ^ {s} \mathbf {A} ^ {s - i} \mathbf {B} C o v \left(\boldsymbol {\xi} _ {t + i}, \mathbf {z} _ {t}\right)} \\ & {\Updownarrow} \\ & {C o v \left(\mathbf {z} _ {t + s}, \mathbf {z} _ {t}\right) = \mathbf {A} ^ {s} V a r \left(\mathbf {z} _ {t}\right) + \sum_ {j = 0} ^ {s - 1} \mathbf {A} ^ {s - (j + 1)} \mathbf {B} C o v \left(\boldsymbol {\xi} _ {t + j + 1}, \mathbf {z} _ {t}\right)} \\ & {i = j + 1 \mathrm{so} j = i - 1} \\ & {\Updownarrow} \\ & {C o v \left(\mathbf {z} _ {t + s}, \mathbf {z} _ {t}\right) = \mathbf {A} ^ {s} V a r \left(\mathbf {z} _ {t}\right) + \sum_ {j = 0} ^ {s - 1} \mathbf {A} ^ {s - 1 - j} \mathbf {B} C o v \left(\boldsymbol {\xi} _ {t + j + 1}, \mathbf {z} _ {t}\right)} \end{array}\]

\[\begin{array}{l} \text { or } \\ C o v \left(\mathbf {z} _ {t + s}, \mathbf {z} _ {t}\right) = \mathbf {A} C o v \left(\mathbf {z} _ {t + s - 1}, \mathbf {z} _ {t}\right) + \mathbf {B} C o v \left(\boldsymbol {\xi} _ {t + s}, \mathbf {z} _ {t}\right) \end{array}\]

\[\begin{array}{r l} & {\mathrm{Forthecontrolvariables:}} \\ & {C o v \left(\mathbf {y} _ {t + s} ^ {r d}, \mathbf {y} _ {t} ^ {r d}\right) = C o v \left(\mathbf {D z} _ {t + s} + \frac {1}{2} \mathbf {g} _ {\sigma \sigma} \sigma^ {2}, \mathbf {D z} _ {t} + \frac {1}{2} \mathbf {g} _ {\sigma \sigma} \sigma^ {2}\right)} \\ & {\qquad = C o v \left(\mathbf {D z} _ {t + s}, \mathbf {D z} _ {t}\right)} \\ & {\qquad = \mathbf {D C o v} \left(\mathbf {z} _ {t + s}, \mathbf {z} _ {t}\right) \mathbf {D} ^ {\prime}} \end{array}\]

Thus we only need to compute

4.5.3 Computing Cov

We consider and note that .

\[\begin{array} { r l } & E \left[ \mathbf { z } _ { t } \pmb { \xi } _ { t + 1 + s } ^ { \prime } \right] = E \left[ \left[ \begin{array} { c } \mathbf { x } _ { t } ^ { f } \\ \mathbf { x } _ { t } ^ { s } \\ \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \\ \mathbf { x } _ { t } ^ { r d } \\ \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { s } \\ \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \end{array} \right] \right. \\ & \quad \times \left[ \begin{array} { c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c } \pmb { \epsilon } _ { t + 1 + s } ^ { \prime } & ( \pmb { \epsilon } _ { t + 1 + s } \otimes \pmb { \epsilon } _ { t + 1 + s } - v e c ( \mathbf { I } _ { n _ { e} } ) ) ^ { \prime } & ( \pmb { \epsilon } _ { t + 1 + s } \otimes \mathbf { x } _ { t + s } ^ { f } ) ^ { \prime } \\ ( \mathbf { x } _ { t + s } ^ { f } \otimes \pmb { \epsilon } _ { t + 1 + s } ) ^ { \prime } & ( \pmb { \epsilon } _ { t + 1 + s } \otimes \mathbf { x } _ { t + s } ^ { s } ) ^ { \prime } & ( \pmb { \epsilon } _ { t + 1 + s } \otimes \mathbf { x } _ { t + s } ^ { f } \otimes \mathbf { x } _ { t + s } ^ { f } ) ^ { \prime } \\ ( \mathbf { x } _ { t + s } ^ { f } \otimes \mathbf { x } _ { t + s } ^ { f } \otimes \pmb { \epsilon } _ { t + 1 + s } ) ^ { \prime } & ( \mathbf { x } _ { t + s } ^ { f } \otimes \pmb { \epsilon } _ { t + 1 + s } \otimes \mathbf { x } _ { t + s } ^ { f } ) ^ { \prime } & ( \mathbf { x } _ { t + s } ^ { f } \otimes \pmb { \epsilon } _ { t + 1 + s } \otimes \pmb { \epsilon } _ { t + 1 + s } ) ^ { \prime } & ( \pmb { x } _ { t + s } ^ { f } \otimes \pmb { \epsilon } _ { t + 1 + s } \otimes \pmb { \epsilon } _ { t + 1 + s } ) ^ { \prime } & ( \pmb { \epsilon } _ { t + 1 + s } \otimes \mathbf { x } _ { t + s } ^ { f } \otimes \pmb { \epsilon } _ { t + 1 + s } ) ^ { \prime } \\ ( \pmb { \epsilon } _ { t + 1 + s } \otimes \pmb { \epsilon } _ { t + 1 + s } \otimes \mathbf { x } _ { t + s } ^ { f } ) ^ { \prime } & ( ( ( \pmb { \epsilon } _ { t + 1 + s } \otimes \pmb { \epsilon } _ { t + 1 + s } \otimes \pmb { \epsilon } _ { t + 1 + s } ) - E [ ( ( \pmb { \epsilon } _ { t + 1 } \otimes \pmb { \epsilon } _ { t + 1 } \otimes ( [ [ {\pmb {\epsilon}} _ { t + 1} ] ] ) ^ { \prime} ] & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & n _ { x } & n _ { e } & n _ { x } & n _ { e } & n _ { x } & n _ { x } & n _ { e } & n _ { x } & n _ { e } & n _ { x } & n _ { e } & n _ { x } & n _ { x } & n _ { e } & n _ { x } & n _ { e } & n _ { x } & n _ { e } & n _ { x } & n _ { e } & n _ { x } & n _ { e } & n _ { x } & n _ { e } & n _ { x } & n _ { e } & n _ { x } & n _ { e } & n _ { x } & n _ { e } & n \\ 0 & n _ { x } & n _ { e } & n _ { x } ^ { 2 } & n _ { x } ^ { 2 } & n _ { e } ^ { 2 } & n _ { x } ^ { 2 } & n _ { x } ^ { 2 } & n _ { e } ^ { 2 } & n _ { x } ^ { 2 } & n _ { x } ^ { 2 } & n _ { x } ^ { 2 } & n _ { x } ^ { 2 } & n _ { x } ^ { 2 } & n _ { x } ^ { 2 } & n _ { e } ^ { 2 } & n _ { x } ^ { 2 } & n _ { x } ^ { 2 } & n _ { e } ^ { 2 } & n _ { x } ^ { 2 } & n _ { x } ^ { 2 } & n _ { e} ^ { 2 }\]

We now compute the non-zero elements in this matrix

\[\begin{array}{r l} & {\mathrm{1)} \mathrm{Thevalueof} r _ {1, 9}} \\ & {r _ {1, 9} = E \left[ \mathbf {x} _ {t} ^ {f} \left(\mathbf {x} _ {t + s} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1 + s} \otimes \boldsymbol {\epsilon} _ {t + 1 + s}\right) ^ {\prime} \right]} \\ & {\qquad = E \left[ \left\{x _ {t} ^ {f} (\gamma_ {1}, 1) \right\} _ {\gamma_ {1} = 1} ^ {n _ {x}} \left\{x _ {t + s} ^ {f} (\gamma_ {2}, 1) \left\{\epsilon_ {t + 1 + s} (\phi_ {1}, 1) \left\{\epsilon_ {t + 1 + s} (\phi_ {2}, 1) \right\} _ {\phi_ {2} = 1} ^ {n _ {e}} \right\} _ {\phi_ {1} = 1} ^ {n _ {e}} \right\} _ {\gamma_ {2} = 1} ^ {n _ {x}} \right]} \end{array}\]

Thus, the quasi Matlab codes are

\[\begin{array} { r l } & \text {We know all the required moments, except } E \left[ \mathbf { x } _ { t } ^ { f } \left( \mathbf { x } _ { t + s } ^ { f } \right) ^ { \prime } \right] , E \left[ \mathbf { x } _ { t } ^ { s } \left( \mathbf { x } _ { t + s } ^ { f } \right) ^ { \prime } \right] , E \left[ \left( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \right) \left( \mathbf { x } _ { t + s } ^ { f } \right) ^ { \prime } \right] , E \left[ \mathbf { x } _ { t } ^ { r d } \left( \mathbf { x } _ { t + s } ^ { f } \right) ^ { \prime } \right] , \\ & { E \left[ \left( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { s } \right) \left( \mathbf { x } _ { t + s } ^ { f } \right) ^ { \prime } \right] , \text {and} E \left[ \left( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \right) \left( \mathbf { x } _ { t + s } ^ { f } \right) ^ { \prime } \right] } \\ & {\quad \text {a) For} E \left[ \mathbf { x } _ { t } ^ { f } \left( \mathbf { x } _ { t + s } ^ { f } \right) ^ { \prime } \right] } \\ & \text {Recall that} \mathbf { x } _ { t + s } ^ { f } = \mathbf { h } _ { \mathbf { x } } ^ { s } \mathbf { x } _ { t } ^ { f } + \sum _ { i = 1 } ^ { s } \mathbf { h } _ { \mathbf { x } } ^ { s - i } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + i } . \\ & { S o } \\ & { E \left[ \mathbf { x } _ { t } ^ { f } \left( \mathbf { x } _ { t + s } ^ { f } \right) ^ { \prime } \right] = E \left[ \mathbf { x } _ { t } ^ { f } \left( \mathbf { h } _ { \mathbf { x } } ^ { s } \mathbf { x } _ { t } ^ { f } + \sum _ { i = 1 } ^ { s } \mathbf { h } _ { \mathbf { x } } ^ { s - i } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + i } \right) ^ { \prime } \right] = E \left[ \mathbf { x } _ { t } ^ { f } \left( \mathbf { x } _ { t } ^ { f } \right) ^ { \prime } \right] ( \mathbf { h } _ { \mathbf { x } } ^ { s ) ' }} \\ & \quad b ) F o r E [ [ \mathbf { x } _ { t } ^ { s } (\mathbf { x } _ { t + s } ^ { f}) ] ] \\ & { E [ [ \mathbf { x } _ { t } ^ { s } (\mathbf { x } _ { t + s } ^ { f}) ] ] = E [ [ \mathbf { x } _ { t } ^ { s } (\mathbf { h } _ { \mathbf { x }} ^ { s } \mathbf { x } _ { t } ^ { f } + \sum _ { i = 1 } ^ { s } \mathbf { h } _ { \mathbf { x }} ^ { s - i } \sigma \boldsymbol { \eta} \boldsymbol { \epsilon } _ { t + i} ) ] ] = E [ [ \mathbf { x } _ { t } ^ { s } (\mathbf { x } _ { t } ^ { f}) ] ] ( \mathbf { h } _ { \mathbf { x } } ^ { s ) ' },}\\ & \quad c ) F o r E [ [ (\mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f }) (\mathbf { x} _ { t + s} ^ { f}) ] ] \\ & { E [ [ (\mathbf { x} _ { t } ^ { f } \otimes \mathbf {\hat{x}} _ { t} ^{ f}) (\mathbf {\hat{x}} _{t + s} ^{f}) ] ] = E [ [ (\mathbf {\hat{x}} _{t} ^{f} \otimes \mathbf {\hat{x}} _{t} ^{f}) (\mathbf {\hat{h}} _{x} ^{s} \mathbf {\hat{x}} _{t} ^{f} + \sum _{i = 1}^{s} \mathbf {\hat{h}} _{x} ^{s - i} \sigma \boldsymbol{ {\eta} }\epsilon_{t + i}) ] ] = E [ [ (\mathbf {\hat{x}} _{t} ^{f} \otimes \mathbf {\hat{x}} _{t} ^{f}) (\mathbf {\hat{x}} _{t} ^{f}) ] ] ( \mathbf {\hat{h}} _{x} ^{s ) ' },}\\ & \quad d ) F o r E [ [ \mathbf {\hat{x}} _{t} ^{r d} (\mathbf {\hat{x}} _{t + s} ^{f}) ] ] \\ & E [ [ \mathbf {\hat{x}} _{t} ^{r d} (\mathbf {\hat{x}} _{t + s} ^{f}) ] ] = E [ [ \mathbf {\hat{x}} _{t} ^{r d} (\mathbf {\hat{h}} _{x}^{s} \mathbf {\hat{x}} _{t} ^{f} + \sum _{i = 1}^{s} \mathbf {\hat{h}} _{x}^{s - i} σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Σ Λ , \\ & {\quad = E [ [ \mathbf {\hat{x}} _{t} ^{r d} (\mathbf {\hat{h}} _{x}^{s} \mathbf {\hat{x}} _{t} ^{f}) ] ] ,}\\ & {\quad = E [ [ \mathbf {\hat{x}} _{t} ^{r d} (\mathbf {\hat{x}} _{t} ^{f}) ] ] ( \mathbf {\hat{h}} _{x}^{s})^{\prime }} \\ & {S o w e n o l y n e e d t o f i n d E [ [ \mathbf {\hat{x}} _{t} ^{r d} (\mathbf {\hat{x}} _{t} ^{f}) ] ] . R e c a l l t h a t} \\ & {\texttt X}, E = 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 , \\ & {{S o}} \\ & E [ [ {\texttt X}, E ] , E ] = E [ [ ({\texttt X}, E ] , E ] , E ] , E ] , E ] , E ] , E ] , E ] , E ] , E ] , E ] , E ] , E ] , E ] , E ] , E ] , E ] , E ] , E ] , E ] , E ] , E ] , E ] , E ] , E ] , E ] , E ] , E ] , E ] , E ] , E ] , E ] , E ] , E ] , K, K, K, K, K, K, K, K, K, K, K, K, K, K, K, K, K, K, K, K, K, K, K, K, K, K, K, K, K, K, K, K, K, K, K, K, K, K, K, K, K, K, K, K, K, K, K, K, K, K, K , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. , \\ & E [ [ {\texttt X}, E ] , E ] = E [ [ ({\texttt X}, E ] , E ] , E ] , E ] , E ] , E ] , E ] , E ] , E ] , E ] , E ] , E ] , E ] , E ] , E ] , E ] , E ] , E ] , E ] , E ] , E ] , E ] , E ] , E ] , E ] \end{array}\]

\[E \left[ \mathbf {x} _ {t} ^ {r d} \left(\mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \right] = \left(\mathbf {I} - \mathbf {h} _ {\mathbf {x}}\right) ^ {- 1} \left[ 2 \tilde {\mathbf {H}} _ {\mathbf {x x}} E \left[ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {s}\right) \left(\mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \right] + \tilde {\mathbf {H}} _ {\mathbf {x x x}} E \left[ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \right] + \frac {3}{6} \mathbf {h} _ {\sigma \sigma \mathbf {x}} \sigma^ {2} E \left[ \mathbf {x} _ {t} ^ {f} \left(\mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \right] \right]\]

\[\begin{array}{l} \text {e)} \text {For} E \left[ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {s}\right) \left(\mathbf {x} _ {t + s} ^ {f}\right) ^ {\prime} \right] \\ E \left[ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {s}\right) \left(\mathbf {x} _ {t + s} ^ {f}\right) ^ {\prime} \right] \\ = E \left[ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {s}\right) \left(\mathbf {h} _ {\mathbf {x}} ^ {s} \mathbf {x} _ {t} ^ {f} + \sum_ {i = 1} ^ {s} \mathbf {h} _ {\mathbf {x}} ^ {s - i} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + i}\right) ^ {\prime} \right] \\ = E \left[ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {s}\right) \left(\mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \right] (\mathbf {h} _ {\mathbf {x}} ^ {s}) ^ {\prime} \end{array}\]

\[\begin{array}{l} \text {f)} \text {For} E \left[ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\mathbf {x} _ {t + s} ^ {f}\right) ^ {\prime} \right] \\ E \left[ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\mathbf {x} _ {t + s} ^ {f}\right) ^ {\prime} \right] \\ = E \left[ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\mathbf {h} _ {\mathbf {x}} ^ {s} \mathbf {x} _ {t} ^ {f} + \sum_ {i = 1} ^ {s} \mathbf {h} _ {\mathbf {x}} ^ {s - i} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + i}\right) ^ {\prime} \right] \\ = E \left[ \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) \left(\mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \right] (\mathbf {h} _ {\mathbf {x}} ^ {s}) ^ {\prime} \end{array}\]

5 The Dynare++ notation

This section presents the pruning method up to third order using the notation in Dynare and Dyanare++. The solution to DSGE models are in Dynare and Dynare++ given by

\[\mathbf {z} _ {t} = \mathbf {f} (\mathbf {z} _ {t - 1}, \mathbf {u} _ {t}, \sigma)\tag{39}\]

where contains all the endogenous variables (i.e. control variables and all state variables), and with size is the vector of disturbances with the property . It is convenient to express this more general solution in a notation that is similar to the one used above. We therefore write (39) as

\[\mathbf {y} _ {t} = \mathbf {g} (\mathbf {x} _ {t - 1}, \mathbf {u} _ {t}, \sigma)\tag{40}\]

\[\mathbf {x} _ {t + 1} = \mathbf {h} \left(\mathbf {x} _ {t}, \mathbf {u} _ {t + 1}, \sigma\right)\tag{41}\]

where and are as defined above. The key difference compared to the notation in Schmitt-Grohé & Uribe (2004) is that the function g depends on the innovations . Note also that the innovations may enter in a non-linear fashion in the h function. Below, it is useful to define

\[\mathbf {v} _ {t, t + 1} \equiv \left[ \begin{array}{c} \mathbf {x} _ {t} \\ \mathbf {u} _ {t + 1} \end{array} \right]\tag{42}\]

where has dimensions . The first subscript of refers to the time index of and the second to the time index of .

A first-order approximation (40) and (41) around the deterministic steady state is

\[\mathbf {y} _ {t} = \mathbf {g} _ {\mathbf {v}} \mathbf {v} _ {t - 1, t}\tag{43}\]

\[\mathbf {x} _ {t + 1} = \mathbf {h} _ {\mathbf {v}} \mathbf {v} _ {t, t + 1}\tag{44}\]

A second-order approximation is

\[\mathbf {y} _ {t} = \mathbf {g} _ {\mathbf {v}} \mathbf {v} _ {t - 1, t} + \frac {1}{2} \mathbf {G} _ {\mathbf {v v}} \left(\mathbf {v} _ {t - 1, t} \otimes \mathbf {v} _ {t - 1, t}\right) + \frac {1}{2} \mathbf {g} _ {\sigma \sigma} \sigma^ {2}\tag{45}\]

and

\[\mathbf {x} _ {t + 1} = \mathbf {h} _ {\mathbf {v}} \mathbf {v} _ {t, t + 1} + \frac {1}{2} \mathbf {H} _ {\mathbf {v v}} (\mathbf {v} _ {t, t + 1} \otimes \mathbf {v} _ {t, t + 1}) + \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}\tag{46}\]

A third-order approximation is

\[\begin{array}{r c l} \mathbf {y} _ {t} & = & \mathbf {g} _ {\mathbf {v}} \mathbf {v} _ {t - 1, t} + \frac {1}{2} \mathbf {G} _ {\mathbf {v v}} (\mathbf {v} _ {t - 1, t} \otimes \mathbf {v} _ {t - 1, t}) + \frac {1}{2} \mathbf {g} _ {\sigma \sigma} \sigma^ {2} \\ & & + \frac {1}{6} \mathbf {G} _ {\mathbf {v v v}} (\mathbf {v} _ {t - 1, t} \otimes \mathbf {v} _ {t - 1, t} \otimes \mathbf {v} _ {t - 1, t}) + \frac {3}{6} \mathbf {g} _ {\sigma \sigma \mathbf {v}} \sigma^ {2} \mathbf {v} _ {t - 1, t} + \frac {1}{6} \mathbf {g} _ {\sigma \sigma \sigma} \sigma^ {3} \end{array}\tag{47}\]

and

\[\begin{array}{r c l} \mathbf {x} _ {t + 1} & = & \mathbf {h} _ {\mathbf {v}} \mathbf {v} _ {t, t + 1} + \frac {1}{2} \mathbf {H} _ {\mathbf {v v}} (\mathbf {v} _ {t, t + 1} \otimes \mathbf {v} _ {t, t + 1}) + \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2} \\ & & + \frac {1}{2} \mathbf {H} _ {\mathbf {v v v}} (\mathbf {v} _ {t, t + 1} \otimes \mathbf {v} _ {t, t + 1} \otimes \mathbf {v} _ {t, t + 1}) + \frac {3}{6} \mathbf {h} _ {\sigma \sigma \mathbf {v}} \sigma^ {2} \mathbf {v} _ {t, t + 1} + \frac {1}{6} \mathbf {h} _ {\sigma \sigma \sigma} \sigma^ {3} \end{array}\tag{48}\]

6 Pruning scheme in Dynare++:

6.1 Second order approximation:

We start considering (46) which we write as

\[\begin{array}{l} x _ {t + 1} \left(j, 1\right) = \mathbf {h} _ {\mathbf {v}} \left(j,:) \right. \mathbf {v} _ {t, t + 1} + \left(\mathbf {v} _ {t, t + 1}\right) ^ {\prime} \mathbf {h} _ {\mathbf {v v}} \left(j,:,:) \right. \mathbf {v} _ {t, t + 1} + \frac {1}{2} h _ {\sigma \sigma} \left(j, 1\right) \sigma^ {2} \\ \Updownarrow \\ x _ {t + 1} \left(j, 1\right) = \mathbf {h} _ {\mathbf {v}} \left(j,:)\right) \left[ \begin{array}{c} \mathbf {x} _ {t} \\ \mathbf {u} _ {t + 1} \end{array} \right] + \left[ \begin{array}{c} \mathbf {x} _ {t} \\ \mathbf {u} _ {t + 1} \end{array} \right] ^ {\prime} \mathbf {h} _ {\mathbf {v v}} \left(j,:,:) \right. \left[ \begin{array}{c} \mathbf {x} _ {t} \\ \mathbf {u} _ {t + 1} \end{array} \right] + \frac {1}{2} h _ {\sigma \sigma} \left(j, 1\right) \sigma^ {2} \\ f o r i n e d i n e f o r e f o r e f o r e f o r e f o r e f o r e f o r e f o r e f o r e f o r e f o r e f o r e f o r e f o r e f o r e f o r e f o r e f o r e f o r e f o r e f o r e f o r e f o r e f o r e f o r e f o l u e a d i n c e s t a t i o n. \end{array}\]

Let us now decompose the state vector as

\[\mathbf {x} _ {t} = \mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s}\tag{49}\]

Notice, that we do not need to compose the innovations as they are a first order effect in the system. For the subsequent decomposition let

\[\mathbf {h} _ {\mathbf {v}} \equiv \left[ \begin{array}{c c} \mathbf {h} _ {\mathbf {x}} (j,:) & \mathbf {h} _ {\mathbf {u}} (j,:) \end{array} \right]\tag{50}\]

\[\mathbf {h} _ {\mathbf {v v}} \left(j,:,:\right) = \left[ \begin{array}{c c} \mathbf {h} _ {\mathbf {x x}} \left(j,:,:\right) & \mathbf {h} _ {\mathbf {x u}} \left(j,:,:\right) \\ \mathbf {h} _ {\mathbf {u x}} \left(j,:,:\right) & \mathbf {h} _ {\mathbf {u u}} \left(j,:,:\right) \end{array} \right]\tag{51}\]

for .

Hence,

\[\begin{array}{r l r} & & {x _ {t + 1} ^ {f} (j, 1) + x _ {t + 1} ^ {s} (j, 1) = \left[ \begin{array}{c c} {\mathbf {h} _ {\mathbf {x}} (j,:)} & {\mathbf {h} _ {\mathbf {u}} (j,:)} \end{array} \right] \left[ \begin{array}{c} {\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s}} \\ {\mathbf {u} _ {t + 1}} \end{array} \right]} \\ & & {+ \frac {1}{2} \left[ \begin{array}{c} {\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s}} \\ {\mathbf {u} _ {t + 1}} \end{array} \right] ^ {\prime} \left[ \begin{array}{c c} {\mathbf {h} _ {\mathbf {x x}} (j,:,:)} & {\mathbf {h} _ {\mathbf {x u}} (j,:,:)} \\ {\mathbf {h} _ {\mathbf {u x}} (j,:,:)} & {\mathbf {h} _ {\mathbf {u u}} (j,:,:)} \end{array} \right] \left[ \begin{array}{c} {\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s}} \\ {\mathbf {u} _ {t + 1}} \end{array} \right] + \frac {1}{2} h _ {\sigma \sigma} (j, 1) \sigma^ {2}} \end{array}\]

↓

\[\begin{array}{l}x _ {t + 1} ^ {f} \left(j, 1\right) + x _ {t + 1} ^ {s} \left(j, 1\right) = \mathbf {h} _ {\mathbf {x}} \left(j,:) \left(\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s}\right) + \mathbf {h} _ {\mathbf {u}} \left(j,:) \mathbf {u} _ {t + 1} \right. \right.\\\quad \left. \right. + \frac {1}{2} \left[\left(\mathbf {x} _ {t} ^ {f}\right) ^ {\prime} + (\mathbf {x} _ {t} ^ {s}) ^ {\prime} \quad \mathbf {u} _ {t + 1} ^ {\prime} \right]\left[\begin{array}{c c}\mathbf {h} _ {\mathbf {x x}} \left(j,:,:) \right.&\mathbf {h} _ {\mathbf {x u}} \left(j,:,:) \right.\\\mathbf {h} _ {\mathbf {u x}} \left(j,:,:) \right.&\mathbf {h} _ {\mathbf {u u}} \left(j,:,:) \right.\end{array}\right]\left[\begin{array}{c}\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s}\\\mathbf {u} _ {t + 1}\end{array}\right]\\+ \frac {1}{2} h _ {\sigma \sigma} \left(j, 1\right) \sigma^ {2}\end{array}\]

↓

↓

\[\begin{array}{r l}x _ {t + 1} ^ {f} (j, 1) + x _ {t + 1} ^ {s} (j, 1) = \mathbf {h} _ {\mathbf {x}} (j,:) (\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s}) + \mathbf {h} _ {\mathbf {u}} (j,:) \mathbf {u} _ {t + 1}\\&+ \frac {1}{2} \left[\left(\left(\mathbf {x} _ {t} ^ {f}\right) ^ {\prime} + (\mathbf {x} _ {t} ^ {s}) ^ {\prime}\right) \mathbf {h} _ {\mathbf {x x}} (j,:,:) + \mathbf {u} _ {t + 1} ^ {\prime} \mathbf {h} _ {\mathbf {u x}} (j,:,:) \right. \quad \left(\left(\mathbf {x} _ {t} ^ {f}\right) ^ {\prime} + (\mathbf {x} _ {t} ^ {s}) ^ {\prime}\right) \mathbf {h} _ {\mathbf {x u}} (j,:,:) + \mathbf {u} _ {t + 1} ^ {\prime} \mathbf {h} _ {\mathbf {u u}} (j,:,:) \left. \right]\left[\begin{array}{c}\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s}\\\mathbf {u} _ {t + 1}\end{array}\right]\\&+ \frac {1}{2} h _ {\sigma \sigma} (j, 1) \sigma^ {2}\end{array}\]

\[\begin{array}{l} x _ {t + 1} ^ {f} (j, 1) + x _ {t + 1} ^ {s} (j, 1) = \mathbf {h} _ {\mathbf {x}} (j,:) (\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s}) + \mathbf {h} _ {\mathbf {u}} (j,:) \mathbf {u} _ {t + 1} \\ \qquad + \frac {1}{2} \left(\left(\mathbf {x} _ {t} ^ {f}\right) ^ {\prime} + (\mathbf {x} _ {t} ^ {s}) ^ {\prime}\right) \mathbf {h} _ {\mathbf {x x}} (j,:,:) (\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s}) + \frac {1}{2} \mathbf {u} _ {t + 1} ^ {\prime} \mathbf {h} _ {\mathbf {u x}} (j,:,:) (\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s}) \\ \qquad + \frac {1}{2} \left(\left(\mathbf {x} _ {t} ^ {f}\right) ^ {\prime} + (\mathbf {x} _ {t} ^ {s}) ^ {\prime}\right) \mathbf {h} _ {\mathbf {x u}} (j,:,:) \mathbf {u} _ {t + 1} + \frac {1}{2} \mathbf {u} _ {t + 1} ^ {\prime} \mathbf {h} _ {\mathbf {u u}} (j,:,:) \mathbf {u} _ {t + 1} \\ \qquad + \frac {1}{2} h _ {\sigma \sigma} (j, 1) \sigma^ {2} \end{array}\]

A law of motion for the first-order terms is thus

\[x _ {t + 1} ^ {f} (j, 1) = \mathbf {h} _ {\mathbf {x}} (j,:) \mathbf {x} _ {t} ^ {f} + \mathbf {h} _ {\mathbf {u}} (j,:) \mathbf {u} _ {t + 1}\tag{52}\]

for .

A law of motion for the second-order terms is thus

\[\begin{array}{r c l} x _ {t + 1} ^ {s} (j, 1) & = & \mathbf {h} _ {\mathbf {x}} (j,:) \mathbf {x} _ {t} ^ {s} + \frac {1}{2} (\mathbf {x} _ {t} ^ {f}) ^ {\prime} \mathbf {h} _ {\mathbf {x x}} (j,:,:) (\mathbf {x} _ {t} ^ {f}) + \frac {1}{2} \mathbf {u} _ {t + 1} ^ {\prime} \mathbf {h} _ {\mathbf {u x}} (j,:,:) \mathbf {x} _ {t} ^ {f} \\ & & + \frac {1}{2} (\mathbf {x} _ {t} ^ {f}) ^ {\prime} \mathbf {h} _ {\mathbf {x u}} (j,:,:) \mathbf {u} _ {t + 1} + \frac {1}{2} \mathbf {u} _ {t + 1} ^ {\prime} \mathbf {h} _ {\mathbf {u u}} (j,:,:) \mathbf {u} _ {t + 1} + \frac {1}{2} h _ {\sigma \sigma} (j, 1) \sigma^ {2} \end{array}\tag{53}\]

for . Note here, that non-linear shocks will imply that we have innovations to , i.e. if and .

For the control variables, we introduce the following notation

\[\mathbf {g} _ {\mathbf {v}} \equiv \left[ \begin{array}{c c} \mathbf {g} _ {\mathbf {x}} (i,:) & \mathbf {g} _ {\mathbf {u}} (i,:) \end{array} \right]\tag{54}\]

\[\mathbf {g _ {v v}} \left(i,:,:\right) = \left[ \begin{array}{c c} \mathbf {g _ {x x}} \left(i,:,:\right) & \mathbf {g _ {x u}} \left(i,:,:\right) \\ \mathbf {g _ {u x}} \left(i,:,:\right) & \mathbf {g _ {u u}} \left(i,:,:\right) \end{array} \right]\tag{55}\]

\[\begin{array}{l} \text {for} i = 1, 2,..., n _ {y}. \text {Thus} \\ y _ {t} (i, 1) = \mathbf {g _ {v}} (i,:) \mathbf {v} _ {t - 1, t} + \frac {1}{2} (\mathbf {v} _ {t - 1, t}) ^ {\prime} \mathbf {g _ {v v}} (i,:,:) \mathbf {v} _ {t - 1, t} \ddot {} + \frac {1}{2} \mathbf {g} _ {\sigma \sigma} (i, 1) \sigma^ {2} \\ \Updownarrow \end{array}\]

\[\begin{array}{l} y _ {t} \left(i, 1\right) = \left[ \begin{array}{c c} \mathbf {g} _ {\mathbf {x}} \left(i,: \right) & \mathbf {g} _ {\mathbf {u}} \left(i,: \right) \end{array} \right] \left[ \begin{array}{c} \mathbf {x} _ {t - 1} ^ {f} + \mathbf {x} _ {t - 1} ^ {s} \\ \mathbf {u} _ {t} \end{array} \right] \\ + \frac {1}{2} \left[ \begin{array}{c} \mathbf {x} _ {t - 1} ^ {f} + \mathbf {x} _ {t - 1} ^ {s} \\ \mathbf {u} _ {t} \end{array} \right] ^ {\prime} \left[ \begin{array}{c c} \mathbf {g} _ {\mathbf {x x}} \left(i,:,:) \right. & \mathbf {g} _ {\mathbf {x u}} \left(i,:,:) \right. \\ \mathbf {g} _ {\mathbf {u x}} \left(i,:,:) \right. & \mathbf {g} _ {\mathbf {u u}} \left(i,:,:) \right. \end{array} \right] \left[ \begin{array}{c} \mathbf {x} _ {t - 1} ^ {f} + \mathbf {x} _ {t - 1} ^ {s} \\ \mathbf {u} _ {t} \end{array} \right] + \frac {1}{2} g _ {\sigma \sigma} \left(i, 1\right) \sigma^ {2} \\ \Updownarrow \end{array}\]

\[\begin{array} { l } y _ { t } \left( i , 1 \right) = \mathbf { g } _ { \mathbf { x } } \left( i , : \right) \left( \mathbf { x } _ { t - 1 } ^ { f } + \mathbf { x } _ { t - 1 } ^ { s } \right) + \mathbf { g } _ { \mathbf { u } } \left( i , : \right) \mathbf { u } _ { t } \\ + \frac { 1 } { 2 } \left[ \begin{array} { c c } \left( \mathbf { x } _ { t - 1 } ^ { f } \right) ^ { \prime } + \left( \mathbf { x } _ { t - 1 } ^ { s } \right) ^ { \prime } & \mathbf { u } _ { t } ^ { \prime } \end{array} \right] \left[ \begin{array} { c c } \mathbf { g } _ { \mathbf { x x } } \left( i , : , : \right) & \mathbf { g } _ { \mathbf { x u } } \left( i , : , : \right) \\ \mathbf { g } _ { \mathbf { u x } } \left( i , : , : \right) & \mathbf { g } _ { \mathbf { u u } } \left( i , : , : \right) \end{array} \right] \left[ \begin{array} { c } \mathbf { x } _ { t - 1 } ^ { f } + \mathbf { x } _ { t - 1 } ^ { s } \\ \mathbf { u } _ { t } \end{array} \right] + \frac { 1 } { 2 } g _ { \sigma \sigma } \left( i , 1 \right) \sigma ^ { 2 } \\ \Updownarrow \\ y _ { t } \left( i , 1 \right) = \mathbf { g } _ { \mathbf { x } } \left( i , : \right) \left( \mathbf { x } _ { t - 1 } ^ { f } + \mathbf { x } _ { t - 1 } ^ { s } \right) + \mathbf { g } _ { \mathbf { u } } \left ( i , : \right) \mathbf { u } _ { t } \\ + \frac { 1 } { 2 } \left[ \begin{array} { c c } \left( \left( \mathbf { x } _ { t - 1 } ^ { f } \right) ^ { \prime } + \left( \mathbf { x } _ { t - 1 } ^ { s } \right) ^ { \prime } \right) & \mathbf { g } _ { \mathbf { x x } } \left( i , : , : \right) + \mathbf { u } _ { t } ^ { \prime } \mathbf { g } _ { \mathbf { u x } } \left( i , : , : \right) \\ & ( ( ( x _ { t - 1 } ^ { f } ) ^ { \prime } + ( x _ { t - 1 } ^ { s } ) ^ { \prime } ) ^ { 2 } ) \\ x = [ ( x _ { t - 1 } ^ { f } + x _ { t - 1 } ^ { s } ] & \\ + \frac { 1 } { 2 } g _ { \sigma \sigma } ( i , 1 ) \sigma ^ { 2 } \\ \Updownarrow \\ y _ { t } \left( i , 1 \right) = \mathbf { g } _ { \mathbf { x } } \left( i , : \right) ( x _ { t - 1 } ^ { f } + x _ { t - 1 } ^ { s } ) + g _ { u } ( i , : ) u _ { t } \\ + \frac { 1 } { 2 } ( ( x _ { t - 1 } ^ { f } ) ^ { \prime } + ( x _ { t - 1 } ^ { s } ) ^ { \prime }) g _ { x x } ( i , : , : ) ( x _ { t - 1 } ^ { f } + x _ { t - 1 } ^ { s } ) + \frac { 1 } { 2 } u _ { t } ^ { \prime } g _ { u x } ( i , : , : ) ( x _ { t - 1 } ^ { f } + x _ { t - 1 } ^ { s } ) \\ + \frac { 1 } { 2 } ( ( x _ { t - 1 } ^ { f } ) ^ { \prime } + ( x _ { t - 1 } ^ { s } ) ^ { \prime }) g _ { x u } ( i , : , : ) u _ { t } + \frac { 1 } { 2 } u _ { t } ^ { \prime } g _ { u u } ( i , : , : ) u _ { t } \\ + \frac { 1 } { 2 } g _ { \sigma \sigma } ( i , 1 ) \sigma ^ { 2 } \\ \end{array} .\]

for . We want to preserve terms up to second order, hence the pruned approximation is

\[\begin{array}{r c l} y _ {t} \left(i, 1\right) & = & \mathbf {g _ {x}} \left(i,:)\right) \left(\mathbf {x} _ {t - 1} ^ {f} + \mathbf {x} _ {t - 1} ^ {s}\right) + \mathbf {g _ {u}} \left(i,:)\right) \mathbf {u} _ {t} \\ & & + \frac {1}{2} \left(\mathbf {x} _ {t - 1} ^ {f}\right) ^ {\prime} \mathbf {g _ {x x}} \left(i,:,:) \right. \mathbf {x} _ {t - 1} ^ {f} + \frac {1}{2} \mathbf {u} _ {t} ^ {\prime} \mathbf {g _ {u x}} \left(i,:,:) \right. \mathbf {x} _ {t - 1} ^ {f} \\ & & + \frac {1}{2} \left(\mathbf {x} _ {t - 1} ^ {f}\right) ^ {\prime} \mathbf {g _ {x u}} \left(i,:,:) \right. \mathbf {u} _ {t} + \frac {1}{2} \mathbf {u} _ {t} ^ {\prime} \mathbf {g _ {u u}} \left(i,:,:) \right. \mathbf {u} _ {t} \\ & & + \frac {1}{2} g _ {\sigma \sigma} \left(i, 1\right) \sigma^ {2} \end{array}\tag{56}\]

for

6.2 Second order approximation: a convenient representation

When coding the derived formulas it is convenient to use directly, and the corresponding derivatives of g and h, because this is how the output from Dynare and Dynare++ is stored. Hence, we can write

\[\mathbf {x} _ {t + 1} ^ {f} = \mathbf {h} _ {\mathbf {v}} \left[ \begin{array}{c} \mathbf {x} _ {t} \\ \mathbf {u} _ {t + 1} \end{array} \right]\tag{57}\]

and

\[x _ {t + 1} ^ {s} (j,:) = \mathbf {h _ {v}} (j,:) \left[ \begin{array}{c} \mathbf {x} _ {t} ^ {s} \\ \mathbf {0} \end{array} \right] + \frac {1}{2} \left[ \begin{array}{c} \mathbf {x} _ {t} ^ {f} \\ \mathbf {u} _ {t + 1} \end{array} \right] ^ {\prime} \mathbf {h _ {v v}} (j,:,:) \left[ \begin{array}{c} \mathbf {x} _ {t} ^ {f} \\ \mathbf {u} _ {t + 1} \end{array} \right] + \frac {1}{2} h _ {\sigma \sigma} (j,:) \sigma^ {2}\tag{58}\]

for . For the control variables we have

\[y _ {t} \left(i, 1\right) = \mathbf {g _ {v}} \left(i,:)\right) \left(\left[ \begin{array}{c} \mathbf {x} _ {t - 1} ^ {f} \\ \mathbf {u} _ {t} \end{array} \right] + \left[ \begin{array}{c} \mathbf {x} _ {t - 1} ^ {s} \\ \mathbf {0} \end{array} \right]\right) + \frac {1}{2} \left[ \begin{array}{c} \mathbf {x} _ {t - 1} ^ {f} \\ \mathbf {u} _ {t} \end{array} \right] ^ {\prime} \mathbf {g _ {v v}} \left(i,:,:)\right) \left[ \begin{array}{c} \mathbf {x} _ {t - 1} ^ {f} \\ \mathbf {u} _ {t} \end{array} \right] + \frac {1}{2} g _ {\sigma \sigma} \left(i, 1\right) \sigma^ {2}\tag{59}\]

for .

Using the kronecker representation (fast for MATLAB) we have

\[\mathbf {x} _ {t + 1} ^ {s} = \mathbf {h} _ {\mathbf {v}} \left[ \begin{array}{c} \mathbf {x} _ {t} ^ {s} \\ \mathbf {0} \end{array} \right] + \tilde {\mathbf {H}} _ {\mathbf {v v}} \left(\left[ \begin{array}{c} \mathbf {x} _ {t} ^ {f} \\ \mathbf {u} _ {t + 1} \end{array} \right] \otimes \left[ \begin{array}{c} \mathbf {x} _ {t} ^ {f} \\ \mathbf {u} _ {t + 1} \end{array} \right]\right) + \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}\tag{60}\]

where

\[\tilde {\mathbf {H}} _ {\mathbf {v v}} \equiv \frac {1}{2} r e s h a p e \left(\mathbf {h} _ {\mathbf {v v}}, n _ {x}, n _ {v} ^ {2}\right)\tag{61}\]

And

\[\mathbf {y} _ {t} = \mathbf {g} _ {\mathbf {v}} \left(\left[ \begin{array}{c} \mathbf {x} _ {t - 1} ^ {f} \\ \mathbf {u} _ {t} \end{array} \right] + \left[ \begin{array}{c} \mathbf {x} _ {t - 1} ^ {s} \\ \mathbf {0} \end{array} \right]\right) + \tilde {\mathbf {G}} _ {\mathbf {v v}} \left(\left[ \begin{array}{c} \mathbf {x} _ {t - 1} ^ {f} \\ \mathbf {u} _ {t} \end{array} \right] \otimes \left[ \begin{array}{c} \mathbf {x} _ {t - 1} ^ {f} \\ \mathbf {u} _ {t} \end{array} \right]\right) + \frac {1}{2} \mathbf {g} _ {\sigma \sigma} \sigma^ {2}\tag{62}\]

where

\[\tilde {\mathbf {G}} _ {\mathbf {v v}} \equiv \frac {1}{2} r e s h a p e \left(\mathbf {g} _ {\mathbf {v v}}, n _ {y}, n _ {v} ^ {2}\right)\tag{63}\]

6.3 Third order approximation:

Let us now decompose the state vector as

\[\mathbf {x} _ {t} = \mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d}\tag{64}\]

Thus

\[\begin{array}{r l}&{x _ {t + 1} \left(j, 1\right) = \mathbf {h} _ {\mathbf {v}} \left(j,:) \mathbf {v} _ {t, t + 1} + \frac {1}{2} \left(\mathbf {v} _ {t, t + 1}\right) ^ {\prime} \mathbf {h} _ {\mathbf {v v}} \left(j,:,:) \mathbf {v} _ {t, t + 1} + \frac {1}{2} h _ {\sigma \sigma} \left(j, 1\right) \sigma^ {2} \right. \right.}\\&{\qquad \left. \right. + \frac {1}{6} \left(\mathbf {v} _ {t, t + 1}\right) ^ {\prime} \left[\begin{array}{c}(\mathbf {v} _ {t, t + 1}) ^ {\prime} \mathbf {h} _ {\mathbf {v v v}} (j, 1,:,:) \mathbf {v} _ {t, t + 1}\\...\\(\mathbf {v} _ {t, t + 1}) ^ {\prime} \mathbf {h} _ {\mathbf {v v v}} (j, n _ {v},:,:) \mathbf {v} _ {t, t + 1}\end{array}\right] + \frac {3}{6} \mathbf {h} _ {\sigma \sigma \mathbf {v}} \left(j,:\right) \sigma^ {2} \mathbf {v} _ {t, t + 1} + \frac {1}{6} h _ {\sigma \sigma \sigma} \left(j, 1\right) \sigma^ {3}}\end{array}\]

\[\begin{array}{r l} & x _ {t + 1} ^ {f} (j, 1) + x _ {t + 1} ^ {s} (j, 1) + x _ {t + 1} ^ {r d} (j, 1) = \left[ \begin{array}{c c} \mathbf {h} _ {\mathbf {x}} (j,:) & \mathbf {h} _ {\mathbf {u}} (j,:) \end{array} \right] \left[ \begin{array}{c} \mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d} \\ \mathbf {u} _ {t + 1} \end{array} \right] \\ & \qquad + \frac {1}{2} \left[ \begin{array}{c} \mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d} \\ \mathbf {u} _ {t + 1} \end{array} \right] ^ {\prime} \left[ \begin{array}{c c} \mathbf {h} _ {\mathbf {x x}} (j,:,:) & \mathbf {h} _ {\mathbf {x u}} (j,:,:) \\ \mathbf {h} _ {\mathbf {u x}} (j,:,:) & \mathbf {h} _ {\mathbf {u u}} (j,:,:) \end{array} \right] \left[ \begin{array}{c} \mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d} \\ \mathbf {u} _ {t + 1} \end{array} \right] + \frac {1}{2} h _ {\sigma \sigma} (j, 1) \sigma^ {2} \\ & + \frac {1}{6} \left[ \begin{array}{c c} (\mathbf {x} _ {t} ^ {f}) ^ {\prime} + (\mathbf {x} _ {t} ^ {s}) ^ {\prime} + (\mathbf {x} _ {t} ^ {r d}) ^ {\prime} & \mathbf {u} _ {t + 1} \end{array} \right] \left[ \begin{array}{c c} [ (\mathbf {x} _ {t} ^ {f}) ^ {\prime} + (\mathbf {x} _ {t} ^ {s}) ^ {\prime} + (\mathbf {x} _ {t} ^ {r d}) ^ {\prime} & \mathbf {u} _ {t + 1} ] \mathbf {h _ {v v v}} (j, 1,:,:) [ \begin{array}{c} \mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d} \\ \mathbf {u} _ {t + 1} \end{array} ] \\ & ... \\ [ (\mathbf {x} _ {t} ^ {f}) ^ {\prime} + (\mathbf {x} _ {t} ^ {s}) ^ {\prime} + (\mathbf {x} _ {t} ^ {r d}) ^ {\prime} & \mathbf {u} _ {t + 1} ] \mathbf {h _ {v v v}} (j, n _ {v},:,:) [ \begin{array}{c} \mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d} \\ \mathbf {u} _ {t + 1} \end{array} ] \\ & + \frac {3}{6} \mathbf {h _ {\sigma \sigma v}} (j,:) \sigma^ {2} [ \begin{array}c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c & \mathbf {x _ {\sigma}}. \\ & \mathbf {\Sigma_ {\sigma}}. \\ & \mathbf {\Sigma_ {\sigma}}. \\ & \mathbf {\Sigma_ {\sigma}}. \\ & \mathbf {\Sigma_ {\sigma}}. \\ & \mathbf {\Sigma_ {\sigma}}. \\ & \mathbf {\Sigma_ {\sigma}}. \\ & \mathbf {\Sigma_ {\sigma}}. \\ & \mathbf {\Sigma_ {\sigma}}. \\ & \mathbf {\Sigma_ {\sigma}} . \\ & \mathbf {\Sigma_ {\sigma}}. \\ & \mathbf {\Sigma_ {\sigma}}. \\ & \mathbf {\Sigma_ {\sigma}}. \\ & \mathbf {\Sigma_ {\sigma}}. \\ & \mathbf {\Sigma_ {\sigma}}. \\ & \mathbf {\Sigma_ {\sigma}}. \\ & \mathbf {\Sigma_ {\sigma}}. \\ & \mathbf {\Sigma_ {\sigma}}. \\ & \end{array}\]

\[\begin{array}{l} x _ {t + 1} ^ {f} (j, 1) + x _ {t + 1} ^ {s} (j, 1) + x _ {t + 1} ^ {r d} (j, 1) = \mathbf {h} _ {\mathbf {x}} (j,:) (\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d}) + \mathbf {h} _ {\mathbf {u}} (j,:) \mathbf {u} _ {t + 1} \\ \qquad + \frac {1}{2} \left(\left(\mathbf {x} _ {t} ^ {f}\right) ^ {\prime} + (\mathbf {x} _ {t} ^ {s}) ^ {\prime} + (\mathbf {x} _ {t} ^ {r d}) ^ {\prime}\right) \mathbf {h} _ {\mathbf {x x}} (j,:,:) (\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d}) + \frac {1}{2} \mathbf {u} _ {t + 1} ^ {\prime} \mathbf {h} _ {\mathbf {u x}} (j,:,:) (\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d}) \\ \qquad + \frac {1}{2} \left(\left(\mathbf {x} _ {t} ^ {f}\right) ^ {\prime} + (\mathbf {x} _ {t} ^ {s}) ^ {\prime} + (\mathbf {x} _ {t} ^ {r d}) ^ {\prime}\right) \mathbf {h} _ {\mathbf {x u}} (j,:,:) \mathbf {u} _ {t + 1} + \frac {1}{2} \mathbf {u} _ {t + 1} ^ {\prime} \mathbf {h} _ {\mathbf {u u}} (j,:,:) \mathbf {u} _ {t + 1} \\ \qquad + \frac {1}{2} h _ {\sigma \sigma} (j, 1) \sigma^ {2} \end{array}\]

\[\begin{array}{r l} & {+ \frac {1}{6} \left[ \begin{array}{l l} \left(\mathbf {x} _ {t} ^ {f}\right) ^ {\prime} + (\mathbf {x} _ {t} ^ {s}) ^ {\prime} + \left(\mathbf {x} _ {t} ^ {r d}\right) ^ {\prime} & \mathbf {u} _ {t + 1} \end{array} \right] \left[ \begin{array}{l l} \left(\mathbf {x} _ {t} ^ {f}\right) ^ {\prime} + (\mathbf {x} _ {t} ^ {s}) ^ {\prime} + \left(\mathbf {x} _ {t} ^ {r d}\right) ^ {\prime} & \mathbf {u} _ {t + 1} \end{array} \right] \mathbf {h} _ {\mathbf {v v v}} (j, 1,:,:): \left[ \begin{array}{c} \mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d} \\ \mathbf {u} _ {t + 1} \end{array} \right]} \\ & {\qquad \qquad \qquad \dots} \\ & {\left[ \begin{array}{l l} \left(\mathbf {x} _ {t} ^ {f}\right) ^ {\prime} + (\mathbf {x} _ {t} ^ {s}) ^ {\prime} + \left(\mathbf {x} _ {t} ^ {r d}\right) ^ {\prime} & \mathbf {u} _ {t + 1} \end{array} \right] \mathcal {h} _ {\mathbf {v v v}} (j, n _ {v},:,:): \left[ \begin{array}{c} \mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d} \\ \mathbf {u} _ {t + 1} \end{array} \right]} \\ & {+ \frac {3}{6} \mathbf {h} _ {\sigma \sigma \mathbf {v}} (j,:) \sigma^ {2} \left[ \begin{array}{c} \mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d} \\ \mathbf {u} _ {t + 1} \end{array} \right] + \frac {1}{6} h _ {\sigma \sigma \sigma} (j, 1) \sigma^ {3}} \end{array}\]

Without reducing the large term for with , it is straightforward to see that the law of motion for that only preserves third order terms is

\[\begin{array}{r l} & x _ {t + 1} ^ {r d} (j, 1) = \mathbf {h} _ {\mathbf {x}} (j,:) \mathbf {x} _ {t} ^ {r d} + \\ & \qquad + \frac {1}{2} (\mathbf {x} _ {t} ^ {s}) ^ {\prime} \mathbf {h} _ {\mathbf {x x}} (j,:,: ) \mathbf {x} _ {t} ^ {f} + \frac {1}{2} (\mathbf {x} _ {t} ^ {f}) ^ {\prime} \mathbf {h} _ {\mathbf {x x}} (j,:,: ) \mathbf {x} _ {t} ^ {s} + \frac {1}{2} \mathbf {u} _ {t + 1} ^ {\prime} \mathbf {h} _ {\mathbf {u x}} (j,:,: ) \mathbf {x} _ {t} ^ {s} \\ & \qquad + \frac {1}{2} (\mathbf {x} _ {t} ^ {s}) ^ {\prime} \mathbf {h} _ {\mathbf {x u}} (j,:,: ) \mathbf {u} _ {t + 1} \\ & \qquad + \frac {1}{6} \left[ \begin{array}{c c} (\mathbf {x} _ {t} ^ {f}) ^ {\prime} & \mathbf {u} _ {t + 1} \end{array} \right] \left[ \begin{array}{c c} [ (\mathbf {x} _ {t} ^ {f}) ^ {\prime} & \mathbf {u} _ {t + 1} ] \mathbf {h} _ {\mathbf {v v v}} (j, 1,:,: ) [ \begin{array}{c c} \mathbf {x} _ {t} ^ {f} \\ \mathbf {u} _ {t + 1} \end{array} ] \\ & \qquad \dots \\ & [ (\mathbf {x} _ {t} ^ {f}) ^ {\prime} & \mathbf {u} _ {t + 1} ] \mathbf {h} _ {\mathbf {v v v}} (j, n _ {v},,:,: ) [ \begin{array}{c c} \mathbf {x} _ {t} ^ {f} \\ \mathbf {u} _ {t + 1} \end{array} ] \end{array} \right] \\ & + \frac {3}{6} \mathbf {h} _ {\sigma \sigma \tilde {\mathbf {x}}} (j,:) \sigma^ {2} \left[ \begin{array}{c} \mathbf {x} _ {t} ^ {f} \\ \mathbf {u} _ {t + 1} \end{array} \right] + \frac {1}{6} h _ {\sigma \sigma \sigma} (j, 1) \sigma^ {3} \end{array}\]

Using the convenient representation we thus have

\[\begin{array}{r c l} \mathbf {x} _ {t + 1} ^ {r d} & = & \mathbf {h} _ {\mathbf {v}} \left[ \begin{array}{c} \mathbf {x} _ {t} ^ {r d} \\ \mathbf {0} \end{array} \right] \\ & & + 2 \tilde {\mathbf {H}} _ {\mathbf {v v}} \left(\left[ \begin{array}{c} \mathbf {x} _ {t} ^ {f} \\ \mathbf {u} _ {t + 1} \end{array} \right] \otimes \left[ \begin{array}{c} \mathbf {x} _ {t} ^ {s} \\ \mathbf {0} \end{array} \right]\right) \\ & & + \tilde {\mathbf {H}} _ {\mathbf {v v v}} \left(\left[ \begin{array}{c} \mathbf {x} _ {t} ^ {f} \\ \mathbf {u} _ {t + 1} \end{array} \right] \otimes \left[ \begin{array}{c} \mathbf {x} _ {t} ^ {f} \\ \mathbf {u} _ {t + 1} \end{array} \right] \otimes \left[ \begin{array}{c} \mathbf {x} _ {t} ^ {f} \\ \mathbf {u} _ {t + 1} \end{array} \right]\right) \\ & & + \frac {3}{6} \mathbf {h} _ {\sigma \sigma \mathbf {v}} \sigma^ {2} \left[ \begin{array}{c} \mathbf {x} _ {t} ^ {f} \\ \mathbf {u} _ {t + 1} \end{array} \right] + \frac {1}{6} \mathbf {h} _ {\sigma \sigma \sigma} \sigma^ {2} \end{array}\tag{65}\]

It is by now straightforward to see that an expression for which only preserves up to third order terms are:

\[\begin{array}{r c l} \mathbf {y} _ {t} & = & \mathbf {g} _ {\mathbf {v}} \left(\left[ \begin{array}{c} \mathbf {x} _ {t - 1} ^ {f} \\ \mathbf {u} _ {t} \end{array} \right] + \left[ \begin{array}{c} \mathbf {x} _ {t - 1} ^ {s} \\ \mathbf {0} \end{array} \right] + \left[ \begin{array}{c} \mathbf {x} _ {t - 1} ^ {r d} \\ \mathbf {0} \end{array} \right]\right) \\ & & + \tilde {\mathbf {G}} _ {\mathbf {v v}} \left(\left(\left[ \begin{array}{c} \mathbf {x} _ {t - 1} ^ {f} \\ \mathbf {u} _ {t} \end{array} \right] \otimes \left[ \begin{array}{c} \mathbf {x} _ {t - 1} ^ {f} \\ \mathbf {u} _ {t} \end{array} \right]\right) + 2 \left(\left[ \begin{array}{c} \mathbf {x} _ {t - 1} ^ {f} \\ \mathbf {u} _ {t} \end{array} \right] \otimes \left[ \begin{array}{c} \mathbf {x} _ {t - 1} ^ {s} \\ \mathbf {0} \end{array} \right]\right)\right) \\ & & + \tilde {\mathbf {G}} _ {\mathbf {v v v}} \left(\left[ \begin{array}{c} \mathbf {x} _ {t - 1} ^ {f} \\ \mathbf {u} _ {t} \end{array} \right] \otimes \left[ \begin{array}{c} \mathbf {x} _ {t - 1} ^ {f} \\ \mathbf {u} _ {t} \end{array} \right] \otimes \left[ \begin{array}{c} \mathbf {x} _ {t - 1} ^ {f} \\ \mathbf {u} _ {t} \end{array} \right]\right) \\ & & + \frac {1}{2} \mathbf {g} _ {\sigma \sigma} \sigma^ {2} + \frac {3}{6} \mathbf {g} _ {\sigma \sigma \mathbf {v}} \sigma^ {2} \left[ \begin{array}{c} \mathbf {x} _ {t - 1} ^ {f} \\ \mathbf {u} _ {t} \end{array} \right] + \frac {1}{6} \mathbf {g} _ {\sigma \sigma \sigma} \sigma^ {2} \end{array}\tag{66}\]

7 Dynare++ notation and statistical properties: second order

7.1 Co-variance stationarity

We start with the state variables. From above we have for the first-order effects that

\[\begin{array} { r l } & x _ { t + 1 } ^ { f } \left( j , 1 \right) = \mathbf { h _ { x } } \left( j , : \right) \mathbf { x } _ { t } ^ { f } + \mathbf { h _ { u } } \left( j , : \right) \mathbf { u } _ { t + 1 } \\ & \Downarrow \\ & \mathbf { x } _ { t + 1 } ^ { f } = \mathbf { h _ { x } } \mathbf { x } _ { t } ^ { f } + \mathbf { h _ { u } } \mathbf { u } _ { t + 1 } \\ & \text {For the second - order effects} \\ & x _ { t + 1 } ^ { s } \left( j , 1 \right) = \mathbf { h _ { x } } \left( j , : \right) \mathbf { x } _ { t } ^ { s } + \frac { 1 } { 2 } \left( \mathbf { x } _ { t } ^ { f } \right) ^ { \prime } \mathbf { h _ { x x } } \left( j , : , : \right) \left( \mathbf { x } _ { t } ^ { f } \right) + \frac { 1 } { 2 } \mathbf { u } _ { t + 1 } ^ { \prime } \mathbf { h _ { u x } } \left( j , : , : \right) \mathbf { x } _ { t } ^ { f } \\ & \qquad + \frac { 1 } { 2 } \left( \mathbf { x } _ { t } ^ { f } \right) ^ { \prime } \mathbf { h _ { x u } } \left( j , : , : \right) \mathbf { u } _ { t + 1 } + \mathbf { u } _ { t + 1 } ^ { \prime } \mathbf { h _ { u u } } \left( j , : , : \right) \mathbf { u } _ { t + 1 } + \frac { 1 } { 2 } h _ { \sigma \sigma } \left( j , 1 \right) \sigma ^ { 2 } \\ & \Downarrow \\ & \mathbf { x } _ { t + 1 } ^ { s } = \mathbf { h _ { x } } \mathbf { x } _ { t } ^ { s } + \tilde { \mathbf { H _ { x x } } } \left( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \right) + \mathbf { u } _ { t + 1 } ^ { \prime } \mathbf { h _ { u x } } \left( j , : , : \right) \left( \mathbf { x } _ { t } ^ { f } \right) + \left( \mathbf { x } _ { t } ^ { f } \right) ^ { \prime } \mathbf { h _ { x u } } \left( j , : , : \right) \mathbf { u } _ { t + 1 } + \tilde { \mathbf { H _ { u u } } } ( \mathbf { u } _ { t + 1 } \otimes \mathbf { u } _ { t + 1 } ) + \frac { 1 } { 2 } \mathbf { h _ { \sigma \sigma } }\sigma ^ { 2 } \\ & \Updownarrow \\ & \mathbf { x } _ { t + 1 } ^ { s } = \mathbf { h _ { x } } \mathbf { x } _ { t } ^ { s } + \tilde { \mathbf { H _ { x x } } } ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) + \tilde { \mathbf { H _ { u x } } } ( \mathbf { u } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } ) + \tilde { \mathbf { H _ { x u } } } ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { u } _ { t + 1 } ) + \tilde { \mathbf { H _ { u u } } } ( \mathbf { u } _ { t + 1 } \otimes \mathbf { u } _ { t + 1 } ) + \frac { 1 } { 2 } \mathbf { h _ { \sigma \sigma }\sigma ^ { 2} } \\ & \Updownarrow \\ & \mathbf { x } _ { t + 1 } ^ { s } = \mathbf { h _ { x } } \mathbf { x } _ { t } ^ { s } + \tilde { \mathbf { H _ { x x } } } ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f )} ) + \tilde { \mathbf { H _ { u x} } } ( \mathbf { u} _ { t + 1 } \otimes \mathbf { x} _ { t } ^ { f )} ) + \tilde { \mathbf { H _ { x u} } } ( \mathbf { x} _ { t } ^ { f } \otimes \mathbf { u} _ { t + 1 )} ) + \tilde { \mathbf { H _ { u u} } } ( \mathbf { u} _ { t + 1 } \otimes \mathbf { u} _ { t + 1 - v e c ( | | | ) - v e c ( | | | ) - v e c ( | | | ) ) ) + \frac { 1}{ 2} | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | & \\ & + {\tilde {\mathbf {\Delta H}}} _ {\mathrm{uu}} v e c ( \| S \| ) + {\frac 12} {\tilde {\mathbf {\Delta H}}} _ {\mathrm{uuc}} v e c ( \| S \| ) . \\ & \\ & + {\tilde {\mathbf {\Delta H}}} _ {\mathrm{uu}} v e c ( \| S \| ) + {\frac 12} {\tilde {\mathbf {\Delta H}}} _ {\mathrm{uuc}} v e c ( \| S \| ) . \\ & \\ & + {\tilde {\mathbf {\Delta H}}} _ {\mathrm{uu}} v e c ( \| S \| ) + {\frac 12} {\tililde {\mathbf {\Delta H}}} _ {\mathrm{uuc}} v e c ( \| S \| ) . \\ & \\ & + {\tilde {\mathbf {\Delta H}}} _ {\mathrm{uu}} v e c ( \| S \| ) + {\frac 12} {\tilde {\mathbf {\Delta H}}} _ {\mathrm{uuc}} v e c ( \| S \| ) . \\ & \\ & + {\tilde {\mathbf {\Delta H}}}. \\ & \\ & + {\tilde {\mathbf {\Delta H}}} _ {\mathrm{uu}} v e c ( \| S \| ) + {\frac 12} {\tilde {\mathbf {\Delta H}}} _ {\mathrm{uuc}} v e c ( \| S \| ) . \\ & \\ & + {\tilde {\mathbf {\Delta H}}} _ {\mathrm{uu}} v e c ( \| S \| ) + {\frac 12} {\hat {\boldsymbol {\chi}}} _ {\mathrm{uuc}} v e c ( \| S \| ) . \\ & \\ & + {\tilde {\boldsymbol {\chi}}} _ {\mathrm{uuc}} v e c ( \| S \| ) + {\frac 12} {\hat {\boldsymbol {\chi}}} _ {\mathrm{uuc}} v e c ( \| S \| ) . \\ & \\ & + {\tilde {\boldsymbol {\chi}}} _ {\mathrm{uuc}} v e c ( \| S \| ) + {\frac 12} {\hat {\boldsymbol {\chi}}} _ {\mathrm{uuc}} v e c ( \| S \| ) , \\ & \\ & + {\tilde {\boldsymbol {\chi}}} _ {\mathrm{uuc}} v e c ( \| S \| ) + {\frac 12} {\hat {\boldsymbol {\chi}}} _ {\mathrm{uuc}} v e c ( \| S \| ) . \\ & \\ & + {\tilde {\boldsymbol {\chi}}} _ {\mathrm{uuc}} v e c ( \| S \| ) + {\frac 12} {\hat {\bm {{\chi}}}} _ {\mathrm{uuc}} v e c ( \| S \| ) . \\ & \\ & + {\tilde {\boldsymbol {\chi}}} _ {\mathrm{uuc}} v e c ( \| S \| ) + {\frac 12} {\hat {\bm {{\chi}}}} _ {\mathrm{uuc}} v e c ( \| S \| ) . \\ & \\ & + {\tilde {\boldsymbol {\chi}}} _ {\mathrm{uuc}} v i n g (\| S \| ) + {\frac 12} {\hat {\bm {{\chi}}}} _ {\mathrm{uuc}} v i n g (\| S \| ) . \\ & \\ & + {\tilde {\boldsymbol {\chi}}} _ {\mathrm{uuc}} v i n g (\| S \| ) + {\frac 12} {\hat {\bm {{\chi}}}} _ {\mathrm{uuc}} v i n g (\| S \| ) . \\ & \\ & + {\tilde {\boldsymbol {\chi}}} _ {\mathrm{uuc}} v i n g (\| S \| ) + {\hat {{\bm {{\chi}}}}} _ {\mathrm{uuc}} v i n g (\| S \| ) . \\ & \\ & + {\tilde {{\bm {{\chi}}}}} _ {\mathrm{uuc}} v i n g (\| S \| ) + {\hat {{\bm {{\chi}}}}} _ {\mathrm{uuc}} v i n g (\| S \| ) . \\ & \\ & + {\tilde {{\bm {{\chi}}}}} _ {\mathrm{uuc}} v i n g (\| S \| ) + {\hat {{\bm {{\chi}}}}} _ {\mathrm{uuc}} v i n g (| / / - v e c (\| S \|)) + {\hat {{\bm {{\chi}}}}} _ {\mathrm{uuc}} v i n g (\| S \| ) . \\ & \\ & + {\tilde {{\bm {{\chi}}}}} _ {\mathrm{uuc}} v i n g (\| S \| ) + {\hat {{\bm {{\chi}}}}} _ {\mathrm{uuc}} v i n g (\| S \| ) . \\ & \\ & + {\tilde {{\bm {}^*}}} _ {\mathrm{uuc}} v i n g (\| S \| ) + {\hat {{\bm {}^*}}} _ {\mathrm{uuc}} v i n g (\| S \| ) . \\ & \\ & + {\tilde {{\bm {}^*}}} _ {\mathrm{uuc}} v i n g (\| S \| ) + {\hat {{\bm {}^*}}} _ {\mathrm{uuc}} v i n g (\| S \| ) . \\ & \\ & + {\tilde {{\bm {}^*}}} _ {\mathrm{uuc}} v / i n g (\| S \| ) + {\hat {{\bm {}^*}}} _ {\mathrm{uuc}} v / i n g (\| S \| ) . \\ & \\ & + {\tilde {{\bm {}^*}}} _ {\mathrm{uuc}} v / i n g (\| S \| ) + {\hat {{\bm {}^*}}} _ {\mathrm{uuc}} v / i n g (\| S \| ) . \\ & \\ & + B. \\ & \\ & - B. \\ & \\ & - B. \\ & \\ & - B. \\ & \\ & - B. \\ & \\ & - B. \\ & \\ & - B. \\ & \\ & - B. \\ & \\ & - B. \\ & \\ & - B. \\ & \\ & - B. \\ & \\ & - B. \\ & \\ & - B. \\ & \\ & - B. \\ & \\ & - B. \\ & \\ & - B . \\ & \\ & - B . \\ & \\ & - B . \\ & \\ & - B . \\ & \\ & - B . \\ & \\ & - B . \\ & \\ & - B . \\ & \\ & - B . \\ & \\ & - B . \\ & \\ & - B . \\ & \\ & - B . \\ & \\ & - B . \\ & \\ & - B . \\ & \\ & - B . \\ & \\ & - B . \\ * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *} \\ & \\ & - B. \\ & \\ & - B. \\ & \\ & - B. \\ & \\ & - B. \\ & \\ & - B. \\ & \\ & - B. \\ & \\ & - B. \\ & \\ & - B. \\ & \\ & - B. \\ & \\ & - B. \\ & \\ & - B. \\ & \\ & - B. \\ & \\ & - B. \\ & \\ & - B. \\ and a b d i s i n d o w i n d o w i n d o w i n d o w i n d o w i n d o w i n d o w i n d o w i n d o w i n d o w i n d o w i n d o w i n d o w i n d o w i n d o w i n d o w i n d o w i n d o w i n d o w i n d o w i n d o W [ I ] = I [ I ] = I [ I ] = I [ I ] = I [ I ] = I [ I ] = I [ I ] = I [ I ] = I [ I ] = I [ I ] = I [ I ] = I [ I ] = I [ I ] = I [ I ] = I [ I ] = I [ I ] = I [ I ] = I [ I ] = I [ I ] = I [ I ] = I [ I ] =I [ I ] = I [ I ] = I [ I ] = I [ I ] = I [ I ] = I [ I ] = I [ I ] = I [ I ] = I [ I ] = I [ I ] = I [ I ] = I [ I ] = I [ I ] = I [ I ] = I [ I ] = I [ I ] = I [ I ] = I [ I ] = I [ I ] = I [ I ] = N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,M ,N ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M , M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ; M a l l a m a l l a m a l l a m a l l a m a l l a m a l l a m a l l a m a l l a m a l l a m a l l a m a l l a m a l l a m a l l a m a l l a m a l l a m a l l a m a l l a m a l l a m a l l a m a l l a m a l l a l l a m a l l a m a l l a m a l l a m a l l a m a l l a m a l l a m a l l a m a l l a m a l l a m a l l a m a l l a m a l l a m a l l a m a l l a m a l l a m a l l a m a l l a m a l l a m a l l a M A L A M A L A M A L A M A L A M A L A M A L A M A L A M A L A M A L A M A L A M A L A M A L A M A L A M A L A M A L A M A L A M A L A M A L A M A L A M A L A M A L A M A L A M A L A M A L A M A L A M A LA M A L A M A L A M A L A M A L A M A L A M A L A M A L A M A L A M A L A M A L A M A L A M A L A M A L A M A L A M A L A M A L A M A L A M A L A M A L A M A L A M A L A M A L A M A L A M A L A M A L y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y yy< nl>\]

where we have defined

\[\begin{array}{r l} & {\tilde {\mathbf {H}} _ {\mathbf {x x}} \equiv \frac {1}{2} r e s h a p e (\mathbf {h} _ {\mathbf {x x}}, n _ {x}, n _ {x} ^ {2})} \\ & {\tilde {\mathbf {H}} _ {\mathbf {u u}} \equiv \frac {1}{2} r e s h a p e (\mathbf {h} _ {\mathbf {u u}}, n _ {x}, n _ {u} ^ {2})} \\ & {\tilde {\mathbf {H}} _ {\mathbf {u x}} \equiv \frac {1}{2} r e s h a p e (\mathbf {h} _ {\mathbf {u x}}, n _ {x}, n _ {u} n _ {x})} \\ & {\tilde {\mathbf {H}} _ {\mathbf {x u}} \equiv \frac {1}{2} r e s h a p e (\mathbf {h} _ {\mathbf {x u}}, n _ {x}, n _ {x} n _ {u})} \end{array}\]

\[\begin{array} { r l } & { \text {Hence, we need to find the law of motions for } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } } \\ & { \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } = \left( \mathbf { h } _ { \mathbf { x } } \mathbf { x } _ { t - 1 } ^ { f } + \mathbf { h } _ { \mathbf { u } } \mathbf { u } _ { t } \right) \otimes \left( \mathbf { h } _ { \mathbf { x } } \mathbf { x } _ { t - 1 } ^ { f } + \mathbf { h } _ { \mathbf { u } } \mathbf { u } _ { t } \right) } \\ & { = \mathbf { h } _ { \mathbf { x } } \mathbf { x } _ { t - 1 } ^ { f } \otimes \left( \mathbf { h } _ { \mathbf { x } } \mathbf { x } _ { t - 1 } ^ { f } + \mathbf { h } _ { \mathbf { u } } \mathbf { u } _ { t } \right) + \mathbf { h } _ { \mathbf { u } } \mathbf { u } _ { t } \otimes \left( \mathbf { h } _ { \mathbf { x } } \mathbf { x } _ { t - 1 } ^ { f } + \mathbf { h } _ { \mathbf { u } } \mathbf { u } _ { t} \right) } \\ & { = \mathbf { h } _ { \mathbf { x } } \mathbf { x } _ { t - 1 } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } \mathbf { x } _ { t - 1 } ^ { f } + \mathbf { h } _ { \mathbf { x } } \mathbf { x } _ { t - 1 } ^ { f } \otimes \mathbf { h } _ { \mathbf { u } } \mathbf { u } _ { t } + \mathbf { h } _ { \mathbf { u } } \mathbf { u } _ { t } \otimes \mathbf { h } _ { \mathbf { x } } \mathbf { x } _ { t - 1 } ^ { f } + \mathbf { h } _ { \mathbf { u } } \mathbf { u } _ { t } \otimes \mathbf { h } _ { \mathbf { u } } \mathbf { u } _ { t } } \\ & { = ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x} ) } ( \mathbf { x } _ { t - 1 } ^ { f } \otimes \mathbf { x } _ { t - 1 } ^ { f} ) + ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { u} ) } ( \mathbf { x } _ { t - 1 } ^ { f } \otimes \mathbf { u } _ { t} ) } \\ & { + ( \mathbf { h } _ { \mathbf { u } } \otimes \mathbf { h } _ { \mathbf { x} ) } ( \mathbf { u } _ { t } \otimes \mathbf { x } _ { t - 1 } ^ { f} ) + ( \mathbf { h } _ { \mathbf { u } } \otimes \mathbf { h } _ { \mathbf { u} ) } ( \mathbf { u } _ { t } \otimes \mathbf { u } _ { t} ) } \\ & { = ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x} ) } ( \mathbf { x } _ { t - 1 } ^ { f } \otimes \mathbf { x } _ { t - 1 } ^ { f} ) + ( \mathbf { h } _ { \mathbf { x } } \otimes {\mathbf h _ { u} ) ( x _ { t - 1} ^ {f} \otimes {\boldsymbol u} _ { t} ) }} \\ & + ( \mathbf { h _ { u} }\otimes\]

\[\begin{array}{l} = (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) + (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {u}}) (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {u} _ {t + 1}) \\ \quad + (\mathbf {h} _ {\mathbf {u}} \otimes \mathbf {h} _ {\mathbf {x}}) (\mathbf {u} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}) + (\mathbf {h} _ {\mathbf {u}} \otimes \mathbf {h} _ {\mathbf {u}}) ((\mathbf {u} _ {t + 1} \otimes \mathbf {u} _ {t + 1}) - v e c (\boldsymbol {\Sigma})) \\ \quad + (\mathbf {h} _ {\mathbf {u}} \otimes \mathbf {h} _ {\mathbf {u}}) v e c (\boldsymbol {\Sigma}) \end{array}\]

Thus we can set up the following system

\[\begin{array}{r l} & {\left[ \begin{array}{c} \mathbf {x} _ {t + 1} ^ {f} \\ \mathbf {x} _ {t + 1} ^ {s} \\ \mathbf {x} _ {t + 1} ^ {f} \otimes \mathbf {x} _ {t + 1} ^ {f} \end{array} \right] = \left[ \begin{array}{c} \mathbf {0} \\ \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2} + \tilde {\mathbf {H}} _ {\mathbf {u u}} v e c (\boldsymbol {\Sigma}) \\ (\mathbf {h} _ {\mathbf {u}} \otimes \mathbf {h} _ {\mathbf {u}}) v e c (\boldsymbol {\Sigma}) \end{array} \right] + \left[ \begin{array}{c c c} \mathbf {h} _ {\mathbf {x}} & \mathbf {0} & \mathbf {0} \\ \mathbf {0} & \mathbf {h} _ {\mathbf {x}} & \tilde {\mathbf {H}} _ {\mathbf {x x}} \\ \mathbf {0} & \mathbf {0} & \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} \end{array} \right] \left[ \begin{array}{c} \mathbf {x} _ {t} ^ {f} \\ \mathbf {x} _ {t} ^ {s} \\ \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \end{array} \right]} \\ & {\quad + \left[ \begin{array}{c c c c} \mathbf {h} _ {\mathbf {u}} & \mathbf {0} & \mathbf {0} & \mathbf {0} \\ \mathbf {0} & \tilde {\mathbf {H}} _ {\mathbf {u u}} & \tilde {\mathbf {H}} _ {\mathbf {u x}} & \tilde {\mathbf {H}} _ {\mathbf {x u}} \\ \mathbf {0} & \mathbf {h} _ {\mathbf {u}} \otimes \mathbf {h} _ {\mathbf {u}} & (\mathbf {h} _ {\mathbf {u}} \otimes \mathbf {h} _ {\mathbf {x}}) & (\mathbf {h} _ {\mathbf {u}} \otimes \mathbf {h} _ {\mathbf {u}}) \end{array} \right] \left[ \begin{array}{c} \mathbf {u} _ {t + 1} \\ \mathbf {u} _ {t + 1} \otimes \mathbf {u} _ {t + 1} - v e c (\boldsymbol {\Sigma}) \\ \mathbf {u} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \\ \mathbf {x} _ {t} ^ {f} \otimes \mathbf {u} _ {t + 1} \end{array} \right]} \end{array}\]

↓

\[\mathbf {z} _ {t + 1} = \mathbf {c} + \mathbf {A} \mathbf {z} _ {t} + \mathbf {B} \boldsymbol {\xi} _ {t + 1}\]

where we have defined

\[\begin{array}{r l} & {\mathbf {c} \equiv \left[ \begin{array}{c} \mathbf {0} \\ \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2} + \tilde {\mathbf {H}} _ {\mathbf {u u}} v e c (\boldsymbol {\Sigma}) \\ (\mathbf {h} _ {\mathbf {u}} \otimes \mathbf {h} _ {\mathbf {u}}) v e c (\boldsymbol {\Sigma}) \end{array} \right]} \\ & {\mathbf {A} \equiv \left[ \begin{array}{c c c} \mathbf {h} _ {\mathbf {x}} & \mathbf {0} & \mathbf {0} \\ \mathbf {0} & \mathbf {h} _ {\mathbf {x}} & \tilde {\mathbf {H}} _ {\mathbf {x x}} \\ \mathbf {0} & \mathbf {0} & \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} \end{array} \right]} \\ & {\mathbf {B} \equiv \left[ \begin{array}{c c c c} \mathbf {h} _ {\mathbf {u}} & \mathbf {0} & \mathbf {0} & \mathbf {0} \\ \mathbf {0} & \tilde {\mathbf {H}} _ {\mathbf {u u}} & \tilde {\mathbf {H}} _ {\mathbf {u x}} & \tilde {\mathbf {H}} _ {\mathbf {x u}} \\ \mathbf {0} & \mathbf {h} _ {\mathbf {u}} \otimes \mathbf {h} _ {\mathbf {u}} & (\mathbf {h} _ {\mathbf {u}} \otimes \mathbf {h} _ {\mathbf {x}}) & (\mathbf {h} _ {\mathbf {u}} \otimes \mathbf {h} _ {\mathbf {u}}) \end{array} \right]} \\ & {\pmb {\xi} _ {t + 1} \equiv \left[ \begin{array}{c} \pmb {\mathrm{u}} _ {t + 1} \\ \pmb {\mathrm{u}} _ {t + 1} \otimes \pmb {\mathrm{u}} _ {t + 1} - v e c (\pmb {\Sigma}) \\ \pmb {\mathrm{u}} _ {t + 1} \otimes \pmb {\mathrm{x}} _ {t} ^ {f} \\ \pmb {\mathrm{x}} _ {t} ^ {f} \otimes \pmb {\mathrm{u}} _ {t + 1} \end{array} \right]} \end{array}\]

Similar arguments as presented above ensure that all eigenvalues of have modulus less than one provided the same holds for .

For the control variables we have

\[\begin{array}{r l} & y _ {t} (i, 1) = \mathbf {g} _ {\mathbf {x}} (i,:) (\mathbf {x} _ {t - 1} ^ {f} + \mathbf {x} _ {t - 1} ^ {s}) + \mathbf {g} _ {\mathbf {u}} (i,:) \mathbf {u} _ {t} \\ & \qquad + \frac {1}{2} (\mathbf {x} _ {t - 1} ^ {f}) ^ {\prime} \mathbf {g} _ {\mathbf {x x}} (i,:,:) \mathbf {x} _ {t - 1} ^ {f} + \frac {1}{2} \mathbf {u} _ {t} ^ {\prime} \mathbf {g} _ {\mathbf {u x}} (i,:,:) \mathbf {x} _ {t - 1} ^ {f} \\ & \qquad + \frac {1}{2} (\mathbf {x} _ {t - 1} ^ {f}) ^ {\prime} \mathbf {g} _ {\mathbf {x u}} (i,:,:) \mathbf {u} _ {t} + \frac {1}{2} \mathbf {u} _ {t} ^ {\prime} \mathbf {g} _ {\mathbf {u u}} (i,:,:) \mathbf {u} _ {t} \\ & \qquad + \frac {1}{2} g _ {\sigma \sigma} (i, 1) \sigma^ {2} \\ & \Downarrow \\ & \mathbf {y} _ {t} = \mathbf {g} _ {\mathbf {x}} (\mathbf {x} _ {t - 1} ^ {f} + \mathbf {x} _ {t - 1} ^ {s}) + \mathbf {g} _ {\mathbf {u}} \mathbf {u} _ {t} + \tilde {\mathbf {G}} _ {\mathbf {x x}} (\mathbf {x} _ {t - 1} ^ {f} \otimes \mathbf {x} _ {t - 1} ^ {f}) \\ & \qquad + \tilde {\mathbf {G}} _ {\mathbf {u x}} (\mathbf {u} _ {t} \otimes \mathbf {x} _ {t - 1} ^ {f}) + \tilde {\mathbf {G}} _ {\mathbf {x u}} (\mathbf {x} _ {t - 1} ^ {f} \otimes \mathbf {u} _ {t}) + \tilde {\mathbf {G}} _ {\mathbf {u u}} (\mathbf {u} _ {t} \otimes \mathbf {u} _ {t}) + \frac {1}{2} \mathbf {g} _ {\sigma \sigma} \sigma^ {2} \\ & \qquad = \mathbf {g} _ {\mathbf {x}} (\mathbf {x} _ {t - 1} ^ {f} + \mathbf {x} _ {t - 1} ^ {s}) + \mathbf {g} _ {\mathbf {u}} \mathbf {u} _ {t} + \tilde {\mathbf {G}} _ {\mathbf {x x}} (\mathbf {x} _ { t - 1} ^ {f} \otimes \mathbf {x} _ {t - 1} ^ {f}) \end{array}\]

\[\begin{array}{r l} & {+ \tilde {\mathbf {G}} _ {\mathbf {u x}} (\mathbf {u} _ {t} \otimes \mathbf {x} _ {t - 1} ^ {f}) + \tilde {\mathbf {G}} _ {\mathbf {x u}} (\mathbf {x} _ {t - 1} ^ {f} \otimes \mathbf {u} _ {t}) + \tilde {\mathbf {G}} _ {\mathbf {u u}} ((\mathbf {u} _ {t} \otimes \mathbf {u} _ {t}) - v e c (\boldsymbol {\Sigma}))} \\ & {+ \frac {1}{2} \mathbf {g} _ {\sigma \sigma} \sigma^ {2} + \tilde {\mathbf {G}} _ {\mathbf {u u}} v e c (\boldsymbol {\Sigma})} \\ & {\mathrm{wherewehavedefined}} \\ & {\tilde {\mathbf {G}} _ {\mathbf {x x}} \equiv \frac {1}{2} r e s h a p e (\mathbf {g} _ {\mathbf {x x}}, n _ {y}, n _ {x} ^ {2})} \\ & {\tilde {\mathbf {G}} _ {\mathbf {u u}} \equiv \frac {1}{2} r e s h a p e (\mathbf {g} _ {\mathbf {u u}}, n _ {y}, n _ {u} ^ {2})} \\ & {\tilde {\mathbf {G}} _ {\mathbf {u x}} \equiv \frac {1}{2} r e s h a p e (\mathbf {g} _ {\mathbf {u x}}, n _ {y}, n _ {u} n _ {x})} \\ & {\tilde {\mathbf {G}} _ {\mathbf {x u}} \equiv \frac {1}{2} r e s h a p e (\mathbf {g} _ {\mathbf {x u}}, n _ {y}, n _ {x} n _ {u})} \end{array}\]

Thus

\[\begin{array}{r l} & {\mathbf {y} _ {t} = \frac {1}{2} \mathbf {g} _ {\sigma \sigma} \sigma^ {2} + \tilde {\mathbf {G}} _ {\mathbf {u u}} v e c (\boldsymbol {\Sigma}) + \left[ \begin{array}{l l l} \mathbf {g} _ {\mathbf {x}} & \mathbf {g} _ {\mathbf {x}} & \tilde {\mathbf {G}} _ {\mathbf {x x}} \end{array} \right] \left[ \begin{array}{c} \mathbf {x} _ {t - 1} ^ {f} \\ \mathbf {x} _ {t - 1} ^ {s} \\ \mathbf {x} _ {t - 1} ^ {f} \otimes \mathbf {x} _ {t - 1} ^ {f} \end{array} \right]} \\ & {\qquad + \left[ \begin{array}{l l l l} \mathbf {g} _ {\mathbf {u}} & \tilde {\mathbf {G}} _ {\mathbf {u u}} & \tilde {\mathbf {G}} _ {\mathbf {u x}} & \tilde {\mathbf {G}} _ {\mathbf {x u}} \end{array} \right] \left[ \begin{array}{c} \mathbf {u} _ {t} \\ \mathbf {u} _ {t} \otimes \mathbf {u} _ {t} - v e c (\boldsymbol {\Sigma}) \\ \mathbf {u} _ {t} \otimes \mathbf {x} _ {t - 1} ^ {f} \\ \mathbf {x} _ {t - 1} ^ {f} \otimes \mathbf {u} _ {t} \end{array} \right]} \\ & {\Updownarrow} \end{array}\]

\[\mathbf {y} _ {t} = \mathbf {d} + \mathbf {E z} _ {t} + \mathbf {F} \boldsymbol {\xi} _ {t}\]

7.2 First and second moments

We also see that . Hence, the first and second moments for are:

\[E \left[ \mathbf {z} _ {t} \right] = \left(\mathbf {I} - \mathbf {A}\right) ^ {- 1} \mathbf {c}\]

and for the variances we have that

\[\begin{array} { r l }& \textnormal { d } \mathrm{forthevarianceswehavethat}\\&{ E \left[ \mathbf { z } _ { t + 1 } \mathbf { z } _ { t + 1 } ^ { \prime } \right] = E \left[ \left( \mathbf { c } + \mathbf { A } \mathbf { z } _ { t } + \mathbf { B } \pmb { \xi } _ { t + 1 } \right) \left( \mathbf { c } + \mathbf { A } \mathbf { z } _ { t } + \mathbf { B } \pmb { \xi } _ { t + 1 } \right) ^ { \prime } \right] }\\&{ = E \left[ \left( \mathbf { c } + \mathbf { A } \mathbf { z } _ { t } + \mathbf { B } \pmb { \xi } _ { t + 1 } \right) \left( \mathbf { c } ^ { \prime } + \mathbf { z } _ { t } ^ { \prime } \mathbf { A } ^ { \prime } + \pmb { \xi } _ { t + 1 } ^ { \prime } \mathbf { B } ^ { \prime } \right) \right] }\\&{ = E \left[ \mathbf { c } \left( \mathbf { c } ^ { \prime } + \mathbf { z } _ { t } ^ { \prime } \mathbf { A } ^ { \prime } + \pmb { \xi } _ { t + 1 } ^ { \prime } \mathbf { B } ^ { \prime } \right) \right] }\\&{ + E \left[ \mathbf { A } \mathbf { z } _ { t } \left( \mathbf { c } ^ { \prime } + \mathbf { z } _ { t } ^ { \prime } \mathbf { A } ^ { \prime } + \pmb { \xi } _ { t + 1 } ^ { \prime } \mathbf { B } ^ { \prime } \right) \right] }\\&{ + E \left[ \mathbf { B } \pmb { \xi } _ { t + 1 } \left( \mathbf { c } ^ { \prime } + \mathbf { z } _ { t } ^ { \prime } \mathbf { A } ^ { \prime } + \pmb { \xi } _ { t + 1 } ^ { \prime } \mathbf { B } ^ { \prime } \right) \right] }\\&{ = E \left[ \mathbf { c c } ^ { \prime } + \mathbf { c z } _ { t } ^ { \prime } \mathbf { A } ^ { \prime } + \mathbf { c } \pmb { \xi } _ { t + 1 } ^ { \prime } \mathbf { B } ^ { \prime } \right] }\\&{ + E \left[ \mathbf { A z } _ { t } \mathbf { c } ^ { \prime } + \mathbf { A z } _ { t } z _ { t } ^ { \prime } \mathbf { A } ^ { \prime } + \mathbf { A z } _ { t } \pmb { \xi } _ { t + 1 } ^ { \prime } \mathbf { B } ^ { \prime } \right] }\\&{ + E \left[ \right. \mathbf { B x y a l l a b s i n g e r m a l y} [ z ] [ z ] [ z ] [ z ] [ z ] [ z ] [ z ] [ z ] [ z ] [ z ] [ z ] [ z ] [ z ] [ z ] [ z ] [ z ] [ z ] [ z ] [ z ] [ z ] [ z ] [ z ] [ z ] [ z ] [ z ] [ z ] [ z ] [ z ] [ z ] [ z ] [ z ] [ z ] [ z ] [ z ] , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . , . . . , . . . , . . . , . . . , . . , . . , . . , : , : , : , : , : , : , : , : , : , : , : , : , : , : , : , : , : , : , : , : , : , : , : , : , : , : , : , : , : , : , : , : , : , : , : , : , : , : , : , : , : , : , : , : , : , : , : | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |&{\mathrm{}}\\&{\quad + E [ A Z (z) - Z (z) ] = E [ A Z (z) - Z (z) ] = E [ A Z (z) - Z (z) ] = E [ A Z (z) - Z (z) ] = E [ A Z (z) - Z (z) ] = E [ A Z (z) - Z (z) ] = E [ A Z (z) - Z (z) ] = E [ A Z (z) - Z (z) ] = E E [ Z (z) - Z (z) ] = E E [ Z (z) - Z (z) ] = E E [ Z (z) - Z (z) ] = E E [ Z (z) - Z (z) ] = E E [ Z (z) - Z (z) ] = E E [ Z (z) - Z (z) ] = E E [ Z (z) - Z (z) ] = E E [ Z (\mathrm{的}) - Z (z) ] = E E [ Z (z) - Z (z) ] = E E [ Z (z) - Z (z) ] = E E [ Z (z) - Z (z) ] = E E [ Z (z) - Z (z) ] = E E [ Z (z) - Z (z) ] = E E [ Z (z) - Z (z) ] = E E [ Z (z) -Z (z) ] = E E [ Z (z) -Z (z) ] = E E [ Z (z) -Z (z) ] = E E [ Z (z) -Z (z) ] = E E [ Z (z) -Z (z) ] = E E [ Z (z) -Z (z) ] = E E [ Z (z) -Z (z) ] = E E [ Z (z) -Z (s) - Z (s) ] = E E [ Z (s) -Z (s) ] = E E [ Z (s) -Z (s) ] = E E [ Z (s) -Z (s) ] = E E [ Z (s) -Z (s) ] = E E [ Z (s) -Z (s) ] = E E [ Z (s) -Z (s) ] = E E [ Z (s) -Z (s) ] = E E [ Z (s) -Z (t) - Z (t) ] = E E [ Z (s) -Z (t) - Z (t) ] = E E [ Z (s) -Z (t) - Z (t) ] = E E [ Z (s) -Z (t) - Z (t) ] = E E [ Z (s) -Z (t) - Z (t) ] = E E [ Z (s) -Z (t) - Z (t) ] = E C e f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f w o r e n d e r e n d e r e n d e r e n d e r e n d e r e n d e r e n d e r e n d e r e n d e r e n d e r e n d e r e n d e r e n d e r e n d e r e n d e r e n d e r e n d e r e n d e r e n d e r e n d e r e n d i n g h a r e n d i n g h a r e n d i n g h a r e n d i n g h a r e n d i n g h a r e n d i n g h a r e n d i n g h a r e n d i n g h a r e n d i n g h a r e n d i n g h a r e n d i n g h a r e n d i n g h a w h a r e n d i n g h a w h a r e n d i n g h a w h a r e n d i n g h a w h a r e n d i n g h a w h a r e n d i n g h a w h a r e n d i n g h a w h a r e n d i n g h a w h a r e n d i n g h a w h a r u p h a w h a r e n d i n g h a w h a r e n d i n g h a w h a r e n d i n g h a w h a r e n d i n g h a w h a r e n d i n g h a w h a r e n d i n g h a w h a r e n d i n g h a w h a r e n d i n g h a w t h a w t h a w t h a w t h a w t h a w t h a w t h a w t h a w t h a w t h a w t h a w t h a w t h a w t h a w t h a w t h a w t h a w t h a w t h a w t h a w t h a w t h a w t h a w t h a w t h a w s u p p l o s i v i s u p p l o s i v i s u p p l o s i v i s u p p l o s i v i s u p p l o s i v i s u p p l o s i v i s u p p l o s i v i s u p p l o s i v i s u p p l o s i v i s u p p l o s i v i s u p p l o s i v s u p p l o s i v s u p p l o s i v s u p p l o s i v s u p p l o s i v s u p p l o s i v s u p p l o s i v s u p p l o s i v s u p p l o s i v s u p p l o s i v s u p p l o s i v s u p p l o s i v i s u p p l o s i v i s u p p l o s i v i s u p p l o s i v i s u p p l o s i v i s u p p l o s i v i s u p p l o s i v i s u p p l o s i v i s u p p l o s i v i s u p p l o s i v j u v j u v j u v j u v j u v j u v j u v j u v j u v j u v j u v j u v j u v j u v j u v j u v j u v j u v j u v j u v j u v j u v j u v j u v j u v j u v j u v j u v j u v j u v j u v j u v j u v j u u j u u j u u j u u j u u j u u j u u j u u j u u j u u j u u j u u j u u j u u j u u j u u j u u j u u j u u j u u j u u j u u j u u j u u j u u j u u j u u j u u j u u j u u j u u j u u j u u j u u k y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y yy kow kow kow kow kow kow kow kow kow kow kow kow kow kow kow kow kow kow kow kow kow kow kow kow kow kow kow kow kow kow kow kow kow kow kow kow kow kow kow kow kow kow kow kow kow kow kow kow kow kow kowerkowkowkowkowkowkowkowkowkowkowkowkowkowkowkowkowkowkowkowkowkowkowkowkowkowkowkowkowkowkowkowkowkowkowkowkowkowkowkowkowkowkowkowkowkowkowkowkowkowkowkoweon, on, on, on, on, on, on, on, on, on, on, on, on, on, on, on, on, on, on, on, on, on, on, on, on, on, on, on, on, on, on, on, on, on, on, on, on, on, on, on, on, on, on, on, on, on, on, on, on, on, on, and then. We then note that}\\&{\cal W}\\&{\cal E}\\&{\cal E}\\&{\cal W}\\&{\cal U}\\&{\cal U}\\&{\cal U}\\&{\cal U}\\&{\cal U}\\&{\cal U}\\&{\cal U}\\&{\cal U}\\&{\cal U}\\&{\cal U}\\&{\cal U}\\&{\cal U}\\&{\cal U}\\&{\cal U}\\&{\cal U}\\&{\cal U}\\&{\cal U}\\&{{\cal U}}\\&{{\cal U}}\\&{{\cal U}}\\&{{\cal U}}\\&{{\cal U}}\\&{{\cal U}}\\&{{\cal U}}\\&{{\cal U}}\\&{{\cal U}}\\&{{\cal U}}\\&{{\cal U}}\\&{{\cal U}}\\&{{\cal U}}\\&{{\cal U}}\\&{{\cal U}}\\{{\cal U}}\\&{{\cal U}}\\&{{\cal U}}\\&{{\cal U}}\\&{{\cal U}}\\&{{\cal U}}\\&{{\cal U}}\\&{{\cal U}}\\&{{\cal U}}\\&{{\cal U}}\\&{{\cal U}}\\&{{\cal U}}\\&{{\cal U}}\\&{{\cal U}}\\&{{- {\cal U}}}\\&{{\cal U}}\\&{{\cal U}}\\&{{\cal U}}\\&{{\cal U}}\\&{{\cal U}}\\&{{\cal U}}\\&{{\cal U}}\\&{{\cal U}}\\&{{\cal U}}\\&{{\cal U}}\\&{{\cal U}}\\&{{\cal U}}\\&{{\cal U}}\\&{{\cal U}}\\/ / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / )/2;}\\&{{\cal L} ({\bf R}, {\bf R}) < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 1 ;}\\&{{\cal L} ({\bf R}, {\bf R}) < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 >.}\\&{{\cal L} ({\bf R}, {\bf R}) < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 < 0 >.}\\&{{\cal L} ({\bf R}, {\bf R}) < 0 < 0 < 0 < 0 < 0 < >.}\\&{{\cal L} ({\bf R}, {\bf R}) < 0 < >.}\\&{{\cal L} ({\bf R}, {\bf R}) < >.}\\&{{\cal L} ({\bf R}, {\bf R}) < >.}\\&{{\cal L} ({\bf R}, {\bf R}) < >.}\\&{{\cal L} ({\bf R}, {\bf R}) < >.}\\&{{\cal L} ({\bf R}, {\bf R}) < >.}\\&{{\cal L} ({\bf R}, {\bf R}),}\\&{{\cal L} ({\bf R}, {\bf R}),}\\&{{\cal L} ({\bf R}, {\bf R}),}\\&{{\cal L} ({\bf R}, {\bf R}),}\\&{{\cal L} ({\bf R}, {\bf R}),}\\&{{\cal L} ({\bf R}, {\bf R}),}\\&{{\cal L} ({\bf R}, {\b m}),}\\&{{\cal L} ({\bf R}, {\b m}),}\\&{{\cal L} ({\bf R}, {\b m}),}\\&{{\cal L} ({\bf R}, {\b m}),}\\&{{\cal L} ({\bf R}, {\b m}),}\\&{{\cal L} ({\bf R}, {\b m}),}\\&{{\cal L} ({\bf R},{\b m}),}\\&{{\cal L} ({\bf R}, {\b m}),}\\&{{\cal L} ({\bf R}, {\b m}),}\\&{{\cal L} ({\bf R}, {\b m}),}\\&{{\cal L} ({\bf R}, {\b m}),}\\&{{\cal L} ({\bf R}, {\b m}),}\\&{{\cal L} ({\bf P}, {\b m}),}\\&{{\cal L} ({\bf P}, {\b m}),}\\&{{\cal L} ({\bf P}, {\b m}),}\\&{{\cal L} ({\bf P}, {\b m}),}\\&{{\cal L} ({\bf P}, {\b m}),}\\&{{\cal L} ({\bf P}, {\b m}),}\\&{\cal L} ({8), {\b m}},\\&{8), {\b m}},\\&{8), {\b m}},\\&{8), {\b m}},\\&{8), {\b m}},\\&{8), {\b m}},\\&{8), {\b m}},\\&{8), {\b m}},\\&{8), {\b m}},\\&{8), {\b m}},\\&{8), {\b m}},\\&{8), {\b m}},\\&{9), {\b m}},\\&{9), {\b m}},\\&{9), {\b m}},\\&{9), {\b m}},\\&{9), {\b m}},\\&{9), {\b m}},\\&{9), {\b m}},\\&{9), {\b m}},\\&{9), {\b m}},\\&{9), {\b m}},\\&{9), {\b m}},\\&{9)},\\&{9), {\b m}},\\&{9), {\b m}},\\&{9), {\b m}},\\&{9), {\b m}},\\&{9), {\b m}},\\&{9), {\b m}},\\&{9), {\b m}},\\&{9), {\b m}},\\&{9), {\b m}},\\&{9), {\b m}},\\&{9), {\b m}},\\&{}^ {- {\cal O}}, {}^ {- {\cal O}}, {}^ {- {\cal O}}, {}^ {- {\cal O}}, {}^ {- {\cal O}}, {}^ {- {\cal O}}, {}^ {- {\cal O}}, {}^ {- {\cal O}}, {}^ {- {\cal O}}, {}^ {- {\cal O}}, {}^ {- {\cal O}}, {}^ {- {\cal O}}, {}^ {- {\cal O}}, {}^ {- {\cal O}}, {}^ {- {\cal O}}, {}^{+}{8}, {}^{+}{8}, {}^{+}{8}, {}^{+}{8}, {}^{+}{8}, {}^{+}{8}, {}^{+}{8}, {}^{+}{8}, {}^{+}{8}, {}^{+}{8}, {}^{+}{8}, {}^{+}{8}, {}^{+}{8}, {}^{+}{8}, {}^{+}{8}, {}^{+}{8}, {}^{+}{8}, {}^{+}\end{array}\]

\[= E \left[ \begin{array}{c c c c} {\bf x} _ {t} ^ {f} {\bf u} _ {t + 1} ^ {\prime} & {\bf x} _ {t} ^ {f} \left({\bf u} _ {t + 1} \otimes {\bf u} _ {t + 1} - v e c \left(\boldsymbol {\Sigma}\right)\right) ^ {\prime} & {\bf x} _ {t} ^ {f} \left({\bf u} _ {t + 1} \otimes {\bf x} _ {t} ^ {f}\right) ^ {\prime} & {\bf x} _ {t} ^ {f} \left({\bf x} _ {t} ^ {f} \otimes {\bf u} _ {t + 1}\right) ^ {\prime} \\ {\bf x} _ {t} ^ {s} {\bf u} _ {t + 1} ^ {\prime} & {\bf x} _ {t} ^ {s} \left({\bf u} _ {t + 1} \otimes {\bf u} _ {t + 1} - v e c \left(\boldsymbol {\Sigma}\right)\right) ^ {\prime} & {\bf x} _ {t} ^ {s} \left({\bf u} _ {t + 1} \otimes {\bf x} _ {t} ^ {f}\right) ^ {\prime} & {\bf x} _ {t} ^ {s} \left({\bf x} _ {t} ^ {f} \otimes {\bf u} _ {t + 1}\right) ^ {\prime} \\ \left({\bf x} _ {t} ^ {f} \otimes {\bf x} _ {t} ^ {f}\right) {\bf u} _ {t + 1} ^ {\prime} & \left({\bf x} _ {t} ^ {f} \otimes {\bf x} _ {t} ^ {f}\right) \left({\bf u} _ {t + 1} \otimes {\bf u} _ {t + 1} - v e c \left(\boldsymbol {\Sigma}\right)\right) ^ {\prime} & \left({\bf x} _ {t} ^ {f} \otimes {\bf x} _ {t} ^ {f}\right) \left({\bf u} _ {t + 1} \otimes {\bf x} _ {t} ^ {f}\right) ^ {\prime} & \left({\bf x} _ {t} ^ {f} \otimes {\bf x} _ {t} ^ {f}\right) \left({\bf x} _ {t} ^ {f} \otimes {\bf u} _ {t + 1}\right) ^ {\prime} \end{array} \right]\]

\[= \left[ \begin{array}{c c c c} \mathbf {0} & \mathbf {0} & \mathbf {0} & \mathbf {0} \\ \mathbf {0} & \mathbf {0} & \mathbf {0} & \mathbf {0} \\ \mathbf {0} & \mathbf {0} & \mathbf {0} & \mathbf {0} \end{array} \right]\]

\[\begin{array}{r l} & {\mathrm{Thus}} \\ & {E \left[ \mathbf {z} _ {t + 1} \mathbf {z} _ {t + 1} ^ {\prime} \right] = \mathbf {c c} ^ {\prime} + \mathbf {c} E \left[ \mathbf {z} _ {t} ^ {\prime} \right] \mathbf {A} ^ {\prime} + \mathbf {A} E \left[ \mathbf {z} _ {t} \right] \mathbf {c} ^ {\prime} + \mathbf {A} E \left[ \mathbf {z} _ {t} \mathbf {z} _ {t} ^ {\prime} \right] \mathbf {A} ^ {\prime} + \mathbf {B} E \left[ \boldsymbol {\xi} _ {t + 1} \boldsymbol {\xi} _ {t + 1} ^ {\prime} \right] \mathbf {B} ^ {\prime}} \\ & {\qquad = \mathbf {c} E \left[ \mathbf {z} _ {t} ^ {\prime} \right] \mathbf {A} ^ {\prime} + \left(\mathbf {c} + \mathbf {A} E \left[ \mathbf {z} _ {t} \right]\right) \mathbf {c} ^ {\prime} + \mathbf {A} E \left[ \mathbf {z} _ {t} \mathbf {z} _ {t} ^ {\prime} \right] \mathbf {A} ^ {\prime} + \mathbf {B} E \left[ \boldsymbol {\xi} _ {t + 1} \boldsymbol {\xi} _ {t + 1} ^ {\prime} \right] \mathbf {B} ^ {\prime}} \end{array}\]

Note also that

\[\begin{array}{r l} & {\mathrm{So}} \\ & {E \left[ \mathbf {z} _ {t + 1} \mathbf {z} _ {t + 1} ^ {\prime} \right] - E \left[ \mathbf {z} _ {t} \right] E \left[ \mathbf {z} _ {t} \right] ^ {\prime} = \mathbf {c} E \left[ \mathbf {z} _ {t} ^ {\prime} \right] \mathbf {A} ^ {\prime} + \left(\mathbf {c} + \mathbf {A} E \left[ \mathbf {z} _ {t} \right]\right) \mathbf {c} ^ {\prime} + \mathbf {A} E \left[ \mathbf {z} _ {t} \mathbf {z} _ {t} ^ {\prime} \right] \mathbf {A} ^ {\prime} + \mathbf {B} E \left[ \pmb {\xi} _ {t + 1} \pmb {\xi} _ {t + 1} ^ {\prime} \right] \mathbf {B} ^ {\prime}} \\ & {\qquad - \left(\mathbf {c} + \mathbf {A} E \left[ \mathbf {z} _ {t} \right]\right) \mathbf {c} ^ {\prime} - \mathbf {c} E \left[ \mathbf {z} _ {t} ^ {\prime} \right] \mathbf {A} ^ {\prime} - \mathbf {A} E \left[ \mathbf {z} _ {t} \right] E \left[ \mathbf {z} _ {t} ^ {\prime} \right] \mathbf {A} ^ {\prime}} \\ & {\qquad = \mathbf {A} E \left[ \mathbf {z} _ {t} \mathbf {z} _ {t} ^ {\prime} \right] \mathbf {A} ^ {\prime} + \mathbf {B} E \left[ \pmb {\xi} _ {t + 1} \pmb {\xi} _ {t + 1} ^ {\prime} \right] \mathbf {B} ^ {\ast} - \mathbf {A} E \left[ \mathbf {z} _ {t} \right] E \left[ \mathbf {z} _ {t} ^ {\prime} \right] \mathbf {A} ^ {\ast}} \\ & {\qquad = \mathbf {A} \left(E \left[ \mathbf {z} _ {t} \mathbf {z} _ {t} ^ {\prime} \right] - E \left[ \mathbf {z} _ {t} \right] E \left[ \mathbf {z} _ {t} ^ {\prime} \right]\right) \mathbf {A} ^ {\ast} + \mathbf {B} E \left[ \pmb {\xi} _ {t + 1} \pmb {\xi} _ {t + 1} ^ {\prime} \right] \mathbf {B} ^ {\ast}} \\ & {\hat {\boldsymbol {\lambda}}} \\ & {- E (\boldsymbol {\xi}, t) = (0, 1) (0, t).} \end{array}\]

\[V a r \left(\mathbf {z} _ {t}\right) = \mathbf {A} V a r \left(\mathbf {z} _ {t}\right) \mathbf {A} ^ {\prime} + \mathbf {B} V a r \left(\boldsymbol {\xi} _ {t + 1}\right) \mathbf {B} ^ {\prime}\]

For the control we have directly that

\[\begin{array}{c} E \left[ \mathbf {y} _ {t} \right] = \mathbf {d} + \mathbf {E} E \left[ \mathbf {z} _ {t} \right] \\ V a r \left[ \mathbf {y} _ {t} \right] = \mathbf {E} V a r \left[ \mathbf {z} _ {t} \right] \mathbf {E} ^ {\prime} + \mathbf {F} V a r \left(\boldsymbol {\xi} _ {t}\right) \mathbf {F} ^ {\prime} \end{array}\]

where we use that . Note that we trivially have which we will use below. Hence, we only need to compute .

7.2.1 Computing Var

We have

\[\begin{array}{l} V a r \left(\boldsymbol {\xi} _ {t + 1}\right) = E \left[ \boldsymbol {\xi} _ {t + 1} \boldsymbol {\xi} _ {t + 1} ^ {\prime} \right] \\ = E \left[ \left[ \begin{array}{c} \mathbf {u} _ {t + 1} \\ \mathbf {u} _ {t + 1} \otimes \mathbf {u} _ {t + 1} - v e c (\boldsymbol {\Sigma}) \\ \mathbf {u} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \\ \mathbf {x} _ {t} ^ {f} \otimes \mathbf {u} _ {t + 1} \end{array} \right] \quad \mathbf {u} _ {t + 1} ^ {\prime} \quad \left(\mathbf {u} _ {t + 1} \otimes \mathbf {u} _ {t + 1}\right) ^ {\prime} - v e c (\mathbf {I}) ^ {\prime} \quad \left(\mathbf {u} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \quad \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {u} _ {t + 1}\right) ^ {\prime} \right] \end{array}\]

\[\begin{array} { r l } & = E \left[ \begin{array} { c c } \mathbf { u } _ { t + 1 } \mathbf { u } _ { t + 1 } ^ { \prime } & \mathbf { u } _ { t + 1 } \left( ( \mathbf { u } _ { t + 1 } \otimes \mathbf { u } _ { t + 1 } ) - v e c \left( \boldsymbol { \Sigma } \right) \right) ^ { \prime } \\ ( \mathbf { u } _ { t + 1 } \otimes \mathbf { u } _ { t + 1 } - v e c \left( \boldsymbol { \Sigma } \right) ) \mathbf { u } _ { t + 1 } ^ { \prime } & ( \mathbf { u } _ { t + 1 } \otimes \mathbf { u } _ { t + 1 } - v e c \left( \mathbf { I } \right) ) ( ( \mathbf { u } _ { t + 1 } \otimes \mathbf { u } _ { t + 1 } ) - v e c \left( \boldsymbol { \Sigma } \right) ) ^ { \prime } \\ ( \mathbf { u } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } ) \mathbf { u } _ { t + 1 } ^ { \prime } & ( \mathbf { u } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } ) ( ( \mathbf { u } _ { t + 1 } \otimes \mathbf { u } _ { t + 1 } ) - v e c \left( \boldsymbol { \Sigma } \right) ) ^ { \prime } \\ ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { u } _ { t + 1 } ) \mathbf { u } _ { t + 1 } ^ { \prime } & ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { u } _ { t + 1 } ) ( ( \mathbf { u } _ { t + 1 } \otimes \mathbf { u } _ { t + 1 } ) - v e c \left( \boldsymbol { \Sigma } \right) ) ^ { \prime } \\ \end{array} \right] \\ & = E \\ & = E \\ & = E \\ & = E \\ & = E \\ & = E \\ & = E \\ & = E \\ & = E \\ & = E \\ & = E \\ & = E \\ & = E \\ & = E \\ & = E \\ & = E \\ & = E \\ & = E \\ & = E \\ & = E \\ & = E \\ & = E \\ & = E \\ & = E \\ & = E \\ & = I n d i s e n g ( i n d i s e n g ( i n d i s e n g ( i n d i s e n g ( i n d i s e n g ( i n d i s e n g ( i n d i s e n g ( i n d i s e n g ( i n d i s e n g ( i n d i s e n g ( i n d i s e n g ( i n d i s e n g ( i i n d i s e n g ( i i n d i s e n g ( i i n d i s e n g ( i i n d i s e n g ( i i n d i s e n g ( i i n d i s e n g ( i i n d i s e n g ( i i n d i s e n g ( i i n d i s e n g ( i i n d i s e n g (i i n d i s e n g ( i i n d i s e n g ( i i n d i s e n g ( i i n d i s e n g ( i i n d i s e n g ( i i n d i s e n g ( i i n d i s e n g ( i i n d i s e n g ( i i n d i s e n g ( i i n d i s e n g ( ) ) ) ) ) ) ) ) \\ & = I n d i s e n g ( i n d i s e n g ( i n d i s e n g ( i n d i s e n g ( i n d i s e n g ( i n d i s e n g ( i n d i s e n g ( i n d i s e n g ( i n d i s e n g ( i n d i s e n g ( i n d i s e ) ) ) ) ) ) \\ & = I n d i s e I n d I n d I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I ND I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N< fcel>E [ (\mathbf { u } _ { t + 1 } , \mathbf { u } _ { t + 1 }, \mathbf { u } _ { t + 1 }, \mathbf { u } _ { t + 1 }, \mathbf { u } _ { t + 1 }, \mathbf { u } _ { t + 1 }, \mathbf { u } _ { t + 1 }, \mathbf { u } _ { t + 1 }, \mathbf { u } _ { t + 1 }, \mathbf { u } _ {t + 1 }, \mathbf { u } _ { t + 1 }, \mathbf { u } _ { t + 1 }, \mathbf { u } _ { t + 1 }, \mathbf { u } _ { t + 1 }, \mathbf { u } _ { t + 1 }, \mathbf { u } _ { t + 1 }, \mathbf { u } _ { t + 1 }, \mathbf U A B C A B C A B C A B C A B C A B C A B C A B C A B C A B C A B C A B C A B C A B C A B C A B C A B C A B C A B C A B C A B C A B C A B C A B C A B C A B C A B C A B C A B C A B C A B C A B C A B C A B C a b a l l o w h a l l o w h a l l o w h a l l o w h a l l o w h a l l o w h a l l o w h a l l o w h a l l o w h a l l o w h a l l o w h a l l o w h a l l o w h a l l o w h a l l o w h a l l o w h a l l o w h a l l l o w h a l l o w h a l l o w h a l l o w h a l l o w h a l l o w h a l l o w h a l l o w h a l l o w h a l l o w h a l l o w h a l l o w h a l l o w h a l l o w h a l l o w h a l l o w h a l l ow h a l l o w h a l l o w h a l l o w h a l l o w h a l l o w h a l l o w h a l l o w h a l l o w h a l l o w h a l l o w h a l l o w h a l l o w h a l l o w h a l l o w h a l l o w h a l l o w h a l | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | & = E \\ & = E \\ & = E \\ & = E \\ & = E \\ & = E \\ & = E \\ & = E \\ & = E \\ & = E \\ & = E \\ & = E \\ & = E \\ & = E \\ & = E \\ & = E \\ & = E \\ & = E \\ & = E \\ & = E \\ & = E \\ & = E \\ & = E \\ & = E \\ & = E; \\ & = E; \\ & = E; \\ & = E; \\ & = E; \\ & = E; \\ & = E; \\ & = E; \\ & = E; \\ & = E; \\ & = E; \\ & = E; \\ & = E; \\ & = E; \\ & = E; \\ & = E; \\ & = E; \\ & = E; \\ & = E; \\ & = E; \\ & = E. \\ & = E; \\ & = E; \\ & = E; \\ & = E; \\ & = E; \\ & = E; \\ & = E; \\ & = E; \\ & = E; \\ & = E; \\ & = E; \\ & = E; \\ & = E; \\ & = E; \\ & = E; \\ & = E; \\ & = E; \\ & = E; \\ & = E; \\ & = E, \\ & = E; \\ & = E; \\ & = E; \\ & = E; \\ & = E; \\ & = E; \\ & = E; \\ & = E; \\ & = E; \\ & = E; \\ & = E; \\ & = E; \\ & = E; \\ & = E; \\ & = E; \\ & = E; \\ & = E; \\ & = E; \\ & = E; \\ & = E ; \\ & = E ; \\ & = E ; \\ & = E ; \\ & = E ; \\ & = E ; \\ & = E ; \\ & = E ; \\ & = E ; \\ & = E ; \\ & = E ; \\ & = E ; \\ & = E ; \\ & = E ; \\ & = E ; \\ & = E ; \\ & = E ; \\ & = E ; \\ & = E ; \\ & = E ; \\ & = E . \\ & = I m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p r m p / R M P M P M P M P M P M P M P M P M P M P M P M P M P M P M P M P M P M P M P M P M P M P M P M P M P M P M P M P M P M P M P M P M P M P M P M P M P M P M P M P M P M P M P M P M P M P M P M P M P M P M F O W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W WWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwww wwwwhcma.com / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / :/ :/ :/ :/ :/ :/ :/ :/ :/ :/ :/ :/ :/ :/ :/ :/ :/ :/ :/ :/ :/ :/ :/ :/ :/ :/ :/ :/ :/ :/ :/ :/ :/ :/ :/ :/ :/ :/ :/ :/ :/ :/ :/ :/ :/ :/ :/ :/ :/ :/ :|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|llllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllccc< nl>\]

These elements can be coded directly as shown above.

8 Equivalence between the SGU-notation and the Dynare notation

This section shows the equivalence between the notation by (Schmitt-Grohé & Uribe (2004)), i.e. the SGU-notation, where innovations only enter linearly and the Dynare and Dynare ++ notation where innovations may enter in a non-linear fashion. The key observation is that the SGU-notation actually also includes the Dynare and Dynare ++ notation when extending the state vector accordingly. Recall from above that the SGU-notation reads:

\[\mathbf {y} _ {t} = \mathbf {g} (\mathbf {x} _ {t}, \sigma)\tag{67}\]

\[\mathbf {x} _ {t + 1} = \mathbf {h} (\mathbf {x} _ {t}, \sigma) + \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1}\tag{68}\]

By lagging (68) by one period we get

\[\mathbf {x} _ {t} = \mathbf {h} (\mathbf {x} _ {t - 1}, \sigma) + \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t}\tag{69}\]

and can then be considered as the extended state vector. Hence, for this extended system we thus have

\[\mathbf {y} _ {t} = \mathbf {g} (\mathbf {x} _ {t - 1}, \boldsymbol {\epsilon} _ {t}, \sigma)\tag{70}\]

\[\mathbf {x} _ {t} = \mathbf {h} _ {1} \left(\mathbf {x} _ {t - 1}, \boldsymbol {\epsilon} _ {t}, \sigma\right)\tag{71}\]

\[\boldsymbol {\epsilon} _ {t + 1} = \mathbf {u} _ {t + 1}\tag{72}\]

which is the Dynare and Dynare ++ notation with innovations entering nonlinearly. Accordingly, we can without loss of generality consider the SGU-notation.

We next illustrate how the Dynare-notation can be implemented with SGU-codes for the simple neoclassical model. The standard implementation reads:

\[f \equiv \left[ \begin{array}{c} c _ {t} + k _ {t + 1} - (1 - \delta) k _ {t} - a _ {t} k _ {t} ^ {\alpha} \\ c _ {t} ^ {- \gamma} - \beta c _ {t + 1} ^ {- \gamma} \left(a _ {t + 1} \alpha k _ {t + 1} ^ {\alpha - 1} + 1 - \delta\right) \\ \log a _ {t + 1} - \rho \log a _ {t} \end{array} \right]\]

where and . The equivalent Dynare-notation implementation is given by

\[f \equiv \left[ \begin{array}{c} c _ {t} + k _ {t + 1} - (1 - \delta) k _ {t} - a _ {t} k _ {t} ^ {\alpha} \\ c _ {t} ^ {- \gamma} - \beta c _ {t + 1} ^ {- \gamma} \left(\exp \left\{\rho \log a _ {t} + \sigma \boldsymbol {\epsilon} _ {t + 1} \right\} \alpha k _ {t + 1} ^ {\alpha - 1} + 1 - \delta\right) \\ \log a _ {t} - \rho \log a _ {t - 1} - \sigma \boldsymbol {\epsilon} _ {t} \\ \boldsymbol {\epsilon} _ {t + 1} \end{array} \right]\]

where and .

9 Existence of Skewness and Kurtosis

This section derives conditions for the existence of skewness and kurtosis in a linear system. We consider the system where A is stable and are mean-zero innovations. Thus, the pruned state-space representation for DSGE models belong to this class. For notational convience, the system is re-express in deviation from its mean as and therefore

\[\begin{array}{l} \mathbf {x} _ {t + 1} = (\mathbf {I} - \mathbf {A}) E [ \mathbf {x} ] + \mathbf {A x} _ {t} + \mathbf {v} _ {t + 1} \\ \Updownarrow \\ \mathbf {x} _ {t + 1} - E [ \mathbf {x} ] = \mathbf {A} (\mathbf {x} _ {t} - E [ \mathbf {x} ]) + \mathbf {v} _ {t + 1} \\ \Updownarrow \\ \mathbf {z} _ {t + 1} = \mathbf {A z} _ {t} + \mathbf {v} _ {t + 1} \\ \text {We then have} \\ \mathbf {z} _ {t + 1} \otimes \mathbf {z} _ {t + 1} = (\mathbf {A z} _ {t} + \mathbf {v} _ {t + 1}) \otimes (\mathbf {A z} _ {t} + \mathbf {v} _ {t + 1}) \\ = \mathbf {A z} _ {t} \otimes (\mathbf {A z} _ {t} + \mathbf {v} _ {t + 1}) + \mathbf {v} _ {t + 1} \otimes (\mathbf {A z} _ {t} + \mathbf {v} _ {t + 1}) \\ = \mathbf {A z} _ {t} \otimes \mathbf {A z} _ {t} + \mathbf {A z} _ {t} \otimes \mathbf {v} _ {t + 1} + \mathbf {v} _ {t + 1} \otimes \mathbf {A z} _ {t} + \mathbf {v} _ {t + 1} \otimes \mathbf {v} _ {t + 1} \\ \mathbf {z} _ {t + 1} \otimes \mathbf {z} _ {t + 1} \otimes \mathbf {z} _ {t + 1} = (\mathbf {A z} _ {t} \otimes \mathbf {A z} _ {t} + \mathbf {A z} _ {t} \otimes \mathbf {v} _ {t + 1} + \mathbf {v} _ {t + 1} \otimes \mathbf {A z} _ {t} + \mathbf {v} _ {t + 1} \otimes \mathbb {V v t a l e s t a m b l i c a r e} \\ = \mathbf {A z} _ {t} \otimes \mathbf {A z} _ {t} \otimes (\mathbf {A z} _ {t} + \mathbf {v} _ {t + 1}) + \mathbf {A z} _ {t} \otimes \mathbb {V v t a l e s t a m b l i c a r e} \\ = \mathbf {\Lambda A z t o m p h i n e d a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y. \\ = \mathbf {\Lambda A z t o m p h i n e d a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y. \\ = \mathbf {\Lambda A z t o m p h i n e d a l y a l y a l y a l y a l y a l y a l y a l y a l u s e d}} \\ = \mathbf {\Lambda A z t o m p h i n e d a l y a l y a l y a l y a l y a l y a l y a l y a l y. \\ + \mathbf {\Lambda A z t o m p h i n e d}} \\ = \mathbf {\Lambda A z t o m p h i n e d}} \\ = \mathbf {\Lambda A z t o m p h i n e d} \\ = \mathbf {\Lambda A z t o m p h i n e d} \\ = \mathbf {\Lambda A z t o m p h i n e d} \\ = \mathbf {\Lambda A z t o m p h i n e d} \\ = \mathbf {\Pi_ {\mathrm{max}}} \\ = \mathbf {\Pi_ {\mathrm{max}}} \\ = \mathbf {\Pi_ {\mathrm{max}}} \\ = \mathbf {\Pi_ {\mathrm{max}}} \\ = \mathbf {\Pi_ {\mathrm{max}}} \\ = \mathbf {\Pi_ {\mathrm{max}}} \\ = \mathbf {\Pi_ {\mathrm{max}}} \\ = \mathbf {\Pi_ {\mathrm{max}}} \\ = \mathbf {\Pi_ {{\mathrm{max}}}} \\ = \mathbf {\Pi_ {\mathrm{max}}} \\ = \mathbf {\Pi_ {{\mathrm{max}}}} \\ = \mathbf {\Pi_ {{\mathrm{max}}}} \\ = \mathbf {\Pi_ {{\mathrm{max}}}} \\ = \mathbf {\Pi_ {{\mathrm{max}}}} \\ = \mathbf {\Pi_ {{\mathrm{max}}}} \\ = \mathbf {\Pi_ {{\mathrm{max}}}} \\ = \mathbf {\Pi_ {{\mathrm{max}}}} \\ = \mathbf {\Pi_ {{0.05}}} \\ = \mathbf {\Pi_ {{0.05}}} \\ = \mathbf {\Pi_ {{0.05}}} \\ = \mathbf {\Pi_ {{0.05}}} \\ = \mathbf {\Pi_ {{0.05}}} \\ = \mathbf {\Pi_ {{0.05}}} \\ = \mathbf {\Pi_ {{0.05}}} \\ = \mathbf {\Pi_ {{0.05}}} \\ ^ - 1, 2, 3, 4, 5, 6, 7, 8, 9, 1 0, 1 1, 1 2, 1 3, 1 4, 1 5, 1 6, 1 7, 1 8, 19, 2 0, 2 1, 2 2, 2 3, 2 4, 2 5, 2 6, 2 7, 2 8, 29, 3 0, 3 1, 3 2, 3 3, 3 4, 3 5, 3 6, 3 7, 3 8, 39, 4 0, 4 1, 4 2, 4 3, 4 4, 4 5, 4 6, 4. ^ {- 1}, ^ {- 1}, ^ {- 1}, ^ {- 1}, ^ {- 1}, ^ {- 1}, ^ {- 1}, ^ {- 1}, ^ {- 1}, ^ {- 1}, ^ {- 1}, ^ {- 1}, ^ {- 1}, ^ {- 1}, ^ {- 1}, ^ {- 1}, ^ {- 1}, ^ {- 1}, ^ {- 1}, ^ {- 1}, ^ {- 1}. ^ {- 1}, ^ {- 1}, ^ {- 1}, ^ {- 1}, ^ {- 1}, ^ {- 1}, ^ {- 1}. ^ {- (0.05)} < . ^ {- (0.05)} < . ^ {- (0.05)} < . ^ {- (0.05)} < . ^ {- (0.05)} < . ^ {- (0.05)} < . ^ {- (0.05)} < . ^ {- (0.05)} < . ^ {- (0.05)} < . ^ {- (0.05)} < . ^ {- (0.05)} < .. > ^ {(0.05)} < .. > ^ {(0.05)} < .. > ^ {(0.05)} < .. > ^ {(0.05)} < .. > ^ {(0.05)} < .. > ^ {(0.05)} < .. > ^ {(0.05)} < .. > ^ {(0.05)} < .. > ^ {(0.05)} < .. > ^ {(0.05)} < .. > ^{(0.05)} < .. > ^ {(0.05)} < .. > ^ {(0.05)} < .. > ^ {(0.05)} < .. > ^ {(0.05)} < .. > ^ {(0.05)} < .. > ^ {(0.05)} < .. > ^ {(0.05)} < .. > ^ {(0.05)} < .. > ^ {(0.05)} < .. > ^ (0.05) < .. > ^ {(0.05)} < .. > ^ {(0.05)} < .. > ^ {(0.05)} < .. > ^ {(0.05)} < .. > ^ {(0.05)} < .. > ^ {(0.05)} < .. > ^ {(0.05)} < .. > ^ {(0.05)} < .. > ^ {(0.05)} < .. > ^ {(0.05) < .. > ^ {(0.05)}}< |content_end|>\]

Thus, to solve for the innovations need to have a finite third moment. At second order, is function of , meaning that must have finite sixth moment. At third order, is function of , meaning that must have finite ninth moment.

\[\mathbf {z} _ {t + 1} \otimes \mathbf {z} _ {t + 1} \otimes \mathbf {z} _ {t + 1} \otimes \mathbf {z} _ {t + 1}\]

\[\begin{array} { l } = ( \mathbf { A } \mathbf { z } _ { t } \otimes \mathbf { A } \mathbf { z } _ { t } \otimes \mathbf { A } \mathbf { z } _ { t } + \mathbf { A } \mathbf { z } _ { t } \otimes \mathbf { A } \mathbf { z } _ { t } \otimes \mathbf { v } _ { t + 1 } \\ + \mathbf { A } \mathbf { z } _ { t } \otimes \mathbf { v } _ { t + 1 } \otimes \mathbf { A } \mathbf { z } _ { t } + \mathbf { A } \mathbf { z } _ { t } \otimes \mathbf { v } _ { t + 1 } \otimes \mathbf { v } _ { t + 1 } \\ + \mathbf { v } _ { t + 1 } \otimes \mathbf { A } \mathbf { z } _ { t } \otimes \mathbf { A } \mathbf { z } _ { t } + \mathbf { v } _ { t + 1 } \otimes \mathbf { A } \mathbf { z } _ { t } \otimes \mathbf { v } _ { t + 1 } \\ + \mathbf { v } _ { t + 1 } \otimes \mathbf { v } _ { t + 1 } \otimes \mathbf { A } \mathbf { z } _ { t } + \mathbf { v } _ { t + 1 } \otimes \mathbf { v } _ { t + 1 } \otimes \mathbf { v } _ { t + 1 } ) \otimes ( \mathbf { A } \mathbf { z } _ { t } + \mathbf { v } _ { t + 1 } ) \\ = \mathbf { A } \mathbf { z } _ { t } \otimes \mathbf { A } \mathbf { z } _ { t } \otimes \mathbf { A } \mathbf { z } _ { t } \otimes ( \mathbf { A } \mathbf { z } _ { t } + \mathbf { v } _ { t + 1 } ) + \mathbf { A } \mathbf { z } _ { t } \otimes \mathbf { A } \mathbf { z } _ { t } \otimes \mathbf { v } _ { t + 1 } \otimes ( \mathbf { A } \mathbf { z } _ { t } + \mathbf { v } _ { t + 1 } ) \\ + \mathbf { A } \mathbf { z } _ { t } \otimes \mathbf { v } _ { t + 1 } \otimes \mathbf { A } \mathbf { z } _ { t } \otimes ( \mathbf { A } \mathbf { z } _ { t } + \mathbf { v } _ { t + 1 } ) + \mathbf { A } \mathbf { z } _ { t } \otimes \mathbf { v } _ { t + 1 } \otimes \mathbf { v } _ { t + 1 } \otimes ( \mathbf { A } \mathbf { z } _ { t } + \mathbf { v } _ { t + 1 } ) \\ + \mathbf { v } _ { t + 1 } \otimes \mathbf { A } \mathbf { z } _ { t } \otimes \mathbf { A } \mathbf { z } _ { t } \otimes ( \mathbf { A } \mathbf { z } _ { t } + \mathbf { v } _ { t + 1 } ) + \mathbf { v } _ { t + 1 } \otimes \mathbf { A } \mathbf { z } _ { t } \otimes \mathbf { v } _ { t + 1 } \otimes ( \mathbf { A } \mathbf { z } _ { t } + \mathbf { v } _ { t + 1 } ) \\ + \mathbf { v } _ { t + 1 } \otimes \mathbf { v } _ { t + 1 } \otimes \mathbf { A } \mathbf { z } _ { t } \otimes ( \mathbf { A } \mathbf { z } _ { t } + \mathbf { v } _ { t + 1 } ) + \mathbf { v } _ { t + 1 } \otimes \mathbf { v } _ { t + 1 } \otimes ( \mathbf { A } \mathbf { z } _ { t } + \mathbf { v } _ { t + 1} ) \\ = & = & = \\ & = & = \\ & = & = \\ & = & = \\ & = & = \\ & = & = \\ & = & = \\ & = & = \\ & = & = \\ & = & = \\ & = & = \\ & = & = \\ & = & = \\ & = & = \\ & = & = \\ & = & = \\ & = & = \\ & = & = \\ & = & = \\ & = & = \\ & = & = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - . \\ & = & = \\ & = & = \\ & = & = \\ & = & = \\ & = & = \\ & = & = \\ & = & = \\ & = & = \\ & = & = \\ & = & = \\ & = & = \\ & = & = \\ & = & = \\ & = & = \\ & = & = \\ & = & = \\ & = & = \\ & = & = \\ & = & = \\ & = & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ | | ^ {- 2} | | ^ {- 2} | | ^ {- 2} | | ^ {- 2} | | ^ {- 2} | | ^ {- 2} | | ^ {- 2} | | ^ {- 2} | | ^ {- 2} | | ^ {- 2} | | ^ {- 2} | | ^ {- 2} | | ^ {- 2} | | ^ {- 2} | | ^ {- 2} | > 0. \\ & < 0. ^ {- 3} < 0. ^ {- 3} < 0. ^ {- 3} < 0. ^ {- 3} < 0. ^ {- 3} < 0. ^ {- 3} < 0. ^ {- 3} < 0. ^ {- 3} < 0. ^ {- 3} < 0. ^ {- 3} < 0. ^ {- 3}, < 0. ^ {- 4} < 0. ^ {- 4} < 0. ^ {- 4} < 0. ^ {- 4} < 0. ^ {- 4} < 0. ^ {- 4} < 0. ^ {- 4} < 0. ^ {- 4} < 0. ^ {- 4}, < 0. ^ {- 5} < 0. ^ {- 5} < 0. ^ {- 5} < 0. ^ {- 5} < 0. ^ {- 5} < 0. ^ {- 5}, < 0. ^ {- 6} < 0. ^ {- 6} < 0. ^ {- 6} < 0. ^ {- 6}, < 0. ^ {- 7} < 0. ^ {- 7} < 0. ^ {- 7} < 0. ^ {- 7}, < 0. ^ {- 8} < 0. ^ {- 8} < 0. ^ {- 8} < 0. ^ {- 8}, < 0. ^ {- 9} < >, < [ a ] a b d e r s i n g e f o r e c o m e d o u s e r s i n g e f o r e c o m e d o u s e r s i n g e f o r e c o m e d o u s e r s i n g e f o r e c o m e d o u s e r s i n g e f o r e c o m e d o u s e r s i n g e f o r e c o m e d o u s e r s i s i n g e f o r e c o m e d o u s e r s i n g e f o r e c o m e d o u s e r s i n g e f o r e c o m e d o u s e r s i n g e f o r e c o m e d o u s e r s i n g e f o r e c o m e d o u s e r s i n g y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l k y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a ll y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i in d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n c h e r r e f f o r e f f o r e f f o r e f f o r e f f o r e f f o r e f f o r e f f o r e f f o r e f f o r e f f o r e f f o r e f f o r e f f o r e f f o r e f f o r e f f o r e f f o r e f f o r e f f o r e f f u r p h a c h e r w h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h w / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / : \\ * ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,* ,$ * , * , * , * , * , * , * , * , * , * , * , * , * , * , * , * , * , * , * , * , * , * , * , * , * , * , * , * , * , * , * , * , * , * , * , * , * , * , * , * , * , * , * , * , * , * , * , * , * , * , * | x | x | x | x | x | x | x | x | x | x | x | x | x | x | x | x | x | x | x | x | x | x | x | x | x | x | | :---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---| | :---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---| | :---:|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---| | :---:|---|---|---|---|---|---|---| | :-1: | : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :: | : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :: |: |: |: |: |: |: |: |: |: |: |: |: |: | < img src="a">\]

Thus, to solve for the innovations need to have a finite fourth moment. At second order, is function of , meaning that must have finite eight moment. At third order, is function of , meaning that must have finite twelve moment.

10 Impulse response functions - the definition by Andreasen

This section derives closed-form solutions for the impulse response function in non-linear DSGE models. Note that this section uses the definition of an impulse response function suggested by Andreasen (2012). This definition is

\[\begin{array}{r c l} I R F _ {\mathbf {v a r}} \left(l, \boldsymbol {\nu}, \mathbf {w} _ {t}\right) & = & E _ {t} \left[ \mathbf {v a r} _ {t + l} | \boldsymbol {\epsilon} _ {t + 1} + \boldsymbol {\nu} \right] \\ & & - E _ {t} \left[ \mathbf {v a r} _ {t + l} \right] \end{array}\]

where is stochastic (appologies for the bad notation!). To reduce the notational burden in the derivations below, we adopt the parsimonious notation

\[I R F _ {\mathbf {v a r}} (l, \pmb {\nu}, \mathbf {w} _ {t}) = E _ {t} [ \widetilde {\mathbf {v a r}} _ {t + l} ] - E _ {t} [ \mathbf {v a r} _ {t + l} ]\]

in relation to the conditional expectation operators.

10.1 At first order

Recall that we have:

\[\begin{array}{r l} & {\mathbf {x} _ {t + 1} ^ {f} = \mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t} ^ {f} + \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1}} \\ & {\mathrm{and}} \\ & {\mathbf {x} _ {t + 2} ^ {f} = \mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t + 1} ^ {f} + \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 2}} \\ & {\qquad = \mathbf {h} _ {\mathbf {x}} \left(\mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t} ^ {f} + \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1}\right) + \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 2}} \\ & {\qquad = \mathbf {h} _ {\mathbf {x}} ^ {2} \mathbf {x} _ {t} ^ {f} + \mathbf {h} _ {\mathbf {x}} \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1} + \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 2}} \\ & {\mathrm{and}} \\ & {\mathbf {x} _ {t + 3} ^ {f} = \mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t + 2} ^ {f} + \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 3}} \\ & {\qquad = \mathbf {h} _ {\mathbf {x}} \left(\mathbf {h} _ {\mathbf {x}} ^ {2} \mathbf {x} _ {t} ^ {f} + \mathbf {h} _ {\mathbf {x}} \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1} + \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 2}\right) + \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 3}} \end{array}\]

\[\begin{array}{r l} & {= \mathbf {h} _ {\mathbf {x}} ^ {3} \mathbf {x} _ {t} ^ {f} + \mathbf {h} _ {\mathbf {x}} ^ {2} \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1} + \mathbf {h} _ {\mathbf {x}} \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 2} + \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 3}} \\ & {= \mathbf {h} _ {\mathbf {x}} ^ {3} \mathbf {x} _ {t} ^ {f} + \sum_ {j = 1} ^ {3} \mathbf {h} _ {\mathbf {x}} ^ {3 - j} \sigma \pmb {\eta} \pmb {\epsilon} _ {t + j}} \end{array}\]

In general

\[\mathbf {x} _ {t + l} ^ {f} = \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\epsilon} _ {t + j}\]

With a shock of in period , we have

\[\tilde {\mathbf {x}} _ {t + l} ^ {f} = \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} (\boldsymbol {\epsilon} _ {t + j} + \boldsymbol {\delta} _ {t + j})\]

where we define such that:

\[\begin{array}{r l} & {\delta_ {t + j} = \pmb {\nu} \quad \mathrm{for} j = 1} \\ & {\delta_ {t + j} = \mathbf {0} \quad \mathrm{for} j \neq 1} \\ & {\mathrm{So}} \\ & {E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} ^ {f} - \mathbf {x} _ {t + l} ^ {f} \right] = E _ {t} \left[ \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} (\pmb {\epsilon} _ {t + j} + \pmb {\delta} _ {t + j}) - \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\epsilon} _ {t + j} \right]} \\ & {\quad = \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \delta_ {t + j}} \\ & {\quad = \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \pmb {\eta} \delta_ {t + 1}} \\ & {\quad \mathrm{usingthedefinitionof} \delta_ {t + j}.} \\ & {\quad = \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \pmb {\eta} \pmb {\nu}} \\ & {\quad \mathrm{because} \delta_ {t + 1} = \pmb {\nu}} \end{array}\]

\[\begin{array}{r l} & {\mathrm{and}} \\ & {E _ {t} \left[ \tilde {\mathbf {y}} _ {t + l} ^ {f} - \mathbf {y} _ {t + l} ^ {f} \right] = \mathbf {g _ {x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} ^ {f} - \mathbf {x} _ {t + l} ^ {f} \right]} \end{array}\]

10.2 At second order

We need to consider:

\[\begin{array} { r l } & { \mathbf { x } _ { t + 1 } ^ { s } = \mathbf { h } _ { \mathbf { x } } \mathbf { x } _ { t } ^ { s } + \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } \left( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \right) + \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } } \\ & { \mathbf { x } _ { t + 2 } ^ { s } = \mathbf { h } _ { \mathbf { x } } \mathbf { x } _ { t + 1 } ^ { s } + \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } \left( \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } \right) + \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } } \\ & { = \mathbf { h } _ { \mathbf { x } } \left( \mathbf { h } _ { \mathbf { x } } \mathbf { x } _ { t } ^ { s } + \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } \left( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \right) + \frac { 1} { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } \right) + \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } \left( \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } \right) + \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 3 } } \\ & { = \mathbf { h } _ { \mathbf { x } } ^ { 2 } \mathbf { x } _ { t } ^ { s } + \mathbf { h } _ { \mathbf { x } } \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } \left( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \right) + \mathbf { h } _ { \mathbf { x } } \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } + \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } \left( \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } \right) + \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { - 2 } , } \\ & { \mathbf { x } _ { t + 3 } ^ { s } = \mathbf { h } _ { \mathbf { x } } \mathbf { x } _ { t + 2 } ^ { s } + \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } \left( \mathbf { x } _ { t + 2 } ^ { f } \otimes \mathbf { x } _ { t + 2 } ^ { f } \right) + \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } } \\ & = \mathbf { h } _ { \mathbf { x } } \left( \mathbf { h } _ { \mathbf { x } } ^ { 2 } \mathbf { x } _ { t } ^ { s } + \mathbf { h } _ { \mathbf { x } } \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x} } \left( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f} \right) + \mathbf { h } _ { \mathbf { x } } \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } + \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x} } \left( \mathbf { x} _ { t + 1 } ^ { f } \otimes \mathbf { x} _ { t + 1 } ^ { f} \right) + \frac { 1}{ 2} \mathbf {\delta h} _ { \sigma \sigma }\right ) , \\ & \quad + \frac 1 2 H _ X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y I . ( n ) , ( n ) , ( n ) , ( n ) , ( n ) , ( n ) , ( n ) , ( n ) , ( n ) , ( n ) , ( n ) , ( n ) , ( n ) , ( n ) , ( n ) , ( n ) , ( n ) , ( n ) , ( n ) , ( n ) , ( n ) , ( n ) , ( n ) , ( n ) , ( n ) , ( n ) . ( n ) , ( n ) , ( n ) , ( n ) , ( n ) , ( n ) , ( n ) , ( n ) , ( n ) , ( n ) , ( n ) , ( n ) , ( n ) , ( n ) , ( n ) , ( n ) . ( n ) , ( n ) , ( n ) , ( n ) , ( n ) , ( n ) , ( n ) , ( n ) , ( n ) . ( n ) , ( n ) , ( n ) , ( n ) , ( n ) , ( n ) , ( n ) , ( n ) . ( n ) , ( n ) , ( n ) , ( n ) , ( n ) , ( n ) , ( n ) , ( n ) . ( n ) , ( n ) , ( n ) , ( n ) , ( n ) , ( n ) , ( n ) , ( n ) . ( n ) . ( n ) , ( n ) , ( n ) , ( n ) , ( n ) , ( n ) , ( n ) , ( n ) . ( n ) , ( n ) , ( n ) , ( n ) , ( n ) , ( n ) , ( n ) . ( n ) . ( n ) . ( n ) . ( n ) . ( n ) . ( n ) . ( n ) . ( n ) . ( n ) . ( n ) . ( n ) . ( n ) . ( n ) . ( n ) . ( n ) . ( n ) . ( n ) . ( n ) . ( n ) . ( n ) . ( n ) . ( n ) . ( n ) . ( n ) . ( n ) : o u a d i o m e r e c o m e r e c o m e r e c o m e r e c o m e r e c o m e r e c o m e r e c o m e r e c o m e r e c o m e r e c o m e r e c o m e r e c o m e r e c o m e r e c o m e r e c o m e r e c o m e r e c o m e c o m e r e c o m e r e c o m e r e c o m e r e c o m e r e c o m e r e c o m e r e c o m e r e c o m e r e c o m e r e c o m e r e c o m e r e c o m e r e c o m e r e c o m e r e c o m e r e c o l i v i a l l i v i a l l i v i a l l i v i a l l i v i a l l i v i a l l i v i a l l i v i a l l i v i a l l i v i a l l i v i a l l i v i a l l i v i a l l i v i a l l i v i a l l i v i a l l i v i a l l i v a l l i v i a l l i v i a l l i v i a l l i v i a l l i v i a l l i v i a l l i v i a l l i v i a l l i v i a l l i v i a l l i v i a l l i v i a l l i v i a l l i v i a l l i v i a l l i v i a l l j y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y z u p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W WWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWwWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwww www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www www wevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevevawooowooowooowooowooowooowooowooowooowooowooowooowooowooowooowooowooowooowooowooowooowooowooowooowooowooowooowooowooowooowooowooowooowooowooowooowooowooowooowooowooowooowooowooowooowooowooowooowooowooowoooccecccecccecccecccecccecccecccecccecccecccecccecccecccecccecccecccecccecccecccecccecccecccecccecccecccecccecccecccecccecccecccecccecccecccecccecccecccecccecccecccecccecccecccecccecccecccecccecccecccecccecccacecccacecccacecccacecccacecccacecccacecccacecccacecccacecccacecccacecccacecccacecccacecccacecccacecccacecccacecccacecccacecccacecccacecccacecccacecccacecccacecccacecccacecccacecccacecccacecccacecccacecccacecccacecccacecccacecccacecccacecccacecccacecccacecccacecccacecccacecccacecccacecccacecccacecccaccaccaaccaccaaccaccaaccaccaaccaccaaccaccaaccaccaaccaccaaccaccaaccaccaaccaccaaccaccaaccaccaaccaccaaccaccaaccaccaaccaccaaccaccaaccaccaaccaccaaccaccaaccaccaaccaccaaccaccaaccaccaaccaccaaccaccaaccaccaaccaccaaccaccaaccaccaaccaccaaccaccaaccaccauccaaagaaagaaagaaagaaagaaagaaagaaagaaagaaagaaagaaagaaagaaagaaagaaagaaagaaagaaagaaagaaagaaagaaagaaagaaagaaagaaagaaagaaagaaagaaagaaagaaagaaagaaagaaagaaagaaagaaagaaagaaagaaagaaagaaagaaagaaagaaagaaagaaagaaagaaagggaactcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcactcaccaactcaccaactcaccaactcaccaactcaccaactcaccaactcaccaactcaccaactcaccaactcaccaactcaccaactcaccaactcaccaactcaccaactcaccaactcaccaactcaccaactcaccaactcaccaactcaccaactcaccaactcaccaactcaccaactcaccaactcaccaactcaccaactcaccaactcaccaactcaccaactcaccaactcaccaactcaccaactcaccaactcaccaectcaccaactcaccaactcaccaactcaccaactcaccaactcaccaactcaccaactcaccaactcaccaactcaccaactcaccaactcaccaactcaccaactcaccaactcaccaactcaccaactcaccaactcaccaactcaccaactcaccaactcaccaactcaccaactcaccaactcaccaactcaccaactcaccaactcaccaactcaccaactcaccaactcaccaactcaccaactcaccaactcaccaactCACCACTCACCACTCACCACTCACCACTCACCACTCACCACTCACCACTCACCACTCACCACTCACCACTCACCACTCACCACTCACCACTCACCACTCACCACTCACCACTCACCACTCACCACTCACCACTCACCACTCACCACTCACCACTCACCACTCACCACTCACCACTCACCACTCACCACTCACCACTCACCACTCACCACTCACCACTCACCACTCACCACTCACCACTACCACTACCACTACCACTACCACTACCACTACCACTACCACTACCACTACCACTACCACTACCACTACCACTACCACTACCACTACCACTACCACTACCACTACCACTACCACTACCACTACCACTACCACTACCACTACCACTACCACTACCACTACCACTACCACTACCACTACCACTACCACTACCACTACCACTACCATACCATACCATACCATACCATACCATACCATACCATACCATACCATACCATACCATACCATACCATACCATACCATACCATACCATACCATACCATACCATACCATACCATACCATACCATACCATACCATACCATACCATACCATACCATACCATACCATACCATACCATACCATACCATACCATACCATACCATACCATACCATACCATACCATACCATACCATACCATACCATACCATACCATACATACAATACAATACAATACAATACAATACAATACAATACAATACAATACAATACAATACAATACAATACAATACAATACAATACAATACAATACAATACAATACAATACAATACAATACAATACAATACAATACAATACAATACAATACAATACAATACAATACAATACAATACAATACAATACAATACAATACAATACAATACAATACAATACAATACAATACAATACAATACAATACAATACAATACAATacaataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataataatattttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttaaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaiaaiiianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianianIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIANIAINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIIN IIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIINIIin II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II In II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II IN II In II In II In II In II In II In II In II In II In II In II In II In II In II In II In II In II In II In II In II In II In II In II In II In II In II In II In II In II In II In II In II In II In II In II In II In II In IIIn I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I II< fcel>x,yy,zz,xx,y,z,x,y,z,x,y,z,x,y,z,x,y,z,x,y,z,x,y,z,x,y,z,x,y,z,x,y,z,x,y,z,x,y,z,x,y,z,x,y,z,x,y,z,x,y,z,x,y,z,x,y,z,x,y,z,x,y,z,x,y,z,x,y,z,x,y,z,x,y,z,x,y,z,x,y,z,x,y,z,x,y,z,x,y,z,x,y,z,x,y,z,x,y,z,x,y,z,x,y,z,x,Y,XX,XX,XX,XX,XX,XX,XX,XX,XX,XX,XX,XX,XX,XX,XX,XX,XX,XX,XX,XX,XX,XX,XX,XX,XX,XX,XX,XX,XX,XX,XX,XX,XX,XX,XX,XX,XX,XX,XX,XX,XX,XX,XX,XX,XX,XX,XX,XX,XX,XX,XXX< nl>\]

\[\begin{array}{l} = \mathbf {h} _ {\mathbf {x}} ^ {3} \mathbf {x} _ {t} ^ {s} + \mathbf {h} _ {\mathbf {x}} ^ {2} \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) + \mathbf {h} _ {\mathbf {x}} \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t + 1} ^ {f} \otimes \mathbf {x} _ {t + 1} ^ {f}\right) + \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t + 2} ^ {f} \otimes \mathbf {x} _ {t + 2} ^ {f}\right) \\ + \mathbf {h} _ {\mathbf {x}} ^ {2} \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2} + \mathbf {h} _ {\mathbf {x}} \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2} + \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2} \\ = \mathbf {h} _ {\mathbf {x}} ^ {3} \mathbf {x} _ {t} ^ {s} + \sum_ {j = 0} ^ {2} \mathbf {h} _ {\mathbf {x}} ^ {2 - j} \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t + j} ^ {f} \otimes \mathbf {x} _ {t + j} ^ {f}\right) + \left(\sum_ {j = 0} ^ {2} \mathbf {h} _ {\mathbf {x}} ^ {2 - j}\right) \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2} \end{array}\]

and in general

\[\begin{array}{r l} & {\mathbf {x} _ {t + l} ^ {s} = \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {s} + \sum_ {j = 0} ^ {l - 1} \mathbf {h} _ {\mathbf {x}} ^ {l - 1 - j} \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t + j} ^ {f} \otimes \mathbf {x} _ {t + j} ^ {f}\right) + \left(\sum_ {j = 0} ^ {l - 1} \mathbf {h} _ {\mathbf {x}} ^ {l - 1 - j}\right) \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}} \\ & {\mathrm{for} l = 1, 2, 3, \ldots} \end{array}\]

Thus, to compute , we need to find . Hence, consider:

\[\begin{array}{l} \mathbf {x} _ {t + l} ^ {f} \otimes \mathbf {x} _ {t + l} ^ {f} = \left(\mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j}\right) \otimes \left(\mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j}\right) \\ = \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} \\ + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} \end{array}\]

and

\[\begin{array}{l} \tilde {\mathbf {x}} _ {t + l} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + l} ^ {f} = \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} (\epsilon_ {t + j} + \delta_ {t + j}) \\ \qquad + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} (\epsilon_ {t + j} + \delta_ {t + j}) \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} (\epsilon_ {t + j} + \delta_ {t + j}) \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} (\epsilon_ {t + j} + \delta_ {t + j}) \\ = \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \odot \left(\sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \epsilon_ {t + j} + \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \delta_ {t + 1}\right) \\ \qquad + \left(\sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \epsilon_ {t + j} + \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \delta_ {t + 1}\right) \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \\ \qquad + \left(\sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \epsilon_ {t + j} + \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \delta_ {t + 1}\right) \otimes \left(\sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \epsilon_ {t + j} + \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \delta_ {t + 1}\right) \end{array}\]

This means that:

\[\begin{array}{l}E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + l} ^ {f} - \mathbf {x} _ {t + l} ^ {f} \otimes \mathbf {x} _ {t + l} ^ {f} \right]\\= E _ {t} [ \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \left(\sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} + \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\delta} _ {t + 1}\right)\\+ \left(\sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} + \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\delta} _ {t + 1}\right) \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f}\\+ \left(\sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} + \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\delta} _ {t + 1}\right)) \otimes \left( \right.\sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} + \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\delta} _ {t + 1}\middle)\\- \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} - \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j}\end{array}\]

\[\begin{array} { r l } & - \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } - \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } ] \\ & = E _ { t } [ \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \delta _ { t + 1 } + \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \delta _ { t + 1 } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \\ & + \left( \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } + \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \delta _ { t + 1 } \right) \otimes \left( \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } + \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \delta _ { t + 1 }\right) \\ & - \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol{ \eta } \boldsymbol { \epsilon } _ { t + j } ] \\ & = E _ { t } [ \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \delta _ { t +1 } + \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \delta _ { t + 1 } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \\ & + ( \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta} \boldsymbol { \epsilon } _ { t + j } + \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta} \delta _ { t + 1 } ) \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta} \boldsymbol { \epsilon } _ { t + j } \\ & + ( \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta} \boldsymbol { \epsilon } _ { t + j } + \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta} \delta _ { t +1 } ) \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta} \delta _ { t + 1 } \\ & - ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (\mathrm{个}) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ), \\ & = E _ { t } [ {\mathbf h} _ { {\mathbf x}} ^ { l } {\mathbf x} _ {\mathbf t} ^ {\mathbf f} \otimes {\mathbf h} _ {\mathbf x} ^ {\mathbf l - 1} {\sigma} {\boldsymbol{0}} {\boldsymbol{0}} {\boldsymbol{0}} {\boldsymbol{0}} {\boldsymbol{0}} {\boldsymbol{0}} {\boldsymbol{0}} {\boldsymbol{0}} {\boldsymbol{0}} {\boldsymbol{0}} {\boldsymbol{0}} {\boldsymbol{0}} {\boldsymbol{0}} {\boldsymbol{0}} {\boldsymbol{0}} {\boldsymbol{0}} {\boldsymbol{0}} {\boldsymbol{\mathrm{的}}} \\ & + (\sum _ { j = 1} ^ { l }) \sum_ {\ell = 1} ^ {\ell - 1} \sum_ {\ell = 1} ^ {\ell - 1} \sum_ {\ell = 1} ^ {\ell - 1} \sum_ {\ell = 1} ^ {\ell - 1} \sum_ {\ell = 1} ^ {\ell - 1} \sum_ {\ell = 1} ^ {\ell - 1} \sum_ {\ell = 1} ^ {\ell - 1} \sum_ (j , i , j , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k ,k , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n ,n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n,n,m,n,m,n,m,n,m,n,m,n,m,n,m,n,m,n,m,n,m,n,m,n,m,n,m,n,m,n,m,n,m,n,m,n,m,n,m,n,m,n,m,n,m,n,m,n,m,n,m,n,m,n,m,n,m,n,m,n,m,n,m,n,m,n,m,n,m,n,m,n,m,n,m,n,m,n,m,n,m,n,m,n,m,n,m,n,m,n,m,n,m,n,m,n,m,n,m,n,m,n,\ldots,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100; \\ & = E _ t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t . \\ & = E _ t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t |t . \\ & = E _ {- 1 / 2} [ e q : e q : e q : e q : e q : e q : e q : e q : e q : e q : e q : e q : e q : e q : e q : e q : e q : e q : e q : e q : e q : e q : e q : e q : e q : e q : e q : e q : e q : e q : e q : e q : e q : e q :e q : e q : e q : e q : e q : e q : e q : e q : e q : e q : e q : e q : e q : e q : e q : e q : e q : e q : e q : e q : e q : e q : e q : e q : e q : e q : e q : e q : e q : e q : e q : e q : e q : eq :e q :e q :e q :e q :e q :e q :e q :e q :e q :e q :e q :e q :e q :e q :e q :e q :e q :e q :e q :e q :e q :e q :e q :e q :e q :e q :e q :e q :e q :e q :e q :e q :e q :e q:e q :e q :e q :e q :e q :e q :e q :e q :e q :e q :e q :e q :e q :e q :e q :e q :e q :e q :e q :e q :eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eq:eaq:senaeq:senaeq:senaeq:senaeq:senaeq:senaeq:senaeq:senaeq:senaeq:senaeq:senaeq:senaeq:senaeq:senaeq:senaeq:senaeq:senaeq:senaeq:senaeq:senaeq:senaeq:senaeq:senaeq:senaeq:senaeq:senaaqq:senaeq:senaeq:senaeq:senaeq:senaeq:senaeq:senaeq:senaeq:senaeq:senaeq:senaeq:senaeq:senaeq:senaeq:senaeq:senaeq:senaeq:senaeq:senaeq:senaeq:senaeq:senaeq:senaeq:senaeq:senaequaseeffeneqqfennnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnccccc; \\ & = E _ {- 1 / 2} [ a b d ] [ a b d ] [ a b d ] [ a b d ] [ a b d ] [ a b d ] [ a b d ] [ a b d ] [ a b d ] [ a b d ] [ a b d ] [ a b d ] [ a b d ] [ a b d ] [ a b d ] [ a b d ] [ a b d ] [ a b d ] [ a b d ] [ a b d ] [ a b d ]. \\ & = E _ {- 1 / 2} [ a b d ] [ a b d ] [ a b d ] [ a b d ] [ a b d ] [ a b d ] [ a b d ] [ a b d ] [ a b d ] [ a b d ] [ a b d ] [ a b d ] [ a b d ] [ a b d ] [ a b d ] [ a b d ] [ a b d ] [ a c m ] [ a c m ] [ a c m ] [ a c m ] [ a c m ] [ a c m ] [ a c m ] [ a c m ] [ a c m ] [ a c m ] [ a c m ] [ a c m ] [ a c m ] [ a c m ] [ a c m ] [ a c m ] [ a c m ] [ a c m ] [ a c m ] [ a c m ] [ a c c m ] [ a c c m ] [ a c c m ] [ a c c m ] [ a c c m ] [ a c c m ] [ a c c m ] [ a c c m ] [ a c c m ] [ a c c m ] [ a c c m ] [ a c c m ] [ a c c m ] [ a c c m ] [ a c c m ] [ a c c m ] [ a c c m ][ a c c m ] [ a c c m ] [ a c c m ] [ a c c m ] [ a c c m ] [ a c c m ] [ a c c m ] [ a c c m ] [ a c c m ] [ a c c m ] [ a c c m ] [ a c c m ][ a c c m ] [ a c c m ] [ a c c m ] [ a c c m ] [ a c cm ] [ a c c m ] [ a c c m ] [ a c c m ] [ a c c m ] [ a c c m ] [ a c c m ] [ a c c m ] [ a c c m ] [ a c c m ] [ a c c m ] [ a c c m ] [ a c c m ] [ a c c m ] [ a c c m ] [ a c c m ] [ a c c m ] [ acscm] [a cscm] [a cscm] [a cscm] [a cscm] [a cscm] [a cscm] [a cscm] [a cscm] [a cscm] [a cscm] [a cscm] [a cscm] [a scctm] [a scctm] [a scctm] [a scctm] [a scctm] [a scctm] [a scctm] [a scctm] [a scctm] [a scctm] [a scctm] [a scctm] [a scctm] [a scctm] [a scctm] [a scctm] [a scctm] [b s i g h i g h i g h i g h i g h i g h i g h i g h i g h i g h i g h i g h i g h i g h i g h i g h i g h i g h i g h i g h i g h i g h i g h i g h i g h i g h i g h i g h i g h i g h i g h i g h i g h i g hi g h i g h i g h i g h i g h i g h i g h i g h i g h i g h i g h i g h i g h i g h i g h i g h i g h i g h i g h i g h i g h i g h i g h i g h i g h i g h i g h i g h i g h i g h i g h i g h i g h i f o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o v o v o v o v o v o v o v o v o v o v o v o v o v o v o v o v o v o v o v o v o v o v o v o v o v o v o v o v o v o v o v o v o v o v o v o v o v o v o v o v o v o v o v o v o v o v o v o v o v o v ov o v o v o v y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y yy. \\ & = E _ {- 1 / 2} E _ {- 1 / 2} E _ {- 1 / 2} E _ {- 1 / 2} E _ {- 1 / 2} E _ {- 1 / 2} E _ {- 1 / 2} E _ {- 1 / 2} E _ {- 1 / 2} E _ {- 1 / 2} E _ {- 1 / 2} E _ {- 1 / 2}E _ {- 1 / 2}E _ {- 1 / 2}E _ {- 1 / 2}E _ {- 1 / 2}E _ {- 1 / 2}E _ {- 1 / 2}E _ {- 1 / 2}E _ {- 1 / 2}E _ {- 1 / 2}E _ {- 1 / 2}E _ {- 1 / 2}E_{- 1 / 2}E_{- 1 / 2}E_{- 1 / 2}E_{- 1 / 2}E_{- 1 / 2}E_{- 1 / 2}E_{- 1 / 2}E_{- 1 / 2}E_{- 1 / 2}E_{- 1 / 2}E_{- 1 / 2}E_- s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u rs u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r su r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u r s u f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f< fcel>+ A B C D E F G H I N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O NO N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N O N S U P U W Y W Y W Y W Y W Y W Y W Y W Y W Y W Y W Y W Y W Y W Y W Y W Y W Y W Y W Y W Y W Y W Y W Y W Y W Y W Y W Y W Y W Y W Y W Y W Y W Y W Y W Y W Y W Y W Y W Y W Y W Y W Y W Y W Y W Y W Y W Y W Y W Y W Y WYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWWYWWYWWYWWYWWYWWYWWYWWYWWYWWYWWYWWYWWYWWYWWYWWYWWYWWYWWYWWYWWYWWYWWYWWYWWYWWYWWYWWYWWYWWYWWYWWYWWYWWYWWYWWYWWYWWYWWYWWYWWYWWYWWYWWYWWYWWYWWYWWYWWYWWYWWWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWYWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwthwthwthwthwthwthwthwthwthwthwthwthwthwthwthwthwthwthwthwthwthwthwthwthwthwthwthwthwthwthwthwthwthwthwthwthwthwthwthwthwthwthwthwthwthwthwthwthwthwthw thwnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnsswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnswnnsnwsnwsnwsnwsnwsnwsnwsnwsnwsnwsnwsnwsnwsnwsnwsnwsnwsnwsnwsnwsnwsnwsnwsnwsnwsnwsnwsnwsnwsnwsnwsnwsnwsnwsnwsnwsnwsnwsnwsnwsnwsnwsnwsnwsnwsnwsnwsnwsnwsnwsnxsnxsnxsnxsnxsnxsnxsnxsnxsnxsnxsnxsnxsnxsnxsnxsnxsnxsnxsnxsnxsnxsnxsnxsnxsnxsnxsnxsnxsnxsnxsnxsnxsnxsnxsnxsnxsnxsnxsnxsnxsnxsnxsnxsnxsnxsnxsnxsnxsnxsnssnssnssnssnssnssnsssnaan#### # ###### ###### ###### ###### ###### ###### ###### ###### ###### ###### ###### ###### ###### ###### ######\]

\[\begin{array}{l} \text {Thus} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + l} ^ {f} - \mathbf {x} _ {t + l} ^ {f} \otimes \mathbf {x} _ {t + l} ^ {f} \right] = \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} + \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} \\ \text {Or (using another index)} \\ E _ {t} \left[ \tilde {\mathbf {x}} _ {t + j} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + j} ^ {f} - \mathbf {x} _ {t + j} ^ {f} \otimes \mathbf {x} _ {t + j} ^ {f} \right] = \mathbf {h} _ {\mathbf {x}} ^ {j} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {j - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} + \mathbf {h} _ {\mathbf {x}} ^ {j - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} \otimes \mathbf {h} _ {\mathbf {x}} ^ {j} \mathbf {x} _ {t} ^ {f} + \mathbf {h} _ {\mathbf {x}} ^ {j - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} \otimes \mathbf {h} _ {\mathbf {x}} ^ {j - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} \\ \text {for j = 1,2,3,...} \end{array}\]

\[\begin{array}{r l} & {\mathrm{Thus,wehaveingeneral}} \\ & {E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} ^ {s} - \mathbf {x} _ {t + l} ^ {s} \right] = E _ {t} \left[ \sum_ {j = 0} ^ {l - 1} \mathbf {h} _ {\mathbf {x}} ^ {l - 1 - j} \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \left(\tilde {\mathbf {x}} _ {t + j} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + j} ^ {f}\right) - \sum_ {j = 0} ^ {l - 1} \mathbf {h} _ {\mathbf {x}} ^ {l - 1 - j} \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t + j} ^ {f} \otimes \mathbf {x} _ {t + j} ^ {f}\right) \right]} \\ & {\quad = \left[ \sum_ {j = 1} ^ {l - 1} \mathbf {h} _ {\mathbf {x}} ^ {l - 1 - j} \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + j} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + j} ^ {f} - \mathbf {x} _ {t + j} ^ {f} \otimes \mathbf {x} _ {t + j} ^ {f} \right] \right]} \\ & {\quad \mathrm{theshokhitsinperiod} t + 1, \mathrm{so} \left(\tilde {\mathbf {x}} _ {t} ^ {f} \otimes \tilde {\mathbf {x}} _ {t} ^ {f}\right) = \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}} \\ & {\quad = \sum_ {j = 1} ^ {l - 1} \mathbf {h} _ {\mathbf {x}} ^ {l - 1 - j} \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {h} _ {\mathbf {x}} ^ {j} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {j - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} + \mathbf {h} _ {\mathbf {x}} ^ {j - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} \otimes \mathbf {h} _ {\mathbf {x}} ^ {j} \mathbf {x} _ {t} ^ {f} + \mathbf {h} _ {\mathbf {x}} ^ {j - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} \otimes \mathbf {h} _ {\mathbf {x}} ^ {j - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu}\right)} \end{array}\]

\[= \sum_ {j = 1} ^ {l - 1} \mathbf {h} _ {\mathbf {x}} ^ {l - 1 - j} \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {h} _ {\mathbf {x}} ^ {j} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {j - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} + \mathbf {h} _ {\mathbf {x}} ^ {j - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} \otimes \mathbf {h} _ {\mathbf {x}} ^ {j} \mathbf {x} _ {t} ^ {f} + \left(\mathbf {h} _ {\mathbf {x}} ^ {j - 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {j - 1}\right) (\sigma \boldsymbol {\eta} \boldsymbol {\nu} \otimes \sigma \boldsymbol {\eta} \boldsymbol {\nu})\right)\]

If we restrict the focus and do the IRF's at the unconditional mean of , then we get

\[E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} ^ {s} - \mathbf {x} _ {t + l} ^ {s} \right] = \sum_ {j = 1} ^ {l - 1} \mathbf {h} _ {\mathbf {x}} ^ {l - 1 - j} \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \left(\left(\mathbf {h} _ {\mathbf {x}} ^ {j - 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {j - 1}\right) (\sigma \boldsymbol {\eta} \boldsymbol {\nu} \otimes \sigma \boldsymbol {\eta} \boldsymbol {\nu})\right)\]

When implementing the IRF, it may be useful to have a recursive expression. Here, it is must convenient to use

\[E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} ^ {s} - \mathbf {x} _ {t + l} ^ {s} \right] = \sum_ {j = 1} ^ {l - 1} \mathbf {h} _ {\mathbf {x}} ^ {l - 1 - j} \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + j} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + j} ^ {f} - \mathbf {x} _ {t + j} ^ {f} \otimes \mathbf {x} _ {t + j} ^ {f} \right]\]

\[E _ {t} \left[ \tilde {\mathbf {x}} _ {t + 1} ^ {s} - \mathbf {x} _ {t + 1} ^ {s} \right] = 0\]

\[\begin{array}{c} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + 2} ^ {s} - \mathbf {x} _ {t + 2} ^ {s} \right] = \sum_ {j = 1} ^ {1} \mathbf {h} _ {\mathbf {x}} ^ {1 - j} \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + j} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + j} ^ {f} - \mathbf {x} _ {t + j} ^ {f} \otimes \mathbf {x} _ {t + j} ^ {f} \right] \\ = \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + 1} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + 1} ^ {f} - \mathbf {x} _ {t + 1} ^ {f} \otimes \mathbf {x} _ {t + 1} ^ {f} \right] \end{array}\]

\[\begin{array}{r l} & E _ {t} \left[ \tilde {\mathbf {x}} _ {t + 3} ^ {s} - \mathbf {x} _ {t + 3} ^ {s} \right] = \sum_ {j = 1} ^ {2} \mathbf {h} _ {\mathbf {x}} ^ {2 - j} \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + j} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + j} ^ {f} - \mathbf {x} _ {t + j} ^ {f} \otimes \mathbf {x} _ {t + j} ^ {f} \right] \\ & \qquad = \mathbf {h} _ {\mathbf {x}} \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + 1} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + 1} ^ {f} - \mathbf {x} _ {t + 1} ^ {f} \otimes \mathbf {x} _ {t + 1} ^ {f} \right] \\ & \qquad + \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + 2} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + 2} ^ {f} - \mathbf {x} _ {t + 2} ^ {f} \otimes \mathbf {x} _ {t + 2} ^ {f} \right] \\ & \qquad = \mathbf {h} _ {\mathbf {x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + 2} ^ {s} - \mathbf {x} _ {t + 2} ^ {s} \right] + \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + 2} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + 2} ^ {f} - \mathbf {x} _ {t + 2} ^ {f} \otimes \mathbf {x}. _ {} ^ f\]

So in general

\[E _ {t} \left[ \tilde {\mathbf {x}} _ {t + k} ^ {s} - \mathbf {x} _ {t + k} ^ {s} \right] = \mathbf {h} _ {\mathbf {x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + k - 1} ^ {s} - \mathbf {x} _ {t + k - 1} ^ {s} \right] + \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + k - 1} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + k - 1} ^ {f} - \mathbf {x} _ {t + k - 1} ^ {f} \otimes \mathbf {x} _ {t + k - 1} ^ {f} \right]\]

For the total state variable:

\[E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} - \mathbf {x} _ {t + l} \right] = E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} ^ {f} - \mathbf {x} _ {t + l} ^ {f} \right] + E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} ^ {s} - \mathbf {x} _ {t + l} ^ {s} \right]\]

For the control variables:

\[\mathbf {y} _ {t + l} ^ {s} = \mathbf {g} _ {\mathbf {x}} \left(\mathbf {x} _ {t + l} ^ {f} + \mathbf {x} _ {t + l} ^ {s}\right) + \frac {1}{2} \mathbf {G} _ {\mathbf {x x}} \left(\mathbf {x} _ {t + l} ^ {f} \otimes \mathbf {x} _ {t + l} ^ {f}\right) + \frac {1}{2} \mathbf {g} _ {\sigma \sigma} \sigma^ {2}\]

\[\tilde {\mathbf {y}} _ {t + l} ^ {s} = \mathbf {g} _ {\mathbf {x}} \left(\tilde {\mathbf {x}} _ {t + l} ^ {f} + \tilde {\mathbf {x}} _ {t + l} ^ {s}\right) + \frac {1}{2} \mathbf {G} _ {\mathbf {x x}} \left(\tilde {\mathbf {x}} _ {t + l} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + l} ^ {f}\right) + \frac {1}{2} \mathbf {g} _ {\sigma \sigma} \sigma^ {2}\]

\[E _ {t} \left[ \tilde {\mathbf {y}} _ {t + l} ^ {s} - \mathbf {y} _ {t + l} ^ {s} \right] = \mathbf {g} _ {\mathbf {x}} \left(E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} ^ {f} - \mathbf {x} _ {t + l} ^ {f} \right] + E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} ^ {s} - \mathbf {x} _ {t + l} ^ {s} \right]\right) + \frac {1}{2} \mathbf {G} _ {\mathbf {x x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + l} ^ {f} - \mathbf {x} _ {t + l} ^ {f} \otimes \mathbf {x} _ {t + l} ^ {f} \right]\]

10.3 At third order

At third order, we additionally need to consider:

\[\mathbf {x} _ {t + 1} ^ {r d} = \mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t} ^ {r d} + \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {s}\right) + \frac {1}{6} \mathbf {H} _ {\mathbf {x x x}} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) + \frac {3}{6} \mathbf {h} _ {\sigma \sigma \mathbf {x}} \sigma^ {2} \mathbf {x} _ {t} ^ {f} + \frac {1}{6} \mathbf {h} _ {\sigma \sigma \sigma} \sigma^ {3}\]

and

\[\mathbf {x} _ {t + 2} ^ {r d} = \mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t + 1} ^ {r d} + \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t + 1} ^ {f} \otimes \mathbf {x} _ {t + 1} ^ {s}\right) + \frac {1}{6} \mathbf {H} _ {\mathbf {x x x}} \left(\mathbf {x} _ {t + 1} ^ {f} \otimes \mathbf {x} _ {t + 1} ^ {f} \otimes \mathbf {x} _ {t + 1} ^ {f}\right) + \frac {3}{6} \mathbf {h} _ {\sigma \sigma \mathbf {x}} \sigma^ {2} \mathbf {x} _ {t + 1} ^ {f} + \frac {1}{6} \mathbf {h} _ {\sigma \sigma \sigma} \sigma^ {3}\]

\[\begin{array} { r l } & = \mathbf { h } _ { \mathbf { x } } \left( \mathbf { h } _ { \mathbf { x } } \mathbf { x } _ { t } ^ { r d } + \mathbf { H } _ { \mathbf { x x } } \left( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { s } \right) + \frac { 1 } { 6 } \mathbf { H } _ { \mathbf { x x x } } \left( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \right) + \frac { 3 } { 6 } \mathbf { h } _ { \sigma \sigma \mathbf { x } } \sigma ^ { 2 } \mathbf { x } _ { t } ^ { f } + \frac { 1 } { 6 } \mathbf { h } _ { \sigma \sigma \sigma } \sigma ^ { 3 } \right) \\ & + \mathbf { H } _ { \mathbf { x x } } \left( \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { s } \right) + \frac { 1 } { 6 } \mathbf { H } _ { \mathbf { x x x } } \left( \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } \right) + \frac { 3 } { 6 } \mathbf { h } _ { \sigma \sigma \mathbf { x } } \sigma ^ { 2 } \mathbf { x } _ { t + 1 } ^ { f } + \frac { 1 } { 6 } \mathbf { h } _ { \sigma \sigma \sigma } \sigma ^ { 3 } \\ & = \mathbf { h } _ { \mathbf { x } } ^ { 2 } \mathbf { x } _ { t } ^ { r d } + \mathbf { h } _ { \mathbf { x } } \mathbf { H } _ { \mathbf { x x } } \left( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { s } \right) + \mathbf { h } _ { \mathbf { x } } \frac { 1 } { 6 } \mathbf { H } _ { \mathbf { x x x } } \left( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \right) + \mathbf { h } _ { \mathbf { x } } \frac { 3 } { 6 } \mathbf { h } _ { \sigma \sigma \mathbf { x } } \sigma ^ { 2 } \mathbf { x } _ { t } ^ { f } + \mathbf { h } _ { \mathbf { x } } \frac { 1 } { 6 } \mathbf { h } _ { \sigma \sigma \sigma } \sigma ^ { 3 } \\ & + \mathbf { H } _ { \mathbf { x x } } \left( \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { s} \right) + \frac { 1 } { 6 } \mathbf { H } _ { \mathbf { x x x } } \left( \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f} \right) + \frac { 3 } { 6 } \mathbf { h } _ { \sigma \sigma \mathbf { x} } \sigma ^ { 2 } \mathbf { x } _ { t + 1 } ^ { f } + \frac { 1 } { 6 } \mathbf { h } _ { \sigma \sigma \sigma } \sigma ^ { 3} ] \\ & { + \mathbf { H } _ { \mathbf { x x} } \left( \mathbf { x } _ { t + 2 } ^ { f } \otimes \mathbf { x } _ { t + 2} ^ { s} \right) + \frac { 1 } { 6 } \mathbf { H } _ { \mathbf { x x x} } \left( \mathbf { x } _ { t + 2 } ^ { f } \otimes \mathbf { x } _ { t + 2 } ^ { f } \otimes \mathbf { x } _ { t + 2} ^ { f} \right) + \frac { 3}{ 6} \mathbf { h } _ { \sigma \sigma \mathbf { x} } \sigma ^ { 2 } \mathbf { x} _ { t + 2 } ^ { f } + \frac { 1}{ 6} \mathbf { h } _ { \sigma \sigma \sigma } \sigma ^ { 3 }} \\ & { = {\mathrm{h}} _ {\mathrm{x}} ^ {\mathrm{3}} {\mathrm{x}} _ {\mathrm{t}} ^ {\mathrm{rd}} + {\mathrm{h}} _ {\mathrm{x}} ^ {\mathrm{2}} {\mathrm{H}} _ {\mathrm{xx} }\left( {\mathrm{x}} _ {\mathrm{t}} ^ {\mathrm{f}} \otimes {\mathrm{x}} _ {\mathrm{t}} ^ {\mathrm{s}}\right) + {\mathrm{h}} _ {\mathrm{x}} ^ {\mathrm{2}} {\frac 16} {\mathrm{H}} _ {\mathrm{xxx} }\left( {\mathrm{x}} _ {\mathrm{t}} ^ {\mathrm{f}} \otimes {\mathrm{x}} _ {\mathrm{t}} ^ {\mathrm{f}} \otimes {\mathrm{x}} _ {\mathrm{t}} ^ {\mathrm{f}}\right) + {\mathrm{h}} _ {\mathrm{x}} ^ {\mathrm{2}} {\frac 36} {\mathrm{h}} _ {\sigma \sigma \mathbf {x} }\sigma ^ {\mathrm{2} }\mathbf {{x}} _ {\mathrm{t}} ^ {\mathrm{f}} + {\mathrm{h}} _ {\mathrm{x}} ^ {\mathrm{2}} {\frac 16} {\mathrm{h}} _ {\sigma \sigma\sigma}\sigma^ {\mathrm{3} }} \\ & { + {\mathrm{h}} _ {\mathrm{x}} {\mathrm{H}} _ {\mathrm{xx} }\left( {\mathrm{x}} _ {\mathrm{t+1}} ^ {\mathrm{f}}\otimes {\mathrm{x}} _ {\mathrm{t+1}} ^ {\mathrm{s}}\right) + {\mathrm{h}} _ {\mathrm{x}} {\frac 16} {\mathrm{H}} _ {\mathrm{xxx} }\left( {\mathrm{x}} _ {\mathrm{t+1}} ^ {\mathrm{f}}\otimes {\mathrm{x}} _ {\mathrm{t+1}} ^ {\mathrm{f}}\otimes {\mathrm{x}} _ {\mathrm{t+1}} ^ {\mathrm{f}}\right) + {\mathrm{h}} _ {\mathrm{x}} {\frac 36} {\mathrm{h}} _ {\sigma \sigma\mathbf {}x}\sigma ^ {\mathrm{2} }\mathbf {{x}} _ {\mathrm{t+1}} ^ {\mathrm{f}} + {\mathrm{h}} _ {\mathrm{x}} {\frac 16} {\mathrm{h}} _ {\sigma\sigma\sigma}\sigma^ {\mathrm{3} }} \\ & + {\mathrm{H}} _ {\mathrm{xx} }\left( {\mathrm{x}} _ {\mathrm{t+2}} ^ {\mathrm{f}}\otimes {\mathrm{x}} _ {\mathrm{t+2}} ^ {\mathrm{s}}\right) + \frac 16{\mathrm{H}_{\mathrm{xxx}}}{\left( {{\mathrm{x}}_{\mathrm{t+2}}}^{ f }}\otimes {\left. {{\mathrm{x}}_{\mathrm{t+2}}}^{ f }}\otimes {\left. {{\mathrm{x}}_{\mathrm{t+2}}}^{ f }}\otimes {\left. {{\mathrm{x}}_{\mathrm{t+2}}}^{ f }}\otimes {\left. {{\mathrm{x}}_{\mathrm{t+2}}}^{ f }}\otimes {\left. {{\mathrm{x}}_{\textup{\scriptsize t+j}}}^{ f }}\otimes {\left. {{\mathrm{x}}_{\textup{\scriptsize t+j}}}^{ f }}\otimes {\left. {{\mathrm{x}}_{\textup{\scriptsize t+j}}}^{ f }}\otimes {\left. {{\mathrm{x}}_{\textup{\scriptsize t+j}}}^{ f }}\otimes \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. \left. {\left/ {{0.000}}}^{ - f - y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y yy}) - [ ( | | | | ) ] ) | | , | | , | | , | | , | | , | | , | | , | | , | | , | | , | | , | | , | | , | | , | | , | | , | | , | | , | | , | | , | | , | | , | | , | | , | | , | | , | | , | | , | | , | | , | | , | | , | | , | | , | | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . : , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,< fcel>and< nl> < fcel>${\boldsymbol{\chi}_{t+3}^{rd}={\boldsymbol{\chi}_{t+2}}}{{\boldsymbol{\chi}_{t+2}}}{{\boldsymbol{\chi}_{t+2}}}{{\boldsymbol{\chi}_{t+2}}}{{\boldsymbol{\chi}_{t+2}}}{{\boldsymbol{\chi}_{t+2}}}{{\boldsymbol{\chi}_{t+2}}}{{\boldsymbol{\chi}_{t+2}}}{{\boldsymbol{\chi}_{t+2}}}{{\boldsymbol{\chi}_{t+2}}}{{\boldsymbol{\chi}_{t+2}}}{{\boldsymbol{\chi}_{t+2}}}{{\boldsymbol{\chi}_{t+2}}}{{\boldsymbol{\chi}_{t+2}}}{{\boldsymbol{\chi}_{t+2}}}{{\boldsymbol{\chi}_{t+2}}}{{\boldsymbol{\chi}_{t+2}}}{{\boldsymbol{\chi}_{t+2}}}{{\boldsymbol{\chi}_{t+2}}}{{\boldsymbol{\chi}_{t+2}}}{{\boldsymbol{\chi}_{t+2}}}{{\boldsymbol{\chi}_{t+2}}}{{\boldsymbol{\chi}_{t+2}}}{{\boldsymbol{\chi}_{t+2}}}{{{{\boldsymbol{\chi}_{t+2}}}{{{{{\boldsymbol{\xi}_{{0.000}}}^{f }}}}\right)}}}{{{{\boldsymbol{\chi}_{0.000}}}^{f }}}{{{{\boldsymbol{\chi}_{0.000}}}^{f }}}{{{{\boldsymbol{\chi}_{0.000}}}^{f }}}{{{{\boldsymbol{\chi}_{0.000}}}^{f }}}{{{{\boldsymbol{\chi}_{0.000}}}^{f }}}{{{{\boldsymbol{\chi}_{0.000}}}^{f }}}{{{{{\boldsymbol{\chi}_{0.000}}}^f }}}{{{{{\boldsymbol{\chi}_{0.000}}}^f }}}{{{{{\boldsymbol{\chi}_{0.000}}}^f }}}{{{{{\boldsymbol{\chi}_{0.000}}}^f }}}{{{{{\boldsymbol{\chi}_{0.000}}}^f }}}{{{{{\boldsymbol{\chi}_{0.000}}}^f }}}{{{{{\boldsymbol{\chi}_{0.005}}}^f }}}{{{{{\boldsymbol{\chi}_{0.005}}}^f }}}{{{{{\boldsymbol{\chi}_{0.005}}}^f }}}{{{{{\boldsymbol{\chi}_{0.005}}}^f }}}{{{{{\boldsymbol{\chi}_{0.005}}}^f }}}{{{{{\boldsymbol{\chi}_{0.005}}}^f }}}{{{{{\boldsymbol{\chi}_{0.48}}}^f }}}{{{{{\boldsymbol{\chi}_{0.48}}}^f }}}{{{{{\boldsymbol{\chi}_{0.48}}}^f }}}{{{{{\boldsymbol{\chi}_{0.48}}}^f }}}{{{{{\boldsymbol{\chi}_{0.48}}}^f }}}{{{{{\boldsymbol{\chi}_{0.48}}}^f }}}{{{{\boldsymbol{\chi}_{0.48}}}^f }}{{{{{\boldsymbol{\chi}_{0.48}}}^f }}}{{{{{\boldsymbol{\chi}_{0.48}}}^f }}}{{{{{\boldsymbol{\chi}_{0.48}}}^f }}}{{{{{\boldsymbol{\chi}_{0.48}}}^f }}}{{{{{\boldsymbol{\chi}_{0.48}}}^f }}}{{{{{\boldsymbol{\chi}_{0.48}}}^f }}}\bigl ( ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ (~~ )~ )~ )~ )~ )~ )~ )~ )~ )~ )~ )~ )~ )~ )~ )~ )~ )~ )~ )~ )~ )~ )~ )~ )~ )~ )~ )~ )~ )~ )~ )~ )~ )~ )~ )~ )~ )~ )~ )~ )~ )~ )~ )~ )~ )~ )~ )~ )~ )~ )~ ), and a large number of large numbers in the algebraic structure is not feasible, so the algebraic structure is not feasible, so the algebraic structure is not feasible, so the algebraic structure is not feasible, so the algebraic structure is not feasible, so the algebraic structure is not feasible, so the algebraic structure is not feasible, so the algebraic structure is not feasible, so the algebraic structure is not feasible, so the algebraic structure is not feasible, so the algebraic structure is not feasible, so the algebraic structure is not feasible, so that it is possible in all cases, but it is possible in all cases, but it is possible in all cases, but it is possible in all cases, but it is possible in all cases, but it is possible in all cases, but it is possible in all cases, but it is possible in all cases, but it is possible in all cases, but it is possible in all cases, but it is possible in all cases, but it is possible in all cases, but it is possible in all cases, but it is possible, but it is possible in all cases, but it is possible in all cases, but it is possible in all cases, but it is possible in all cases, but it is possible in all cases, but it is possible in all cases, but it is possible in all cases, but it is possible in all cases, but it is possible in all cases, but it is possible in all cases, but it is possible in all cases, but it is possible in all cases, but itis possible in all cases, but it is possible in all cases, but it is possible in all cases, but it is possible in all cases, but it is possible in all cases, but it is possible in all cases, but it is possible in all cases, but it is possible in all cases, but it is possible in all cases, but it is possible in all cases, but it is possible in all cases, but it is possible in all cases, but it is possible in all classes, but it is possible in all classes, but it is possible in all classes, but it is possible in all classes, but it is possible in all classes, but it is possible in all classes, but it is possible in all classes, but it is possible in all classes, but it is possible in all classes, but it is possible in all classes, but it is possible in all classes, but it is possible in all classes, but it is possible in all classes, but itis possible in all classes, but it is possible in all classes, but it is possible in all classes, but it is possible in all classes, but it is possible in all classes, but it is possible in all classes, but it is possible in all classes, but it is possible in all classes, but itis possible in all classes,but itis possible in all classes,but itis possible in all classes,but itis possible in all classes,but itis possible in all classes,but itis possible in all classes,but itis possible in all classes,but itis possible in all classes,but itis possible in all classes,but itis possible in all classes,but itis possible in all classes,but itis possible in all classes,but itis possible in all classes,but itis possiblein all classes,but itis possiblein all classes,but itis possiblein all classes,but itis possiblein all classes,but itis possiblein all classes,but itis possiblein all classes,but itis possiblein all classes,but itis possiblein all classes,but itis possiblein all classes,but itis possiblein all classes,but itispossiblein all classes,but itispossiblein all classes,but itispossiblein all classes,but itispossiblein all classes,but itispossiblein all classes,but itispossiblein all classes,but itispossiblein all classes,but itispossiblein all classes,but itispossiblein all classes,butitallclassesolidation of the total number of matrices for each class; and this number of matrices for each class; and this number of matrices for each class; and this number of matrices for each class; and this number of matrices for each class; and this number of matrices for each class; and this number of matrices for each class; and this number of matrices for each class; and this number of matrices for each class; and this number of matrices for each class; and this number of matrices for each class; and this number of matrices for each class; and this numbersolidation of the total number of matrices for each class; and this number of matrices for each class; and this number of matrices for each class; and this number of matrices for each class; and this number of matrices for each class; and this number of matrices for each class; and this number of matrices for each class; and this number of matrices for each class; and this number of matrices for each class; and this number of matrices for each class; and this number of matrices for every class; and this number of matrices for every class; and this number of matrices for every class; and this number of matrices for every class; and this number of matrices for every class; and this number of matrices for every class; and this number of matrices for every class; and this number of matrices for every class; and this number of matrices for every class; and this number of matrices for every class; and this number of matrices for every class; and this number of matrices for every Class I only; and this number of matrices for every Class I only; and this number of matrices for every Class I only; and this number of matrices for every Class I only; and this number of matrices for every Class I only; and this number of matrices for every Class I only; and this number of matrices for every Class I only; and this number of matrices for every Class I only; and this number of matrices for every Class I only; and this number of matrices for every Class I only;\(E_{t}\) \(E_{t+1}\) \(E_{t-1}\) \(E_{t-1}\) \(E_{t-1}\) \(E_{t-1}\) \(E_{t-1}\) \(E_{t-1}\) \(E_{t-1}\) \(E_{t-1}\) \(E_{t-1}\) \(E_{t-1}\) \(E_{t-1}\) \(E_{t-1}\) \(E_{t-1}\)< lcel>< lcel>< nl>\]

\[\begin{array}{r l} & {= \sum_ {j = 1} ^ {l - 1} \mathbf {h} _ {\mathbf {x}} ^ {l - 1 - j} \left(\mathbf {H} _ {\mathbf {x x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + j} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + j} ^ {s} - \mathbf {x} _ {t + j} ^ {f} \otimes \mathbf {x} _ {t + j} ^ {s} \right] + \frac {3}{6} \mathbf {h} _ {\sigma \sigma \mathbf {x}} \sigma^ {2} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + j} ^ {f} - \mathbf {x} _ {t + j} ^ {f} \right]\right)} \\ & {+ \sum_ {j = 1} ^ {l - 1} \mathbf {h} _ {\mathbf {x}} ^ {l - 1 - j} \frac {1}{6} \mathbf {H} _ {\mathbf {x x x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + j} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + j} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + j} ^ {f} - \mathbf {x} _ {t + j} ^ {f} \otimes \mathbf {x} _ {t + j} ^ {f} \otimes \mathbf {x} _ {t + j} ^ {f} \right]} \end{array}\]

as the shock hits the economy in period

A recursive version:

\[E _ {t} \left[ \tilde {\mathbf {x}} _ {t + 1} ^ {r d} - \mathbf {x} _ {t + 1} ^ {r d} \right] = 0\]

\[\begin{array}{r l} & E _ {t} \left[ \tilde {\mathbf {x}} _ {t + 2} ^ {r d} - \mathbf {x} _ {t + 2} ^ {r d} \right] = \sum_ {j = 1} ^ {1} \mathbf {h} _ {\mathbf {x}} ^ {1 - j} \left(\mathbf {H} _ {\mathbf {x x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + j} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + j} ^ {s} - \mathbf {x} _ {t + j} ^ {f} \otimes \mathbf {x} _ {t + j} ^ {s} \right] + \frac {3}{6} \mathbf {h} _ {\sigma \sigma \mathbf {x}} \sigma^ {2} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + j} ^ {f} - \mathbf {x} _ {t + j} ^ {f} \right]\right) \\ & \qquad + \sum_ {j = 1} ^ {1} \mathbf {h} _ {\mathbf {x}} ^ {1 - j} \frac {1}{6} \mathbf {H} _ {\mathbf {x x x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + j} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + j} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + j} ^ {f} - \mathbf {x} _ {t + j} ^ {f} \otimes \mathbf {x} _ {t + j} ^ {f} \otimes \mathbf {x} _ {t + j} ^ {f} \right] \\ & = \mathbf {H} _ {\mathbf {x x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + 1} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + 1} ^ {s} - \mathbf {x} _ {t + 1} ^ {f} \otimes \mathbf {x} _ {t + 1} ^ {s} \right] + \frac {3}{6} \mathbf {h} _ {\sigma \sigma \mathbf {x}} \sigma^ {2} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + 1} ^ {f} - \mathbf {x} _ {t + 1} ^ {f} \right] \\ & \qquad + \frac {1}{6} \mathbf {H} _ {\mathbf {x x x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + 1} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + 1} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + 1} ^ {f} - \mathbf {x} _ {t + 1} ^ {f} \otimes \mathbf {x} _ {t + 1} ^ {f} \otimes \mathbf {x} _ {t + 1} ^ {f} \right] \end{array}\]

\[\begin{array} { r l } & E _ { t } \left[ \tilde { \mathbf { x } } _ { t + 3 } ^ { r d } - \mathbf { x } _ { t + 3 } ^ { r d } \right] = \sum _ { j = 1 } ^ { 2 } \mathbf { h } _ { \mathbf { x } } ^ { 2 - j } \left( \mathbf { H } _ { \mathbf { x x } } E _ { t } \left[ \tilde { \mathbf { x } } _ { t + j } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + j } ^ { s } - \mathbf { x } _ { t + j } ^ { f } \otimes \mathbf { x } _ { t + j } ^ { s } \right] + \frac { 3 } { 6 } \mathbf { h } _ { \sigma \sigma \mathbf { x } } \sigma ^ { 2 } E _ { t } \left[ \tilde { \mathbf { x } } _ { t + j } ^ { f } - \mathbf { x } _ { t + j } ^ { f } \right]\right) \\ & \qquad + \sum _ { j = 1 } ^ { 2 } \mathbf { h } _ { \mathbf { x } } ^ { 2 - j } \frac { 1 } { 6 } \mathbf { H } _ { \mathbf { x x x } } E _ { t } \left[ \tilde { \mathbf { x } } _ { t + j } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + j } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + j } ^ { f } - \mathbf { x } _ { t + j } ^ { f } \otimes \mathbf { x } _ { t + j } ^ { f } \otimes \mathbf { x } _ { t + j } ^ { f } \right] \\ & = \mathbf { h } _ { \mathbf { x } } \left( \mathbf { H } _ { \mathbf { x x } } E _ { t } \left[ \tilde { \mathbf { x } } _ { t + 1 } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + 1 } ^ { s } - \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { s } \right] + \frac { 3 } { 6 } \mathbf { h } _ { \sigma \sigma \mathbf { x } } \sigma ^ { 2 } E _ { t } \left[ \tilde { \mathbf { x } } _ { t + 1 } ^ { f } - \mathbf { x } _ { t + 1 } ^ { f } \right]\right) \\ & + \mathbf { H } _ { \mathbf { x x } } E _ { t } \left[ \tilde { \mathbf { x } } _ { t + 2 } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + 2 } ^ { s } - \mathbf { x } _ { t + 2 } ^ { f } \otimes \mathbf { x } _ { t + 2 } ^ { s } \right] + \frac { 3 } { 6 } \mathbf { h } _ { \sigma \sigma \mathbf { x } } \sigma ^ { 2 } E _ { t } \left[ \tilde { \mathbf { x } } _ { t + 2 } ^ { f } - \mathbf { x } _ { t + 2 } ^ { f } \right] \\ & + \mathbf { h _ { x }} \frac { 1 } { 6 } \mathbf { H _ { x x x }} E _ { t } \left[ \tilde { \mathbf { x } } _ { t + 1 } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + 1 } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + 1 } ^ { f } - \mathbf { x} _ { t + 1 } ^ { f } \otimes \mathbf { x} _ { t + 1 } ^ { f } \otimes \mathbf { x} _ { t + 1 } ^ { f } \right] \\ & + \frac 1 6 \mathbf { H _ { x x x }} E _ { t } \left[ \tilde { \mathbf { x } } _ { t + 2 } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + 2 } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + 2 } ^ { f } - \mathbf { x} _ { t + 2 } ^ { f } \otimes \mathbf { x} _ { t + 2 } ^ { f } \otimes \mathbf { x} _ { t + 2 } ^ { f } \right] \\ & = \mathbf {\Delta h _ {\alpha}} E _ { t } [ \tilde {\mathbf {\Delta x}} _ {\tau , d} ^ {\tau , d} - \mathbf {\Delta x} _ {\tau , d} ^ {\tau , d} ] \\ & + \mathbf {\Delta H _ {\alpha}} E _ { t } [ \tilde {\mathbf {\Delta x}} _ {\tau , d} ^ {\tau , d} - \tilde {\mathbf {\Delta x}} _ {\tau , d} ^ {\tau , d} - \tilde {\mathbf {\Delta x}} _ {\tau , d} ^ {\tau , d} ] + \frac 3 6 [ \tilde {\mathbf {\Delta h}} _ {\sigma , d} ^ {\tau , d} - [ \tilde {\mathbf {\Delta x}} _ {\tau , d} ^ {\tau , d} ] ] \\ & + \frac 1 6 [ \tilde {\mathbf {\Delta H}} _ {\alpha , d} ^ {\tau , d} - [ \tilde {\mathbf {\Delta x}} _ {\tau , d} ^ {\tau , d} ] ] . \\ & = [ ( | | | ) ] . \\ & = [ ( | | | ) ] . \\ & = [ ( | | | ) ] . \\ & = [ ( | | | ) ] . \\ & = [ ( | | | ) ] . \\ & = [ ( | | | ) ] . \\ & = [ ( | | | ) ] . \\ & = [ ( | | | ) ] . \\ & = [ ( | | | ) ] . \\ & = [ ( | | | ) ]. \\ & = [ ( | | | ) ] . \\ & = [ ( | | | ) ] . \\ & = [ ( | | | ) ] . \\ & = [ ( | | | ) ] . \\ & = [ ( | | | ) ] . \\ & = [ ( | | | ) ] . \\ & = [ ( | | | ) ] . \\ & = [ ( | | | ) ] . \\ & = [ ( | | | ) ] . \end{array}\]

So in general

\[\begin{array}{r l} & E _ {t} \left[ \tilde {\mathbf {x}} _ {t + k} ^ {r d} - \mathbf {x} _ {t + k} ^ {r d} \right] = \mathbf {h} _ {\mathbf {x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + k - 1} ^ {r d} - \mathbf {x} _ {t + k - 1} ^ {r d} \right] + \mathbf {H} _ {\mathbf {x x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + k - 1} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + k - 1} ^ {s} - \mathbf {x} _ {t + k - 1} ^ {f} \otimes \mathbf {x} _ {t + k - 1} ^ {s} \right] \\ & \qquad + \frac {3}{6} \mathbf {h} _ {\sigma \sigma \mathbf {x}} \sigma^ {2} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + k - 1} ^ {f} - \mathbf {x} _ {t + k - 1} ^ {f} \right] \\ & \qquad + \frac {1}{6} \mathbf {H} _ {\mathbf {x x x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + k - 1} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + k - 1} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + k - 1} ^ {f} - \mathbf {x} _ {t + k - 1} ^ {f} \otimes \mathbf {x} _ {t + k - 1} ^ {f} \otimes \mathbf {x} _ {t + k - 1} ^ {f} \right] \end{array}\]

Thus, we know . So we only need to compute and . This is done in the next two subsections.

10.3.1 For

Consider:

\[\begin{array} { r l } & \mathbf { x } _ { t + l } ^ { f } \otimes \mathbf { x } _ { t + l } ^ { f } \otimes \mathbf { x } _ { t + l } ^ { f } = \left( \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } + \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } \right) \otimes \left( \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } + \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } \right) \otimes \\ & = ( \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } + \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } + \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } + \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j} ) \\ & \otimes \left( \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } + \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } \right) \\ & = ( \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } + \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x} _ { t } ^ { f } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta} \boldsymbol { \epsilon } _ { t + j } + \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta} \boldsymbol { \epsilon } _ { t + j } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x} _ { t } ^ { f } + \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta} \boldsymbol { \epsilon } _ { t + j} \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta} \boldsymbol { \epsilon } _ { t + j} ) * \\ & {\quad + ( [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ ] ] / 2020202020202020202020202020202020202020202020202020202020202020202020202020 2 ) / 4 ) / 4 ) / 4 ) / 4 ) / 4 ) / 4 ) / 4 ) / 4 ) / 4 ) / 4 ) / 4 ) / 4 ) / 4 ) / 4 ) / 4 ) / 4 ) / 4 ) / 4 ) / 4 ) / 4 ) / 4 ) / 4 ) / 4 ) / 4 ) / 4 ) / 8 ,} \\ & {\quad = ( [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ ] ] / 20202020202020202020202020202020202020202020202020202020202020202020 3 ) / 4 ) / 4 ) / 4 ) / 4 ) / 4 ) / 4 ) / 4 ) / 4 ) / 4 ) / 4 ) / 4 ) / 4 ) / 4 ) / 4 ) / 4 ) / 4 ) / 4 ) / 8 ,} \\ & {\quad + ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. ,} \\ & \quad + ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( | d i n e c o n s i n e c o n s i n e c o n s i n e c o n s i n e c o n s i n e c o n s i n e c o n s i n e c o n s i n e c o n s i n e c o n s i n e c o n s i n e c o n s i n e c o n s i n e c o n s i n e c o n s i m a g e r e d u s i n e c o n s i n e c o n s i n e c o n s i n e c o n s i n e c o n s i n e c o n s i n e c o n s i n e c o n s i n e c o n s i n e c o n s i n e c o n s i n e c o n s i n e c o n s i n e c o n s z e r r e d u s i n e c o n s z e r r e d u s i n e c o n s z e r r e d u s i n e c o n s z e r r e d u s i n e c o n s z e r r e d u s i n e c o n s z e r r e d u s i n e c o n s z e r r e d u s i n e c o n s z a r r e d u s i n e c o n s z a r r e d u s i n e c o n s z a r r e d u s i n e c o n s z a r r e d u s i n e c o n s z a r r e d u s i n e c o n s z a r r e d u s i n e c o n s z a r r e d u s i n e f o r t h a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a v a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a w a k y , k y , k y , k y , k y , k y , k y , k y , k y , k y , k y , k y , k y , k y , k y , k y , k y , k y , k y , k y , k y , k y , k y , k y , k y , k y , k y , k y , k y , k y , k y , k y , k y , k y ,k y , k y , k y , k y , k y , k y , k y , k y , k y , k y , k y , k y , k y , k y , k y , k y , k y , k y , k y , k y , k y , k y , k y , k y , k y , k y , k y , k y , k y , k y , k y , k y , k y , k Y , K Y , K Y , K Y , K Y , K Y , K Y , K Y , K Y , K Y , K Y , K Y , K Y , K Y , K Y , K Y , K Y , K Y , K Y , K Y , K Y , K Y , K Y , K Y , K Y , K Y , K Y , K Y , K Y , K Y , K Y , K Y , K Y , K Y , KY , K Y , K Y ,K Y ,K Y ,K Y ,K Y ,K Y ,K Y ,K Y ,K Y ,K Y ,K Y ,K Y ,K Y ,K Y ,K Y ,K Y ,K Y ,K Y ,K Y ,K Y ,K Y ,K Y ,K Y ,K Y ,K Y ,K Y ,K Y ,K Y ,K Y ,K Y ,K Y ,K Y ,K Y ,K Y ,KY ; K Y ; K Y ; K Y ; K Y ; K Y ; K Y ; K Y ; K Y ; K Y ; K Y ; K Y ; K Y ; K Y ; K Y ; K Y ; K Y ; K Y ; K Y ; K Y ; K Y ; K Y ; K Y ; K Y ; K Y ; K Y ; K Y ; K Y ; K Y ; K Y ; K Y ; K Y ; K Y ; K Y ; K Y; K Y ; K Y ; K Y ; K Y ; K Y ; K Y ; K Y ; K Y ; K Y ; K Y ; K Y ; K Y ; K Y ; K Y ; K Y ; K Y ; K Y ;KY ;KY ;KY ;KY ;KY ;KY ;KY ;KY ;KY ;KY ;KY ;KY ;KY ;KY ;KY ;KY ;KY ;KY ;KY ;KY ;KY ;KY ;KY ;KY ;KY ;KY ;KY ;KY ;KY ;KY ;KY ;KY ;KY ;KY;KY ;KY ;KY ;KY ;KY ;KY ;KY ;KY ;KY ;KY ;KY ;KY ;KY ;KY ;KY ;KY ;KY ;KY ;KY ;KY ;KY ;KY ;KY ;KY ;KY ;KY ;KY ;KY ;KY ;KY ;KY;KY;KY;KY;KY;KY;KY;KY;KY;KY;KY;KY;KY;KY;KY;KY;KY;KY;KY;KY;KY;KY;KY;KY;KY;< fcel>\( (\texttt{S}_\texttt{S}_\texttt{S}_\texttt{S}_\texttt{S}_\texttt{S}_\texttt{S}_\texttt{S}_\texttt{S}_\texttt{S}_\texttt{S}_\texttt{S}_\texttt{S}_\texttt{S}_{\texttt{S}_{\texttt{S}_{\texttt{S}_{\texttt{SL}}}}})_l, q,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r,r-r, and then we have to be the same number of two or more than one number of two or more than one number of two or more than one number of two or more than one number of two or more than one number of two or more than one number of two or more than one number of two or more than one number of two or more than one number of two or more than one number of two or more than one number of two or more than one number of two or more than one number of two or more than one number of two or more than one number, but we have to be the same number of two or more than one number of two or more than one number of two or more than one number of two or more than one number of two or more than one number of two or more than one number of two or more than one number of two or more than one number of two or more than one number of two or more than one number of two or more than one number of two or more than one number of two or more than one number oftwo numbers that are not included in the image. The image is not included in the image. We have to be the same number of two or more than one number of two or more than one number of two or more than one number of two or more than one number of two or more than one number of two or more than one number of two or more than one number of two or more than one number of two or more than one number of two or more than one number of two or more than one number of two or more than one number of two or more than one number of two and then we have to be the same number of two or more than one number of two or more than one number of two or more than one number of two or more than one number of two or more than one number of two or more than one number of two or more than one number of two or more than one number of two or more than one number of two or more than one number of two or more than one number of two or more than one number of two or more than one numbers that are not included in the image. We have to be the same number of two or more than one number of two or more than one number of two or more than one number of two or more than one number of two or more than one number of two or more than one number of two or more than one number of two or more than one number of two or more than one number of two or more than one number of two or more than one number of two or more than one number of two and then we have to be the same number of two or greater than one number of two or greater than one number of two or greater than one number of two or greater than one number of two or greater than one number of two or greater than one number of two or greater than one number of two or greater than one number of two or greater than one number of two or greater than one number of two or greater than one number of two or greater than one number of two or greater than one number of two or greater than one number of two or greater than one number, but we have to be the same number of both the same number as they are not included in the image. We have to be the same number as they are not included in the image. We have to be the same number as they are not included in the image. We have to be the same number as they are not included in the image. We have to be the same number as they are not included in the image. We have to be the same number as they are not included in the image. We have to be the same number as they are not included in the image. We have to be the same number: for example, we can use the original data to create the original data using the formula `np_formula` and it is also provided in the image. However, we can use the original data to create the original data using the formula `np_formula` and it is also provided in the image. We can use the original data to create the original data using the formula `np_formula` and it is also provided in the image. We can use the original data to create the original data using the formula `np_formula` and it is also provided in the image. We can use the original data to create the original data using the formula `np_formula` and it is also provided in the image. We can use the original data to create the original data using the formula `np_formula` and its corresponding values. We can use the original data to create the original data using the formula `np_formula` and its corresponding values with respect to our original data using the formula `np_formula` and its corresponding values with respect to our original data using the formula `np_formula` and its corresponding values with respect to our original data using the formula `np_formula` and its corresponding values with respect to our original data using the formula `np_formula` and its corresponding values with respect to our original data using the formula `np_formula` and its corresponding values with respect to our original data using the formula `np_formula` and its corresponding values with respectively. We can use the original data to create the original data using the formula `np_formula` and its corresponding values using the formula `np_formula` and its corresponding values with respect to our original data using the formula `np_formula` and its corresponding values using the formula `np_formula` and its corresponding values with respect to our original data using the formula `np_formula` and its corresponding values using the formula `np_formula` and its corresponding values with respect to our original data using the formula `np_formula` and its corresponding values using the formula `np_formula` and its corresponding values using the formula `np_formula` and its corresponding values using the formula `np_formula` and its corresponding values using the formula `np_formula` and its corresponding values using the formula `np_formula` and its corresponding values using the formula `np_formula` and its corresponding values using the formula `np_formula` and its corresponding values using the formula `np_formula` and its corresponding values using the formula `np_formula` and its corresponding values with respect to our original data using the formula `np_formula` and its corresponding values using the formula `np_formula` and its corresponding values using the formula `np_formula` and its corresponding values using the formula `np_formula` and its corresponding values using the formula `np_formula` and its corresponding values using the formula `np_formula` and its corresponding values using the formula `npcal*{}^{\prime}{}^{\prime}{}^{\prime}{}^{\prime}{}^{\prime}{}^{\prime}{}^{\prime}{}^{\prime}{}^{\prime}{}^{\prime}{}^{\prime}{}^{\prime}{}^{\prime}{}^{\prime}{}^{\prime}{}^{\prime}{}^{\prime}{}^{\prime}{}^{\prime}{}^{\prime}{}^{\prime},\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\)\]

And we therefore have

\[\begin{array} { l } \tilde { \mathbf { x } } _ { t + l } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + l } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + l } ^ { f } = \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } + \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } ( \boldsymbol { \epsilon } _ { t + j } + \boldsymbol { \delta } _ { t + j } ) \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \\ + \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } ( \boldsymbol { \epsilon } _ { t + j } + \boldsymbol { \delta } _ { t + j } ) \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x} _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x} _ { t } ^ { f } + \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } ( \boldsymbol { \epsilon } _ { t + j } + \boldsymbol { \delta } _ { t + j } ) \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } ( \boldsymbol { \epsilon } _ { t + j } + \boldsymbol { \delta } _ { t + j } ) \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathsf { x} _ { t } ^ { f } \\ + \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x} _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x} _ { t } ^ { f } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } ( \boldsymbol { \epsilon } _ { t + j } + \boldsymbol { \delta } _ { t + j } ) + \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x} _ { t } ^ { f } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta} ( \boldsymbol { \epsilon } _ { t + j } + \boldsymbol { \delta } _ { t + j } ) \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta} ( \boldsymbol { \epsilon } _ { t + j } + \boldsymbol { \delta } _ { t + j } ) \\ + \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta} ( \boldsymbol { \epsilon } _ { t + j } + \boldsymbol { \delta } _ { t + j } ) \otimes \mathbf { h } _ { \mathbf { x }} ^ { l } \mathbf { x} _ { t } ^ { f } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x }} ^ { l - j } \sigma \boldsymbol { \eta} ( \boldsymbol { \epsilon} _ { t + j } + \boldsymbol { \delta} _ { t + j} ) \\ + \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x }} ^ { l - j } \sigma \boldsymbol { \eta} ( \boldsymbol { \epsilon} _ { t + j } + \boldsymbol { \delta} _ { t + j} ) \otimes \sum _ { j = 1 } ^ { l } \mathbf { h} _ {\mathbf {\Delta x}} ^ { l - j} \sigma \boldsymbol {\eta} ( \boldsymbol {\epsilon} _ { t + j} + \boldsymbol {\delta} _ { t + j} ) \\ = [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ]: \\ = [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ & i n g e a d o n s i n g e a d o n s i n g e a d o n s i n g e a d o n s i n g e a d o n s i n g e a d o n s i n g e a d o n s i n g e a d o n s i n g e a d o n s i n g e a d o n s i n g e a d o n s i n g e a d o n s o n g e a d o n s o n g e a d o n s o n g e a d o n s o n g e a d o n s o n g e a d o n s o n g e a d o n s o n g e a d o n s o n g e a d o n s o n g e a d o n s o n g e a d o n s o n g e a d o n s o s o n g e a d o n s o n g e a d o n s o n g e a d o n s o n g e a d o n s o n g e a d o n s o n g e a d o n s o n g e a d o n s o n g e a d o n s o n g e a d o n s o n g e a d o n s o n g e a d o n c u r r y , i m , i m , i m , i m , i m , i m , i m , i m , i m , i m , i m , i m , i m , i m , i m , i m , i m , i m , i m , i m , i m , i m , i m , i m , i m , i m , i m , i m , i m , i m , i m , i m , i m , i m . \\ + ( 2 / 3 ) ( 2 / 3 ) ( 2 / 3 ) ( 2 / 3 ) ( 2 / 3 ) ( 2 / 3 ) ( 2 / 3 ) ( 2 / 3 ) ( 2 / 3 ) ( 2 / 3 ) ( 2 / 3 ) ( 2 / 3 ) ( 2 / 3 ) ( 2 / 3 ) ( 2 / 3 ) ( ( 2 / 3 ) ( 2 / 3 ) ( 2 / 3 ) ( 2 / 3 ) ( 2 / 3 ) ( 2 / 3 ) ( 2 / 3 ) ( 2 / 3 ) ( 2 / 3 ) ( 2 / 3 ) ( 2 / 3 ) ( 2 / 3 ) ( 2 / 3 ) ( 2 / 3 ) ( (\mathrm{的}) ( 2 / 3 ) ( 2 / 3 ) ( 2 / 3 ) ( 2 / 3 ) ( 2 / 3 ) ( 2 / 3 ) ( 2 / 3 ) ( 2 / 3 ) ( (\mathrm{的}) ( 2 / 3 ) ( (\mathrm{的}) ( 2 / 3 ) ( (\mathrm{的}) ( 2 / 3 ) ( (\mathrm{的}) ( 2 / 3 ) ( (\mathrm{的}) ( 2 / 3 ) ( (\mathrm{的}) ( (\mathrm{的}) ( (\mathrm{的}) ( (\mathrm{的}) ( (\mathrm{的}) ( (\mathrm{的}) ( (\mathrm{的}) ( (\mathrm{的}) ( (\mathrm{的}) ( (\mathrm{的}) ( (\mathrm{的}) ( (\mathrm{的}) ( (\mathrm{的}) ( (\mathrm{的}) ( (\mathrm{的}) ( (\mathrm{的}) ( (\mathrm{的}) ( (\mathrm{\Delta}) | (\mathrm{的}) | (\mathrm{的}) | (\mathrm{的}) | (\mathrm{的}) | (\mathrm{的}) | (\mathrm{的}) | (\mathrm{的}) | (\mathrm{的}) | (\mathrm{的}) | (\mathrm{的}) | (\mathrm{的}) | (\mathrm{的}) | (\mathrm{的}) | (\mathrm{的}) | (\mathrm{的}) | (\mathrm{的}) | (\mathrm{的}) | (\textit {\Delta}) | (\mathrm{的}) | (\mathrm{的}) | (\mathrm{的}) | (\mathrm{的}) | (\mathrm{的}) | (\textit {\Delta}) | (\textit {\Delta}) | (\textit {\Delta}) | (\textit {\Delta}) | (\textit {\Delta}) | (\textit {\Delta}) | (\textit {\Delta}) | (\textit {\Delta}) | (\textit {\Delta}) | (\textit {\Delta}) | (\textit {\Delta}) | (\textit {\Delta}) | (\textit {\Delta}) | (\textit {\Delta}) | (\textit {\Delta}) | + (5 / 3) * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * \\ + (5 / 3) * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * & + (5 / 3) {* *} {*} \left( \left( \left( \left( \left( \left( \left( \left( \left( \left( \left( \left( \left( \left( \left( \left( \left( \left( \left( \left( \left( \left( \left( \left( \left( \left(\left( \left( {\left( {{\left( {{\left( {{\left( {{\left( {{\left( {{\left( {{\left( {{\left( {{\left( {{\left( {{\left( {{\left( {{\left( {{\left( {{\left( {{\left( {{\left( {{\left( {{\left( {{\left( {{\left( {{\left( {}_{i}}}}}}}}}}}}}}|{}^{\prime }\right) }\right) }\right) }\right) }\right) }\right) }\right) }\right) }\right) }\right) }\right) }\right) }\right) }\right) }\right) }\right) }\right) }\right) }\right) }\right) }\right) }\right) }\right) }\right) }\right) }\right) }\right) }\right) }\right) }\right) }\right) }\right) }\right) *\]

\[\begin{array}{l}+ \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \left(\sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} + \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\delta} _ {t + 1}\right) \otimes \left( \right.\sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} + \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\delta} _ {t + 1}\middle)\\+ \left(\sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} + \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\delta} _ {t + 1}\right) \odot \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \left(\sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} + \mathbf {h} _ {\mathbf x} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\delta} _ {t + 1}\right)\\+ \left(\sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} + \mathbf {h} _ {\mathbf x} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\delta} _ {t + 1}\right) \otimes \left(\sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf x} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} + \mathbf {h} _ {\mathbf x} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\delta} _ {t + 1}\right) \otimes \left( \right.\sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf x}^{l - j}\sigma\boldsymbol{\eta}\boldsymbol{\epsilon}_{t + j}\\+ \mathbf {h}_{\mathbf x}\mathbf{d}_{t}\end{array}\]

This means that:

\[\begin{array} { l } \text {is means that:} \\ \left[ \tilde { \mathbf { x } } _ { t + l } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + l } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + l } ^ { f } - \mathbf { x } _ { t + l } ^ { f } \otimes \mathbf { x } _ { t + l } ^ { f } \otimes \mathbf { x } _ { t + l } ^ { f } \right] \\ = E _ { t } [ \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } + \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \left( \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } + \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \delta _ { t + 1 } \right) \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \\ + \left( \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } + \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \delta _ { t + 1} \right) \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \\ + \left( \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } + \mathrm{h} _ { \mathrm{x} } ^ { l - 1 } \sigma \boldsymbol { \eta } \delta _ { t + 1} \right) \otimes \left( \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathrm{x} } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } + \mathrm{h} _ {\mathrm{x} } ^ { l - 1 } \sigma \boldsymbol { \eta } \delta _ { t + 1} \right) \otimes \mathrm{h} _ {\mathrm{x} } ^ { l } \mathbf { x } _ { t } ^ { f } \\ + \mathrm{h} _ {\mathrm{x} } ^ { l } \mathbf { x} _ { t } ^ { f } \otimes \mathrm{h} _ {\mathrm{x} } ^ { l } \mathbf { x} _ { t } ^ { f } \otimes \left( \sum _ { j = 1 } ^ { l } \mathrm{h} _ {\mathrm{x} } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } + \mathrm{h} _ {\mathrm{x} } ^ { l - 1 } \sigma \boldsymbol { \eta } \delta _ { t + 1} \right) \\ + E _ { t } [ \mathbf { h } _ {\mathrm{x} } ^ { l } \mathbf { x} _ { t } ^ { f } \otimes \mathbf { h } _ {\mathrm{x} } ^ { l } \mathbf { x} _ { t } ^ { f } ] \\ + E _ {\mathrm{t}} [ E _ {\mathrm{t}} [ E _ {\mathrm{t}} ] ] \\ = E _ {\mathrm{t}} [ E _ {\mathrm{t}} ] \\ = E _ {\mathrm{t}} [ E _ {\mathrm{t}} ] \\ = E _ {\mathrm{t}} [ E _ {\mathrm{t}} ] \\ = E _ {\mathrm{t}} [ E _ {\mathrm{t}} ] \\ = E _ {\mathrm{t}} [ E _ {\mathrm{t}} ] \\ = E _ {\mathrm{t}} [ E _ {\mathrm{t}} ] \\ = E _ {\mathrm{t}}, \\ = E _ {\mathrm{t}} [ E _ {\mathrm{t}} ] \\ = E _ {\mathrm{t}} [ E _ {\mathrm{t}} ] \\ = E _ {\mathrm{t}} [ E _ {\mathrm{t}} ] \\ = E _ {\mathrm{t}} [ E _ {\mathrm{t}} ] \\ = E _ {\mathrm{t}} [ E _ {\mathrm{t}} ] \\ = E _ {\mathrm{t}} ] \\ = E _ {\mathrm{t}} [ E _ {\mathrm{t}} ] \\ = E _ {\mathrm{t}} [ E _ {\mathrm{t}} ] \\ = E _ {\mathrm{t}} [ E _ {\mathrm{t}} ] \\ = E _ {\mathrm{t}} [ E _ {\mathrm{t}} ] \\ = E _ {\mathrm{t}} [ E _ {\mathrm{t}} ] \\ . \\ = E _ {\mathrm{t}} [ E _ {\mathrm{t}} ] \\ = E _ {\mathrm{t}} [ E _ {\mathrm{t}} ] \\ = E _ {\mathrm{t}} [ E _ {\mathrm{t}} ] \\ = E _ {\mathrm{t}} [ E _ {\mathrm{t}} ] \\ = E _ {\mathrm{t}} [ E _ {\mathrm{t}} ] \\ = E _ {\mathrm{t}} ! , \\ = E _ {\mathrm{t}} ! , \\ = E _ {\mathrm{t}} ! , \\ = E _ {\mathrm{t}} ! , \\ = E _ {\mathrm{t}} ! , \\ = E _ {\mathrm{t}} ! , \\ = E _ {\mathrm{t}} ! , \\ = E _ {\mathrm{t}} ! , \\ = E _ {\mathrm{t}} ! , \\ = E, \\ = E, \\ = E, \\ = E, \\ = E, \\ = E, \\ = E, \\ = E, \\ = E, \\ = E, \\ = E, \\ = E, \\ = E, \\ = E, \\ = E, \\ = E, \\ = E, \\ = E, \\ = E, \\ = E, \\ = E, \\ = E, \\ = E, \\ = E, \\ = E, \\ = E. \\ = E. \\ = E. \\ = E. \\ = E. \\ = E. \\ = E. \\ = E. \\ = E. \\ = E. \\ = E. \\ = E. \\ = E. \\ = E. \\ = E. \\ = E. \\ = E. \\ = E. \\ = E. \\ = E. \\ = E. \\ = E. \\ = E. \\ = E. \\ = E. \\ = E . \\ = E . \\ = E . \\ = E . \\ = E . \\ = E . \\ = E . \\ = E . \\ =E .\\ =E .\\ =E .\\ =E .\\ =E .\\ =E .\\ =E .\\ =E .\\ =E .\\ =E .\\ =E .\\ =E .\\ =E .\\ =E .\\ =E .\\ =E .\\ =E .\\ =E .\\ =E .\\ =E .\\ =E .\\ =E .\\ =E .\\ =E .\\ =E .\\ =E .\\ =E .\\ =E .\\ =E .\\ =E .\\ =E .\\ =E .\\ =E .\\ =E .\\ =E .\\ =E .\\ =E .\\ =E .\\ =E .\\ =E .\\ =E .\\ =E .\\ =E .\\ =E .\\ =E .\\ =E .\\ =E .\\ =E .\\ =E .\\ =E .\\ =E .< lcel> \end{array}\]

\[\begin{array} { l } + \left( \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } + \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \delta } _ { t + 1 } \right) \otimes \left( \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } + \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \delta } _{ t + 1 } \right) \otimes \left( \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } + \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { {\eta } } \boldsymbol { \delta } _ { t + 1 } \right) ] \\ - E _ { t } [ \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \\ + \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { {\eta } } \boldsymbol { \epsilon } _ { t + j } \\ + \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { {\eta} } \boldsymbol { \epsilon } _ { t + j } \\ + \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta} \boldsymbol { \epsilon } _ { t + j } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { {\eta} } \boldsymbol { \epsilon } _ { t + j } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { {\eta} } \boldsymbol { {\epsilon} } _ { t + j} ] \\ = E _ { t } [ \mathbf { h } _ { \mathbf { x }} ^ { l } \mathbf { x} _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x }} ^ { l - 1 } \sigma \boldsymbol { {\eta} }\delta _ { t + 1 } \otimes \mathbf { h } _ { \mathbf { x }} ^ { l } \mathbf { x} _ { t } ^ { f } + \mathbf { h } _ { \mathbf { x }} ^ { l - 1 } \sigma \boldsymbol { {\eta} }\delta _ { t + 1 } \otimes \mathbf { h } _ { \mathbf { x }} ^ { l } \mathbf { x} _ { t } ^ { f } ] \\ + ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (\ ( ( ( ( ( (\ ( ( (\ ( ( (\ ( ( (\ ( (\ ( (\ ( (\ ( (\ ( (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ (\ ) ,\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\]

\[\begin{array}{l} + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \left(\sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} + \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\delta} _ {t + 1}\right) \\ + \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\delta} _ {t + 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \left(\sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} + \mathbf {h} {_ {\mathbf {x}}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\delta} _ {t + 1}\right) \\ + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} \otimes \left(\sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} + \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\delta} _ {t + 1}) \otimes \left(\sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} + \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\delta} _ {t + 1}\middle) \\ + \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\delta} _ {t + 1} \otimes \left(\sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} + \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\delta} _ {t + 1)}\right) \otimes \left(\sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} + \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\delta} _ {t + 1}\]

\[\begin{array} { r l } & - E _ { t } [ \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \epsilon _ { t + j } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \epsilon _ { t + j } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \\ & + \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \epsilon _ { t + j } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \epsilon_ { t + j } \\ & + \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \epsilon _ { t + j } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \epsilon _ { t + j } \\ & + \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \epsilon _ { t + j } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \epsilon_{ t + j } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \epsilon _ { t + j } ] \\ & = E _ { t } [ \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \delta _ { t + 1 } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } + \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \delta _ { t + 1 } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \\ & + \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \epsilon _ { t + j } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \pi }\epsilon _ { t + j }\otimes\]

\[\begin{array} { l } + \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { 9 } \boldsymbol { \epsilon } _ { t + j } ] \\ = E _ { t } [ \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { 9 } \delta _ { t + 1 } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } + \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { 9 } \delta _ { t + 1 } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \\ + 0 + \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { 9 } \boldsymbol { 9 } \epsilon _ { t + j } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { 9 } \delta _ { t + 1 } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \\ + \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { 9 } \delta _ { t + 1 } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { 9 } \epsilon _ { t + j } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } + \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { 9 } \delta _ { t + 1 } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { 9 } \delta _ { t + 1 } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \\ + \mathbf { h } _ { \mathbf { x } } ^ { I } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { I } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { I - 1 } \sigma \boldsymbol { 9 } \delta _ { t + 1 } \\ + 0 + \mathbf { h } _ { \mathbf { x } } ^ { I } \mathbf { x } _ { t } ^ { f } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { 9 } \epsilon _ { t + j } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { 9 } \delta _ { t + 1 } \\ + \mathbf { h } _ { \mathbf { x }} ^ { I } \mathbf { x} _ { t } ^ { f } \otimes \mathbf { h} _ { \mathbf { x }} ^ { I - 1 } \sigma \boldsymbol { 9 } \delta _ { t + 1 } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h} _ { \mathbf { x }} ^ { l - j } \sigma \boldsymbol { 9 } \epsilon _ { t + j } + \mathbf { h} _ { \mathbf { x }} ^ { I } \mathbf { x} _ { t } ^ { f } \otimes \mathbf { h} _ { \mathbf { x }} ^ { I - 1 } \sigma \boldsymbol { 9 } \delta _ { t + 1 } \otimes \mathbf { h} _ {\mathrm{x}} ^ {( l - 1)} s o n g e r a d i s c o n d i s c o n d i s c o n d i s c o n d i s c o n d i s c o n d i s c o n d i s c o n d i s c o n d i s c o n d i s c o n d i s c o n d i s c o n d i s c o n d i s c o n d i s c o n d i s c o n d i s , \\ + 0 + \sum _ { j = 1 } ^ { l } \mathbf { h} _ { \mathbf { x }} ^ { l - j } s o n g e r a d i s c o n d i s c o n d i s c o n d i s c o n d i s c o n d i s c o n d i s c o n d i s c o n d i s c o n d i s c o n d i s , \\ + h _ {\mathrm{x}} ^ {( l - 1)} s o n g e r a d i s c o n d i s c o n d i s c o n d i s c o n d i s c o n d i s c o n d i s c o n d i s , \\ + (0 + \sum _ { j = 1 } ^ { l } h _ {\mathrm{x}} ^ {( l - j)} s o n g e r a d i s c o n d i s c o n d i s , \\ + (2 (2 (2 (2 (2 (2 (2 (2 (2 (2 (2 (2 (2 (2 (2 (2 (2 (2 (2 (2 (2 (2 (2 (2 (2 (2 (2 (2 (2 (2 (2 (2 (2 (2 (2 (2 (2 (2 (2 (2 (2 (2 (2 (2 (2 (2 (2 (2 (2 (2 (3 (\mathrm{次}) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ), \\ - E _ {\mathrm{t}} [ 0 \\ + 0 \\ + 0 \\ + 0 ] \\ T e r m s c a n n e l o u t \\ & = E _ {\mathrm{t}} [ 0 \\ & = E _ {\mathrm{t}} [ 0 \\ & = E _ {\mathrm{t}} [ 0 \\ & = E _ {\mathrm{t}} [ 0 \\ & = E _ {\mathrm{t}} [ 0 \\ & = E _ {\mathrm{t}} [ 0 \\ & = E _ {\mathrm{t}} [ 0 \\ & = E _ {\mathrm{t}} [ 0 \\ & = E _ {\mathrm{f}} [ 0 \\ & = E _ {\mathrm{f}} [ 0 \\ & = E _ {\mathrm{f}} [ 0 \\ & = E _ {\mathrm{f}} [ 0 \\ & = E _ {\mathrm{f}} [ 0 \\ & = E _ {\mathrm{f}} [ 0 \\ & = E _ {\mathrm{f}} [ 0 \\ & = E _ {\mathrm{f}} [ 0 \\ & = E_{\mathrm{f}} [ 0 \\ & = E_{\mathrm{f}} [ 0 \\ & = E_{\mathrm{f}} [ 0 \\ & = E_{\mathrm{f}} [ 0 \\ & = E_{\mathrm{f}} [ 0 \\ & = E_{\mathrm{f}} [ 0 \\ & = E_{\mathrm{f}} [ 0 \\ & = E_{\mathrm{f}} [ 0 \\ & = E_{\mathrm{f}} [ 0 \\ & = E_{\mathrm{f}} [ 0 \\ & = E_{\mathrm{f}} [ 0 \\ & = E_{\mathrm{f}} [ 0 \\ & = E_{\mathrm{f}} [ 0 \\ & = E_{\mathrm{f}} [ 0 \\ & = E_{\mathrm{f}} [ 0 \\ & < E_{\mathrm{f}} [ 0 \\ & < E_{\mathrm{f}} [ 0 \\ & < E_{\mathrm{f}} [ 0 \\ & < E_{\mathrm{f}} [ 0 \\ & < E_{\mathrm{f}} [ 0 \\ & < E_{\mathrm{f}} [ 0 \\ & < E_{\mathrm{f}} [ 1 ] \\ & < E_{\mathrm{f}} [ 1 ] \\ & < E_{\mathrm{f}} [ 1 ] \\ & < E_{\mathrm{f}} [ 1 ] \\ & < E_{\mathrm{f}} [ 1 ] \\ & < E_{\mathrm{f}} [ 1 ] \\ & < E_{\mathrm{f}} [ 1 ] \\ & > E_{\mathrm{f}} [ k ] \\ & > K | k | | k | | | k | | | k | | | k | | | k | | | k | | | k | | | k | | | k | | | k | | | k | | | k | | | k | | | k | | | k | | | k | | | k | | | k | | | k | | | k | | | k | | | k | | | k | | | k | | | k | | | k | .\\\]

\[\begin{array}{l} = E _ {t} [ \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \delta_ {t + 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \delta_ {t + 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \\ + 0 \\ + 0 + \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \delta_ {t + 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \delta_ {t + 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \\ + \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \delta_ {t + 1} \\ + 0 \\ + 0 + \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \delta_ {t + 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \delta_ {t + 1} \\ + 0 \\ + 0 + \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \delta_ {t + 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \delta_ {t + 1} \\ + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \epsilon_ {t + j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \delta_ {t + 1} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \epsilon_ {t + j} \\ + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \epsilon_ {t + j} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \epsilon_ {t + j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \delta_ {t + 1} + 0. \end{array}\]

\[\begin{array}{l} + \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \pmb {\delta} _ {t + 1} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \pmb {\epsilon} _ {t + j} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \pmb {\epsilon} _ {t + j} + \mathbf {0} \\ + \mathbf {0} + \left. \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \pmb {\delta} _ {t + 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \pmb {\delta} _ {t + 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \pmb {\delta} _ {t + 1} \right] \\ \text {using the zero - mean property of} \pmb {\epsilon} _ {t + j} \end{array}\]

\[\begin{array}{r l} & {= \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta \pmb {\delta} _ {t + 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f}} \\ & {+ \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta \pmb {\delta} _ {t + 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f}} \\ & {+ \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta \pmb {\delta} _ {t + 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta \pmb {\delta} _ {t + 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f}} \\ & {+ \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta \pmb {\delta} _ {t + 1}} \\ & {+ \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta \pmb {\delta} _ {t + 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta \pmb {\delta} _ {t + 1}} \\ & {+ \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta \pmb {\delta} _ {t + 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta \pmb {\delta} _ {t + 1}} \\ & {+ E _ {t} \left[ \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \eta \pmb {\epsilon} _ {t + j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta \pmb {\delta} _ {t + 1} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \eta \pmb {\epsilon} _ {t + j} \right]} \\ & {+ E _ {t} \left[ \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \eta \pmb {\epsilon} _ {t + j} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \eta \pmb {\epsilon} _ {t + j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta \pmb {\delta} _ {t + 1} \right]} \\ & {+ E _ {t} \left[ \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta \pmb {\delta} _ {t + 1} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \eta \pmb {\epsilon} _ {t + j} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \eta \pmb {\epsilon} _ {t + j} + E _ {\mathrm{d}} (\mathrm{d}) E _ {\mathrm{d}} (\mathrm{d}) E _ {\mathrm{d}} (\mathrm{d}) E _ {\mathrm{d}} (\mathrm{d}) E _ {\mathrm{d}} (\mathrm{d}) E _ {\mathrm{d}} (\mathrm{d}) E _ {\mathrm{d}} (\mathrm{d}) E _ {\mathrm{d}} (\mathrm{d}) E _ {\mathrm{d}} (\mathrm{d}) E _ {\texttt {(d)}} E _ {\texttt {(d)}} E _ {\texttt {(d)}} E _ {\texttt {(d)}} E _ {\texttt {(d)}} E _ {\texttt {(d)}} E _ {\texttt {(d)}} E _ {\texttt {(d)}} E _ {\texttt {(d)}} E _ {\texttt {(d)}} E _ {\texttt {(d)}} E _ {\texttt {(d)}} E _ {\texttt {(d)}} E _ {\mathrm{d}}} \\ & + H (\boldsymbol {\xi}) (H (\boldsymbol {\xi})) (H (\boldsymbol {\xi})) (H (\boldsymbol {\xi})) (H (\boldsymbol {\xi})) (H (\boldsymbol {\xi})) (H (\boldsymbol {\xi})) (H (\boldsymbol {\xi})) (H (\boldsymbol {\xi})) (H (\boldsymbol {\xi})) (H (\boldsymbol {\xi})) (H (\boldsymbol {\xi})) (H (\boldsymbol {\xi})) (H (\boldsymbol {\xi})) (H (\overline {{\boldsymbol {\xi}}})) (H (\overline {{\boldsymbol {\xi}}})) (H (\overline {{\boldsymbol {\xi}}})) (H (\overline {{\boldsymbol {\xi}}})) (H (\overline {{\boldsymbol {\xi}}})) (H (\overline {{\boldsymbol {\xi}}})) (H (\overline {{\boldsymbol {\xi}}})) (H (\overline {{\boldsymbol {\xi}}})) (H (\overline {{\overline {{\boldsymbol {\xi}}}}})) (H (\overline {{\overline {{\boldsymbol {\xi}}}}})) (H (\overline {{\overline {{\boldsymbol {\xi}}}}})) (H (\overline {{\overline {{\boldsymbol {\xi}}}}})) (H (\overline {{\overline {{\boldsymbol {\xi}}}}})) (H (\overline {{\overline {{\boldsymbol {\xi}}}}})) (H (\overline {{\boldsymbol {\xi}}})) (H (\overline {{\overline {{\boldsymbol {\xi}}}}})) (H (\overline {{\overline {{\boldsymbol {\xi}}}}})) (H (\overline {{\overline {{\boldsymbol {\xi}}}}})) (H (\overline {{\overline {{\boldsymbol {\xi}}}}})) (H (\overline {{\overline {{\boldxi}}}})) (H (\overline {{\overline {{\boldxi}}}})) (H (\overline {{\overline {{\boldxi}}}})) (H (\overline {{\overline {{\boldxi}}}})) (H (\overline {{\overline {{\boldxi}}}})) (H (\overline {{\overline {{\boldxi}}}})) (H (\overline {{\overline {{\boldxi}}}})) (H (\underline \underline \underline \underline \underline \underline \underline \underline \underline \underline \underline \underline \underline \underline \underline \underline \underline \underline \underline \underline \underline \underline \underline \underline \underline {\underline {{\underline {{\underline {{\underline {{\underline {{\underline {{\underline {{\underline {{\underline {{\Sigma}}} / 2}}} / 2}}} / 2}}} / 2}}} / 2}}} / 2) / 4) / 2) / 2) / 2) / 2) / 2) / 2) / 2) / 2) / 2) / 2) / 2) / 2) / 2) / 2) / 2) / 2) / 2) / 2) / 2) / 2) / 2) / 2) / 2) / 2) / 2) / 3. H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z, W, X, Y, Z, Y, Z, Y, Z, Y, Z, Y, Z, Y, Z, Y, Z, Y, Z, Y, Z, Y, Z, Y, Z, Y, Z, Y, Z, Y, Z, Y, Z, Y, Z, Y, Z, Y, Z, Y, Z, Y, Z, Y, Z, Y, Z, Y, Z, Y, Z, Y, Z, Y, Y, Z, Y, Z, Y, Z, Y, Z, Y, Z, Y, Z, Y, Z, Y, Z, Y, Z, Y, Z, Y, Z, Y, Z, Y, Z, Y, Z, Y, Z, Y, Z, Y, Z, Y, Z, Y, Z, Y, Z, Y, Z, Y, Z, Y, Z, Y, Z, Y, z),}\]

\[\begin{array}{r l} & {= \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta \pmb {\nu} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f}} \\ & {+ \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta \pmb {\nu} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f}} \\ & {+ \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta \pmb {\nu} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta \pmb {\nu} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f}} \\ & {+ \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta \pmb {\nu}} \\ & {+ \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta \pmb {\nu} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta \pmb {\nu}} \\ & {+ \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta \pmb {\nu} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta \pmb {\nu}} \\ & {+ E _ {t} \left[ \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \eta \pmb {\epsilon} _ {t + j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta \pmb {\nu} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \eta \pmb {\epsilon} _ {t + j} \right]} \\ & {+ E _ {t} \left[ \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \eta \pmb {\epsilon} _ {t + j} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \eta \pmb {\epsilon} _ {t + j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta \pmb {\nu} \right]} \\ & {+ E _ {t} \left[ \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta \pmb {\nu} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \eta \pmb {\epsilon} _ {t + j} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \eta \pmb {\epsilon} _ {t + j} + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + . } \\ & {{+ h _ {\mathbf {x}} ^ {l - 1}} \sigma \eta \pmb {\nu} \otimes h _ {\mathbf {x}} ^ {l - 1} \sigma \eta \pmb {\nu} \otimes h _ {\mathbf {x}} ^ {l - 1} \sigma \eta \pmb {\nu}} \\ & {{u s i n g t h a t d e t a t h a t e r m a x t e r m a x t e r m a x t e r m a x t e r m a x t e r m a x t e r m a x t e r m a x t e r m a x t e r m a x t e r m a x t e r m a x t e r m a x t e r m a x t e r m a x t e r m a x t e r m a x t e r s u p p l o w.}} \\ & {{+ h _ {\mathbf {x}} ^ {(i)} (i) (i) (i) (i) (i) (i) (i) (i) (i) (i) (i) (i) (i) (i) (i) (i) (i) (i) (i) (i) (i) (i) (i) (i) (i) (i) (i) (i) (i) (i) (i) (i) (i) (i) = 0, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z, w, y, z, z)}} \\ & {{+ h _ {\mathbf {x}} ^ {(j)} (j) (j) (j) (j) (j) (j) (j) (j) (j) (j) (j) (j) (j) (j) (j) (j) (j) (j) (j) (j) (j) (j) (j) (j) (j) (j) (j) (j) (j) (j) (j) (j) (j) (j) = 0, j, k, l, m, n, o, p, q, r, s, t, u, v, w, y, z, z)}} \\ & {{+ h _ {\mathbf {x}} ^ {(k)} (k) (k) (k) (k) (k) (k) (k) (k) (k) (k) (k) (k) (k) (k) (k) (k) (k) (k) (k) (k) (k) (k) (k) (k) (k) (k)}} \\ & {{+ h _ {\mathbf {x}} ^ {(l)} (\mathrm{的}) (\mathrm{的}) (\mathrm{的}) (\mathrm{的}) (\mathrm{的}) (\mathrm{的}) (\mathrm{的}) (\mathrm{的}) (\mathrm{的}) (\mathrm{的}) (\mathrm{的}) (\mathrm{的}) (\mathrm{的}) (\mathrm{的}) (\mathrm{的}) (\mathrm{的}) (\mathrm{的}) (\mathrm{的}) (\mathrm{的}) (\mathrm{的}) (\mathrm{的}).}} \\ & {{+ h _ {\mathbf {x}} ^ {(m)} (\mathrm{的}) (\mathrm{的}) (\mathrm{的}) (\mathrm{的}) (\mathrm{的}) (\mathrm{的}) (\mathrm{的}) (\mathrm{的}) (\mathrm{的}) (\mathrm{的}) (\mathrm{的}) (\mathrm{的}) (\mathrm{的}) (\mathrm{的}) (\mathrm{的}) (\mathrm{的}) (\mathrm{的}) (\mathrm{的}).}} \\ & {{+ h _ {\mathcal {\mathrm{大}}} ^ {(n)} (\mathrm{的}) (\mathrm{n}) (\mathrm{n}) (\mathrm{n}) (\mathrm{n}) (\mathrm{n}) (\mathrm{n}) (\mathrm{n}) (\mathrm{n}) (\mathrm{n}) (\mathrm{n}) (\mathrm{n}) (\mathrm{n}) (\mathrm{n}) (\mathrm{n}) (\mathrm{n}) (\mathrm{n}) (\mathrm{n}) (\mathrm{n}) (\mathrm{n}) (\mathrm{n}) (\mathrm{n}) (\mathrm{n}) (\mathrm{n}).}} \\ & {{+ h _ {\mathcal {\mathrm{大}}} ^ {(n)} (\mathrm{n}) (\mathrm{n}) (\mathrm{n}) (\mathrm{n}) (\mathrm{n}) (\mathrm{n}) (\mathrm{n}) (\mathrm{n}) (\mathrm{n}) (\mathrm{n}) (\mathrm{n}) (\mathrm{n}) (\mathrm{n}) (\mathrm{n}) (\mathrm{n}) (\mathrm{n}) (\mathrm{n}) (\mathrm{n}) (\mathrm{n}).}} \\ & {{+ h _ {\mathcal {\mathrm{大}}} ^ {(n)} (\mathrm{n}) ((n)) ((n)) ((n)) ((n)) ((n)) ((n)) ((n)) ((n)) ((n)) ((n)) ((n)) ((n)) ((n)) ((n)) ((n)) ((n)) ((n)) ((n)}} \\ & {{+ h _ {\mathcal {\mathrm{大}}} ^ {(n)} ((n)) ((n)) ((n)) ((n)) ((n)) ((n)) ((n)) ((n)) ((n)) ((n)) ((n)) ((n)) ((n)) ((n)) ((n)}} \\ & {{+ h _ {\mathcal {\mathrm{大}}} ^ {(n)} ((n)) ((n)) ((n)) ((n)) ((n)) ((n)) ((n)) ((n)) ((n)) ((n)) ((n)) ((n)) (((n)) ((n)}} \\ & {{+ h _ {\mathcal {\mathrm{大}}} ^ {(n)} ((n)) ((n)) ((n)) ((n)) ((n)) ((n)) ((n)) ((n)) ((n)) ((n)) ((n)) ((n)) ((n)}} \\ & {{+ h _ {\mathcal {\mathrm{大}}} ^ {(n)} ((n)) ((n)) ((n)) ((n)) ((n)) ((n)) ((n)) ((n)) ((n)) ((n)) ((n)}} \\ & {{+ h _ {\mathcal {\mathrm{大}}} ^ {(n)} ((n)) ((n)) ((n)) ((n)) ((n)) ((n)) ((n)) ((n)) ((n)) ((n)) ((n)}} \\ & {{+ h _ {\mathcal {\mathrm{大}}} ^ {- 1} ((n)} ([ n ] ([ n ] ([ n ] ([ n ] ([ n ] ([ n ] ([ n ] ([ n ] ([ n ] ([ n ] ([ n ] ([ n ] ([ n ] ([ n ] ([ n ] ([ n ] ([ n ] ([ n ] ([ n ] ([ n ] ([ n ] ([ n ] ([ n ] ([ n ] ([ n ] ([ n ] ([ n ] ([ n ] ([ n ] ([ n ] ([ n ] ([ n ] ([ n ] ([ n ]([ n ] ([ n ] ([ n ] ([ n ] ([ n ] ([ n ] ([ n ] ([ n ] ([ n ] ([ n ] ([ n ] ([ n ] ([ n ] ([ n ] ([ n ] ([ n ] ([ n ] ([ n ] ([ n ] ([ n ] ([ n ] ([ n ] ([ n ] ([ n ] ([ n ] ([ n ] ([ n ] ([ n ] ([ n ] ([ n ] ([ n ] ([ n ] ([ n ] ([ N ]) ([ i ]) ([ i ]) ([ i ]) ([ i ]) ([ i ]) ([ i ]) ([ i ]) ([ i ]) ([ i ]) ([ i ]) ([ i ]) ([ i ]) ([ i ]) ([ i ]) ([ i ]) ([ i ]) ([ i ]) ([ i ]) ([ i ]) ([ i ]) ([ i ]) ([ i ]) ([ i ]) ([ i ]) ([ i ]) ([ i ]) ([ i ]) ([ i ]) ([ i ]) ([ i ]) ([ i ]) ([ i ]) ([ i ]) ([ i ])).} \\ & {{+ h _ {\mathcal {\mathrm{d}}} [ h _ {\mathcal {\mathrm{d}}} [ h _ {\mathcal {\mathrm{d}}} [ h _ {\mathcal {\mathrm{d}}} [ h _ {\mathcal {\mathrm{d}}} [ h _ {\mathcal {\mathrm{d}}} [ h _ {\mathcal {\mathrm{d}}} [ h _ {\mathcal {\mathrm{d}}} [ h _ {\mathcal {\mathrm{d}}} [ h _ {\mathcal {\mathrm{d}}} [ f ] [ f ] [ f ] [ f ] [ f ] [ f ] [ f ] [ f ] [ f ] [ f ] [ f ] [ f ] [ f ] [ f ] [ f ],}} \\ & {{+ h _ {\mathcal {\mathrm{d}}} [ h _ {\mathcal {\mathrm{d}}} [ h _ {\mathcal {\mathrm{d}}} [ h _ {\mathcal {\mathrm{d}}} [ h _ {\mathcal {\mathrm{d}}} [ h _ {\mathcal {\mathrm{d}}} [ h _ {\mathcal {\mathrm{d}}} [ h _ {\mathcal {\mathrm{d}}} [ h _ {\mathcal {\textit}} [ h _ {\textit}} [ h _ {\textit}} [ h _ {\textit} [ h _ {\textit} [ h _ {\textit} [ h _ {\textit} [ h _ {\textit} [ h _ {\textit} [ h _ {\textit} [ h _ {\textit} [ h _ {\textit} [ h _ {\textit} [ h _ {\textit} [ h _ {\textit} [ h _ {\boldsymbol x}, [ h _ {\boldsymbol x}, [ h _ {\boldsymbol x}, [ h _ {\boldsymbol x}, [ h _ {\boldsymbol x}, [ h _ {\boldsymbol x}, [ h _ {\boldsymbol x}, [ h _ {\boldsymbol x}, [ h _ {\boldsymbol x}, [ h _ {\boldsymbol x}, [ h _ {\boldsymbol x}, [ h _ {\boldsymbol x}, [ h _ {\boldsymbol x}, [ h _{\boldsymbol x}, [ h_{\boldsymbol x}, [ h_{\boldsymbol x}, [ h_{\boldsymbol x}, [ h_{\boldsymbol x}, [ h_{\boldsymbol x}, [ h_{\boldsymbol x}, [ h_{\boldsymbol x}, [ h_{\boldsymbol x}, [ h_{\boldsymbol x}, [ h_{\boldsymbol x}, [ h_{\boldsymbol x}, [ h_{\boldsymbol x}, [ h_{\boldsymbol x}, [ h_{\boldsymbol x}, [ h_{\boldsymbol x}, [ h_{\boldsymbol x}, [ h_{\boldsymbol x}, [ h_{\boldsymbol x}, [ h_{\boldsymbol x}, [ h_{\boldsymbol x}, [ h_{\boldsymbol x}, [ h_{\boldsymbol x}, [ h_{\boldsymbol x}, [ h_{\boldsymbol x}, [ h_{\boldsymbol x}, [ h_{- 1}), [ h_{- 1}), [ h_{- 1}), [ h_{- 1}), [ h_{- 1}), [ h_{- 1}), [ h_{- 1}), [ h_{- 1}), [ h_{- 1}), [ h_{- 1}), [ h_{- 1}), [ h_{- 1}), [ h_{- 1}), [ h_{- 1}), [ h_{- 1}), [ H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H . \\ & + E _ {(i)} (\frac {(i)}{(i)} (\frac {(i)}{(i)} (\frac {(i)}{(i)} (\frac {(i)}{(i)} (\frac {(i)}{(i)} (\frac {(i)}{(i)} (\frac {(i)}{(i)} (\frac {(i)}{(i)} (\frac {(i)}{(i)} (\frac {(i)}{(i)} (\frac {(i)}{(i)} (\frac {(i)}{(I)} (\frac {(i)}{(I)} (\frac {(i)}{(I)} (\frac {(i)}{(I)} (\frac {(i)}{(I)} (\frac {(i)}{(I)} (\frac {(i)}{(I)} (\frac {(i)}{(I)} (\frac {(i)}{(I)} (\frac {(i)}{(I)} (\frac {(i)}{(I)} ({\frac {(i)}{(I)}} (\frac {(i)}{(I)} (\frac {(i)}{(I)} (\frac {(i)}{(I)} (\frac {(i)}{(I)} (\frac {(i)}{(I)} (\frac {(i)}{(I)}(\frac {(i)}{(I)} (\frac {(i)}{(I)} (\frac {(i)}{(I)} (\frac {(i)}{(I)} (\frac {(i)}{(I)} (\frac {(i)}{(I)} (\frac {(i)}{(I)} (\frac {(i)}{(I)} (\frac {(i)}{(I)} ({\frac {(i)}{(I)} ({\frac {(i)}{(I)} ({{\bar {}^{\prime}})^{\prime}} ({\bar {}^{\prime}})^{\prime}}({{\bar {}^{\prime}})^{\prime}}({{\bar {}^{\prime}})^{\prime}}({{\bar {}^{\prime}})^{\prime}}({{\bar {}^{\prime}})^{\prime}}({{\bar {}^{\prime}})^{\prime}}({{\bar {}^{\prime}})^{\prime}}({{\bar {}^{\prime}})^{\prime}}({{\bar {}^{\prime}})^{\prime}}({{\bar {}^8^{\prime}})^{\prime}}({{\bar {}^8^{\prime}})^{\prime}}({{\bar {}^8^{\prime}})^{\prime}}({{\bar {}^8^{\prime}})^{\prime}}({{\bar {}^8^{\prime}})^{\prime}}({{\bar {}^8^{\prime}})^{\prime}}({{\bar {}^8^{\prime}})^{\prime}}({{\bar {}^8^8^{\prime}})^{\prime}}({{\bar {}^8^8^{\prime}})^{\prime}}({{\bar {}^8^8^{\prime}})^{\prime}}({{\bar {}^8^8^{\prime}})^{\prime}}({{\bar {}^8^8^{\prime}})^{\prime}}({{{\bar {}^8^8^{\prime}}}^{(k)}}({{{\bar {}^8^8^{\prime}}}^{(k)}}({{{\bar {}^8^8^{\prime}}}^{(k)}}({{{\bar {}^8^8^{\prime}}}^{(k)}}({{{\bar {}^8^8^{\prime}}}^{(k)}}({{{\bar {}^8^8^{\prime}}}^{(k)}}({{{\bar {}^8^8^{\prime}}}^{(\bar {}^8^8^{\prime}}}^{(k)}}({{{\bar {}^8^8^{\prime}}}^{(k)}}({{{\bar {}^8^8^{\prime}}}^{(k)}}({{{\bar {}^8^8^{\prime}}}^{(k)}}({{{\bar {}^8^8^{\prime}}}^{(k)}}({{{\bar {}^8^8^{\prime}}}^{(k)}}{{{\bar {}^8^8^{\prime}}}^{(k)}}({{{\bar {}^8^8^{\prime}}}^{(k)}}({{{\bar {}^8^8^{\prime}}}^{(k)}}({{{\bar {}^8^8^{\prime}}}^{(k)}}({{{\bar {}^8^8^{\prime}}}^{(k)}}({{{\bar {}^8^8^{\prime}}}^{(K)}}({{{\bar {}^8^8^{\prime}}}^{(K)}}({{{\bar {}^8^8^{\prime}}}^{(K)}}({{{\bar {}^8^8^{\prime}}}^{(K)}}({{{\bar {}^8^8^{\prime}}}^{(K)}}({{{\bar {}^8^8^{\prime}}}^{(K)}}({{{\bar {}^8^8^{\prime}}} ^{(K)}}({{{\bar {}^8^8^{\prime}}}^{(K)}}({{{\bar {}^8^8^{\prime}}}^{(K)}}({{{\bar {}^8^8^{\prime}}}^{(K)}}({{{\bar {}^8^8^{\prime}}}^{(K)}}({{{\bar {}^8^8^{\prime}}}^{(K)}}({\bar {}^8^8^{\prime}_{}^{(K)}}({{\bar {}^8^8^{\prime}_{}^{(K)}}(^{(K)}}[{\bar {}^8^8^{\prime}_{}^{(K)}}[{\bar {}^8^8^{\prime}_{}^{(K)}}[{\bar {}^8^8^{\prime}_{}^{(K)}}[{\bar {}^8^8^{\prime}_{}^{(K)}}[{\bar {}^8^8^{\prime}_{}^{(K)}}[{\bar {}^8^8^{\prime}_{}^{(K)}}[{\bar {}^8_{}^{(K)}}[{\bar {}^8_{}^{(K)}}[{\bar {}^8_{}^{(K)}}[{\bar {}^8_{}^{(K)}}[{\bar {}^8_{}^{(K)}}[{\bar {}^8_{}^{(K)}}[{\bar {}^8_{}^{(K)}}[{\bar {}^8_{}^{(K)}}[\overline H_4 K_2 K_2 K_2 K_2 K_2 K_2 K_2 K_2 K_2 K_2 K_2 K_2 K_2 K_2 K_2 K_2 K_2 K_2 K_2 K_2 K_2 K_2 K_2 K_2 K_2 K_2 K_2 K_2 K_2 K_2 K_2 K_2 K_2 K_2 C_ {- | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D | D |D | D |D |D |D |D |D |D |D |D |D |D |D |D |D |D |D |D |D |D |D |D |D |D |D |D |D |D |D |D |D |D |D |D |D |D |D |D |D |D |D |D |D |D |D |D |D |D |D |D |D |D |C | |}\]

\[\begin{array}{l} \text {We then note that:} \\ \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta \boldsymbol {\nu} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \\ \quad + \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta \boldsymbol {\nu} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \\ \quad + \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta \boldsymbol {\nu} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta \boldsymbol {\nu} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \\ \quad + \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta \boldsymbol {\nu} \\ \quad + \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta \boldsymbol {\nu} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta \boldsymbol {\nu} \\ \quad + \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta \boldsymbol {\nu} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta \boldsymbol {\nu} \\ = \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes (\mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta \boldsymbol {\nu} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta \boldsymbol {\nu} + \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta \boldsymbol {\nu} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta \boldsymbol {\nu}) \\ + \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta \boldsymbol {\nu} \otimes (\mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta \boldsymbol {\nu} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathrm{h}_{\mathrm{x}} ^{l - 1}\sigma\eta\boldsymbol{\nu}) \end{array}\]

\[\begin{array} { r l } & = \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \left( \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \nu } + \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \nu } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \nu } \right) \\ & + \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \left( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \nu } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \right) \\ & + \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \nu } \otimes \left( \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } + \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \nu } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \right) \\ & + \mathbf { h } _ { \mathbf { x } } ^ { l - n } \sigma \boldsymbol { \eta } \boldsymbol { \nu } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \nu } \\ & = \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \left( \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } + \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \nu} \right) \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \nu } \\ & + \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \left( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \nu} \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f} \right) \\ & + \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \nu} \otimes \left( \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } + \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta} \boldsymbol { \nu} \right) \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f} \\ & + \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta} \boldsymbol { \nu} \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta} \boldsymbol { \nu} \\ & . \\ & E _ { t }\left[ {\tilde {\mathbf {\Delta x}}} _ {\ell + l} ^ {\ell} {\otimes} {\tilde {\mathbf {\Delta x}}} _ {\ell + l} ^ {\ell} {\otimes} {\tilde {\mathbf {\Delta x}}} _ {\ell + l} ^ {\ell} - {\tilde {\mathbf {\Delta x}}} _ {\ell + l} ^ {\ell} {\otimes} {\tilde {\mathbf {\Delta x}}} _ {\ell + l} ^ {\ell} {\otimes} {\tilde {\mathbf {\Delta x}}} _ {\ell + l} ^ {\ell} {\otimes} {\tilde {\mathbf {\Delta x}}} _ {\ell + l} ^ {\ell} \right] \\ & = {\tilde {\mathbf {\Delta x}}} _ {\ell + l} ^ {\ell} {\tilde {\mathbf {\Delta x}}} _ {\ell + l} ^ {\ell} {\otimes} {\tilde {\mathbf {\Delta x}}} _ {\ell + l} ^ {\ell} {\otimes} {\tilde {\mathbf {\Delta x}}} _ {\ell + l} ^ {\ell} - {\tilde {\mathbf {\Delta x}}} _ {\ell + l} ^ {\ell} {\otimes} {\tilde {\mathbf {{x}}} _ {{t + l}} ^ {\ell}} {\otimes} {\tilde {\mathbf {{x}}} _ {{t + l}} ^ {\ell}} {\otimes} {\tilde {\mathbf {{x}}} _ {{t + l}} ^ {\ell}} \\ & + E _ {\ell}\left[ 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 4. \\ & + E _ {\ell}\left[ 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8. \\ & + E _ {\ell}\left[ 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0, j = k (k , m ) | k , m ) | k , m ) | k , m ) | k , m ) | k , m ) | k , m ) | k , m ) | k , m ) | k , m ) | k , m ) | k , m ) | k , m ) | k , m ) | k , m ) | k , m ) | k , m ) | k , m ) | k , m ) | k , m ) | k , m ) | n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n . \\ & + E _ {\ell}\left[ e q u a d i s e c o r r e s e q u a d i s e c o r r e s e q u a d i s e c o r r e s e q u a d i s e c o r r e s e q u a d i s e c o r r e s e q u a d i s e c o r r e s e q u a d i s e c o r r e s e q u a d i s e c o r r e s eq u a d i s e c o r r e s e q u a d i s e c o r r e s e q u a d i s e c o r r e s e q u a d i s e c o r r e s e q u a d i s e c o r r e s e q u a d i s e c o r r e s e q u a d i s e c o r r e s e q u u a d i s e c o r r e s e q u a d i s e c o r r e s e q u a d i s e c o r r e s e q u a d i s e c o r r e s e q u a d i s e c o r r e s e q u a d i s e c o r r e s e q u a d i s e c o r r e s e q u w a d i s e c o r r e s e q u a d i s e c o r r e s e q u a d i s e c o r r e s e q u a d i s e c o r r e s e q u a d i s e c o r r e s e q u a d i s e c o r r e s e q u a d i s e c o r r e s e q u a c d i s e c o r t h y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b yb y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y b y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y BY B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B Y B YB YBYBYBYBYBYBYBYBYBYBYBYBYBYBYBYBYBYBYBYBYBYBYBYBYBYBYBYBYBYBYBYBYBYBYBYBYBYBYBYBYBYBYBYBYBYBYBYBYBYBYB< fcel>E_{t}\left[{\tilde{\mathbf{x}}}_{t+ l}\right]\otimes\left(\mathrm{h}_{\mathrm{x}}^{\mathrm{l}}\mathrm{x}_{t}^{\mathrm{f}}+\mathrm{h}_{\mathrm{x}}^{\mathrm{l}-1}\mathrm{c}_{\mathrm{m}}\mathrm{c}_{\mathrm{m}}\mathrm{c}_{\mathrm{m}}\mathrm{c}_{\mathrm{m}}\mathrm{c}_{\mathrm{m}}\mathrm{c}_{\mathrm{m}}\mathrm{c}_{\mathrm{m}}\mathrm{c}_{\mathrm{m}}\mathrm{c}_{\mathrm{m}}\mathrm{c}_{\mathrm{m}}\mathrm{c}_{\mathrm{m}}^{\prime}\mathrm{c}_{\mathrm{m}}^{\prime}\mathrm{c}_{\mathrm{m}}^{\prime}\mathrm{c}_{\mathrm{m}}^{\prime}\mathrm{c}_{\mathrm{m}}^{\prime}\mathrm{c}_{\mathrm{m}}^{\prime}\mathrm{c}_{\mathrm{m}}^{\prime}\mathrm{c}_{\mathrm{m}}^{\prime}\mathrm{c}_{\mathrm{m}}, j = k (k, m ) | k, m ) | k, m ) | k, m ) | k, m ) | k, m ) | k, m ) | k, m ) | k, m ) | k, m ) | k, m ) | k, m ) | k, m ) | k, m ) | k, m ) | k, m ) | k, m ) | k, m ) | k, m ) | k, m ) | k, m ) |k, m ) |k, m ) |k, m ) |k, m ) |k, m ) |k, m ) |k, m ) |k, m ) |k, m ) |k, m ) |k, m ) |k, m ) |k, m ) |k, m ) |k, m ) |k, m ) |k, m ) |k, m ) |k, m ) |k, m ) |n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m;m,k,l,k,l,k,l,k,l,k,l,k,l,k,l,k,l,k,l,k,l,k,l,k,l,k,l,k,l,k,l,k,l,k,l,k,l,k,l,k,l,k,l,k,l,k,l,k,l,k,l,k,l,k,l,k,l,k,l,k,l,k,l,k,l,k,l,k,l,k,l,k,l,k,l,k,l,k,l,k,l,k,l,k,l,k,l,k,l,k,l,k,l,k,l,k,l,k,l,k,l,klllkllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllll< ecel>< nl>\]

The final three terms can be computed as follows:

\[\begin{array}{r l} & E _ {t} \left[ \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} \right] \\ & \quad = E _ {t} [ \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} \\ & \qquad + \mathbf {h} _ {\mathbf {x}} ^ {l - 2} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 2} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} \\ & \qquad +... \\ & \qquad + \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + l} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j ]} \\ & = E _ {t} [ \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} \\ & \qquad + \mathbf {h} _ {\mathbf {x}} ^ {l - 2} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 2} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 2} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 2} \\ & \qquad +... \\ & \qquad + \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + l} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} \otimes \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + l ]} \\ & b e c a u s e t h e i n n o v a t i o n s a r e i n d e p e n d e n t a c r o s s t i m e. \end{array}\]

\[\begin{array} { l } = \sum _ { j = 1 } ^ { l } E _ { t } [ \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \nu } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } ] \\ = \sum _ { j = 1 } ^ { l } \Omega _ { j } \\ \text {Let} \Omega _ { j } \equiv E _ { t } \left[ \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \nu } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + 1 } \right] . \text {So} \\ \Omega _ { j } \equiv E _ { t } \left[ \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + 1 } \otimes \left\{ \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } ( \gamma _ { 2 }, : ) \sigma \boldsymbol { \eta } \boldsymbol { \nu } \times \left\{ \mathbf { h } _ { \mathbf { x } } ^ { l - j } ( \gamma _ { 1 }, : ) \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + 1 } \right\} _ { \gamma _ { 1 } = 1 } ^ { n _ { x }} \right\} _ { \gamma _ { 2 } = 1 } ^ { n _ { x }} \\ = E _ { t } \left\{ \mathbf { h } _ { \mathbf { x } } ^ { l - j } ( \gamma _ { 3 , : }) \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + 1 } \times \left\{ \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } ( \gamma _ { 2 }, : ) \sigma \boldsymbol { \eta } \boldsymbol { \nu } \times \left\{ \mathbf { h } _ { \mathbf { x } } ^ { l - j } ( \gamma _ { 1 }, : ) \sigma \boldsymbol { \eta} \boldsymbol { \epsilon } _ { t + 1 } \right\} _ { \gamma _ { 1 } = 1 } ^ { n _ { x }} \right\} _ { \gamma _ { 2 } = 1 } ^ { n _ { x }} \\ = E _ { t } \left\{ {\mathbf h} _ { {\mathbf x}} ^ { l - j} ( {\boldsymbol {\gamma}} _ { 3 , :}) {\boldsymbol {\sigma}} \sum_ {\phi_ { 2} = 1} ^ {{n _ { e}}} {\boldsymbol {\eta}} ( {:}, {\phi_ { 2}}) {\boldsymbol {\epsilon}} _ { t + 1} ( {\phi_ { 2}}, 1) {\times} \sum_ {\phi_ { 1} = 1} ^ {{n _ { e}}} {\boldsymbol {\eta}} ( {:}, {\phi_ { 1}}) {\boldsymbol {\epsilon}} _ { t + 1} ( {\phi_ { 1}}, 1) \sum_ {\phi_ { 2} = 1} ^ {{n _ { e}}} {\boldsymbol {\eta}} ( {:}, {\phi_ { 2}}) {\boldsymbol {\epsilon}} _ { t + 1} ( {\phi_ { 2}}, 1) \sum_ {\phi_ { 1} = 1} ^ {{n _ { e}}} {\boldsymbol {\eta}} ( {:}, {\phi_ { 1}}) {\boldsymbol {\epsilon}} _ { t + 1} ( {\phi_ { 1}}, 1) \sum_ {\phi_ { 2} = 1} ^ {{n _ { e}}} {\boldsymbol {\eta}} ( {:}, {\phi_{ 2}}) {\boldsymbol {\epsilon}} _ { t + 1} ( {\phi_ { 2}}, 1) \sum_ {\phi_ { 1} = 1} ^ {{n _ { e}}} {\boldsymbol {\eta}} ( {:}, {\phi_{ 1}}) {\boldsymbol {\epsilon}} _{ t + 1} ( {\phi_{ 1}}, 1) \sum_ {\phi_{ 2} = 1} ^ {{n _ { e}}} {\boldsymbol {\eta}} ( {:}, {\phi_{ 2}}) {\boldsymbol {\epsilon}}_{ t + 1} ( {\phi_{ 2}}, 1) \sum_ {\phi_{ 2} = 1} ^ {{n _ { e}}} {\boldsymbol {\eta}} ( {:}, {\phi_{ 2}}) {\boldsymbol {\epsilon}}_{ t + 1} ( {\phi_{ 2}}, 1) \sum_ {\phi_{ 2} = 1} ^ {{n _ { e}}} {\boldsymbol {\eta}} ( {:}, {\phi_ {-}}) {\boldsymbol{\epsilon}}_{ t + 1} ( {\phi_{ - }} {\sum_ {\phi_{ - }}} {\boldsymbol{\eta}} ( {:}, {\phi_{ - }}) {\boldsymbol{\epsilon}}_{ t + 1} ( {\phi_{ - }} {\sum_ {\phi_{ - }}} {\boldsymbol{\eta}} ( {:}, {\phi_{ - }}) {\boldsymbol{\epsilon}}_{ t + 1} ( {\phi_{ - }} {\sum_ {\phi_{ - }}} {\boldsymbol{\eta}} ( {:}, {\phi_{ - }}) {\boldsymbol{\epsilon}_ {-}} ) \\ = E _ { t } \left\{ {\mathbf h} _ { {\mathbf x}} ^ { l - j} ( {\boldsymbol {\gamma}} _ { 3 , :}) {\boldsymbol {\sigma}} {\sum_ {{\phi_ { 2} = 1}}} ^ {n _ {{e}}} {\sum_ {{\phi_ {{1}} = 1}}} {\sum_ {{\phi_ {{1}} = 1}}} {\sum_ {{\phi_ {{1}} = 1}}} {\sum_ {{\phi_ {{1}} = 1}}} {\sum_ {{\phi_ {{1}} = 1}}} {\sum_ {{\phi_ {{1}} = 1}}} {\sum_ {{\phi_ {{1}} = 1}}} {{- }} {{\sum_ {{\phi_ {{1}} = 1}}} {{\sum_ {{\phi_ {{1}} = 1}}} {{\sum_ {{\phi_ {{1}} = 1}}} {{\sum_ {{\phi_ {{1}} = 1}}} {{\sum_ {{\phi_ {{1}} = 1}}} {{- }}} {{\sum_ {{\phi_ {{1}} = 1}}} {{\sum_ {{\phi_ {{1}} = 1}}} {{- }}} {{\sum_ {{\phi_ {{1}} = 1}}} {{- }}} {{\sum_ {{\phi_ {{1}} = 1}}} {{- }}} {{\sum_ {{\phi_ {{0}} = 2}}} {{- }}} {{\sum_ {{\phi_ {{0}} = 2}}} {{- }}} {{\sum_ {{\phi_ {{0}} = 2}}} {{- }}} {{\sum_ {{\phi_ {{0}} = 2}}} {{- }}} {{\sum_ {{\phi_ {{0}} = 2}}} {{- }}} {{\sum_ {{\phi_ {{{0}}}} = 2}}} {{- }}} {{\sum_ {{\phi_ {{{0}}}} = 2}}} {{- }}} {{\sum_ {{\phi_ {{{0}}}} = 2}}} {{- }}} {{\sum_ {{\phi_ {{{0}}}} = 2}}} {{- }}} {{\sum_ {{\phi_ {{{0}}}} = 2}}} {{- }}} {} \\ = E _ { t } \left\{ {\mathbf h} _ { {\mathbf x}} ^ { l - j} ( {\boldsymbol {\gamma}} _ {3, :}) {\boldsymbol {\sigma}} {\sum_ {{\phi_ {_2} = 1}}} ^{{{n _{{e}}}}} [ {\boldsymbol {\eta}} ( :, {\phi_ {_2})} ] {\boldsymbol {\sigma}} [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ ] ] / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / | ^ {- k} | ^ {- k} | ^ {- k} | ^ {- k} | ^ {- k} | ^ {- k} | ^ {- k} | ^ {- k} | ^ {- k} | ^ {- k} | ^ {- k} | ^ {- k} | ^ {- k} | ^ {- k} | ^ {- k} | ^ {- k} | ^ {- k} | ^{- k} | ^ {- k} | ^ {- k} | ^ {- k} | ^ {- k} | ^ {- k} | ^ {- k} | ^ {- k} | ^ {- k} | ^ {- k} | ^ {- k} | ^ {- k} | ^ {- k} | ^ {- k} | ^ {- k} | ^ {- k} | ^ {- k} | | = E _ { t } \left\{ {\mathbf h} _ { {\mathbf x}} ^ { l - j} ( {\boldsymbol {\gamma}} _ {3, :}) {\boldsymbol {\sigma}} {\sum_ {{{\phi_ {_2}}} = 1}}{{\phi_ {_2}}} [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ ] / ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] | ^ {- k} | ^ {- k} | ^ {- k} | ^ {- k} | ^ {- k} | ^ {- k} | ^ {- k} | ^ {- k} | ^ {- k} | ^ {- k} | ^ {- k} | ^ {- k} | ^ {- k} | ^ {- k} | ^ {- k} | ^ {- k} | ^ {-k} | ^ {-k} | ^ {-k} | ^ {-k} | ^ {-k} | ^ {-k} | ^ {-k} | ^ {-k} | ^ {-k} | ^ {-k} | ^ {-k} | ^ {-k} | ^ {-k} | ^ {-k} | ^ {-k} | ^ {-k} | ^ {-k} |^ {-k}. \\ = E _ { t } E _ { t } E _ { t } E _ { t } E _ { t } E _ { t } E _ { t } E _ { t } E _ { t } E _ { t } E _ { t } E _ { t } E _ { t } E _ { t } E _ { t } E _ { t } E _ { t } E _ {t + 1 }\]

because the innovations are independent. This expression is directly implementable.

\[\begin{array}{r l} & {\mathrm{And}} \\ & {E _ {t} \left[ \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\epsilon} _ {t + j} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\epsilon} _ {t + j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \pmb {\eta} \pmb {\nu} \right]} \\ & {\quad = \sum_ {j = 1} ^ {l} E _ {t} [ \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\epsilon} _ {t + j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\epsilon} _ {t + j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \pmb {\eta} \pmb {\nu} ]} \\ & {\quad = \sum_ {j = 1} ^ {l} E _ {t} [ (\mathbf {h} _ {\mathbf {x}} ^ {l - j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - j}) (\sigma \pmb {\eta} \pmb {\epsilon} _ {t + j} \otimes \sigma \pmb {\eta} \pmb {\epsilon} _ {t + j}) \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \pmb {\eta} \pmb {\nu} ]} \\ & {\quad = \sum_ {j = 1} ^ {l} (\mathbf {h} _ {\mathbf {x}} ^ {l - j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - j}) E _ {t} [ \sigma \pmb {\eta} \pmb {\epsilon} _ {t + j} \otimes \sigma \pmb {\eta} \pmb {\epsilon} _ {t + j} ] \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \pmb {\eta} \pmb {\nu}} \\ & {\quad = \sum_ {j = 1} ^ {l} (\mathbf {h} _ {\mathbf {x}} ^ {l - j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - j}) \pmb {\Lambda} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \pmb {\eta} \pmb {\nu}} \\ & {\mathrm{where} \pmb {\Lambda} \equiv E _ {t} [ \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1} \otimes \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1} ]. \mathrm{Tocompute} \pmb {\Lambda} \mathrm{wenotethat}} \\ & {\pmb {\Lambda} = E _ {t} [ \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1} \otimes \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1} ]} \\ & {\quad = E _ {t} [ \{\sigma \pmb {\eta} (\gamma_ {2},:) \pmb {\epsilon} _ {t + 1} \sigma \pmb {\eta} (\gamma_ {1},:) \pmb {\epsilon} _ {t + 1} \} _ {\gamma_ {1} = 1} ^ {n _ {x}} ] _ {\gamma_ {2} = 1} ^ {n _ {x}}} \\ & {\quad = E _ {t} [ [ \{\sigma \sum_ {\phi_ {1} = 1} ^ {n _ {e}} \pmb {\eta} (\gamma_ {2}, \phi_ {1}) \pmb {\epsilon} _ {t + 1} (\phi_ {1}, 1) \sum_ {\phi_ {2} = 1} ^ {n _ {e}} \sigma \pmb {\eta} (\gamma_ {1}, \phi_ {2}) \pmb {\epsilon} _ {t + 1} (\phi_ {2}, 1) ] _ {\gamma_ {1} = 1} ^ {n _ {x}} ] _ {\gamma_ {2} = 1} ^ {n _ {x}}} \end{array}\]

\[\begin{array}{l} = E _ {t} \left\{\left\{\sigma \sum_ {\phi_ {1} = 1} ^ {n _ {e}} \sum_ {\phi_ {2} = 1} ^ {n _ {e}} \boldsymbol {\eta} (\gamma_ {2}, \phi_ {1}) \boldsymbol {\epsilon} _ {t + 1} (\phi_ {1}, 1) \sigma \boldsymbol {\eta} (\gamma_ {1}, \phi_ {2}) \boldsymbol {\epsilon} _ {t + 1} (\phi_ {2}, 1) \right\} _ {\gamma_ {1} = 1} ^ {n _ {x}} \right\} _ {\gamma_ {2} = 1} ^ {n _ {x}} \\ = \left\{\left\{\sigma \sum_ {\phi_ {1} = 1} ^ {n _ {e}} \boldsymbol {\eta} (\gamma_ {2}, \phi_ {1}) \sigma \boldsymbol {\eta} (\gamma_ {1}, \phi_ {1}) \right\} _ {\gamma_ {1} = 1} ^ {n _ {x}} \right\} _ {\gamma_ {2} = 1} ^ {n _ {x}} \\ = \left\{\left\{\sigma^ {2} \boldsymbol {\eta} (\gamma_ {2},:) \boldsymbol {\eta} (\gamma_ {1},:) ^ {\prime} \right\} _ {\gamma_ {1} = 1} ^ {n _ {x}} \right\} _ {\gamma_ {2} = 1} ^ {n _ {x}} \end{array}\]

\[\begin{array}{l} \text {And finally} \\ E _ {t} \left[ \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} \right] \\ = \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} \otimes E _ {t} [ \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} ] \\ = \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} \otimes (\mathbf {h} _ {\mathbf {x}} ^ {l - j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - j}) E _ {t} [ \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} \otimes \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} ] \\ = \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} \otimes (\mathbf {h} _ {\mathbf {x}} ^ {l - j} \otimes \mathbf { h } _ {\mathbf {x}} ^ {l - j}) \boldsymbol {\Lambda} \end{array}\]

10.3.2 For

\[\begin{array} { r l } & { } \mathrm{recallfromabovethat} \\ & { } \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { s } = \left( \mathbf { h } _ { \mathbf { x } } \mathbf { x } _ { t } ^ { f } + \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + 1 } \right) \otimes \left( \mathbf { h } _ { \mathbf { x } } \mathbf { x } _ { t } ^ { s } + \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } \left( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \right) + \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } \right) \\ & { } = ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) \left( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { s } \right) + ( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } ) \left( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \right) + ( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } ) \mathbf { x } _ { t } ^ { f } \\ & { } + ( \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x } } ) ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { s } ) + ( \sigma \boldsymbol { \eta } \otimes \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } ) ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) + ( \sigma \boldsymbol { \eta } \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } ) \boldsymbol { \epsilon } _ { t + 1 } \\ & \\ & ~ T h e r e f o r e : \\ & { } \mathbf { x } _ { t + 2 } ^ { f } \otimes \mathbf { x } _ { t + 2 } ^ { s } = ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ( \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { s } ) + ( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } ) ( \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } ) + ( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } ) \mathbf { x } _ { t + 1 } ^ { f } \\ & ~ ~ + ( \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x } } ) ( \boldsymbol { \epsilon } _ { t + 2 } \otimes \mathbf { x } _ { t + 1 } ^ { s} ) + ( \sigma \boldsymbol { \eta } \otimes \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x} } ) ( \boldsymbol { \epsilon} _ { t + 2 } \otimes \mathbf { x} _ { t + 1 } ^ { f } \otimes \mathbf { x} _ { t + 1 } ^ { f} ) + ( \sigma \boldsymbol { \eta } \otimes \frac { 1 } { 2 } \mathbf { h} _ { \sigma \sigma } \sigma ^ { 2} ) \boldsymbol { \epsilon} _ { t + 2 } \\ & \\ & = ( {\mathbf h} _ {\mathbf x} \otimes {\mathbf h} _ {\mathbf x}) [ ( {\mathbf h} _ {\mathbf x} \otimes {\mathbf h} _ {\mathbf x}) ( {\mathbf x} _ { t } ^ { f } \otimes {\mathbf x} _ { t } ^ { s} ) + ( {\mathbf h} _ {\mathbf x} \otimes \frac { 1 } { 2 } {\mathbf H} _ {\mathbf x x} ) ( {\mathbf x} _ { t } ^ { f } \otimes {\mathbf x} _ { t } ^ { f } \otimes {\mathbf x} _ { t } ^ { f} ) + ( {\mathbf h} _ {\mathbf x} \otimes \frac { 1 } { 2 } {\mathbf h} _ { \sigma \sigma } \sigma ^ { 2} ) {\mathbf x} _ { t } ^ { f } \\ & ~ + ( \sigma \boldsymbol { \eta } \otimes {\mathbf h} _ {\mathbf x}) ( {\boldsymbol {\epsilon}} _ { t + 1 } \otimes {\boldsymbol {\chi}} _ { t } ^ { s} ) + ( \sigma \boldsymbol { \eta } \otimes \frac { 1 } { 2 } {\mathbf H} _ {\mathbf x x} ) ( {\boldsymbol {\epsilon}} _ { t + 1 } \otimes {\boldsymbol {\chi}} _ { t } ^ { f } \otimes {\boldsymbol {\chi}} _ { t } ^ { f} ) + ( \sigma \boldsymbol { \eta } \otimes \frac { 1 } { 2 } {\boldsymbol {\chi}} _ { \sigma \sigma } \sigma ^ { 2} ) {\boldsymbol {\epsilon}} _ { t + 1 ]} \\ & \\ & + ( {\mathbf h} _ {\mathbf x} \otimes \frac { 1 } { 2 } {\mathbf H} _ {\mathbf x x}) ( {\boldsymbol {\chi}} _ { t + 1 } ^ {} ) ( {\boldsymbol {\chi}} _ {{t + 1}} ^ {} ) ( {\boldsymbol {\chi}} _ {{t + 1}} ^ {} ) + ( {\boldsymbol {\chi}} _ {{\boldsymbol {\chi}}} ^ {} ) ( {\boldsymbol {\chi}} _ {{t + 1}} ^ {} ) - ( {\boldsymbol {\chi}} _ {{t + 1}} ^ {} ) - ( {\boldsymbol {\chi}} _ {{t + 1}} ^ {} ) - ( {\boldsymbol {\chi}} _ {{t + 1}} ^ {} ) - ( {\boldsymbol {\chi}} _ {{t + 1}} ^ {} ) - ( {\boldsymbol {\chi}} _ {{t + 1}} ^ {} ) - ( {\boldsymbol {\chi}} _ {{\boldsymbol {\chi}}} ^ {} ) - ( {\boldsymbol {\chi}} _ {{t + 1}} ^ {} ) - ( {\boldsymbol {\chi}} _ {{t + 1}} ^ {} ) - ( {\boldsymbol {\chi}} _ {{t + 1}} ^ {} ) - ( {\boldsymbol {\chi}} _ {{t + 1}} ^ {} ) - ( {\boldsymbol {\chi}} _ {{t + 1}} ^ {} ) - ( {\boldsymbol {\xi}} _ {{t + 1}} ^ {} ) - ( {\boldsymbol {\xi}} _ {{t + 1}} ^ {} ) - ( {\boldsymbol {\xi}} _ {{t + 1}} ^ {} ) - ( {\boldsymbol {\xi}} _ {{t + 1}} ^ {} ) - ( {\boldsymbol {\xi}} _ {{t + 1}} ^ {} ) - ( {\boldsymbol {\xi}} _ {{t + 1}} ^{} ) - ( {\boldsymbol {\xi}} _ {{t + 1}} ^{} ) - ( {\boldsymbol {\xi}} _ {{t + 1}} ^{} ) - ( {\boldsymbol {\xi}} _ {{t + 1}} ^{} ) - ( {\boldsymbol {\xi}} _ {{t + 1}} ^{} ) - ( {\boldsymbol {\xi}} _ {{t + 1}} ^{} ) - ( {\boldsymbol {\xi}} . \\ & ~ + (\sigma n g m a l i c k e r e d i n e s s i n g e d i n e s s i n g e d i n e s s i n g e d i n e s s i n g e d i n e s s i n g e d i n e s s i n g e d i n e s s i n g e d i n e s s i n g e d i n e s s i n g e d i n e s s i n g e d i n e s s i ng e d i n e s s i n g e d i n e s s i n g e d i n g e d i n g e d i n g e d i n g e d i n g e d i n g e d i n g e d i n g e d i n g e d i n g e d i n g e d i n g e d i n g e d i n g e d i n g e d i n g e d i n g e d i n g e d i n g e d i n g e l o w a r y , ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ & > \\ & = ( h a b a l l o u p l o r a b l o u p l o r a b l o u p l o r a b l o u p l o r a b l o u p l o r a b l o u p l o r a b l o u p l o r a b l o u p l o r a b l o u p l o r a b l o u p l o r a b l o u p l o r a b l o u p l o r a b j o w a l l o u p l o r a b j o w a l l o u p l o r a b j o w a l l o u p l o r a b j o w a l l o u p l o r a b j o w a l l o u p l o r a b j o w a l l o u p l o r a b j o w a l l o u p l o r a b j o w a l l l o u p l o r a b j o w a l l o u p l o r a b j o w a l l o u p l o r a b j o w a l l l o u p l o r a b j o w a l l l o u p l o r a b j o w a l l l o u p l o r a b j o w a l l l o u p l o r a b j o w a l l l o u p l o r a b j o w a l l l o u p l o r a b j o w a l l l o u p l o r a c k e r e d i n g e d i n g e d i n g e d i n g e d i n g e d i n g e d i n g e d i n g e d i n g e d i n g e d i n g e d i n g e d i n g e d i n g e d i n g e d i n g e d i n g e d i n g e d i n g e d i n k e r c e c t i v e r s s i v e r s s i v e r s s i v e r s s i v e r s s i v e r s s i v e r s s i v e r s s i v e r s s i v e r s s i v e r s s i v e r s s i v e r s s i v e r s s i v e r s s i v e r s s i v e r s s i v e v e r c k e r c t i v e r c k e r c t i v e r c k e r c t i v e r c k e r c t i v e r c k e r c t i v e r c k e r c t i v e r c k e r c t i v e r c k e r c t i v e r c k e r c t i v e r c k e r c t i v e r c k e r c t t i v e r c k e r c t t i v e r c k e r c t t i v e r c k e r c t t i v e r c k e r c t t i v e r c k e r c t t i v e r c k e r c t t i v e r c k e r c t t i v e r c k e r c t t i v e r c k e r c t . > \\ & = (\alpha_ {-} | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | < |txt_contd|>\]

\[\begin{array} { l } + \left( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } \right) \left( \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } \right) + \left( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } \right) \mathbf { x } _ { t + 1 } ^ { f } \\ + \left( \sigma \eta \otimes \mathbf { h } _ { \mathbf { x } } \right) \left( \boldsymbol { \epsilon } _ { t + 2 } \otimes \mathbf { x } _ { t + 1 } ^ { s } \right) + \left( \sigma \eta \otimes \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } \right) \left( \boldsymbol { \epsilon } _ { t + 2 } \otimes \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } \right) + \left( \sigma \eta \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } \right) \boldsymbol { \epsilon } _ { t + 2 } \\ a n d \\ \mathbf { x } _ { t + 3 } ^ { f } \otimes \mathbf { x } _ { t + 3 } ^ { s } = ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) \left( \mathbf { x } _ { t + 2 } ^ { f } \otimes \mathbf { x } _ { t + 2 } ^ { s } \right) + ( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } ) \left( \mathbf { x } _ { t + 2 } ^ { f } \otimes \mathbf { x } _ { t + 2 } ^ { f } \otimes \mathbf { x } _ { t + 2 } ^ { f } \right) + ( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } ) \mathbf { x } _ { t + 2 } ^ { f } \\ + ( \sigma \eta \otimes \mathbf { h } _ { \mathbf { x } } ) ( \boldsymbol { \epsilon } _ { t + 3 } \otimes \mathbf { x } _ { t + 2 } ^ { s } ) + ( \sigma \eta \otimes \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } ) ( \boldsymbol { \epsilon } _ { t + 3 } \otimes \mathbf { x } _ { t + 2 } ^ { f } \otimes \mathbf { x } _ { t + 2 } ^ { f } ) + ( \sigma \eta \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } ) \boldsymbol { \epsilon } _ { t + 3 } \\ = ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } ) } [ ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } ) ) ^ { 2 } ( [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ ] i ] )} ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) \\ + ( {\cal H} _ {\mathrm{ex}}) ( {\cal X} _ {\mathrm{ex}}) ( {\cal X} _ {\mathrm{ex}}) ( {\cal X} _ {\mathrm{ex}}) ( {\cal X} _ {\mathrm{ex}}) ( {\cal X} _ {\mathrm{ex}}) ( {\cal X} _ {\mathrm{ex}}) ( {\cal X} _ {\mathrm{ex}}) ( {\cal X} _ {\mathrm{ex}}) ( {\cal X} _ {\ell}) \\ + ( {\cal H} _ {\mathrm{ex}}) ( {\cal X} _ {\mathrm{ex}}) ( {\cal X} _ {\ell}) + ( {\cal X} _ {\mathrm{ex}}) ( {\cal X} _ {\ell}) + ( {\cal X} _ {\ell}) + ( {\cal X} _ {\ell}) - ( {\cal X} _ {\ell}) - ( {\cal X} _ {\ell}) - ( {\cal X} _ {\ell}) - ( {\cal X} _ {\ell}) - ( {\cal X} _ {\ell}) - ( {\cal X} _ {\ell}) - ( {\cal X} _ {\ell}) - ( {\cal X} _ {\ell}) - ( {\cal X} _ {\ell}) - ( {\cal X} _ {\ell}) + ( {\cal X} _ {\ell}) - ( {\cal X} _ {\ell}) - ( {\cal X} _ {\ell}) - ( {\cal X} _ {\ell}) - ( {\cal X} _ {\ell}) - ( {\cal X} _ {\ell}) - ( {\cal X} _ {\ell}) - ( {\cal X} _ {\ell}) - ( {\cal X} _ {\ell}) + ( {\cal X} _ {\ell}) + ( {\cal X} _ {\ell}) - ( {\cal X} _ {\ell}) - ( {\cal X} _ {\ell}) - ( {\cal X} _ {\ell}) - ( {\cal X} _ {\ell}) - ( {\cal X} _ {\ell}) - ( {\cal X} _ {\ell}) - ( {\cal X} _ {\ell}) + ( {\cal X} _ {\ell}) - ( {\cal X} _ {\ell}) + ( {\cal X} _ {\ell}) - ( {\cal X} _ {\ell}) - ( {\cal X} _ {\ell}) - ( {\cal X} _ {\ell}) - ( {\cal X} _ {\ell}) - ( {\cal X} _ {\ell}) - ( {\cal X} _ {\ell}) - ( {\cal X} _ {\ell}) + ( {\cal X} _ {\ell}) + ( {\cal X} _ {\ell}) + ( {\cal X} _ {\ell}) - ( {\cal X} _ {\ell}) - ( {\cal X} _ {\ell}) - ( {\cal X} _ {\ell}) - ( {\cal X} _ {\ell}) - ( {\cal X} _ {\ell}) - ( {\cal X} _ {\ell}) - ( {\cal X} _ {\ell}) , \\ + (\sigma_ {{\cal H}}) (\sigma_ {{\cal H}}) (\sigma_ {{\cal H}}) (\sigma_ {{\cal H}}) (\sigma_ {{\cal H}}) (\sigma_ {{\cal H}}) (\sigma_ {{\cal H}}) (\sigma_ {{\cal H}}) (\sigma_ {{\cal H}}) (\sigma_ {{\cal H}}) (\sigma_ {{\cal H}}) (\sigma_ {{\cal H}}) (+ (\sigma_ {{\cal H}}) (\sigma_ {{\cal H}}) (\sigma_ {{\cal H}}) (\sigma_ {{\cal H}}) (\sigma_ {{\cal H}}) (\sigma_ {{\cal H}}) (\sigma_ {{\cal H}}) (\sigma_ {{\cal H}}) (\sigma_ {{\cal H}}) (\sigma_ {{\cal H}}) (\sigma_ {{\cal H}}) (- (\sigma_ {{\cal H}}) (\sigma_ {{\cal H}}) (\sigma_ {{\cal H}}) (\sigma_ {{\cal H}}) (\sigma_ {{\cal H}}) (\sigma_ {{\cal H}}) (\sigma_ {{\cal H}}) (\sigma_ {{\cal H}}) (\sigma_ {{\cal H}}) (\sigma_ {{\cal H}}) (- (\sigma_ {{\cal H}})) \\ + (\sigma_ {{\cal H}}) (\sigma_ {{\cal H}}) (\sigma_ {{\cal H}}) (\sigma_ {{\cal H}}) (\sigma_ {{\cal H}}) (\sigma_ {{\cal H}}) (\sigma_ {{\cal H}}) (\sigma_ {{\cal H}}) (\sigma_ {{\cal H}}) (\sigma_ {{\cal H}}) (\sigma_ {{\cal H}}, \\ + (\sigma_ {{\cal H}}) (\sigma_ {{\cal H}}) (\sigma_ {{\cal H}}) (\sigma_ {{\cal H}}) (\sigma_ {{\cal H}}, \\ + (\sigma_ {{\cal H}}) (\sigma_ {{\cal H}}) (\sigma_ {{\cal H}}, \\ + (\sigma_ {{\cal H}}) (\sigma_ {{\cal H}}, \\ + (\sigma_ {{\cal H}}) (\sigma_ {{\cal H}}, \\ + (\sigma_ {{\cal H}}, \\ + (\sigma_ {{\cal H}}, \\ + (\sigma_ {{\cal H}}, \\ + (\sigma_ {{\cal H}}, \\ + (\sigma_ {{\cal H}}, \\ + (\sigma_ {{\cal H}}, \\ + (\sigma_ {{\cal H}}, \\ + (\sigma_ {{\cal H}}, \\ + (\sigma_ {{\cal H}}, \\ + (\sigma_ {{\cal H}}, \\+ (\sigma_ {{\cal H}}, \\+ (\sigma_ {{\cal H}}, \\+ (\sigma_ {{\cal H}}, \\+ (\sigma_ {{\cal H}}, \\+ (\sigma_ {{\cal H}}, \\+ (\sigma_ {{\cal H}}, \\+ (\sigma_ {{\cal H}}, \\+ (\sigma_ {{\cal H}}, \\+ (\sigma_ {{\cal H}}, \\+ (\sigma_ {{\cal H}}, \\- (\sigma_ {{\cal H}}, \\+ (\sigma_ {{\cal H}}, \\+ (\sigma_ {{\cal H}}, \\+ (\sigma_ {{\cal H}}, \\+ (\sigma_ {{\cal H}}, \\+ (\sigma_ {{\cal H}}, \\+ (\sigma_ {{\cal H}}, \\+ (\sigma_ {{\cal H}}, \\+ (\sigma_ {{\cal H}}, \\+ (\sigma_ {{\cal H}}, \\+\infty , \\+ (\infty , \\+ (\infty , \\+ (\infty , \\+ (\infty , \\+ (\infty , \\+ (\infty , \\+ (\infty , \\+ (\infty , \\+ (\infty , \\+ (\infty , \\+ (\infty , \\+ (\infty , \\+ (\infty , \\+ (\infty , \\+ (\infty , \\+ (\infty , \\+ (\infty ,\\+ (\infty , \\+ (\infty , \\+ (\infty , \\+ (\infty , \\+ (\infty , \\+ (\infty , \\+ (\infty , \\+ (\infty , \\+ (\infty , \\+ (\infty , \\+ (\infty , \\+ (\infty , \\+ (\infty , \\+ (\infty , \\+ (\infty , \\+ (\infty , \\+ (\inemptyset , \\+ (\bf e c c o n s p o r s p o r s p o r s p o r s p o r s p o r s p o r s p o r s p o r s p o r s p o r s p o r s p o r s p o r s p o r s p o r s p o r s p o r s p o r s p o r s p o r s p o r s p o r s p o r s p o r s p o p e r s p o r s p o r s p o r s p o r s p o r s p o r s p o r s p o r s p o r s p o r s p o r s p o r s p o r s p o r s p o r s p o r s p o r s p o r s p o r s p o r s p o r s p o r s p o r s p o r s p e r s p o r s p o r s p o r s p o r s p o r s p o r s p o r s p o r s p o r s p o r s p o r s p o r s p o r s p o r s p o r s p o r s p o r s p o r s p o r s p e r s p e r s p o r s p o r s p o r s p e r s p e r s p e r s p e r s p e r s p e r s p e r s p e r s p e r s p e r s p e r s p e r s p e r s p e r s p e r s p e r s p e r s p e r s p e r s p e r s p e r s p e r s p e r s p e r s p e r s p q u a l y a n d i n g e n e r a l \\ & A n d i n g e n e r a l \\ & x ^ f_{t + l} ⓧ x ^s_{t + l}= (x ^f_{t}, x ^l_{t}, x ^f_{t}, x ^s_{t}, x ^l_{t}, x ^l_{t}, x ^l_{t}, x ^l_{t}, x ^l_{t}, x ^l_{t}, x ^l_{t}, x ^l_{t}, x ^l_{t}, x ^l_{t}, x ^l_{t}, x ^l_{t}, x ^l_{t}, x ^l_{t}, x ^l_{t}, x ^l_{t}, x ^l_{t}, y = y^{\prime}\end{array}\]

\[\begin{array}{r l} & {+ \sum_ {i = 0} ^ {l - 1} \left(\mathbf {h _ {x}} \otimes \mathbf {h _ {x}}\right) ^ {l - 1 - i} \left(\sigma \pmb {\eta} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}\right) \pmb {\epsilon} _ {t + 1 + i}} \\ & {+ \sum_ {i = 0} ^ {l - 1} \left(\mathbf {h _ {x}} \otimes \mathbf {h _ {x}}\right) ^ {l - 1 - i} \left(\sigma \pmb {\eta} \otimes \mathbf {h _ {x}}\right) \left(\pmb {\epsilon} _ {t + 1 + i} \otimes \mathbf {x} _ {t + i} ^ {s}\right)} \\ & {+ \sum_ {i = 0} ^ {l - 1} \left(\mathbf {h _ {x}} \otimes \mathbf {h _ {x}}\right) ^ {l - 1 - i} \left(\sigma \pmb {\eta} \otimes \frac {1}{2} \mathbf {H _ {x x}}\right) \left(\pmb {\epsilon} _ {t + 1 + i} \otimes \mathbf {x} _ {t + i} ^ {f} \otimes \mathbf {x} _ {t + i} ^ {f}\right)} \end{array}\]

We therefore have

\[\begin{array}{r l} & {\tilde {\mathbf {x}} _ {t + l} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + l} ^ {s} = (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {l} (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {s}) + \sum_ {i = 0} ^ {l - 1} (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {l - 1 - i} (\mathbf {h} _ {\mathbf {x}} \otimes \frac {1}{2} \mathbf {H} _ {\mathbf {x x}}) (\tilde {\mathbf {x}} _ {t + i} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + i} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + i} ^ {f})} \\ & {\quad + \sum_ {i = 0} ^ {l - 1} (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {l - 1 - i} (\mathbf {h} _ {\mathbf {x}} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}) \tilde {\mathbf {x}} _ {t + i} ^ {f}} \\ & {\quad + \sum_ {i = 0} ^ {l - 1} (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {l - 1 - i} (\sigma \pmb {\eta} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}) (\pmb {\epsilon} _ {t + 1 + i} + \pmb {\delta} _ {t + 1 + i})} \\ & {\quad + \sum_ {i = 0} ^ {l - 1} (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {l - 1 - i} (\sigma \pmb {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) ((\pmb {\epsilon} _ {t + 1 + i} + \pmb {\delta} _ {t + 1 + i}) \otimes \tilde {\mathbf {x}} _ {t + i} ^ {s})} \\ & {\quad + \sum_ {i = 0} ^ {l - 1} (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {l - 1 - i} (\sigma \pmb {\eta} \otimes \frac {1}{2} \mathbf {H} _ {\mathbf {x x}}) ((\pmb {\epsilon} _ {t + 1 + i} + \pmb {\delta} _ {t + 1 + i}) \otimes \tilde {\mathbf {x}} _ {t + i} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + i} ^ {f})} \end{array}\]

\[\begin{array} { r l } & { \mathrm{Thus} } \\ & { E _ { t } \left[ \tilde { \mathbf { x } } _ { t + l } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + l } ^ { s } - \mathbf { x } _ { t + l } ^ { f } \otimes \mathbf { x } _ { t + l } ^ { s } \right] } \\ & { = E _ { t } [ ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { l } ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { s } ) + \sum _ { i = 0 } ^ { l - 1 } ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { l - 1 - i } ( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } ) ( \tilde { \mathbf { x } } _ { t + i } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + i } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + i } ^ { f } ) } \\ & { + \sum _ { i = 0 } ^ { l - 1 } ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { l - 1 - i } ( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } ) \tilde { \mathbf { x } } _ { t + i } ^ { f } } \\ & { + \sum _ { i = 0 } ^ { l - 1 } ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { l - 1 - i } ( \sigma \eta \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } ) ( \epsilon _ { t + 1 + i } + \delta _ { t + 1 + i } ) } \\ & { + \sum _ { i = 0 } ^ { l - 1 } ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { l - 1 - i } ( \sigma \eta \otimes \mathbf { h } _ { \mathbf { x } } ) ( ( \epsilon _ { t + 1 + i } + \delta _ { t + 1 + i } ) \otimes \tilde { \mathbf { x } } _ { t + i } ^ { s } ) } \\ & { + \sum _ { i = 0 } ^ { l - 1 } ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { l - 1 - i } ( \sigma \eta \otimes \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x} } ) ( ( \epsilon _ { t + 1 + i } + \delta _ { t + 1 + i } ) \otimes \tilde { \mathbf { x } } _ { t + i } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + i } ^ { f } ) } \\ & - \{ ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { l } ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { s} ) + \sum _ { i = 0 } ^ { l - 1 } ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { l - 1 - i } ( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { H } _ {\tt x} {\tt x} {\tt x} {\tt x} {\tt x} {\tt x} {\tt x} {\tt x} {\tt x} {\tt x} {\tt x} {\tt x} {\tt x} {\tt x} {\tt x} {\tt x} {\tt x} {\tt x} {\tt x} {\tt x} {\tt x} {\tt x} {\tt x} {\tt x} {\tt x} {\tt x}) , \\ & { + \sum _ { i = 0 } ^ { l - 1 } ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { l - 1 - i } ( \mathbf { h } _ { \mathbf { x } } \otimes \frac 1 2 h _ { \sigma \sigma }\sigma ^ 2 ) {\tt X} _ { t + i } ^ { f }} \\ & { + \sum _ { i = 0 } ^ { l - 1 } ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x} }\right) ^ { l - 1 - i } ( \sigma \eta \otimes \frac 1 2 h _ {\sigma\sigma }\sigma ^ 2 ) {\tt E} _ { t + 1 + i }} \\ & + \sum _ { i = 0 } ^ { l - 1 } ( \mathbf { h } _ { \mathbf { x} }\otimes\]

\[\begin{array} { r l } & { + \sum _ { i = 0 } ^ { l - 1 } ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { l - 1 - i } \left( \sigma \boldsymbol { \eta } \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } \right) \boldsymbol { \delta } _ { t + 1 + i } } \\ & { + \sum _ { i = 0 } ^ { l - 1 } ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { l - 1 - i } \left( \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x } } \right) \left( ( \boldsymbol { \epsilon } _ { t + 1 + i } + \boldsymbol { \delta } _ { t + 1 + i } ) \otimes \tilde { \mathbf { x } } _ { t + i } ^ { s } \right) } \\ & { + \sum _ { i = 0 } ^ { l - 1 } ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { l - 1 - i } \left( \sigma \boldsymbol { \eta } \otimes \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } \right) \left( ( \boldsymbol { \epsilon } _ { t + 1 + i } + \boldsymbol { \delta } _ { t + 1 + i } ) \otimes \tilde { \mathbf { x } } _ { t + i } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + i } ^ { f } \right) } \\ & { - \{ \sum _ { i = 0 } ^ { l - 1 } ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { l - 1 - i } ( \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x } } ) ( \boldsymbol { \epsilon } _ { t + 1 + i } \otimes \mathbf { x } _ { t + i } ^ { s } ) } \\ & { + \sum _ { i = 0 } ^ { l - 1 } ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { l - 1 - i } ( \sigma \boldsymbol { \eta } \otimes \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } ) ( ( \boldsymbol { \epsilon } _ { t + 1 + i } \otimes \mathbf { x } _ { t + i } ^ { f } \otimes \mathbf { x } _ { t + i } ^ { f } ) \} ] } \\ & = E _ { t } [ \sum _ { i = 0 } ^ { l - 1 } ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { l - 1 - i } ( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } ) ( \tilde { \mathbf { x } } _ { t + i } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + i } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + i } ^ { f } - \mathbf { x } _ { t + i } ^ { f } \otimes \mathbf { x } _ { t + i } ^ { f } \otimes \mathbf { x } _ { t + i } ^ { f } ) \\ & + \sum _ { i = 0 } ^ { l - 1 } ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { l - 1 - i } ( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } ) ( \tilde { \mathbf { x } } _ { t + i } ^ { f } - \mathbf { x } _ { t + i } ^ { f } ) \\ & + \sum _ { i = 0 } ^ { l - 1 } ( \mathbf { h } _ { \mathbf { x} } \otimes \mathbf { h } _ { \mathbf { x} }) ^ { l - 1 - i } ( \sigma \boldsymbol { \eta} \otimes \frac { 1 } { 2 } \mathbf { h} _ { \sigma \sigma } \sigma ^ { 2} ) [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .\]

because is a function of which is a function of . The zero-mean iid innovations therefore implies that, and

The same argument implies that

\[\text { and } E _ {t} \left[ \left(\boldsymbol {\epsilon} _ {t + 1 + i} \otimes \tilde {\mathbf {x}} _ {t + i} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + i} ^ {f}\right) \right] = \mathbf {0}\]

\[\begin{array}{l} = \sum_ {i = 1} ^ {l - 1} \left(\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}\right) ^ {l - 1 - i} \left(\mathbf {h} _ {\mathbf {x}} \otimes \frac {1}{2} \mathbf {H} _ {\mathbf {x x}}\right) E _ {t} \left[ \tilde {\mathbf {x}} _ {t + i} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + i} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + i} ^ {f} - \mathbf {x} _ {t + i} ^ {f} \otimes \mathbf {x} _ {t + i} ^ {f} \otimes \mathbf {x} _ {t + i} ^ {f} \right] \\ + \sum_ {i = 1} ^ {l - 1} \left(\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}\right) ^ {l - 1 - i} \left(\mathbf {h} _ {\mathbf {x}} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}\right) E _ {t} \left[ \tilde {\mathbf {x}} _ {t + i} ^ {f} - \mathbf {x} _ {t + i} ^ {f} \right] \\ + \left(\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}\right) ^ {l - 1} \left(\sigma \boldsymbol {\eta} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}\right) \boldsymbol {\nu} \\ + \left(\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}\right) ^ {l - 1} \left(\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}\right) (\boldsymbol {\nu} \otimes E _ {t} [ \tilde {\mathbf {x}} _ {t} ^ {s} ]) \\ + \left(\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}\right) ^ {l - 1} \left(\sigma \boldsymbol {\eta} \otimes \frac {1}{2} \mathbf {H} _ {\mathbf {x x}}\right) (\boldsymbol {\nu} \otimes E _ {t} [ \tilde {\mathbf {x}} _ {t} ^ {f} \otimes \tilde {\mathbf {x}} _ {t} ^ {f} ]) \\ u s i n g \delta_ {t + 1} = \boldsymbol {\nu} f o r e l s e \delta_ {t + 1 + i} = 0 \\ = k (k) ^ {- 1}, k (k) = k (k). \end{array}\]

and therefore the index for starts at 1

\[\begin{array}{r l} & {= \sum_ {i = 1} ^ {l - 1} \left(\mathbf {h _ {x}} \otimes \mathbf {h _ {x}}\right) ^ {l - 1 - i} \left(\mathbf {h _ {x}} \otimes \frac {1}{2} \mathbf {H _ {x x}}\right) E _ {t} \left[ \tilde {\mathbf {x}} _ {t + i} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + i} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + i} ^ {f} - \mathbf {x} _ {t + i} ^ {f} \otimes \mathbf {x} _ {t + i} ^ {f} \otimes \mathbf {x} _ {t + i} ^ {f} \right]} \\ & {+ \sum_ {i = 1} ^ {l - 1} \left(\mathbf {h _ {x}} \otimes \mathbf {h _ {x}}\right) ^ {l - 1 - i} \left(\mathbf {h _ {x}} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}\right) E _ {t} \left[ \tilde {\mathbf {x}} _ {t + i} ^ {f} - \mathbf {x} _ {t + i} ^ {f} \right]} \\ & {+ \left(\mathbf {h _ {x}} \otimes \mathbf {h _ {x}}\right) ^ {l - 1} \left(\sigma \pmb {\eta} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}\right) \pmb {\nu}} \\ & {+ \left(\mathbf {h _ {x}} \otimes \mathbf {h _ {x}}\right) ^ {l - 1} \left(\sigma \pmb {\eta} \otimes \mathbf {h _ {x}}\right) (\pmb {\nu} \otimes E _ {t} [ \tilde {\mathbf {x}} _ {t} ^ {s} ])} \\ & + \left(\mathbf {h _ {x}} \otimes \mathbf {h _ {x}}\right) ^ {l - 1} \left(\sigma \pmb {\eta} \otimes \frac {1}{2} \mathbf {H _ {x x}}\right) (\pmb {\nu} \otimes E _ {t} [ \tilde {\mathbf {x}} _ {t} ^ {f} \otimes \tilde {\mathbf {x}} _ {t} ^ {f} ])\]

\[\begin{array} { l } = \sum _ { i = 1 } ^ { l - 1 } ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { l - 1 - i } ( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } ) E _ { t } \left[ \tilde { \mathbf { x } } _ { t + i } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + i } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + i } ^ { f } - \mathbf { x } _ { t + i } ^ { f } \otimes \mathbf { x } _ { t + i } ^ { f } \otimes \mathbf { x } _ { t + i } ^ { f } \right] \\ + \sum _ { i = 1 } ^ { l - 1 } ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { l - 1 - i } ( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } ) E _ { t } \left[ \tilde { \mathbf { x } } _ { t + i } ^ { f } - \mathbf { x } _ { t + i } ^ { f } \right] \\ + ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { l - 1 } ( \sigma \boldsymbol { \eta } \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } ) \boldsymbol { \nu } \\ + ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { l - 1 } ( \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x } } ) ( \boldsymbol { \nu } \otimes \mathbf { x } _ { t } ^ { s } ) \\ + ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { l - 1 } ( \sigma \boldsymbol { \eta } \otimes \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } ) ( \boldsymbol { \nu } \otimes \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) \\ b e c a u s e t h e s h o c k h i t s i n p e r i o d t + 1 , s o E _ { t } [ \tilde { \mathbf { x } } _ { t } ^ { s} ] = \mathbf { x } _ { t } ^ { s} a n d E _ { t } [ \tilde { \mathbf { x } } _ { t } ^ { f } \otimes \tilde { \mathbf { x } } _ { t } ^ { f} ] = \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ t + i + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 . \\ = \sum _ { i = 1 } ^ { l - 1 } ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { l - 1 - i } ( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { H } _ { \texttt ~ x x x} ) E _ { t } [ \tilde {\mathbf { x }} _ { t + i} ^ { f} \otimes \tilde {\mathbf { x }} _ { t + i} ^ { f} \otimes \tilde {\mathbf { x }} _ { t + i} ^ { f} - \mathbf {\Delta x} _ { t + i} ^ { f} \otimes \mathbf {\Delta x} _ { t + i} ^ { f} \otimes \mathbf {\Delta x} _ { t + i} ^ { f} ] \\ + \sum _ { i = 1 } ^ { l - 1 } ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { l - 1 - i } ( \mathbf {\Delta h} _ {\mathbf {\Delta x}}) E _ { t } [ \tilde {\mathbf {\Delta x}} _ { t + i} ^ { f} - \mathbf {\Delta x} _ {\texttt {\scriptsize f}} ^ { f} ] \\ + ( \mathbf {\Delta h} _ {\texttt {\scriptsize x}}) ^ {- l - 1} ( \sigma \boldsymbol {\eta} | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | + ( (\mathbf {\Delta h} _ {\texttt {\scriptsize x}}) ^ {- l - 1} ( \sigma \boldsymbol {\eta} | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | + ( (\mathbf {\Delta h} _ {\texttt {\scriptsize x}}) ^ {- l - 1} ( (\sigma \boldsymbol {\eta} | | | | | | | | | | | | | | | | | | | | | | | | | | | | | + ( (\mathbf {\Delta h} _ {\texttt {\scriptsize x}}) ^ {- l - 1} ( (\sigma \boldsymbol {\eta} | | | | | | | | | | ] ) \\ = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = \\ + & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = \\ + & = & = & = & = & = & = & = & = & = & = & = & ? \\ + & = & = & = & = & = & = & = & = & = & = & ? \\ + & = & = & = & = & ? \\ + & = & = & = & = & ? \\ + & = & = & = & ? \\ + & = & = & = & ? \\ + & = & = & ? \\ + & = & ? \\ + & = & ? \\ + & = & ? \\ + & = & ? \\ + & = & ? \\ + & = & ? \\ + & = & ? \\ + & = & ? \\ + & = & ? \\ + & = & ? \\ + & = & ? \\ + & = & ? \\ + & = & ? \\ + & = & ? \\ + & = & ? \\ + & = & ? \\ + & = & . \\ X _ {\textit {\scriptsize l}} := X _ {\textit {\scriptsize l}} := X _ {\textit {\scriptsize l}} := X _ {\textit {\scriptsize l}} := X _ {\textit {\scriptsize l}} := X _ {\textit {\scriptsize l}} := X _ {\textit {\scriptsize l}} := X _ {\textit {\scriptsize l}} := X _ {\textit {\scriptsize l}} := X _ {\textit {\scriptsize l}} := Y _ {\textit {\scriptsize l}} := Y _ {\textit {\scriptsize l}} := Y _ {\textit {\scriptsize l}} := Y _ {\textit {\scriptsize l}} := Y _ {\textit {\scriptsize l}} := Y _ {\textit {\scriptsize l}} := Y _ {\textit {\scriptsize l}} := Y _ {\textit {\scriptsize l}} := Y _ {\textit {\scriptsize l}} := Y, \\ X _ {\textit {\scriptsize l}} := X _ {\textit {\scriptsize l}} := X _ {\textit {\scriptsize l}} := X _ {\textit {\scriptsize l}} := X _ {\textit {\scriptsize l}} := X _ {\textit {\scriptsize l}} := X _ {\textit {\scriptsize l}} := X _ {\textit {\scriptsize l}} := X _ {\textit {\scriptsize l}} : \\ X _ {\textit {\scriptsize l}} := X _ {\textit {\scriptsize l}} := X _ {\textit {\scriptsize l}} := X _ {\textit {\scriptsize l}} := X _ {\textit {\scriptsize l}} := X _ {\textit {\scriptsize l}} := X _ {\textit {\scriptsize l}} := X _ {\textit {\scriptsize l}} := X _ {\textit {\scriptsize l}} ; \\ X _ {\textit {\scriptsize l}} := X _ {\textit {\scriptsize l}} := X _ {\textit {\scriptsize l}} := X _ {\textit {\scriptsize l}} := X _ {\textit {\scriptsize l}} := X _ {\textit {\scriptsize l}} := X _ {\textit {\scriptsize l}} := X _ {\textit {\scriptsize l}} := X _ {\textit {\scriptsize l}}:= \\ X _ {\textit {\scriptsize l}} := X _ {\textit {\scriptsize l}} := X _ {\textit {\scriptsize l}} := X _ {\textit {\scriptsize l}} := X _ {\textit {\scriptsize l}} := X _ {\textit {\scriptsize l}} := X _ {\textit {\scriptsize l}} := X _ {\textit {\scriptsize l}} := X _ {\textit {\scriptsize l}}:: \\ X _ {\textit {\scriptsize l}} := X _ {\textit {\scriptsize l}} := X _ {\textit {\scriptsize l}} := X _ {\textit {\scriptsize l}} := X _ {\textit {\scriptsize l}} := X _ {\textit {\scriptsize l}} := X _ {\textit {\scriptsize l}} := X _ {\textit {\scriptsize l}} := X _ {\textit {\scriptsize l}} :: \\ X _{\textit{\scriptsize l}} := X_{\textit{\scriptsize l}} := X_{\textit{\scriptsize l}} := X_{\textit{\scriptsize l}} := X_{\textit{\scriptsize l}} := X_{\textit{\scriptsize l}} := X_{\textit{\scriptsize l}} := X_{\textit{\scriptsize l}} := X_{\textit{\scriptsize l}} := X_{\textit{\scriptsize l}} : \\ X_{\textit{\scriptsize l}} := X_{\textit{\scriptsize l}} := X_{\textit{\scriptsize l}} := X_{\textit{\scriptsize l}} := X_{\textit{\scriptsize l}} := X_{\textit{\scriptsize l}} := X_{\textit{\scriptsize l}} := X_{\textit{\scriptsize l}} := X_{\textit{\scriptsize l}} := X_\mathrmt~i~i~i~i~i~i~i~i~i~i~i~i~i~i~i~i~i~i~i~i~i~i~i~i~i~i~i~i~i~i~i~i~i~i~i~i~i~i~i~i~i~i~i~i~i~i~i~i~i~i~n\\ + Z_0^* Z_0^* Z_0^* Z_0^* Z_0^* Z_0^* Z_0^* Z_0^* Z_0^* Z_0^* Z_0^* Z_0^* Z_0^* Z_0^* Z_0^* Z_0^* Z_0^* Z_0^* Z_0^* Z_0^* Z_0^*, \\ X_0^* Z_0^* Z_0^* Z_0^* Z_0^* Z_0^* Z_0^* Z_0^* Z_0^* Z_0^* Z_0^* Z_0^* Z_0^* Z_0^* Z_0^* Z_0^* Z_0^* Z_0^* Z_0^* Z_0^* BZ_0^* BZ_0^* BZ_0^* BZ_0^* BZ_0^* BZ_0^* BZ_0^* BZ_0^* BZ_0^* BZ_0^* BZ_0^* BZ_0^* BZ_0^* BZ_0^* BZ_0^* BZ_0^* BZ_0^*, \\ X_2^* Z_2^* Z_2^* Z_2^* Z_2^* Z_2^* Z_2^* Z_2^* Z_2^* Z_2^* Z_2^* Z_2^* Z_2^* Z_2^* Z_2^* Z_2^* Z_2^* Z_2^* Z_2^* Z_2^* Z_2^* BZ_2^* BZ_2^* BZ_2^* BZ_2^* BZ_2^* BZ_2^* BZ_2^* BZ_2^* BZ_2^* BZ_2^* BZ_2^* BZ_2^*, \\ X_\textt-3.5.7.8.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.9.8^{T / T},\\ X_\textt-3.6.7.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8.8,\\ X_\textt-3.7.7.7.7.7.7.7.7.7.7.7.7.7.7.7.7.7.7.7.7.7.7.7.7.7,\\ X_\textt-3.8.7.7.7-7-7-7-7-7-7-7-7-7-7-7-7-7-7-7-7-7-7-7-7-7-7-7-7-7-7-7-7-7-7-7-7-7-7-7-7-7-7-7,\\ X_\textt-3.9,7-9,7-9,7-9,7-9,7-9,7-9,7-9,7-9,7-9,7-9,7-9,7-9,7-9,7-9,7-9,7-9,7-9,7-9,7-9,7-9,7-9,7-9,7-9,7-9,7-9,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,6^{T / T},\\ X_\textt-3,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53; \\ X_\text{t-3,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29},\\ X_\text{t-3,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28},\\ X_\textt-3,8,9,10,-5,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-6,-5,.\\ X_\textt-3,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,x,u,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,m,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,u,v,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,w,m,u,v,u,v,u,v,u,v,u,v,u,v,u,v,u,v,u,v,u,v,u,v,u,v,u,v,u,v,u,v,u,v,u,v,u,v,u,v,u,v,u,v,u,v,u,v,u,v,u,v,u,v,u,v,u,v,u,v,u,v,u,v,u,v,u,v,u,v,u,v,u,v,u,v,u,v,u,v,u,v,u,v,u,v,u,v,u,v,u,v,u,v,u,v,u,v,u,v,u,v,u-v,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t:t|,\end{array}\]

\[\begin{array} { r l } & { + ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) \left( \sigma \eta \pmb { \nu } \otimes \left( \mathbf { h } _ { \mathbf { x } } \mathbf { x } _ { t } ^ { s } + \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } \left( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \right) + \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } \right) \right) } \\ & { = ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) X _ { 1 } + ( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } ) E _ { t } \left[ \tilde { \mathbf { x } } _ { t + 1 } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + 1 } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + 1 } ^ { f } - \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } \right] } \\ & { + ( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } ) E _ { t } \left[ \tilde { \mathbf { x } } _ { t + 1 } ^ { f } - \mathbf { x } _ { t + 1 } ^ { f } \right] } \\ & { X _ { 3 } = \sum _ { i = 1 } ^ { 2 } ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { 2 - i } ( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } ) E _ { t } \left[ \tilde { \mathbf { x } } _ { t + i } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + i } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + i } ^ { f } - \mathbf { x } _ { t + i } ^ { f } \otimes \mathbf { x } _ { t + i } ^ { f } \otimes \mathbf { x } _ { t + i } ^ { f } \right] } \\ & { + \sum _ { i = 1 } ^ { 2 } ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { 2 - i } ( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } ) E _ { t } [ [ \tilde { \mathbf { x } } _ { t + i } ^ { f } - \mathbf { x } _ { t + i } ^ { f} ] ] } \\ & { + ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { 2 } ( \sigma \eta \pmb { \nu } \otimes ( [ \mathbf { h } _ {\mathbf { x }} \mathbf { x} _ { t } ^ { s } + \frac 1 2 H _ {\mathbf { x x }} ( [ \mathbf { x} _ { t } ^ { f } \otimes [ \mathbf { x} _ { t } ^ { f} ] + \frac 1 2 h _ {\sigma \sigma} \sigma ^ { 2} ] ) ) } \\ & = ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ \mathbf { x } ) ) ( [ \mathbf { h} _ {\mathbf { x}} \otimes [ {\frac 1 2 H _ {\mathbf { x x }} ] E _ { t }} [ [ {\tilde {\mathbf {\Delta x}}} _ { t + 1} ^ {} , [ {\tilde {\mathbf {\Delta x}}} _ { t + 1} ^ {} , [ {\tilde {\mathbf {\Delta x}}} _ { t + 1} ^ {} , - [ {\tilde {\mathbf {\Delta x}}} _ { t + 1} ^ {} , [ {\tilde {\mathbf {\Delta x}}} _ { t + 1} ^ {} , ] E _ {[ t ] , k ] , k ] , k ] , k ] , k ] , k ] , k ] , k ] , k ] , k ] , k ] , k , k ] , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , k , m , n ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . : X = X / K ; X = Y / K ; Y = Z / K ; Z = Z / K ; Z = Z / K ; Z = Z / K ; Z = Z / K ; Z = Z / K ; Z = Z / K ; Z = Z / K ; Z = Z / K ; Z = Z / K ; Z = Z / K ; Z = Z / K ; Z = Z / K ; Z = Z / K ; Z = Z / K ; Z = Z / K ; Z = Z / K ; Z = Z / N ; Z = Z / N ; Z = Z / N ; Z = Z / N ; Z = Z / N ; Z = Z / N ; Z = Z / N ; Z = Z / N ; Z = Z / N ; Z = Z / N ; Z = Z / N ; Z = Z / N ; Z = Z / N ; Z = Z / N ; Z = Z / N ; Z = Z / N ; Z = Z / N ; Z = Y / K ; Y = Y / K ; Y = Y / K ; Y = Y / K ; Y = Y / K ; Y = Y / K ; Y = Y / K ; Y = Y / K ; Y = Y / K ; Y = Y / K ; Y = Y / K ; Y = Y / K ; Y = Y / K ; Y = Y / K ; Y = Y / K ; Y = Y / N ; Y = Y / N ; Y = Y / N ; Y = Y / N ; Y = Y / N ; Y = Y / N ; Y = Y / N ; Y = Y / N ; Y = Y / N ; Y = Y / N ; Y = Y / N ; Y = Y / N ; Y = Y / N ; Y = Y / N ; Y = Y / N ; Y = Y / N ; Y = Y / N ; Y = | y | | | y | | | y | | | y | | | y | | | y | | | y | | | y | | | y | | | y | | | y | | | y | | | y | | | y | | | y | | | y | | | y | | | y | | | y | | | y | | | y | | | y | | | y | | | y | | | y | | | y | | . }\]

10.3.3 Summarizing

At third order, the total effect on the state variables is:

\[\begin{array} { r l } & E _ { t } \left[ \tilde { \mathbf { x } } _ { t + l } - \mathbf { x } _ { t + l } \right] = E _ { t } \left[ \tilde { \mathbf { x } } _ { t + l } ^ { f } - \mathbf { x } _ { t + l } ^ { f } \right] + E _ { t } \left[ \tilde { \mathbf { x } } _ { t + l } ^ { s } - \mathbf { x } _ { t + l } ^ { s } \right] + E _ { t } \left[ \tilde { \mathbf { x } } _ { t + l } ^ { r d } - \mathbf { x } _ { t + l } ^ { r d } \right] \\ & \text {For the control variables:} \\ & \mathbf { y } _ { t + l } ^ { r d } = \mathbf { g _ { x } } \left( \mathbf { x } _ { t + l } ^ { f } + \mathbf { x } _ { t + l } ^ { s } + \mathbf { x } _ { t + l } ^ { r d } \right) + \frac 1 2 \mathbf { G _ { x x } } \left( \left( \mathbf { x } _ { t + l } ^ { f } \otimes \mathbf { x } _ { t + l } ^ { f } \right) + 2 \left( \mathbf { x } _ { t + l } ^ { f } \otimes \mathbf { x } _ { t + l } ^ { s } \right) \right) \\ & \quad + \frac 1 6 \mathbf { G _ { x x x } } \left( \mathbf { x } _ { t + l } ^ { f } \otimes \mathbf { x } _ { t + l } ^ { f } \otimes \mathbf { x } _ { t + l } ^ { f } \right) + \frac 1 2 \mathbf { g _ { \sigma \sigma } } \sigma ^ { 2 } + \frac 3 6 \mathbf { g _ { \sigma \sigma x } } \sigma ^ { 2 } \mathbf { x _ { t + l } ^ { f } + \frac 1 6 g _ { \sigma \sigma \sigma } \sigma ^ { 3 } } \\ & \tilde { \mathbf { y } } _ { t + l } ^ { r d } = \mathbf { g _ { x } } \left( \tilde { \mathbf { x } } _ { t + l } ^ { f } + \tilde { \mathbf { x } } _ { t + l } ^ { s } + \tilde { \mathbf { x } } _ { t + l } ^ { r d } \right) + \frac 1 2 \mathbf { G _ { x x } } \left( \left( \tilde { \mathbf { x } } _ { t + l } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + l } ^ { f } \right) + 2 \left( \tilde { \mathbf { x } } _ { t + l } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + l } ^ { s} \right) \right) \\ & \quad + \frac 1 6 \mathbf { G _ { x x x } } \left( \tilde { \mathbf { x } } _ { t + l } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + l } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + l } ^ { f} \right) + \frac 1 2 \mathbf { g _ { \sigma \sigma } } \sigma ^ { 2 } + \frac 3 6 \mathbf { g _ { \sigma \sigma x } } \sigma ^ { 2 } \tilde { \mathbf { x } } _ { t + l } ^ { f } + \frac 1 6 g _ { \sigma \sigma \sigma } \sigma ^ { 3 } \\ & {\mathrm{So:}} \\ & E _ { t } \left[ \tilde { \mathbf { y } } _ { t + l } ^ { r d } - \mathbf { y} _ { t + l } ^ { r d } \right] \\ & = \mathbf { g _ { x } } \left( E _ { t } [ \tilde { \mathbf { x } } _ { t + l } ^ { f } - \mathbf { x} _ { t + l ]} ^ { f} ] + E _ { t } [ \tilde { \mathbf { x } } _ { t + l ]} ^ { s} - \mathbf { x} _ { t + l ]} ^ { s} ] + E _ { t } [ \tilde { \mathbf { x } } _ { t + l ]} ^ { r d} - \mathbf { x} _ { t + l ]} ^ { r d} ]\right) \\ & + \frac 1 2 \mathbf { G _ { x x }} ( E _ { t } [ \tilde { \mathbf { x } } _ { t + l ]} ^ { f} \otimes \tilde { \mathbf { x } } _ { t + l ]} ^ { f} - \mathbf { x} _ { t + l ]} ^ { f} \otimes \mathbf { x} _ { t + l ]} ^ { f} ] + 2 E _ { t } [ \tilde { \mathbf { x } } _ { t + l ]} ^ { f} \otimes \tilde { \mathbf { x } } _ { t + l ]} ^ { s} - \mathbf { x} _ { t + l ]} ^ { f} \otimes \mathbf { x} _ { t + l ]} ^ { s} ] ) \\ & + \frac 1 6 \mathbf { G _ { x x x }} E _ { t } [ \tilde { \mathbf { x } } _ { t + l ]} ^ { f} \otimes \tilde { \mathbf { x } } _ { t + l ]} ^ { f} \otimes \tilde { \mathbf { x } } _ { t + l ]} ^ {-} - {\mathbf {\Delta x}} _ {{t + l ]}} ^ {-} {\mathbf {\Delta x}} _ {{t + l ]}} ^ {-} {\mathbf {\Delta x}} _ {{t + l ]}} ^ {-} {\mathbf {\Delta x}} _ {{t + l ]}} ^ {-} {\mathbf {\Delta x}} _ {{t + l ]}} ^ {-} {\mathbf {\Delta x}} _ {{t + l ]}} ^ {-} {\mathbf {\Delta x}} _ {{t + l ]}}. \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & .; \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . \\ & . ; \\ & . \\ & . \\ & . \\ & . \\ & .; \\ & . \\ & .; \\ & .; \\ & .; \\ & .; \\ & .; \\ & .; \\ & .; \\ & .; \\ & .; \\ & .; \\ & .; \\ & .; \\ & .; \\ & .; \\ & .; \\ & .; \\ & .; \\ & .; \\ & .; \\ & .; \\ & .; \\ & .; \\ & .; \\ & .; \\ & .; \\ & . ; \\ & .; \\ & .; \\ & .; \\ & .; \\ & .; \\ & .; \\ & .; \\ & .; \\ & .; \\ & .; \\ & .; \\ & .; \\ & .; \\ & .; \\ & .; \\ & .; \\ & .; \\ & .; \\ & .; \\ & .; \\ & .; \\ & .; \\ & .; \\ & .; \\ & .,; \\ & ,; \\ & ,; | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | I = I, J = I, K = I, L = I, M = I, N = I, O = I, P = I, Q = I, R = I, S = I, T = I, U = I, V = I, W = I, X = I, Y = I, Z = I, AA = I, AB = I, AC = I, AD = I, AE = I, AF = I, AG = I, AH = I, AI = I, AJ = I, AK = I, AL = I, AM = I, AN = I, AO = I, AP = I, AQ = I, AR = I, AS = I, AT = I, AU = I, AV = I, AW = I, AX = I, AY = I, AZ = I, BA = I, BB = I, BC = I, BD = I, BE = I, BF = I, BG = I, BH = I, BI = I, BJ = I, BK = I, BL = I, BM = I, BN = I, BO = I, BP = I, BZ = I, CA = I, CB = I, CCB = I, CDB = I, CEB = I, CFB = I, DDB = I, EFB = I, FGB = I, GHB = I, GGB = I, HHB = I, IDB = I, EBCB = I, FBCB = I, GGBCB = I, HBCB = I, EIACB = I, LACB = I, MLCB = I, MFCB = I, MFGBCB = I, MHCBCB = I, MRTBCB = I, MRTBCB = I, MRTBCB = I, MRTBCB = I, MRTBCB = I, MRTBCB = I, MRTBCB = I, MRTBCB = I, MRTBCB = I, MRTBCB = I, MRTBCB = I, MRTBCB = I, MRTBCB = I, MRTBCB = I, MRTBCS , MRTBCS , MRTBCS , MRTBCS , MRTBCS , MRTBCS , MRTBCS , MRTBCS , MRTBCS , MRTBCS , MRTBCS , MRTBCS , MRTBCS , MRTBCS , MRTBCS , MRTBCS , MRTBCS , MRTBCS , MRTBCS , MRTBCS , MRTBCS< nl>\]

11 Impulse response functions - GIRF

This section derives closed-form solutions for the generalized impulse response function in non-linear DSGE models when defined as the GIRF. That is

\[\begin{array}{r c l} G I R F _ {\mathbf {v a r}} \left(l, \boldsymbol {\nu}, \mathbf {w} _ {t}\right) & = & E _ {t} \left[ \mathbf {v a r} _ {t + l} | \boldsymbol {\epsilon} _ {t + 1} = \boldsymbol {\nu} \right] \\ & & - E _ {t} \left[ \mathbf {v a r} _ {t + l} \right] \end{array}\]

To reduce the notational burden in the derivations below, we adopt the parsimonious notation

\[I R F _ {\mathbf {v a r}} (l, \pmb {\nu}, \mathbf {w} _ {t}) = E _ {t} [ \widetilde {\mathbf {v a r}} _ {t + l} ] - E _ {t} [ \mathbf {v a r} _ {t + l} ]\]

in relation to the conditional expectation operators.

11.1 At first order

Recall that we have:

\[\mathbf {x} _ {t + 1} ^ {f} = \mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t} ^ {f} + \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1}\]

and

\[\begin{array}{r l} & {\mathbf {x} _ {t + 2} ^ {f} = \mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t + 1} ^ {f} + \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 2}} \\ & {\qquad = \mathbf {h} _ {\mathbf {x}} \left(\mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t} ^ {f} + \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1}\right) + \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 2}} \\ & {\qquad = \mathbf {h} _ {\mathbf {x}} ^ {2} \mathbf {x} _ {t} ^ {f} + \mathbf {h} _ {\mathbf {x}} \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1} + \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 2}} \end{array}\]

and

\[\begin{array}{r l} & {\mathbf {x} _ {t + 3} ^ {f} = \mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t + 2} ^ {f} + \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 3}} \\ & {\qquad = \mathbf {h} _ {\mathbf {x}} \left(\mathbf {h} _ {\mathbf {x}} ^ {2} \mathbf {x} _ {t} ^ {f} + \mathbf {h} _ {\mathbf {x}} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} + \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 2}\right) + \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 3}} \\ & {\qquad = \mathbf {h} _ {\mathbf {x}} ^ {3} \mathbf {x} _ {t} ^ {f} + \mathbf {h} _ {\mathbf {x}} ^ {2} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} + \mathbf {h} _ {\mathbf {x}} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 2} + \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 3}} \\ & {\qquad = \mathbf {h} _ {\mathbf {x}} ^ {3} \mathbf {x} _ {t} ^ {f} + \sum_ {j = 1} ^ {3} \mathbf {h} _ {\mathbf {x}} ^ {3 - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j}} \end{array}\]

In general

\[\mathbf {x} _ {t + l} ^ {f} = \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\epsilon} _ {t + j}\]

With a shock of in period , we have

\[\tilde {\mathbf {x}} _ {t + l} ^ {f} = \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\delta} _ {t + j}\]

where we define such that:

\[\begin{array}{r l} \pmb {\delta} _ {t + j} = \pmb {\nu} & \mathrm{for} j = 1 \\ \pmb {\delta} _ {t + j} = \pmb {\epsilon} _ {t + j} & \mathrm{for} j \neq 1 \end{array}\]

Hence, agents know the size of the shock at time , and it is therefore in agents' information set.

\[\begin{array}{l} \text {So} \\ E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} ^ {f} - \mathbf {x} _ {t + l} ^ {f} \right] = E _ {t} \left[ \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \pmb {\delta} _ {t + j} - \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \pmb {\epsilon} _ {t + j} \right] \\ = \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \pmb {\delta} _ {t + 1} \end{array}\]

\[\begin{array}{l} = \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} \\ \text {because} \delta_ {t + 1} = \boldsymbol {\nu} \end{array}\]

\[\begin{array}{r l} & {\mathrm{and}} \\ & {E _ {t} \left[ \tilde {\mathbf {y}} _ {t + l} ^ {f} - \mathbf {y} _ {t + l} ^ {f} \right] = \mathbf {g _ {x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} ^ {f} - \mathbf {x} _ {t + l} ^ {f} \right]} \end{array}\]

11.2 At second order

We need to consider:

\[\mathbf {x} _ {t + 1} ^ {s} = \mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t} ^ {s} + \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) + \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}\]

\[\begin{array}{r l} & {\mathbf {x} _ {t + 2} ^ {s} = \mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t + 1} ^ {s} + \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t + 1} ^ {f} \otimes \mathbf {x} _ {t + 1} ^ {f}\right) + \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}} \\ & {\quad = \mathbf {h} _ {\mathbf {x}} \left(\mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t} ^ {s} + \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) + \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}\right) + \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t + 1} ^ {f} \otimes \mathbf {x} _ {t + 1} ^ {f}\right) + \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}} \\ & {\quad = \mathbf {h _ {x} ^ {2}} \mathbf {x} _ {t} ^ {s} + \mathbf {h _ {x}} \frac {1}{2} \mathbf {H _ {x x}} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) + \mathbf {h _ {x}} \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2} + \frac {1}{2} \mathbf {H _ {x x}} \left(\mathbf {x} _ {t + 1} ^ {f} \otimes \mathbf {x} _ {t + 1} ^ {f}\right) + \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}} \end{array}\]

\[\begin{array} { r l } & { \mathbf { x } _ { t + 3 } ^ { s } = \mathbf { h } _ { \mathbf { x } } \mathbf { x } _ { t + 2 } ^ { s } + \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } \left( \mathbf { x } _ { t + 2 } ^ { f } \otimes \mathbf { x } _ { t + 2 } ^ { f } \right) + \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } } \\ & { = \mathbf { h } _ { \mathbf { x } } \left( \mathbf { h } _ { \mathbf { x } } ^ { 2 } \mathbf { x } _ { t } ^ { s } + \mathbf { h } _ { \mathbf { x } } \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } \left( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \right) + \mathbf { h } _ { \mathbf { x } } \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } + \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } \left( \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } \right) + \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } \right) } \\ & { + \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } \left( \mathbf { x } _ { t + 2 } ^ { f } \otimes \mathbf { x } _ { t + 2 } ^ { f } \right) + \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 3 } } \\ & { = \mathbf { h } _ { \mathbf { x } } ^ { 3 } \mathbf { x } _ { t } ^ { s } + \mathbf { h } _ { \mathbf { x } } ^ { 2 } \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } \left( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \right) + \mathbf { h } _ { \mathbf { x } } ^ { 2 } \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } + \mathbf { h } _ { \mathbf { x } } \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } \left( \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } \right) + \mathbf { h } _ { \mathbf { x } } \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } } \\ & { + \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } \left( \mathbf { x } _ { t + 2 } ^ { f } \otimes \mathbf { x } _ { t + 2 } ^ { f } \right) + \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \tau ^ { 2 } } \\ & = \mathbf { h } _ { \mathbf { x } } ^ { 3 } \mathbf { x } _ { t } ^ { s } + \mathbf { h } _ { \mathbf { x } } ^ { 2 } \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x} } ( [ \mathbf { x} ] _ t ) + [ \mathbf { h} ] _ \alpha , j , k , l , m , n , o , p , q , r , s , t , u , v , w , x , y , z , w , u , v , w , z , w , u , v , w , z , w , u , v , w , z , w , u , v , w , z , w , u , v , w , z , w , u , v , w , u , v , w , u , v , w , u , v , w , u , v , w , u , v , w , u , v , w , u , v , w , u , v , w , u , v , w , u , v , w , u , v , w , u , v , w , u , v , w , u , v , w , u , v , w , u , v , w , w , u , v , w , u , v , w , u , v , w , u , v , w , u , v , w , u , v , w , u , v , w , u , v , w , u , v , w , u , v , w , u , v , w , u , v , w , u , v , w , u , v , w , u , v , w , u , v , w , v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . : - [ a n d i a n g e n t i o n e ] [ a n d i a n g e n t i o n e ] [ a n d i a n g e n t i o n e ] [ a n d i a n g e n t i o n e ] [ a n d i a n g e n t i o n e ] [ a n d i a n g e n t i o n e ] [ a n d i a n g e n t i o n e ] [ a n c e n t i o n e ] [ a n c e n t i o n e ] [ a n c e n t i o n e ] [ a n c e n t i o n e ] [ a n c e n t i o n e ] [ a n c e n t i o n e ] [ a n c e n t i o n e ] [ a n c e n t i o n e ] [ a n c e n t t i o n e ] [ a n c e n t t i o n e ] [ a n c e n t t i o n e ] [ a n c e n t t i o n e ] [ a n c e n t t i o n e ] [ a n c e n t t i o n e ] [ a n c e n t t i o n e ] [ a n c e n t t i o n e ] [ a m m a l l o r s i s s i s s i s s i s s i s s i s s i s s i s s i s s i s s i s s i s s i s s i s s i s s i s s i s s i s s i s s i s s i s s i s s i s s i s s i s s i s s i s s i s s i s s i s s i s s i s s i s s i s c o r r e d b o l l o r d i m e d d o r d d o r d d o r d d o r d d o r d d o r d d o r d d o r d d o r d d o r d d o r d d o r d d o r d d o r d d o r d d o r d d o r d d o r d d o r d d o r d d o r d d o r d d o r d d o r d d o r d d o l l o r d d o r d d o r d d o r d d o r d d o r d d o r d d o r d d o r d d o r d d o r d d o r d d o r d d o r d d o r d d o r d d o r d d o r d d o r d d o r d d o r d d o r d d o r d d o r d d o r d t h a l l o r l i m e l i m e l i m e l i m e l i m e l i m e l i m e l i m e l i m e l i m e l i m e l i m e l i m e l i m e l i m e l i m e l i m e l i m e l i m e l i m e l i m e l i m e l i m e l i m e l i m e l i m a l l o r l i m e l i m e l i m e l i m e l i m e l i m e l i m e l i m e l i m e l i m e l i m e l i m e l i m e l i m e l i m e l i m e l i m e l i m e l i m e l i m e l i m e l i m e l i m e l i f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f ~ ; ~ - [ I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I ] [ A B C D E F A S E W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T OW T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O WT O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W T O W TOWTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTOTATUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUTUT UT U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U V E N G R E M M E L E M M E L E M M E L E M M E L E M M E L E M M E L E M M E L E M M E L E M M E L E M M E L E M M E L E M M E L E M M E L E M M E L E M M E L E M M E L E M M E L E M M E L E M M E L E M M E L E M M E l E M M E L E M M E L E M M E L E M M E L E M M E L E M M E L E M M E L E M M E L E M M E L E M M E L E M M E L E M M E L E M M E L E M M E L E M M E L E M M E L E M M E L E M M E L E M M E L E M M E K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K KK K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k kkappa, p, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, q, and p < 0.05\]

and in general

\[\begin{array}{l} \mathbf {x} _ {t + l} ^ {s} = \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {s} + \sum_ {j = 0} ^ {l - 1} \mathbf {h} _ {\mathbf {x}} ^ {l - 1 - j} \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t + j} ^ {f} \otimes \mathbf {x} _ {t + j} ^ {f}\right) + \left(\sum_ {j = 0} ^ {l - 1} \mathbf {h} _ {\mathbf {x}} ^ {l - 1 - j}\right) \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2} \\ \text {for l = 1,2,3,...} \end{array}\]

Thus, to compute , we need to find . Hence, consider:

\[\begin{array}{l} \mathbf {x} _ {t + l} ^ {f} \otimes \mathbf {x} _ {t + l} ^ {f} = \left(\mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j}\right) \otimes \left(\mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j}\right) \\ = \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} \\ + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} \end{array}\]

and

\[\tilde {\mathbf {x}} _ {t + l} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + l} ^ {f} = \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\delta} _ {t + j}\]

\[+ \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\delta} _ {t + j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\delta} _ {t + j} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\delta} _ {t + j}\]

This means that:

\[\begin{array}{l} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + l} ^ {f} - \mathbf {x} _ {t + l} ^ {f} \otimes \mathbf {x} _ {t + l} ^ {f} \right] \\ = E _ {t} [ \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \delta_ {t + j} \\ + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \delta_ {t + j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \delta_ {t + j} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \delta_ {t + j} \\ - \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} - \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \epsilon_ {t + j} \\ - \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \epsilon_ {t + j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} - \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \epsilon_ {t + j} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \epsilon_ {t + j} ] \\ = E _ {t} [ \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ i t o f t o r s t o r s t o r s t o r s t o r s t o r s t o r s t o r s t o r s t o r s t o r s t o r s t o r s t o r s t o r s t o r s t o r s t o r s t o r s t o r s t o r s t o r s t o r s t o r s t o r s t o r i n g e d i n e d i n e d i n e d i n e d i n e d i n e d i n e d i n e d i n e d i n e d i n e d i n e d i n e d i n e d i n e d i n e d i n e d i n e d i n e d i n e d i n e d i n e d i n e d i n e d i n e d i d u a l y a l l e m a x e l y a l l e m a x e l y a l l e m a x e l y a l l e m a x e l y a l l e m a x e l y a l l e m a x e l y a l l e m a x e l y a l l e m a x e l y a l l e m a x e l y a l l e m a x e l y a l l e m a x i n g E _ {t} [ \epsilon_ {t + j} ] = 0 \\ = 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, \\ = E _ {\mathrm{e}} [ A (\epsilon) ] = A (\epsilon) A (\epsilon) A (\epsilon) A (\epsilon) A (\epsilon) A (\epsilon) A (\epsilon) A (\epsilon) A (\epsilon) A (\epsilon) A (\epsilon) A (\epsilon) A (\epsilon) A (\epsilon) A (\epsilon) A (\epsilon) A (\epsilon) A (\epsilon) A (\epsilon) A (\epsilon) A (\epsilon) A (\epsilon) A (\epsilon) A (\epsilon) A (\epsilon) A (\epsilon), \\ = E _ {\mathrm{e}} [ A (\epsilon) ] = E _ {\mathrm{e}} [ A (\epsilon) ] = E _ {\mathrm{e}} [ A (\epsilon) ] = E _ {\mathrm{e}} [ A (\epsilon) ] = E _ {\mathrm{e}} [ A (\epsilon) ] = E _ {\mathrm{e}} [ A (\epsilon) ] = E _ {\mathrm{e}} [ A (\epsilon) ] = E _ {\mathrm{e}} [ A (\epsilon) ] = F (A (\epsilon)) F (A (\epsilon)) F (A (\epsilon)) F (A (\epsilon)) F (A (\epsilon)) F (A (\epsilon)) F (A (\epsilon)) F (A (\epsilon)) F (A (\epsilon)) F (A (\epsilon)) F (A (\epsilon)) F (A (\epsilon)) F (A (\epsilon)) F (A (\epsilon)) F (A (\epsilon)) F (A (\epsilon)) F (A (\epsilon)) F (A (E)) F (A (E)) F (A (E)) F (A (E)) F (A (E)) F (A (E)) F (A (E)) F (A (E)) F (A (E)) F (A (E)) F (A (E)) F (A (E)) F (A (E)) F (A (E)) F (A (E)) F (A (E)) F (A (E)) F (F) F (F) F (F) F (F) F (F) F (F) F (F) F (F) F (F) F (F) F (F) F (F) F (F) F (F) F (F) F (F) F (F) F (F) F (F) F (F) F (F) F (F) F (F) F (F) F (F) F (F). \\ = E _ {\mathrm{e}} [ A (\epsilon) ] = E _ {\mathrm{e}} [ A (\epsilon) ] = E _ {\mathrm{e}} [ A (\epsilon) ] = E _ {\mathrm{e}} [ A (\epsilon) ] = E _ {\mathrm{e}} [ A (\epsilon) ] = E _ {\mathrm{e}} [ A (\epsilon) ] = E _ {\mathrm{e}} [ A (\epsilon) ] \\ = E _ {\mathrm{e}} [ A (\epsilon) ] = E _ {\mathrm{e}} [ A (\epsilon) ] = E _ {\mathrm{e}} [ A (\epsilon) ] = E _ {\mathrm{e}} [ A (\epsilon) ] = E _ {\mathrm{e}} [ A (\epsilon) ] = E _ {\mathrm{e}} [ A (\epsilon) ] = E _ {\mathrm{e}} [ A (\epsilon) ] & . \\ = E _ {\mathrm{e}} [ A (\epsilon) ] = E _ {\mathrm{e}} [ A (\epsilon) ] = E _ {\mathrm{e}} [ A (\epsilon) ] = E _ {\mathrm{e}} [ A (\epsilon) ] = E _ {\mathrm{e}} [ A (\epsilon) ] = E _ {\mathrm{e}} [ A (\epsilon) ] \\ = E _ {\mathrm{e}} [ A (\epsilon) ] & . \\ = E _ {\mathrm{e}} [ A (\epsilon) ] & . \\ = E _ {\mathrm{e}} [ A (\epsilon) ] & . \\ = E _ {\mathrm{e}} [ A (\epsilon) ] & . \\ = E _ {\mathrm{e}} [ A (\epsilon) ] & . \\ = E _ {\mathrm{e}} [ A (\epsilon) ] & . \\ = E _ {\mathrm{e}} [ A (\epsilon) ], & . \\ = E _ {\mathrm{e}} [ A (\epsilon), I ] & . \\ = E _ {\mathrm{e}} [ A (\epsilon), I ] & . \\ = E _ {\mathrm{e}} [ A (\epsilon), I ] & . \\ = E _ {\mathrm{e}} [ A (\epsilon), I ] & . \\ = E _ {\mathrm{e}} [ A (\epsilon), I ] & . \\ = E _ {\mathrm{e}} [ A (\epsilon), I ] & . \\ = E _ {\mathrm{d}} [ A (\epsilon), I ] & . \\ = E _ {\mathrm{d}} [ A (\epsilon), I ] & . \\ = E _ {\mathrm{d}} [ A (\epsilon), I ] & . \\ = E _ {\mathrm{d}} [ A (\epsilon), I ] & . \\ = E _ {\mathrm{d}} [ A (\epsilon), I ] & . \\ = E _ {\mathrm{d}} [ A (\epsilon), I ] & .. \\ = E _ {\mathrm{d}} [ A (\epsilon), I ] & .. \\ = E _ {\mathrm{d}} [ A (\epsilon), I ] & .. \\ = E _ {\mathrm{d}} [ A (\epsilon), I ] & .. \\ = E _ {\mathrm{d}} [ A (\epsilon), I ] & .. \\ = E _ {\mathrm{d}} [ A (\epsilon), I ] & .. \\ = E _ {\mathrm{d}} {[ B ({\mu})}, & .. \\ = E _ {\mathrm{d}} {[ B ({\mu})}, & .. \\ = E _ {\mathrm{d}} {[ B ({\mu})}, & .. \\ = E _ {\mathrm{d}} {[ B ({\mu})}, & .. \\ = E _ {\mathrm{d}} {[ B ({\mu})}, & .. \\ = E _ {\mathrm{d}} {[ B ({\mu})}, & .. \\ = E _ {\mathrm{c}} {[ B ({\mu})}, & .. \\ = E _ {\mathrm{c}} {[ B ({\mu})}, & .. \\ = E _ {\mathrm{c}} {[ B ({\mu})}, & .. \\ = E _ {\mathrm{c}} {[ B ({\mu})}, & .. \\ = E _ {\mathrm{c}} {[ B ({\mu})}, & .. \\ = E _ {\mathrm{c}} {[ B ({\mu})}, & ... \\ = E _ {\mathrm{c}} {[ B ({\mu})}, & .. \\ = E _ {\mathrm{c}} {[ B ({\mu})}, & .. \\ = E _ {\mathrm{c}} {[ B ({\mu})}, & .. \\ = E _ {\mathrm{c}} {[ B ({\mu})}, & .. \\ = E _ {\mathrm{c}} {[ B ({\mu})}, & .. \\ = E _ {\mathrm{c}} {(B ({\mu})}, & .. \\ = E _ {\mathrm{c}} {[ B ({\mu})}, & .. \\ = E _ {\mathrm{c}} {[ B ({\mu})}, & .. \\ = E _ {\mathrm{c}} {[ B ({\mu})}, & .. \\ = E _ {\mathrm{c}} {[ B ({\mu})}, & .. \\ = E _ {\mathrm{c}} {[ B ({\mu})}, & .. \\ =E q u a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y a l y b i f f i c h. \\ - 2 p h i c k w i f f i c h. & . \\ - 2 p h i c k w i f f i c h. & . \\ - 2 p h i c k w i f f i c h. & . \\ - 2 p h i c k w i f f i c h. & . \\ - 2 p h i c k w i f f i c h. & . \\ - 2 p h i c k w i f f i c h. & . \\ - 2 P h i c k w i f f i c h. & . \\ - 2 P h i c k w i f f i c h. & . \\ - 2 P h i c k w i f f i c h. & . \\ - 2 P h i c k w i f f i c h. & . \\ - 2 P h i c k w i f f i c h. & . \\ - 2 P h i c k W i f f f i c h. & . \\ - 2 P h i c k W i f f f i c h. & . \\ - 2 P h i c k W i f f f i c h. & . \\ - 2 P h i c k W i f f f i c h. & . \\ - 2 P h i c k W i f f f i c h. & . \\ - 2 P h i c k W i for f i c h. & . \\ - 2 P h i c k W i for f i c h. & . \\ - 2 P h i c k W i for f i c h. & . \\ - 2 P h i c k W i for f i c h. & . \\ - 2 P h i c k W i for f i c h. & . \\ - 2 P h i c k W i for f i c h , & . \\ - 2 P h i c k W i for f i c h , & . \\ - 2 P h i c k W i for f i c h , & . \\ - 2 P h i c k W i for f i c h , & . \\ - 2 P h i c k W i for f i c h , & . \\ - 2 P h i c k W I for f i c h , & . \\ - 2 P h i c k W I for f i c h , & . \\ - 2 P h i c k W I for f i c h , & . \\ - 2 P h i c k W I for f i c h , & . \\ - 2 P h i c k W I for f i c h , & . \\ - 2 P h i c k W Y U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U V I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I N O D S O D S O D S O D S O D S O D S O D S O D S O D S O D S O D S O D S O D S O D S O D S O D S O D S O D S O D S O D S O D S O D S O D S O D S O D S O D S O D S O D S O D S O D S O D S O D S O D S O D S S O D S S O D S S O D S S O D S S O D S S O D S S O D S S O D S S O D S S O D S S O D S S O D S S O D S S O D S S O D S S O D S S O D S S O D S S O D S S O D S S O D S S O D S S O D S S O D S S O D s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s.s u n g. & . \\ - 2 p H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H/H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H /H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H / H /H /H /H /H /H /H /H /H /H /H /H /H /H /H /H /H /H /H /H /H /H /H /H /H /H /H /H /H /H /H /H /H /H /H /H /H /H /H /H /H /H /H /H /H /H /H /H /H /H /H /< ecel>< nl>\]

\[\begin{array} { l } = E _ { t } [ \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \delta } _ { t + 1 } + \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \delta } _ { t + 1 } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \\ + \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \delta } _ { t + j } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \delta } _ { t + j } \\ - \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } ] \\ = E _ { t } [ \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \delta } _ { t + 1 } + \mathbf { h } _ { \mathcal { X }} ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \delta } _ { t + 1 } \otimes \mathbf { h } _ { \mathbf { x }} ^ { l } \mathbf { x } _ { t } ^ { f } \\ + ( \mathbf { h } _ { \mathbf { x }} ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \delta } _ { t + 1 } + \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x }} ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } ) \otimes ( ( \mathbf { h } _ { \mathbf { x }} ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \delta } _ { t + 1 } + \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x }} ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t +j } ) \\ - ( ( \mathbf { h } _ { \mathbf {\Delta x}} ^ { l - 1 } \sigma \boldsymbol { \eta} \boldsymbol { \epsilon } _ { t + 1 } + \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf {\Delta x}} ^ { l - j } \sigma \boldsymbol { \eta} \boldsymbol { \epsilon } _ { t + j } ) ) \\ = E _ { t } [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ ] )\]

\[\begin{array} { r l } & = E _ { t } [ \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \delta _ { t + 1 } + \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \delta _ { t + 1 } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \\ & + \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \delta _ { t + 1 } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \delta _ { t + 1 } + 0 \\ & + 0 + \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \epsilon _ { t + j } \otimes \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \epsilon _ { t + j } \\ & - \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \epsilon _ { t + 1 } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \epsilon _ { t + 1 } + 0 \\ & - 0 - \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \epsilon _ { t + j } \otimes \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \epsilon_ { t + j } ] \\ & = E _ { t } [ \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \delta _ { t + 1 } + \mathbf { h } _ { \mathbf { x } ) } ^ { l - 1 } \sigma \boldsymbol { \eta } \delta _ { t + 1 } \otimes \mathbf { h } _ { \mathbf { x }} ^ { l } \mathbf { x } _ { t } ^ { f } \\ & + \mathbf { h } _ { \mathbf { x } ) } ^ { l - 1 } \sigma \boldsymbol { \eta } \delta _ { t + 1 } \otimes \mathbf { h } _ { \mathbf { x } ) } ^ { l - 1 } \sigma \boldsymbol { \eta } \delta _ { t + 1 } \\ & + 0 \\ & - \mathbf { h } _ { \mathbf { x } ) } ^ { l - 1 } \sigma \boldsymbol { \eta } \epsilon _ { t + 1 } \otimes \mathbf { h } _ { \mathbf { x } ) } ^ { l - 1 } \sigma \boldsymbol { \eta } \epsilon _ { t + 1 } \\ & - 0 ] \\ & = E _ { t } [ \mathbf { h } _ { \mathbf { x } ) } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } ) } ^ { l - 1 } \sigma \boldsymbol { \eta } \delta _ { t + 1 } + \mathbf { h } _ { \mathbf { x ) }} ^ { l - 1 } \sigma \boldsymbol { \eta } \delta _ { t + 1 } \otimes \mathbf { h } _ { \mathbf { x ) }} ^ { l } \mathbf { x } _ { t } ^ { f } \\ & + \mathbf { h } _ { \mathbf { x ) }} ^ { l - 1 } \sigma \boldsymbol { \eta } \delta _ { t + 1 } \otimes \mathbf { h } _ { \mathbf { x ) }} ^ { l - 1 } \sigma \boldsymbol { \eta } \delta _ { t + 1 } \\ & - \mathbf { h } _ { \mathbf { x ) }} ^ { l - 1 } \sigma \boldsymbol { \eta } \epsilon _ { t + 1 } \otimes \mathbf { h } _ { \mathbf { x ) }} ^ { l - 1 } \sigma \boldsymbol { \eta } \epsilon _ { t + 1} ] \\ & = {\textbf {\Delta}} {\textbf {\Delta}} {\textbf {\Delta}} {\textbf {\Delta}} {\textbf {\Delta}} {\textbf {\Delta}} {\textbf {\Delta}} {\textbf {\Delta}} {\textbf {\Delta}} {\textbf {\Delta}} {\textbf {\Delta}} {\textbf {\Delta}} {\textbf {\Delta}} {\textbf {\Delta}} {\textbf {\Delta}} {\textbf {\Delta}} {\textbf {\Delta}} {\textbf{\Delta}} {\textbf{\Delta}} {\textbf{\Delta}} {\textbf{\Delta}} {\textbf{\Delta}} {\textbf{\Delta}} {\textbf{\Delta}} {\textbf{\Delta}} {\textbf{\Delta}} {\textbf{\Delta}} {\textbf{\Delta}} {\textbf{\Delta}} {\textbf{\Delta}} {\textbf{\Delta}} {\textbf{\Delta}} {\textbf{\Delta}} {\textbf{\Delta}} {\mathrm{d} t} \\ & = {\textbf {\Delta}} {\textbf {\Delta}} {\textbf {\Delta}} {\textbf {\Delta}} {\textbf {\Delta}} {\textbf {\Delta}} {\textbf {\Delta}} {\textbf {\Delta}} {\textbf {\Delta}} {\textbf {\Delta}} {\textbf {\Delta}} {\textbf {\Delta}} {\textbf{\Delta}} {\textbf{\Delta}} {\textbf{\Delta}} {\textbf{\Delta}} {\textbf{d} t} \\ & + {\textbf {\Delta}} {\textbf {\Delta}} {\textbf{\Delta}} {\textbf{\Delta}} {\textbf{\Delta}} {\textbf{\Delta}} {\textbf{\Delta}} {\textbf{\Delta}} {\textbf{\Delta}} {\textbf{\Delta}} {\textbf{\Delta}} {\textbf{\Delta}} {\textbf{\Delta}} {\textbf{\Delta}} {\textbf{\Delta}} {\textbf{\Delta}} {\textbf{d} t} \\ & - E _ { t }\left[ {{\textbf{h}_{\mathbf{x}}} ^{l - 1} }\sigma {{\boldsymbol\eta }} {{\boldsymbol\epsilon}_{t + 1}}\otimes {{\textbf{h}_{\mathbf{x}}} ^{l - 1} }\sigma {{\boldsymbol\eta }} {{\boldsymbol\epsilon}_{t + 1}} ] \\ & = E _ {-} ({\texttt{d} s}) [ A (x) ] , \\ & = E _ {-} ({\texttt{d} s}) [ B (x) ] . \\ & = E _ {-} ({\texttt{d} s}) [ C (x) ] . \\ & = E _ {-} ({\texttt{d} s}) [ D (x) ] . \\ & = E _ {-} ({\texttt{d} s}) [ E (x) ] . \\ & = E _ {-} ({\texttt{d} s}) [ F (x) ] . \\ & = E _ {-} ({\texttt{d} s}) [ G (x) ] . \\ & = E _ {-} ({\texttt{d} s}) [ H (x) ] . \\ & = E _ {-} ({\texttt{d} s}) [ I (x) ] . \\ & = E _ {-} ({\texttt{d} s}) [ J (x) ] . \\ & = E _ {-} ({\texttt{d} s}) [ K (x) ] . \\ & = E _ {-} ({\texttt{d} s}) [ L (x) ] . \\ & = E _ {-} ({\texttt{d} s}) [ M (x) ] . \\ & = E _ {-} ({\texttt{d} s}) [ N (x) ] . \\ & = E _ {-} ({\texttt{d} s}) [ O (x) ] . \\ & = E _ {-} ({\texttt{d} s}) [ P (x) ] . \\ & = E _ {-} ({\texttt{d} s}) [ Q (x) ] . \\ & = E _ {-} ({\texttt{d} s}) [ R (x) ] . \\ & = E _ {-} ({\texttt{d} s}) [ S (x) ] . \\ & = E _ {-} ({\texttt{d} s}) [ T (x) ] . \\ & = E _ {-} ({\texttt{d} s}) [ U (x) ] . \\ & = E _ {-} ({\texttt{d} s}) [ V (x) ] . \\ & = E _ {-} ({\texttt{d} s}) [ W (x) ] . \\ & = E _ {-} ({\texttt{d} s}) [ X (x) ] . \\ & = E _ {-} ({\texttt{d} s}) [ Y (x) ] . \\ & = E _ {-} ({\texttt{d} s}) [ Z (x) ] . \\ & = E _ {-} ({\texttt{d} s}) [ Z (x) ] . \\ & = E _ {-} ({\texttt{d} s}) [ Z (x) ] . \\ & = E _ {-} ({\texttt{d} s}) [ Z (x) ] . \\ & = E _ {-} ({\texttt{d} s}) [ Z (x) ] . \\ &= E_{-}\left[ Z_{-}\right]\left[ Z_{-}\right]. \\ &= Z_{-}\left[ Z_{+}\right]\left[ Z_{+}\right]. \\ &= Z_{-}\left[ Z_{-}\right]\left[ Z_{-}\right]. \\ &= Z_{-}\left[ Z_{+}\right]\left[ Z_{+}\right]. \\ &= Z_{-}\left[ Z_{-}\right]\left[ Z_{-}\right]. \\ &= Z_{-}\left[ Z_{+}\right]\left[ Z_{+}\right]. \\ &= Z_{-}\left[ Z_{+}\right]\left[ Z_{+}\right]. \\ &= Z_{-}\left[ Z_{+}\right]\left[ Z_{+}\right]. \\ &= Z_{-}\left[ Z_{+}\right]\left[ Z_{+}\right]. \\ &= Z_{-}\left[ Z_{+}\right]\left[ Z_{+}\right]. \\ &= Z_ {-}\left[ Z_{+}\right]\left[ Z_{+}\right]. \\ &= Z_ {-}\left[ Z_{+}\right]\left[ Z_{+}\right]. \\ &= Z_ {-}\left[ Z_{+}\right]\left[ Z_{+}\right]. \\ &= Z_ {-}\left[ Z_{+}\right]\left[ Z_{+}\right]. \\ &= Z_ {-}\left[ Z_{+}\]

So we only need to compute . We then note that

\[\begin{array} { r l } & = E _ { t } \left[ \left( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \right) \left( \sigma \pmb { \eta } \pmb { \epsilon } _ { t + 1 } \otimes \sigma \pmb { \eta } \pmb { \epsilon } _ { t + 1 } \right) \right] \\ & \text {using} ( \mathbf { A } \otimes \mathbf { B } ) ( \mathbf { C } \otimes \mathbf { D } ) = \mathbf { A C } \otimes \mathbf { B D } \text {if} \mathbf { A C} \text {and} \mathbf { B D} \text {are defined} \\ & = \left( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \right) E _ { t } \left[ \sigma \pmb { \eta } \pmb { \epsilon } _ { t + 1 } \otimes \sigma \pmb { \eta } \pmb { \epsilon } _ { t + 1 } \right] \\ & = \left( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \right) \pmb { \Lambda } \\ & \text {where} \pmb { \Lambda } \equiv E _ { t } \left[ \sigma \pmb { \eta } \pmb { \epsilon } _ { t + 1 } \otimes \sigma \pmb { \eta } \pmb { \epsilon } _ { t + 1 } \right] . \text {We then have} \\ & \pmb { \Lambda } = E _ { t } \left[ \sigma \pmb { \eta } \pmb { \epsilon } _ { t + 1 } \otimes \sigma \pmb { \eta } \pmb { \epsilon } _ { t + 1 } \right] \\ & = E _ { t } \left\{ \left\{ \sigma \pmb { \eta } ( \gamma _ { 2 } , : ) \pmb { \epsilon } _ { t + 1 } \sigma \pmb { \eta } ( \gamma _ { 1 } , : ) \pmb { \epsilon } _ { t + 1 } \right\} _ { \gamma _ { 1 } = 1 } ^ { n _ { x }} \right\} _ { \gamma _ { 2 } = 1 } ^ { n _ { x }} \\ & = E _ { t } \left\{ \left\{ \sigma \sum _ { \phi _ { 1 } = 1 } ^ { n _ { e } } \pmb { \eta } ( \gamma _ { 2 } , \phi _ { 1 } ) \pmb { \epsilon } _ { t + 1 } ( \phi _ { 1 } , 1 ) \sum _ { \phi _ { 2 } = 1 } ^ { n _ { e } } \sigma \pmb { \eta } ( \gamma _ { 1 } , \phi _ { 2 } ) \pmb { \epsilon } _ { t + 1 } ( \phi _ { 2 } , 1 ) \right\} _ { \gamma _ { 1 } = 1 } ^ { n _ { x }} \right\} _ { \gamma _ { 2 } = 1 } ^ { n _ { x }} \\ & = E _ { t } \left\{ \left\{ \sigma \sum _ { \phi _ { 1 } = 1 } ^ { n _ { e } } \sum _ { \phi _ { 2 } = 1 } ^ { n _ { e } } \pmb { \eta } ( \gamma _ { 2 } , \phi _ { 1 } ) \pmb { \epsilon } _ { t + 1 } ( \phi _ { 1 } , 1 ) \sigma \pmb { \eta } ( \gamma _ { 1 } , \phi _ { 2 } ) \pmb { \epsilon } _ { t + 1 } ( \phi _ { 2 } , 1 ) \right\} _ { \gamma _ { 1 } = 1 } ^ { n _ { x }} \right\} _ { \gamma _ { 2 } = 1 } ^ { n _ {\mathrm{x}}} \\ & = \\ & = \\ & = \\ & = \\ & = \\ & = \\ & = \\ & = \\ & = \\ & = \\ & = \\ & = \\ & = \\ & = \\ & = \\ & = \\ & = \\ & = \\ & = \\ & = \\ & = \\ & = \\ & = \\ & = \\ & = \\ & = \\ & = \\ & = \\ & = \\ & = \\ & = \\ & = \\ & = \\ & =; \\ & = \\ & = \\ & = \\ & = \\ & = \\ & = \\ & = \\ & = \\ & = \\ & = \\ & = \\ & = \\ & = \\ & = \\ & = \\ & = \\ & = \\ & = \\ & = \\ & = \\ & = \\ & = \\ & = \\ & = \\ & = \\ & = \\ & = \\ & = \\ & = \\ & = \\ & = \\ & = \\ & = ; \\ & = ; \\ & = ; \\ & = ; \\ & = ; \\ & = ; \\ & = ; \\ & = ; \\ & = ; \\ & = ; \\ & = ; \\ & = ; \\ & = ; \\ & = ; \\ & = ; \\ & = ; \\ & = ; \\ & = ; \\ & = ; \\ & = ; \\ & = ; \\ & = ; \\ & = ; \\ & = ; \\ & = ; \\ & = ? ; \\ & = ? ; \\ & = ? ; \\ & = ? ; \\ & = ? ; \\ & = ? ; \\ & = ? ; \\ & = ? ; \\ & = ? ; \\ & = ? ; \\ & = ? ; \\ & = ? ; \\ & = ? ; \\ & = ? ; \\ & = ? ; \\ & = ? ; \\ & = ? ; \\ & = ? ; \\ & = ? ; \\ & = ? ; \\ & =? ; \\ & = ? ; \\ & = ? ; \\ & = ? ; \\ & = ? ; \\ & = ? ; \\ & = ? ; \\ & = ? ; \\ & = ? ; \\ & = ? ; \\ & = ? ; \\ & = ? ; \\ & = ? ; \\ & = ? ; \\ & = ? ; \\ & = ? ; \\ & = ? ; \\ & = ? ; \\ & = ? ; \\ & = ? ; \\ & = ?? ^ [ n ] ^ [ n ] ^ [ n ] ^ {[ n ] [ n ] ]}. \\ ( n ) ^ [ n ] ^ {[ n ] [ n ]}. I n i n g (\tau) [ n ] ^ {[ n ] [ n ]}. I n j o w i s e r e (\tau) [ n ] ^ {[ n ] [ n ]}. I n j o w i s e r e (\tau) [ n ] ^ {[ n ] [ n ]}. I n j o w i s e r e (\tau) [ n ] ^ {[ n ] [ n ]}. I n j o w i s e r e (\tau) [ n ] ^ {[ n ] [ n ]}. I n j o w i s e r e (\tau) I n j o w i s e r e (\tau) I n j o w i s e r e (\tau) I n j o w i s e r e (\tau) I n j o w i s e r e (\tau) I n j o w i s e r e (\tau) I n j o w i s e r e (\tau) I n j o w i s e r e (\tau) I n j o w i s e l o f i n d i a c t i o n. I n j o w i s e r e (\tau) I n j o w i s e r e (\tau) I n j o w i s e r e (\tau) I n j o w i s e r e (\tau) I n j o w i s e r e (\tau) I n j o w i s e r e (\tau) I n j o w i s e r e (\tau) I n j o w i s e t h a r e. I n j o w i s e r e (\tau) I n j o w i s e r e (\tau) I n j o w i s e r e (\tau) I n j o w i s e r e (\tau) I n j o w i s e r e (\tau) I n j o w i s e r e (\tau) I n j o w i s e r e (\tau) I n j o w i s i v e r e (\tau) I n j o w i s e r e (\tau) I n j o w i s i v e r e (\tau) I n j o w i s i v e r e (\tau) I n j o w i s i v e r e (\tau) I n j o w i s i v e r e (\tau) I n j o w i s i v e r e (\tau) I n j o w i s i v e r e (\tau) I n j o w i s i v e r e (\tau) I n j y o l l a c k a l l a c k a l l a c k a l l a c k a l l a c k a l l a c k a l l a c k a l l a c k a l l a c k a l l a c k a l l a c k a l l a c k a l l a c k a l l a c k a l l a c k a l l a c k a l l a c k a l l a b u l d . I n j o w i s i v e r e (\tau) I n j o w i s i v e r e (\tau) I n j o w i s i v e r e (\tau) I n j o w i s i v e r e (\tau) I n j o w i s i v e r e (\tau) I n j o w i s i v e r e (\tau) I n j o w i s i v i v e r e (\tau) I n j o w i s i v e r e (\tau) I n j o w i s i v e r e (\tau) I n j o w i s i v e r e (\tau) I n j o w i s i v e r e (\tau) I n j o w i s i v e r e (\tau) I n j o w i s i v e r e (\tau) I N O S O S O S O S O S O S O S O S O S O S O S O S O S O S O S O S O S O S O S O S O S O S O S O S O S O S O S O S O S O S O S O S O S O S O S O S O S O S O S O S O S O S O S O S O S O S O S O S O S O S O T h a r e. I n J o w i s i v e r e (\tau) I N J o w i s i v e r e (\tau) I N J o w i s i v e r e (\tau) I N J o w i s i v e r e (\tau) I N J o w i s i v e r e (\tau) I N J o w i s i v e r e (\tau) I N J o w i s i v e r e (\tau) I N J o w i s i v e r [ N ] ^ {[ N ] [ N ]}. I N J o w i s i v e r [ N ] ^ {[ N ] [ N ]}. I N J o w i s i v [ N ] ^ {[ N ] [ N ]}. I N J o w i s [ N ] ^ {[ N ] [ N ]}. I N J o w [ N ] ^ {[ N ] [ N ]}. I N J o w [ N ] ^ {[ N ] [ N ]}. I N J o w [ N ] ^ {[ N ] [ N ]}. I N J o w [ N ] ^ {[ N ] [ N ]}. I N J o w [ N ] ^ {[ N ] [ N ]}. I N J o w [ N ] ^ {[ N ] [ N ]}. I NJ o w [ N ] ^ {[ N ] [ N ]}. I NJ o w [ N ] ^ {[ N ] [ N ]}. I NJ o w [ N ] ^ {[ N ] [ N ]}. I NJ o w [ N ] ^ {[ N ] [ N ]}. I NJ o w [ N ] ^ {[ N ] [ N ]}. I NJ o w [ N ] ^ {[ N ] [ N ]}.I NJ o w [ N ] ^ {[ N ] [ N ]}.I NJ o w [ N ] ^ {[ N ] [ N ]}.I NJ o w [ N ] ^ {[ N ] [ N ]}.I NJ o w [ N ] ^ {[ N ] [ N ]}.I NJ o w [ N ] ^ {[ N ] [ N ]}.I NJ o w [ N ] ^ {[ N ] [ N ], q.t.} .I NJ o w [ N ] ^ {[ N ] [ N ], q.t.} .I NJ o w [ N ] ^ {[ N ] [ N ], q.t.} .I NJ o w [ N ] ^ {[ N ] [ N ], q.t.} .I NJ o w [N ] ^ {[N ] [N ], q.t.} .I NJ o w [N ] ^ {[N ] [N ], q.t.} .I NJ o w [N ] ^ {[N ] [N ], q.t.} .I NJ o w [N ] ^ {[N ] [N ], q.t.} .I NJ o w [N ] ^ {[N ] [N ], q.t.} .I NJ o w [N ]^ {[N ] [N ], q.t.} .I NJ o w [N ] ^ {[N ] [N ], q.t.} .I NJ o w [N ] ^ {[N ] [N ], q.t.} .I NJ o w [N ] ^ {[N ] [N ], q.t.} .I NJ o w [N ] ^ {[N ] [N ], q.t.} .I NJ o W , K , L , M , M , M , M , M , M , M , M , M , M , M , M , M , M , M , M , M , M , M , M , M , M , M , M , M , M , M , M , M , M , M , M , M , M , M , M , M , M , M , M , M , M , M , M , M , M , M , M , M , M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,M ,T h a r d . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H .H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H . H .H .H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H/H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H.H,H.H.H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H,H, p.m., m.p.m., c.m., d.m., and m.p.m., m.p.m., c.m., d.m., m.p.m., m.p.m., m.p.m., m.p.m., m.p.m., m.p.m., m.p.m., m.p.m., m.p.m., m.p.m., m.p.m., m.p.m., m.p.m., m.p.m., m.p.m., m.p.m., m.p.m., m.p.m., m.p.m., m.p.m., m.p.m., m.p.m., m.p.m., m.p.m., m.p.m., m.P.M.S.O.S.O.S.O.S.O.S.O.S.O.S.O.S.O.S.O.S.O.S.O.S.O.S.O.S.O.S.O.S.O.S.O.S.O.S.O.S.O.S.O.S.O.S.O.S.O.S.O.S.O.S.O.S.O.S.O.S.O.S.O.S.O.S.O.S.O.S.O.S.O.S.O.S.O.S.O.S.O.S.O.S.O.S.O.S.O.S.O.S.O.S.O.S.O.S.O.S.O.S.\]

\[\begin{array}{l} \text {Thus} \\ E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + l} ^ {f} - \mathbf {x} _ {t + l} ^ {f} \otimes \mathbf {x} _ {t + l} ^ {f} \right] = \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} + \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} \\ - (\mathbf {h} _ {\mathbf {x}} ^ {l - 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1}) \boldsymbol {\Lambda} \\ = \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} + \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \nu \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + (\mathbf {h} _ {\mathbf {x}} ^ {l - 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1}) (\sigma \boldsymbol {\eta} \boldsymbol {\nu} \otimes \sigma \boldsymbol {\eta} \boldsymbol {\nu}) \\ - (\mathbf {h} _ {\mathbf {x}} ^ {l - 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1}) \boldsymbol {\Lambda} \\ = \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x i}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} + \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + (\mathbf {h} _ {\mathbf {x}} ^ {l - 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1}) ((\sigma \boldsymbol {\eta} \boldsymbol {\nu} \otimes \sigma \boldsymbol {\eta} \boldsymbol {\nu}) - \boldsymbol {\Lambda}) \end{array}\]

\[\begin{array}{l} \text {Or (using another index)} \\ E _ {t} \left[ \tilde {\mathbf {x}} _ {t + j} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + j} ^ {f} - \mathbf {x} _ {t + j} ^ {f} \otimes \mathbf {x} _ {t + j} ^ {f} \right] = \mathbf {h} _ {\mathbf {x}} ^ {j} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {j - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} + \mathbf {h} _ {\mathbf {x}} ^ {j - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} \otimes \mathbf {h} _ {\mathbf {x}} ^ {j} \mathbf {x} _ {t} ^ {f} + (\mathbf {h} _ {\mathbf {x}} ^ {j - 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {j - 1}) ((\sigma \boldsymbol {\eta} \boldsymbol {\nu} \otimes \sigma \boldsymbol {\eta} \boldsymbol {\nu}) - \boldsymbol {\Lambda}) \\ \text {for j = 1,2,3,...} \end{array}\]

\[\begin{array}{l} \text {Thus, we have in general} \\ E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} ^ {s} - \mathbf {x} _ {t + l} ^ {s} \right] = E _ {t} \left[ \sum_ {j = 0} ^ {l - 1} \mathbf {h} _ {\mathbf {x}} ^ {l - 1 - j} \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \left(\tilde {\mathbf {x}} _ {t + j} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + j} ^ {f}\right) - \sum_ {j = 0} ^ {l - 1} \mathbf {h} _ {\mathbf {x}} ^ {l - 1 - j} \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t + j} ^ {f} \otimes \mathbf {x} _ {t + j} ^ {f}\right) \right] \\ = \sum_ {j = 1} ^ {l - 1} \mathbf {h} _ {\mathbf {x}} ^ {l - 1 - j} \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + j} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + j} ^ {f} - \mathbf {x} _ {t + j} ^ {f} \otimes \mathbf {x} _ {t + j} ^ {f} \right] \\ \text {the shock hits in period t + 1, so } \left(\tilde {\mathbf {x}} _ {t} ^ {f} \otimes \tilde {\mathbf {x}} _ {t} ^ {f}\right) = \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \\ = \sum_ {j = 1} ^ {l - 1} \mathbf {h} _ {\mathbf {x}} ^ {l - 1 - j} \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {h} _ {\mathbf {x}} ^ {j} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {j - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} + \mathbf {h} _ {\mathbf {x}} ^ {j - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} \otimes \mathbf {h} _ {\mathbf {x}} ^ {j} \mathbf {x} _ {t} ^ {f} + (\mathbf {h} _ {\mathbf {x}} ^ {j - 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {j - 1}) ((\sigma \boldsymbol {\eta} \boldsymbol {\nu} \otimes \sigma \boldsymbol {\eta} \boldsymbol {\nu}) - \boldsymbol {\Lambda})\right) \end{array}\]

If we restrict the focus and do the GIRF's at the unconditional mean of , then we get

\[E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} ^ {s} - \mathbf {x} _ {t + l} ^ {s} \right] = \sum_ {j = 1} ^ {l - 1} \mathbf {h} _ {\mathbf {x}} ^ {l - 1 - j} \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \left(\left(\mathbf {h} _ {\mathbf {x}} ^ {j - 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {j - 1}\right) (\sigma \boldsymbol {\eta} \boldsymbol {\nu} \otimes \sigma \boldsymbol {\eta} \boldsymbol {\nu} - \boldsymbol {\Lambda})\right)\]

When implementing the GIRF, it may be useful to have a recursive expression. Here, it is must convenient to use

\[\begin{array}{l} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} ^ {s} - \mathbf {x} _ {t + l} ^ {s} \right] = \sum_ {j = 1} ^ {l - 1} \mathbf {h} _ {\mathbf {x}} ^ {l - 1 - j} \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + j} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + j} ^ {f} - \mathbf {x} _ {t + j} ^ {f} \otimes \mathbf {x} _ {t + j} ^ {f} \right] \\ \mathrm{So} \\ E _ {t} \left[ \tilde {\mathbf {x}} _ {t + 1} ^ {s} - \mathbf {x} _ {t + 1} ^ {s} \right] = 0 \end{array}\]

\[\begin{array}{r} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + 2} ^ {s} - \mathbf {x} _ {t + 2} ^ {s} \right] = \sum_ {j = 1} ^ {1} \mathbf {h} _ {\mathbf {x}} ^ {1 - j} \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + j} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + j} ^ {f} - \mathbf {x} _ {t + j} ^ {f} \otimes \mathbf {x} _ {t + j} ^ {f} \right] \\ = \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + 1} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + 1} ^ {f} - \mathbf {x} _ {t + 1} ^ {f} \otimes \mathbf {x} _ {t + 1} ^ {f} \right] \end{array}\]

\[\begin{array}{r l} & E _ {t} \left[ \tilde {\mathbf {x}} _ {t + 3} ^ {s} - \mathbf {x} _ {t + 3} ^ {s} \right] = \sum_ {j = 1} ^ {2} \mathbf {h} _ {\mathbf {x}} ^ {2 - j} \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + j} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + j} ^ {f} - \mathbf {x} _ {t + j} ^ {f} \otimes \mathbf {x} _ {t + j} ^ {f} \right] \\ & \qquad = \mathbf {h} _ {\mathbf {x}} \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + 1} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + 1} ^ {f} - \mathbf {x} _ {t + 1} ^ {f} \otimes \mathbf {x} _ {t + 1} ^ {f} \right] \\ & \qquad + \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + 2} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + 2} ^ {f} - \mathbf {x} _ {t + 2} ^ {f} \otimes \mathbf {x} _ {t + 2} ^ {f} \right] \\ & \qquad = \mathbf {h} _ {\mathbf {x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + 2} ^ {s} - \mathbf {x} _ {t + 2} ^ {s} \right] + \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + 2} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + 2} ^ {f} - \mathbf {x} _ {t + 2} ^ {f} \otimes \mathbf {x}. \right. \\ & \qquad \end{array}\]

\[\begin{array}{r l} & {\mathrm{Soingeneral}} \\ & {E _ {t} \left[ \tilde {\mathbf {x}} _ {t + k} ^ {s} - \mathbf {x} _ {t + k} ^ {s} \right] = \mathbf {h} _ {\mathbf {x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + k - 1} ^ {s} - \mathbf {x} _ {t + k - 1} ^ {s} \right] + \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + k - 1} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + k - 1} ^ {f} - \mathbf {x} _ {t + k - 1} ^ {f} \otimes \mathbf {x} _ {t + k - 1} ^ {f} \right]} \end{array}\]

For the total state variable:

\[E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} - \mathbf {x} _ {t + l} \right] = E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} ^ {f} - \mathbf {x} _ {t + l} ^ {f} \right] + E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} ^ {s} - \mathbf {x} _ {t + l} ^ {s} \right]\]

For the control variables:

\[\begin{array}{r l} & {\mathbf {y} _ {t + l} ^ {s} = \mathbf {g} _ {\mathbf {x}} \left(\mathbf {x} _ {t + l} ^ {f} + \mathbf {x} _ {t + l} ^ {s}\right) + \frac {1}{2} \mathbf {G} _ {\mathbf {x x}} \left(\mathbf {x} _ {t + l} ^ {f} \otimes \mathbf {x} _ {t + l} ^ {f}\right) + \frac {1}{2} \mathbf {g} _ {\sigma \sigma} \sigma^ {2}} \\ & {\tilde {\mathbf {y}} _ {t + l} ^ {s} = \mathbf {g} _ {\mathbf {x}} \left(\tilde {\mathbf {x}} _ {t + l} ^ {f} + \tilde {\mathbf {x}} _ {t + l} ^ {s}\right) + \frac {1}{2} \mathbf {G} _ {\mathbf {x x}} \left(\tilde {\mathbf {x}} _ {t + l} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + l} ^ {f}\right) + \frac {1}{2} \mathbf {g} _ {\sigma \sigma} \sigma^ {2}} \\ & {E _ {t} \left[ \tilde {\mathbf {y}} _ {t + l} ^ {s} - \mathbf {y} _ {t + l} ^ {s} \right] = \mathbf {g} _ {\mathbf {x}} \left(E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} ^ {f} - \mathbf {x} _ {t + l} ^ {f} \right] + E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} ^ {s} - \mathbf {x} _ {t + l} ^ {s} \right]\right) + \frac {1}{2} \mathbf {G} _ {\mathbf {x x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + l} ^ {f} - \mathbf {x} _ {t + l} ^ {f} \otimes \mathbf {x} _ {t + l} ^ {f} \right]} \end{array}\]

11.3 At third order

At third order, we additionally need to consider:

\[\begin{array} { r l } & { \mathbf { x } _ { t + 1 } ^ { r d } = \mathbf { h } _ { \mathbf { x } } \mathbf { x } _ { t } ^ { r d } + \mathbf { H } _ { \mathbf { x x } } \left( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { s } \right) + \frac { 1 } { 6 } \mathbf { H } _ { \mathbf { x x x } } \left( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \right) + \frac { 3 } { 6 } \mathbf { h } _ { \sigma \sigma \mathbf { x } } \sigma ^ { 2 } \mathbf { x } _ { t } ^ { f } + \frac { 1 } { 6 } \mathbf { h } _ { \sigma \sigma \sigma } \sigma ^ { 3 } } \\ & { \mathrm{and} } \\ & { \mathbf { x } _ { t + 2 } ^ { r d } = \mathbf { h } _ { \mathbf { x } } \mathbf { x } _ { t + 1 } ^ { r d } + \mathbf { H } _ { \mathbf { x x } } \left( \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { s } \right) + \frac { 1 } { 6 } \mathbf { H } _ { \mathbf { x x x } } \left( \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } \right) + \frac { 3 } { 6 } \mathbf { h } _ { \sigma \sigma \mathbf { x } } \sigma ^ { 2 } \mathbf { x } _ { t + 1 } ^ { f } + \frac { 1 } { 6 } \mathbf { h } _ { \sigma \sigma \sigma } \sigma ^ { 3 } } \\ & { = \mathbf { h } _ { \mathbf { x } } \left( \mathbf { h } _ { \mathbf { x } } \mathbf { x } _ { t } ^ { r d } + \mathbf { H } _ { \mathbf { x x } } \left( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { s } \right) + \frac { 1 } { 6 } \mathbf {H } _ { \mathbf { x x x } } \left( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \right) + \frac { 3 } { 6 } \mathbf { h } _ { \sigma \sigma \mathbf { x } } \sigma ^{ 2 } \mathbf { x } _ { t } ^ { f } + \frac { 1 } { 6 } \mathbf { h } _ { \sigma \sigma \sigma } \sigma ^{ 3} \right) \\ & { + \mathbf { H } _ { \mathbf { x x } } \left( \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { s} \right) + \frac { 1 } { 6 } \mathbf { H } _ { \mathbf { x x x } } \left( \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f} \right) + \frac { 3 } { 6 } \mathbf { h } _ { \sigma \sigma \mathbf { x } } \sigma ^{ 2 } \mathbf { x } _ { t + 1 } ^ { f } + \frac { 1 } { 6 } \mathbf { h } _ { \sigma \sigma \sigma } \sigma ^{ 3} \\ & { = {\mathbf h} _ { {\mathbf x}} ^ { 2} {\mathbf x} _ { t } ^ { r d } + {\mathbf h} _ {\mathbf x} {\mathbf H} _ { {\mathbf x x }} ( {\mathbf x} _ { t } ^ { f} \otimes {\mathbf x} _ { t } ^ { s}) + {\mathbf h} _ {\mathbf x} {\frac 16} {\mathbf H} _ { {\mathbf x x x }} ( {\mathbf x} _ { t } ^ { f} \otimes {\mathbf x} _ { t } ^ { f} \otimes {\mathbf x} _ { t ) f ) + {\mathbf h} _ {\mathbf x} {\frac 36} {\mathbf h} _ { {\sigma \sigma x}} {\sigma^ 2} {\mathbf x} _ { t f ) f ) + {\mathbf h} _ {\mathbf x} {\frac 16} {\mathbf h} _ { {\sigma \sigma \sigma}} {\sigma^ 3} \\ & { + {\mathbf H} _ { {\mathbf x x }} ( {\mathbf x} _ { t + 1 } ^ { f} \otimes {\mathbf x} _ { t + 1 ) f ) + \frac 16 {\mathbf H} _ { {\mathbf x x x }} ( {\mathbf x} _ { t + 1 ] f ) + {\frac 36} {\mathbf h} _ { {\sigma \sigma x}} {\sigma^ 2} {\mathbf x} _ { t + 1 ] f ) + {\frac 36} {\mathbf h} _ { {\sigma \sigma \sigma}} {\sigma^ 3} , \\ & { + {\mathbf H} _ { {\mathbf x x }} ( {\mathbf x} _ { t + 2 ] f ) + {\frac 16} {\mathbf H} _ { {\mathbf x x x }} ( {\mathbf x} _ { t + 2 ] f ) + {\frac 36} {\mathbf h} _ { {\sigma \sigma x}} {\sigma^ 2} {\mathbf x} _ { t + 2 ] f ) + {\frac 36} {\mathbf h} _ { {\sigma \sigma \sigma}} {\sigma^ 3} , \\ & = {{\texttt h a b j e}} [ {{\texttt h a b j e}} [ {{\texttt h a b j e}} [ {{\texttt h a b j e}} ] + {{\texttt h a b j e}} [ {{\texttt h a b j e}} ] - {{\texttt h a b j e}} [ {{\texttt h a b j e}} ] - {{\texttt h a b j e}} [ {{\texttt h a b j e}} ] - {{\texttt h a b j e}} [ {{\texttt h a b j e}} ] - {{\texttt h a b j e}} [ {{\texttt h a b j e}} ] - {{\texttt h a b J e}}} ] \\ & = {{\texttt h a b j e}} [ {{\texttt h a b j e}} [ {{\texttt h a b j e}} ] - {{\texttt h a b j e}} [ {{\texttt h a b j e}} ] - {{\texttt h a b j e}} [ {{\texttt h a b j e}} ] - {{\texttt h a b j e}} [ {{\texttt h a b j e}} ] - {{\texttt h a b k e}}} ] \\ & = {{\texttt h a b j e}} [ {{\texttt h a b j e}} [ {{\texttt h a b j e}} ] - {{\texttt h a b j e}} [ {{\texttt h a b j e}} ] - {{\texttt h a b j e}} [ {{\texttt h a b j e}} ] - {{\texttt h a b k e}}} ] \\ & = {{\texttt h a b j e}} [ {{\texttt h A B E}}} [ {{\texttt h A B E}}} ] - {{\texttt h A B E}}} [ {{\texttt h A B E}}} ] - {{\texttt h A B E}}} [ {{\texttt h A B E}}} ] - {{\texttt h A B E}}} [ {{\texttt h A B E}}} ] - {{\texttt h A B E}}} [ {{\texttt h A B E}}} ] \\ & = {{\texttt h A B E}} [ {{\texttt h A B E}} ] - {{\texttt h A B E}} [ {{\texttt h A B E}} ] - {{\texttt h A B E}} [ {{\texttt h A B E}} ] - {{\texttt h A B E}} [ {{\texttt h A B E}} ] - {{\texttt h A B E}} [ {{\texttt hA B E}} ] \\ & = {{\texttt h A B E}} [ {{\texttt hA B E}} ] - {{\texttt hA B E}} [ {{\texttt hA B E}} ] - {{\texttt hA B E}} [ {{\texttt hA B E}} ] - {{\texttt hA B E}} [ {{\texttt hA B E}} ] - {{\texttt hA B E}} [ {{\texttt hA B E}} ] \\ & = {{\texttt hA B E}} [ {{\texttt hA B E}} ] - {{\texttt hA B E}} [ {{\texttt hA B E}} ] - {{\texttt hA B E}} [ {{\texttt hA B E}} ] - {{\texttt hA B E}} [ {{\texttt hA B E}} ] - {{\texttt hA B E}} [ {{\texttt hA B E}} ] \\ \\ & = {{\texttt hA B E}} [ {{\texttt hA B E}} ] - {{\texttt hA B E}} [ {{\texttt hA B E}} ] - {{\texttt hA B E}} [ {{\texttt hA B E}} ] - {{\texttt hA B E}} [ {{\texttt hA B E}} ] - {{\texttt hA B E}} [ \mathrmhavonlavonlavonlavonlavonlavonlavonlavonlavonlavonlavonlavonlavonlavonlavonlavonlavonlavonlavonlavonlavonlavonlavonlavonlavonlavonlavonlavonlavonlavonlavonlavonlavonlavonlvaren l varen l varen l varen l varen l varen l varen l varen l varen l varen l varen l varen l varen l varen l varen l varen l varen l varen l varen l varen l varen l varen l varen l varen l varen l varen l w i n l v i n l w i n l w i n l w i n l w i n l w i n l w i n l w i n l w i n l w i n l w i n l w i n l w i n l w i n l w i n l w i n l w i n l w i n l w i n l w i n l w i n l w i n l w i n l w i n l w i n l w i n lw i n l w i n l w i n l w i n l w i n l w i n l w i n l w i n l w i n l w i n l w i n l w i n l w i n l w i n l w i n l w i n l w i n l w i n l w i n l w i n l w i n l w i n l w i n l w i n l w i n l< nl>\]

\[\begin{array}{r l} & {+ \sum_ {j = 0} ^ {2} \mathbf {h} _ {\mathbf {x}} ^ {2 - j} \frac {3}{6} \mathbf {h} _ {\sigma \sigma \mathbf {x}} \sigma^ {2} \mathbf {x} _ {t + j} ^ {f}} \\ & {+ \sum_ {j = 0} ^ {2} \mathbf {h} _ {\mathbf {x}} ^ {2 - j} \frac {1}{6} \mathbf {h} _ {\sigma \sigma \sigma} \sigma^ {3}} \\ & {= \mathbf {h} _ {\mathbf {x}} ^ {3} \mathbf {x} _ {t} ^ {r d} + \sum_ {j = 0} ^ {2} \mathbf {h} _ {\mathbf {x}} ^ {2 - j} \left[ \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t + j} ^ {f} \otimes \mathbf {x} _ {t + j} ^ {s}\right) + \frac {1}{6} \mathbf {H} _ {\mathbf {x x x}} \left(\mathbf {x} _ {t + j} ^ {f} \otimes \mathbf {x} _ {t + j} ^ {f} \otimes \mathbf {x} _ {t + j} ^ {f}\right) + \frac {3}{6} \mathbf {h} _ {\sigma \sigma \mathbf {x}} \sigma^ {2} \mathbf {x} _ {t + j} ^ {f} \right]} \\ & {+ \left(\sum_ {j = 0} ^ {2} \mathbf {h} _ {\mathbf {x}} ^ {2 - j}\right) \frac {1}{6} \mathbf {h} _ {\sigma \sigma \sigma} \sigma^ {3}} \end{array}\]

In general

\[\begin{array}{r l} & {\mathbf {x} _ {t + l} ^ {r d} = \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {r d} + \sum_ {j = 0} ^ {l - 1} \mathbf {h} _ {\mathbf {x}} ^ {l - 1 - j} \left[ \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t + j} ^ {f} \otimes \mathbf {x} _ {t + j} ^ {s}\right) + \frac {1}{6} \mathbf {H} _ {\mathbf {x x x}} \left(\mathbf {x} _ {t + j} ^ {f} \otimes \mathbf {x} _ {t + j} ^ {f} \otimes \mathbf {x} _ {t + j} ^ {f}\right) + \frac {3}{6} \mathbf {h} _ {\sigma \sigma \mathbf {x}} \sigma^ {2} \mathbf {x} _ {t + j} ^ {f} \right]} \\ & {\quad + \left(\sum_ {j = 0} ^ {l - 1} \mathbf {h} _ {\mathbf {x}} ^ {l - 1 - j}\right) \frac {1}{6} \mathbf {h} _ {\sigma \sigma \sigma} \sigma^ {3}} \end{array}\]

Thus

\[\begin{array}{l} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} ^ {r d} - \mathbf {x} _ {t + l} ^ {r d} \right] = E _ {t} [ \sum_ {j = 0} ^ {l - 1} \mathbf {h} _ {\mathbf {x}} ^ {l - 1 - j} \left[ \mathbf {H} _ {\mathbf {x x}} \left(\tilde {\mathbf {x}} _ {t + j} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + j} ^ {s}\right) + \frac {1}{6} \mathbf {H} _ {\mathbf {x x x}} \left(\tilde {\mathbf {x}} _ {t + j} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + j} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + j} ^ {f}\right) + \frac {3}{6} \mathbf {h} _ {\sigma \sigma \mathbf {x}} \sigma^ {2} \tilde {\mathbf {x}} _ {t + j} ^ {f} \right] \\ - \{\sum_ {j = 0} ^ {l - 1} \mathbf {h} _ {\mathbf {x}} ^ {l - 1 - j} \left[ \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t + j} ^ {f} \otimes \mathbf {x} _ {t + j} ^ {s}\right) + \frac {1}{6} \mathbf {H} _ {\mathbf {x x x}} \left(\mathbf {x} _ {t + j} ^ {f} \otimes \mathbf {x} _ {t + j} ^ {f} \otimes \mathbf {x} _ {t + j} ^ {f}\right) + \frac {3}{6} \mathbf {h} _ {\sigma \sigma \mathbf {x}} \sigma^ {2} \mathbf {x} _ {t + j} ^ {f} \right] \} ] \\ = \sum_ {j = 1} ^ {l - 1} \mathbf {h} _ {\mathbf {x}} ^ {l - 1 - j} \left(\mathbf {H} _ {\mathbf {x x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + j} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + j} ^ {s} - \mathbf {x} _ {t + j} ^ {f} \otimes \mathbf {x} _ {t + j} ^ {s} \right] + \frac {3}{6} \mathbf {h} _ {\sigma \sigma \mathbf {x}} \sigma^ {2} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + j} ^ {f} - \mathbf {x} _ {t + j} ^ {f} \right]\right) \\ + \sum_ {j = 1} ^ {l - 1} \mathbf {h} _ {\mathbf {x}} ^ {l - 1 - j} \frac {1}{6} \mathbf {H} _ {\mathbf {x x x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + j} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + j} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + j} ^ {f} - \mathbf {x} _ {t + j} ^ {f} \otimes \mathbf {x} _ {t + j} ^ {f} \otimes \mathbf {x} _ {t + j} ^ {f} \right] \\ = e t h e c h a c k b i t s t h e a c o n v e r i n p e r i o d t + 1. \end{array}\]

as the shock hits the economy in period

A recursive version:

\[\begin{array} { r l }&{ \text {A recursive version:} }\\&{ E _ { t } \left[ \tilde { \mathbf { x } } _ { t + 1 } ^ { r d } - \mathbf { x } _ { t + 1 } ^ { r d } \right] = 0 }\\&{ E _ { t } \left[ \tilde { \mathbf { x } } _ { t + 2 } ^ { r d } - \mathbf { x } _ { t + 2 } ^ { r d } \right] = \sum _ { j = 1 } ^ { 1 } \mathbf { h } _ { \mathbf { x } } ^ { 1 - j } \left( \mathbf { H } _ { \mathbf { x x } } E _ { t } \left[ \tilde { \mathbf { x } } _ { t + j } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + j } ^ { s } - \mathbf { x } _ { t + j } ^ { f } \otimes \mathbf { x } _ { t + j } ^ { s } \right] + \frac { 3 } { 6 } \mathbf { h } _ { \sigma \sigma \mathbf { x } } \sigma ^ { 2 } E _ { t } \left[ \tilde { \mathbf { x } } _ { t + j } ^ { f } - \mathbf { x } _ { t + j } ^ { f } \right]\right) }\\&{\qquad + \sum _ { j = 1 } ^ { 1 } \mathbf { h } _ { \mathbf { x } } ^ { 1 - j } \frac { 1 } { 6 } \mathbf { H } _ { \mathbf { x x x } } E _ { t } \left[ \tilde { \mathbf { x } } _ { t + j } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + j } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + j } ^ { f } - \mathbf { x } _ { t + j } ^ { f } \otimes \mathbf { x } _ { t + j } ^ { f } \otimes \mathbf { x } _ { t + j } ^ { f } \right]}\\&{\qquad = \mathbf { H } _ { \mathbf { x x } } E _ { t } \left[ \tilde { \mathbf { x } } _ { t + 1 } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + 1 } ^ { s } - \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { s } \right] + \frac { 3 } { 6 } \mathbf { h } _ { \sigma \sigma \mathbf { x } } \sigma ^ { 2 } E _ { t } \left[ \tilde { \mathbf { x } } _ { t + 1 } ^ { f } - \mathbf { x } _ { t + 1 } ^ { f } \right]}\\&{\qquad + \frac { 1 } { 6 } \mathbf { H } _ { \mathbf { x x x } } E _ { t } \left[ \tilde { \mathbf { x } } _ { t + 1 } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + 1 } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + 1 } ^ { f } - \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } \right]}\\& E _ { t } \left[ \tilde { \mathbf { x } } _ { t + 3 } ^ { r d } - \mathbf { x } _ { t + 3 } ^ { r d } \right] = \sum _ { j = 1 } ^ { 2 } \mathbf { h _ { x }} ^ { 2 - j } \left( \mathbf { H _ { x x }} E _ { t } \left[ \tilde { \mathbf { x } } _ { t + j } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + j } ^ { s} - \mathbf { x} _ { t + j } ^ { f} \otimes \mathbf { x} _ { t + j} ^ { s} \right] + \frac { 3 } { 6 } \mathbf { h _ { \sigma \sigma x }} \sigma ^ { 2 } E _ { t } \left[ \tilde { \mathbf { x } } _ { t + j } ^ { f } - \mathbf { x} _ { t + j } ^ { f} \right]\right)\\&{\qquad + \sum _ { j = 1 } ^ { 2 } \mathbf { h _ { x }} ^ { 2 - j } \frac 1 6 \mathbf H _ { \mathbf x x x} E _ { t } \left[ \tilde { \mathbf {\Delta x}} _ { t + j} ^ { f} \otimes \tilde { \mathbf {\Delta x}} _ { t + j} ^ { f} \otimes \tilde { \mathbf {\Delta x}} _ { t + j} ^ { f} - \mathbf {\Delta x} _ { t + j} ^ { f} \otimes \mathbf {\Delta x} _ { t + j} ^ { f} \otimes \mathbf {\Delta x} _ { t + j} ^ { f} \right]}\\&\qquad = \mathbf h _ { x }\left( \mathbf H _ { x x }\right) E _ { t }\left[ \tilde {\mathbf {\Delta x}} _{ t + 1} ^{ f }\otimes\tilde {\mathbf {\Delta x}} _{ t + 1} ^{ s}-\mathbf {\Delta x}_{t+ 1}^{f}\otimes\mathbf {\Delta x}_{t+ 1}^{s}\right] + \frac 3 6\mathbf h _ { \sigma o x }\sigma ^{ 2 }\boldsymbol E_{t}\left[ \right. \tilde {\mathbf {\Delta x}} _{ t + 1} ^{ f}-\mathbf {\Delta x} _{ t + 1} ^{ f}\left. \right)\left. \right)\\&{\qquad + H _ {\mathbf x}\boldsymbol E_{t}\left[ \tilde {\mathbf {\Delta x}} _{ t + 2} ^{f}\otimes\tilde {\mathbf {\Delta x}} _{ t + 2} ^{s}-\mathbf {\Delta x} _{ t + 2} ^{f}\otimes\mathbf {\Delta x} _{ t + 2} ^{s}\right] +\frac 3 6\mathbf h _ {\sigma o x }\sigma ^{ 2 }\boldsymbol E_{t}\left[ \tilde {\mathbf {\Delta x}} _{ t + 2} ^{f}-\mathbf {\Delta x} _{ t + 2} ^{f}\right]}\\&\qquad + h _ {\mathrm{x}}\frac 1 6 H _ {\mathrm{xxx}} E _{t}\left[ \tilde {\mathbf {\Delta x}} _{ t + 1} ^{ f }\otimes\tilde {\mathbf {\Delta x}} _{ t + 1} ^{ f }\otimes\tilde {\mathbf {\Delta x}} _{ t + 1} ^{ f}-\mathbf {\Delta x} _{ t + 1} ^{ f }\otimes\mathbf {\Delta x} _{ t + 1} ^{ f }\otimes\mathbf {\Delta x} _{ t + 1} ^{ f }\right] .\end{array}\]

\[\begin{array}{r l} & + \frac {1}{6} \mathbf {H} _ {\mathbf {x x x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + 2} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + 2} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + 2} ^ {f} - \mathbf {x} _ {t + 2} ^ {f} \otimes \mathbf {x} _ {t + 2} ^ {f} \otimes \mathbf {x} _ {t + 2} ^ {f} \right] \\ & = \mathbf {h} _ {\mathbf {x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + 2} ^ {r d} - \mathbf {x} _ {t + 2} ^ {r d} \right] \\ & + \mathbf {H} _ {\mathbf {x x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + 2} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + 2} ^ {s} - \mathbf {x} _ {t + 2} ^ {f} \otimes \mathbf {x} _ {t + 2} ^ {s} \right] + \frac {3}{6} \mathbf {h} _ {\sigma \sigma \mathbf {x}} \sigma^ {2} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + 2} ^ {f} - \mathbf {x} _ {t + 2} ^ {f} \right] \\ & + \frac {1}{6} \mathbf {H} _ {\mathbf {x x x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + 2} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + 2} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + 2} ^ {f} - {\mathbf {x}} _ {t + 2} ^ {f} \otimes {\mathbf {x}} _ {t + 2} ^ {f} \otimes {\mathbf {x}} _ {t + 2} ^ {f} \right] \end{array}\]

So in general

\[\begin{array}{r l} & E _ {t} \left[ \tilde {\mathbf {x}} _ {t + k} ^ {r d} - \mathbf {x} _ {t + k} ^ {r d} \right] = \mathbf {h} _ {\mathbf {x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + k - 1} ^ {r d} - \mathbf {x} _ {t + k - 1} ^ {r d} \right] + \mathbf {H} _ {\mathbf {x x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + k - 1} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + k - 1} ^ {s} - \mathbf {x} _ {t + k - 1} ^ {f} \otimes \mathbf {x} _ {t + k - 1} ^ {s} \right] \\ & \qquad + \frac {3}{6} \mathbf {h} _ {\sigma \sigma \mathbf {x}} \sigma^ {2} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + k - 1} ^ {f} - \mathbf {x} _ {t + k - 1} ^ {f} \right] \\ & \qquad + \frac {1}{6} \mathbf {H} _ {\mathbf {x x x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + k - 1} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + k - 1} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + k - 1} ^ {f} - \mathbf {x} _ {t + k - 1} ^ {f} \otimes \mathbf {x} _ {t + k - 1} ^ {f} \otimes \mathbf {x} _ {t + k - 1} ^ {f} \right] \end{array}\]

Thus, we know . So we only need to compute

and . This is done in the next two subsections. For these derivations recall that we define such that:

\[\begin{array}{r l} \boldsymbol {\delta} _ {t + j} = \boldsymbol {\nu} & \text {for} j = 1 \\ \boldsymbol {\delta} _ {t + j} = \boldsymbol {\epsilon} _ {t + j} & \text {for} j \neq 1 \end{array}\]

11.3.1 For

Consider:

\[\begin{array} { l } \mathbf { x } _ { t + l } ^ { f } \otimes \mathbf { x } _ { t + l } ^ { f } \otimes \mathbf { x } _ { t + l } ^ { f } = \left( \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } + \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } \right) \otimes \left( \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } + \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } \right) \otimes \\ = ( \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } + \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } + \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } + \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } ) \\ \otimes \left( \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } + \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } \right) \\ = ( \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } + \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ {\mathrm{t}} ^ { f } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x }} ^ { l - j } \sigma \boldsymbol { \eta }\boldsymbol { \epsilon}_{ t + j} + \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x }} ^ { l - j }\sigma\boldsymbol{ \eta }\boldsymbol{ \epsilon}_{ t + j}\otimes\mathbf{h}_{\mathbf{x}}^{l}\mathbf{x}_{t}^{f}+\sum _{j = 1} ^{l }\mathbf{h}_{\mathbf{x}}^{l - j}\sigma\boldsymbol{ \eta }\boldsymbol{ \epsilon}_{ t + j}\otimes\sum _{j = 1} ^{l }\mathbf{h}_{\mathbf{x}}^{l - j}\sigma\boldsymbol{ \eta }\boldsymbol{ \epsilon}_{ t + j})\otimes\mathbf{h}_{\mathbf{x}}^{l}\mathbf{x}_{t}^{f}\\ + ( \mathbf{ h}_{\mathbf{x}}^{l }\mathbf{x}_{t}^{f}\otimes\mathbf{h}_{\mathbf{x}}^{l }\mathbf{x}_{t}^{f}+ \mathbf{h}_{\mathbf{x}}^{l }\mathbf{x}_{t}^{f}\otimes\sum _{j = 1} ^{l }\mathbf{h}_{\mathbf{x}}^{l - j}\sigma\boldsymbol{ \eta }\boldsymbol{ \epsilon}_{ t + j}+\sum _{j = 1} ^{l }\mathbf{h}_{\mathbf{x}}^{l - j}\sigma\boldsymbol{ \eta }\boldsymbol{ \epsilon}_{ t + j}\otimes\mathbf{h}_{\mathbf{x}}^{l }\mathbf{x}_{t}^{f}+\sum _{j = 1} ^{l }\mathbf{h}_{\mathbf{x}}^{l - j}\sigma\boldsymbol{ \eta }\boldsymbol{ \epsilon}_{ t + j}\otimes\sum _{j = 1} ^{l }\mathbf{h}_{\mathbf{x}}^{l - j}\sigma\boldsymbol{ \eta }\boldsymbol{ \epsilon}_{ t + j})\otimes\sum _{j = 1} ^{l }\mathbf{h}_{\mathbf{x}}^{l - j}\sigma\boldsymbol{ \eta }\boldsymbol{ \epsilon}_{ t + j}\\ = \\ = \\ [ ] = \\ [ ] = \\ [ ] = \\ [ ] = \\ [ ] = \\ [ ] = \\ [ ] = \\ [ ] = \\ [ ] = \\ [ ] = \\ [ ] = \\ [ ] = \\ [ ] = \\ [ ] = \\ [ ] = \\ [ ] = \\ [ ] = \\ [ ] = \\ [ ] = \\ [ ] = \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ], \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]). \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ . \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ]. \\ [ ] .\\\]

And we therefore have

\[\tilde {\mathbf {x}} _ {t + l} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + l} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + l} ^ {f} = \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \delta_ {t + j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f}\]

\[\begin{array}{r l} & {+ \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\delta} _ {t + j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\delta} _ {t + j} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\delta} _ {t + j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f}} \\ & {+ \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\delta} _ {t + j} + \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\delta} _ {t + j} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\δ} _ {t + j}} \\ & {+ \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\delta} _ {t + j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\delta} _ {t + j} + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\delta} _ {t + j} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmod {\pmod {\delta}},} \end{array}\]

\[\begin{array} { r l } & E _ { t } \left[ \tilde { \mathbf { x } } _ { t + l } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + l } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + l } ^ { f } - \mathbf { x } _ { t + l } ^ { f } \otimes \mathbf { x } _ { t + l } ^ { f } \otimes \mathbf { x } _ { t + l } ^ { f } \right] \\ & { = E _ { t } [ \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } + \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta _ { t + j } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } } \\ & { + \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta _ { t + j } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } + \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta _ { t + j } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta _ { t + j } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } } \\ & { + \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta _ { t + j } + \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta _ { t + j } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta_ { t + j }} \\ & { + \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta _ { t + j } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta _ { t + j } + \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j} \sigma \boldsymbol { \eta} \delta _ { t + j } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j} \sigma \boldsymbol { \eta} \delta _ { t + j }} \\ & \\ & - E _ { t } [ \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x} _ t f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f w e r d i s e n d i s e n d i s e n d i s e n d i s e n d i s e n d i s e n d i s e n d i s e n d i s e n d i s e n d i s e n d i s e n d i s e n d i s e n d i s e n d i s e n d i s e n d i s e n d i s e n d i s e m a g e r e d i s e n d i s e n d i s e n d i s e n d i s e n d i s e n d i s e n d i s e n d i s e n d i s e n d i s e n d i s e n d i s e n d i s e n d i s e n d i s e n d i s e n d i s e n d i s e n d i s e w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w o w y , \\ & \\ & + E _ {\mathrm{t}} [ 0 + H _ {\mathrm{x}} ^ {\prime} X ] (0 + H _ {\mathrm{x}}) (0 + H _ {\mathrm{x}}) (0 + H _ {\mathrm{x}}) (0 + H _ {\mathrm{x}}) (0 + H _ {\mathrm{x}}) (0 + H _ {\mathrm{x}}) (0 + H _ {\mathrm{x}}) (0 + H _ {\mathrm{x}}) (0 + H _ {\mathrm{x}}) (0 + H _ {\mathrm{x}}) (0 + H _ {\mathrm{x}} ) (0 + H _ {\mathrm{x}}) (0 + H _ {\mathrm{x}}) (0 + H _ {\mathrm{x}}) (0 + H _ {\mathrm{x}}) (0 + H _ {\mathrm{x}}) (0 + H _ {\mathrm{x}}) (0 + H _ {\mathrm{x}}) (0 + H _ {\mathrm{x}}) (0 + H _ {\mathrm{x}}) (0 + H _ {\mathrm{x}}), \\ & \\ & + E _ {\mathrm{t}} [ 0 + H _ {\mathrm{x}} ^ {\prime} X ] (0 + H _ {\mathrm{x}}) (0 + H _ {\mathrm{x}}) (0 + H _ {\mathrm{x}}) (0 + H _ {\mathrm{x}}) (0 + H _ {\mathrm{x}}) (0 + H _ {\mathrm{x}}) (0 + H _ {\mathrm{x}}) ( 0 + H _ {\mathrm{x}}) (0 + H _ {\mathrm{x}}) (0 + H _ {\mathrm{x}}) (0 + H _ {\mathrm{x}}) ( 0 + H _ {\mathrm{x}} ) ( 0 + H _ {\mathrm{x}} ) ( 0 + H _ {\mathrm{x}} ) ( 0 + H _ {\mathrm{x}} ) ( 0 + H _ {\mathrm{x}} ) ( 0 + H _ {\mathrm{x}} ) ( 0 + H _ {\mathrm{x}} ) ( 0 + H _ {\mathrm{x}} ) ( 0 + H _ {\mathrm{x}} ) ( 0 + H _ {\mathrm{x}} ) . \\ & \\ & + E _ {\mathrm{t}} [ 0 + H _ {\mathrm{x}} ^ {\prime} X ] (0 + H _ {\mathrm{x}}) (0 + H _ {\mathrm{x}}) (0 + H _ {\mathrm{x}}) (0 + H _ {\mathrm{x}}) (0 + H _ {\mathrm{x}}) ( 0 + H _ {\mathrm{x}}) ( 0 + H _ {\mathrm{x}} ) ( 0 + H _ {\mathrm{x}} ) ( 0 + H _ {\mathrm{x}} ) ( 0 + H _ {\mathrm{x}} ) ( 0 + H _ {\mathrm{x}} ) ( 0 + H _ {\mathrm{x}} ) ( 0 + H _ {\mathrm{x}} ) ( 0 + H _ {\mathrm{x}} ) ( 1 ) , \\ & \\ & + E _ {\mathrm{t}} [ 0 + H _ {\mathrm{x}} ^ {\prime} X ] (0 + H _ {\mathrm{x}}) (0 + H _ {\mathrm{x}}) (0 + H _ {\mathrm{x}}) (0 + H _ {\mathrm{x}}) ( 0 + H _ {\mathrm{x}}) ( 0 + H _ {\mathrm{x}} ) ( 1 ) , \\ & \\ & - E _ {\mathrm{t}} [ 1 + 1 ] (\texttt * * O b a c k a b a c k a b a c k a b a c k a b a c k a b a c k a b a c k a b a c k a b a c k a b a c k a b a c k a b a c k a b a c k a b a c k a b a c k a b a c k a b a c k a b a c k a b a c k a b a c k a b a c k A B A C A B A C A B A C A B A C A B A C A B A C A B A C A B A C A B A C A B A C A B A C A B A C A B A C A B A C A B A C A B A C A B A C A B A C A B A C A B A C A B A C A B A C A B A C A B A C A B A C A B A c a b a c k a b a c k a b a c k a b a c k a b a c k a b a c k a b a c k a b a c k a b a c k a b a c k a b a c k a b a c k a b a c k a b a c k a b a c k a b a c k a b a c k a b a c k a b a c k a b a c , \\ & \\ & - E [ 1 ] (\texttt * * O b a c k a b a c k A B A C A B A C A B A C A B A C A B A C A B A C A B A C A B A C A B A C A B A C A B A C A B A C A B A C A B A C A B A C A B A C A B A C A B A C A B A C A B A C A B A C A B A C A B I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N DI N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I N D I U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V V v u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v . \\ & \\ & - E [ 1 ] (\texttt * * O b a c k a b a c k a b a c k a b a c k a b a c k a b a c k a b a c k a b a c k a b a c k a b a c k a b a c k a b a c k a b a c k a b a c k a b a c k a b a c k a b a c k a b a c k a c k a b a c k a b a c k a b a c k a b a c k a b a c k a b a c k a b a c k a b a c k K S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S,S \\ & \\ & - E [ 1 ] (\texttt * * O b a c k a b a c k K S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S-S \\ & \\ & - E [ 1 ] (\texttt * * O b a c k K S S ) (\texttt * * O b a c k K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K KK M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m mmumommumommumommumommumommumommumommumommumommumommumommumommumommumommumommumommumommumommumommumommumommumommumommumommumommumommumommumommumommumommumommumommumommumomnunlunlunlunlunlunlunlunlunlunlunlunlunlunlunlunlunlunlunlunlunlunlunlunlunlunlunlunlunlunlunlunlunlunlunlunlunlunlunlunlunlunlunlunlunlunlunlunlunlunlunrulnulnulnulnulnulnulnulnulnulnulnulnulnulnulnulnulnulnulnulnulnulnulnulnulnulnulnulnulnulnulnulnulnulnulnulnulnulnulnulnulnulnulnulnulnulnulnulnulnulnulsulteal, and then the term of the term of the term of the term of the term of the term of the term of the term of the term of the term of the term of the term of the term of the term of the term of the term of the term of the term of the term of the term of the term of the term of the term of the term of the term of the term of the term of the term of the term of the term of the term of the term of the term of the term of that region of the region of the region of the region of the region of the region of the region of the region of the region of the region of the region of the region of the region of the region of the region of the region of the region of the region of the region of the region of the region of the region of the region of the region of the region of the region of the region of the region of the region of the region of the region of the region of the region of the region of the regions of the regions of the regions of the regions of the regions of the regions of the regions of the regions of the regions of the regions of the regions of the regions of the regions of the regions of the regions of the regions of the regions of the regions of the regions of the regions of the regions of the regions of the regions of the regions of the regions of the regions of the regions of the regions of the regions of the regions of the regions of the regions of the regions of the regions and in this case, and then there is no value for any single point in this case. The values are estimated based on an estimate from each other to which there is no value for any single point in this case. The values are estimated based on an estimate from each other to which there is no value for any single point in this case. The values are estimated based on an estimate from each other to which there is no value for any single point in this case. The values are estimated based on an estimate from each other to which there is no value for any single point in this case. The values are estimated based on an estimate from each other to each other to which there is no value for any single point in this case. The values are estimated based on an estimate from each other to which there is no value for any single point in this case. The values are estimated based on an estimate from each other to which there is no value for any single point in this case. The values are estimated based on an estimate from each other to which there is no value for any single point in this case. The values are estimated based on an estimatefrom each other to which there is no value for any single point in this case. The values are estimated based on an estimate from each other to which there is no value for any single point in this case. The values are estimated based on an estimate from each other to which there is no value for any single point in this case. The values are estimated based on an estimate from each other to which there is no value for any single point in this case. The values are estimated based on an estimateFrom each other to which there is no value for any single point in this case. The values are estimated based on an estimate from each other to which there is no value for any single point in this case. The values are estimated based on an estimate from each other to which there is no value for any single point in this case. The values are estimated based on an estimate from each other to which there is no value for any single point in this case. The values are estimated based on an estimate From each other to which there is no value for any single point in this case. The values are estimated based on an estimate from each other to which there is no value for any single point in this case. The values are estimated based on an estimate from each other to which there is no value for any single point in this case. The values are estimated based on an estimate from each other to which there is no value for any single point in this case. The values are estimated based on an estimate Infor All States, and then there is no value for any single point in this case. The values are estimated based on an estimate from each other to which there is no value for any single point in this case. The values are estimated based on an estimate from each other to which there is no value for any single point in this case. The values are estimated based on an estimate from each other to which there is no value for any single point in this case. The values are estimated based on an estimate from Each Other's Note-4, and then there is no value for any single point in this case. The values are estimated based on an estimate from each other to which there is no value for any single point in this case. The values are estimated based on an estimate from each other to which there is no value for any single point in this case. The values are estimated based on an estimate from Each Other's Note-4, and then there is no value for any single point in this case. The values are estimated based on an outcome from each other to which there is no value for any single point in this case. The values are estimated based on an outcome from each other to which there is no value for any single point in this case. The values are estimated based on an outcome from each other to which there is no value for any single point in this case. The values are estimated based on an outcome from each other to which there is no value for any single point in this case. The values are estimated based on an outcome from each other towhich one or two points between them, and then there is no value for any single point between them, and then there is no value for any single point between them, and then there is no value for any single point between them, and then there is no value for any single point between them, and then there is no value for any single point between them, and then there is no value for any single point between them, and then there is no value for any single point between them, and then there is no value for any single point between them, and then there is novalue for all states, and then there is no value for all states, and then there is no value for all states, and then there is no value for all states, and then there is no value for all states, and then there is no value for all states, and then there is no value for all states, and then there is no value for all states, and then there is no value for all states, and then there is no value for all states, and then there is no value for all states, and then there is novalue for all states, and then there is no value for all states, and then there is no value for all states, and then there is no value for all states, and then there is no value for all states, and then there is no value for all states, and then there is no value for all states, and then there is no value for all states, and then there is no value for all states, and then there is no value for all States, and then there is no value for all States, and then there is no value for all States, and then there is no value for all States, and then there is no value for all States, and then there is no value for all States, and then there is no value for all States, and then there is no value for all States, and then there is no value for all States, and then there is no value for all States, and then Thereinenceof All States, and then there is no value for any single point between\]

\[= E _ {t} [ \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \pmb {\eta} \pmb {\delta} _ {t + 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \pmb {\eta} \pmb {\delta} _ {t + 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \pmb {\eta} \pmb {\delta} _ {t + 1}\]

\[\begin{array} { r l } & { + \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta _ { t + j } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta _ { t + j } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } } \\ & { + \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta _ { t + j } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta_ { t + j } } \\ & { + \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta _ { t + j } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta _ { t + j } } \\ & { + \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta _ { t + j } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta} \delta _ { t + j } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta _ { t + j } } \\ & { - E _ { t } [ \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \epsilon _ { t + j } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \epsilon _ { t + j } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } } \\ & + \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \sum _ { j = 1 } ^ { l } \mathbf {\Delta h} _ \mathbf {\Delta x} ) i n g a d e s s o u p h a b e d i s s o u p h a b e d i s s o u p h a b e d i s s o u p h a b e d i s s o u p h a b e d i s s o u p h a b e d i s s o u p h a b e d i s s o u p h a b e d i s s o u p h a b e d i s s o u p h a b e d i i n g , m a x , m a y , m a z , m a x , m a y , m a z , m a x , m a y , m a z , m a x , m a y , m a z , m a x , m a y , m a z , m a x , m a y , m a z , m a x , m a y , m a z , m a x , m a y , m a z , m a x , m a y , m a z , m a y , m a z , m a x , m a y , m a z , m a x , m a y , m a z , m a x , m a y , m a z , m a x , m a y , m a z , m a x , m a y , m a z , m a x , m a y , m a z , m a x , m a y , m a z , m a x , m ay , m a z , m a x , m a y , m a z , m a x , m a y , m a z , m a x , m a y , m a z , m a x , m a y , m a z , m a x , m a y , m a z , m a x , m a y , m a z , m a x , m a y , m a z , m a x , m a y , m az , m a x , m a y , m a z , m a x , m a y , m a z , m a x , m a y , m a z , m a x , m a y , m a z , m a x , m a y , m a z , m a x , m a y , m a z , m a x , m a y , m a z , m a x , m a y , m a z , m ax , m a y , m a z , m a x , m a y , m a z , m a x , m a y , m a z , m a x , m a y , m a z , m a x , m a y , m a z , m a x , m a y , m a z , m a x , m a y , m a z , m a x , m a y , m a z , m a x , m a_y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . : M. A. c. c. c. c. c. c. c. c. c. c. c. c. c. c. c. c. c. c. c. c. c. c. c. c. c. c. c. c. c. c. c. c. c. c. c. c. c. c. c. c. c. c. c. c. c. c. c. c. c. c. (c) ; M. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. (c) ; M.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.C.A.M.A.M.A.M.A.M.A.M.A.M.A.M.A.M.A.M.A.M.A.M.A.M.A.M.A.M.A.M.A.M.A.M.A.M.A.M.A.M.A.M.A.M.A.M.A.M.A.M.A.M.A.M.A.M.A.M.A.M.A.M.A.M.A.M.A.M.A.M.A.M.A.M.A.M.A.M.A.M.A.M.A.M.A.M.A.M.A.M.A.M.A.M.A.M.A.M.A.M.A,M.A.M.A,M.A.M.A,M.A.M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M,A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M.A,M,A,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N,N-N,\mathrm{and}N,\mathrm{and}N,\mathrm{and}N,\mathrm{and}N,\mathrm{and}N,\mathrm{and}N,\mathrm{and}N,\mathrm{and}N,\mathrm{and}N,\mathrm{and}N,\mathrm{and}N,\mathrm{and}N,\mathrm{and}N,\mathrm{and}N,\mathrm{and}N,\mathrm{and}N,\mathrm{and}N,\mathrm{ and}N,\mathrm{and}N,\mathrm{and}N,\mathrm{and}N,\mathrm{and}N,\mathrm{and}N,\mathrm{and}N,\mathrm{and}N,\mathrm{and}N,\mathrm{and}N,\mathrm{and}N,\mathrm{and}N,\mathrm{and}N,\mathrm{and}N,\mathrm{and}N,\mathrm{and}N,\mathrm{and}N,\end{array}\]

We now evaluate the expressions on each of the four last lines:

\[\begin{array} { r l } & A _ { 1 } \equiv E _ { t } \left[ \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta _ { t + j } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta _ { t + j } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } - \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \epsilon _ { t + j } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \epsilon _ { t + j } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \right] \\ & = E _ { t } [ \left( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \delta _ { t + 1 } + \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta _ { t + j } \right) \otimes \left( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \delta _ { t + 1 } + \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta _ { t + j } \right) \times \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \\ & - ( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \epsilon _ { t + 1 } + \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \epsilon _ { t + j } ) \otimes ( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \epsilon _ { t + 1 } + \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \epsilon _ { t + j } ) \otimes ( \texttt {h} _ { \mathbf { x } } ^ { l } \texttt {x} _ { t } ^ { f ] } \\ & = E _ { t } [ \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \delta _ { t + 1 } \otimes ( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \delta _ { t + 1 } + \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta _ { t + j } ) \otimes ( \texttt {h} _ { \mathbf { x } } ^ { l } ) [ \\ & + \sum _ { j = 2 } ^ { l } [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ ] , i n d ] , i n d ] , i n d ] , i n d ] , i n d ] , i n d ] , i n d ] , i n d ] , i n d ] , i n d ] , i n d ] , i n d ] , i n d ] , i n d ] , i n d ] , i n d ] , i n d ] , i n d ] , i n d ] , \\ & - ( [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ ] , i n d ] , i n d ] , i n d ] , i n d ] , i n d ] , i n d ] , i n d ] , i n d ] , i n d ] , i n d ] , i n d ] , i n d ] , i n d ] , i n d ] , i n d ] , i n d ] , \\ & - ( | | | ) ( | | | ) ( | | | ) ( | | | ) ( | | | ) ( | | | ) ( | | | ) ( | | | ) ( | | | ) ( | | | ) ( | | | ) ( | | | ) ( | | | ) ( | | | ) ( | | | ) ( | | | ) ( | | | ) ( | | | ) ( | | | ) ( | | | ) ( | | | ). \\ & = E _ { t - 1} (\mathbf {\Sigma}) . \\ & = E _ {\mathrm{d}} (\mathbf {\Sigma}) . \\ & = E _ {\mathrm{d}} (\mathbf {\Sigma}) . \\ & = E _ {\mathrm{d}} (\mathbf {\Sigma}) . \\ & = E _ {\mathrm{d}} (\mathbf {\Sigma}) . \\ & = E _ {\mathrm{d}} (\mathbf {\Sigma}) . \\ & = E _ {\mathrm{d}} (\mathbf {\Sigma}) . \\ & = E _ {\mathrm{s}} (\mathbf {\Sigma}) . \\ & = E _ {\mathrm{s}} (\mathbf {\Sigma}) . \\ & = E _ {\mathrm{s}} (\mathbf {\Sigma}) . \\ & = E _ {\mathrm{s}} (\mathbf {\Sigma}) . \\ & = E _ {\mathrm{s}} (\mathbf {\Sigma}) . \\ & = E _ {\mathrm{s}} (\mathbf {\Sigma}) . \\ & = E _ {\mathrm{s}} (\mathcal {\Omega}) . \\ & = E _ {\mathrm{s}} (\mathcal {\Omega}) . \\ & = E _ {\mathrm{s}} (\mathcal {\Omega}) . \\ & = E _ {\mathrm{s}} (\mathcal {\Omega}) . \\ & = E _ {\mathrm{s}} (\mathcal {\Omega}) . \\ & = E _ {\mathrm{s}} (\mathcal {\Omega}) . \\ & = E _ {\mathrm{s}} (\mathcal {\Omega}) . \\ * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * . \\ & = E _ {\mathrm{s}} (\mathcal {\Omega}) . \\ & = E _ {\mathrm{s}} (\mathcal {\Omega}) . \\ & = E _ {\mathrm{s}} (\mathcal {\Omega}) . \\ & = E _ {\mathrm{s}} (\mathcal {\Omega}) . \\ & = E _ {\mathrm{s}} (\mathcal {\Omega}) . \\ & = E _ {\mathrm{s}} (\mathcal {\Omega}) . \\ & > E _ {\mathrm{s}} (\mathcal {\Omega}) . \\ & > E _ {\mathrm{s}} (\mathcal {\Omega}) . \\ & > E _ {\mathrm{s}} (\mathcal {\Omega}) . \\ & > E _ {\mathrm{s}} (\mathcal {\Omega}) . \\ & > E _ {\mathrm{s}} (\mathcal {\Omega}) . \\ & > E _ {\mathrm{s}} (\mathcal {\Omega}) . \\ & > E _ \texttt {} ^ {[ 3 ]} ^ {- 1} ; ^ {[ 3 ]} ; ^ {[ 3 ]} ; ^ {[ 3 ]} ; ^ {[ 3 ]} ; ^ {[ 3 ]} ; ^ {[ 3 ]} ; ^ {[ 3 ]} ; ^ {[ 3 ]} ; ^ {[ 3 ]} ; ^ {[ 3 ]} ; ^ {[ 3 ]} ; ^ {[ 3 ]} ; ^ {[ 3 ]} ; ^ {[ 3 ]} ; ^ {[ 3 ]} , \\ & > E _ \texttt {} ^ {[ 3 ]} ^ {- 1} ; ^ {[ 3 ]} ; ^ {[ 3 ]} ; ^ {[ 3 ]} ; ^ {[ 3 ]} ; ^ {[ 3 ]} ; ^ {[ 3 ]} ; ^ {[ 3 ]} ; ^ {[ 3 ]} ; ^ {[ 3 ]} ; ^ {[ 3 ]} ; ^ {[ 3 ]} ; ^[[ 3 ] ; ^[[ 3 ] ; ^[[ 3 ] ; ^[[ 3 ] ; ^[[ 3 ] ; ^[[ 3 ] ; ^[[ 3 ] ; ^[[ 3 ] ; ^[[ 3 ] ; ^[[ 3 ] ; ^[[ 3 ] ; ^[[ 3 ] ; ^[[ 3 ] ; ^[[ 3 ] ; ^[[ 3 ] ; ^[[ 3 ] ; ^[[ 3 ] ; ^[[\]

\[\begin{array}{l} = E _ {t} [ \left(\mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\delta} _ {t + 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\delta} _ {t + 1} + \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\delta} _ {t + 1} \otimes \sum_ {j = 2} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\delta} _ {t + j}\right) \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \\ \quad + \left(\sum_ {j = 2} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\delta} _ {t + j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\delta} _ {t + 1} + \sum_ {j = 2} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\delta} _ {t + j} \otimes \sum_ {j = 2} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\delta} _ {t + j}\right) \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \\ \quad - \left(\mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} + \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} \otimes \sum_ {j = 2} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j}\right) \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \\ \quad - \left(\sum_ {j = 2} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} + \sum_ {j = 2} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} \otimes \sum_ {j = 2} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j}\right) \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f ]}. \end{array}\]

\[\mathrm{Note} A _ {1} \equiv E _ {t} \left[ \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\delta} _ {t + j} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\delta} _ {t + j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} - \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\epsilon} _ {t + j} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\epsilon} _ {t + j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \right]\]

Therefore we immediately see from the structure of the terms that

\[\begin{array}{l} A _ {2} \equiv \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\delta} _ {t + j} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\delta} _ {t + j} - \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} \\ = E _ {t} [ \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} - \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} ] \\ = E _ {t} [ \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes (\mathbf {h} _ {\mathbf {x}} ^ {l - 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1}) (\sigma \boldsymbol {\eta} \boldsymbol {\nu} \otimes \sigma \boldsymbol {\eta} \boldsymbol {\nu}) - \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes (\mathbf {h} _ {\mathbf {x}} ^ {l - 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1}) (\sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} \otimes \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1}) ] \\ = \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes (\mathbf {h} _ {\mathbf {x}} ^ {l - 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1}) ((\sigma \boldsymbol {\eta} \boldsymbol {\nu} \otimes \sigma \boldsymbol {\eta} \boldsymbol {\nu}) - \boldsymbol {\Lambda}) \end{array}\]

\[\begin{array}{r l} & {\mathrm{Forthethirdterm:}} \\ & {A _ {3} \equiv \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\delta} _ {t + j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\delta} _ {t + j} - \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\epsilon} _ {t + j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\epsilon} _ {t + j}} \\ & {\qquad = E _ {t} [ \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \pmb {\eta} \pmb {\nu} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \pmb {\eta} \pmb {\nu} - \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1} ]} \\ & {\qquad = \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \pmb {\eta} \pmb {\nu} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \pmb {\eta} {\pmb {\nu}} - {\boldsymbol {\Gamma}} (l)} \end{array}\]

The only element which is not directly computable is which must be computed element by element. Hence, consider

\[\begin{array}{l}\boldsymbol {\Gamma} (l) \equiv E _ {t} \left[ \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} \otimes \left\{\mathbf {h} _ {\mathbf {x}} ^ {l} (\gamma_ {2},:) \mathbf {x} _ {t} ^ {f} \times \left\{\mathbf {h} _ {\mathbf {x}} ^ {l - j} (\gamma_ {1},:) \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} \right\} _ {\gamma_ {1} = 1} ^ {n _ {x}} \right\} _ {\gamma_ {2} = 1} ^ {n _ {x}} \right]\\= E _ {t} \left\{\mathbf {h} _ {\mathbf {x}} ^ {l - j} (\gamma_ {3,:}) \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} \times \left\{\mathbf {h} _ {\mathbf {x}} ^ {l} (\gamma_ {2},:) \mathbf {x} _ {t} ^ {f} \times \left\{\mathbf {h} _ {\mathbf {x}} ^ {l - j} (\gamma_ {1},:) \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} \right\} _ {\beta_ {1} = 1} ^ {n _ {x}} \right\} _ {\gamma_ {2} = 1} ^ {n _ {x}} \right\} _ {\gamma_ {3} = 1} ^ {n _ {x}}\\= E _ {t} \left\{\mathbf {h} _ {\mathbf {x}} ^ {l - j} (\gamma_ {3,:}) \sigma \sum_ {\phi_ {2} = 1} ^ {n _ {e}} \boldsymbol {\eta} (:, \phi_ {2}) \boldsymbol {\epsilon} _ {t + 1} (\phi_ {2}, 1) \times \left\{\mathbf {h} _ {\mathbf {x}} ^ {l} (\gamma_ {2},:) \mathbf {x} _ {t} ^ {f} \times \left\{\mathbf {h} _ {\mathbf {x}} ^ {l - j} (\gamma_ {1},:) \sigma \sum_ {\phi_ {1} = 1} ^ {n _ {e}} \boldsymbol {\eta} (:, \phi_ {1}) \boldsymbol {\epsilon} _ {t + 1} (\phi_ {1}, 1) \right\} _ {\gamma_ {1} = 1} ^ {n _ {x}} \right\} _ {\gamma_ {2} = 1} ^ {n _ {x}} \right\} _ {\gamma_ {3} = 1} ^ {n _ {x}}\\= E _ {t} \left\{\mathbf {h} _ {\mathbf {x}} ^ {l - j} (\gamma_ {3,:}) \sigma \sum_ {\phi_ {2} = 1} ^ {n _ {e}} \sum_ {\phi_ {1} = 1} ^ {n _ {e}} \boldsymbol {\eta} (:, \phi_ {2}) \boldsymbol {\epsilon} _ {t + 1} (\phi_ {2}, 1) \times \left\{\mathbf {h} _ {\mathbf {x}} ^ {l} (\gamma_ {2},:) \mathbf {x} _ {t} ^ {f} \times \left\{\mathbf {h} _ {\mathbf {x}}^{l - j} (\gamma_ {1},:) \sigma \boldsymbol {\eta} (:, \phi_ {1}) \boldsymbol {\epsilon} _ {t + 1} (\phi_ {1}, 1) \right\}_{\gamma_ {1} = 1} ^ {n _ {x}} \right\}_{\gamma_ {2} = 1} ^ {n _ {x}} \right\}_{\gamma_ {3} = 1} ^ {n _ {x}}\\= \left\{ \right.\mathbf {h} _ {\mathbf {x}} ^ {l - j} (\gamma_ {3,:}) \sigma \sum_ {\phi_ {1} = 1} ^ {n _ {e}} \boldsymbol {\eta} (:, \phi_ {1}) \times \left\{\mathbf {h} _ {\mathbf {x}} ^ {l} (\gamma_ {2},:) \mathbf {x} _ {t} ^ {f} \times \left\{\mathbf {h} _ {\mathbf {x}} ^ {l - j} (\gamma_ {1},:) \sigma \boldsymbol {\eta} (:, \phi_ {1}) \right\}_{\gamma_ {1} = 1} ^ {n _ {x}} \right\}_{\gamma_ {2} = 1} ^ {n _ {x}}\\b e c a u s e t h e i n n o v a t i o n s a r e i n d e p d e n d e n t. T h i s e x p r e s s i o n i s d i r e c t l y i m p l e m e n t a b l e.\end{array}\]

\[\begin{array} { r l } & A _ { 4 } = E _ { t } \left[ \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \pmb { \delta } _ { t + j } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \pmb { \delta } _ { t + j } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { 9 } \pmb { \delta } _ { t + j } - \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { 9 } \pmb { \epsilon } _ { t + j } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { 9 } \pmb { \epsilon } _ { t + j } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \pmb { \epsilon } _ { t + j } \right] \\ & = E _ { t } [ ( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { 9 } \pmb { \delta } _ { t + 1 } + \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { 9 } \pmb { \delta } _ { t + j } ) \otimes ( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { 9 } \pmb { \delta } _ { t + 1 } + \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { 9 } \pmb { \delta } _ {t + j } ) \otimes ( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { 9 } \pmb { \delta } _ { t + 1 } + \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { 9 } \pmod {\delta _ { t + j }} ) \\ & - ( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { 9 } \pmb { \epsilon } _ { t + 1 } + \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { 9 } \pmb { \epsilon } _ { t + j } ) \otimes ( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { 9 } \pmb { \epsilon } _ { t + 1 } + \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { 9 } \pmb { \epsilon } _ {t + j } ) \otimes ( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { 9 } \pmb { \epsilon } _ { t + 1 } + \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { 9 } \pmod {\delta _ { t + j }} ] \\ & = E _ { t } [ \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { 9 } \pmb { \delta } _ { t + 1 } \otimes ( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { 9 } \pmb { \delta } _ { t + 1 } + \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { 9 } \pmb { \delta} _ { t + j} ) \otimes ( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { 9 } \pmb { \delta } _ { t + 1 } + \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { 9} \pmb {\delta} _ { t + j} ) \\ & + ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (\mathrm{个}) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ), \\ & = E _ t [ (\mathbf {\Sigma} [ (\mathbf {\Sigma} [ (\mathbf {\Sigma} [ (\mathbf {\Sigma} [ (\mathbf {\Sigma} [ (\mathbf {\Sigma} [ (\mathbf {\Sigma} [ (\mathbf {\Sigma} [ (\mathbf {\Sigma} [ (\mathbf {\Sigma} [ (\mathbf {\Sigma} [ (\mathbf {\Sigma} [ (\mathbf {\Sigma} [ (\mathbf {\Sigma} [ (\mathbf {\Sigma} [ ( | | . | | . | | . | | . | | . | | . | | . | | . | | . | | . | | . | | . | | . | | . | | . | | . | | . | | . | | . | | . | | . | | . | | . | | . | | .) , i n , n , m , n , n , m , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , n , N , N , N , N , N , N , N , N , N , N , N , N , N , N , N , N , N , N , N , N , N , N , N , N , N , N , N , N , N , N , N , N , N , N , N , N , N , N , N , N , N , N , N , N , N , N , N , N , N , N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N ,N, N ; i n a d i o n e f i o n c o m e r s e r e f i o n c o m e r e f i o n c o m e r e f i o n c o m e r e f i o n c o m e r e f i o n c o m e r e f i o n c o m e r e f i o n c o m e r e f i o n c o m e r e f i o n c o m e r e f i o n c o m e r e f i o n c o c o m e r e f i o n c o c o m e r e f i o n c o c o m e r e f i o n c o c o m e r e f i o n c o c o m e r e f i o n c o c o m e r e f i o n c o c o m e r e f i o n c o c o m e r e f i o n c o c o m c o c o m e r e f i o n c o c o m e r e f i o n c o c o m e r e f i o n c o c o m e r e f i o n c o c o m e r e f i o n c o c o m e r e f i o n c o c o m e r e f i o n c o c o m e r e f i o n c o c a p p a b a l y a b a l y a b a l y a b a l y a b a l y a b a l y a b a l y a b a l y a b a l y a b a l y a b a l y a b a l y a b a l y a b a l y a b a l y a b a l y a b a l y a b a l y a b a l y a b a l y a b a l u v i d i g u t h i s w i d i g u t h i s w i d i g u t h i s w i d i g u t h i s w i d i g u t h i s w i d i g u t h i s w i d i g u t h i s w i d i g u t h i s w i d i g u t h i s w i d i g u t h i s w i d i g u t h i s v i d i g u t h i s v i d i g u t h i s v i d i g u t h i s v i d i g u t h i s v i d i g u t h i s v i d i g u t h i s v i d i g u t h i s v i d i g u t h i s v i d i g u t h i s v i d i g u t h i s w i d i g u t h i s w i d i g u t h i s w i d i g u t h i s w i d i g u t h i s w i d i g u t h i s w i d i g u t h i s w i d i g u t h i s w i d i g u t h i s w i d i g u t h i s w i d i g u t h i s w i d i g u t h i s w i d i g u t h i s w i d i g u t h i s w i d i g u t h i s w i d i g u t h i s w i d i g u t h i s w i d i g u t h i s w i d i g u t h i is w i d i g u t h i is w i d i g u t h i is w i d i g u t h i is w i d i g u t h i is w i d i g u t h i is w i d i g u t h i is w i d i g u t h i is w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W WWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWW< fcel>I am the case of the case of the case of the case of the case of the case of the case of the case of the case of the case of the case of the case of the case of the case of the case of the case of the case of the case of the case of the case of the case of the case of the case of the case of the case of the case of the case of the case of the case of the case of the case of the case of the case of the case of that one or two cases in which there are no cases in each case, and there are no cases in each case, but there are no cases in each case, but there are no cases in each case, but there are no cases in each case, but there are no cases in each case, but there are no cases in each case, but there are no cases in each case, but there are no cases in each case, but there are no cases in each case, but there are no cases in each case, but there are no cases in each case, but there are no cases in each case, but There is no cases in each case, but there are no cases in each case, but there are no cases in each case, but there are no cases in each case, but there are no cases in each case, but there are no cases in each case, but there are no cases in each case, but there are no cases in each case, but there are no cases in each case, but there are no cases in each case, but there are no cases in each case, but there areno cases in each case, but there are no cases in each case, but there are no cases in each case, but there are no cases in each case, but there are no cases in each case, but there are no cases in each case, but there are no cases in each case, but there are no cases in each case, but there are no cases in each case, but there are no cases in each case, but there are no cases in each case, but there are nocases in all cases. I can't have an outcome that would be given if you want to perform better than what you want to perform better than what you want to perform better than what you want to perform better than what you want to perform better than what you want to perform better than what you want to perform better than what you want to perform better than what you want to perform better than what you want to perform better than what you want to perform better than what you want to perform better than what you want to perform better than what you want to perform better than what you want to perform better than what you support this way. I can't have an outcome that would be given if you want to perform better than what you want to perform better than what you want to perform better than what you want to perform better than what you want to perform better than what you want to perform better than what you want to perform better than what you want to perform better than what you want to perform better than what you want to perform better than what you want to perform better than what you want to perform better than what you want to produce this way. I can't have an outcome that would be given if you want to perform better than what you want to perform better than what you want to perform better than what you want to perform better than what you want to perform better than what you want to perform better than what you want to produce this way. I can't have an outcome that would be given if you want to perform better than what you want to perform better than what you want to perform better than what you want to perform better than what you want to produce this way. I can't have an outcome that would be given if you want to perform better than what you want to perform better than what you want to perform better than what you want to produce this way. I can't have an outcome that would be given if you want to perform better than what you want to perform better than what you want to perform better than what you want to produce this way. I can't have an outcome that would be given if you want to perform better than what you want to perform better than what you want to perform better than what your own or more people who will not have any effect on how many people will not have any effect on how many people will not have any effect on how many people will not have any effect on how many people will not have any effect on how many people will not have any effect on how many people will not have any effect on how many people will not have any effect on how many people will not have any effect on how many people will not have any effect on how many people will not have any effect on how many people will not have any effect on how many people may be able to make it up at all time points. I can't have an outcome that would be given if you want to perform better than what you want to perform better than what you want to perform better than what you want to perform better than what you want to perform better than what you want to perform better than what you want to perform better than what you want to perform better than what you want to perform better than what you want to produce this way. I can't have an outcome that would be given if you want to perform better than what you want to perform better than what you want to perform Better than this way. I can't have an outcome that would be given if you want to perform better than what you want to perform Better than what you want to perform Better than what you want to perform Better than what you want to perform Better than what you want to perform Better than what you want to perform Better than what you want to perform Better than what you want to perform Better than what you want to perform Better than what you want to perform Better than what you want to produce this way. I can't have an outcome that would be given if you want to perform better than what you want to perform Better than what you want to perform Better than what you want to perform Better than what you want to perform Better than what you want to perform Better than what you want to perform Better than what you want to produce this way. I can't have an outcome that would be given if you want to perform better than what you want to perform Better than what you want to perform Better than what you want to perform Better than what you want to perform Better than what you wantto our own or more people who will not have any effect on how many people will not have any effect on how many people will not have any effect on how many people will not have any effect on how many people will not have any effect on how many people will not have any effect on how many people will not have any effect on how many people will not have any effect on how many people will not have any effect on how many people will not have any effect on how many people will not have any impact on how many people will not have any impact on how many people will not have any impact on how many people will not have any impact on how many people will not have any impact on how many people will not have any impact on how many people will not have any impact on how many people will not have any impact on how many people will not have any impact on how many people will not have any impact on how many people will not have any impact on how many people will not have any impacton how many people will not have any impact on how many people will not have any impact on how many people will not have any impact on how many people will not have any impact on how many people will not have any impact on how many people will not have any impact on how many people will not have any impact on how many people will not have any impact on how many people will not have any impact on how many people will not have any impact on how many people will not have any impact on How Many Way's Way's Way's Way's Way's Way's Way's Way's Way's Way's Way's Way's Way's Way's Way's Way's Way's Way's Way's Way's Way's Way's Way's Way's Way's Way's Way's Way's Way's Way's Way's Way's Way's Way's Way's Way's Way's Way's Way's Way's Way's Way's Way's Way's Way's Way's Way's Way's Way's Way's Way'< nl>\]

\[\begin{array} { r l } & { + \left( \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta _ { t + j } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \delta _ { t + 1 } + \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta _ { t + j } \otimes \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta _ { t + j } \right) \otimes \left( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \delta _ { t + 1 } + \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta _ { t + j } \right) \\ & { - \left( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \epsilon _ { t + 1 } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \epsilon _ { t + 1 } + \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \epsilon _ { t + 1 } \otimes \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \epsilon _ { t + j } \right) \otimes \left( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \epsilon _ { t + 1 } + \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \epsilon _ { t + j } \right) \\ & { - \left( \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \epsilon _ { t + j } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \epsilon _ { t + 1 } + \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \epsilon _ { t + j } \otimes \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \epsilon _ { t + j } \right) \otimes \left( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \epsilon _ {t + 1 } + \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \epsilon _ { t + j } \right) ] \\ & { = E _ { t } [ ( {\mathbf h} _ { {\mathbf x}} ^ { l - 1 } \sigma {\boldsymbol {\eta}} {\delta} _ { t + 1 } \otimes {\mathbf h} _ { {\mathbf x}} ^ { l - 1 } \sigma {\boldsymbol {\eta}} {\delta} _ { t + 1 } + {\mathbf h} _ { {\mathbf x}} ^ { l - 1 } \sigma {\boldsymbol {\eta}} {\delta} _ { t + 1 } \otimes {\sum _ { j = 2 } ^ { l }} {\mathbf h} _ { {\mathbf x}} ^ { l - j } {\sigma} {\boldsymbol {\eta}} {\delta} _ { t + j} ) \otimes {\mathbf h} _ { {\mathbf x}} ^ { l - 1 } {\sigma} {\boldsymbol {\eta}} {\delta} _ { t + 1 } \\ & { + ( {\mathbf h} _ { {\mathbf x}} ^ { l - 1 } \sigma {\boldsymbol {\eta}} {\delta} _ { t + 1 } \otimes {\mathbf h} _ { {\mathbf x}} ^ { l - 1 } {\sigma} {\boldsymbol {\eta}} {\delta} _ { t + 1 } + {\mathbf h} _ { {\mathbf x}} ^ { l - 1 } {\sigma} {\boldsymbol {\eta}} {\delta} _ { t + 1 } ) \otimes {\sum _ { j = 2 }} {\mathbf h} _ { {\mathbf x}} ^ { l - j } {\sigma} {\boldsymbol {\eta}} {\delta} _ { t + j} \\ & { + ( \sum _ { j = 2 } ^ { l }} {\mathbf h} _ { {\mathbf x}} ^ { l - j } {\sigma} {\boldsymbol {\eta}} {\delta} _ { t + j } \otimes {\mathbf h} _ { {\mathbf x}} ^ { l - 1 } {\sigma} {\boldsymbol {\eta}} {\delta} _ { t + 1 } + ( \sum _ { j = 2 }} {\mathbf h} _ { {\mathbf x}} ^ { l - j } {\sigma} {\boldsymbol {\eta}} {\delta} _ { t + j } ) \otimes {\sum _ { j = 2 }} {\mathbf h} _ { {\mathbf x}} ^ { l - j } {\sigma} {\boldsymbol {\eta}} {\delta} _ { t + j} \\ & { + ( \sum _ { j = 2 }} {\mathbf h} _ { {\mathbf x}} ^ { l - j } {\sigma} {\boldsymbol {\eta}} {\delta} _ { t + j } \otimes {\mathbf h} _ { {\mathbf x}} ^ { l - 1 } {\sigma} {\boldsymbol {\eta}} {\delta} _ { t + 1 } + ( \sum _ { j = 2 }} {\\mathbf h} _ { {\mathbf x}} ^ {- l - j} {\sigma} {\boldsymbol{\eta}} {\delta} _{ t + j} ) , \\ & {- ( {\mathbf h} _ { {\mathbf x}} ^ {- l - 1} {\sigma} {\boldsymbol{\eta}} {\epsilon} _ { t + 1 } (\otimes) {\mathbf h} _ {{\mathbf x}} ^ {- l - 1} {\sigma} {\boldsymbol{\eta}} {\epsilon} _{ t + 1} + ( {\mathbf h} _ {{\mathbf x}} ^ {- l - 1} {\sigma} {\boldsymbol{\eta}} {\epsilon} _{ t + j} ) ) , \\ & {- ( ({\mathbf h} _ {{\mathbf x}} ^ {- l - 1} {\sigma} {\boldsymbol{\eta}} {\epsilon} _{ t + 1 } + ( ({\mathbf h} _ {{\mathbf x}} ^ {- l - 1} {\sigma} {\boldsymbol{\eta}} {\epsilon} _{ t + j} ) ) , \\ & {- ( ({\mathbf h} _ {{\mathbf x}} ^ {- l - 1} {\sigma} {\boldsymbol{\eta}} {\epsilon} _{ t + 1 } + ( ({\mathbf h}_{\mathrm{eff}} ^ {- l - 1} ) ) , \\ & {- ( ( ( ({\mathbf h}_{\mathrm{eff}} ^ {- l - 1} ) ) ) , \\ & {- ( ( ( ({\mathbf h}_{\mathrm{eff}} ^ {- l - 1} ) ) ) , \\ & {- ( ( ( ({\mathbf h}_{\mathrm{eff}} ^ {- l - 1} ) ) ) , \\ & {- ( ( ( ({\mathbf h}_{\mathrm{eff}} ^ {- l - 1} ) ) ) ,}\\ & {- ( ( ( ({\mathbf h}_{\mathrm{eff}} ^ {- l - 1} ) ) ) , \\ & {- ( ( ({\mathbf h}_{\mathrm{eff}} ^ {- l - 1} ) ) ) , \\ & {- ( ( ({\mathbf h}_{\mathrm{eff}} ^ {- l - 1} ) ) ) , \\ & {- ( ( ({\mathbf h}_{\mathrm{eff}} ^ {- l - 1} ) ) ) , \\ & {- ( ( ({\mathbf h}_{{\mathrm{eff}}} ^ {- l - 1} ) ) ) , \\ & {- ( ( ({\mathbf h}_{{\mathrm{eff}}} ^ {- l - 1} ) ) ) , \\ & {- ( ( ({\mathbf h}_{{\mathrm{eff}}} ^ {- l - 1} ) ) ) , \\ & {- ( ( ({\mathbf h}_{{\mathrm{eff}}} ^ {- l - 1}) ) ) , \\ & {- ( ( ({\mathbf h}_{{\mathrm{eff}}} ^ {- l - 1} ) ) ) , \\ & {- ( ( ({\mathbf h}_{{\mathrm{eff}}} ^ {- l - 1} ) ) ) , \\ & {- ( ( ({\mathbf h}_{{\mathrm{eff}}} ^ {- l - 1} ) ) ) , \\ & {. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .}. \\ & {. = E _ { t }\big [ (\{{\mathbf h}_{\mathrm{x}}^{l - 1}\sigma{\boldsymbol{\eta}}{\delta}_{t+ 1}\otimes{\mathbf h}_{\mathrm{x}}^{l - 1}\sigma{\boldsymbol{\eta}}{\delta}_{t+ 1}\oplus0) , \\ & {. = E_{ t }\big [ (\{{\mathbf h}_{\mathrm{x}}^{l - 1}\sigma{\boldsymbol{\eta}}{\delta}_{t+ 1}\otimes{\mathbf h}_{\mathrm{x}}^{l - 1}\sigma{\boldsymbol{\eta}}{\delta}_{t+ 1}\oplus0) , \\ & {. = E_{ t }\big [ (\{{\mathbf h}_{\mathrm{x}}^{l - 2}\sigma{\boldsymbol{\eta}}{\delta}_{t+ 2}\otimes{\mathbf h}_{\mathrm{x}}^{l - 2}\sigma{\boldsymbol{\eta}}{\delta}_{t+ 2}\oplus0) , \\ & {. = E_{ t }\big [ (\{{\mathbf h}_{\mathrm{x}}^{l - 2}\sigma{\boldsymbol{\eta}}{\delta}_{t+ 2}\otimes{\mathbf h}_{\mathrm{x}}^{l - 2}\sigma{\boldsymbol{\eta}}{\delta}_{t+ 2}\oplus0) , \\ & {. = E_{ t }\big [ (\{{{\mathrm{h}_{x}}}^ {- l - j}\sigma{\boldsymbol{\eta}}^ {- k}\sigma{\boldsymbol{\eta}}^ {- k}\sigma{\boldsymbol{\eta}}^ {- k}\sigma{\boldsymbol{\eta}}^ {- k}\sigma{\boldsymbol{\eta}}^ {- k}\sigma{\boldsymbol{\eta}}^ {- k}\sigma{\boldsymbol{\eta}}^ {- k}\sigma{\boldsymbol{\eta}}^ {- k}\sigma{\boldsymbol{\eta}}^ {- k}\sigma{\boldsymbol{\eta}}^ {- k}|,}\\ & {. = E_{ t }\big [ (\{{{\mathrm{h}_{x}}}^ {- l - j}\sigma{\boldsymbol{\eta}}^ {- k}\sigma{\boldsymbol{\eta}}^ {- k}\sigma{\boldsymbol{\eta}}^ {- k}\sigma{\boldsymbol{\eta}}^ {- k}\sigma{\boldsymbol{\eta}}^ {- k}\sigma{\boldsymbol{\eta}}^ {- k}\sigma{\boldsymbol{\eta}}^ {-k}|,}\\ & {. = E_{ t }\big [ (\{{{\mathrm{h}_{x}}}^ {- l - j}\sigma{\boldsymbol{\eta}}^ {- k}\sigma{\boldsymbol{\eta}}^ {- k}\sigma{\boldsymbol{\eta}}^ {- k}\sigma{\boldsymbol{\eta}}^ {- k}\sigma{\boldsymbol{\eta}}^ {- k}\sigma{\boldsymbol{\eta}}^ {- k}\sigma{\boldsymbol{\eta}},}\\ & {. = E_{ t }\big [ (\{{{\mathrm{h}_{x}}}^ {- l - j}\sigma{\boldsymbol{\eta}}^ {- k}\sigma{\boldsymbol{\eta}}^ {- k}\sigma{\boldsymbol{\eta}}^ {- k}\sigma{\boldsymbol{\eta}}^ {- k}\sigma{\boldsymbol{\eta}}^ {- k}\sigma{\boldsymbol{\eta}}^ {- k}\sigma{\boldsymbol{\eta}},}\\ \\ & {. = E_{ t }\big [ (\{{{\mathrm{h}_{x}}}^ {- l - j}\sigma{\boldsymbol{\eta}}^ {- k}\sigma{\boldsymbol{\eta}}^ {- k}\sigma{\boldsymbol{\eta}}^ {- k}\sigma{\boldsymbol{\eta}}^ {- k}\sigma{\boldsymbol{e}_{x}}}\\ & {. = E_{ t }\big [ (\{{{\mathrm{h}_{x}}}^ {- l - j}\sigma{\boldsymbol{e}_{x}}}\\ & {. = E_{ t }\big [ (\{{{\mathrm{h}_{x}}}^ {- l - j}\sigma{\boldsymbol{e}_{x}}}\\ & {. = E_{ t }\big [ (\{{{\mathrm{h}_{x}}}^ {- l - j}\sigma{\boldsymbol{e}_{x}}}\\ & {. = E_{ t }\big [ (\{{{\mathrm{h}_{x}}}^ {- i- j}\sigma{\boldsymbol{e}_{x}}}\\ & {. = E_{ t }\big [ (\{{{\mathrm{h}_{x}}}^ {- i- j}\sigma{\boldsymbol{e}_{x}}}\\ & {. = E_{ t }\big [ (\{{{\mathrm{h}_{x}}}^ {- i- j}\sigma{\boldsymbol{e}_{x}}}\\ & {. = E_{ t }\big [ (\{{{\mathrm{h}_{x}}}^{- i- j}\sigma{\boldsymbol{e}_{x}}}\\ & {. = E_{ t }\big [ (\{{{\mathrm{h}_{x}}}^{- i- j}\sigma{\boldsymbol{e}_{x}}}\\ & {. = E_{ t }\big [ (\{{{\mathrm{h}_{x}}}^{- i- j}\sigma{\boldsymbol{e}_{x}}}\\ & {. = E_{ t }\big [ (\{{{\mathrm{h}_{x}(i)^{-j}-i,j},i,j},i,j},i,j},i,j},i,j},i,j},i,j},i,j},i,j},i,j},i,j},i,j},i,j},i,j},i,j},i,j},i,j},i,j},i,j},i,j},i,j},i,j},i,j},i,j},i,j},i,j},i,j,i,j,i,j,i,j,i,j,i,j,i,j,a. \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {} \\ & {}\\ & {} \\ & {}\\ & {}\\ & {}\\ & {}\\ & {}\\ & {}\\ & {}\\ & {}\\ & {}\\ & {}\\ & {}\\ & {}\\ & {}\\ & {}\\ & {}\\ & {}\\ & {}\\ & {}\\ & {}\\ & {}\\ & {}\\ & {}\\ & {}\\ & {}\\ & {}\\ & {}\\ & {}\\ & {}\\ & {}\\ & {}\\ & {}\\ & {}\\ & {}\\ & {}\\< fcel>= E_t [\({\mathrm{h}_{x}^{l-1}\sigma}\mathrmn~o~o~o~o~o~o~o~o~o~o~o~o~o~o~o~o~o~o~o~o~o~o~o~o~o~o~o~o~o~o~o~o~o~o~o~o~o~o~o~o~o~o~o~o~o~o~o~o~o~o~n~o~n~o~n~o~n~o~n~o~n~o~n~o~n~o~n~o~n~o~n~o~n~o~n~o~n~o~n~o~n~o~n~o~n~o~n~o~n ~ o n ~ o n ~ o n ~ o n ~ o n ~ o n ~ o n ~ o n ~ o n ~ o n ~ o n ~ o n ~ o n ~ o n ~ o n ~ o n ~ o n ~ o n ~ o n ~ o n ~ o n ~ o n ~ o n ~ o n ~ o n ~ o n ~ o n ~ o n ~ o n ~ o n ~ o n ~ o n ~ o n ~ o n > a b d a c e s e f f e r e d i g e a d i g e a d i g e a d i g e a d i g e a d i g e a d i g e a d i g e a d i g e a d i g e a d i g e a d i g e a d i g e a d i g e a d i g e a d i g e a d i g e a d i g e a d i g e a d i g e a d i g e e a d i g e a d i g e a d i g e a d i g e a d i g e a d i g e a d i g e a d i g e a d i g e a d i g e a d i g e a d i g e a d i g e a d i g e a d i g e a d i g e a d i g e a d i g e a d i g e a d i f f e r e d i g e a d i f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w v w v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v vvvi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi vi viviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviviaviivviiviiviiviiviiviiviiviiviiviiviiviiviiviiviiviiviiviiviiviiviiviiviiviiviiviiviiviiviiviiviiviiviiviiviiviiviiviiviiviiviiviiviiviiviiviiviiviiviiiiviiiiviiiiviiiiviiiiviiiiviiiiviiiiviiiiviiiiviiiiviiiiviiiiviiiiviiiiviiiiviiiiviiiiviiiiviiiiviiiiviiiiviiiiviiiiviiiiviiiiviiiiviiiiviiiiviiiiviiiiviiiiviiiiviiiiviiiiviiiiviiiiviiiiviiiiviiiiviiiiviiiiviiiiviiiiviiiiviiiiviiiiviiiiviiiiviiiiviiiixiviiiixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixixiaaiuiuiuiuiuiuiuiuiuiuiuiuiuiuiuiuiuiuiuiuiuiuiuiuiuiuiuiuiuiuiuiuiuiuiuiuiuiuiuiuiuiuiuiuiuiuiuiuiuiuiuiuiuiuiuiuiuiuiuiuiuiuiuiuiuiuiuiuiuiuiuiuuaaiuiuuaaiuuaaiuuaaiuuaaiuuaaiuuaaiuuaaiuuaaiuuaaiuuaaiuuaaiuuaaiuuaaiuuaaiuuaaiuuaaiuuaaiuuaaiuuaaiuuaaiuuaaiuuaaiuuaaiuuaaiuuaaiuuaaiuuaaiuuaaiuuaaiuuaaiuuuaaiuuuaaiuuuaaiuuuaaiuuuaaiuuuaaiuuuaaiuuuaaiuuuaaiuuuaaiuuuaaiuuuaaiuuuaaiuuuaaiuuuaaiuuuaaiuuuaaiuuuaaiuuuaaiuuuaaiuuuaaiuuuaaiuuuaaiuuuaaiuuuaaiuuuaaiuuuaaiuuuaaiuuuaaiuuuaaiuuuaaiuuuaaiuuuaaiuuucauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauauaaaaineeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaeaueeeaueeeaueeeaueeeaueeeaueeeaueeeaueeeaueeeaueeeaueeeaueeeaueeeaueeeaueeeaueeeaueeeaueeeaueeeaueeeaueeeaueeeaueeeaueeeaueeeaueeeaueeeaueeeaueeeaueeeaueeeaueeeaueeeaueeeaaueeeaaueeeaaueeeaaueeeaaueeeaaueeeaaueeeaaueeeaaueeeaaueeeaaueeeaaueeeaaueeeaaueeeaaueeeaaueeeaaueeeaaueeeaaueeeaaueeeaaueeeaaueeeaaueeeaaueeeaaueeeaaueeeaaueeeaaueeeaaueeeaaueeeaaueeeaaueeeaaueeeaaeaueeeaaeaueeeaaeaueeeaaeaueeeaaeaueeeaaeaueeeaaeaueeeaaeaueeeaaeaueeeaaeaueeeaaeaueeeaaeaueeeaaeaueeeaaeaueeeaaeaueeeaaeaueeeaaeaueeeaaeaueeeaaeaueeeaaeaueeeaaeaueeeaaeaueeeaaeaueeeaaeaueeeaaeaueeeaaeareaend{array}< |content_end|>\]

\[\begin{array}{l} + \left(\sum_ {j = 2} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \delta_ {t + j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \delta_ {t + 1} \otimes \sum_ {j = 2} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \delta_ {t + j} + 0\right) \\ - \left(\mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \epsilon_ {t + 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \epsilon_ {t + 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \epsilon_ {t + 1}\right) \\ + (0) ] \\ = \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} \\ + E _ {t} \left[ \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} \otimes \sum_ {j = 2} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \delta_ {t + j} \otimes \sum_ {j = 2} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \delta_ {t + j} \right] \\ + E _ {t} \left[ \sum_ {j = 2} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \delta_ {t + j} \otimes \sum_ {j = 2} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \delta_ {t + j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} \right] \\ + E _ {t} \left[ \sum_ {j = 2} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \delta_ {t + j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} \otimes \sum_ {j = 2} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \delta_ {t + j} \right] \\ - E _ {t} \left[ \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \epsilon_ {t + 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \epsilon_ {t + 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \epsilon_ {t + 1} \right] \end{array}\]

How consider the following terms:

\[\begin{array} { r l } & A _ { 4 , 1 } \equiv E _ { t } \left[ \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \nu } \otimes \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \pmb { \delta } _ { t + j } \otimes \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \pmb { \delta } _ { t + j } \right] \\ & \qquad = E _ { t } [ \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \nu } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 2 } \sigma \boldsymbol { \eta } \pmb { \epsilon } _ { t + 2 } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \pmb { \epsilon } _ { t + j } \\ & \qquad + \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \nu } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 3 } \sigma \boldsymbol { \eta } \pmb { \epsilon } _ { t + 3 } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \pmb { \epsilon } _ { t + j } \\ & \qquad + ... \\ & \qquad + \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \nu } \otimes \sigma \boldsymbol { \eta } \pmb { \epsilon } _ { t + l } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \pmb { \epsilon } _ { t + j } ] \\ & = E _ { t } [ \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \nu } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 2 } \sigma \boldsymbol { \eta } \pmb { \epsilon } _ { t + 2 } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 2 } \sigma \boldsymbol { \eta } \pmb { \epsilon } _ { t + 2 } \\ & \qquad + \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \nu } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 3 } \sigma \boldsymbol { \eta } \pmb { \epsilon } _ { t + 3 } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 3 } \sigma \boldsymbol { \eta } \pmb { \epsilon } _ { t + 3 } \\ & \qquad + ... \\ & \qquad + \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \nu } \otimes \sigma \boldsymbol { \eta } \pmb { \epsilon } _ { t + l } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - l } \sigma \boldsymbol { \eta } \pmb { \epsilon } _ { t + l } ] \\ & b e c a u s e t h e i n n o v a t i o n s a r e i n d e p e n d e n t a c r o s s t i m e \\ & = {\sum _ { j = 2 } ^ { l }} E _ { t } [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) ] ) \\ & = {\sum _ { j = 2 } ^ { l }} {\mathbf h} _ { {\mathbf x}} ^ { l - 1} {\sigma} {\pmb {\eta}} {\pmb {\nu}} {\otimes} ({\mathbf h} _ {\mathbf x} ^ {l - j} {\otimes} {\mathbf h} _ {\mathbf x} ^ {l - j}) E _ { t } [ ( {\sigma} {\pmb {\eta}} {\pmb {\epsilon}} _ { t + j} {\otimes} {\sigma} {\pmb {\eta}} {\pmb {\epsilon}} _ { t + j}) ] \\ & = {\sum _ { j = 2 } ^ { l }} {\mathbf h} _ {\mathbf x} ^ {l - 1} {\sigma} {\pmb {\eta}} {\pmb {\nu}} {\otimes} ({\mathbf h} _ {\mathbf x} ^ {l - j} {\otimes} {\mathbf h} _ {\mathbf x} ^ {l - j}) E _ { t } [ ( {\sigma} {\pmb {\eta}} {\pmb {\epsilon}} _ { t + 1} {\otimes} {\sigma} {\pmb {\eta}} {\pmb {\epsilon}} _ { t + 1}) ] \\ & b e c a u s e t h e i n n o v a t i o n s a r e i d e n t i c a l d i s t r i b u t e d a c r o s s t i m e \\ & = {\sum _ { j = 2 } ^ { l }} {\mathbf h} _ {\mathbf x} ^ {l - 1} {\sigma} {\pmb {\eta}} {\pmb {\nu}} {\otimes} ({\mathbf h} _ {\mathbf x} ^ {l - j} {\otimes} {\mathbf h} _ {\mathbf x} ^ {l - j}) {\Lambda} \\ & = 8. 6 7 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 . 6 . 6 . 6 . 6 . 6 . 6 . 6 . 6 . 6 . 6 . 6 . 6 . 6 . 6 . 6 . 6 . 6 . 6 . 6 . 6 . 6 . 6 . 6 . 6 . 6 . 6 . 6 . 6 . 6 . 6 . 6 . 6 . 6 . 6 ~ , ~ , ~ , ~ , ~ , ~ , ~ , ~ , ~ , ~ , ~ , ~ , ~ , ~ , ~ , ~ , ~ , ~ , ~ , ~ , ~ , ~ , ~ , ~ , ~ , ~ , ~ , ~ , ~ , ~ , ~ , ~ , ~ , ~ , ~ , ~ , ~ , ~ , ~ , ~ , ~ , ~ , ~ , ~ , ~ , ~ , ~ , ~ , ~ , ~ , ~, ~ , ~ , ~ , ~ , ~ , ~ , ~ , ~ , ~ , ~ , ~ , ~ , ~ , ~ , ~, ~, ~, ~, ~, ~, ~, ~, ~, ~, ~, ~, ~, ~, ~, ~, ~, ~, ~, ~, ~, ~, ~, ~, ~, ~, ~, ~, ~, ~, ~, ~, ~, ~, ~, ~, ~, ~, ~, ~, ~, ~, ~, ~, ~, ~, ~, ~, ~, ~, ~,,~ | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |\]

We therefore immediately have for the second term:

\[A _ {4, 2} \equiv E _ {t} \left[ \sum_ {j = 2} ^ {l} {\bf h} _ {\bf x} ^ {l - j} \sigma \pmb {\eta} \pmb {\delta} _ {t + j} \otimes \sum_ {j = 2} ^ {l} {\bf h} _ {\bf x} ^ {l - j} \sigma \pmb {\eta} \pmb {\delta} _ {t + j} \otimes {\bf h} _ {\bf x} ^ {l - 1} \sigma \pmb {\eta} \pmb {\nu} \right] = \sum_ {j = 2} ^ {l} \Lambda \left({\bf h} _ {\bf x} ^ {l - j} \otimes {\bf h} _ {\bf x} ^ {l - j}\right) \otimes {\bf h} _ {\bf x} ^ {l - 1} \sigma \pmb {\eta} \pmb {\nu}\]

\[\begin{array} { r l } & { \text {For the third term (when using the results from above)} } \\ & { A _ { 4 , 3 } \equiv E _ { t } \left[ \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \pmb { \eta } \pmb { \delta } _ { t + j } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \pmb { \eta } \pmb { \nu } \otimes \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \pmb { \eta } \pmb { \delta } _ { t + j } \right] } \\ & { = \sum _ { j = 2 } ^ { l } E _ { t } \left[ \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \pmb { \eta } \pmb { \epsilon } _ { t + j } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \pmb { \eta } \pmb { \nu } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \pmb { \eta } \pmb { \epsilon } _ { t + j } \right] } \\ & { = \sum _ { j = 2 } ^ { l } \pmb { \Omega } _ { j } } \\ & { \text {where} \pmb { \Omega } _ { j } \equiv E _ { t } \left[ \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \pmb { \eta } \pmb { \epsilon } _ { t + 1 } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \pmb { \eta } \pmb { \nu } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \pmb { \eta } \pmb { \epsilon } _ { t + 1 } \right] . S o } \\ & { \pmb { \Omega } _ { j } \equiv E _ { t } \left[ \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \pmb { \eta } \pmb { \epsilon } _ { t + 1 } \otimes \left\{ \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } ( \gamma _ { 2 }, : ) \sigma \pmb { \eta } \pmb { \nu } \times \left\{ \mathbf { h } _ { \mathbf { x } } ^ { l - j } ( \gamma _ { 1 }, : ) \sigma \pmb { \eta } \pmb { \epsilon } _ { t + 1} \right\} _ { \gamma _ { 1 } = 1 } ^ { n _ { x }} \right\} _ { \gamma _ { 2 } = 1 } ^ { n _ { x }}\right] } \\ & { = E _ { t } \left\{ \mathbf { h } _ { \mathbf { x } } ^ { l - j } ( \gamma _ { 3 , : }) \sigma \pmb { \eta } \pmb { \epsilon } _ { t + 1 } \times \left\{ \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } ( \gamma _ { 2 }, : ) \sigma \pmb { \eta } \pmb { \nu } \times \left\{ \mathbf { h } _ { \mathbf { x } } ^ { l - j } ( \gamma _ { 1 }, : ) \sigma \pmb { \eta} \pmb { \epsilon } _ { t + 1} \right\} _ { \gamma _ { 1 } = 1 } ^ { n _ { x }} \right\} _ { \gamma _ { 2 } = 1 } ^ { n _ { x }}\right\} _ { \gamma _ { 3 } = 1 } ^ { n _ { x }}} \\ & = E _ { t } \left\{ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ $ [ ] $ , then, then, then, then, then, then, then, then, then, then, then, then, then, then, then, then, then, then, then, then, then, then, then, then, then, then, then, then, then, then, then, then, then, then, then, then, then, then, then, then, then, then, then, then, then, then, then, then, then, then, then ,then ,then ,then ,then ,then ,then ,then ,then ,then ,then ,then ,then ,then ,then ,then ,then ,then ,then ,then ,then ,then ,then ,then ,then ,then ,then ,then ,then ,then ,then ,then ,then ,then ,then ,then ,then ,then ,then ,then ,then ,then ,then ,then ,then ,then ,then ,then ,then ,then ,then ,\\ & = E _ { t } \left\{~ [ h _ { x } ^ { l - j } ( y ) ( y ) * h _ { x i n g a d e s s i n g a d e s s i n g a d e s s i n g a d e s s i n g a d e s s i n g a d e s s i n g a d e s s i n g a d e s s i n g a d e s s i n g a d e s s i n g a d e s s i n g a d e s s i n g a d e s s i n g e f o r t h e t h e p r o r e (y) > y < y < y < y < y < y < y < y < y < y < y < y < y < y < y < y < y < y < y < y < y < y < y < y < y < y < y < y < y < y < y < y < y < y < | y | > | y | > | y | > | y | > | y | > | y | > | y | > | y | > | y | > | y | > | y | > | y | > | y | > | y | > | y | > | y | > | y | > | y | > | y | > | y | > | y | > | y | > | y | > | y | > | y | > | y |> | y | > | y | > | y | > | y | > | y | > | y | > | y | > | y | > | y | > | y | > | y | > | y | > | y | > | y | > | y | > | y | > | y | > | y | > | y | > | y | > | y | > | y | > | y | > | y | > | y |>\infty .} \\ & = E _ { t } \left\{~ [ h _ x i n g a d e s s i n g a d e s s i n g a d e s s i n g a d e s s i n g a d e s s i n g a d e s s i n g a d e s s i n g a d e s s i n g a d e s s i n g a d e s s i n g a d e s s i n g a d e s c e r m : [ v ] ; [ w ] ; [ w ] ; [ w ] ; [ w ] ; [ w ] ; [ w ] ; [ w ] ; [ w ] ; [ w ] ; [ w ] ; [ w ] ; [ w ] ; [ w ] ; [ w ] ; [ w ] ; [ w ] ; [ w ] ; [ w ] ; [ w ] ; [ w ] ; [ w ] ; [ w ] ; [ w ] ; [ w ] ; [ w ] ; [ u ] ; [ u ] ; [ u ] ; [ u ] ; [ u ] ; [ u ] ; [ u ] ; [ u ] ; [ u ] ; [ u ] ; [ u ] ; [ u ] ; [ u ] ; [ u ] ; [ u ] ; [ u ] ; [ u ] ; [ u ] ; [ u ] ; [ u ] ; [ u ] ; [ u ] ; [ u ] ; [ u ] ; [ u ] ; [ v ] ; [ v ] ; [ v ] ; [ v ] ; [ v ] ; [ v ] ; [ v ] ; [ v ] ; [ v ] ; [ v ] ; [ v ] ; [ v ] ; [ v ] ; [ v ] ; [ v ] ; [ v ] ; [ v ] ; [ v ] ; [ v ] ; [ v ] ; [ v ] ; [ v ] ; [ v ] ; [ v ] ; [ v ] ; [ u ] ; [ u ] ; [ u ] ; [ u ] ; [ u ] ; [ u ] ; [ u ] ; [ u ] ; [ u ] ; [ u ] ; [ u ] ; [ u ] ; [ u ] ; [ u ] ; [ u ] ; [ u ] ; [ u ] ; [ u ] ; [ u ] ; [ u ] ; [ u ] ; [ u ] ; [ u ] ; [ u ] ; [ w ] ; [ w ] ; [ w ] ; [ w ] ; [ w ] ; [ w ] ; [ w ] ; [ w ] ; [ w ] ; [ w ] ; [ w ] ; [ w ] ; [ w ] ; [ w ] ; [ w ] ; [ w ] ; [ w ] ; [ w ] ; [ w ] ; [ w ] ; [ w ] ; [ w ] ; [ w ] ; [ w ] ; [ v ] ; [ v ] ; [ v ] ; [ v ] ; [ v ] ; [ v ] ; [ v ] ; [ v ] ; [ v ] ; [ v ] ; [ v ] ; [ v ] ; [ v ] ; [ v ] ; [ v ] ; [ v ] ; [ v ] ; [ v ] ; [ v ] ; [ v ] ; [ v ]; {[ v i r c e r t i o n t i o n t i o n t i o n t i o n t i o n t i o n t i o n t i o n t i o n t i o n t i o n t i o n t i o n t i o n t i o n t i o n t i o n t i o n t i o n t i o n t i o n t i o n t i o n t i o n t i on t i o n t i o n t i o n t i o n t i o n t i o n t i o n t i o n t i o n t i o n t i o n t i o n t i o n t i o n t i o n t i o n t i o n t i o n t i o n t i o n t i o n t i o n t i o n t i o n t i o m e r e.} \\ & = \left\{~ h _ {\mathbf {\Delta}} ^ {\mathbf {\Delta}} (x) - h _ {\mathbf {\Delta}} ^ {\mathbf {\Delta}} (y) - h _ {\mathbf {\Delta}} ^ {\mathbf {\Delta}} (z) - h _ {\mathbf {\Delta}} ^ {\mathbf {\Delta}} (w) - h _ {\mathbf {\Delta}} ^ {\mathbf {\Delta}} (x) - h _ {\mathbf {\Delta}} ^ {\mathbf {\Delta}} (y) - h _ {\mathbf {\Delta}} ^ {\mathbf {\Delta}} (z) - h _ {\mathbf {\Delta}} ^ {\mathbf {\Delta}} (y) - h _ {\mathbf {\Delta}} ^ {\mathbf {\Delta}} (z) - h _ {\mathbf {\Delta}} ^ {\mathbf {\Delta}} (y) - h _ {\mathbf {\Delta}} ^ {\mathbf {\Delta}} (z) - h _ {\mathbf {\Delta}} ^ {\mathbf {\Delta}} (y) - h _{\mathbf {\Delta}} ^ {\mathbf {\Delta}} (z) - h _ {\mathbf {\Delta}} ^ {\mathbf {\Delta}} (y) - h _{\mathbf {\Delta}} ^ {\mathbf {\Delta}} (z) - h _{\mathbf {\Delta}} ^ {\mathbf {\Delta}} (y) - h _{\mathbf {\Delta}} ^ {\mathbf {\Delta}} (z) - h_{\mathbf {\Delta}} ^ {\mathbf {\Delta}} (y) - h_{\mathbf {\Delta}} ^ {\mathbf {\Delta}} (z) - h_{\mathbf {\Delta}} ^ {\mathbf {\Delta}} (y) - h_{\mathbf {\Delta}} ^ {\mathbf {\Delta}} (z) - h_{\mathbf {\Delta}} ^ {\mathbf {\Delta}} (y) - h_{\mathbf {\Delta}} ^ {\mathbf {\Delta}} (z) - h_{\mathbf {\Delta}}, {[ v} ) + H _{v} ^{l - j} (\mathrm{d} ) + H _{v} ^{l - j} (\mathrm{d} ) + H _{v} ^{l - j} (\mathrm{d} ) + H _{v} ^{l - j} (\mathrm{d} ) + H _{v} ^{l - j} (\mathrm{d} ) + H _{v} ^{l - j} (\mathrm{d} ) + H _{v} ^{\prime}, {[ v} ) + H _{v} ^{l - j} (\mathrm{d} ) + H _{v} ^{l - j} (\mathrm{d} ) + H _{v} ^{l - j} (\mathrm{d} ) + H _{v} ^{l - j} (\mathrm{d} ) + H _{v} ^{l - j} (\mathrm{d} ) + H _{v}\]

\[\begin{array} { r l } & \text {For the fourth term must be computed element by element} \\ & { A _ { 4 , 4 } \equiv E _ { t } \left[ \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \pmb { \eta } \pmb { \epsilon } _ { t + 1 } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \pmb { \eta } \pmb { \epsilon } _ { t + 1 } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \pmb { \eta } \pmb { \epsilon } _ { t + 1 } \right] } \\ & { = E _ { t } \left[ \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \pmb { \eta } \pmb { \epsilon } _ { t + 1 } \otimes \left\{ \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } ( \gamma _ { 2 } , : ) \sigma \pmb { \eta } \pmb { \epsilon } _ { t + 1 } \times \left\{ \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } ( \gamma _ { 1 } , : ) \sigma \pmb { \eta } \pmb { \epsilon } _ { t + 1 } \right\} _ { \gamma _ { 1 } = 1 } ^ { n _ { x } } \right\} _ { \gamma _ { 2 } = 1 } ^ { n _ { x } } \right] } \\ & { = E _ { t } \left[ \left\{ \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } ( \gamma _ { 3 , : } ) \sigma \pmb { \eta } \pmb { \epsilon } _ { t + 1 } \times \left\{ \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } ( \gamma _ { 2 } , : ) \sigma \pmb { \eta } \pmb { \epsilon } _ { t + 1 } \times \left\{ \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } ( \gamma _ { 1 } , : ) \sigma \pmb { \eta } \pmb { \epsilon } _ { t + 1 } \right\} _ { \gamma _ { 1 } = 1 } ^ { n _ { x } } \right\} _ { \gamma _ { 2 } = 1 } ^ { n _ { x } } \right\} _ { \gamma _ { 3 } = 1 } ^ { n _ { x } } \right] } \\ & = E _ { t } [ \{ \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } ( \gamma _ { 3 , : } ) \sigma \sum _ { \phi _ { 3 } = 1 } ^ { n _ { e } } \pmb { \eta } ( : , \phi _ { 3} ) \pmb { \epsilon } _ { t + 1 } ( \phi _ { 3 } , 1 ) \times \\ & {\qquad [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ ] ) ] }\]

Thus, we finally have:

\[\begin{array}{r l} & {\mathrm{Thus,wefinallyhave:}} \\ & {E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + l} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + l} ^ {f} - \mathbf {x} _ {t + l} ^ {f} \otimes \mathbf {x} _ {t + l} ^ {f} \otimes \mathbf {x} _ {t + l} ^ {f} \right]} \\ & {\qquad = \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta \pmb {\nu} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f}} \\ & {\qquad + \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta \pmb {\nu} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f}} \\ & {\qquad + \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta \pmb {\nu}} \\ & {A _ {1}) \qquad + (\mathbf {h} _ {\mathbf {x}} ^ {l - 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1}) ((\sigma \eta \pmb {\nu} \otimes \sigma \eta \pmb {\nu}) - \pmb {\Lambda}) \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f}} \\ & {A _ {2}) \qquad + \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes (\mathbf {h} _ {\mathbf {x}} ^ {l - 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1}) ((\sigma \eta \pmb {\nu} \otimes \sigma \eta \pmb {\nu}) - \pmb {\Lambda})} \\ & {A _ {3}) \qquad + \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta \pmb {\nu} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta \pmb {\nu} - \pmb {\Gamma (l)}} \\ & {A _ {4}) \qquad + \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta \pmb {\nu} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta \pmb {\nu} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta \pmb {\nu}} \\ & {A _ {4, 1}) \qquad + \sum_ {j = 2} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta \pmb {\nu} \otimes (\mathbf {h} _ {\mathbf {x}} ^ {l - j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - j}) \Lambda} \\ & {A _ {4, 2}) \qquad + \sum_ {j = 2} ^ {l} \Lambda (\mathbf {h} _ {\mathbf {x}} ^ {l - j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - j}) \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta \pmb {\nu}} \\ & {A _ {4, 3}) \qquad + \sum_ {j = 2} ^ {l} \Omega_ {j}} \\ & {A _ {4, 4}) \qquad + E _ {t} [ \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta \pmb {\epsilon} _ {{t + 1}} \otimes \mathbf {h} _ {\mathbf {x}} ^ {{l - 1}} \sigma \eta \pmb {\epsilon} _ {{t + 1}} \otimes \mathbf {h} _ {\mathbf {x}} ^ {{l - 1}} \sigma \eta \pmb {\epsilon} _ {{t + 1}} ]} \end{array}\]

11.3.2 For

Recall from above that

\[\begin{array}{r l} & {\mathbf {x} _ {t + 1} ^ {f} \otimes \mathbf {x} _ {t + 1} ^ {s} = (\mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t} ^ {f} + \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1}) \otimes (\mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t} ^ {s} + \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) + \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2})} \\ & {\qquad = (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {s}) + (\mathbf {h} _ {\mathbf {x}} \otimes \frac {1}{2} \mathbf {H} _ {\mathbf {x x}}) (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) + (\mathbf {h} _ {\mathbf {x}} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}) \mathbf {x} _ {t} ^ {f}} \\ & {\qquad + (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) (\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {s}) + (\sigma \boldsymbol {\eta} \otimes \frac {1}{2} \mathbf {H} _ {\mathbf {x x}}) (\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) + (\sigma \boldsymbol {\eta} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}) \boldsymbol {\epsilon} _ {t + 1}} \end{array}\]

Therefore:

\[\begin{array} { r l } & { } \text {Therefore:} \\ & \mathbf { x } _ { t + 2 } ^ { f } \otimes \mathbf { x } _ { t + 2 } ^ { s } = ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ( \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { s } ) + ( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } ) ( \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } ) + ( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } ) \mathbf { x } _ { t + 1 } ^ { f } \\ & { } \quad + ( \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x } } ) ( \boldsymbol { \epsilon } _ { t + 2 } \otimes \mathbf { x } _ { t + 1 } ^ { s } ) + ( \sigma \boldsymbol { \eta } \otimes \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } ) ( \boldsymbol { \epsilon } _ { t + 2 } \otimes \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } ) + ( \sigma \boldsymbol { \eta } \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } ) \boldsymbol { \epsilon } _ { t + 2 } \\ & { } = ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) [ ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { s } ) + ( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } ) ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) + ( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } ) \mathbf { x } _ { t } ^ { f } \\ & { } \quad + ( \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x } } ) ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { s } ) + ( \sigma \boldsymbol { \eta } \otimes \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } ) ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) + ( \sigma \boldsymbol { \eta } \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } ) \boldsymbol { \epsilon } _ { t + 1 ]} \\ & { } + ( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x} } ) ( \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } ) + ( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { h } _ {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} , \\ & { } + ( \sigma \boldsymbol { \eta } \otimes \mathbf { h _ { x }} ) ( \boldsymbol { \epsilon _ { t + 2 }} \otimes \mathbf { x _ { t + 1 }} ^ { s} ) + ( \sigma \boldsymbol { \eta } \otimes \frac { 1 } { 2 } \mathbf { H _ { x x }} ) ( \boldsymbol { \epsilon _ { t + 2 }} \otimes \mathbf { x _ { t + 1 }} ^ { f} \otimes \mathbf { x _ { t + 1 }} ^ { f} ) + ( \sigma \boldsymbol { \eta } \otimes \frac { 1 } { 2 } \mathbf h _ s o r o n c o u a d i n g e a l l o w e r d i n g e a l l o w e r d i n g e a l l o w e r d i n g e a l l o w e r d i n g e a l l o w e r d i n g e a l l o w e r d i n g e a l l o w e r d i n g e a l l o w e r d i n g e a l l o w e r d i n k , \\ & \\ & = ( [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ ] o u v a l l o w e r d i n g e a l l o w e r d i n g e a l l o w e r d i n g e a l l o w e r d i n g e a l l o w e r d i n g e a l l o w e r d i n g e a l l o w e r d i n g e a l l o w e r d i n g e a l l u r d i n g e a l l u r d i n g e a l l u r d i n g e a l l u r d i n g e a l l u r d i n g e a l l u r d i n g e a l l u r d i n g e a l l u r d i n g e a l l u r d i n g e a l l u r d i n g e a l l . \\ & \\ & = ( [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ ] o u v a l l o w e r d i n g e a l l o w e r d i n g e a l k u m p o p p o p p o p p o p p o p p o p p o p p o p p o p p o p p o p p o p p o p p o p p o p p o p p o p p o p p o p p o p p o p p o p p o p p o p p o p p o p p o p p o p p o p p o p p o p p o p p o p p o p s , \\ & \\ & = ( [ [ [ [ [ [ [ [ [ [ [ [ [ [ ] o u v a l l o w e r d i n g e a l k u m p o p p o p p o p p o p p o p p o p p o p p o p p o p s , \\ & \\ & = ( [ [ [ [ [ ] o u v a l l o w e r d i n g e a l k u m p o p s , \\ & \\ & = ( [ [ ] o u v a l l o w e r d i n g e a l k u m p o p s , \\ & \\ & = ( [ [ ] o u v a l l o w e r d i n g e a l k u m p o p s , \\ & \\ & = ( [ ] o u v a l l o w e r d i n g e a l k u m p o p s , \\ & \\ & = ( [ ] o u v a l l o w e r d i n g e a l k u m p o p s , \\ & \\ & = ( [ ] o u v a l l o w e r d i n g e a l k u m p o p s , - 1 , - 1 , - 1 , - 1 , - 1 , - 1 , - 1 , - 1 , - 1 , - 1 , - 1 , - 1 , - 1 , - 1 , - 1 , - 1 , - 1 , - 1 , - 1 , - 1 , - 1 , - 1 , - 1 , - 1 , - 1 , - 3 , - 3 , - 3 , - 3 , - 3 , - 3 , - 3 , - 3 , - 3 , - 3 , - 3 , - 3 , - 3 , - 3 , - 3 , - 3 , - 3 , - 3 , - 3 , - 3 , - 3 , - 3 , - 3 , - 3 , - 3 , - 4 , - 4 , - 4 , - 4 , - 4 , - 4 , - 4 , - 4 , - 4 , - 4 , - 4 , - 4 , - 4 , - 4 , - 4 , - 4 , - 4 , - 4 , - 4 , - 4 , - 4 , - 4 , - 4 , - 4 , - 4 , - 5 , - 5 , - 5 , - 5 , - 5 , - 5 , - 5 , - 5 , - 5 , - 5 , - 5 , - 5 , - 5 , - 5 , - 5 , - 5 , - 5 , - 5 , - 5 , - 5 , - 5 , - 5 , - 5 , - 5 , - 5 , - 6 , - 6 , - 6 , - 6 , - 6 , - 6 , - 6 , - 6 , - 6 , - 6 , - 6 , - 6 , - 6 , - 6 , - 6 , - 6 , - 6 , - 6 , - 6 , - 6 , - 6 , - 6 , - 6 , - 6 , - 6 , - 7 , - 7 , - 7 , - 7 , - 7 , - 7 , - 7 , - 7 , - 7 , - 7 , - 7 , - 7 , - 7 , - 7 , - 7 , - 7 , -\]

\[\begin{array} { l } + ( \sigma \boldsymbol { \eta } \otimes \mathbf { h _ { x } } ) \left( \boldsymbol { \epsilon } _ { t + 3 } \otimes \mathbf { x } _ { t + 2 } ^ { s } \right) + ( \sigma \boldsymbol { \eta } \otimes \frac { 1 } { 2 } \mathbf { H _ { x x } } ) \left( \boldsymbol { \epsilon } _ { t + 3 } \otimes \mathbf { x } _ { t + 2 } ^ { f } \otimes \mathbf { x } _ { t + 2 } ^ { f } \right) + ( \sigma \boldsymbol { \eta } \otimes \frac { 1 } { 2 } \mathbf { h _ { \sigma \sigma } \sigma ^ { 2 } } ) \boldsymbol { \epsilon } _ { t + 3 } \\ = ( \mathbf { h _ { x } } \otimes \mathbf { h _ { x } } ) [ ( \mathbf { h _ { x } } \otimes \mathbf { h _ { x } } ) ^ { 2 } \left( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { s } \right) + ( \mathbf { h _ { x } } \otimes \mathbf { h _ { x } } ) \left( \mathbf { h _ { x } } \otimes \frac { 1 } { 2 } \mathbf { H _ { x x } } \right) \left( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \right) + ( \mathbf { h _ { x } } \otimes \mathbf { h _ { x } } ) \left( \mathbf { h _ { x } } \otimes \frac { 1 } { 2 } \mathbf { h _ { \sigma \sigma } \sigma ^ { 2 } } \right) \mathbf { x } _ { t } ^ { f } \\ + ( \mathbf { h _ { x } } \otimes \mathbf { h _ { x } } ) ( \sigma \boldsymbol { \eta } \otimes \mathbf { h _ { x } } ) ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { s } ) + ( \mathbf { h _ { x } } \otimes \mathbf { h _ { x } } ) ( \sigma \boldsymbol { \eta } \otimes \frac { 1 } { 2 } \mathbf { H _ { x x } } ) ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) + ( \mathbf { h _ { x } } \otimes \mathbf { h _ { x } } ) ( \sigma \boldsymbol { \eta } \otimes \frac { 1 } { 2 } \mathbf { h _ { \sigma \sigma } \sigma ^ { 2 } } ) \boldsymbol { \epsilon } _ { t + 1 } \\ + ( \mathbf { h _ { x } } \otimes \frac { 1 } { 2 } \mathbf { H _ { x x } } ) ( \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } ) + ( (\boldsymbol {\sigma} , ( ) ^ {\prime}) ^ {\prime} ( ) ^ {\prime} ( ) ^ {\prime} ( ) ^ {\prime} ( ) ^ {\prime} ( ) ^ {\prime} ( ) ^ {\prime} ( ) ^ {\prime} ( ) ^ {\prime} ( ) ^ {\prime} ( ) ^ {\prime} ( ) ^ {\prime} ( ) ^ {\prime} ( ) ^ {\prime} ( ) ^ {\prime} ( ) ^ {\prime} ( ) ^ {\prime} ( ) ^ {\prime}, \\ + ( (\sigma , ( ) ^ {\prime}) ^ {\prime} ( ) ^ {\prime} ( ) ^ {\prime} ( ) ^ {\prime} ( ) ^ {\prime} ( ) ^ {\prime} ( ) ^ {\prime} ( ) ^ {\prime} ( ) ^ {\prime} ( ) ^ {\prime} ( ) ^ {\prime} ( ) ^ {\prime} ( ) ^ {\prime} ( ) ^ {\prime} ( ) ^ {\prime} ( ) ^ {\infty} ] \\ + ( (\boldsymbol {\mathrm{h}} _ { x } \otimes \frac { 1 } { 2 } \mathbf { H _ { x x }} ) ( (\boldsymbol {\mathrm{x}} _ { t + 2 } ^ { f } (\boldsymbol {\mathrm{x}} _ { t + 2 } ^ { f (\boldsymbol {\mathrm{x}} _ { t + 2 }} (\boldsymbol {\mathrm{x}} _ { t + 2 } ^ { f (\boldsymbol {\mathrm{x}} _ { t + 2 }} (\boldsymbol {\mathrm{x}} _ { t + 2 } ^ { f (\boldsymbol {\mathrm{x}} _ { t + 2 }} (\boldsymbol {\mathrm{x}} _ { t + 2 (\boldsymbol {\mathrm{x}} _ { t + 2 }} (\boldsymbol {\mathrm{x}} _ { t + 2 (\boldsymbol {\mathrm{x}} _ { t + 2 }} (\boldsymbol {\mathrm{x}} _ { t + 2 (\boldsymbol {\mathrm{x}} _ { t + 2 }} (\boldsymbol {\mathrm{x}} _ { t + 2 (\boldsymbol {\mathrm{x}} _ { t + n - 1 }} (\boldsymbol {\mathrm{x}} _ { t + n - 1 (\boldsymbol {\mathrm{x}} _ { t + n - 1 }} (\boldsymbol {\mathrm{x}} _ t + n - n - 1 (\boldsymbol {\mathrm{x}} _ t + n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n - n , \\ + ( (\sigma , ( ) ^ {\prime}) ^ {\prime} ( ) ^ {\prime} ( ) ^ {\prime} ( ) ^ {\prime} ( ) ^ {\prime} ( ) ^ {\prime} ( ) ^ {\prime} ( ) ^ {\prime} ( ) ^ {\prime} ( ) ^ {\prime} ( ) ^ {\prime} ( ) ^ {\prime} ( ) ^ {\prime} ( ) ^ {\prime} ( ) ^ {\prime}, \\ + ( (\sigma , ( ) ^ {\prime} ( ) ^ {\prime} ( ) ^ {\prime} ( ) ^ {\prime} ( ) ^ {\prime} ( ) ^ {\prime} ( ) ^ {\prime} ( ) ^ {\prime} ( ) ^ {\prime} ( ) ^ {\prime} ( ) ^ {\prime} ( ) ^ {\prime}, \\ + ( (\sigma , ( ) ^ {\prime} ( ) ^ {\prime} ( ) ^ {\prime} ( ) ^{\prime} ( )^{\prime} ( )^{\prime} ( )^{\prime} ( )^{\prime} ( )^{\prime} ( )^{\prime} ( )^{\prime}, \\ + (\sigma , ( ) ^{\prime} ( )^{\prime} ( )^{\prime} ( )^{\prime} ( )^{\prime} ( )^{\prime} ( )^{\prime} ( )^{\prime}, \\ + (\sigma , ( )^{\prime} ( )^{\prime} ( )^{\prime} ( )^{\prime} ( )^{\prime} ( )^{\prime} ( )^{\prime}, \\ + (\sigma , ( )^{\prime} ( )^{\prime} ( )^{\prime} ( )^{\prime} ( )^{\prime}, \\ + (\sigma , ( )^{\prime} ( )^{\prime} ( )^{\prime} ( )^{\prime}, \\ + (\sigma , ( )^{\prime} ( )^{\prime} ( )^{\prime}, \\ + (\sigma , ( )^{\prime} ( )^{\prime}, \\ + (\sigma , ( )^{\prime}, \\ + (\sigma , ( )^{\prime}, \\ + (\sigma ,( )^{\prime}, \\ + (\sigma ,( )^{\prime}, \\ + (\sigma ,( )^{\prime}, \\ + (\sigma ,( )^{\prime}, \\ + (\sigma ,( )^{\prime}, \\ + (\sigma ,( )^{\prime}, \\ + (\sigma ,( )^{\prime}, \\ + (\sigma ,( )^{\prime}, \\ + (\sigma ,( )^{\prime}, \\ + (\sigma ,( ), \\ + (\sigma ,( ), \\ + (\sigma ,( ), \\ + (\sigma ,( ), \\ + (\sigma ,( ), \\ + (\sigma ,( ), \\ + (\sigma ,( ), \\ + (\sigma ,( ), \\ + (\sigma ,( ), \\ + (\sigma ,( ), \\ + (\sigma ,( ), \\ + (\sigma ,( ), \\ + (\sigma ,( ), \\ + (\bar{c}_{t+1, i})^{l-1}\text{ and in general}\\ = (\bar{c}_{t+1, i})^{l-1}\text{ Note for l=0,1,2,3....Note for l=0 we have $\bar{c}_{t}$ $^{l-1}$ $\text{ and in general}$}\\ = (\bar{c}_{t+1, i})^{l-1}\text{ Note for l=0 we have $\bar{c}_{t}$ $^{l-1}$ $\text{ and in general}$}.\\ = (\bar{c}_{t+1, i})^{l-1}\text{ Note for l=0 we have $\bar{c}_{t}$ $^{l-1}$ $\text{ and in general}$}.\\ = (\bar{c}_{t+1, i})^{l-1}\text{ Note for l=0 we have $\bar{c}_{t}$ $^{l-1}$ $\text{ and in general}$}.\\ = (\bar{i}_{t+1, i})^{l-1}\text{ Note for l=0 we have $\bar{c}_{t}$ $^{l-1}$ $\text{ and in general}$}.\\ = (\bar{i}_{t+1, i})^{l-1}\text{ Note for l=0 we have $\bar{c}_{t}$ $^{l-1}$ $\text{ and in general}$}.\\ = (\bar{i}_{t+1, j})^{l-1}\text{ Note for l=0 we have $\bar{c}_{t}$ $^{l-1}$ $\text{ and in general}$}.\\ = (\bar{i}_{t+1, j})^{l-1}\text{ Note for l=0 we have $\bar{c}_{t}$ $^{l-1}$ $\text{ and in general}$}.\\ = (\bar{i}_{t+1, j})^{l-1},\\ = (\bar{i}_{t+1, j})^{l-1}\text{ Note for l=0 we have $\bar{c}_{t}$ $^{l-1}$ $\text{ and in general}$}.\\ = (\bar{i}_{t+1, j})^{l-1}\text{ Note for l=0 we have $\bar{c}_{t}$ $^{l-1}$ $\text{ and in general}$}.\\ = (\bar{i}_t\text{ and }\text{and in general}\\ = (\bar{i}_t\text{ and }\text{and in general}\\ = (\bar{i}_t\text{ and }\text{and in general}\\ = (\bar{i}_t\text{ and }\text{and in general}\\ = (\bar{i}_t\text{ and }\text{and in general}\\ = (\bar{i}_t\text{ and }\text{and in general}\\ = (\bar{i}_t\text{ and }\textand\nuon,\quad(\bar{i}_t\text{ and }\textnun,\quad(\bar{i}_t\text{ and }\text{nun,\quad(\bar{i}_t\text{ and }\text{nun,\quad(\bar{i}_t\text{ and }\text{nun,\quad(\bar{i}_t\text{ and }\text{nun,\quad(\bar{i}_t\text{ and }\text{nun,\quad(\bar{i}_t\text{ and }\text{nun,\quad(\bar{i}_t\text.})\quad(\bar{i}_t\text.})\quad(\bar{i}_t\text.})\quad(\bar{i}_t\text.})\quad(\bar{i}_t\text.})\quad(\bar{i}_t\text.))}\\ = (\bar{i}_t\text{ and }\text{nun,\quad(\bar{i}_t\text.})\quad(\bar{i}_t\text{ and }\text{nun,\quad(\bar{i}_t\text.})\quad(\bar{i}_t\text.))\quad(\bar{i}_t\text.))\\ = (\bar{i}_t\text{ and }\text{nun,\quad(\bar{i}_t\text.})\quad(\bar{i}_t\text.))\\ = (\bar{i}_t\text{ and }\text{nun,\quad(\bar{i}_t\text.})\quad(\bar{i}_t\text.))\\ = (\bar{i}_t\text{ and }\text{nun,\quad(\bar{i}_t\text.})\quad(\bar{i}_t\text.))\\ = (\bar{i}_t\text{ and }\text{nul,\quad(\bar{i}_t\text.})\quad(\bar{i}_t\text.))\\ = (\bar{i}_t\text{ and }\text{nul,\quad(\bar{i}_t\text.})\quad(\bar{i}_t\text.))\\ = (\bar{i}_t\text{ and }\text{nul,\quad(\bar{i}_t\text.})\quad(\bar{i}_t\text.))\\ = (\overline{{i_{j_0}}})^{l-1}\overline{{i_{j_0}}}\overline{{i_{j_0}}}\overline{{i_{j_0}}}\overline{{i_{j_0}}}\overline{{i_{j_0}}}\overline{{i_{j_0}}}\overline{{i_{j_0}}}\overline{{i_{j_0}}}\overline{{i_{j_0}}}\overline{{i_{j_0}}}\overline{{i_{j_0}}}\\ = (\overline{{i_{j_0}}})^{l-1}\overline{{i_{j_0}}}\overline{{i_{j_0}}}\overline{{i_{j_0}}}\overline{{i_{j_0}}}\overline{{i_{j_0}}}\overline{{i_{j_0}}}\overline{{i_{j_0}}}\\ = (\overline{{i_{j_0}}})^{l-1}\overline{{i_{j_0}}}\overLine{{i_{j_0}}}\overline{{i_{j_0}}}\overline{{i_{j_0}}}\overline{{i_{j_0}}}\overline{{i_{j_0}}}\\ = (\overline{{i_{j_0}}})^{l-1}\overline{{i_{j_0}}}\overline{{i_{j_0}}}\overline{{i_{j_0}}}\overline{{i_0}}\\ = (\overline{{i_{j_0}}})^{l-1}\overline{{i_{j_0}}}\overline{{i_{j_0}}}\overline{{i_{j_0}}}\overline{{i_{j_0}}}\overline{{i_{j_0}}}\\ = (\overline{{i_{j_0}}})^{l-1}\overline{{i_{j_0}}}\overline{{i_{j_0}}}\overline{{i_{j_0}}}\overLine{{i_{j_0}}}\overline{{i_{j_0}}}\overline{{i_{j_0}}}\\ = (\overline{{i_{j_0}}})^{l-1}\overline{{i_{j_0}}}\overline{{i_{j_0}}}\overline{{i_{j_0}}}\overline{{i_{j_0}}}\\ = (\overline{{i_{j_0}}})^{l-1}\overline{{i_{j_0}}}\overline{{i_{j_0}}}\overline{{i_{j_0}}}\overline{{i_{j_0}}}\\ = (\overline{{i_{j_0}}})^{l+1}\overline{{i_{j_0}}}\overline{{i_{j_0}}}\overline{{i_{j_0}}}\overline{{i_{j_0}}}\\ = (\overline{{i_{j_0}}})^{l+1}\overline{{i_{j_0}}}\overline{{i_{j_0}}}\overline{{i_{j_0}}}\\ = (\overline{{i_{j_0}}})^{l+1}\overline{{i_{j_0}}}\overline{{i_{j_0}}}\overline{{i_{j_0}}}\\ = (\overline{{i_{j_0}}})^{l+1}\overline{{i}_{j_0}}\overline{{i_{j_0}}}\\ = (\overline{{i_{j_0}}})^{l+1}\overline{{i_{j_0}}}\\ = (\overline{{i_{j_0}}})^{l+1}\overline{{i_{j_0}}}\\ = (\overline{{i_{j_0}}})^{l+1}\overline{{i_{j_0}}}\\ = (\overline{{i_{j_0}}})^{l+1}\overline{{i_{j_0}}}\\ = (\overline{{i_{j_n}}})^{l+1}\overline{{i_{j_n}}}\\ = (\overline{{i_{j_n}}})^{l+1}\overline{{i_{j_n}}}\\ = (\overline{{i_{j_n}}})^{l+1}\overline{{i_{j_n}}}\\ = (\overline{{i_{j_n}}})^{l+1}\overline{{i_{j_n}}}\\ = (\overline{{i_{j_n}}})^{l+1}\overline{{i_{j,n}}}\\ = (\overline{{i_{j_n}}})^{l+1}\overline{{i_{j,n}}}\\ = (\overline{{i_{j_n}}})^{l+1}\overline{{i_{j,n}}}\\ = (\overline{{i_{j_n}}})^{l+1}\overline{{i_{j,n}}}\\ = (\overline{{i_{j_n}}})^{l+1}\overline{{i_{j_n}}}\\ = (\overline{{i_{j_n}}})^{l+1}\overline{{i_{j,n}}}\\ = (\overline{{i_{j_n}}})^{l+1}\overline{{i_{j,n}}}\\ = (\overline { i}_{k,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l, l, l, l, l, l, l, l, l, l, l, l, l, l, l, l, l, l, l, l, l, l, l, l, l, l, l, l, l, l, l, l, l, l, l, l, l, l, l, l, l, l, l, l, l, l, l, l, l, l, l, .}\\ = (-6)^{-4}(x)\\ = (-6)^{-4}(x)\\ = (-6)^{-4}(x)\\ = (-6)^{-4}(x)\\ = (-6)^{-4}(x)\\ = (-6)^{-4}(x)\\ = (-6)^{-4}(x)\\ = (-6)^{-4}(x)\\ = (-6)^{-4}(x)\\ = (-6)^{-4}(x)\\ = (-6)^{-4}(x) \\ = (-6)^{-4}(x)\\ = (-6)^{-4}(x)\\ = (-6)^{-4}(x)\\ = (-6)^{-4}(x)\\ = (-6)^{-4}(x)\\ = (-6)^{-4}(x)\\ = (-6)^{-4}(x)\\ = (-6)^{-4}(x)\\ = (-6)^{-4}(x)\\ = (-6)^{-4}(x) \(;\\ = (-6)^{-4}(x)\\ = (-6)^{-4}(x)\\ = (-6)^{-4}(x)\\ = (-6)^{-4}(x)\\ = (-6)^{-4}(x)\\ = (-6)^{-4}(x)\\ = (-6)^{-4}(x)\\ = (-6)^{-4}(x)\\ = (-6)^{-4}(x)\\ = (-6)^{-4}(x)\end{array}\]

We therefore have

\[\begin{array} { l } \tilde { \mathbf { x } } _ { t + l } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + l } ^ { s } = ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { l } \left( \tilde { \mathbf { x } } _ { t } ^ { f } \otimes \tilde { \mathbf { x } } _ { t } ^ { s } \right) + \sum _ { i = 0 } ^ { l - 1 } ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { l - 1 - i } ( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } ) \left( \tilde { \mathbf { x } } _ { t + i } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + i } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + i } ^ { f } \right) \\ + \sum _ { i = 0 } ^ { l - 1 } ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { l - 1 - i } ( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } ) \tilde { \mathbf { x } } _ { t + i } ^ { f } \\ + \sum _ { i = 0 } ^ { l - 1 } ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { l - 1 - i } ( \sigma \eta \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } ) \delta _ { t + 1 + i } \\ + \sum _ { i = 0 } ^ { l - 1 } ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { l - 1 - i } ( \sigma \eta \otimes \mathbf { h } _ { \mathbf { x } } ) ( \delta _ { t + 1 + i } \otimes \tilde { \mathbf { x } } _ { t + i } ^ { s } ) \\ + \sum _ { i = 0 } ^ { l - 1 } ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { l - 1 - i } ( \sigma \eta \otimes \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x} } ) ( \delta _ { t + 1 + i } \otimes \tilde { \mathbf { x } } _ { t + i } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + i } ^ { f } ) \\ \\ + \sum _ { i = 0 } ^ { l - 1 } ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { l - 1 - i } ( \sigma \eta \otimes \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x} } ) ( \tilde { \mathbf { x } } _ { t + i } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + i } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + i } ^ { f } ) \\ \\ E _ { t }\left[ \tilde { \mathbf { x } } _ { t + l } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + l } ^ { s} - \mathbf { x} _ { t + l } ^ { f} \otimes \mathbf { x} _ { t + l } ^ { s} \right] \\ = E _ { t }\left[ ( \mathbf { h } _ {\mathbf { x}} \otimes \mathbf { h _ {\alpha }} ) ^ { l }\right) ( \tilde {\mathbf {\Delta}} _ {\boldsymbol {\tau}} ^ {\boldsymbol {\tau}} (\boldsymbol {\tau}) (\tilde {\mathbf {\Delta}} _ {\boldsymbol {\tau}} ^ {\boldsymbol {\tau}} (\boldsymbol {\tau})) + \sum _ { i = 0} ^ { l - 1 } ( \mathbf {\Delta h} _ {\boldsymbol {\tau}} (\boldsymbol {\tau}) (\tilde {\mathbf {\Delta}} _ {\boldsymbol {\tau}} ^ {\boldsymbol {\tau}} (\boldsymbol {\tau})) (\tilde {\mathbf {\Delta}} _ {\boldsymbol {\tau}} ^ {\boldsymbol {\tau}} (\boldsymbol {\tau})) (\tilde {\mathbf {\Delta}} _ {\boldsymbol {\tau}} ^ {\boldsymbol {\tau}} (\boldsymbol {\tau})) (\tilde {\mathbf {\Delta}} _ {\boldsymbol {\tau}} ^ {\boldsymbol {\tau}} (\boldsymbol {\tau})) (\tilde {\mathbf {\Delta}} _ {\boldbm {\tau}} ^ {\boldsymbol {\tau}} (\boldsymbol {\tau})) (\tilde {\mathbf {\Delta}} _ {\boldsymbol {\tau}} ^ {\boldsymbol {\tau}} (\boldsymbol {\tau})) (\tilde {\mathbf {\Delta}} _ {\boldsymbol {\tau}} ^ {\boldsymbol {\tau}} (\boldsymbol {\tau})) (\tilde {\mathbf {\Delta}} _ {\boldsymbol {\tau}} ^ {\boldsymbol {\tau}} (\boldbm {\tau})) (\tilde {\mathbf {\Delta}} _ {\boldsymbol {\tau}} ^ {\boldsymbol {\tau}} (\boldsymbol {\tau})) (\tilde {\mathbf {\Delta}} _ {\boldsymbol {\tau}} ^ {\boldsymbol {\tau}} (\boldsymbol {\tau})) (\tilde {\mathbf {\Delta}} _ {\boldsymbol {\tau}} ^ {\boldsymbol {\tau}} (\boldsymbol {\tau})) (\tilde {\mathbf {\Delta}}_{ t + i} ^ {- 1}) \\ + & \\ + & \\ + & \\ + & \\ + & \\ + & \\ + & \\ + & \\ + & \\ + & \\ + & \\ + & \\ + & \\ + & \\ + & \\ + & \\ + & \\ + & \\ + & \\ + & \\ + & \\ + & \\ + & \\ + & \\ + & \\ + & \\ + & \\ + & \\ + & \\ + & \\ + & \\ + & \\ + & \\ + &\\ + & \\ + & \\ + & \\ + & \\ + & \\ + & \\ + & \\ + & \\ + & \\ + & \\ + & \\ + & \\ + & \\ + & \\ + & \\ + & \\ + & \\ + & \\ + & \\ + & \\ + & \\ + & \\ + & \\ + & \\ + & \\ + & \\ + & \\ + & \\ + & \\ + & \\ + & \\ + & \\ + & \\ - [ ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( | v a r e c o n s t a r e s t a r e s t a r e s t a r e s t a r e s t a r e s t a r e s t a r e s t a r e s t a r e s t a r e s t a r e s t a r e s t a r e s t a r e s t a r e s t a r e s t a r e s t a r e s t a r e s t a r / v a r e d u n g e n d y n o w k e n d y n o w k e n d y n o w k e n d y n o w k e n d y n o w k e n d y n o w k e n d y n o w k e n d y n o w k e n d y n o w k e n d y n o w k e n d y n o w k e n d y n o w k e n d y n o w k e n d z o w k e n d z o w k e n d z o w k e n d z o w k e n d z o w k e n d z o w k e n d z o w k e n d z o w k e n d z o w k e n d z o w k e n d z o w k e n d z o w k e n d z o w k e n d z o w k e n d z o o w k e n d z o w k e n d z o w k e n d z o w k e n d z o w k e n d z o w k e n d z o w k e n d z o w k e n d z o w k e n d z o w k e n d z o w k e n d z o w k e n d z o w k e n d z o w k e n d z o v a w k e n d z o v a w k e n d z o v a w k e n d z o v a w k e n d z o v a w k e n d z o v a w k e n d z o v a w k e n d z o v a w k e n d z o v a w k e n d z o v a w k e n d z o v a w k e n d z o v a v k e n d z o v a v k e n d z o v a v k e n d z o v a v k e n d z o v a v k e n d z o v a v k e n d z o v a v k e n d z o v a v k e n d z o v a v k e n d z o v a v k e n d z o v a v k e n d z o v a v , v a v , v a v , v a v , v a v , v a v , v a v , v a v , v a v , v a v , v a v , v a v , v a v , v a v , v a v , v a v , v a v , v a v , v a v , v a v , v a v , v a v , v a v , v a v , v a v , v a v, v a v , v a v , v a v , v a v , v a v , v a v , v a v , v a v , v a v , v a v , v a v ,v a . \\ = E _ { t }\left[ (\sum_ {(i = 0)} ^ {(l - 1)} (\sum_ {(i = 0)} ^ {(l - 1)} (\sum_ {(i = 0)} ^ {(l - 1)} (\sum_ {(i = 0)} ^ {(l - 1)} (\sum_ {(i = 0)} ^ {(l - 1)} (\sum_ {(i = 0)} ^ {(l - 1)} (\sum_ {(i = 0)} ^ {(l - 1)} (\sum_ {| i | > 0} ^ {(l - 1)} (\sum_ {| i | > 0} ^ {(l - 1)} (\sum_ {| i | > 0} ^ {(l - 1)} (\sum_ {| i | > 0} ^ {(l - 1)} (\sum_ {| i | > 0} ^ {(l - 1)} (\sum_ {| i | > 0} ^ {(l - 1)} (\sum_ {| i | >0} ^ {(l - 1)} (\sum_ {| i | > 0} ^ {(l - 1)} (\sum_ {| i | > 0} ^ {(l - 1)} (\sum_ {| i | > 0} ^ {(l - 1)} (\sum_ {| i | > 0} ^ {(l - 1)} (\sum_ {| i | > 0} ^ {(l - 1)} (\sum_ {| i |\to 0} ^ {(l - 1)} (\sum_ {| i | > 0} ^ {(l - 1)} (\sum_ {| i | > 0} ^ {(l - 1)} (\sum_ {| i | > 0} ^ {(l - 1)} (\sum_ {| i | > 0} ^ {(l - 1)} (\sum_ {| i | > 0} ^ {(l - 1)} (\sum_ {| j | > j} ^ {(l - 1)} (\sum_ {| j | > j} ^ {(l - 1)} (\sum_ {| j | > j} ^ {(l - 1)} (\sum_ {| j | > j} ^ {(l - 1)} (\sum_ {| j | > j} ^ {(l - 1)} (\sum_ {| j | > j} ^ {(l - 1)} (\sum_ {| j | > j} ^ {(j | j)} (\sum_ {| j | > j} ^ {(j | j)} (\sum_ {| j | > j} ^ {(j | j)} (\sum_ {| j | > j} ^ {(j | j)} (\sum_ {| j | > j} ^ {(j | j)} (\sum_ {| j | > j} ^ {(j | j)} (\sum_ {| j | > j} ^ {(j | j)} (\sum_ | j | > j.5) | V A R S U T O N D I N G U N D I N G U N D I N G U N D I N G U N D I N G U N D I N G U N D I N G U N D I N G U N D I N G U N D I N G U N D I N G U N D I N G U N D I N G U N D I N G U N D I N G U N D I N G U N D I N G U N D IN G U N D I N G U N D I N G U N D I N G U N D I N G U N D I N G U N D I N G U N D I N G U N D I N G U N D I N G U N D I N G U N D I N G U N D I N G U N D I N G U N D I N G U N D I N G U N D I N G U N M O W K L O W K L O W K L O W K L O W K L O W K L O W K L O W K L O W K L O W K L O W K L O W K L O W K L O W K L O W K L O W K L O W K L O W K L O W K L O W K L O W K L O W K L O W K L O W K L O W K L O W KL O W K L O W K L O W K L O W K L O W K L O W K L O W K L O W K L O W K L O W K L O W K L O W K L O W K L O W K L O W K L O W K L O W K L O W K L O W K L O W K L O W K L O W K L O W K L O W K L O W K l O W K l O W K l O W K l O W K l O W K l O W K l O W K l O W K l O W K l O W K l O W K l O W K l O W K l O W K l O W K l O W K l O W K l O W K l O W K l O W K l O W K l O W K l O W K l O W K l O W K | V A R S U T O N D I N G U N D I N G U N D I N G U N D I N G U N D I N G U N D I N G U N D I N G U N D I N G U N D I N G U N D I N G U N D I N G U N D I N G U N D I N G U N D I N G U N D I N G U N C | V A R S U T O N D I N G U N C | V A R S U T O N D I N G U N C | V A R S U T O N D I N G U N C | V A R S U T O N D I N G U N C | V A R S U T O N D I N G U N C | V A R S U T O N D I N G U N C | V A R S U T O N D IN G U N C | V A R S U T O N D IN G U N C | V A R S U T O N D IN G U N C | V A R S U T O N D IN G U N C | V A R S U T O N D IN G U N C | V A R S U T O N D IN G U N C | V A R S U T O N D IN G U N c | V A R S U T O N D IN G U N c | V A R S U T O N D IN G U N c | V A R S U T O N D IN G U N c | V A R S U T O N D IN G U N c | V A R S U T O N D IN G UN c | V A R S U T O N D IN G UN c | V A R S U T O N D IN GUN c | V A R S U T O N D IN GUN c | V A R S U T O N D IN GUN c | V A R S U T O N D IN GUN c | V A R S U T O N D IN GUN c | V A R S U T O N D IN g u n g u n g u n g u n g u n g u n g u n g u n g u n g u n g u n g u n g u n g u n g u n g u n g u n g u n g u n g u n g u n g u n g u n g u n g u n g u n g u n g u n g u n g u n g u n g u n g u n g u ng u n g u n g u n g u n g u n g u n g u n g u n g u n g u n g u n g u n g u n g u n g u n g u n g u n g u n g u n g u n g u n g u n g u n g u n g u n g u n g u n g u n g u n g u n g u n g u m p h i m p h i m p h i m p h i m p h i m p h i m p h i m p h i m p h i m p h i m p h i m p h i m p h i m p h i m p h i m p h i m p h i m p h i m p h i m p h i m p h i m p h i m p h i m p h i m p h i m p h /m p h /m p h /m p h /m p h /m p h /m p h /m p h /m p h /m p h /m p h /m p h /m p h /m p h /m p h /m p h /m p h /m p h /m p h /m p h /m p h /m p h /m p h /m p h /m p h /m p h /n m p h /m p h /m p h /m p h /m p h /m p h /m p h /m p h /m p h /m p h /m p h /m p h /m p h /m p h /m p h /m p h /m p h /m p h /m p h /m p h /m p h /m p h /m p h /m p h /m p h/ m p h/ m p h/ m p h/ m p h/ m p h/ m p h/ m p h/ m p h/ m p h/ m p h/ m p h/ m p h/ m p h/ m p h/ m p h/ m p h/ m p h/ m p h/ m p h/ m p h/ m p h/ m p h/ m p h/ m p h/ m p h/ M P H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H HH H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H H HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHhI ) ] = E _ {}^ {} [ E _ {}^ {} ] = E _ {}^ {} [ E _ {}^ {} ] = E _ {}^ {} [ E _ {}^ {} ] = E _ {}^ {} [ E _ {}^ {} ] = E _ {}^ {} [ E _ {}^ {} ] = E _ {}^ {} [ E _ {}^ {} ] = E _ {}^ {} [ E _ {}^ {} ] = E _ {}^ {} [ E _ {}^ {} ] = E _ {}^ {} [E _ {}^ {} ] = E _ {}^ {} [ E _ {}^ {} ] = E _ {}^ {} [ E _ {}^ {} ] = E _ {}^ {} [ E _ {}^ {} ] = E _ {}^ {} [ E _ {}^ {} ] = E _ {}^ {} [ E _ {}^ {} ] = E _ {}^ {} [ E _ {}^ {} ] = E _ {}^{} [ E _ {}^{} ] = E _ {}^{} [ E _ {}^{} ] = E _ {}^{} [ E _ {}^{} ] = E _ {}^{} [ E _ {}^{} ] = E _ {}^{} [ E _ {}^{} ] = E _ {}^{} [ E _ {}^{} ] = E _ {}^{} [ E _ {}^{} ] = E _ {}^{} [ E _ {}^{} ] = E _ {}^{} [E _ {}^{} ] = E _ {}^{} [E _ {}^{} ] = E _ {}^{} [E _ {}^{} ] = E _ {}^{} [E _ {}^{} ] = E _ {}^{} [E _ {}^{} ] = E _ {}^{} [E _ {}^{} ] = E _ {}^{} [E _ {}^{} ] = E _ {}^{} [E _ {}^{} ] = E _ {{\cal M}} [ B ] [ B ] [ B ] [ B ] [ B ] [ B ] [ B ] [ B ] [ B ] [ B ] [ B ] [ B ] [ B ] [ B ] [ B ] [ B ] [ B ] [ B ] [ B ] [ B ] [ B ] [ B ] [ B ] [ B ] [ B ] [ B ] [ B ] [ B ] [ B ] [ B ] [ B ] [ B ] [ B ] [ B ][ B ] [ B ] [ B ][ B ][ B ][ B ][ B ][ B ][ B ][ B ][ B ][ B ][ B ][ B ][ B ][ B ][ B ][ B ][ B ][ B ][ B ][ B ][ B ][ B ][ B ][ B ][ B ][ B ][ B ][ B ][ B ][ B ][ B ][ B ][ B ][ B ][ b ][ B ], [ b ], [ b ], [ b ], [ b ], [ b ], [ b ], [ b ], [ b ], [ b ], [ b ], [ b ], [ b ], [ b ], [ b ], [ b ], [ b ], [ b ], [ b ], [ b ], [ b ], [ b ], [ b ], [ b ], [ b ], [ b ], [ b ], [ b ], [ b ], [ b ], [ b ], [ b ], [ b ], [ b ], [ b ].\]

\[\begin{array} { l } = E _ { t } [ \sum _ { i = 0 } ^ { l - 1 } ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { l - 1 - i } ( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } ) ( \tilde { \mathbf { x } } _ { t + i } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + i } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + i } ^ { f } - \mathbf { x } _ { t + i } ^ { f } \otimes \mathbf { x } _ { t + i } ^ { f } \otimes \mathbf { x } _ { t + i } ^ { f } ) \\ + \sum _ { i = 0 } ^ { l - 1 } ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { l - 1 - i } ( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } ) ( \tilde { \mathbf { x } } _ { t + i } ^ { f } - \mathbf { x } _ { t + i } ^ { f } ) \\ + ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { l - 1 } ( \sigma \boldsymbol { \eta } \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } ) \delta _ { t + 1 } \\ + ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { l - 1 } ( \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x } } ) ( \delta _ { t + 1 } \otimes \tilde { \mathbf { x } } _ { t } ^ { s } - \epsilon _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { s } ) \\ + \sum _ { i = 1 } ^ { l - 1 } ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { l - 1 - i } ( \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x } } ) ( \delta _ { t + 1 + i } \otimes \tilde { \mathbf { x } } _ { t + i } ^ { s } - \epsilon _ { t + 1 + i } \otimes \mathbf { x } _ { t + i } ^ { s } ) \\ + ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { l - 1 } ( \sigma \boldsymbol { \eta } \otimes \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x} } ) ( \delta _ { t + 1 } \otimes \tilde { \mathbf { x } } _ { t } ^ { f } \otimes \tilde { \mathbf { x } } _ { t } ^ { f } - \epsilon _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f} ) \\ + \sum _ { i = 1 } ^ { l - 1 } ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { l - 1 - i } ( \sigma \boldsymbol { \eta } \otimes \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x} } ) ( \delta _ { t + 1 + i } \otimes \tilde { \mathbf { x } } _ { t + i } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + i } ^ { f } - \epsilon _ { t + 1 + i } \otimes \mathbf { x} _ { t + i } ^ { f } \otimes \mathbf { x} _ { t + i} ^ { f} ) ] \\ = \\ = \\ [ 2 ] = \\ [ 2 ] = \\ [ 2 ] = \\ [ 2 ] = \\ [ 2 ] = \\ [ 2 ] = \\ [ 2 ] = \\ [ 2 ] = \\ [ 2 ] = \\ [ 2 ] = \\ [ 2 ] = \\ [ 2 ] = \\ [ 2 ] = \\ [ 2 ] = \\ [ 2 ] = \\ [ 2 ] = \\ [ 2 ] = \\ [ j ] = \\ [ j ] = \\ [ j ] = \\ [ j ] = \\ [ j ] = \\ [ j ] = \\ [ j ] = \\ [ j ] = \\ [ j ] = \\ [ j ] = \\ [ j ] = \\ [ j ] = \\ [ j ] = \\ [ j ] = \\ [ j ] = \\ [ j ] = \\ [ j ] = \\ [ j ] = \\ [ j ] = \\ [ j ] = \\ [ k ] = \\ [ k ] = \\ [ k ] = \\ [ k ] = \\ [ k ] = \\ [ k ] = \\ [ k ] = \\ [ k ] = \\ [ k ] = \\ [ k ] = \\ [ k ] = \\ [ k ] = \\ [ k ] = \\ [ k ] = \\ [ k ] = \\ [ k ] = \\ [ k ] = \\ [ k ] = \\ [ k ] = \\ [ k ] = \\ [ j ] = \\ [ j ] = \\ [ j ] = \\ [ j ] = \\ [ j ] = \\ [ j ] = \\ [ j ] = \\ [ j ] = \\ [ j ] = \\ [ j ] = \\ [ j ] = \\ [ j ] = \\ [ j ] = \\ [ j ] = \\ [ j ] = \\ [ j ] = \\ [ j ] = \\ [ j ] = \\ [ j ] = \\ [ n ] = \\ [ n ] = \\ [ n ] = \\ [ n ] = \\ [ n ] = \\ [ n ] = \\ [ n ] = \\ [ n ] = \\ [ n ] = \\ [ n ] = \\ [ n ] = \\ [ n ] = \\ [ n ] = \\ [ n ] = \\ [ n ] = \\ [ n ] = \\ [ n ] = \\ [ n ] = \\ [ n ] = \\ [ n ] = \\ [ m ] = \\ [ m ] = \\ [ m ] = \\ [ m ] = \\ [ m ] = \\ [ m ] = \\ [ m ] = \\ [ m ] = \\ [ m ] = \\ [ m ] = \\ [ m ] = \\ [ m ] = \\ [ m ] = \\ [ m ] = \\ [ m ] = \\ [ m ] = \\ [ m ] = \\ [ m ] = \\ [ m ] = \\ [ m ] = \\ [ n ] = \\ [ n ] = \\ [ n ] = \\ [ n ] = \\ [ n ] = \\ [ n ] = \\ [ n ] = \\ [ n ] = \\ [ n ] = \\ [ n ] = \\ [ n ] = \\ [ n ] = \\ [ n ] = \\ [ n ] = \\ [ n ] = \\ [ n ] = \\ [ n ] = \\ [ n ] = \\ [ n ] = \\ [ q u a d o r e c o n s i o n g e r e c o n s i o n g e r e c o n s i o n g e r e c o n s i o n g e r e c o n s i o n g e r e c o n s i o n g e r e c o n s i o n g e r e c o n s i o n g e r e c o n s i o n g e r e c o n s i o n g e l o w i v e r y , p o r a l l o w i v e r y , p o r a l l o w i v e r y , p o r a l l o w i v e r y , p o r a l l o w i v e r y , p o r a l l o w i v e r y , p o r a l l o w i v e r y , p o r a l l o w i v e r y , p o r a l l l o w i v e r y , p o r a l l o w i v e r y , p o r a l l l o w i v e r y , p o r a l l l o w i v e r y , p o r a l l l o w i v e r y , p o r a l l l o w i v e r y , p o r a l l l o w i v e r y , p o r a l l l o w i v e r y , p o r a l l l o w i v e r y . p o r a l l l o w i v e r y , p o r a l l l o w i v e r y , p o r a l l l o w i v e r y , p o r a l l l o w i v e r y , p o r a l l l o w i v e r y , p o r a l l l o w i v e r y , p o r a l l l o w u v e r y , p o r a l l l o w u v e v e r y , p o r a l l l o w u v e v e r y , p o r a l l l o w u v e v e r y , p o r a l l l o w u v e v e v e r y , p o r a l l l o w u v e v e v e v e r y , p o r a l l l o w u v e v e v e v e v e v e v e v e v e v e v e v e v e v e v e v e v e v e v e v e v e v e v e v e v e v e v e v e v e v e v e v e v e v e v e v e v e v e v e v e v e v e v e v e v e v e v e v e v e v e v & p o r a l l l o w i v e r y , p o r a l l l o w i v e r y , p o r a l l l o w i v e r y , p o r a l l l o w i v e r y , p o r a l l l o w i v e r y , p o r a l l l o w i v e r y , p o r a l l l o w j u a d u s s i g h t h a b s t h a b s t h a b s t h a b s t h a b s t h a b s t h a b s t h a b s t h a b s t h a b s t h a b s t h a b s t h a b s t h a b s t h a b s t h a b s t h a b s t h a b s t h a b s t h a b s t h a b c u s s i g h t h a b c u s s i g h t h a b c u s s i g h t h a b c u s s i g h t h a b c u s s i g h t h a b c u s s i g h t h a b c u s s i g h t h a b c u s s i g h t h a b c u s s i g h t h a b c u s s i g h . p o r a l l | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | :---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|j|\kappa|\kappa|\kappa|\kappa|\kappa|\kappa|\kappa|\kappa|\kappa|\kappa|\kappa|\kappa|\kappa|\kappa|\kappa|\kappa|\kappa|\kappa|\kappa|\kappa|\kappa|\kappa|\kappa|\kappa|\kappa|\kappa|\kappa|\kappa|\kappa|\kappa|\kappa|\kappa|\kappa|\kappa |\kappa|\kappa|\kappa|\kappa|\kappa|\kappa|\kappa|\kappa|\kappa|\kappa|\kappa|\kappa|\kappa|\kappa|\kappa|\kappa|\kappa|\kappa|\kappa|\kappa|\kappa|\kappa|\kappa|\kappa|\kappa|\kappa|\kappa|\kappa|\kappa|\kappa|\kappa|\kappa|\kappa|\mbox{\rm j} / {\rm j} / {\rm j} / {\rm j} / {\rm j} / {\rm j} / {\rm j} / {\rm j} / {\rm j} / {\rm j} / {\rm j} / {\rm j} / {\rm j} / {\rm j} / {\rm j} / {\rm j} / {\rm j} / {\rm j} / {\rm j} / {\rm j} / {\rm j} / {{\rm j}} / {{\rm j}} / {{\rm j}} / {{\rm j}} / {{\rm j}} / {{\rm j}} / {{\rm j}} / {{\rm j}} / {{\rm j}} / {{\rm j}} / {{\rm j}} / {{\rm j}} / {{\rm j}} / {{\rm j}} / {{\rm j}} / {{\rm j}} / {{\rm j}} / {{\bm {\mu}}} / {{\bm {\mu}}} / {{\bm {\mu}}} / {{\bm {\mu}}} / {{\bm {\mu}}} / {{\bm {\mu}}} / {{\bm {\mu}}} / {{\bm {\mu}}} / {{\bm {\mu}}} / {{\bm {\mu}}} / {{\bm {\mu}}} / {{\bm {\mu}}} / {{\bm {\mu}}} / {{\bm {\mu}}} / {{\bm {\mu}}} /{{\bm {\mu}}} / {{\bm {\mu}}} / {{\bm {\mu}}} / {{\bm {\mu}}} / {{\bm {\mu}}} / {{\bm {\mu}}} / {{\bm {\mu}}} / {{\bm {\mu}}} / {{\bm {\mu}}} / {{\bm {\mu}}} / {{\bm {\mu}}} / {{\bm {\mu}}} / {{\bm {\mu}}} / {{\bm {\mu}}} / {{\bmod k}} / {{\bmod k}} / {{\bmod k}} / {{\bmod k}} / {{\bmod k}} / {{\bmod k}} / {{\bmod k}} / {{\bmod k}} / {{\bmod k}} / {{\bmod k}} / {{\bmod k}} / {{\bmod k}} / {{\bmod k}} / {{\bmod k}} / {{\bmod k}} / {}^i ) & ; & ; & ; & ; & ; & ; & ; & ; & ; & ; & ; & ; & ; & ; & ; & ; & ; & ; & ; & ; & ; & ; & ; & ; & ; & ; & ; & ; & ; & ; & ; & ; & ; & ; & ; & ; & ; & ; & ; & ; & ; & ; & ; & ; & ; & ; & ; & ; & ; & ; & ? & ? & ? & ? & ? & ? & ? & ? & ? & ? & ? & ? & ? & ? & ? & ? & ? & ? & ? & ? & ? & ? & ? & ? & ? & ? & ? & ? & ? & ? & ? & ? & ? & ? & ? & ? & ? & ? & ? & ? & ? & ? & ? & ? & ? & ? & ? & ? & ? & ? & !\\n& n& n& n& n& n& n& n& n& n& n& n& n& n& n& n& n& n& n& n& n& n& n& n& n& n& n& n& n& n& n& n& n& n& n& n& n& n& n& n& n& n& n& n& n& n& n& n& n& n& n& .\\n& m& m& m& m& m& m& m& m& m& m& m& m& m& m& m& m& m& m& m& m& m& m& m& m& m& m& m& m& m& m& m& m& m& m& m& m& m& m& m\\n& q& q& q&& q&& q&& q&& q&& q&& q&& q&& q&& q&& q&& q&& q&& q&& q&& q&& q&& q&& q&& q&& q&& q&& q&& q&& q&& q&& q&& q&& q&& q&& q&& q&& q&& q&& q&& q&& q\\n& z& z& z&& z&& z&& z&& z&& z&& z&& z&& z&& z&& z&& z&& z&& z&& z&& z&& z&& z&& z&& z&& z&& z&& z&& z&& z&& z&& z&& z&& q\\n& c& c& c& c& c& c& c& c& c& c& c& c& c& c& c& c& c& c& c& c& c& c& c& c& c& c)& c& c& c& c& c& c& c& c & c & c & c & c & c & c & c & c & c & c & c & c & c & c & c & q\\n& d& d & d & d & d & d & d & d & d & d & d & d & d & d & d & d & d & d & d & d & d & d & d & d & d & d & d & d & d & d & d & d & d & d & d & d & q\\n& g&( g)& g&( g)& g&( g)& g&( g)& g&( g)& g&( g)& g&( g)& g&( g)& g&( g)& g&( g)& g&( g)& g&( g)& g&( g)& g&( g)& g&( g)& g&( g)& g&( g)& g&( g)& g&( g)& g&( g)& g&( g)& g&( g)& g&( g)& g&( g)& g&( g)& g&( g)\\n& f(j)\\n &= f(j)\\n &= f(j)\\n &= f(j)\\n &= f(j)\\n &= f(j)\\n &= f(j)\\n &= f(j)\\n &= f(j)\\n &= f(j)\\n &= f(j)\\n &= f(j)\\n &= f(j)\\n &= f(j)\\n &= f(j)\\n &= f(j)\\n &= f(j)\\n &= f(j) \( ^{i,j}{}^{\prime}\cdots{}^{\prime}\cdots{}^{\prime}\cdots{}^{\prime}\cdots{}^{\prime}\cdots{}^{\prime}\cdots{}^{\prime}\cdots{}^{\prime}\cdots{}^{\prime}\cdots{}^{\prime}\cdots{}^{\prime}\cdots{}^{\prime}\cdots{}^{\prime}\cdots{}^{\prime}\cdots{}^{\prime}\cdots{}^{i,j}{}^{\prime}\cdots{}^{\prime}\cdots{}^{\prime}\cdots{}^{\prime}\cdots{}^{\prime}\cdots{}^{\prime}\cdots{}^{\prime}\cdots{}^{\prime}\cdots{}^{\prime}\cdots{}^{\prime}\cdots{}^{\prime}\cdots{}^{\prime}\cdots{}^{\prime}\cdots{}^{\prime}\end{array}\]

because is a function of which is a function of . The zero-mean iid innovations therefore implies that, and

The same argument implies that

\[E _ {t} \left[ \left(\boldsymbol {\epsilon} _ {t + 1 + i} \otimes \tilde {\mathbf {x}} _ {t + i} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + i} ^ {f}\right) \right] = \mathbf {0}\]

\[\begin{array}{l} = \sum_ {i = 0} ^ {l - 1} \left(\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}\right) ^ {l - 1 - i} \left(\mathbf {h} _ {\mathbf {x}} \otimes \frac {1}{2} \mathbf {H} _ {\mathbf {x x}}\right) \left(\tilde {\mathbf {x}} _ {t + i} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + i} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + i} ^ {f} - \mathbf {x} _ {t + i} ^ {f} \otimes \mathbf {x} _ {t + i} ^ {f} \otimes \mathbf {x} _ {t + i} ^ {f}\right) \\ + \sum_ {i = 0} ^ {l - 1} \left(\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}\right) ^ {l - 1 - i} \left(\mathbf {h} _ {\mathbf {x}} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}\right) \left(\tilde {\mathbf {x}} _ {t + i} ^ {f} - \mathbf {x} _ {t + i} ^ {f}\right) \\ + \left(\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}\right) ^ {l - 1} \left(\sigma \boldsymbol {\eta} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}\right) \boldsymbol {\nu} \\ + \left(\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}\right) ^ {l - 1} \left(\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}\right) (\boldsymbol {\nu} \otimes \mathbf {x} _ {t} ^ {s}) \\ + \left(\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}\right) ^ {l - 1} \left(\sigma \boldsymbol {\eta} \otimes \frac {1}{2} \mathbf {H} _ {\mathbf {x x}}\right) (\boldsymbol {\nu} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) \\ \text {the shock hitting in period t + 1}\Longrightarrow \tilde {\mathbf {x}} _ {t} ^ {f} = \mathbf {x} _ {t} ^ {f} \text {and}\tilde {\mathbf {x}} _ {t} ^ {s} = \mathbf {x} _ {t} ^ {s} \end{array}\]

\[\begin{array}{r l} & {= \sum_ {i = 0} ^ {l - 1} \left(\mathbf {h _ {x}} \otimes \mathbf {h _ {x}}\right) ^ {l - 1 - i} \left(\mathbf {h _ {x}} \otimes \frac {1}{2} \mathbf {H _ {x x}}\right) \left(\tilde {\mathbf {x}} _ {t + i} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + i} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + i} ^ {f} - \mathbf {x} _ {t + i} ^ {f} \otimes \mathbf {x} _ {t + i} ^ {f} \otimes \mathbf {x} _ {t + i} ^ {f}\right)} \\ & {+ \sum_ {i = 0} ^ {l - 1} \left(\mathbf {h _ {x}} \otimes \mathbf {h _ {x}}\right) ^ {l - 1 - i} \left(\mathbf {h _ {x}} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}\right) \left(\tilde {\mathbf {x}} _ {t + i} ^ {f} - \mathbf {x} _ {t + i} ^ {f}\right)} \\ & {+ \left(\mathbf {h _ {x}} \otimes \mathbf {h _ {x}}\right) ^ {l - 1} \left(\sigma \pmb {\eta} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}\right) (\pmb {\nu} \otimes 1)} \\ & {+ \left(\mathbf {h _ {x}} \otimes \mathbf {h _ {x}}\right) ^ {l - 1} \left(\sigma \pmb {\eta} \otimes \mathbf {h _ {x}}\right) (\pmb {\nu} \otimes \mathbf {x} _ {t} ^ {s})} \\ & {+ \left(\mathbf {h _ {x}} \otimes \mathbf {h _ {x}}\right) ^ {l - 1} \left(\sigma \pmb {\eta} \otimes \frac {1}{2} \mathbf {H _ {x x}}\right) (\pmb {\nu} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f})} \end{array}\]

\[\begin{array} { r l } & = \sum _ { i = 0 } ^ { l - 1 } \left( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } \right) ^ { l - 1 - i } \left( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } \right) \left( \tilde { \mathbf { x } } _ { t + i } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + i } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + i } ^ { f } - \mathbf { x } _ { t + i } ^ { f } \otimes \mathbf { x } _ { t + i } ^ { f } \otimes \mathbf { x } _ { t + i } ^ { f } \right) \\ & + \sum _ { i = 0 } ^ { l - 1 } \left( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } \right) ^ { l - 1 - i } \left( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } \right) \left( \tilde { \mathbf { x } } _ { t + i } ^ { f } - \mathbf { x } _ { t + i } ^ { f } \right) \\ & + ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { l - 1 } \left( \sigma \eta \nu \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } \right) \\ & + ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { l - 1 } \left( \sigma \eta \nu \otimes \mathbf { h } _ { \mathbf { x } } \mathbf { x } _ { t } ^ { s } \right) \\ & + ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { l - 1 } \left( \sigma \eta \nu \otimes \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) \right) \\ & = \sum _ { i = 0 } ^ { l - 1 } ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { l - 1 - i } ( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } ) ( \tilde { \mathbf { x } } _ { t + i } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + i } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + i } ^ { f } - \mathbf { x } _ { t + i } ^ { f } \otimes \mathbf { x } _ { t +i} ^ { f } \otimes \mathbf { x } _ { t + i} ^ { f } ) \\ & + \sum _ { i = 0 } ^ { l - 1 } ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { l - 1 - i } ( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } ) ( \tilde { \mathbf { x } } _ { t + i} ^ { f } - \mathbf { x } _ { t + i} ^ { f} ) \\ & + ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { l - 1 } ( \sigma \eta \nu \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } + \sigma \eta \nu \otimes \mathbf { h _ { x }} {\mathbf x} _ { t s} ^ { s} + \sigma \eta \nu \otimes \frac { 1 } { 2} {\mathbf H} _ { {\mathrm{xx}}} ( {\mathbf x} _ { t s} ^ { f} {\otimes} {\mathbf x} _ {{t s}} ^ {{f}} ) ) \\ & = \sum _ { i = 1 } ^ { l - 1 } ( {\mathrm{h} _{x}} {\otimes} {\mathrm{h} _{x}} ) ^ { l - 1 - i } ( {\mathrm{h} _{x}} {\otimes} {\frac 12} {\mathrm{H} _{xx}} ) ( {\tilde {\mathrm{x}}} _ {{t + i}} ^ {{f}} {\otimes} {\tilde {\mathrm{x}}} _ {{t + i}} ^ {{f}} {\otimes} {\tilde {\mathrm{x}}} _ {{t + i}} ^ {{f}} - {\mathrm{x} _ {{t + i}} ^ {{f}}} {\otimes} {\mathrm{x} _ {{t + i}} ^ {{f}}} {\otimes} {\mathrm{x} _ {{t + i}} ^ {{f}}} ) \\ & + [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] \\ & + ( [ [ [ ] ] ] ] ^ {- 1 - i} ( [ [ [ ] ] ] ^ {- 1 - i} ( [ [ ] ] ] ^ {- 1 - i} ( [ [ ] ] ] ^ {- 1 - i} ( [ [ ] ] ] ^ {- 1 - i} ( [ [ ] ] ] ^ {- 1 - i} ( [ [ ] ] ] ^ {- 1 - i} ( [ [ ] ] ) \\ & + ( [ [ ] ] ] ^ {- 1 - i} ( [ [ ] ] ] ^ {- 1 - i} ( [ [ ] ] ] ^ {- 1 - i} ( [ [ ] ] ] ^ {- 1 - i} ( [ [ ] ] ] ^ {- 1 - i} ( [ [ ] ] ) \\ & + ( [ [ ] ] ] ^ {- 1 - i} ( [ [ ] ] ] ^ {- 1 - i} ( [ [ ] ] ] ^ {- 1 - i} ( [ [ ] ] ] ^ {- 1 - i} ( [ [ ] ] ) \\ & + ( [ [ ] ] ] ^ {- l - 1 - i} ( [ [ ] ] ] ^ {- l - 1 - i} ( [ [ ] ] ] ^ - l - l - l - l - l - l - l - l - l - l - l - l - l - l - l - l - l - l - l - l - l - l - l - l - l - l - l - l - l - l - l - l - l - l - l - l - l - l - l - l - l - l - l - l - l - l - l - l - l - l - l , \\ & + ( [ [ ] ] ] ^ {- l - 1 - i} ( [ [ ] ] ] ^ {- l - l - i} ( [ [ ] ] ^ {- l - l - i} ( [ [ ] ^ {- l - l - i} ( ) ) ) ) \\ & + ( [ [ [ ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ) \\ & + ( [ [ ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ) \\ & + ( [ [ ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ) \\ & + ( [ [ ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ) \\ & + ( [ [ ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ) \\ & + ( [ [ ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ) \\ & + ( [ [ ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ) \\ & + ( [ [ ]. ) \\ & + ( [ ]. ) \\ & + ( . ) \\ & + ( . ) \\ & + ( . ) \\ & + ( . ) \\ & + ( . ) \\ & + ( . ) \\ & + ( . ) \\ & + ( . ) \\ & + ( . ) \\ & + ( . ) \\ & + ( . ) \\ & + ( . ) \\ & + ( . ) \\ & > L. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C. C.< nl>\]

To derive a recursive version for this sum, we let

\[\begin{array} { l } X _ { l } = \sum _ { i = 1 } ^ { l - 1 } ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { l - 1 - i } ( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } ) ( \tilde { \mathbf { x } } _ { t + i } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + i } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + i } ^ { f } - \mathbf { x } _ { t + i } ^ { f } \otimes \mathbf { x } _ { t + i } ^ { f } \otimes \mathbf { x } _ { t + i } ^ { f } ) \\ + \sum _ { i = 1 } ^ { l - 1 } ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { l - 1 - i } ( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } ) ( \tilde { \mathbf { x } } _ { t + i } ^ { f } - \mathbf { x } _ { t + i } ^ { f } ) \\ + ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { l - 1 } ( \sigma \eta \nu \otimes ( \mathbf { h } _ { \mathbf { x } } \mathbf { x } _ { t } ^ { s } + \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) + \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } ) ) \\ \textnormal { S O } \\ X _ { 1 } = \sigma \eta \nu \otimes ( \mathbf { h } _ { \mathbf { x } } \mathbf { x } _ { t } ^ { s } + \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) \\ + X _ { 2 } = \sum _ { i = 1 } ^ { 1 } ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { 1 - i } ( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } ) ( \tilde { \mathbf { x } } _ { t + i } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + i } ^ { f } \ottimes \tilde { \mathbf { x } } _ { t + i } ^ { f } - \mathbf { x } _ { t + i } ^ { f } \otimes \mathbf { x } _ { t + i } ^ { f } \otimes \mathbf { x } _ { t + i } ^ { f } ) \\ + ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ( \sigma \eta \nu \otimes ( \mathbf { h } _ { \mathbf { x } } \mathbf { x } _ { t } ^ { s } + \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x} } ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) + \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2} ) ) \\ = ( [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ) \\ + ( [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ) \\ + ( [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ ] ] ] ] ] ] ] ] ] ] ] ] ) \\ + ( [ [ [ [ [ [ [ ] ] ] ] ] ) \\ + ( [ [ [ ] ] ] ) \\ = ( | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | \end{array}\]

\[\begin{array} { r l } & X _ { 3 } = \sum _ { i = 1 } ^ { 2 } \left( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } \right) ^ { 2 - i } \left( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } \right) \left( \tilde { \mathbf { x } } _ { t + i } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + i } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + i } ^ { f } - \mathbf { x } _ { t + i } ^ { f } \otimes \mathbf { x } _ { t + i } ^ { f } \otimes \mathbf { x } _ { t + i } ^ { f } \right) \\ & + \sum _ { i = 1 } ^ { 2 } \left( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } \right) ^ { 2 - i } \left( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } \right) \left( \tilde { \mathbf { x } } _ { t + i } ^ { f } - \mathbf { x } _ { t + i } ^ { f } \right) \\ & + ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { 2 } \left( \sigma \eta \pmb { \nu } \otimes \left( \mathbf { h } _ { \mathbf { x } } \mathbf { x } _ { t } ^ { s } + \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } \left( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \right) + \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } \right) \right) \\ & \\ & = ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } ) ( \tilde { \mathbf { x } } _ { t + 1 } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + 1 } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + 1 } ^ { f } - \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } ) \\ & + ( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } ) ( \tilde { \mathbf { x } } _ { t + 2 } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + 2 } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + 2 } ^ { f } - \mathbf { x } _ { t + 2 } ^ { f } \otimes \mathbf { x } _ { t + 2 } ^ { f } \otimes \mathbf { x } _ { t + 2 } ^ { f } ) \\ & + ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x} ) } ( \mathbf { h } _ { \mathbf { x} } \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2} ) ( \tilde { \mathbf { x } } _ { t + 1 } ^ { f } - \mathbf { x} _ { t + 1 } ^ { f} ) \\ & + ( \mathbf { h } _ { \mathbf { x} } \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2} ) ( \tilde { \mathbf { x } } _ { t + 2 } ^ { f } - \mathbf { x} _ { t + 2 } ^ { f} ) \\ & + ( \mathbf { h } _ { \mathbf { x} } \otimes \mathbf { h _ {\alpha} ) ^ 2} ( \sigma \eta v) ( [ - [ ] [ ] [ - [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ], [ ]. \\ & \\ & = ( X , X ) X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X . \\ & \\ & = ( X , X ) X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X , X . \\ & \\ & = ( A , A ) A + ( A ) A + ( A ) A + ( A ) A + ( A ) A + ( A ) A + ( A ) A + ( A ) A + ( A ) A + ( A ) A + ( A ) A + ( A ) A + ( A ) A + ( A ) A + ( A ) A + ( A ) A + ( A ) A + ( A ) A + ( A ) A + ( A ) A + ( A ) A + (\mathrm{的}) A + (\mathrm{的}) A + (\mathrm{的}) A + (\mathrm{的}) A + (\mathrm{的}) A + (\mathrm{的}) A + (\mathrm{的}) A + (\mathrm{的}) A + (\mathrm{的}) A + (\mathrm{的}) A + (\mathrm{的}) A + (\mathrm{的}) A + (\mathrm{的}) A + (\mathrm{的}) A + (\mathrm{的}) A - (\mathrm{的}) A - (\mathrm{的}) A - (\mathrm{的}) A - (\mathrm{的}) A - (\mathrm{的}) A - (\mathrm{的}) A - (\mathrm{的}) A - (\mathrm{的}) A - (\mathrm{的}) A - (\mathrm{的}) A - (\mathrm{的}) A - (\mathrm{的}) A - (\mathrm{的}) A - (\mathrm{的}) A - (\textit {\Delta} ) B - (\textit {\Delta} ) B - (\textit {\Delta} ) B - (\textit {\Delta} ) B - (\textit {\Delta} ) B - (\textit {\Delta} ) B - (\textit {\Delta} ) B - (\textit {\Delta} ) B - (\textit {\Delta} ) B - (\textit {\Delta} ) B - (\textit {\Delta} ) B - (\textit {\Delta} ) B . \\ & \\ & = ( B , B ) B - ( B ) B - ( B ) B - ( B ) B - ( B ) B - ( B ) B - ( B ) B - ( B ) B - ( B ) B - ( B ) B - ( B ) B - ( B ) B - ( B ) B - ( B ) B - ( B ) B - ( B ) B - ( B ) B - ( B ) B - ( B ) B - ( B ) B - ( B ) B - (\textit {\Delta} ) C - (\textit {\Delta} ) C - (\textit {\Delta} ) C - (\textit {\Delta} ) C - (\textit {\Delta} ) C - (\textit {\Delta} ) C - (\textit {\Delta} ) C - (\textit {\Delta} ) C - (\textit {\Delta} ) C - (\textit {\Delta} ) C - (\textit {\Delta} ) C - (\textit {\Delta} ) C = C . \\ & \\ & = ( C , C ) C - ( C ) C - ( C ) C - ( C ) C - ( C ) C - ( C ) C - ( C ) C - ( C ) C - ( C ) C - ( C ) C - ( C ) C - ( C ) C - ( C ) C - ( C ) C = C . \\ & \\ & = ( D , D ) D - D = D = D = D = D = D = D = D = D = D = D = D = D = D = D = D = D = D = D = D = D = D = D = D = D = D = D = D = D = D = D = D = D = D = D = D = D = D = D = D = D = D = D = D = D = D = D = D = D = D = D < T < T < T < T < T < T < T < T < T < T < T < T < T < T < T < T < T < T < T < T < T < T < T < T < T < T < T < T < T < T < T < T < T < T< |content_end|>\]

11.3.3 Summarizing

At third order, the total effect on the state variables is:

\[E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} - \mathbf {x} _ {t + l} \right] = E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} ^ {f} - \mathbf {x} _ {t + l} ^ {f} \right] + E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} ^ {s} - \mathbf {x} _ {t + l} ^ {s} \right] + E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} ^ {r d} - \mathbf {x} _ {t + l} ^ {r d} \right]\]

For the control variables:

\[\begin{array}{r l} & {\mathrm{forthecontrolvariables:}} \\ & {\mathbf {y} _ {t + l} ^ {r d} = \mathbf {g} _ {\mathbf {x}} \left(\mathbf {x} _ {t + l} ^ {f} + \mathbf {x} _ {t + l} ^ {s} + \mathbf {x} _ {t + l} ^ {r d}\right) + \frac {1}{2} \mathbf {G} _ {\mathbf {x x}} \left(\left(\mathbf {x} _ {t + l} ^ {f} \otimes \mathbf {x} _ {t + l} ^ {f}\right) + 2 \left(\mathbf {x} _ {t + l} ^ {f} \otimes \mathbf {x} _ {t + l} ^ {s}\right)\right)} \\ & {\quad + \frac {1}{6} \mathbf {G} _ {\mathbf {x x x}} \left(\mathbf {x} _ {t + l} ^ {f} \otimes \mathbf {x} _ {t + l} ^ {f} \otimes \mathbf {x} _ {t + l} ^ {f}\right) + \frac {1}{2} \mathbf {g} _ {\sigma \sigma} \sigma^ {2} + \frac {3}{6} \mathbf {g} _ {\sigma \sigma \mathbf {x}} \sigma^ {2} \mathbf {x} _ {t + l} ^ {f} + \frac {1}{6} \mathbf {g} _ {\sigma \sigma \sigma} \sigma^ {3}} \\ & {\tilde {\mathbf {y}} _ {t + l} ^ {r d} = \mathbf {g} _ {\mathbf {x}} \left(\tilde {\mathbf {x}} _ {t + l} ^ {f} + \tilde {\mathbf {x}} _ {t + l} ^ {s} + \tilde {\mathbf {x}} _ {t + l} ^ {r d}\right) + \frac {1}{2} \mathbf {G} _ {\mathbf {x x}} \left(\left(\tilde {\mathbf {x}} _ {t + l} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + l} ^ {f}\right) + 2 \left(\tilde {\mathbf {x}} _ {t + l} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + l} ^ {s}\right)\right)} \\ & {\quad + \frac {1}{6} \mathbf {G} _ {\mathbf {x x x}} \left(\tilde {\mathbf {x}} _ {t + l} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + l} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + l} ^ {f}\right) + \frac {1}{2} \mathbf {g} _ {\sigma \sigma} \sigma^ {2} + \frac {3}{6} \mathbf {g} _ {\sigma \sigma \mathbf {x}} \sigma^ {2} \tilde {\mathbf {x}} _ {t + l} ^ {f} + \frac {1}{6} \mathbf {g} _ {\sigma \sigma \sigma} \sigma^ {3}} \end{array}\]

So:

\[\begin{array}{r l} & E _ {t} \left[ \tilde {\mathbf {y}} _ {t + l} ^ {r d} - \mathbf {y} _ {t + l} ^ {r d} \right] \\ & \quad = \mathbf {g} _ {\mathbf {x}} \left(E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} ^ {f} - \mathbf {x} _ {t + l} ^ {f} \right] + E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} ^ {s} - \mathbf {x} _ {t + l} ^ {s} \right] + E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} ^ {r d} - \mathbf {x} _ {t + l} ^ {r d} \right]\right) \\ & \quad + \frac 12 \mathbf {G} _ {\mathbf {x x}} \left(E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + l} ^ {f} - \mathbf {x} _ {t + l} ^ {f} \otimes \mathbf {x} _ {t + l} ^ {f} \right] + 2 E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + l} ^ {s} - \mathbf {x} _ {t + l} ^ {f} \otimes \mathbf {x} _ {t + l} ^ {s} \right]\right) \\ & \quad + \frac 16 \mathbf {G} _ {\mathbf {x x x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + l} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + l} ^ {f} - \mathbf {x} _ {t + l} ^ {f} \otimes \mathbf {x} _ {t + l} ^ {f} \otimes \mathbf {x} _ {t + l} ^ {f} \right] + \frac 36 \mathbf {g} _ {\sigma \sigma \mathbf {x}} \sigma^ {2} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} ^ {f} - \mathbf {x} _ {t + l} ^ {f} \right] \end{array}\]

12 Impulse response functions - GIRF version 2

In our derivations of the GIRF's above, we condition on the entire vector even though only one or two elements in may differ from zero. This interpretation of the definition of GIRF's has some unfortunate properties - for instance, following a disturbance to shock we may actually see changes in shock if shock follows a non-linear law of motion! To avoid this odd behaviour we will now adopt another interpretation of the definition of a GIRF where we only condition on the one-zero shocks hitting the economy.

Hence, this section derives closed-form solutions for the generalized impulse response function in non-linear DSGE models defined as

\[G I R F _ {\mathbf {v a r}} (l, \nu_ {i}, \mathbf {w} _ {t}) = E _ {t} [ \mathbf {v a r} _ {t + l} | \nu_ {i} ] - E _ {t} [ \mathbf {v a r} _ {t + l} ]\]

for a disturbance to innovation . To reduce the notational burden in the derivations below, we adopt the parsimonious notation

\[I R F _ {\mathbf {v a r}} \left(l, \nu_ {i}, \mathbf {w} _ {t}\right) = E _ {t} \left[ \widetilde {\mathbf {v a r}} _ {t + l} \right] - E _ {t} \left[ \mathbf {v a r} _ {t + l} \right]\]

in relation to the conditional expectation operators. The formulas we derive below also apply if we want to explore the joint effects of more than one shock - for instance simultaneous shocking disturbances and , i.e. .

12.1 The model for the conditional information

This subsection explains how we will compute conditional expectations by conditioning on - and possible more disturbances. Let S be diagonal selection matrix with either 1 or zeros on the diagonal, and let the shock sizes appear in the vector of dimension . For shocks which are not hit by a disturbance, we simply put them to zero.

As an example, consider an economy with three shocks and we want to condition our expectations on the first shock. Hence, we need the vector

\[\left[ \begin{array}{c} \nu_ {1} \\ \epsilon_ {2, t + 1} \\ \epsilon_ {3, t + 1} \end{array} \right]\]

We can form this vector by letting

\[\mathbf {S} = \left[ \begin{array}{l l l} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array} \right]\]

and

\[\boldsymbol {\nu} = \left[ \begin{array}{c} \nu_ {1} \\ 0 \\ 0 \end{array} \right].\]

Then we have

\[\begin{array}{r c l} \mathbf {S} \boldsymbol {\nu} + (\mathbf {I} - \mathbf {S}) \boldsymbol {\epsilon} _ {t + 1} & = & \left[ \begin{array}{c c c} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array} \right] \left[ \begin{array}{c} \nu_ {1} \\ 0 \\ 0 \end{array} \right] + \left[ \begin{array}{c c c} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right] \left[ \begin{array}{c} \epsilon_ {1, t + 1} \\ \epsilon_ {2, t + 1} \\ \epsilon_ {3, t + 1} \end{array} \right] \\ & = & \left[ \begin{array}{c} \nu_ {1} \\ \epsilon_ {2, t + 1} \\ \epsilon_ {3, t + 1} \end{array} \right] \end{array}\]

Similarly, if we want to condition on the first two shocks, then we let

\[\mathbf {S} = \left[ \begin{array}{c c c} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{array} \right]\]

and

\[\boldsymbol {\nu} = \left[ \begin{array}{c} \nu_ {1} \\ \nu_ {2} \\ 0 \end{array} \right].\]

meaning that

\[\mathbf {S} \pmb {\nu} + (\mathbf {I} - \mathbf {S}) \pmb {\epsilon} _ {t + 1} = \left[ \begin{array}{c} \nu_ {1} \\ \nu_ {2} \\ \epsilon_ {3, t + 1} \end{array} \right]\]

12.2 At first order

Recall that we have:

\[\mathbf {x} _ {t + 1} ^ {f} = \mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t} ^ {f} + \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1}\]

and

\[\begin{array}{r l} & {\mathbf {x} _ {t + 2} ^ {f} = \mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t + 1} ^ {f} + \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 2}} \\ & {\qquad = \mathbf {h} _ {\mathbf {x}} \left(\mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t} ^ {f} + \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1}\right) + \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 2}} \\ & {\qquad = \mathbf {h} _ {\mathbf {x}} ^ {2} \mathbf {x} _ {t} ^ {f} + \mathbf {h} _ {\mathbf {x}} \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1} + \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 2}} \end{array}\]

and

\[\begin{array}{r l} & {\mathbf {x} _ {t + 3} ^ {f} = \mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t + 2} ^ {f} + \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 3}} \\ & {\quad = \mathbf {h} _ {\mathbf {x}} \left(\mathbf {h} _ {\mathbf {x}} ^ {2} \mathbf {x} _ {t} ^ {f} + \mathbf {h} _ {\mathbf {x}} \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1} + \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 2}\right) + \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 3}} \\ & {\quad = \mathbf {h} _ {\mathbf {x}} ^ {3} \mathbf {x} _ {t} ^ {f} + \mathbf {h} _ {\mathbf {x}} ^ {2} \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1} + \mathbf {h} _ {\mathbf {x}} \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 2} + \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 3}} \\ & {\quad = \mathbf {h} _ {\mathbf {x}} ^ {3} \mathbf {x} _ {t} ^ {f} + \sum_ {j = 1} ^ {3} \mathbf {h} _ {\mathbf {x}} ^ {3 - j} \sigma \pmb {\eta} \pmb {\epsilon} _ {t + j}} \end{array}\]

In general

\[\mathbf {x} _ {t + l} ^ {f} = \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\epsilon} _ {t + j}\]

With a shock of in period , we have

\[\tilde {\mathbf {x}} _ {t + l} ^ {f} = \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\delta} _ {t + j}\]

where we define such that

\[\boldsymbol {\delta} _ {t + j} = \left\{ \begin{array}{c l} \boldsymbol {\nu} + (\mathbf {I} - \mathbf {S}) \boldsymbol {\epsilon} _ {t + 1} & \text { for } j = 1 \\ \boldsymbol {\epsilon} _ {t + j} & \text { for } j \neq 1 \end{array} \right.\]

Agents know the size of the shock at time , and it is therefore in agents' information set. I.e. is non-stochastic.

So

\[E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} ^ {f} - \mathbf {x} _ {t + l} ^ {f} \right] = E _ {t} \left[ \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\delta} _ {t + j} - \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\epsilon} _ {t + j} \right]\]

\[E _ {t} \left[ \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} (\boldsymbol {\nu} + (\mathbf {I} - \mathbf {S}) \boldsymbol {\epsilon} _ {t + 1}) \right]\]

\[\mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta \nu\]

\[\begin{array}{r l} & {\mathrm{and}} \\ & {E _ {t} \left[ \tilde {\mathbf {y}} _ {t + l} ^ {f} - \mathbf {y} _ {t + l} ^ {f} \right] = \mathbf {g} _ {\mathbf {x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} ^ {f} - \mathbf {x} _ {t + l} ^ {f} \right]} \end{array}\]

12.3 At second order

We need to consider:

\[\begin{array}{r l} & {\mathbf {x} _ {t + 1} ^ {s} = \mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t} ^ {s} + \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) + \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}} \\ & \\ & {\mathbf {x} _ {t + 2} ^ {s} = \mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t + 1} ^ {s} + \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t + 1} ^ {f} \otimes \mathbf {x} _ {t + 1} ^ {f}\right) + \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}} \\ & {\quad = \mathbf {h} _ {\mathbf {x}} \left(\mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t} ^ {s} + \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) + \frac {1}{2} \mathbf {h} _ {\boldsymbol {\sigma \sigma}} \sigma^ {2}\right) + \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t + 1} ^ {f} \otimes \mathbf {x} _ {t + 1} ^ {f}\right) + \frac {1}{2} \mathbf {h} _ {\boldsymbol {\sigma \sigma}} \sigma^ {2}} \\ & {\quad = \mathbf {h} _ {\mathbf {x}} ^ {2} \mathbf {x} _ {t} ^ {s} + \mathbf {h} _ {\mathbf {x}} \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) + \mathbf {h} _ {\mathbf {x}} \frac {1}{2} \mathbf {h} _ {\boldsymbol {\sigma \sigma}} \sigma^ {2} + \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t + 1} ^ {f} \otimes \mathbf {x} _ {t + 1} ^ {f}\right) + \frac {1}{2} \mathbf {h} _ {\boldsymbol {\sigma \sigma}} \sigma^ {2}} \end{array}\]

\[\begin{array} { r l } & { \mathbf { x } _ { t + 3 } ^ { s } = \mathbf { h } _ { \mathbf { x } } \mathbf { x } _ { t + 2 } ^ { s } + \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } \left( \mathbf { x } _ { t + 2 } ^ { f } \otimes \mathbf { x } _ { t + 2 } ^ { f } \right) + \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } } \\ & { = \mathbf { h } _ { \mathbf { x } } \left( \mathbf { h } _ { \mathbf { x } } ^ { 2 } \mathbf { x } _ { t } ^ { s } + \mathbf { h } _ { \mathbf { x } } \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } \left( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \right) + \mathbf { h } _ { \mathbf { x } } \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } + \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } \left( \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } \right) + \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } \right) } \\ & { + \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } \left( \mathbf { x } _ { t + 2 } ^ { f } \otimes \mathbf { x } _ { t + 2 } ^ { f } \right) + \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 3 } } \\ & { = \mathbf { h } _ { \mathbf { x } } ^ { 3 } \mathbf { x } _ { t } ^ { s } + \mathbf { h } _ { \mathbf { x } } ^ { 2 } \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } \left( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \right) + \mathbf { h } _ { \mathbf { x } } ^ { 2 } \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } + \mathbf { h } _ { \mathbf { x } } \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } \left( \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } \right) + \mathbf { h } _ { \mathbf { x } } \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } } \\ & { + \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } \left( \mathbf { x } _ { t + 2 } ^ { f } \otimes \mathbf { x } _ { t + 2 } ^ { f } \right) + \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \tau ^ { 2 } } \\ & = \mathbf { h } _ { \mathbf { x } } ^ { 3 } \mathbf { x } _ { t } ^ { s } + \mathbf { h } _ { \mathbf { x } } ^ { 2 } \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x} } ( [ \mathbf { x} ] _ t ) + [ \mathbf { h} ] _ {\alpha , j , k , l , m , n , o , p , q , r , s , t , u , v , w , x , y , z , w , u , v , w , z , u , v , w , z , u , v , w , z , u , v , w , z , u , v , w , z , u , v , w , z , u , v , w , z , u , v , w , z , u , v , w , z , u , v , w , z , u , v , w , z , u , v , w , z , u , v , w , z , u , u , v , w , z , u , v , w , z , u , v , w , z , u , v , w , z , u , v , w , z , u , v , w , z , u , v , w , z , u , v , w , z , u , v , w , z , u , v , w , z , u , v , w , z , u , v , w , z , u , w , w , z , u , v , w , z , u , v , w , z , u , v , w , z , u . }\end{array}\]

and in general

\[\begin{array}{r l} & {\mathbf {x} _ {t + l} ^ {s} = \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {s} + \sum_ {j = 0} ^ {l - 1} \mathbf {h} _ {\mathbf {x}} ^ {l - 1 - j} \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t + j} ^ {f} \otimes \mathbf {x} _ {t + j} ^ {f}\right) + \left(\sum_ {j = 0} ^ {l - 1} \mathbf {h} _ {\mathbf {x}} ^ {l - 1 - j}\right) \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}} \\ & {\mathrm{for} l = 1, 2, 3, \ldots} \end{array}\]

Thus, to compute , we need to find . Hence, consider:

\[\begin{array}{l} \mathbf {x} _ {t + l} ^ {f} \otimes \mathbf {x} _ {t + l} ^ {f} = \left(\mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j}\right) \otimes \left(\mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j}\right) \\ = \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} \\ + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} \end{array}\]

and

\[\begin{array}{l} \tilde {\mathbf {x}} _ {t + l} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + l} ^ {f} = \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\delta} _ {t + j} \\ \qquad + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\delta} _ {t + j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\delta} _ {t + j} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\delta} _ {t + j} \end{array}\]

where we define such that:

\[\boldsymbol {\delta} _ {t + j} = \left\{ \begin{array}{c l} \boldsymbol {\nu} + (\mathbf {I} - \mathbf {S}) \boldsymbol {\epsilon} _ {t + 1} & \mathrm{for} j = 1 \\ \boldsymbol {\epsilon} _ {t + j} & \mathrm{for} j \neq 1 \end{array} \right.\]

This means that:

\[E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + l} ^ {f} - \mathbf {x} _ {t + l} ^ {f} \otimes \mathbf {x} _ {t + l} ^ {f} \right]\]

\[\begin{array}{l} = E _ {t} [ \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\delta} _ {t + j} \\ + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\delta} _ {t + j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\delta} _ {t + j} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\delta} _ {t + j} \\ - \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} - \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} \\ - \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} - \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j ]} \\ = E _ {t} [ \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ t. f. f. f. f. f. f. f. f. f. f. f. f. f. f. f. f. f. f. f. f. f. f. f. f. f. f. f. f. f. f. f. f. f. f. f. f. f. f. f. f. f. f. f. f. f. f. f. f. f. f. \\ + \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\delta} _ {t + 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ t. f. f. f. f. f. f. f. f. f. f. f. f. f. f. f. f. f. f. f. f. f. f. f. f. f. f. \\ - \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ t. f. f. f. f. f. f. f. f. f. f. f. f. f. f. f. f. \\ - 0 - \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {{\epsilon}} _ {{t + j}} ] \\ u s i n g E _ {{t}} [ {\epsilon_ {{t + j}}} ] = 0 \end{array}\]

\[\begin{array} { r l } & = E _ { t } [ \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \delta _ { t + 1 } + \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \delta _ { t + 1 } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \\ & + \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta _ { t + j } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta _ { t + j } \\ & - \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \epsilon _ { t + j } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \epsilon _ { t + j } ] \\ & \text {cancelling} \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \\ & = E _ { t } [ \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } ( \boldsymbol { \nu} + ( I - S ) \epsilon _ { t + 1 } ) + \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } ( \boldsymbol { \nu} + ( I - S ) \epsilon _ { t + 1 } ) \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \\ & + ( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } ( \boldsymbol { \nu} + ( I - S ) \epsilon _ { t + 1 } ) + \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \epsilon _ { t + j } ) \otimes ( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } ( \boldsymbol { \nu} + ( I - S ) \epsilon _ { t + 1 } ) + \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \theta } \epsilon _ { t + j } ) \\ & - ( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \epsilon _ { t + j } + \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \epsilon _ { t + j } ) ) \otimes ( ( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta} \epsilon _ { t + j } + \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta} \epsilon _ { t + j } ) ] \\ & = E _ { t } [ \mathbf { h } _ { \mathbf { x } } ^ { l } {\mathbf x} _ { t } ^ { f } \otimes {\mathbf h} _ {\mathbf x} ^ { l - 1} {\sigma} {\boldsymbol {\eta}} {\boldsymbol {\nu}} + {\mathbf h} _ {\mathbf x} ^ { l - 1} {\sigma} {\boldsymbol {\eta}} {\boldsymbol {\nu}} {\otimes} {\mathbf h} _ {\mathbf x} ^ { l} {\mathbf x} _ {\mathrm{t}} ^ { f } \\ & + ( {\mathbf h} _ {\mathbf x} ^ { l - 1} {\sigma} {\boldsymbol {\eta}} {\boldsymbol {\nu}} + {\mathbf h} _ {\mathbf x} ^ { l - 1} {\sigma} {\boldsymbol {\eta}} ( I - S) {\epsilon} _ { t + 1} ) {\otimes} ( ( {\mathbf h} _ {\mathbf x} ^ { l - 1} {\sigma} {\boldsymbol {\eta}} ( {\boldsymbol {\nu}} + ( I - S) {\epsilon} _ { t + 1} ) + \sum _ { j = 2 } ^ { l } {\mathbf h} _ {\mathbf x} ^ { l - j} {\sigma} {\boldsymbol {\eta}} {\epsilon} _ { t + j} ) \\ & + ( [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ) \\ & + ( [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ) \\ & + ( ( [ [ ] ] ] ) \\ & - ( ( [ ] ] ) \\ & - ( ( [ ] ] ) \\ & - ( ( [ ] ] ) \\ & - ( ( [ ] ] ) \\ & - ( ( [ ] ] ) \\ & - ( ( [ ] ] ) \\ & - ( ( [ ] ] ) \\ & - ( ( [ ]: ] ) \\ & - ( ( [ ]: ] ) \\ & - ( ( [ ]: ] ) \\ & - ( ( [ ]: ] ) \\ & - ( ( [ ]: ] ) \\ & - ( ( [ ]: ] ) \\ & - ( ( [ ]: ] ) \\ & - ( ( [ ]: ]: ) \\ & - ( ( [ ]: ]: ) \\ & - ( ( [ ]: ]: ) \\ & - ( ( [ ]: ]: ) \\ & - ( ( [ ]: ]: ) \\ & - ( ( [ ]: ]: ) \\ & - ( ( [ ]: ]: ) \\ & - ( ( [ ]: ]: ) | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | & = E _ { t } [ {\mathbf h} _ {\mathbf x} ^ { l} {\mathbf x} _ { t} ^ { f } \otimes {\mathbf h} _ {\mathbf x} ^ { l - 1} {\sigma} {\boldsymbol {\eta}} {\boldsymbol {\nu}} + {\mathbf h} _ {\mathbf x} ^ { l - 1} {\sigma} {\boldsymbol {\eta}} {\boldsymbol {\nu}} {\otimes} {\mathbf h} _ {\mathbf x} ^ { l} {\mathbf x} _ {\mathrm{t}} ^ { f }\]

\[\begin{array} { r l } & { + \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \left( \mathbf { I } - \mathbf { S } \right) \boldsymbol { \epsilon } _ { t + 1 } \otimes \left( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \left( \boldsymbol { \nu } + ( \mathbf { I } - \mathbf { S } ) \boldsymbol { \epsilon } _ { t + 1 } \right) + \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } \right) } \\ & { + \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } \otimes \left( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \nu } + \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \left( \mathbf { I } - \mathbf { S } \right) \boldsymbol { \epsilon } _ { t + 1 } \right) } \\ & { + \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } \otimes \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } } \\ & { - \left( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + 1 } \right) \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + 1 } } \\ & { - \left( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + 1 } \right) \otimes \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } } \\ & { - \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + 1 } } \\ & - \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } \otimes \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { {\eta} }\]

\[\begin{array} { r l } & { = E _ { t } [ \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \eta \pmb { \nu } + \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \eta \pmb { \nu } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } + \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \eta \pmb { \nu } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \eta \pmb { \nu } } \\ & { + \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \eta ( \mathbf { I } - \mathbf { S } ) \epsilon _ { t + 1 } \otimes ( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \eta ( \mathbf { I } - \mathbf { S } ) \epsilon _ { t + 1 } ) } \\ & { - ( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \eta \epsilon _ { t + 1 } ) \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \eta \epsilon _ { t + 1 } ] } \\ & { c a n c e l i n g t e r m s \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \eta \epsilon _ { t + j } \otimes \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \eta \epsilon _ { t + j } } \\ & = E _ { t } [ \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \eta \pmb { \nu } + \mathbf { h } _ { \mathbf { x } } ^ { l - 2 } \sigma \eta \pmb { \nu } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } + \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \eta \pmb { \nu } \otimes \mathbf { h } _ { \mathbf { x } }\]

Hence, we only need to compute and . We know using if and are defined

and

\[E _ {t} \left[ \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S}) \boldsymbol {\epsilon} _ {t + 1} \otimes \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S}) \boldsymbol {\epsilon} _ {t + 1} \right] = E _ {t} \left[ \left(\sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S}) \otimes \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S})\right) \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) \right] = \left(\sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S}) \otimes \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S})\right) v e c (\mathbf {I})\]

Thus, we have

\[\begin{array} { r l } & { \text {Thus, we have} } \\ & { E _ { t } \left[ \tilde { \mathbf { x } } _ { t + l } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + l } ^ { f } - \mathbf { x } _ { t + l } ^ { f } \otimes \mathbf { x } _ { t + l } ^ { f } \right] = \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \nu } + \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \nu } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } + \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \nu } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \nu }} \\ & {\qquad + \left( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \right) \left( ( \sigma \boldsymbol { \eta } ( \mathbf { I } - \mathbf { S } ) \otimes \sigma \boldsymbol { \eta } ( \mathbf { I } - \mathbf { S } ) ) v e c ( \mathbf { I } ) - ( \sigma \boldsymbol { \eta } \otimes \sigma \boldsymbol { \eta } ) v e c ( \mathbf { I } ) \right) } \\ & { = \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \nu } + \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \boldsymbol { \sigma } \boldsymbol { \eta } \boldsymbol { \nu } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } + \left( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \right) ( \sigma \boldsymbol { \eta } \boldsymbol { \nu } \otimes \sigma \boldsymbol { \eta } \boldsymbol { \nu } ) \\ & {\qquad + \left( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \right) ( ( \sigma \boldsymbol { \eta } ( \mathbf { I } - \mathbf { S } ) \otimes \sigma \boldsymbol { \eta } ( \mathbf { I } - \mathbf { S } ) ) - ( \sigma \boldsymbol { \eta } \otimes \sigma \boldsymbol { \eta} ) ) v e c ( \mathbf { I } ) } \\ & { = \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \nu } + \mathbf { h } _ { \mathbf { x } } ^ { l - i } \sigma \boldsymbol { \eta } \boldsymbol { \nu } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f }} \\ & {\qquad + ( [ ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (\mathrm{})} ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , |, |, |, |, |, |, |, |, |, |, |, |, |, |, |, |, |, |, |, |, |, |, |, |, |, |, |, |, |, |, |, |, |, |, |, |, |, |, |, |, |, |, |, |, |, |, |, |, |, |, |} \\ & = [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ ] o n d ] o n d ] o n d ] o n d ] o n d ] o n d ] o n d ] o n d ] o n d ] o n d ] o n d ] o n d ] o n d ] o n d ] o n d ] o n d ] o n d ] o n d ] o n d ] o n d ] o n d ] o n d ] o n d ] o n d ] o n d ] o n n a s e r e s e r e s e r e s e r e s e r e s e r e s e r e s e r e s e r e s e r e s e r e s e r e s e r e s e r e s e r e s e r e s e r e s e r e s e r e s e r e s e r e s e r e s e r e s e r e s e r e s e r a t i o n i n i n i n i n i n i n i n i n i n i n i n i n i n i n i n i n i n i n i n i n i n i n i n i n i n i n i n i n i n i n i n i n i n i n i n i n i n i n i n i n i n i n i n i n i n i n i n i n i n i n j u a t h a t h a t h a t h a t h a t h a t h a t h a t h a t h a t h a t h a t h a t h a t h a t h a t h a t h a t h a t h a t h a t h a t h a t h a t h a t h a t h a t h a t h a t h a t h a t h a t h a t h b u l l u m p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w s y m u l l u m p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p u l l u m p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p o w p u l l u m p o w p o w p o w p o w p o w p o w p u l l u m q u a l l u m q u a l l u m q u a l l u m q u a l l u m q u a l l u m q u a l l u m q u a l l u m q u a l l u m q u a l l u m q u a l l u m q u a l l u m q u a l l u m q u a l l u m q u a l l u m q u a l ll u m q u a l l u m q u a l l u m q u a l l u m q u a l l u m q u a l l u m q u a l l u m q u a l l u m q u a l l u m q u a l l u m q u a l l u m q u a l l u m q u a l l u m q u a l l u m q u a l l um q u a l l u m q u a l l u m q u a l l u m q u a l l u m q u a l l u m q u a l l u m q u a l l u m q u a l l u m q u a l l u m q u a l l u m q u a l l u m q u a l l u m q u a l l u m q u a l l u m q v a c k y c k y c k y c k y c k y c k y c k y c k y c k y c k y c k y c k y c k y c k y c k y c k y c k y c k y c k y c k y c k y c k y c k y c k y c k y c k y c k y c k y c k y c k y c k y c k y c k y c k y & = [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ @ ] @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ # : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :: / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / \\ & = H _ {\mathrm{in}} ^ {\prime} H _ {\mathrm{in}} ^ {\prime} H _ {\mathrm{in}} ^ {\prime} H _ {\mathrm{in}} ^ {\prime} H _ {\mathrm{in}} ^ {\prime} H _ {\mathrm{in}} ^ {\prime} H _ {\mathrm{in}} ^ {\prime} H _ {\mathrm{in}} ^ {\prime} H _ {\mathrm{in}} ^ {\prime} H _ {\mathrm{in}} ^ {\prime}H _ {\mathrm{in}} ^ {\prime} H _ {\mathrm{in}} ^ {\prime} H _ {\mathrm{in}} ^ {\prime} H _ {\mathrm{in}} ^ {\prime} H _ {\mathrm{in}} ^ {\prime} H _ {\mathrm{in}} ^ {\prime} H _ {\mathrm{in}} ^ {\prime} H _ {\mathrm{in}} ^ {\prime} H _ {\mathrm{in}} ^ {\prime} H_{\mathrm{in}} ^ {\prime} H_{\mathrm{in}} ^ {\prime} H_{\mathrm{in}} ^ {\prime} H_{\mathrm{in}} ^ {\prime} H_{\mathrm{in}} ^ {\prime} H_{\mathrm{in}} ^ {\prime} H_{\mathrm{in}} ^ {\prime} H_{\mathrm{in}} ^ {\prime} H_{\mathrm{in}} ^ {\prime} H_{\textit{a} }\]

where

\[\boldsymbol {\Lambda} \equiv ((\sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S}) \otimes \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S})) - (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta})) v e c (\mathbf {I})\]

\[\begin{array}{l} \text {Or (using another index)} \\ E _ {t} \left[ \tilde {\mathbf {x}} _ {t + j} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + j} ^ {f} - \mathbf {x} _ {t + j} ^ {f} \otimes \mathbf {x} _ {t + j} ^ {f} \right] = \mathbf {h} _ {\mathbf {x}} ^ {j} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {j - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} + \mathbf {h} _ {\mathbf {x}} ^ {j - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} \otimes \mathbf {h} _ {\mathbf {x}} ^ {j} \mathbf {x} _ {t} ^ {f} + \left(\mathbf {h} _ {\mathbf {x}} ^ {j - 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {j - 1}\right) [ \sigma \boldsymbol {\eta} \boldsymbol {\nu} \otimes \sigma \boldsymbol {\eta} \boldsymbol {\nu} + \boldsymbol {\Lambda} ] \\ \text {for j = 1,2,3,...} \end{array}\]

Thus, we have in general

\[\begin{array}{r l} & E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} ^ {s} - \mathbf {x} _ {t + l} ^ {s} \right] = E _ {t} \left[ \sum_ {j = 0} ^ {l - 1} \mathbf {h} _ {\mathbf {x}} ^ {l - 1 - j} \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \left(\tilde {\mathbf {x}} _ {t + j} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + j} ^ {f}\right) - \sum_ {j = 0} ^ {l - 1} \mathbf {h} _ {\mathbf {x}} ^ {l - 1 - j} \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t + j} ^ {f} \otimes \mathbf {x} _ {t + j} ^ {f}\right) \right] \\ & \quad = \sum_ {j = 1} ^ {l - 1} \mathbf {h} _ {\mathbf {x}} ^ {l - 1 - j} \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + j} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + j} ^ {f} - \mathbf {x} _ {t + j} ^ {f} \otimes \mathbf {x} _ {t + j} ^ {f} \right] \\ & \quad \mathrm{theshockhitsinperiod} t + 1, \mathrm{so} \left(\tilde {\mathbf {x}} _ {t} ^ {f} \otimes \tilde {\mathbf {x}} _ {t} ^ {f}\right) = \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \end{array}\]

\[= \sum_ {j = 1} ^ {l - 1} \mathbf {h} _ {\mathbf {x}} ^ {l - 1 - j} \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {h} _ {\mathbf {x}} ^ {j} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {j - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} + \mathbf {h} _ {\mathbf {x}} ^ {j - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} \otimes \mathbf {h} _ {\mathbf {x}} ^ {j} \mathbf {x} _ {t} ^ {f} + \left(\mathbf {h} _ {\mathbf {x}} ^ {j - 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {j - 1}\right) [ \sigma \boldsymbol {\eta} \boldsymbol {\nu} \otimes \sigma \boldsymbol {\eta} \boldsymbol {\nu} + \boldsymbol {\Lambda} ]\right)\]

When implementing the GIRF, it may be useful to have a recursive expression. Here, it is must convenient to use the general expression

\[\begin{array}{l} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} ^ {s} - \mathbf {x} _ {t + l} ^ {s} \right] = \sum_ {j = 1} ^ {l - 1} \mathbf {h} _ {\mathbf {x}} ^ {l - 1 - j} \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + j} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + j} ^ {f} - \mathbf {x} _ {t + j} ^ {f} \otimes \mathbf {x} _ {t + j} ^ {f} \right] \\ \mathrm{So} \\ E _ {t} \left[ \tilde {\mathbf {x}} _ {t + 1} ^ {s} - \mathbf {x} _ {t + 1} ^ {s} \right] = 0 \end{array}\]

\[\begin{array}{r} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + 2} ^ {s} - \mathbf {x} _ {t + 2} ^ {s} \right] = \sum_ {j = 1} ^ {1} \mathbf {h} _ {\mathbf {x}} ^ {1 - j} \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + j} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + j} ^ {f} - \mathbf {x} _ {t + j} ^ {f} \otimes \mathbf {x} _ {t + j} ^ {f} \right] \\ = \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + 1} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + 1} ^ {f} - \mathbf {x} _ {t + 1} ^ {f} \otimes \mathbf {x} _ {t + 1} ^ {f} \right] \end{array}\]

\[\begin{array}{r} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + 3} ^ {s} - \mathbf {x} _ {t + 3} ^ {s} \right] = \sum_ {j = 1} ^ {2} \mathbf {h} _ {\mathbf {x}} ^ {2 - j} \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + j} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + j} ^ {f} - \mathbf {x} _ {t + j} ^ {f} \otimes \mathbf {x} _ {t + j} ^ {f} \right] \\ = \mathbf {h} _ {\mathbf {x}} \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + 1} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + 1} ^ {f} - \mathbf {x} _ {t + 1} ^ {f} \otimes \mathbf {x} _ {t + 1} ^ {f} \right] \end{array}\]

\[\begin{array}{r l} & + \frac {1}{2} \mathbf {H _ {x x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + 2} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + 2} ^ {f} - \mathbf {x} _ {t + 2} ^ {f} \otimes \mathbf {x} _ {t + 2} ^ {f} \right] \\ & = \mathbf {h _ {x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + 2} ^ {s} - \mathbf {x} _ {t + 2} ^ {s} \right] + \frac {1}{2} \mathbf {H _ {x x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + 2} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + 2} ^ {f} - \mathbf {x} _ {t + 2} ^ {f} \otimes \mathbf {x} _ {t+ 2} ^ {f} \right] \end{array}\]

So in general

\[E _ {t} \left[ \tilde {\mathbf {x}} _ {t + k} ^ {s} - \mathbf {x} _ {t + k} ^ {s} \right] = \mathbf {h} _ {\mathbf {x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + k - 1} ^ {s} - \mathbf {x} _ {t + k - 1} ^ {s} \right] + \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + k - 1} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + k - 1} ^ {f} - \mathbf {x} _ {t + k - 1} ^ {f} \otimes \mathbf {x} _ {t + k - 1} ^ {f} \right]\]

For the total state variable:

\[E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} - \mathbf {x} _ {t + l} \right] = E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} ^ {f} - \mathbf {x} _ {t + l} ^ {f} \right] + E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} ^ {s} - \mathbf {x} _ {t + l} ^ {s} \right]\]

For the control variables:

\[\begin{array}{r l} & {\mathbf {y} _ {t + l} ^ {s} = \mathbf {g} _ {\mathbf {x}} \left(\mathbf {x} _ {t + l} ^ {f} + \mathbf {x} _ {t + l} ^ {s}\right) + \frac {1}{2} \mathbf {G} _ {\mathbf {x x}} \left(\mathbf {x} _ {t + l} ^ {f} \otimes \mathbf {x} _ {t + l} ^ {f}\right) + \frac {1}{2} \mathbf {g} _ {\sigma \sigma} \sigma^ {2}} \\ & {\tilde {\mathbf {y}} _ {t + l} ^ {s} = \mathbf {g} _ {\mathbf {x}} \left(\tilde {\mathbf {x}} _ {t + l} ^ {f} + \tilde {\mathbf {x}} _ {t + l} ^ {s}\right) + \frac {1}{2} \mathbf {G} _ {\mathbf {x x}} \left(\tilde {\mathbf {x}} _ {t + l} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + l} ^ {f}\right) + \frac {1}{2} \mathbf {g} _ {\sigma \sigma} \sigma^ {2}} \end{array}\]

\[E _ {t} \left[ \tilde {\mathbf {y}} _ {t + l} ^ {s} - \mathbf {y} _ {t + l} ^ {s} \right] = \mathbf {g} _ {\mathbf {x}} \left(E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} ^ {f} - \mathbf {x} _ {t + l} ^ {f} \right] + E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} ^ {s} - \mathbf {x} _ {t + l} ^ {s} \right]\right) + \frac {1}{2} \mathbf {G} _ {\mathbf {x x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + l} ^ {f} - \mathbf {x} _ {t + l} ^ {f} \otimes \mathbf {x} _ {t + l} ^ {f} \right]\]

12.4 Second order: at the steady state with shock size of unity

If we restrict the focus and do the GIRF's at the unconditional mean of , then we get

\[E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} ^ {s} - \mathbf {x} _ {t + l} ^ {s} \right] = \sum_ {j = 1} ^ {l - 1} \mathbf {h} _ {\mathbf {x}} ^ {l - 1 - j} \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \left(\left(\mathbf {h} _ {\mathbf {x}} ^ {j - 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {j - 1}\right) [ \sigma \boldsymbol {\eta} \boldsymbol {\nu} \otimes \sigma \boldsymbol {\eta} \boldsymbol {\nu} + \boldsymbol {\Lambda} ]\right)\]

If we further assume that there is only one shock to the ith innovation and that the size of this shock is one, i.e. and for . In this case we have

\[\begin{array}{r l} & {\sigma \eta \nu \otimes \sigma \eta \nu + \Lambda} \\ & {\quad = \sigma \eta \nu \otimes \sigma \eta \nu + ((\sigma \eta (\mathbf {I} - \mathbf {S}) \otimes \sigma \eta (\mathbf {I} - \mathbf {S})) - (\sigma \eta \otimes \sigma \eta)) v e c (\mathbf {I})} \\ & {\quad = (\sigma \eta \otimes \sigma \eta) (\nu \otimes \nu) + ((\sigma \eta \otimes \sigma \eta) ((\mathbf {I} - \mathbf {S}) \otimes (\mathbf {I} - \mathbf {S})) - (\sigma \eta \otimes \sigma \eta)) v e c (\mathbf {I})} \\ & {\quad = (\sigma \eta \otimes \sigma \eta) (\mathbf {S} \otimes \mathbf {S}) v e c (\mathbf {I}) + ((\sigma \eta \otimes \sigma \eta) ((\mathbf {I} - \mathbf {S}) \otimes (\mathbf {I} - \mathbf {S})) - (\sigma \eta \otimes \sigma \eta)) v e c (\mathbf {I})} \\ & {\quad b e c a u s e \nu \otimes \nu = (\mathbf {S} \otimes \mathbf {S}) v e c (\mathbf {I})} \\ & {\quad = (\sigma \eta \otimes \sigma \eta) \{\mathbf {S} \otimes \mathbf {S} + ((\mathbf {I} - \mathbf {S}) \otimes (\mathbf {I} - \mathbf {S})) - \mathbf {I} \otimes \mathbf {I} \} v e c (\mathbf {I})} \\ & {\quad b e c a u s e I _ {n _ {e} ^ {2}} = I \otimes I w h e r e I h a s d i m e n s i o n n _ {e} \times n _ {e}} \end{array}\]

\[\left(\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}\right) \left\{2 \left(\mathbf {S} \otimes \mathbf {S}\right) - \mathbf {I} \otimes \mathbf {S} - \mathbf {S} \otimes \mathbf {I} \right\} v e c (\mathbf {I})\]

We now only need to realize that the term in the curly bracket is zero. To do so, let us introduce the matrix , defined as with all remaining elements being zero. Hence, with dimension can be written as . Furthermore, by assumption. Thus, we have for the expression in the curly bracket:

\[2 (\mathbf {S} \otimes \mathbf {S}) - \mathbf {I} \otimes \mathbf {S} - \mathbf {S} \otimes \mathbf {I}\]

\[\begin{aligned} & = 2 \left(\mathbf{D}_{i}\otimes \mathbf{D}_{i}\right) - \left(\sum_{j = 1}^{n_{\varepsilon}}\mathbf{D}_{j}\right)\otimes \mathbf{D}_{i} - \mathbf{D}_{i}\otimes \left(\sum_{j = 1}^{n_{\varepsilon}}\mathbf{D}_{j}\right)\\ & = 2 \left(\mathbf{D}_{i}\otimes \mathbf{D}_{i}\right) - \left(\sum_{j = 1}^{n_{\varepsilon}}\mathbf{D}_{j}\otimes \mathbf{D}_{i}\right) - \left(\sum_{j = 1}^{n_{\varepsilon}}\mathbf{D}_{i}\otimes \mathbf{D}_{j}\right)\\ & = 2 \left(\mathbf{D}_{i}\otimes \mathbf{D}_{i}\right) - \left(\sum_{\substack{j = 1\\ i\neq j}}^{n_{\varepsilon}}\mathbf{D}_{j}\otimes \mathbf{D}_{i} + \mathbf{D}_{i}\otimes \mathbf{D}_{i}\right) - \left(\sum_{\substack{j = 1\\ i\neq j}}^{n_{\varepsilon}}\mathbf{D}_{i}\otimes \mathbf{D}_{j} + \mathbf{D}_{i}\otimes \mathbf{D}_{i}\right) \end{aligned}\]

\[= -\sum_{\substack{j = 1\\ i\neq j}}^{n_{\varepsilon}}\mathbf{D}_{j}\otimes \mathbf{D}_{i} - \sum_{\substack{j = 1\\ i\neq j}}^{n_{\varepsilon}}\mathbf{D}_{i}\otimes \mathbf{D}_{j}\]

Hence,

\[\begin{array}{l}\sigma \boldsymbol{\eta}\boldsymbol{\nu}\otimes \sigma \boldsymbol{\eta}\boldsymbol {\nu} + \boldsymbol {\Lambda} = (\sigma \boldsymbol {\eta}\otimes \sigma \boldsymbol {\eta}) \left\{2 \left(\mathbf{S}\otimes \mathbf{S}\right) - \mathbf{I}\otimes \mathbf{S} - \mathbf{S}\otimes \mathbf{I}\right\} vec \left(\mathbf{I}\right)\\ \\ = (\sigma \boldsymbol {\eta}\otimes \sigma \boldsymbol {\eta})\left\{-\sum_{\substack{j = 1\\ i\neq j}}^{n_{\varepsilon}}\mathbf{D}_{j}\otimes \mathbf{D}_{i} - \sum_{\substack{j = 1\\ i\neq j}}^{n_{\varepsilon}}\mathbf{D}_{i}\otimes \mathbf{D}_{j}\right\} vec \left(\sum_{k = 1}^{n_{\varepsilon}}\mathbf{D}_{k}\right)\\ \\ = (\sigma \boldsymbol {\eta}\otimes \sigma \boldsymbol {\eta})\left\{-\sum_{\substack{j = 1\\ i\neq j}}^{n_{\varepsilon}}\mathbf{D}_{j}\otimes \mathbf{D}_{i} - \sum_{\substack{j = 1\\ i\neq j}}^{n_{\varepsilon}}\mathbb{D}_{i}\otimes \mathbf{D}_{j}\right\} (\sum_{k = 1}^{n_{\varepsilon}}vec \left(\mathbf{D}_{k}\right))\\ \\ = (\sigma \boldsymbol {\eta}\otimes \sigma \boldsymbol {\eta})\left\{-\sum_{\substack{j = 1\\ i\neq j}}^{n_{\varepsilon}}\sum_{k = 1}^{n_{\varepsilon}} \left(\mathbf{D}_{j}\otimes \mathbf{D}_{i}\right) vec \left(\mathbf{D}_{k}\right) - \sum_{\substack{j = 1\\ i\neq j}}^{n_{\varepsilon}}\sum_{k = 1}^{n_{\varepsilon}} \left(\mathbf{D}_{i}\otimes \mathbf{D}_{j}\right) vec \left(\mathbf{D}_{k}\right)\right\} \\ \\ = (\sigma \boldsymbol {\eta}\otimes \sigma \boldsymbol {\eta})\left\{-\sum_{\substack{j = 1\\ i\neq j}}^{n_{\varepsilon}}\sum_{k = 1}^{n_{\varepsilon}} vec \left(\mathbf{D}_{i}\mathbf{D}_{k}\mathbf{D}_{j}\right) - \sum_{\substack{j = 1\\ i\neq j}}^{n_{\varepsilon}}\sum_{k = 1}^{n_{\varepsilon}} vec \left(\mathbf{D}_{j}\mathbf{D}_{k}\mathbf{D}_{i}\right)\right\} \\ because vec (\mathbf{A}\mathbf{B}\mathbf{C}) = (\mathbf{C}'\otimes \mathbf{A}) vec (\mathbf{B}) and \mathbf{D}_{i} is symmetric\\ \\ = (\sigma \boldsymbol {\eta}\otimes \sigma \boldsymbol {\eta})\left\{-\sum_{\substack{j = 1\\ i\neq j}}^{n_{\varepsilon}}\sum_{k = 1}^{n_{\varepsilon}} vec (\mathbf{0}) - \sum_{\substack{j = 1\\ i\neq j}}^{n_{\varepsilon}}\sum_{k = 1}^{n_{\varepsilon}} vec (\mathbf{0})\right\} = \mathbf{0} \end{array}\]

because is only different from the zero matrix when i = k = j, but we have . As a result, and from above, this also implies . Thus, we have that

For the states:

\[\begin{array}{r l} & {\mathrm{Forthestates:}} \\ & {E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} - \mathbf {x} _ {t + l} \right] = E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} ^ {f} - \mathbf {x} _ {t + l} ^ {f} \right] + E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} ^ {s} - \mathbf {x} _ {t + l} ^ {s} \right]} \\ & {\qquad = E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} ^ {f} - \mathbf {x} _ {t + l} ^ {f} \right]} \\ & {\mathrm{Forthecontrolvariables:}} \\ & {E _ {t} \left[ \tilde {\mathbf {y}} _ {t + l} ^ {s} - \mathbf {y} _ {t + l} ^ {s} \right] = \mathbf {g _ {x}} \left(E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} ^ {f} - \mathbf {x} _ {t + l} ^ {f} \right] + E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} ^ {s} - \mathbf {x} _ {t + l} ^ {s} \right]\right) + \frac 12 \mathbf {G _ {x x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + l} ^ {f} - \mathbf {x} _ {t + l} ^ {f} \otimes \mathbf {x} _ {t + l} ^ {f} \right]} \\ & {\qquad = \mathbf {g _ {x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} ^ {f} - \mathbf {x} _ {t + l} ^ {f} \right]} \end{array}\]

which are precisely the impulse response functions at first order.

12.5 At third order

At third order, we additionally need to consider:

\[\begin{array}{r l} & {\mathbf {x} _ {t + 1} ^ {r d} = \mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t} ^ {r d} + \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {s}\right) + \frac {1}{6} \mathbf {H} _ {\mathbf {x x x}} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) + \frac {3}{6} \mathbf {h} _ {\sigma \sigma \mathbf {x}} \sigma^ {2} \mathbf {x} _ {t} ^ {f} + \frac {1}{6} \mathbf {h} _ {\sigma \sigma \sigma} \sigma^ {3}} \\ & {\mathrm{and}} \\ & {\mathbf {x} _ {t + 2} ^ {r d} = \mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t + 1} ^ {r d} + \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t + 1} ^ {f} \otimes \mathbf {x} _ {t + 1} ^ {s}\right) + \frac {1}{6} \mathbf {H} _ {\mathbf {x x x}} \left(\mathbf {x} _ {t + 1} ^ {f} \otimes \mathbf {x} _ {t + 1} ^ {f} \otimes \mathbf {x} _ {t + 1} ^ {f}\right) + \frac {3}{6} \mathbf {h} _ {\sigma \sigma \mathbf {x}} \sigma^ {2} \mathbf {x} _ {t + 1} ^ {f} + \frac {1}{6} \mathbf {h} _ {\sigma \sigma \sigma} \sigma^ {3}} \\ & {\qquad = \mathbf {h} _ {\mathbf {x}} \left(\mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t} ^ {r d} + \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {s}\right) + \frac {1}{6} \mathbf {H} _ {\mathbf {x x x .}} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) + \frac {3}{6} \mathbf {h} _ {\sigma \sigma \mathbf {x}} \sigma^ {2} \mathbf {x} _ {t} ^ {f} + {\frac {1}{6}} \mathbf {h} _ {\sigma \sigma \sigma} \sigma^ {3}\right)} \\ & {\qquad + \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t + 1} ^ {f} \otimes \mathbf {x} _ {t + 1} ^ {s}\right) + \frac {1}{6} \mathbf {H} _ {\mathbf {x x x}} \left(\mathbf {x} _ {t + 1} ^ {f} \odot \mathbf {x} _ {t + 1} ^ {f} \otimes \mathbf {x} _ {t + 1} ^ {f}\right) + \frac {3}{6} \mathbf {h} _ {\sigma \sigma \mathbf {x}} \sigma^ {2} \mathbf {x} _ {t + 1} ^ {f} + {\frac {1}{6}} \mathbf {h} _ {\sigma \sigma \sigma} \sigma^ {3}} \\ & {\qquad = \mathbf {h} _ {\mathbf {x}} ^ {2} \mathbf {x} _ {t} ^ {r d} + \mathbf {h} _ {\mathbf {x}} \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {s}\right) + \mathbf {h} _ {\mathbf {x}} {\frac 16 H _ {\mathbf {x x x}}} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) + \mathbf {{h}} _ {\mathbf {{x}}} {\frac 36 h _ {\sigma \sigma x}} \sigma^ {2} \mathbf {{x}} _ {{t}} ^ {{f}} + \mathbf {{h}} _ {\mathbf {{x}}} {\frac 16 h _ {\sigma \sigma \sigma}} \sigma^ {{3}}} \end{array}\]

\[\begin{array} { r l } & + \mathbf { H } _ { \mathbf { x x } } \left( \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { s } \right) + \frac { 1 } { 6 } \mathbf { H } _ { \mathbf { x x x } } \left( \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } \right) + \frac { 3 } { 6 } \mathbf { h } _ { \sigma \sigma \mathbf { x } } \sigma ^ { 2 } \mathbf { x } _ { t + 1 } ^ { f } + \frac { 1 } { 6 } \mathbf { h } _ { \sigma \sigma \sigma } \sigma ^ { 3 } \\ & { \mathrm{and} } \\ & \mathbf { x } _ { t + 3 } ^ { r d } = \mathbf { h } _ { \mathbf { x } } \mathbf { x } _ { t + 2 } ^ { r d } + \mathbf { H } _ { \mathbf { x x } } \left( \mathbf { x } _ { t + 2 } ^ { f } \otimes \mathbf { x } _ { t + 2 } ^ { s } \right) + \frac { 1 } { 6 } \mathbf { H } _ { \mathbf { x x x } } \left( \mathbf { x } _ { t + 2 } ^ { f } \otimes \mathbf { x } _ { t + 2 } ^ { f } \otimes \mathbf { x } _ { t + 2 } ^ { f } \right) + \frac { 3 } { 6 } \mathbf { h } _ { \sigma \sigma \mathbf { x } } \sigma ^ { 2 } \mathbf { x } _ { t + 2 } ^ { f } + \frac { 1 } { 6 } \mathbf { h } _ { \sigma \sigma \sigma } \sigma ^ { 3 } \\ & = \mathbf { h } _ { \mathbf { x } } [ \mathbf { h } _ { \mathbf { x } } ^ { 2 } \mathbf { x } _ { t } ^ { r d } + \mathbf { h } _ { \mathbf { x } } \mathbf { H } _ { \mathbf { x x } } ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { s } ) + \mathbf { h } _ { \mathbf { x } } \frac { 1 } { 6 } \mathbf { H } _ { \mathbf { x x x } } ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) + \mathbf { h } _ { \mathbf { x } } \frac { 3 } { 6 } \mathbf { h } _ { \sigma \sigma \mathbf { x } } \sigma ^ { 2 } \mathbf { x } _ { t } ^ { f } + \mathbf { h } _ { \mathbf { x } } \frac { 1 } { 6 } \mathbf { h } _ { \sigma \sigma \sigma } \sigma ^ { 3 } \\ & { + \mathbf { H } _ { \mathbf { x x } } ( \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { s } ) + \frac { 1 } { 6 } \mathbf { H } _ { \mathbf { x x x } } ( \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } ) + \frac { 3 } { 6 } \mathbf { h } _ { \sigma \sigma \mathbf { x } } \sigma ^ { 2 } \mathbf { x} _ { t + 1 } ^ { f } + \frac { 1 } { 6 } \mathbf { h } _ { \sigma \sigma \sigma } \sigma ^ { 3 ] }} \\ & + \mathbf { H } _ { \mathbf { x x} } ( \mathbf { x } _ { t + 2 } ^ { f } \otimes \mathbf { x } _ { t + 2 } ^ { s} ) + \frac { 1 } { 6 } \mathbf { H } _ { \mathbf { x x x} } ( \mathbf { x } _ { t + 2 } ^ { f } \otimes \mathbf { x } _ { t + 2 } ^ { f } \otimes \mathbf { x} _ { t + 2} ^ { f )} + \frac { 3}{ 6} {\mathbf h} _ {\sigma \sigma {\mathbf x}} {\sigma^ {\prime}} ^ {\sigma^ {\prime}} {\mathbf x} _ {\tau , j} ^ {\sigma^ {\prime}} {\sigma^ {\prime}} ^ {\sigma^ {\prime}} \\ & = {\mathbf h} _ {\mathbf x} ^ {\mathrm{3}} {\mathbf x} _ {\tau} ^ {\mathrm{rd}} + {\sum_ {\substack {\boldsymbol j = 0 \\ }} ^ {\boldsymbol j = - j}} {\mathbf h} _ {\boldsymbol x} ^ {\mathrm{2-j}} {\mathbf H} _ {\boldsymbol xx} ( {\boldsymbol x} _ {\tau , j} ^ {\mathrm{f}} {\otimes} {\boldsymbol x} _ {\tau , j} ^ {\mathrm{s}} ) \\ & + {\sum_ {\substack {\boldsymbol j = 0 \\ }} ^ {\boldsymbol j = - j}} {\mathbf h} _ {\boldsymbol x} ^ {\mathrm{2-j}} {\frac {\boldsymbol j}{6}} {\mathbf H} _ {\boldsymbol xxx} ( {\boldsymbol x} _ {\tau , j} ^ {\mathrm{f}} {\otimes} {\boldsymbol x} _ {\tau , j} ^ {\mathrm{f}} {\otimes} {\boldsymbol x} _ {\tau , j} ^ {\mathrm{f}} ) \\ & + {\sum_ {\substack {\boldsymbol j = 0 \\ }} ^ {\boldsymbol j = - j}} {\mathbf h} _ {\boldsymbol x} ^ {\mathrm{2-j}} {\frac {\boldsymbol j}{6}} {\mathbf h} _ {\sigma \sigma {\boldsymbol x}} {\sigma^ {\prime}} ^ {\sigma^ {\prime}} {\boldsymbol x} _ {\tau , j} ^ {\mathrm{f}} \\ & + {\sum_ {\substack {\boldsymbol j = 0 \\ }} ^ {\boldsymbol j = - j}} {\mathbf h} _ {\boldsymbol x} ^ {\mathrm{2-j}} {\frac {\boldsymbol j}{6}} {\mathbf h} _ {\sigma \sigma \sigma} {\sigma^ {\prime}} ^ {\sigma^ {\prime}} \\ & = {{\mathbf h} _ {\boldsymbol x} ^ {\mathrm{3}}} {{\boldsymbol x} _ {\tau} ^ {\mathrm{rd}}} + {\sum_ {\substack{\boldsymbol j = 0 \\ }} ^{\boldsymbol j = - j}} {{\boldsymbol h} _{\boldsymbol x} ^{\mathrm{2-j}}} [ [ {{\texttt H} _{{\texttt xx}}} ( {{\texttt x} _{\tau ,j}} ^{\mathrm{f}} {\otimes} {{\texttt x} _{\tau ,j}} ^{\mathrm{s}} ) + {\frac{1}{6}} {{\texttt H} _{{\texttt xx}}} ( {{\texttt x} _{\tau ,j}} ^{\mathrm{f}} {\otimes} {{\texttt x} _{\tau ,j}} ^{\mathrm{f}} {\otimes} {{\texttt x} _{\tau ,j}} ^{\mathrm{f}} ) + {\frac{3}{6}} {{\texttt h} _{{\sigma}\sigma\chi}} {{\sigma^{\prime}}} ^{\sigma^{\prime}} {{\texttt x} _{\tau ,j}} ^{\mathrm{f}} ] \\ & + ( | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | \end{array}\]

as the shock hits the economy in period

A recursive version:

\[E _ {t} \left[ \tilde {\mathbf {x}} _ {t + 1} ^ {r d} - \mathbf {x} _ {t + 1} ^ {r d} \right] = 0\]

\[\begin{array}{r l} & E _ {t} \left[ \tilde {\mathbf {x}} _ {t + 2} ^ {r d} - \mathbf {x} _ {t + 2} ^ {r d} \right] = \sum_ {j = 1} ^ {1} \mathbf {h} _ {\mathbf {x}} ^ {1 - j} \left(\mathbf {H} _ {\mathbf {x x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + j} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + j} ^ {s} - \mathbf {x} _ {t + j} ^ {f} \otimes \mathbf {x} _ {t + j} ^ {s} \right] + \frac {3}{6} \mathbf {h} _ {\sigma \sigma \mathbf {x}} \sigma^ {2} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + j} ^ {f} - \mathbf {x} _ {t + j} ^ {f} \right]\right) \\ & \qquad + \sum_ {j = 1} ^ {1} \mathbf {h} _ {\mathbf {x}} ^ {1 - j} \frac {1}{6} \mathbf {H} _ {\mathbf {x x x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + j} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + j} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + j} ^ {f} - \mathbf {x} _ {t + j} ^ {f} \otimes \mathbf {x} _ {t + j} ^ {f} \otimes \mathbf {x} _ {t + j} ^ {f} \right] \\ & = \mathbf {H} _ {\mathbf {x x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + 1} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + 1} ^ {s} - \mathbf {x} _ {t + 1} ^ {f} \otimes \mathbf {x} _ {t + 1} ^ {s} \right] + \frac {3}{6} \mathbf {h} _ {\sigma \sigma \mathbf {x}} \sigma^ {2} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + 1} ^ {f} - \mathbf {x} _ {t + 1} ^ {f} \right] \\ & \qquad + \frac {1}{6} \mathbf {H} _ {\mathbf {x x x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + 1} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + 1} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + 1} ^ {f} - \mathbf {x} _ {t + 1} ^ {f} \otimes \mathbf {x} _ {t + 1} ^ {f} \otimes \mathbf {x} _ {t + 1} ^ {f} \right] \end{array}\]

\[\begin{array} { r l }&E _ { t } \left[ \tilde { \mathbf { x } } _ { t + 3 } ^ { r d } - \mathbf { x } _ { t + 3 } ^ { r d } \right] = \sum _ { j = 1 } ^ { 2 } \mathbf { h } _ { \mathbf { x } } ^ { 2 - j } \left( \mathbf { H } _ { \mathbf { x x } } E _ { t } \left[ \tilde { \mathbf { x } } _ { t + j } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + j } ^ { s } - \mathbf { x } _ { t + j } ^ { f } \otimes \mathbf { x } _ { t + j } ^ { s } \right] + \frac { 3 } { 6 } \mathbf { h } _ { \sigma \sigma \mathbf { x } } \sigma ^ { 2 } E _ { t } \left[ \tilde { \mathbf { x } } _ { t + j } ^ { f } - \mathbf { x } _ { t + j } ^ { f } \right]\right)\\&\qquad + \sum _ { j = 1 } ^ { 2 } \mathbf { h } _ { \mathbf { x } } ^ { 2 - j } \frac { 1 } { 6 } \mathbf { H } _ { \mathbf { x x x } } E _ { t } \left[ \tilde { \mathbf { x } } _ { t + j } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + j } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + j } ^ { f } - \mathbf { x } _ { t + j } ^ { f } \otimes \mathbf { x } _ { t + j } ^ { f } \otimes \mathbf { x } _ { t + j } ^ { f } \right]\\&= \mathbf { h } _ { \mathbf { x } } \left( \mathbf { H } _ { \mathbf { x x } } E _ { t } \left[ \tilde { \mathbf { x } } _ { t + 1 } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + 1 } ^ { s } - \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { s } \right] + \frac { 3 } { 6 } \mathbf { h } _ { \sigma \sigma \mathbf { x } } \sigma ^ { 2 } E _ { t } \left[ \tilde { \mathbf { x } } _ { t + 1 } ^ { f } - \mathbf { x } _ { t + 1 } ^ { f } \right]\right)\\&+ \mathbf { H } _ { \mathbf { x x } } E _ { t } \left[ \tilde { \mathbf { x } } _ { t + 2 } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + 2 } ^ { s } - \mathbf { x } _ { t + 2 } ^ { f } \otimes \mathbf { x } _ { t + 2 } ^ { s } \right] + \frac { 3 } { 6 } \mathbf { h } _ { \sigma \sigma \mathbf { x } } \sigma ^ { 2 } E _ { t } \left[ \tilde { \mathbf { x } } _ { t + 2 } ^ { f } - \mathbf { x } _ { t + 2 } ^ { f } \right]\\&+ \mathbf { h } _ { \mathbf { x } } \frac { 1 } { 6 } \mathbf { H } _ { \mathbf { x x x } } E _ { t } \left[ \tilde { \mathbf { x } } _ { t + 1 } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + 1 } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + 1 } ^ { f } - \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } \right]\\&+ \frac 1 6 \mathbf H _ ~ x x x ~ E ~ t ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ ] ) ~ ] ~ ] ~ ] ~ ] ~ ] ~ ] ~ ] ~ ] ~ ] ~ ] ~ ] ~ ] ~ ] ~ ] ~ ] ~ ] ~ ] ~ ] ~ ] ~ ] ~ ] , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .\\&= {\bf h} _ {\bf x} E _ { t } \left[ {\tilde {\bf x}} _ { t + 2} ^ { r d} - {\bf x} _ { t + 2} ^ { r d} \right]\\&+ {\bf H} _ {\mathrm{xx}} E _ { t } \left[ {\tilde {\bf x}} _ { t + 2} ^ { f} \otimes {\tilde {\bf x}} _ { t + 2} ^ { s} - {\bf x} _ { t + 2} ^ { f} \otimes {\bf x} _ { t + 2} ^ { s} \right] + \frac 3 6 {\bf h} _ {\sigma \sigma {\bf x}} \sigma ^ { 2} E _ { t } \left[ {\tilde {\bf x}} _ { t + 2} ^ { f} - {\bf x} _ { t + 2} ^ { f} \right]\\&+ \frac 1 6 {\bf H _ ~ x x x~ E~ t} \left[ \right. {\tilde {\bf x}} _ { t + 2} ^ { f} \otimes {\tilde {\bf x}} _ { t + 2} ^ { f} \otimes {\tilde {\bf x}} _ { t + 2} ^ {-} - {\bf x} _ { t + 2} ^ {-} \otimes {\bf x} _ { t + 2} ^ {-} {\bf x} _ { t + 2} ^ {-} {\bf x} _ {{t + 2}} ^ {-} {\bf x} _ {{t + 2}} ^ {-} {\bf x} _ {{t + 2}} ^ {-} {\bf x} _ {{t + 2}} ^ {-} {\bf x} _ {{t + 2}} ^ {-} {\bf x} _ {{t + 2}} ^ {-} {\bf x} _ {{t + 2}} ^ {-} {\bf x} _ {{t + 2}} ^{-} {\bf x} _ {{t + 2}} ^ {-} {\bf x} _ {{t + 2}} ^ {-} {\bf x} _ {{t + 2}} ^ {-} {\bf x} _ {{t + 2}} ^ {-} {\bf x} _ {{t + 2}} ^ {-} {\bf x} _ {{t + 2}} ^ {-} {\bf x} _ {{t + 2}} ^ {-} ,\\&= {\bf h} _ {\mathrm{x}} E _ { t }\left[ {\tilde {\bf x}} _ { t + 2} ^ { r d} - {\bf x} _ { t + 2} ^ { r d} \right]\\&+ \bf H _ ~ xx~ E~ t~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ~ [ ] )~ ]~ ]~ ]~ ]~ ]~ ]~ ]~ ]~ ]~ ]~ ]~ ]~ ]~ ]~ ]~ ]~ ]~ ]~ ]~ ]~ ]~ ]~ ]~ ]~ ]~ ]~ ]~ ]~ ]~ ]~ ]~ ]~ ]~ ]~ ]~ ]~ ]~ ]~ ]~ ]~ ]~ ]~ ]~ ]~ ]~ ]~ ]~ ]~ ]~ ]~ ].\\&= {\bf h} _ {\mathrm{x}} E _ { t }\left[ {\tilde {\bf x}} _ {{t + 2}} ^ {{r d}} - {\bf x} _ {{t + 2}} ^ {{r d}} \right]\\&+ \bf H _ ~ xx~ E~ t~ [ ~[ ~[ ~[ ~[ ~[ ~[ ~[ ~[ ~[ ~[ ~[ ~[ ~[ ~[ ~[ ~[ ~[ ~[ ~[ ▲ y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y yy a n d e r m a l i o n,\\&= \frac 16 {\cal H}_ {}^ {}^ {}^ {}^ {}^ {}^ {}^ {}^ {}^ {}^ {}^ {}^ {}^ {}^ {}^ {}^ {}^ {}^ {}^ {}^ {}^ {}^ {}^ {}^ {}^ {}^ {}^ {}^ {}^ {}^ {}^ {}^ {}^ {}^ {}^ {}^ {}^ {}^ {}^ {}^ {}^ {}^ {}^ {}^ {}^ {}^ {}^ {}^ {}^ {}^ {} ^ {},\\&= {\cal H}_ {\mathrm{xx}} E_{t}\left[ \right. B_{t}\left(1 - B_{t}\right) B_{t}\left(1 - B_{t}\right) B_{t}\left(1 - B_{t}\right) B_{t}\left(1 - B_{t}\right) B_{t}\left(1 - B_{t}\right) B_{t}\left(1 - B_{t}\right) B_{t}\left(1 - B_{t}\right) B_{t}\left(1 - B_{t}\right) B_{t},B_{t},B_{t},B_{t},B_{t},B_{t},B_{t},B_{t},B_{t},B_{t},B_{t},B_{t},B_{t},B_{t},B_{t},B_{t},B_{t},B_{t},B_{t},B_{t},B_{t},B_{t},B_{t},B_{t},B_{t},B_{t},B_{t},B_{t},B_{t},B_{t},B_{t},B_{t},B_{t},B_{t},B_{t},B_{t},B_{t},B_{t},B_{t},B_{t},B_{t},B_{t},B_{t},B_{t},B_{t},B_{t},B_{t},B_{t},B_{t},B_{t},B_{t}.\\&= B_{{\cal H}_{\mathrm{xx}}}E_{{\cal H}_{\mathrm{xx}}}E_{{\cal H}_{\mathrm{xx}}}E_{{\cal H}_{\mathrm{xx}}}E_{{\cal H}_{\mathrm{xx}}}E_{{\cal H}_{\mathrm{xx}}}E_{{\cal H}_{\mathrm{xx}}}E_{{\cal H}_{\mathrm{xx}}}E_{{\cal H}_{\mathrm{xx}}}E_{\partial / |{\cal H}_{\mathrm{xx}}} |{\partial / |{\cal H}_{\mathrm{xx}}} |{\partial / |{\cal H}_{\mathrm{xx}}} |{\partial / |{\cal H}_{\mathrm{xx}}} |{\partial / |{\cal H}_{\mathrm{xx}}} |{\partial / |{\cal H}_{\mathrm{xx}}} |{\partial / |{\cal H}_{\mathrm{xx}}} |{\partial / |{\cal H}_{\mathrm{xx}}} |{\partial/ |{\cal H}_{\mathrm{xx}}} |{\partial / |{\cal H}_{\mathrm{xx}}} |{\partial / |{\cal H}_{\mathrm{xx}}} |{\partial / |{\cal H}_{\mathrm{xx}}} |{\partial / |{\cal H}_{\mathrm{xx}}} |{\partial / |{\cal H}_{\mathrm{xx}}} |{\partial / |{\cal H}_{\mathrm{xx}}} |{\partial / |}{\cal H}_{\mathrm{xx}}.\\&= B_{{\cal H}_{\mathrm{xx}}}E_{{\cal H}_{\mathrm{xx}}}E_{{\cal H}_{\mathrm{xx}}}E_{{\cal H}_{\mathrm{xx}}}E_{{\cal H}_{\mathrm{xx}}}E_{{\cal H}_{\mathrm{xx}}}E_{{\cal H}_{\mathrm{xx}}}E_{{\cal H}_{\mathrm{xx}}} E_{{\cal H}_{\mathrm{xx}}}E_{{\cal H}_{\mathrm{xx}}}E_{{\cal H}_{\mathrm{xx}}}E_{{\cal H}_{\mathrm{xx}}}E_{{\cal H}_{\mathrm{xx}}}E_{{\cal H}_{\mathrm{xx}}}E_{{\cal H}_{\mathrm{xx}}}E_{{\cal H}_{\mathrm{xx}}}E_{\delta / |{\cal H}_{\mathrm{xx}}} |{\delta / |{\cal H}_{\mathrm{xx}}} |{\delta / |{\cal H}_{\mathrm{xx}}} |{\delta / |{\cal H}_{\mathrm{xx}}} |{\delta / |{\cal H}_{\mathrm{xx}}} |{\delta / |{\cal H}_{\mathrm{xx}}} |{\delta / |{\cal H}_{\mathrm{xx}}} |{\delta / |{\cal H}_{\mathrm{xx}}} |{\delta/ |{\cal H}_{\mathrm{xx}}} |{\delta/ |{\cal H}_{\mathrm{xx}}} |{\delta/ |{\cal H}_{\mathrm{xx}}} |{\delta/ |{\cal H}_{\mathrm{xx}}} |{\delta/ |{\cal H}_{\mathrm{xx}}} |{\delta/ |{\cal H}_{\mathrm{xx}}} |{\delta/ |{\cal H}_{\mathrm{xx}}} |{\delta/ |}{\delta/ |}{\delta/ |}{\delta/ |}{\delta/ |}{\delta/ |}{\delta/ |}{\delta/ |}{\delta/ |}{\delta/ |}{\delta/ |}{\delta/ |}{\delta/ |}{\delta/ |}{\delta/ |}{\delta/ |}{\delta/ |}{\delta/ |}{\delta/ |}{\delta/ |}{\delta/ |}{8.5}\end{array}\]

So in general

\[\begin{array}{r l} & E _ {t} \left[ \tilde {\mathbf {x}} _ {t + k} ^ {r d} - \mathbf {x} _ {t + k} ^ {r d} \right] = \mathbf {h} _ {\mathbf {x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + k - 1} ^ {r d} - \mathbf {x} _ {t + k - 1} ^ {r d} \right] + \mathbf {H} _ {\mathbf {x x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + k - 1} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + k - 1} ^ {s} - \mathbf {x} _ {t + k - 1} ^ {f} \otimes \mathbf {x} _ {t + k - 1} ^ {s} \right] \\ & \quad + \frac {3}{6} \mathbf {h} _ {\sigma \sigma \mathbf {x}} \sigma^ {2} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + k - 1} ^ {f} - \mathbf {x} _ {t + k - 1} ^ {f} \right] \\ & \quad + \frac {1}{6} \mathbf {H} _ {\mathbf {x x x}} E _ {t} \left[ \tilde {\mathbf {x}} _ {t + k - 1} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + k - 1} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + k - 1} ^ {f} - \mathbf {x} _ {t + k - 1} ^ {f} \otimes \mathbf {x} _ {t + k - 1} ^ {f} \otimes \mathbf {x} _ {t + k - 1} ^ {f} \right] \end{array}\]

Thus, we know . So we only need to compute

and . This is done in the next two subsections. For these derivations recall that we define as above, i.e.

\[\boldsymbol {\delta} _ {t + j} = \left\{ \begin{array}{c l} \boldsymbol {\nu} + (\mathbf {I} - \mathbf {S}) \boldsymbol {\epsilon} _ {t + 1} & \text { for } j = 1 \\ \boldsymbol {\epsilon} _ {t + j} & \text { for } j \neq 1 \end{array} \right..\]

12.5.1 For

Consider

\[\begin{array}{r l} & {\mathbf {x} _ {t + l} ^ {f} \otimes \mathbf {x} _ {t + l} ^ {f} \otimes \mathbf {x} _ {t + l} ^ {f} = (\mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\epsilon} _ {t + j}) \otimes (\mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\epsilon} _ {t + j}) \otimes (\mathbf {h} _ {\mathbf {xi}} ^ {l} \mathbf {x} _ {t} ^ {f} + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\epsilon} _ {t + j})} \\ & {\qquad = (\mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\epsilon} _ {t + j} + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\epsilon} _ {t + j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\epsilon} _ {t + j} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\epsilon} _ {t + j})} \end{array}\]

\[\begin{array} { r l } & { \otimes \left( \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } + \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } \right) } \\ & { = ( \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } + \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } + \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } + \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } ) \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } } \\ & + ( \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } + \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t} ^ { f } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } + \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \boldsymbol { \sigma } \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } + \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta} \boldsymbol { \epsilon } _ { t + j } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta} \boldsymbol { \epsilon } _ { t + j ) ) }\otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta} \boldsymbol { \epsilon } _ t + j ) j , j , j , j , j , j , j , j , j , j , j , j , j , j , j , j , j , j , j , j , j , j , j , j , j , j , j , j , j , j , j , j , j , j , j , j , j , j , j , j , j , j , j , j , j , j , j , j , j , j , j . \\ & = [ {\bf h} _ { {\bf x} {\bf x} {\bf x} {\bf f}} ] [ {\bf h} _ {\mathrm{x} {\bf x} {\bf f}} ] [ {\bf h} _ {\mathrm{x} {\bf x} {\bf f}} ] [ {\bf h} _ {\mathrm{x} {\bf x} {\bf f}} ] [ {\bf h} _ {\mathrm{x} {\bf x} {\bf f}} ] [ {\bf h} _ {\mathrm{x} {\bf x} {\bf f}} ]. \\ & { + [ 2 i n g ( 1 - i n g ( 1 - i n g ( 1 - i n g ( 1 - i n g ( 1 - i n g ( 1 - i n g ( 1 - i n g ( 1 - i n g ( 1 - i n g ( 1 - i n g ( 1 - i n g ( 1 - i n g ( 1 - i n g ( 1 - i n g ( 1 - i i n g ( 1 - i n g ( 1 - i i n g ( 1 - i i n g ( 1 - i i n g ( 1 - i i n g ( 1 - i i n g ( 1 - i i n g ( 1 - i i n g ( 1 - i i n g ( 1 - i i n g ( 1 - i i n g ( 1 - i i n g ( 1 - i i n g ( 1 - i i n g( 1 - i i n g ( 1 - i i n g ( 1 - i i n g ( 1 - i i n g ( 1 - i i n g ( 1 - i i n g ( 1 - i i n g ( 1 - i i n g ( 1 - i i n g ( 1 - i i n g ( 1 - i i n g ( 1 - i i n g ( 1 - e q u a d ) ) ) ) ) ) ) ) ] [ (\mathrm{的} _ {\mathrm{的}} ) ] [ (\mathrm{的} _ {\mathrm{的}} ) ] [ (\mathrm{的} _ {\mathrm{的}} ) ] [ (\mathrm{的} _ {\mathrm{的}} ) ] [ (\mathrm{的} _ {\mathrm{的}} ) ] [ (\mathrm{的} _ {\mathrm{的}} ) ] [ (\mathrm{的} _ {\mathrm{的}} ) ] [ (\mathrm{的} ~ ) ] [ (\mathrm{的} ~ ) ] [ (\mathrm{的} ~ ) ] [ (\mathrm{的} ~ ) ] [ (\mathrm{的} ~ ) ] [ (\mathrm{的} ~ ) ] [ (\mathrm{的} ~ ) ] [ (\mathrm{的} ~ ) ] [ (\mathrm{的} ~ ) ] [ (\mathrm{的} ~ ) ] [ (\mathrm{的}~ ) ] [ (\mathrm{的} ~ ) ] [ (\mathrm{的} ~ ) ] [ (\mathrm{的} ~ ) ] [ (\mathrm{的} ~ ) ] [ (\mathrm{的} ~ ) ] [ (\mathrm{的} ~ ) ] [ (\mathrm{的} ~ ) ] [ (\mathrm{的} ~ ) ] [ (\mathrm{的} ~ ) ] [ (\mathrm{的} . ) ] [ (\mathrm{的} ~ ) ] [ (\mathrm{的} ~ ) ] [ (\mathrm{的} ~ ) ] [ (\mathrm{的} ~ ) ] [ (\mathrm{的} ~ ) ] [ (\mathrm{的} ~ ) ] [ (\mathrm{的} ~ ) ] [ (\mathrm{的} ~ ) ] [ (\mathrm{的} ~ ) ] [ (\mathrm{的} · ) ] [ (\mathrm{的} · ) ] [ (\mathrm{的} · ) ] [ (\mathrm{的} · ) ] [ (\mathrm{的} · ) ] [ (\mathrm{的} · ) ] [ (\mathrm{的} · ) ] [ (\mathrm{的} · ) ] [ (\mathrm{的} · ) ] [ (\mathrm{的} · ) ] [ (\mathrm{的} ~ ) ] [ (\mathrm{的} ~ ) ] [ (\mathrm{的} ~ ) ] [ (\mathrm{的} ~ ) ] [ (\mathrm{的} ~ ) ] [ (\mathrm{的} ~ ) ] [ (\mathrm{的} ~ ) ] [ (\mathrm{的} ~) ] [ (\mathrm{的} ~) ] [ (\mathrm{的} ~) ] [ (\mathrm{的} ~) ] [ (\mathrm{的} ~) ] [ (\mathrm{的} ~) ] [ (\mathrm{的} ~) ] [ (\mathrm{的} ~) ] [ (\mathrm{的} ~) ] [ (\mathrm{的} ~) ] [ (\mathrm{的} ~)} \\ & + [ 2 i n g ( 2 i n g ( 2 i n g ( 2 i n g ( 2 i n g ( 2 i n g ( 2 i n g ( 2 i n g ( 2 i n g ( 2 i n g ( 2 i n g ( 2 i n g ( 2 i n g ( 2 i n g ( 2 i n g ( 2 i n g ( 2 i n g ( 2 i m a s s c o m p o r s s c o m p o r s s c o m p o r s s c o m p o r s s c o m p o r s s c o m p o r s s c o m p o r s s c o m p o r s s c o m p o r s s c o m p o r s s c o m p o r s s c o m p o r s s c o m p o r s s c a c k e r d e f f e r e d e f f e r e d e f f e r e d e f f e r e d e f f e r e d e f f e r e d e f f e r e d e f f e r e d e f f e r e d e f f e r e d e f f e r e d e f f e r e d e f f e r e d e f f e r e d e f f e r e d f e r e d e f f e r e d e f f e r e d e f f e r e d e f f e r e d e f f e r e d e f f e r e d e f f e r e d e f f e r e d e f f e r e d e f f e r e d e f f e r e d e f f e r e d e f f e r e d e f f a c k e r d e f f a c k e r d e f f a c k e r d e f f a c k e r d e f f a c k e r d e f f a c k e r d e f f a c k e r d e f f a c k e r d e f f a c k e r d e f f a c k e r d e f f a c k e r d e f f a c k e r k e r d e f f a c k e r d e f f a c k e r d e f f a c k e r d e f f a c k e r d e f f a c k e r d e f f a c k e r d e f f a c k e r d e f f a c k e r d e f f a c k e r d e f f a c k e r d e f f a c K E F A S U P O I N G E F A S U P O I N G E F A S U P O I N G E F A S U P O I N G E F A S U P O I N G E F A S U P O I N G E F A S U P O I N G E F A S U P O I N G E F A S U P O I N G E F A S U P O I N G E F A S U P O I NG E F A S U P O I N G E F A S U P O I N G E F A S U P O I N G E F A S U P O I N G E F A S U P O I N G E F A S U P O I N G E F A S U P O I N G E F A S U P O I N G E F A S U P O I N G E F A S U P O I N N E F A S U P O I N N E F A S U P O I N N E F A S U P O I N N E F A S U P O I N N E F A S U P O I N N E F A S U P O I N N E F A S U P O I N N E F A S U P O I N N E F A S U P O I N N E F A S U P O I NN E F A S U P O I NN E F A S U P O I NN E F A S U P O I NN E F A S U P O I NN E F A S U P O I NN E F A S U P O I NN E F A S U P O I NN E F A S U P O I NN E F A S U P O I NN E F A S U P O I N N E F A S U P O I N N E F A S U P O I N N E F A S U P O I N N E F A S U P O I N N E F A S U P O I N N E F A S U P O I N N E F A S U P O I N N E F A S U P O I N N E F A S U P O I N M E T H W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W WWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWW< fcel>[0,0]\]

And we therefore have

\[\begin{array}{r l} & {\tilde {\mathbf {x}} _ {t + l} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + l} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + l} ^ {f} = \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \delta_ {t + j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f}} \\ & {\quad + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \delta_ {t + j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \delta_ {t + j} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \delta_ {t + j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f}} \\ & {\quad + \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \delta_ {t + j} + \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \delta_ {t + j} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \delta_ {t + j}} \\ & {\quad + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \delta_ {t + j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \delta_ {t + j} + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \delta_ {t + j} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \delta_ {t + i j},} \end{array}\]

This means that:

\[\begin{array} { r l } & E _ { t } \left[ \tilde { \mathbf { x } } _ { t + l } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + l } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + l } ^ { f } - \mathbf { x } _ { t + l } ^ { f } \otimes \mathbf { x } _ { t + l } ^ { f } \otimes \mathbf { x } _ { t + l } ^ { f } \right] \\ & { = E _ { t } [ \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } + \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta _ { t + j } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } } \\ & { + \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta _ { t + j } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } + \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta _ { t + j } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta _ { t + j } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } } \\ & { + \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta _ { t + j } + \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta _ { t + j } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta_ { t + j }} \\ & { + \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta _ { t + j } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta} \delta _ { t + j } + \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta} \delta _ { t + j } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta} \delta _ { t + j }} \\ & \\ & - E _ { t } [ \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathx , s , t ] + H ( s , t ) = H ( s , t ) \\ & + E ( s , t ) = H ( s , t ) \\ & + E ( s , t ) = H ( s , t ) \\ & + E ( s , t ) = H ( s , t ) \\ & + E ( s , t ) = H ( s , t ) \\ & + E ( s , t ) = H ( s , t ) \\ & + E ( s , t ) = H ( s , t ) \\ & + E ( s , s ) = H ( s , s ) \\ & + E ( s , s ) = H ( s , s ) \\ & + E ( s , s ) = H ( s , s ) \\ & + E ( s , s ) = H ( s , s ) \\ & + E ( s , s ) = H ( s , s ) \\ & + E ( s , s ) = H ( s , s ) \\ & + E ( s , s ) = H ( s , s ) \\ & - E ( s , s ) = H ( s , s ) \\ & + E ( s , s ) = H ( s , s ) \\ & + E ( s , s ) = H ( s , s ) \\ & + E ( s , s ) = H ( s , s ) \\ & + E ( s , s ) = H ( s , s ) \\ & + E ( s , s ) = H ( s , s ) \\ & + E ( s . s ) = H ( s . s ) \\ & + E ( s . s ) = H ( s . s ) \\ & + E ( s . s ) = H ( s . s ) \\ & + E ( s . s ) = H ( s . s ) \\ & + E ( s . s ) = H ( s . s ) \\ & + E ( s . s ) = H ( s . s ) \\ & + E ( s . s ) = H ( s . s ) \\ * p o n d e c o m e n d e n d e n d e n d e n d e n d e n d e n d e n d e n d e n d e n d e n d e n d e n d e n d e n d e n d e n d e n d e n d e n d e n d e n d e n d e n d e n d e n d e n d e n d e n d e n d e n d e n d i n d e n d e n d e n d e n d e n d e n d e n d e n d e n d e n d e n d e n d e n d e n d e n d e n d e n d e n d e n d e n d e n d e n d e n d e n d e n d e n d e n d e n d e n d e n d e n d e n d e r a c k i g u a r i o u r b i v i c k i g u a r i o u r b i v i c k i g u a r i o u r b i v i c k i g u a r i o u r b i v i c k i g u a r i o u r b i v i c k i g u a r i o u r b i v i c k i g u a r i o u r b i v i c k i g u a r i o w i r y w i r y w i r y w i r y w i r y w i r y w i r y w i r y w i r y w i r y w i r y w i r y w i r y w i r y w i r y w i r y w i r y w i r y w i r y w i r y w i r y w i r y w i r y w i r y w i r y w i r | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I II / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *) & \\ & + E (\mathrm{d} ;) = E (\mathrm{d} ;) = E (\mathrm{d} ;) = E (\mathrm{d} ;) = E (\mathrm{d} ;) = E (\mathrm{d} ;) = E (\mathrm{d} ;) = E (\mathrm{d} ;) = E (\mathrm{d} ;) = E (\mathrm{d} ;) = E (\mathrm{d} ;) = E (\mathrm{d} ;) = F (\mathrm{d} ;) = F (\mathrm{d} ;) = F (\mathrm{d} ;) = F (\mathrm{d} ;) = F (\mathrm{d} ;) = F (\mathrm{d} ;) = F (\mathrm{d} ;) = F (\mathrm{d} ;) = F (\mathrm{d} ;) = F (\mathrm{d} ;) = F (\mathrm{d} ;) = F ({\mathrm{d}} ; {\mathrm{d}} ; {\mathrm{d}} ; {\mathrm{d}} ; {\mathrm{d}} ; {\mathrm{d}} ; {\mathrm{d}} ; {\mathrm{d}} ; {\mathrm{d}} ; {\mathrm{d}} ; {\mathrm{d}} ; {\mathrm{d}} ; {\mathrm{d}} ; {\mathrm{d}} ; {\mathrm{d}} ; {\mathrm{d}} ; {\mathrm{d}} ; {\mathrm{d}}, \\ & + E (\mathrm{d} ;) = E (\mathrm{d} ;) = E (\mathrm{d} ;) = E (\mathrm{d} ;) = E (\mathrm{d} ;) = E (\mathrm{d} ;) = E (\mathrm{d} ;) = E (\mathrm{d} ;) = E (\mathrm{d} ;) = F (\mathrm{d} ; {\mathrm{d}} ; {\mathrm{d}} ; {\mathrm{d}} ; {\mathrm{d}} ; {\mathrm{d}} ; {\mathrm{d}} ; {\mathrm{d}} ; {\mathrm{d}} ; {\mathrm{d}} ; {\mathrm{d}} ; {\mathrm{d}} ; {\mathrm{d}} ; {\mathrm{d}}; \\ & + E (\mathrm{d} ;) = F (\mathrm{d} ;) = F (\mathrm{d} ;) = F (\mathrm{d} ;) = F (\mathrm{d} ;) = F (\mathrm{d} ;) = F (\mathrm{d} ;) = F (\mathrm{d} ;) = F (\mathrm{d} ;) = F (\mathrm{d} ;) = F (\mathrm{d} ;) = G (\mathrm{d} ; {\mathrm{d}} ; {\mathrm{d}} ; {\mathrm{d}} ; {\mathrm{d}} ; {\mathrm{d}} ; {\mathrm{d}} ; {\mathrm{d}} ; {\mathrm{d}} ; {\mathrm{d}} ; {\mathrm{d}} ; {\mathrm{d}} ; {\mathrm{d}} ; {\mathrm{d}}; \\ & + E (\mathrm{d} ;) = F (\mathrm{\Delta} S, S, S, S, S, S, S, S, S, S, S, S, S, S, S, S, S, S, S, S, S, S, S, S, S, S, S, S, S, S, S, S, S, S, S, S, S, S, S, S, S, S, S, S, S, S, S, S, S, S, S , S , S , S , S , S , S , S , S , S , S , S , S , S , S , S , S , S , S , S , S , S , S , S , S , S , S , S , S , S , S , S , S , S , S , S , S , S , S , S , S , S , S , S , S , S , S , S ,S ,S ,S ,S ,S ,S ,S ,S ,S ,S ,S ,S ,S ,S ,S ,S ,S ,S ,S ,S ,S ,S ,S ,S ,S ,S ,S ,S ,S ,S ,S ,S ,S ,S ,S ,S ,S ,S ,S ,S ,S ,S ,S ,S ,S ,S ,S ,S ,S ,S ,T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T >T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T > T ] < [a ] < [a ] < [a ] < [a ] < [a ] < [a ] < [a ] < [a ] < [a ] < [a ] < [a ] < [a ] < [a ] < [a ] < [a ] < [a ] < [a ] < [a ] < [a ] < [a ] < [b ] < [b ] < [b ] < [b ] < [b ] < [b ] < [b ] < [b ] < [b ] < [b ] < [b ] < [b ] < [b ] < [b ] < [b ] < [b ] < [b ] < [b ] < [b ] < [b ] < [c ] < [c ] < [c ] < [c ] < [c ] < [c ] < [c ] < [c ] < [c ] < [c ] < [c ] < [c ] < [c ] < [c ] < [c ] < [c ] < [c ] < [c ] < [c ] < [c ] < [ c ] < [c ] < [c ] < [c ] < [c ] < [c ] < [c ] < [c ] < [c ] < [c ] < [c ] < [c ] < [c ] < [c ] < [c ] < [c ] < [c ] < [c ] < [c ] < [c ] < [s i f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f f w i r a c k a m a b a c k a m a b a c k a m a b a c k a m a b a c k a m a b a c k a m a b a c k a m a b a c k a m a b a c k a m a b a c k a m a b a c k a m a b a c k a m a b a c k a m a b a c k a m a b a c k a m a b a b a c k a m a b a c k a m a b a c k a m a b a c k a m a b a c k a m a b a c k a m a b a c k a m a b a c k a m a b a c k a m a b a c k a m a b a c k a m a b a c k a m a b a c k a m a b a c k a m a B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B C O P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W WWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwimwimwimwimwimwimwimwimwimwimwimwimwimwimwimwimwimwimwimwimwimwimwimwimwimwimwimwimwimwimwimwimwimwimwimwimwimwimwimwimwimwimwimwimwimwimwimwimwimwimwumomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomomonononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononononsion on one or more than one or more than one or more than one or more than one or more than one or more than one or more than one or more than one or more than one or more than one or more than one or more than one or more than one or more than one or more than one or more than one or more than one or more than one or more than one or more than one or more than one or more than one or more than one or more than one or more than one or more. The full name of the name is “*” and the full name of the name is “*” and the full name of the name is “*” and the full name of the name is “*” and the full name of the name is “*” and the full name of the name is “*” and the full name of the name is “*” and the full name of the name is “*” and the full name of the name is “*” and the full name of the name is “*” and the Full Name of the Name of the Name of the Name of the Name of the Name of the Name of the Name of the Name of the Name of the Name of the Name of the Name of the Name of the Name of the Name of the Name of the Name of the Name of the Name of the Name of the Name of the Name of the Name of the Name of the Name of the Name of the Name of the Name of the Name of the Name of the Name of the Name of the Name of the N/A. This is an important part in this way that it is not possible to be used in this way to create an important part in this way to create an important part in this way to create an important part in this way to create an important part in this way to create an important part in this way to create an important part in this way to create an important part in this way to create an important part in this way to create an important part in this way to create an important part in this way to create an important part in this way to create an important part in this way to create an important part in this way for which it is not possible to be used in this way to create an important part in this way to create an important part in this way to create an important part in this way to create an important part in this way to create an important part in this way to create an important part in this way to create an important part in this way to create an important part in this way to create an important part in this way to create an important part in this way to create an important part in this way to generate an important part in this way to generate an important part in this way to generate an important part in this way to generate an important part in this way to generate an important part in this way to generate an important part in this way to generate an important part in this way to generate an important part in this way to generate an important part in this way to generate an important part in this way to generate an important part in this way to generate an important part in this way to generate an important part from its own source. The following table includes: For example, it is not possible to be used in this way to create an important part in this way to create an important part in this way to create an important part in this way to create an important part in this way to create an important part in this way to create an important part in this way to create an important part in this way to create an important part in this way to create an important part in this way to create an important part in this way to create an important part in this way to produce an important part from its own source. The following table provides it as follows: - $E_{t}$ - $E_{t}$ - $E_{t}$ - $E_{t}$ - $E_{t}$ - $E_{t}$ - $E_{t}$ - $E_{t}$ - $E_{t}$ - $E_{t}$ - $E_{t}$ - $E_{t}$ - $E_{t}$ - $E_{t}$ - $E_{t}$ - $(E_{t})$\]

\[\begin{array}{r l} & {= E _ {t} [ 0 + \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \pmb {\eta} \pmb {\delta} _ {t + 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f}} \\ & {+ \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \pmb {\eta} \pmb {\delta} _ {t + 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\delta} _ {t + j} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\delta} _ {t + j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f}} \\ & {+ \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \pmb {\eta} \pmb {\delta} _ {t + 1} + \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\delta} _ {t + j} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\δ} _ {t + j}} \\ & {+ \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\delta} _ {t + j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\delta} _ {t + j} + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\delta} _ {t + j} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmod [ 2 ] [ 2 ] [ 2 ] [ 2 ] [ 2 ] [ 2 ] [ 2 ] [ 2 ] [ 2 ] [ 2 ] [ 2 ] [ 2 ] [ 2 ] [ 2 ] [ 2 ] [ 2 ] [ 2 ] [ 2 ] [ 2 ] [ 2 ] [ 2 ] [ 2 ] [ 2 ] [ 2 ] [ 2 ] [ 2 ].} \end{array}\]

\[\begin{array}{r l} & {- E _ {t} [ + \mathbf {0} + \mathbf {0}} \\ & {+ \mathbf {0} + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\epsilon} _ {t + j} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\epsilon} _ {t + j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f}} \\ & {+ \mathbf {0} + \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\epsilon} _ {t + j} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\alpha} _ {t + j}} \\ & {+ \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\epsilon} _ {t + j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\epsilon} _ {t + j} + \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\epsilon} _ {t + j} \otimes \sum_ {j = 1} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmod {\mathbf {x}}} \\ & {\mathrm{using} E _ {t} [ \pmb {\epsilon} _ {t + i} ] = 0 \mathrm{andcancellingtheterm} \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f}} \end{array}\]

\[\begin{array} { r l } & = E _ { t } [ \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \delta _ { t + 1 } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } + \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \delta _ { t + 1 } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } + \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \delta _ { t + 1 } \\ & + \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta _ { t + j } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta _ { t + j } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \\ & + \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta _ { t + j } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta_ { t + j } \\ & + \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta _ { t + j } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta _ { t + j } \\ & + \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta _ { t + j } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \xi } _ { t + j } \\ & - E _ { t } [ \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta} \epsilon _ { t + j } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta} \epsilon _ { t + j } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x} _ { t } ^ { f } \\ & + \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x} _ { t } ^ { f } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta} \epsilon _ { t + j } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h} _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol {\eta} \epsilon _ { t + j } \\ & + \sum _ { j = 1 } ^ { l } \mathbf { h} _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol {\eta} \epsilon _ { t + j } \otimes \mathbf {{ h} _ {\mathbf {{x}}} ^ {\ell}} ^ {\ell - j} {\mathbf {{x} _{t}}} ^ {\ell - j} {\mathbf {{x} _{t}}} ^ {\ell - j} {\mathbf {{x} _{t}}} ^ {\ell - j} {\mathbf {{x} _{t}}} ^ {\ell - j} \\ & + [ ( | | | ) ] [ ( | | | ) ] [ ( | | | ) ] [ ( | | | ) ] [ ( | | | ) ] [ ( | | | ) ] [ ( | | | ) ] [ ( | | | ) ] [ ( | | | ) ] [ ( | | | ) ] [ ( | | | ) ] [ ( | | | ) ] [ ( | | | ) ] [ ( | | | ) ] [ ( | | | ) ] [ (\mathrm{的}) ] [ (\mathrm{的}) ] [ (\mathrm{的}) ] [ (\mathrm{的}) ] [ (\mathrm{的}) ] [ (\mathrm{的}) ] [ (\mathrm{的}) ] [ (\mathrm{的}) ] [ (\mathrm{的}) ] [ (\mathrm{的}) ] [ (\mathrm{的}) ] [ (\mathrm{的}) ] [ (\mathrm{的}) ] [ (\mathrm{的}) ] [ (\mathrm{的}) ] , \\ & = E _ { t } [ {\mathbf {{h} _ {\mathbf {{x}}} ^ {\ell}}} {\mathbf {{x} _{t}}} ^ {\ell} \otimes {\mathbf {{h} _ {\mathbf {{x}}} ^ {\ell - 1}}} {\sigma} {\boldsymbol {\eta}} {\boldsymbol {\nu}} \otimes {\mathbf {{h} _ {\mathbf {{x}}} ^ {\ell}}} {\mathbf {{x} _{t}}} ^ {\ell} + {\mathbf {{h} _ {\mathbf {{x}}} ^ {\ell - 1}}} {\sigma} {\boldsymbol {\eta}} {\boldsymbol {\nu}} \otimes ((| | |) [ ( | | |) ] [ ( | | |) ] [ ( | | |) ] [ ( | | |) ] [ ( | | |) ] [ ( | | |) ] [ ( | | |) ] [ ( | | |) ] [ ( | | |) ] [ ( | | |) ] [ ( | | |) ] [ ( | | |) ] [ ( | | |) ] [ ( | | |) ] [ ( | | |) ] [ (\mathrm{的}) ] [ (\mathrm{的}) ] [ (\mathrm{的}) ] [ (\mathrm{的}) ] [ (\mathrm{的}) ] [ (\mathrm{的}) ] [ (\mathrm{的}) ] [ (\mathrm{的}) ] [ (\mathrm{的}) ] [ (\mathrm{的}) ] [ (\mathrm{的}) ] [ (\mathrm{的}) ] [ (\mathrm{的}) ] [ (\mathrm{的}) ] . \\ & + [ ( | | | ) ] [ ( | | | ) ] [ ( | | | ) ] [ ( | | | ) ] [ ( | | | ) ] [ ( | | | ) ] [ ( | | | ) ] [ ( | | | ) ] [ ( | | | ) ] [ ( | | | ) ] [ ( | | | ) ] [ ( | | | ) ] [ ( | | | ) ] [ ( |\nabla ) ! n a b d i n e s e n c o n s e n c o n s e n c o n s e n c o n s e n c o n s e n c o n s e n c o n s e n c o n s e n c o n s e n c o n s e n c o n s e n c o n s e n c o n s e n c o n s e n c o n s e n c o n s e n c o n s e n . \\ & + [ {\mathbf {{h}} _ {\mathbf {{x}}} ^ {\ell}} {\mathbf {{x}} _ {{t}}} ^ {\ell} + [ {\sum_ {| j = 1} ^ {| l}} {\sum_ {| k = 1} ^ {| l}} {\sum_ {| m = 1} ^ {| l}} {\sum_ {| n = 1} ^ {| l}} {\sum_ {| m = 1} ^ {| l}} {\sum_ {| n = 1} ^ {| l}} {\sum_ {| m = 1} ^ {| l}} {\sum_ {| n = 1} ^ {| l}} {\sum_ {| m = 1} ^ {| l}} {\sum_ {| m = 1} ^ {| l}} {\sum_ {| m = 1} ^ {| l}} {\sum_ {| m = 1} ^ {| l}} {\sum_ {| m = 1} ^ {| l}} {\sum_ {| m = 1} ^ {| l}} {\sum_ {| m = 1} ^ {| l}} {\sum_ {| m = 1} ^ {| l}} [ p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q :p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p p : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q: p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : P Q P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P T A N S O M O M O M O M O M O M O M O M O M O M O M O M O M O M O M O M O M O M O M O M O M O M O M O M O M O M O M O M O M O M O M O M O M O M O M O M O M O M O M O M O M O M O M O M O M O M O M O M O M O M O m o m o m o m o m o m o m o m o m o m o m o m o m o m o m o m o m o m o m o m o m o m o m o m o m o m o m o m o m o m o m o m o m o m o m o m o m o m o m o m o m o m o m o m o m o m o m o m o m o m o m e d i n e s e n c e n c e n c e n c e n c e n c e n c e n c e n c e n c e n c e n c e n c e n c e n c e n c e n c e n c e n c e n c e n c e n c e n c e n c e n c e n c e n c e n c e n c e n c e n c e n c e n c e n c i v i a r y i a r y i a r y i a r y i a r y i a r y i a r y i a r y i a r y i a r y i a r y i a r y i a r y i a r y i a r y i a r y i a r y i a r y i a r y i a r y i a r y i a r y i a r y i a r y i a r y i a r w i d u a r w i d u a r w i d u a r w i d u a r w i d u a r w i d u a r w i d u a r w i d u a r w i d u a r w i d u a r w i d u a r w i d u a r w i d u a r w i d u a r w i d u a r w i d u a r w i d u a r w w i d u a r w w i d u a r w w i d u a r w w i d u a r w w i d u a r w w i d u a r w w i d u a r w w i d u a r w w i d u a r w w i d u a r w w i d u a r w w i d u a r w w i d u a r w w i d u a r w w w i d u a r w w w i d u a r w w w i d u a r w w w i d u a r w w w i d u a r w w w i d u a r w w w i d u a r w w w i d u a r w w w i d u a r w w w i d u a r w w w w i d u a r w w w w i d u a r w w w w i d u a r w w w w i d u a r w w w w i d u a r w w w w i d u a r w w w w w i d u a r w w w w w i d u a r w w w w w i d u a r w w w w w i d u a r w w w w w i d u a r w w w w w i d u a r w w w w w i d u a r w w w w w i d u a r w w w w w i d u a r w w w w w i d u a r W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W WWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWwE I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I II , \\ & + [ H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H ; + [ H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H , H ; + [ H , G , G , G , G , G , G , G , G , G , G , G , G , G , G , G , G , G , G , G , G , G , G , G , G , G , G , G , G , G , G , G , G , G , G , G , G , G , G , G , G , G , G , G + [ G , G + G + G + G + G + G + G + G + G + G + G + G + G + G + G + G + G + G + G + G + G + G + G + G + G + G + + [ g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g g gg / 2 . \\ & + [ R / C ], \\ & = E _ {- L} E _ {- L} E _ {- L} E _ {- L} E _ {- L} E _ {- L} E _ {- L} E _ {- L} E _ {- L} E _ {- L} E _ {- L} E _ {- L} E _ {- L} E _ {- L} E _ {- L} E _ {- L} E _ {- L} E _ {- L} E _ {- L} E _ {- L} E _ {- L}\]

\[\begin{array} { l } + \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta _ { t + j } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta _ { t + j } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta_ { t + j } - \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { {\eta } } \boldsymbol { \epsilon } _ { t + j } \\ u s i n g \delta _ { t + 1 } = \boldsymbol { \nu } + ( \mathbf { I } - \mathbf { S} ) \boldsymbol { \epsilon } _ { t + 1 } a n d E _ { t } [ \boldsymbol { \epsilon } _ { t + 1 } ] = 0 . W e n o w e v a l u a t e t h e e x p r e s s i o n s o n e a c h o f t h e four l a s t l i n e s : \\ A _ { 1 } \equiv E _ { t } \left[ \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta _ { t + j } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta_{ t + j } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } - \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } \otimes \sum _ { j = 1 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldbm { \eta } \boldsymbol { \epsilon } _ { t + j } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \right] \\ = E _ { t } [ ( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \delta _ { t + 1 } + \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta _ { t + j} ) \otimes ( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \delta _ { t + 1 } + \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta _ { t +j} ) \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \\ - ( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \epsilon _ { t + 1 } + \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \epsilon _ { t + j} ) \otimes ( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta} \epsilon _ { t + 1 } + \sum _ { j = 2 } ^ { l } {\mathbf h} _ {\mathbf x} , {\mathbf x} , {\mathbf x} , {\mathbf x} , {\mathbf x} , {\mathbf x} , {\mathbf x} , {\mathbf x} , {\mathbf x} , {\mathbf x} , {\mathbf x} , {\mathbf x} , {\mathbf x} , {\mathbf x} , {\mathbf x} , {\mathbf x} , {\mathbf x} , {\mathbf y} ] \\ = E _ { t } [ ( {\mathbf h} _ {\mathbf x} ^ { l - 1} \sigma \boldsymbol { \eta } \delta _ { t + 1 } + ( {\mathbf h} _ {\mathbf x} ^ { l - 1} \sigma \boldsymbol { \eta} {\delta_ { t + 1 }} + {\mathbf h} _ {\mathbf x} ^ { l - 1} {\sigma} ) + ( {\mathbf h} _ {\mathbf x} ^ {{l - 1}} {\sigma} ) + ( {\mathbf h} _ {\mathbf x} ^ {{l - 1}} {\sigma} ) + ( {\mathbf h} _ {\mathbf x} ^ {{l - 1}} {\sigma} ) + ( {\mathbf h} _ {\mathbf x} ^ {{l - 1}} {\sigma} ) + ( {\mathrm{h}} _ {\mathrm{x}} ^ {{l - 1}} {\sigma} ) + ( {\mathrm{h}} _ {\mathrm{x}} ^ {{l - 1}} {\sigma} ) + ( {\mathrm{h}} _ {\mathrm{x}} ^ {{l - 1}} {\sigma} ) + ( {\mathrm{h}} _ {\mathrm{x}} ^ {{l - 1}} {\sigma} ) + \\ = E _ { t } [ ( {\mathbf h} _ {\mathbf x} ^ {{l - 1}} {\sigma} ) + ( {\mathbf h} _ {\mathbf x} ^ {{l - 1}} {\sigma} ) + ( {\mathbf h} _ {\mathbf x} ^ {{l - 1}} {\sigma} ) + ( {\mathrm{h}} _ {\mathrm{x} , {\mathrm{f}}} ^ {{l - 1}} {\sigma} ) + ( {\mathrm{h}} _ {\mathrm{x} , {\mathrm{f}}} ^ {{l - 1}} {\sigma} ) + ( {\mathrm{h}} _ {\mathrm{x} , {\mathrm{f}}} ^ {{l - 1}} {\sigma} ) + ( {\mathrm{h}} _ {\mathrm{x} , {\mathrm{s}}} ^ {{l - 1}} {\sigma} ) + \\ = E [ ( {\mathbf h} _ {\mathbf x} ^ {{l - 1}} {\sigma} ) + ( {\mathbf h} _ {\mathbf x} ^ {{l - 1}} {\sigma} ) + ( {\mathrm{h}} _ {\mathrm{x}} ^ {{l - 1}} {\sigma} ) + ( {\mathrm{h}} _ {\mathrm{x}}^ {{l - 1}} {\sigma} ) + ( {\mathrm{h}} _ {\mathrm{x}}^ {{l - 1}} {\sigma} ) + \\ = E [ ( {\mathrm{h}} _ {\mathrm{x}} ^ {{l - 1}} {\sigma} ) + ( {\mathrm{h}} _ {\mathrm{x}} ^ {{l - 1}} {\sigma} ) + ( {\mathrm{h}} _ {\mathrm{x}} ^ {{l - 1}} {\sigma} ) + ( {\mathrm{h}} _ {\mathrm{x}} ^ {{\ell - 1}} {\sigma} ) + \\ = E [ ( {\mathrm{h}} _ {\mathrm{x}} ^ {{l - 1}} {\sigma} ) + ( {\mathrm{h}} _ {\mathrm{x}} ^ {{l - 1}} {\sigma} ) + ( {\mathrm{h}} _ {\mathrm{x}} ^ {{l - 1}} {\sigma} ) - \\ = E [ ] \\ = E [ ] \\ = E [ ] \\ = E [ ] \\ = E [ ] \\ = E [ ] \\ = E [ ] \\ = E [ ] \\ = E [ ] \\ = E [ ] \\ = E [ ] \\ = E [ ] \\ = E [ ] \\ = E [ ] \\ = E [ ] \\ = E [ ] \\ = E [ ] \\ = E [ ] \\ = E [ ] \\ = E [ ] \\ = E [ ], \\ = E [ ], \\ = E [ ], \\ = E [ ], \\ = E [ ], \\ = E [ ], \\ = E [ ], \\ = E [ ], \\ = E [ ], \\ = E [ ], \\ = E [ ], \\ = E [ ], \\ = E [ ], \\ = E [ ], \\ = E [ ], \\ = E [ ], \\ = E [ ], \\ = E [ ], \\ = E [ ], \\ = E [ ], \\ = E [ ]. \\ = E [ ]. \\ = E [ ]. \\ = E [ ]. \\ = E [ ]. \\ = E [ ]. \\ = E [ ]. \\ = E [ ]. \\ = E [ ]. \\ = E [ ]. \\ = E [ ]. \\ = E [ ]. \\ = E [ ]. \\ = E [ ]. \\ = E [ ]. \\ = E [ ]. \\ = E [ ]. \\ = E [ ]. \\ = E [ ]. \\ = E [ ]. \\ = E [ ] \\ = E [ ]. \\ = E [ ]. \\ = E [ ]. \\ = E [ ]. \\ = E [ ]. \\ = E [ ]. \\ = E [ ]. \\ = E [ ]. \\ = E [ ]. \\ = E [ ]. \\ = E [ ]. \\ = E [ ]. \\ = E [ ]. \\ = E [ ]. \\ = E [ ]. \\ = E [ ]. \\ = E [ ]. \\ = E [ ]. \\ = E [ ]. \\ = E [ ], \\ = E [ ], \\ = E [ ], \\ = E [ ], \\ = E [ ], \\ = E [ ], \\ = E [ ], \\ =E [ ] \\ =E [ ] \\ =E [ ] \\ =E [ ] \\ =E [ ] \\ =E [ ] \\ =E [ ] \\ =E [ ] \\ =E [ ] \\ =E [ ] \\ =E [ ] \\ =E [ ] \\ =E [ ] \\ =E [ ] \\ =E [ ] \\ =E [ ] \\ =E [ ] \\ =E [ ] \\ =E [ ] \\ =E [ ] \\ =E [ ] =E[ ]\\=E[ ]\\=E[ ]\\=E[ ]\\=E[ ]\\=E[ ]\\=E[ ]\\=E[ ]\\=E[ ]\\=E[ ]\\=E[ ]\\=E[ ]\\=E[ ]\\=E[ ]\\=E[ ]\\=E[ ]\\=E[ ]\\=E[ ]\\=E[ ]\\=E[ ]\\=E[ ]{\cal H},\\=E[ ]{\cal H}.\\=E[ ]{\cal H}.\\=E[ ]{\cal H}.\\=E[ ]{\cal H}.\\=E[ ]{\cal H}.\\=E[ ]{\cal H}.\\=E[ ]{\cal H}.\\=E[ ]{\cal H}.\\=E[ ]{\cal H}.\\=E[ ]{\cal H}.\\=E[ ]{\cal H}.\\=E[ ]{\cal H}.\\\=E[]\end{array}\]

\[\begin{array} { r l } & { } \quad - \left( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + 1 } + \mathbf { 0 } \right) \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \\ & { } \quad - \left( 0 + \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } \otimes \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } \right) \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } ] \\ & { } \text {using} \boldsymbol { \epsilon } _ { t + 1 } \text {has mean zero and is independent across time} \\ & = E _ { t } [ ( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \nu } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \nu } + \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } ( \mathbf { I - S} ) \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } ( \mathbf { I - S} ) \boldsymbol { \epsilon } _ { t + 1 } ) \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \\ & + 0 \\ & - ( 0 ) \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } ] \\ & c a n c e l l i n g ( \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } \otimes \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { {\eta} } _ { t + j } ) \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \\ & = E _ { t } [ ( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1} ) ( \sigma \boldsymbol { \eta } \boldsymbol { \nu } \otimes \sigma \boldsymbol { {\eta} } \boldsymbol { \nu} ) \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \\ & + ( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1} ) ( \sigma \boldsymbol { \eta} ( I - S ) \boldsymbol { \epsilon } _ { t + 1 } \otimes \sigma \boldsymbol { {\eta} } ( I - S ) \boldsymbol { \epsilon } _ { t + 1} ) \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \\ & - ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ A ) | B D ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )\]

\[\begin{array} { r l } & = E _ { t } [ \left( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \delta _ { t + 1 } + \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta _ { t + j } \right) \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \left( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \delta _ { t + 1 } + \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta _ { t + j } \right) \\ & - \left( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + 1 } + \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } \right) \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \left( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + 1 } + \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon }_{ t + j} \right) ] \\ & = E _ { t } [ \left( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \delta _ { t + 1 } + \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta _ {t + j } \right) \otimes \left( \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \delta _ { t + 1 } + \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta _ { t + j} \right) \\ & - \left( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + 1 } + \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \epsilon _ { t + j} \right) \otimes \left( \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \epsilon _ { t + 1 } + \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta} \epsilon _ { t + j} \right) ] \\ & = E _ { t } [ \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \delta _ { t + 1 } \otimes ( ) [ h _ {\mathbf {\Sigma}} ^ {\mathrm{d}} , h _ {\mathbf {\Sigma}} ^ {\mathrm{d}} , h _ {\mathbf {\Sigma}} ^ {\mathrm{d}} , h _ {\mathbf {\Sigma}} ^ {\mathrm{d}} , h _ {\mathbf {\Sigma}} ^ {\mathrm{d}} , h _ {\mathbf {\Sigma}} ^ {\mathrm{d}} , h _ {\mathbf {\Sigma}} ^ {\mathrm{d}} , h _ {\mathbf {\Sigma}} ^{\mathrm{d}} , h _ {\mathbf {\Sigma}} ^{\mathrm{d}} , h _ {\mathbf {\Sigma}} ^{\mathrm{d}} , h _ {\mathbf {\Sigma}} ^{\mathrm{d}} , h _ {\mathbf {\Sigma}} ^{\mathrm{d}} , h _ {\mathbf {\Sigma}} ^{\mathrm{d}} , h _ {\mathbf {\Sigma}} ^{\mathrm{d}} , h _ {\mathbb M} ^ {\mathrm{d}} , h _ {\mathbb M} ^{\mathrm{d}} , h _ {\mathbb M} ^{\mathrm{d}} , h _ {\mathbb M} ^{\mathrm{d}} , h _ {\mathbb M} ^{\mathrm{d}} , h _ {\mathbb M} ^{\mathrm{d}} , h _ {\mathbb M} ^{\mathrm{d}} , h _ {\mathbb M} ^{\mathrm{d}} , h _ {\mathbb M} ^{\mathrm{s}} , h _ {\mathbb M} ^{\mathrm{s}} , h _ {\mathbb M} ^{\mathrm{s}} , h _ {\mathbb M} ^{\mathrm{s}} , h _ {\mathbb M} ^{\mathrm{s}} , h _ {\mathbb M} ^{\mathrm{s}} , h _ {\mathbb M} ^{\mathrm{s}} , h _ {\mathbb M} ^{\mathrm{s}} , h _ {\mathbb M}, h _ {\mathbb M} ^{\mathrm{s}} , h _ {\mathbb M} ^{\mathrm{s}} , h _ {\mathbb M} ^{\mathrm{s}} , h _ {\mathbb M} ^{\mathrm{s}} , h _ {\mathbb M} ^{\mathrm{s}} , h _ {\mathbb M} ^{\mathrm{s}} , h _ {\mathbb M} ^{\mathrm{s}} , h _ {\mathbb M} ^{s} , h _ {\mathbb M} ^{s} , h _ {\mathbb M} ^{s} , h _ {\mathbb M} ^{s} , h _ {\mathbb M} ^{s} , h _ {\mathbb M} ^{s} , h _ {\mathbb M} ^{s} , h _ {\mathbb M} ^{s} , h _ {\mathbb M}^{s} , h _ {\mathbb M}^{s} , h _ {\mathbb M}^{s} , h _ {\mathbb M}^{s} , h _ {\mathbb M}^{s} , h _ {\mathbb M}^{s} , h _ {\mathbb M}^{s} , h _ {\mathbb M}^{s} , h_{\mathbb M}^{s} , h_{\mathbb M}^{s} , h_{\mathbb M}^{s} , h_{\mathbb M}^{s} , h_{\mathbb M}^{s} , h_{\mathbb M}^{s} , h_{\mathbb M}^{s} , h_{\mathbb M}^{s} , h_{\mathbb M}^{s} , h_{\texttt{S}_{\texttt{S}}} , h_{\texttt{S}_{\texttt{S}}} , h_{\texttt{S}_{\texttt{S}}} , h_{\texttt{S}_{\texttt{S}}} , h_{\texttt{S}_{\texttt{S}}} , h_{\texttt{S}_{\texttt{S}}} , h_{\texttt{S}_{\texttt{S}}} , h_\texttt{S}_{t}_j, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, \\ & = E _ { t } [ a b d i s c e n t i o n ( a b d i s c e n t i o n ( a b d i s c e n t i o n ( a b d i s c e n t i o n ( a b d i s c e n t i o n ( a b d i s c e n t i o n ( a b d i s c e n t i o n ( a b d i s c e n t i o n ( a b d i s c e n s t i o n ( a b d i s c e n s t i o n ( a b d i s c e n s t i o n ( a b d i s c e n s t i o n ( a b d i s c e n s t i o n ( a b d i s c e n s t i o n ( a b d i s c e n s t i o n ( a b d i s c e n s t ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ), and the total sum of all elements is zero. The total sum of all elements is calculated by the sum of all elements in each element is the sum of all elements in each element is the sum of all elements in each element is the sum of all elements in each element is the sum of all elements in each element is the sum of all elements in each element is the sum of all elements in each element is the sum of all elements in each element is the sum of all elements in each element is the sum of all elements in each element is the sum of all elements in each element is the sum of all elements in each element is that the sum of all elements is the sum of all elements in each element is the sum of all elements in each element is the sum of all elements in each element is the sum of all elements in each element is the sum of all elements in each element is the sum of all elements in each element is the sum of all elements in each element is the sum of all elements in each element is the sum of all elements in each element is the sum of all elements in each element is the sum of all elements in each elementis the sum of all elements is the sum of all elements in each element is the sum of all elements in each element is the sum of all elements in each element is the sum of all elements in each element is the sum of all elements in each element is the sum of all elements in each element is the sum of all elements in each element is the sum of all elements in each element is the sum of all elements in each element is the sum of all elements in each element is the sum ofall elements in each element is the sum of all elements in each element is the sum of all elements in each element is the sum of all elements in each element is the sum of all elements in each element is the sum of all elements in each element is the sum of all elements in each element is the sum of all elements in each element is the sum of all elements in each element is the sum of all elements in each element is the sum of all elements in each element is the sum of all items in each element is the sum of all items in each element is the sum of all items in each element is the sum of all items in each element is the sum of all items in each element is the sum of all items in each element is the sum of all items in each element is the sum of all items in each element is the sum of all items in each element is the sum of all items in each element is the sum of all items in each element is the sum of all itemsin each item in each item is the sum of all items in each item is the sum of all items in each item is the sum of all items in each item is the sum of all items in each item is the sum of all items in each item is the sum of all items in each item is the sum of all items in each item is the sum of all items in each item is the sum of all items in each item is the sum of all items in each item is the sum of all items in each item is the sum of all itemsin each item in each item is the sum of all items in each item is the sum of all items in each item is the sum of all items in each item is the sum of all items in each item is the sum of all itemsin each item in each item is the sum of all itemsin each item is the sum of all itemsin each item is the sum of all itemsin each item is the sum of all itemsin each item is the sum of all itemsin each item is the sum of all itemsin each item is the sum of all itemsin each item is the sum of all itemsin each item is the sum of all itemsin each item is the sum of all itemsin each item is the sum of all itemsin each item is the sum of all itemsin every one or two times a day after day 30; and then one or two times a day after day 30; and then one or two times a day after day 30; and then one or two times a day after day 30; and then one or two times a day after day 30; and then one or two times a day after day 30; and then one or two times a day after day 30; and then one or two times a day after day 30; and then one/20000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000168999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999988888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888887777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777777766666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666665444444444444444444444444444444444444444444444444444444444444444444444444444444444444444435555555555555555555555555555555555555555555555555555555555555555555555555555555555555555553333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333 22222222222222222222222222222222222222222222222222222222222222222222222222222222222222221111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111< fcel>\( E_{t}\left[a b d\right] = E_a^T(a)b^T(a)b^T(a)b^T(a)b^T(a)b^T(a)b^T(a)b^T(a)b^T(a)b^T(a)b^T(a)b^T(a)b^T(a)b^T(a)b^T(a)b^T(a)b^T(a)b^T(a)b^T(a)b^T(a)b^T(a)b^T(a)b^T(a)b^T(a)b^T(a)b^T(a)c a b d a b d a b d a b d a b d a b d a b d a b d a b d a b d a b d a b d a b d a b d a b d a b d a b d a b d a b d a b d a b d a b d a b d a b d a b d a b d a b d a b d a b d a b d a b d a b d a b d a b d A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A B A< nl>\]

\[\begin{array}{r l} & {\mathrm{cancellingtheterm} \sum_ {j = 2} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\epsilon} _ {t + j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \sum_ {j = 2} ^ {l} \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \pmb {\eta} \pmb {\epsilon} _ {t + j}} \\ & \\ & {= E _ {t} [ \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \pmb {\eta} \pmb {\nu} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \pmb {\eta} \pmb {\nu} + \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \pmb {\eta} (\mathbf {I} - \mathbf {S}) \pmb {\epsilon} _ {t + 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \pmb {\eta} (\mathbf {I} - \mathbf {S}) \pmb {\epsilon} _ {t + 1}} \\ & {- \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1} ]} \end{array}\]

\[\begin{array}{l} \text {So we only need to compute the two terms involving \epsilon_{t + 1} . We next consider:} \\ E _ {t} \left[ \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S}) \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S}) \boldsymbol {\epsilon} _ {t + 1} \right] \\ = E _ {t} \left[ v e c \left(\left(\mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f}\right) \left(\mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S}) \boldsymbol {\epsilon} _ {t + 1}\right) ^ {\prime}\right) \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S}) \boldsymbol {\epsilon} _ {t + 1} \right] \\ \text {using a\otimes b = vec(ba^{\prime})} \\ = E _ {t} \left[ v e c \left(\left(\mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f}\right) \boldsymbol {\epsilon} _ {t + 1} ^ {\prime} \left(\mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S})\right) ^ {\prime}\right) \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S}) \boldsymbol {\epsilon} _ {t + 1} \right] \\ = E _ {t} \left[ \left(\mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S}) \otimes \left(\mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f}\right)\right) v e c (\boldsymbol {\epsilon} _ {t + 1} ^ {\prime}) \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S}) \boldsymbol {\epsilon} _ {t + 1} \right] \\ \text {using vec(ABC)} = (C ^ {\prime} \otimes A) v e c (B) \\ = E _ {t} \left[ \left(\mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S}) \otimes (\mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f})\right) \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S}) \boldsymbol {\epsilon} _ {t + 1} \right] \\ \text {using vec(\epsilon_ {{t + 1}} ^{\prime}) = \epsilon_ {{t + 1}}} \\ = E _ {t} \left[ (\mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S}) \otimes (\mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f}) \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S})) (\boldsymbol {\epsilon_ {{t + 1}}} \otimes \boldsymbol {\epsilon_ {{t + 1}}}) ] \\ = (\mathbf {h} _ {\mathbf {x}} ^ {{l - 1}} \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S}) \otimes (\mathbf {h} _ {\mathbf {x}} ^ {{l}} \mathbf {x} _ {{t}} ^ {{f}}) \otimes \mathbf {h} _ {\mathbf {x}} ^ {{l - 1}} \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S})) v e c (\mathbf {I}) \\ = E _ {{t}} [ (\mathbf {h} _ {\mathbf {x}} ^ {{l - 1}} [ (\mathbf {h} _ {\mathbf {x}} ^ {{l - 1}} [ (\mathbf {h} _ {\mathbf {x}} ^ {{l - 1}} [ (\mathbf {h} _ {\mathbf {x}} ^ {{l - 1}} [ (\mathbf {h} _ {\mathbf {x}} ^ {{l - 1}} [ (\mathbf {h} _ {\mathbf {x}} & [ (\boldsymbol {\epsilon_ {{t + 1}}} ] ]\right)) ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ],\]

and

\[\begin{array}{l} E _ {t} \left[ \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} \right] \\ = E _ {t} \left[ v e c \left(\left(\mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f}\right) \left(\mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1}\right) ^ {\prime}\right) \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} \right] \\ \text {using a\otimes b = vec(ba^{\prime})} \\ = E _ {t} \left[ v e c \left(\left(\mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f}\right) (\boldsymbol {\epsilon} _ {t + 1}) ^ {\prime} (\mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta}) ^ {\prime}\right) \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} \right] \\ = E _ {t} \left[ \left(\mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f}\right) v e c (\boldsymbol {\epsilon} _ {t + 1} ^ {\prime}) \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} \right] \\ \text {using vec(ABC)} = (C ^ {\prime} \otimes A) v e c (B) \\ = E _ {t} \left[ \left(\mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f}\right) \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} \right] \\ \text {using vec (\epsilon_ {t + 1}) = σ\eta\epsilon_ {t + 1}} \\ = E _ {t} \left[ \left(\left(\mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f}\right) \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta}\right) (\boldsymbol {\epsilon_ {t + 1}} \otimes \boldsymbol {\epsilon_ {t + 1}}) \right] \\ = (\mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f}) (\boldsymbol {\epsilon_ {t + 1}}) E _ {t} [ \boldsymbol {\epsilon_ {t + 1}} \otimes \boldsymbol {\epsilon_ {t + 1}} ] \\ = (\mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l} \mathbf {x} _ {t} ^ {f}) (\boldsymbol {\epsilon_ {t + 1}}) v e c (\mathbf {I}) \end{array}\]

\[\begin{array} { l } \text {Thus,} \\ A _ { 3 } = E _ { t } [ \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \eta \boldsymbol { \nu } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \eta \boldsymbol { \nu } + \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \eta ( \mathbf { I } - \mathbf { S } ) \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \eta ( \mathbf { I } - \mathbf { S } ) \boldsymbol { \epsilon } _ { t + 1 } \\ - \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \eta \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \eta \boldsymbol { \epsilon } _ { t + 1 } ] \\ = \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \eta \boldsymbol { \nu } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \eta \nu + (\mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \eta ( \mathbf { I } - \mathbf { S } ) \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \eta ( \mathbf { I } - \mathbf { S} ) ) v e c ( \mathbf { I } ) \\ - (\mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \eta \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \eta) v e c ( \mathbf { I } ) \\ = \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \eta \boldsymbol {\nu} \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \eta \boldsymbol {\nu} + (\mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \eta ( \mathbf { I } - \mathbf { S} ) \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \eta ( \mathbf { I } - \mathbf { S} ) - \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \eta \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \eta) v e c ( \mathbf {I} ) \\ F i n a l l y \\ A _ 4 = E _ { t } [ [ | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | ] \\ = E _ { t } [ ( [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ]: \\ = E _ { t } [ ( [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ & o n d i s t i o n d i s t i o n d i s t i o n d i s t i o n d i s t i o n d i s t i o n d i s t i o n d i s t i o n d i s t i o n d i s t i o n d i s t i o n d i s t i o n d i s t i o n d i s t i o n d i s t i o m a g e r e d e r e d e r e d e r e d e r e d e r e d e r e d e r e d e r e d e r e d e r e d e r e d e r e d e r e d e r e d e r e d e r e d e r e d e r e d e r e d e r e d e r e d e r e d e r e d e r e d e r e c o n t a p p . \\ + E _ { t } [ ( [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ ! o u g l a t i o n d i s t i o n d i s t i o n d i s t i o n d i s t i o n d i s t i o n d i s t i o n d i s t i o n d i s t i o n d i s t i o n d i s t i o n d i s t i o n d i s t i o n d i s t i o n d i s t ) w a b j a b j a b j a b j a b j a b j a b j a b j a b j a b j a b j a b j a b j a b j a b j a b j a b j a b j a b j a b j a b j a b j a b j a b j a b j a b j a b j a b j a b j a b j a b j a b j a b j a b j y a b j y a b j y a b j y a b j y a b j y a b j y a b j y a b j y a b j y a b j y a b j y a b j y a b j y a b j y a b j y a b j y a b j y a b j y a b j y a b j y a b j y a b j y a b j y a b j y a b j z y a b j z y a b j z y a b j z y a b j z y a b j z y a b j z y a b j z y a b j z y a b j z y a b j z y a b j z y a b j z y a b j z y a b j z y a b j z y a b j z y a b j z y a b j z y a b j z y a b j k y w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W WWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWW< fcel>[0]< fcel>[0]< nl>\]

\[\begin{array} { l } + \left( \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta _ { t + j } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \delta _ { t + 1 } + \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta _ { t + j } \otimes \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta _ { t + j } \right) \otimes \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \delta _ { t + j } \\ - \left( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \epsilon _ { t + 1 } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \epsilon _ { t + 1 } + \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \epsilon _ { t + 1 } \otimes \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \epsilon _ { t + j } \right) \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \epsilon _ { t + 1 } \\ - \left( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \epsilon _ { t + 1 } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \epsilon _ { t + 1 } + \mathbf { h } _ { \mathrm{x} } ^ { l - 1 } \sigma \boldsymbol { \eta } \epsilon _ { t + 1 } \otimes \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \epsilon _ { t + j } \right) \otimes \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \epsilon _ { t + j } \\ - \left( \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \epsilon _ { t + j } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \epsilon _ { t + 1 } + \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \epsilon _ { t + j } \otimes \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \epsilon _ { t + j } \right) \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \epsilon _ { t + 2 }\right) \\ - E _ { t } [ ( [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ) * ( [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ) * ( [ [ [ [ [ [ [ [ [ [ [ ] ] ] ] ] ] ] ] ] ] ) * ( [ [ [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( . ) * ( ) * ( ) * ( ) * ( ) * ( ) * ( ) * ( ) * ( ) * ( ) * ( ) * ( ) * ( ) * ( ) * ( ) * ( ) * ( ) * ( ) * ( ) * ( ) * ( ) * ( ) * ( ) * ( ) * ( ) * ( ) * ( 0. 5) * ( 0. 5) * ( 0. 5) * ( 0. 5) * ( 0. 5) * ( 0. 5) * ( 0. 5) * ( 0. 5) * ( 0. 5) * ( 0. 5) * ( 0. 5) * ( 0. 5) * ( 0. 5) * ( 1. 5) * ( 1. 5) * ( 1. 5) * ( 1. 5) * ( 1. 5) * ( 1. 5) * ( 1. 5) * ( 1. 5) * ( 1. 5) * ( 1. 5) * ( 1. 5) * ( 1. 5) * ( 1. 5), \\ + E _ {\mathrm{t}} / (\sum_ {j = 2} ^ {\mathrm{L}} H _ {\mathrm{x}} ^ {\mathrm{L} - j} H _ {\mathrm{x}} ^ {\mathrm{L} - j} H _ {\mathrm{x}} ^ {\mathrm{L} - j} H _ {\mathrm{x}} ^ {\mathrm{L} - j} H _ {\mathrm{x}} ^ {\mathrm{L} - j} H _ {\mathrm{x}} ^ {\mathrm{L} - j} H _ {\mathrm{x}} ^ {\mathrm{L} - j} H _ {\mathrm{x}} ^ {\mathrm{l} - j} H _ {\mathrm{x}} ^ {\mathrm{L} - j} H _ {\mathrm{x}} ^ {\mathrm{L} - j} H _ {\mathrm{x}} ^ {\mathrm{L} - j} H _ {\mathrm{x}} ^ {\mathrm{L} - j} H _ {\mathrm{x}} ^ {\mathrm{L} - j} H _ {\mathrm{x}} ^ {\mathrm{L} - j} H _ \mathrm{x}) , H _ {\mathrm{x}} ^ {\mathrm{L} - j} H _ {\mathrm{x}} ^ {\mathrm{L} - j} H _ {\mathrm{x}} ^ {\mathrm{L} - j} H _ {\mathrm{x}} ^ {\mathrm{L} - j} H _ {\mathrm{x}} ^ {\mathrm{L} - j} H _ {\mathrm{x}} ^ {\mathrm{L} - j} H _ {\mathrm{x}} ^ \mathrm{i} , H _ {\mathrm{x}} ^ {\mathrm{L} - j} H _ {\mathrm{x}} ^ {\mathrm{L} - j} H _ {\mathrm{x}} ^ {\mathrm{L} - j} H _ {\mathrm{x}} ^ {\mathrm{L} - j} H _ {\mathrm{x}} ^ {\mathrm{L} - j} H _ {\mathrm{x}} ^ {\mathrm{L} - j} H _ {\mathrm{x}} ^ \textit{e}, H _ {\mathrm{x}} ^ \textit{e}, H _ {\mathrm{x}} ^ \textit{e}, H _ {\mathrm{x}} ^ \textit{e}, H _ {\mathrm{x}} ^ \textit{e}, H _ {\mathrm{x}} ^ \textit{e}, H _ {\mathrm{x}} ^ \textit{e}, H _ {\mathrm{x}} ^ \textit{e}, H _ {\mathrm{x}} ^ {\prime}, H _ {\mathrm{x}} ^ \textit{e}, H _ {\mathrm{x}} ^ \textit{e}, H _ {\mathrm{x}} ^ \textit{e}, H _ {\mathrm{x}} ^ \textit{e}, H _ {\mathrm{x}} ^ \textit{e}, H _ {\mathrm{x}} ^ \textit{e}, H _ {\mathrm{x}} ^ \textit{e}, H _ {\mathrm{x}} ^\textit{e}, H _ {\mathrm{x}} ^\textit{e}, H _ {\mathrm{x}} ^\textit{e}, H_\textit{e},H_\textit{e},H_\textit{e},H_\textit{e},H_\textit{e},H_\textit{e},H_\textit{e},H_\textit{e},H_\textit{e},H_\textit{e},H_\textit{e},H_\textit{e},H_\textit{e},H_\texti,H_\textit{e},H_\textit{e},H_\textit{e},H_\textit{e},H_\textit{e},H_\textit{e},H_\textit{e},H_\textit{e},H_\textit{e},H_\textit{e},H_\textit{e},H_{\textit{e},H_{\times}\textnormal*{\pm}\pm\pm\pm\pm\pm\pm\pm\pm\pm\pm\pm\pm\pm\pm\pm\pm\pm\pm\pm\pm\pm\pm\pm\pm\pm\pm\pm\pm\pm\pm\pm\pm\pm\pm\pm\pm\pm\pm\pm\pm\pm\pm\pm\pm\pm\pm\pm\pm\pm\pm\pm}\right) \\ + E _ t , i n g , i n g , i n g , i n g , i n g , i n g , i n g , i n g , i n g , i n g , i n g , i n g , i n g , i n g , i n g , i n g , i n g , i n g , i n g , i n g , i n g , i n g , i n g , i n g , i n g , i ng , i n g , i n g , i n g , i n g , i n g , i n g , i n g , i n g , i n g , i n g , i n g , i n g , i n g , i n g , i n g , i n g , i n g , i n g , i n g , i n g , i n g , i n g , i n g , i n g , i n k , i n k , i n k , i n k , i n k , i n k , i n k , i n k , i n k , i n k , i n k , i n k , i n k , i n k , i n k , i n k , i n k , i n k , i n k , i n k , i n k , i n k , i n k , i n k , i n k , i n m , i n m , i n m , i n m , i n m , i n m , i n m , i n m , i n m , i n m , i n m , i n m , i n m , i n m , i n m , i n m , i n m , i n m , i n m , i n m , i n m , i n m , i n m , i n m , i n m , i nm , i nm , i nm , i nm , i nm , i nm , i nm , i nm , i nm , i nm , i nm , i nm , i nm ,i in a . \\ = E _ t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | + E _ t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | t | + E _ T A B C D E F G E F G E F G E F G E F G E F G E F G E F G E F G E F G E F G E F G E F G E F G E F G E F G E F G E F G E F G E F G E F G E F G E F G E F G E F G E F G E F G E F G E F G E F G E F G E F G E F G E F G< fcel>E = E_t[(\sum_ [j,k,l]k,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l,l, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, I, II, III, IVIII, VIII, VIIII, VIIIII, VIIIIII, VIIIIII, VIIIIII, VIIIIII, VIIIIII, VIIIIII, VIIIIII, VIIIIII, VIIIIII, VIIIIII, VIIIIII, VIIIIII, VIIIIII, VIIIIII, VIIIIII, VIIIIII, VIIIIII, VIIIIII, VIIIIII, VIIIIII, VIIIIII, VIIIIII, VIIIIII, VIIIIII, VIIIIII, VIIIIII, VIIIIII, VIIIIII, VIIIIII, VIIIIII, VIIIIII, VIIIIII, VIIIIII, VIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII< nl>\]

\[\begin{array} { l } - \left( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + 1 } + \mathbf { 0 } \right) \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + 1 } \\ - 0 \\ - 0 \\ - \left( 0 + \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } \otimes \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } \right) \otimes \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } ] \\ u s i n g \boldsymbol { \epsilon } _ { t + j } h a s m e a n z e r o a n d i s i n d e p e n d e n t a c r o s s t i m e \\ = E _ { t } [ \left( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } ( \boldsymbol { \nu } + ( \mathbf { I } - \mathbf { S} ) \boldsymbol { \epsilon } _ { t + 1 } ) \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } ( \boldsymbol { \nu } + ( \mathbf { I } - \mathbf { S} ) \boldsymbol { \epsilon } _ { t + 1 } ) + \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } ( \boldsymbol { \nu } + ( \mathbf { I } - \mathbf { S} ) \boldsymbol { \epsilon } _ { t + 1 } ) \otimes \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } \right) \\ \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } ( \boldsymbol { \nu } + ( \mathbf { I } - \mathbf { S} ) \boldsymbol { \epsilon } _ { t + 1 } ) \\ + \left( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \nu } \otimes \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } \right) \otimes \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j} \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } \\ + \left( \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } ( \boldsymbol { \nu } + ( \mathbf { I } - \mathbf { S} ) \boldsymbol { \epsilon } _ { t + 1 } ) + \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } \otimes \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j} \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j} \right) \\ + \\ - 0 ] \\ c a n c e l l i n g {\sum _ { j = 2 }} ^ { l } {\mathbf h} _ {\mathbf x} ^ {{l - j}} {\sigma} {\boldsymbol {\eta}} {\epsilon} _ {{t + j}} {\otimes} {\sum_ {{j = 2}}} ^ {{l}} {\mathbf h} _ {\mathbf x} ^ {{l - j}} {\sigma} {\boldsymbol {\eta}} {\epsilon} _ {{t + j}} {\otimes} {\sum_ {{j = 2}}} ^ {{l}} {\mathbf h} _ {\mathbf x} ^ {{l - j}} {\sigma} {\boldsymbol {\eta}} {\epsilon} _ {{t + j}} \\ = E _ { t } [ {\mathbf h} _ {\mathbf x} ^ {{l - 1}} {\sigma} {\boldsymbol {\eta}} ( {\boldsymbol {\nu}} + ( {\mathbf I} - {\mathbf S} ) {\epsilon} _ {{t + 1}} ) {\otimes} {\mathbf h} _ {\mathbf x} ^ {{l - 1}} {\sigma} {\boldsymbol {\eta}} ( {\boldsymbol {\nu}} + ( {\mathbf I} - {\mathbf S} ) {\epsilon} _ {{t + 1}} ) {\otimes} {\mathbf h} _ {\mathbf x} ^ {{l - 1}} {\sigma} {\boldsymbol {\eta}} ( {\boldsymbol {\nu}} + ( {\mathbf I} - {\mathbf S} ) {\epsilon} _ {{t + 1}} ) \\ + \\ + \\ + \\ + \\ + \\ + \\ + \\ + \\ + \\ + \\ + \\ + \\ + \\ + \\ + \\ + \\ + \\ + \\ + \\ + \\ + \\ + \\ + \\ + \\ + \\ + \\ + \\ + \\ + \\ + \\ + \\ + \\ + \\ + \\ + \\ + \\ + \\ + \\ + \\ + \\ + \\ + \\ + \\ + \\ + \\ + \\ + \\ + \\ + \\ + \\ - & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & \\ - 0 ] \\ c a n c e l l i n g {\sum_ {{j = 2}}} ^ {{l}} {\mathbf h} _ {\mathbf x} ^ {{l - j}} {\sigma} {\boldsymbol {\eta}} {\epsilon} _ {{t + j}} {\otimes} {\sum_ {{j = 2}}} ^ {{l}} {\mathbf h} _ {\mathbf x} ^ {{l - j}} {\sigma} {\boldsymbol {\xi}} {\epsilon} _ {{t + j}} {\otimes} {\sum_ {{j = 2}}} ^ {{l}} {\mathbf h} _ {\mathbf x} ^ {{l - j}} {\sigma} {\boldsymbol {\xi}} {\epsilon} _ {{t + j}} \\ = E _ { t } [ {\mathbf h} _ {\mathbf x} ^ {{l - 1}} {\sigma} {\boldsymbol {\eta}} ( {\boldsymbol {\nu}} + ( {\mathbf I} - {\mathbf S} ) {\epsilon} _ {{t + 1}} ) {\otimes} {\mathbf h} _ {\mathbf x}\]

\[\begin{array} { l } = E _ { t } [ ( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \eta \nu \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \eta \nu + \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \eta \nu \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \eta ( \mathbf { I } - \mathbf { S } ) \epsilon _ { t + 1 } ) \otimes ( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \eta \nu + \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \eta ( \mathbf { I } - \mathbf { S } ) \epsilon _ { t + 1 } ) \\ + ( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \eta ( \mathbf { I } - \mathbf { S } ) \epsilon _ { t + 1 } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \eta \nu + \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \eta ( \mathbf { I } - \mathbf { S } ) \epsilon _ { t + 1 } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \eta ( \mathbf { I } - \mathbf { S } ) \epsilon _ { t + 1 } ) \otimes ( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \eta \nu + \mathbf { h } _ { \mathbf { y } } ^ { l - 1 } \sigma \eta ( \mathbf { I } - \mathbf { S } ) \epsilon _ { t + 1 } \\ + 0 \\ + \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \eta \nu \otimes \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \eta \epsilon _ { t + j } \otimes \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \eta \epsilon _ { t + j } \\ + 0 \\ + \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \eta \epsilon _ { t + j } \otimes \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \eta \epsilon _ { t + j } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \eta \nu \\ + \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \eta \epsilon _ { t + j } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \eta \nu \otimes \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \eta \epsilon _ { t + j } \\ - \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \eta \epsilon _ { t + 1 } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \eta \epsilon _ { t + 1 } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \eta \epsilon _ { t + 1 } ] \\ = E _ { t } [ ( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \eta \nu \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \eta \nu + \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \eta \nu \otimes {\mathbf h} _ {\mathrm{I}} ^ {- 1} (\mathbf {\Sigma} | {\boldsymbol {\Sigma}}) | {\boldsymbol {\Sigma}} | {\boldsymbol {\Sigma}} | {\boldsymbol {\Sigma}} | {\boldsymbol {\Sigma}} | {\boldsymbol {\Sigma}} | {\boldsymbol {\Sigma}} | {\boldsymbol {\Sigma}} | {\boldsymbol {\Sigma}} | {\boldsymbol {\Sigma}} | {\boldsymbol {\Sigma}} | {\boldsymbol {\Sigma}} | {\boldsymbol {\Sigma}} | {\boldsymbol {\Sigma}} | {\boldsymbol {\Sigma}} | {\boldsymbol {\Sigma}} |{\boldsymbol {\Sigma}} | {\boldsymbol {\Sigma}} | {\boldsymbol {\Sigma}} | {\boldsymbol {\Sigma}} | {\boldsymbol {\Sigma}} | {\boldsymbol {\Sigma}} | {\boldsymbol {\Sigma}} | {\boldsymbol {\Sigma}} | {\boldsymbol {\Sigma}} | {\boldsymbol {\Sigma}} | {\boldsymbol {\Sigma}} | {\boldsymbol {\Sigma}} | {\boldsymbol {\Sigma}} | {\boldsymbol {\Sigma}} | {\boldmb u} \\ + ( [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] & \\ + ( [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] & \\ + ( [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ ] ] ] ] ] ] ] ) ) ) ) ) ) ) ) ) ) ) ) & \\ + ( [ [ [ [ [ ! ) ] ] ) ) ) & \\ + ( ) & \\ + ( ) & \\ + ( ) & \\ + ( ) & \\ + ( ) & \\ + ( ) & \\ + ( ) & \\ + ( ) & \\ + ( ) & \\ + ( ) & \\ + ( ) & \\ + ( ) & \\ + ( ) & \\ + ( ) & \\ + ( ) & \\ + ( ) & \\ + ( ) & \\ - ( ) & \\ - ( ) & \\ - ( ) & \\ - ( ) & \\ - ( ) & \\ - ( ) & \\ - ( ) & \\ - ( ) & \\ - ( ) & \\ - ( ) & \\ - ( ) & \\ - ( ) & \\ - ( ) & \\ - ( ) & \\ - ( ) & \\ - ( ) & \\ - ( ) = ( 3. 5) / 4. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . 5 . , 6 . 6 . 6 . 6 . 6 . 6 . 6 . 6 . 6 . 6 . 6 . 6 . 6 . 6 . 6 . 6 . 6 . 6 . 6 . 6 . 6 . 6 . 6 . 6 . 6 . 6 . 6 . 6 . 6 . 6 . 6 . 6 . 6 . 6 . , i n d i n e r e f f o r e f f o r e f f o r e f f o r e f f o r e f f o r e f f o r e f f o r e f f o r e f f o r e f f o r e f f o r e f f o r e f f o r e f f o r e f f o r e f f o r e f f o r e f f o r e f f o r e f f a r e s t i o n, i n d i n e r e f f o r e f f o r e f f o r e f f o r e f f o r e f f o r e f f o r e f f o r e f f o r e f f o r e f f o r e f f o r e f f o r e f f o r e f f o r e f f o r e f f o r e f f o r e f w i c k , i n d i n e r e f f o r e f f o r e f f o r e f f o r e f f o r e f f o r e f f o r e f f o r e f f o r e f f o r e f f o r e f f o r e f f o r e f w i c k , i n d i n e r e f w i c k , i n d i n e r e f w i c k , i n d i n e r e f w i c k , i n d i n e r e f w i c k , i n d i n e r e f w i c k , i n d i n e r e f w i c k , i n d i n e r e f w i c k , i n d i n e r g a t , i n d i n g a t , i n d i n g a t , i n d i n g a t , i n d i n g a t , i n d i n g a t , i n d i n g a t , i n d i n g a t , i n d i n g a t , i n d i n g a t , i n d i n g a t , i n d i n g a t , i n d i ng a t , i n d i n g a t , i n d i n g a t , i n d i n g a t , i n d i n g a t , i n d i n g a t , i n d i n g a t , i n d i n g a t , i n d i n g a t , i n d i n g a t , i n d i n g a t , i n d i n g b a t , i n d i n g b a t , i n d i n g b a t , i n d i n g b a t , i n d i n g b a t , i n d i n g b a t , i n d i n g b a t , i n d i n g b a t , i n d i n g b a t , i n d i n g b a t , i n d i n g c a t , i n d i n g c a t , i n d i n g c a t , i n d i n g c a t , i n d i n g c a t , i n d i n g c a t , i n d i n g c a t ,i n d i n g c a t ,i n d i n g c a t ,i n d i n g c a t ,i n d i n g c a t ,i n d i n g c a t ,i n d i n g c a t ,i in d i n g c a t ,i in d i n g c a t ,i in d i n g c a t ,i in d i n g c a t ,i in d i n g c a t ,i in d i n g c a t ,i in d i n g c a t ,i in d i n g c a t ,i in d i n g c a t ,i in d i n g c a t ,i\nu ; \\ = E _ {t} {[} (h _ {{\mathrm{x}}} ^ {- 1} | s | h _ {{\mathrm{x}}} ^ {- j} | s | h _ {{\mathrm{x}}} ^ {- j} | s | h _ {{\mathrm{x}}} ^ {- j} | s | h _ {{\mathrm{x}}} ^ {- j} | s | h _ {{\mathrm{x}}} ^ {- j} | s | h _ {{\mathrm{x}}} ^ {- j} | s | h _ {{\mathrm{x}}} ^ {- j} | s | h _ {{\mathrm{x}}} ^ {- j} {|} (h _ {{\mathrm{x}}} ^ {- j} | s | h _ {{\mathrm{x}}} ^ {- j} | s | h _ {{\mathrm{x}}} ^ {- j} | s | h _ {{\mathrm{x}}} ^ {- j} | s | h _ {{\mathrm{x}}} ^ {- j} | s | h _ {{\mathrm{x}}} ^ {- j} {|} (h _ {{\mathrm{x}}} ^ {- j} | s | h _{{\mathrm{x}}} ^ {- j} | s | h _ {{\mathrm{x}}} ^ {- j} | s | h _ {{\mathrm{x}}} ^ {- j} {|} (h _ {{\mathrm{x}}} ^ {- j} | s | h _{{\mathrm{x}}} ^ {- j} | s | h _ {{\mathrm{x}}} ^ {- j} {|} (h _ {{\mathrm{x}}} ^ {- j} | s | h _{{\mathrm{x}}} ^ {- j} | s | h _{{\mathrm{x}}} ^ {- j} {|} (h _{{\mathrm{x}}} ^ {- j} | s | h _{{\mathrm{x}}} ^ {- j} | s | h _{{\mathrm{x}}} ^ {- j} {|} (h _{{\mathrm{x}}} ^ {- j} | s | h _{{\mathrm{x}}} ^ {- j} | s | h _{{\mathrm{x}}} ^ {- j} {|} (h _{{\mathrm{x}}} ^ {- j} | s | h _{{\texttt {.x}}} ^ {- j} | s | h _{{\texttt {.x}}} ^ {- j} {|} (h _{{\texttt {.x}}} ^ {- j} | s | h _{{\texttt {.x}}} ^ {- j} {|} (h _{{\texttt {.x}}} ^ {- j} | s | h _{{\texttt {.x}}} ^ {- j} {|} (h _{{\texttt {.x}}} ^ {- j} | s | h _{{\texttt {.x}}} ^ {- j} {|} (h _{{\mathrm{y}}} ^ {- j} | s | h _{{\mathrm{y}}} ^ {- j} {|} (h _{{\mathrm{y}}} ^ {- j} | s | h _{{\mathrm{y}}} ^ {- j} {|} (h _{{\mathrm{y}}} ^ {- j} | s | h _{{\mathrm{y}}} ^ {- j} {|} (h _{{\mathrm{y}}} ^ {- j} | s | h _{{\mathrm{y}}} ^ {- j} {|} (h_{\mathrm{y}}^ {- j} {|} (h_{\mathrm{y}}^ {- j} {|} (h_{\mathrm{y}}^ {- j} {|} (h_{\mathrm{y}}^ {- j} {|} (h_{\mathrm{y}}^ {- j} {|} (h_{\mathrm{y}}^ {- j} {|} (h_{\mathrm{y}}^ {- j} {|} (h_{\mathrm{y}}^ {-j} {|} (h_{\mathrm{y}}^ {-j} {|} (h_{\mathrm{y}}^ {-j} {|} (h_{\mathrm{y}}^ {-j} {|} (h_{\mathrm{y}}^ {-j} {|} (h_{\mathrm{y}}^ {-j} {|} (h_{\mathrm{y}}^ {-j} {|} (h_{\mathrm{y}}^ {-j}{|} (h_{\mathrm{y}}^ {-j} {|} (h_{\mathrm{y}}^ {-j} {|} (h_{\mathrm{y}}^ {-j} {|} (h_{\mathrm{y}}^ {-j} {|} (h_{\mathrm{y}}^ {-j} {|} (h_{\mathrm{y}}^ {-j} {|} (h_{\mathrm{y}}^ {-j} {|} (\texttt {.x})|; \\ + (s) / p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q :p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : pq : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q : p q :p q : p q:\\ A _ {4,1}) & + H _ {\texttt {.x}} ^ {(l)} (\sigma) (\sigma) (\sigma) (\sigma) (\sigma) (\sigma) (\sigma) (\sigma) (\sigma) (\sigma) (\sigma) (\sigma) (\sigma) (\sigma) (\sigma) (\sigma) (\sigma) (\sigma) (\sigma) (\sigma) (\sigma) (\sigma) (\sigma) (\sigma) (\sigma) (\sigma) (\sigma) (\sigma) (\sigma) (\sigma) (\sigma) (\sigma) (\sigma) (\sigma)(\sigma) (\sigma) (\sigma) (\sigma) (\sigma) (\sigma) (\sigma) (\sigma) (\sigma) (\sigma) (\sigma) (\sigma) (\sigma) (\sigma) (\sigma) (\sigma) (\sigma) (\sigma) (\sigma) (\sigma) (\sigma) (\sigma) (\sigma) (\sigma) (\sigma) (\sigma) (\sigma) (\sigma) (\sigma) (\sigma) (\sigma) (\sigma) (\sigma) (\tau)) \\ A _ {4,2}) & + H _ {\texttt {.x}} ^ {(l)} (\sigma) (\delta) (\delta) (\delta) (\delta) (\delta) (\delta) (\delta) (\delta) (\delta) (\delta) (\delta) (\delta) (\delta) (\delta) (\delta) (\delta) (\delta) (\delta) (\delta) (\delta) (\delta) (\delta) (\delta) (\delta) (\delta) (\delta) (\delta) (\delta) (\delta) (\delta) (\delta) (\delta) (\delta) (\delta)(δ)\end{array}\]

\[\left. A _ {4, 8}\right) \qquad - \left. \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1} \right]\]

We can directly compute the first term in this expression. As for all remaining terms, we provide formulas below:

\[\begin{array}{r l} & A _ {4, 1} = E _ {t} [ \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta \boldsymbol {\nu} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta (\mathbf {I} - \mathbf {S}) \epsilon_ {t + 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta (\mathbf {I} - \mathbf {S}) \epsilon_ {t + 1} ] \\ & \qquad = E _ {t} [ \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta \boldsymbol {\nu} \otimes ((\mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta (\mathbf {I} - \mathbf {S}) \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta (\mathbf {I} - \mathbf {S})) (\epsilon_ {t + 1} \otimes \epsilon_ {t + 1})) ] \\ & \qquad = \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta \boldsymbol {\nu} \otimes (\mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta (\mathbf {I} - \mathbf {S}) \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta (\mathbf {I} - \mathbf {S})) v e c (\mathbf {I}) \\ & \qquad = \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta \boldsymbol {\nu} \times 1 \otimes (\mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta (\mathbf {I} - \mathbf {S}) \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta (\mathbf {I} - \mathbf {S})) v e c (\mathbf {I}) \\ & \qquad = (\mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta \boldsymbol {\nu} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta (\mathbf {I} - \mathbf {S}) \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta (\mathbf {I} - \mathbf {S})) (1 \otimes v e c (\mathbf {I})) \\ & \qquad u s i n g (\mathbf {A} \otimes \mathbf {B}) (\mathbf {C} \otimes \mathbf {D}) = \mathbf {A C} \otimes \mathbf {B D} i f \mathbf {A C} a n d \mathbf {B D} a r e d e f i n e d \\ & \qquad = (\mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta \boldsymbol {\nu} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \eta (\mathbf {I} - \mathbf {S}) \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} {\sigma} \eta (\mathbf {I} - {\boldsymbol {\mathrm{S}}})) v e c (\mathbf {I}) \\ & \qquad = (\mathbf {h} _ {\mathbf {x}} ^ {l - 1} {\sigma} {\eta ({\boldsymbol {\nu}}} {\sigma ({\boldsymbol {\mathrm{X}}})}) v e c (\mathbf {I}) \\ & = E _ {t}, E _ {\mathrm{X}} = E _ {\mathrm{X}} ^ {(2)} E _ {\mathrm{X}} ^ {(3)} E _ {\mathrm{X}} ^ {(4)} E _ {\mathrm{X}} ^ {(5)} E _ {\mathrm{X}} ^ {(6)} E _ {\mathrm{X}} ^ {(7)} E _ {\mathrm{X}} ^ {(8)} E _ {\mathrm{X}} ^ {(9)} E _ {\mathrm{X}} ^ {(1 0)} E _ {\mathrm{X}} ^ {(1 1)} E _ {\mathrm{X}} ^ {(1 2)} E _ {\mathrm{X}} ^ {(1 3)} E _ {\mathrm{X}} ^ {(1 4)} E _ {\mathrm{X}} ^ {(1 5)} E _ {\mathrm{X}} ^ {(1 6)} E _ {\mathrm{X}} ^ {(1 7)} E _ {\mathrm{X}} ^ {(1 8)} E _ {\mathrm{X}} ^ {(1 9)} E _ {\mathrm{X}} ^ {(2 0)} E _ {\mathrm{X}} ^ {(2 1)} E _ {\mathrm{X}} ^ {(2 2)} E _ {\mathrm{X}} ^ {(2 3)} E _ {\mathrm{X}} ^ {(2 4)} E _ {\mathrm{X}} ^ {(2 5)} E _ {\mathrm{X}} ^ {(2 6)} E _ {\mathrm{X}} ^ {(2 7)} E _ {\mathrm{X}} ^ {(2 8)} E _ {\mathrm{X}} ^ {(2 9)} E _ {\mathrm{X}} ^ {(3 0)} E _ {\mathrm{X}} ^ {(3 1)} E _ {\mathrm{X}} ^ {(3 2)} E _ {\mathrm{X}} ^ {(3 3)} E _ {\mathrm{X}} ^ {(3 4)} E _ {\mathrm{X}} ^ {(3 5)} E _ {\mathrm{X}} ^ {(3 6)} E _ {\mathrm{X}} ^ {(3 7)} E _ {\mathrm{X}} ^ {(3 8)} E _ {\mathrm{X}} ^ {(3 9)} E _ {\mathrm{X}} ^ {(4 0)} E _ {\mathrm{X}} ^ {(4 1)} E _ {\mathrm{X}} ^ {(4 2)} E _ {\mathrm{X}} ^ {(4 3)} E _ {\mathrm{X}} ^ {(4 4)} E _ {\mathrm{X}} ^ {(4 5)} E _ {\mathrm{X}} ^ {(4 6)} E _ {\mathrm{X}} ^ {(4 7)} E _ {\mathrm{X}} ^ {(4 8)} E _ {\mathrm{X}} ^ {(4 9)} E _ {\mathrm{X}} ^ {(5 0)} E _ {\mathrm{X}} ^ {(5 1)} E _ {\mathrm{X}} ^ {(5 2)} E _ {\mathrm{X}} ^ {(5 3)} E _ {\mathrm{X}} ^ {(5 4)} E _ {\mathrm{X}} ^ {(5 5)} E _ {\mathrm{X}} ^ {(5 6)} E _ {\mathrm{X}} ^ {(5 7)} E _ {\mathrm{X}} ^ {(5 8)} E _ {\mathrm{X}} ^ {(5 9)} E _ {\mathrm{X}} ^ {(6 0)} E _ {\mathrm{X}} ^ {(6 1)} E _ {\mathrm{X}} ^ {(6 2)} E _ {\mathrm{X}} ^ {(6 3)} E _ {\mathrm{X}} ^ {(6 4)} E _ {\mathrm{X}} ^ {(6 5)} E _ {\mathrm{X}} ^ {(6 6)} E _ {\mathrm{X}} ^ {(6 7)} E _ {\mathrm{X}} ^ {(6 8)} E _ {\mathrm{X}} ^ {(6 9)} E _ {\mathrm{X}} ^ {(7 0)} E _ {\mathrm{X}} ^ {(7 1)} E _ {\mathrm{X}} ^ {(7 2)} E _ {\mathrm{X}} ^ {(7 3)} E _ {\mathrm{X}} ^ {(7 4)} E _ {\mathrm{X}} ^ {(7 5)} E _ {\mathrm{X}} ^ {(7 6)} E _ {\mathrm{X}} ^ {(7 7)} E _ {\mathrm{X}} ^ {(7 8)} E _ {\mathrm{X}} ^ {(7 9)} E _ {\mathrm{X}} ^ {(8 0)} E _ {\mathrm{X}} ^ {(8 1)} E _ {\mathrm{X}} ^ {(8 2)} E _ {\mathrm{X}} ^ {(8 3)} E _ {\mathrm{X}} ^ {(8 4)} E _ {\mathrm{X}} ^ {(8 5)} E _ {\mathrm{X}} ^ {(8 6)} E _ {\mathrm{X}} ^ {(8 7)} E _ {\mathrm{X}} ^ {(8 8)} E _ {\mathrm{X}} ^ {(8 9)} E _ {\mathrm{X}} ^ {(9 0)}) \\ & = (E) (E) (E) (E) (E) (E) (E) (E) (E) (E) (E) (E) (E) (E) (E) (E) (E) (E) (E) (E) (E) (E) (E) (E) (E) (E) (E) (E) (E) (E) (E) (E) (E) (E) \\ & = (E) (E) (E) (E) (E) (E) (E) (E) (E) (E) (E) (E) (E) (E) (E) (E) (E) (E) (E) (E) (E) (E) (E) (E) (E) (E) (E). \end{array}\]

\[\begin{array}{l} A _ {4, 2} = E _ {t} [ \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S}) \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S}) \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} ] \\ = E _ {t} [ (\mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S}) \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S})) (\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}) \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} ] \\ = (\mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S}) \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S})) v e c (\mathbf {I}) \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} \\ = (\mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S}) \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S}) \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu}) v e c (\mathbf {I}) \end{array}\]

\[\begin{array}{r l} & A _ {4, 3} = E _ {t} [ \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S}) \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S}) \boldsymbol {\epsilon} _ {t + 1} ] \\ & \qquad = E _ {t} [ \left\{\mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} (\mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S}) \boldsymbol {\epsilon} _ {t + 1}) ^ {\prime} \right\} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S}) \boldsymbol {\epsilon} _ {t + 1} ] \\ & \qquad \text {using a\otimes b = vec(ba^{\prime})} \\ & \qquad = E _ {t} [ \left\{\mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} \boldsymbol {\epsilon} _ {t + 1} ^ {\prime} (\mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S})) ^ {\prime} \right\} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S}) \boldsymbol {\epsilon} _ {t + 1} ] \\ & \qquad = E _ {t} [ \left\{(\mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S}) \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu}) v e c (\boldsymbol {\epsilon} _ {t + 1} ^ {\prime}) \right\} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S}) \boldsymbol {\epsilon} _ {t + 1} ] \\ & \qquad \text {using vec(ABC)} = (C ^ {\prime} \otimes A) v e c (B) \\ & \qquad = E _ {t} [ (\mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S}) \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu}) \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S}) \boldsymbol {\epsilon} _ {t + 1} ] \\ & \qquad = E _ {t} [ (\mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S}) \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu}) \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S}) ] (\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}) ] \\ & \qquad = (\mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S}) \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S})) v e c (\mathbf {I}) \\ & \qquad = E _ {t}, E _ {t + 1}. \end{array}\]

\[\begin{array}{r l} & A _ {4, 4} = E _ {t} [ \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S}) \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S}) \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S}) \pmb {\epsilon} _ {t + 1} ] \\ & \qquad = E _ {t} [ \left\{\mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S}) \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S}) \right\} (\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}) \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S}) \pmb {\epsilon} _ {t + 1} ] \\ & \qquad = E _ {t} [ \left\{\mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\η} (\mathbf {I} - \mathbf {S}) \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S}) \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} (\mathbf {I} - \mathbf {S}) \right\} (\boldsymbol {\epsilon} _ {t + 2} \otimes \boldsymbol {\epsilon} _ {t + 2}) ] \end{array}\]

where has dimension and contains all the third moments of .

\[\begin{array} { l } A _ { 4 , 5 } = E _ { t } [ \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \nu } \otimes \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } \otimes \sum _ { j = 2 } ^ { l } \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } ] \\ = E _ { t } [ \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \nu } \otimes \left( \sum _ { j = 2 } ^ { l } \sum _ { k = 2 } ^ { l } \left( \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - k } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + k } \right) \right) ] \\ = E _ { t } [ \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \nu } \otimes \sum _ { j = 2 } ^ { l } \left( \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + j } \right) ] \\ b e c a u s e \boldsymbol { \epsilon } _ { t + j } a r e i n d e p e n d e n t a c r o s s t i m e \\ = E _ { t } [ \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \nu } \otimes \sum _ { j = 2 } ^ { l } \left( \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - j } \sigma \boldsymbol { \eta} \right) ( \boldsymbol { \epsilon } _ { t + j } \otimes \boldsymbol { \epsilon } _ { t + j} ) ] \\ = {\mathbf { h }} _ { {\mathbf x}} ^ { l - 1} {\sigma} {\boldsymbol {\eta}} {\boldsymbol {\nu}} \otimes \sum _ { j = 2 } ^ { l } ( {\mathbf h} _ {\mathbf x} ^ { l - j} {\sigma} {\boldsymbol {\eta}} \otimes {\mathbf h} _ {\mathbf x} ^ { l - j} {\sigma} {\boldsymbol {\eta}} ) v e c ( {\mathbf I} ) \\ = \sum _ { j = 2 } ^ { l } {\mathbf h} _ {\mathbf x} ^ { l - 1} {\sigma} {\boldsymbol {\eta}} {\boldsymbol {\nu}} \otimes ( {\mathbf h} _ {\mathbf x} ^ { l - j} {\sigma} {\boldsymbol {\eta}} \otimes {\mathbf h} _ {\mathbf x} ^ { l - j} {\sigma} {\boldsymbol {\eta}} ) v e c ( {\mathbf I} ) \\ A _ { 4 , 6 } = E _ { t } \left[ \sum _ { j = 2 } ^ { l } {\mathbf h} _ {\mathbf x} ^ { l - j} {\sigma} {\boldsymbol {\eta}} {\boldsymbol {\epsilon}} _ { t + j } \otimes \sum _ { j = 2 } ^ { l } {\mathbf h} _ {\mathbf x} ^ { l - j} {\sigma} {\boldsymbol {\eta}} {\boldsymbol {\epsilon}} _ { t + j } \otimes {\mathbf h} _ {\mathbf x} ^ { l - 1} {\sigma} {\boldsymbol {\eta}} {\boldsymbol {\nu}} \right] \\ = \sum _ { j = 2 } ^ { l } ( {\mathbf h} _ {\mathbf x} ^ { l - j} {\sigma} {\boldsymbol {\eta}} \otimes {\mathbf h} _ {\mathbf x} ^ { l - j} {\sigma} {\boldsymbol {\eta}} ) v e c ( {\mathbf I} ) \otimes {\mathbf h} _ {\mathbf x} ^ { l - 1} {\sigma} {\boldsymbol {\eta}} {\boldsymbol {\nu}} \\ A _ { 4 , 7 } = E _ { t } \left[ \sum _ { j = 2 } ^ { l } {\mathbf h} _ {\mathbf x} ^ { l - j} {\sigma} {\boldsymbol {\eta}} {\boldsymbol {\epsilon}} _ { t + j } \otimes {\mathbf h} _ {\mathbf x} ^ { l - 1} {\sigma} {\boldsymbol {\eta}} {\boldsymbol {\nu}} \otimes \sum _ { j = 2 } ^ { l } {\mathbf h} _ {\mathbf x} ^ { l - j} {\sigma} {\boldsymbol {\eta}} {{\boldsymbol {\epsilon}} _ { t + j }} ] \\ = \sum _ { j = 2 } ^ { l } E _ { t } [ {\mathbf h} _ {\mathbf x} ^ { l - j} {\sigma} {\boldsymbol {\eta}} {{\boldsymbol {\epsilon}} _ { t + j }} \otimes {\mathbf h} _ {\mathbf x} ^ { l - 1} {\sigma} {\boldsymbol {\eta}} {{\boldsymbol {\nu}}} \otimes {\mathbf h} _ {\mathbf x} ^ { l - j} {\sigma} {{\boldsymbol {\eta}}} {{\boldsymbol {\epsilon}} _ { t + j }} ] \\ b e c a u s e \boldsymbol { \epsilon } _ { t + j } i s i n d e p d e n t a c r o s s t i m e \\ = \sum _ { j = 2 } ^ { l } E _ { t } [ ( [ {{\mathbf h}_{\mathbf x}} ^{ l - 1} ] [ [ {{\boldsymbol{\eta}}} ] ] [ [ {{\mathbf h}_{\mathbf x}} ^{ l - j} ] [ [ {{\boldsymbol{\eta}}} ] ] [ [ {{\boldsymbol{\eta}}} ] ] [ [ {{\boldsymbol{\eta}}} ] ] [ [ {{\boldsymbol{\eta}}} ] ] [ [ {{\boldsymbol{\eta}}} ] ] [ [ {{\boldsymbol{\eta}}} ] ] [ [ {{\boldsymbol{\eta}}} ] ] ] \\ u s i n g a \otimes b = v e c (b a ^ {'}) \\ = \sum _ { j = 2 } ^ { l } E _ { t } [ ( [ {{\mathbf h}_{\mathbf x}} ^{ l - 1} ] [ [ {{\boldsymbol{\eta}}} ] ] [ [ {{\mathbf h}_{\mathbf x}} ^{ l - j} ] [ [ {{\boldsymbol{\eta}}} )^ {'} ] ] [ [ {{\boldsymbol{\eta}}} ] ] [ [ {{\boldsymbol{\eta}}} )^ {'} ] ] \\ = \sum _ { j = 2 } ^ { l } E _ { t } [ ( [ {{\mathbf h}_{\mathbf x}} ^{ l - j} ] [ [ {{\boldsymbol{\eta}}} ] ] [ [ {{\mathbf h}_{\mathbf x}} ^{ l - j} ] [ [ {{\boldsymbol{\eta}}} )^ {'} ] ] [ [ {{\boldsymbol{\eta}}} )^ {'} ] ] \\ u s i n g v e c (A B C) = (C ^ {'} | A) v e c (B) \\ & . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .\]

\[\begin{array}{l} = \sum_ {j = 2} ^ {l} E _ {t} \left[ \left(\mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu}\right) \boldsymbol {\epsilon} _ {t + j} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + j} \right] \\ = \sum_ {j = 2} ^ {l} E _ {t} \left[ \left(\mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta}\right) (\boldsymbol {\epsilon} _ {t + j} \otimes \boldsymbol {\epsilon} _ {t + j}) \right] \\ = \sum_ {j = 2} ^ {l} \left(\mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\nu} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - j} \sigma \boldsymbol {\eta}\right) v e c (\mathbf {I}) \end{array}\]

\[\begin{array}{l} A _ {4, 8} = E _ {t} \left[ \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} \right] \\ = E _ {t} \left[ \left(\mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta}\right) (\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}) \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} \right] \\ = E _ {t} \left[ \left(\mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbb {X}} ^ {l - 1} \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbb {X}} ^ {l - 1} \sigma \boldsymbol {\eta}\right) (\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}) \right] \\ = \left(\mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} ^ {l - 1} \sigma \boldsymbol {\eta}\right) \mathbf {m} ^ {3} (\boldsymbol {\epsilon} _ {t + 1}, \boldsymbol {\epsilon} _ {t + 1}, \boldsymbol {\epsilon} _ {t + 1}) \end{array}\]

where has dimension and contains all the third moments of .

Thus, we finally have

\[\begin{array} { l } E _ { t } \left[ \tilde { \mathbf { x } } _ { t + l } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + l } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + l } ^ { f } - \mathbf { x } _ { t + l } ^ { f } \otimes \mathbf { x } _ { t + l } ^ { f } \otimes \mathbf { x } _ { t + l } ^ { f } \right] = \\ \quad + \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \nu } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \\ \quad + \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \nu } \otimes \left( ( \mathbf { h } _ { \mathbf { x } } ^ { l } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } ) ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) \right) \\ \quad + \left( ( \mathbf { h } _ { \mathbf { x } } ^ { l } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } ) ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) \right) \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \nu } \\ A _ { 1 } ) + ( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } ) [ ( \sigma \boldsymbol { \eta } \boldsymbol { \nu } \otimes \sigma \boldsymbol { \eta } \boldsymbol { \nu} ) + \boldsymbol { \Lambda } ] \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \\ A _ { 2 } ) + \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes ( \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } ) [ ( \sigma \boldsymbol { \eta } \boldsymbol { \nu } \otimes \sigma \boldsymbol { \eta } \boldsymbol { \nu} ) + \boldsymbol { \Lambda } ] \\ A _ { 3 } ) + \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \nu } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l } \mathbf { x } _ { t } ^ { f } \otimes \mathbf { h } _ { \mathbf { x } } ^ { l - 1 } \sigma \boldsymbol { \eta } \boldsymbol { \nu} \\ + ( [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ ] v e c ( I ) ) , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ) ] ] ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ), \\ A _ { 4 , 1 } ) + ( [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ ] v e c ( I ) ) , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v e c ( I ) , a v e c ( I ) , b v e c ( I ) , c v e c ( I ) , d v e c ( I ) , e v e c ( I ) , f v e c ( I ) , g v e c ( I ) , h v e c ( I ) , i v e c ( I ) , j v e c ( I ) , k v e c ( I ) , l v e c ( I ) , m v e c ( I ) , n v e c ( I ) , o v e c ( I ) , p v e c ( I ) , q v e c ( I ) , r v e c ( I ) , s v e c ( I ) , t v e c ( I ) , u v e c ( I ) , v e c ( I ) , w v e c ( I ) , x v e c ( I ) , y v e c ( I ) , z v e c ( I ) , w y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y & A _ { 4 , 2} ) A _ { 4 , 2} ) A _ { 4 , 3} ) A _ { 4 , 4} ) A _ { 4 , 5} ) A _ { 4 , 6} ) A _ { 4 , 7} ) A _ { 4 , 8} ) \end{array}\]

12.5.2 For

\[\begin{array}{r l} & {\mathrm{callfromabovethat}} \\ & {\mathbf {x} _ {t + 1} ^ {f} \otimes \mathbf {x} _ {t + 1} ^ {s} = \left(\mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t} ^ {f} + \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1}\right) \otimes \left(\mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t} ^ {s} + \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) + \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}\right)} \\ & {\qquad = (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {s}\right) + (\mathbf {h} _ {\mathbf {x}} \otimes \frac {1}{2} \mathbf {H} _ {\mathbf {x x}}) \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) + (\mathbf {h} _ {\mathbf {x}} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}) \mathbf {x} _ {t} ^ {f}} \\ & {\qquad + (\sigma \pmb {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) (\pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {s}) + (\sigma \pmb {\eta} \otimes \frac {1}{2} \mathbf {H} _ {\mathbf {x x}}) (\pmb {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) + (\sigma \pmb {\eta} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}) \pmb {\epsilon} _ {t + 1}} \end{array}\]

Therefore:

\[\begin{array} { r l } & \mathbf { x } _ { t + 2 } ^ { f } \otimes \mathbf { x } _ { t + 2 } ^ { s } = ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ( \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { s } ) + ( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } ) ( \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } ) + ( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } ) \mathbf { x } _ { t + 1 } ^ { f } \\ & \qquad + ( \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x } } ) ( \boldsymbol { \epsilon } _ { t + 2 } \otimes \mathbf { x } _ { t + 1 } ^ { s } ) + ( \sigma \boldsymbol { \eta } \otimes \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } ) ( \boldsymbol { \epsilon } _ { t + 2 } \otimes \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } ) + ( \sigma \boldsymbol { \eta } \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } ) \boldsymbol { \epsilon } _ { t + 2 } \\ & = ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) [ ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { s } ) + ( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } ) ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) + ( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } ) \mathbf { x } _ { t } ^ { f } \\ & \qquad + ( \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x } } ) ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { s } ) + ( \sigma \boldsymbol { \eta } \otimes \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x} } ) ( \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) + ( \sigma \boldsymbol { \eta } \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } ) \boldsymbol { \epsilon } _ { t + 1 ]} \\ & + ( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x} } ) ( \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } ) + ( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { h } _ {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} {\sigma} , \\ & + ( \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ \mathbf { x} ) ( ( \boldsymbol { \epsilon } _ { t + 2 } \otimes X ) ^ s ) + ( ( S O U ) ( ( E T ) ( X U ) ) ( X U ) ) + ( S O U ) ( X U ) ) - ( S O U ) ( X U ) ) - ( S O U ) ( X U ) ) - ( S O U ) ( X U ) ) - ( S O U ) ( X U ) ) - ( S O U ) ( X U ) ) - ( S O U ) ( X U ) ) - ( S O U ) ( X U ) ) - ( S O U ) ( X U ) ) - ( S O U ) ( X U ) ) - . \\ & = ( H _ x y z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z z zz \\ & + ( H _ x y z z zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz< fcel>+ (\mathrm{H}) = (\mathrm{H}) = (\mathrm{H}) = (\mathrm{H}) = (\mathrm{H}) = (\mathrm{H}) = (\mathrm{H}) = (\mathrm{H}) = (\mathrm{H}) = (\mathrm{H}) = (\mathrm{H}) = (\mathrm{H}) = (\mathrm{H}) = (\mathrm{H}) = (\mathrm{H}) = (\mathrm{H}) = (\mathrm{H}) = (\mathrm{N}) = (\mathrm{N}) = (\mathrm{N}) = (\mathrm{N}) = (\mathrm{N}) = (\mathrm{N}) = (\mathrm{N}) = (\mathrm{N}) = (\mathrm{N}) = (\mathrm{N}) = (\mathrm{N}) = (\mathrm{N}) = (\mathrm{N}) = (\mathrm{N}) = (\mathrm{N}) = (\mathrm{N}) = (\mathrm{N}) = (\nabla) = (\nabla) = (\nabla) = (\nabla) = (\nabla) = (\nabla) = (\nabla) = (\nabla) = (\nabla) = (\nabla) = (\nabla) = (\nabla) = (\nabla) = (\nabla) = (\nabla) = (\nabla) = (\nabla) = (\nobla) = (\nobla) = (\nobla) = (\nobla) = (\nobla) = (\nobla) = (\nobla) = (\nobla) = (\nobla) = (\nobla) = (\nobla) = (\nobla) = (\nobla) = (\nobla) = (\nobla) = (\nobla) = (\nobla) = ( n o w i n e d i n e d i n e d i n e d i n e d i n e d i n e d i n e d i n e d i n e d i n e d i n e d i n e d i n e d i n e d i n e d i n e d i n e d i n e d i n e d i n e d i n e d i n e d i n e d i n e d i n e c a l l e r e. \\ & + ( H _ x y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y u p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p q u a l l e r e. \\ & + ( H _ x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x . \\ & + ( H _ x y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y yy u v a l l e r e. \\ & + ( H _ x Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y A B I N G R M I N G R M I N G R M I N G R M I N G R M I N G R M I N G R M I N G R M I N G R M I N G R M I N G R M I N G R M I N G R M I N G R M I N G R M I N G R M I N G R M I N G R M I N G R M I N G R M I N G R m i n g r o d i o n g r o d i o n g r o d i o n g r o d i o n g r o d i o n g r o d i o n g r o d i o n g r o d i o n g r o d i o n g r o d i o n g r o d i o n g r o d i o n g r o d i o n g r o d i o n g r o d i o n q u a l l e r e . \\ & + ( H _ {\infty} Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z ZZ a b a l l e r e. \\ & + ( H _ {\infty} Z a b a l l e r e. \\ & + ( H _ {\infty} a b a l l e r e. \\ & + ( H _ {\infty} a b a l l e r e. \\ & + ( H _ {\infty} a b a l l e r e. \\ & + ( H _ {\infty} a b a l l e r e. \\ & + ( H _ {\infty} a b a l l e r e. \\ & + ( H _ {\infty} a b a l ll e r e . \\ & + ( H _ {\infty} a b a l l e r e. \\ & + ( H _ {\infty} a b a l l e r e. \\ & + ( H _ {\infty} a b a l l e r e. \\ & + ( H _ {\infty} a b a l l e r e. \\ & + ( H _ {\infty} a b a l l e r e. \\ & + (H a b a l l e r e . \\ & + ( H a b a l l e r e . \\ & + ( H a b a l l e r e . \\ & + ( H a b a l l e r e . \\ & + ( H a b a l l e r e . \\ & + ( H a b a l l e r e . \\ & + ( H a b a l l e r e . \\ & + ( H a b a l l e r e . \\ & + ( H a c b a l l e r e . \\ & + ( H a b a l l e r e . \\ & + ( H a b a l l e r e . \\ & + ( H a b a l l e r e . \\ & + ( H a b a l l e r e . \\ & + ( H a b a l l e r e . \\ & + ( H a b a l l e r e . \\ & + ( H a b c a l l e r e . \\ & + ( H a b c a l l e r e . \\ & + ( H a b c a l l e r e . \\ & + ( H a b c a l l e r e . \\ & + ( H a b c a l l e r e . \\ & + ( H a b c a l l e r e . \\ & + ( H a b c a l l e r e . \\ & - ( H a b c a l l e r e . \\ & - ( H a b c a l l e r e . \\ & - ( H a b c a l l e r e . \\ & - ( H a b c a l l e r e . \\ & - ( H a b c a l l e r e . \\ & - ( H a b c a l l e r e . \\ & - ( H a b c a l l er e . \\ & - ( H a b c a l l e r e . \\ & - ( H a b c a l l e r e . \\ & - ( H a b c a l l e r e . \\ & - ( H a b c a l l e r e . \\ & - ( H a b c a l l e r e . \\ & - ( H a b c a l l e r e . \\ & - ( H a b s c a l l e r e . \\ & - ( H a b s c a l l e r e . \\ & - ( H a b s c a l l e r e . \\ & - ( H a b s c a l l e r e . \\ & - ( H a b s c a l l e r e . \\ & - ( H a b s c a l l e r e . \\ & - ( H a b s c a ll u v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v j u v i v j u v j u v j u v j u v j u v j u v j u v j u v j u v j u v j u v j u v j u v j u v j u v j u v j u v j u v j u v j u v j u v j u v j u v j u v j u v j u v j u v j u v j u v j u v j u v j u v j u v j u w k w k w k w k w k w k w k w k w k w k w k w k w k w k w k w k w k w k w k w k w k w k w k w k w k w k w k w k w k w k w k w k w k w k w k w k w k w k w k w k w k w k w k w k w k w k w k w k w k w k w K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W KW K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K W K | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F |F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | F | C < |content_end|>\]

and

\[\begin{array} { r l } & \mathbf { x } _ { t + 3 } ^ { f } \otimes \mathbf { x } _ { t + 3 } ^ { s } = ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) \left( \mathbf { x } _ { t + 2 } ^ { f } \otimes \mathbf { x } _ { t + 2 } ^ { s } \right) + ( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } ) \left( \mathbf { x } _ { t + 2 } ^ { f } \otimes \mathbf { x } _ { t + 2 } ^ { f } \otimes \mathbf { x } _ { t + 2 } ^ { f } \right) + ( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } ) \mathbf { x } _ { t + 2 } ^ { f } \\ & \qquad + ( \sigma \eta \otimes \mathbf { h } _ { \mathbf { x } } ) \left( \epsilon _ { t + 3 } \otimes \mathbf { x } _ { t + 2 } ^ { s } \right) + ( \sigma \eta \otimes \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } ) \left( \epsilon _ { t + 3 } \otimes \mathbf { x } _ { t + 2 } ^ { f } \otimes \mathbf { x } _ { t + 2 } ^ { f } \right) + ( \sigma \eta \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } ) \boldsymbol { \epsilon } _ { t + 3 } \\ & = ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) [ ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { 2 } ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { s } ) + ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } ) ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) + ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } ) \mathbf { x } _ { t } ^ { f } \\ & \qquad + ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ( \sigma \eta \otimes \mathbf { h } _ { \mathbf { x } } ) ( \epsilon _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { s } ) + ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ( \sigma \eta \otimes \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } ) ( \epsilon _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } ) + ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x} } ) ( \sigma \eta \otimes \frac { 1 } { 2 } \mathbf { h} _ { \sigma \sigma } \sigma ^ { 2} ) \boldsymbol { \epsilon _ { t + 1 } } \\ & + ( [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] : , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .\]

\[\begin{array}{l} + (\sigma \boldsymbol {\eta} \otimes \mathbf {h _ {x}}) (\boldsymbol {\epsilon} _ {t + 3} \otimes \mathbf {x} _ {t + 2} ^ {s}) + (\sigma \boldsymbol {\eta} \otimes \frac {1}{2} \mathbf {H _ {x x}}) (\boldsymbol {\epsilon} _ {t + 3} \otimes \mathbf {x} _ {t + 2} ^ {f} \otimes \mathbf {x} _ {t + 2} ^ {f}) + (\sigma \boldsymbol {\eta} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}) \boldsymbol {\epsilon} _ {t + 3} \\ = (\mathbf {h _ {x}} \otimes \mathbf {h _ {x}}) ^ {3} (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {s}) + \sum_ {i = 0} ^ {2} (\mathbf {h _ {x}} \otimes \mathbf {h _ {x}}) ^ {2 - i} (\mathbf {h _ {x}} \otimes \frac {1}{2} \mathbf {H _ {x x}}) (\mathbf {x} _ {t + i} ^ {f} \otimes \mathbf {x} _ {t + i} ^ {f} \otimes \mathbf {x} _ {t + i} ^ {f}) \\ + \sum_ {i = 0} ^ {2} (\mathbf {h _ {x}} \otimes \mathbf {h _ {x}}) ^ {2 - i} (\mathbf {h _ {x}} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}) \mathbf {x} _ {t + i} ^ {f} \\ + \sum_ {i = 0} ^ {2} (\mathbf {h _ {x}} \otimes \mathbf {h _ {x}}) ^ {2 - i} (\sigma \boldsymbol {\eta} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}) \boldsymbol {\epsilon} _ {t + 1 + i} \\ + \sum_ {i = 0} ^ {2} (\mathbf {h _ {x}} \otimes \mathbf {h _ {x}}) ^ {2 - i} (\sigma \boldsymbol {\eta} \otimes \mathbf {h _ {x}}) (\boldsymbol {\epsilon} _ {t + 1 + i} \otimes \mathbf {x} _ {t + i} ^ {s}) \\ + \sum_ {i = 0} ^ {2} (\mathbf {h _ {x}} \otimes \mathbf {h _ {x}}) ^ {2 - i} (\sigma \boldsymbol {\eta} \otimes \frac {1}{2} \mathbf {H _ {x x}}) (\boldsymbol {\epsilon} _ {t + 1 + i} \otimes \mathbf {x} _ {t + i} ^ {f} \otimes \mathbf {x} _ {t + i} ^ {f}) \end{array}\]

And in general

\[\begin{array}{l} \mathbf {x} _ {t + l} ^ {f} \otimes \mathbf {x} _ {t + l} ^ {s} = (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {l} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {s}\right) + \sum_ {i = 0} ^ {l - 1} (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {l - 1 - i} \left(\mathbf {h} _ {\mathbf {x}} \otimes \frac {1}{2} \mathbf {H} _ {\mathbf {x x}}\right) \left(\mathbf {x} _ {t + i} ^ {f} \otimes \mathbf {x} _ {t + i} ^ {f} \otimes \mathbf {x} _ {t + i} ^ {f}\right) \\ \quad + \sum_ {i = 0} ^ {l - 1} (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {l - 1 - i} \left(\mathbf {h} _ {\mathbf {x}} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}\right) \mathbf {x} _ {t + i} ^ {f} \\ \quad + \sum_ {i = 0} ^ {l - 1} (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {l - 1 - i} \left(\sigma \boldsymbol {\eta} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}\right) \boldsymbol {\epsilon} _ {t + 1 + i} \\ \quad + \sum_ {i = 0} ^ {l - 1} (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {l - 1 - i} \left(\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}\right) \left(\boldsymbol {\epsilon} _ {t + 1 + i} \otimes \mathbf {x} _ {t + i} ^ {s}\right) \\ \quad + \sum_ {i = 0} ^ {l - 1} (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {l - 1 - i} \left(\sigma \boldsymbol {\eta} \otimes \frac {1}{2} \mathbf {H} _ {\mathbf {x x}}\right) \left(\boldsymbol {\epsilon} _ {t + 1 + i} \otimes \mathbf {x} _ {t + i} ^ {f} \otimes \mathbf {x} _ {t + i} ^ {f}\right) \\ \text {for l = 1,2,3.} \end{array}\]

for

We therefore have

\[\begin{array}{l} \tilde {\mathbf {x}} _ {t + l} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + l} ^ {s} = (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {l} (\tilde {\mathbf {x}} _ {t} ^ {f} \otimes \tilde {\mathbf {x}} _ {t} ^ {s}) + \sum_ {i = 0} ^ {l - 1} (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {l - 1 - i} (\mathbf {h} _ {\mathbf {x}} \otimes \frac {1}{2} \mathbf {H} _ {\mathbf {x x}}) (\tilde {\mathbf {x}} _ {t + i} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + i} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + i} ^ {f}) \\ \quad + \sum_ {i = 0} ^ {l - 1} (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {l - 1 - i} (\mathbf {h} _ {\mathbf {x}} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}) \tilde {\mathbf {x}} _ {t + i} ^ {f} \\ \quad + \sum_ {i = 0} ^ {l - 1} (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {l - 1 - i} (\sigma \boldsymbol {\eta} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}) \delta_ {t + 1 + i} \\ \quad + \sum_ {i = 0} ^ {l - 1} (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {l - 1 - i} (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) (\delta_ {t + 1 + i} \otimes \tilde {\mathbf {x}} _ {t + i} ^ {s}) \\ \quad + \sum_ {i = 0} ^ {l - 1} (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {l - 1 - i} (\sigma \boldsymbol {\eta} \otimes \frac {1}{2} \mathbf {H} _ {\mathbf {x x}}) (\delta_ {t + 1 + i} \otimes \tilde {\mathbf {x}} _ {t + i} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + i} ^ {f}) \end{array}\]

\[\begin{array}{r l} & {\mathrm{Thus}} \\ & {E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + l} ^ {s} - \mathbf {x} _ {t + l} ^ {f} \otimes \mathbf {x} _ {t + l} ^ {s} \right]} \\ & {\qquad = E _ {t} [ (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {l} (\tilde {\mathbf {x}} _ {t} ^ {f} \otimes \tilde {\mathbf {x}} _ {t} ^ {s}) + \sum_ {i = 0} ^ {l - 1} (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {l - 1 - i} (\mathbf {h} _ {\mathbf {x}} \otimes \frac {1}{2} \mathbf {H} _ {\mathbf {x x}}) (\tilde {\mathbf {x}} _ {t + i} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + i} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + i} ^ {f})} \\ & {\qquad + \sum_ {i = 0} ^ {l - 1} (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {l - 1 - i} (\mathbf {h} _ {\mathbf {x}} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}) \tilde {\mathbf {x}} _ {t + i} ^ {f}} \\ & {\qquad + \sum_ {i = 0} ^ {l - 1} (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {l - 1 - i} (\sigma \pmb {\eta} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}) \pmb {\delta} _ {t + 1 + i}} \end{array}\]

\[\begin{array} { l } + \sum _ { i = 0 } ^ { l - 1 } ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { l - 1 - i } ( \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x } } ) ( \boldsymbol { \delta } _ { t + 1 + i } \otimes \tilde { \mathbf { x } } _ { t + i } ^ { s } ) \\ + \sum _ { i = 0 } ^ { l - 1 } ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { l - 1 - i } ( \sigma \boldsymbol { \eta } \otimes \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } ) ( \boldsymbol { \delta } _ { t + 1 + i } \otimes \tilde { \mathbf { x } } _ { t + i } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + i } ^ { f } ) \\ - \{ ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { l } ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { s } ) + \sum _ { i = 0 } ^ { l - 1 } ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { l - 1 - i } ( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } ) ( \mathbf { x } _ { t + i } ^ { f } \otimes \mathbf { x } _ { t + i } ^ { f } \otimes \mathbf { x } _ { t + i } ^ { f } ) \\ + \sum _ { i = 0 } ^ { l - 1 } ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { l - 1 - i } ( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } ) \mathbf { x } _ { t + i } ^ { f } \\ + \sum _ { i = 0 } ^ { l - 1 } ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { l - 1 - i } ( \sigma \boldsymbol { \eta } \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } ) \boldsymbol { \epsilon } _ { t + 1 + i } \\ + \sum _ { i = 0 } ^ { l - 1 } ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { l - 1 - i } ( \sigma \boldsymbol { \eta} \otimes \mathbf { h } _ { \mathbf { x } } ) ( \boldsymbol { \epsilon } _ { t + 1 + i } \otimes \mathbf { x } _ { t + i } ^ { s } ) \\ + \sum _ { i = 0 } ^ { l - 1 } ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { l - 1 - i } ( \sigma \boldsymbol { \eta} \otimes \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x} } ) ( \boldsymbol { \epsilon } _ { t + 1 + i } \otimes \mathbf { x } _ { t + i } ^ { f } \otimes \mathbf { x } _ { t + i } ^ { f } ) \\ ] \\ = E _ { t } [ \sum _ { i = 0 } ^ { l - 1 } ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x} }) ^ { l - 1 - i } ( \mathbf { h} _ {\mathbf { x}} \otimes \frac { 1 } { 2} | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | ] \\ + \sum _ { i = 0 } ^ { l - 1 } ( \mathbf { h} _ {\mathbf { x }} \otimes \mathbf { h} _ {\mathbf { x} }) ^ { l - 1 - i } ( \mathbf { h} _ {\mathbf { x}} \otimes \frac 1 2 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | ] \\ + \sum _ { i = 0 } ^ { l - 1 } ( \mathbf { h} _ {\mathbf { x }} \otimes \mathbf { h} _ {\mathbf { x} }) ^ { l - 1 - i } ( \sigma \boldsymbol { \eta} \otimes \frac 1 2 | | | | | | | | | | | | | | | | | | | | | | ] \\ + \sum _ { i = 0 } ^ { l - 1 } ( (\boldsymbol {\mathrm{h}} _ {\mathrm{x}}) ^ {( l - 1 - i)} (\sigma) (\boldsymbol {\eta} + (\boldsymbol {\mathrm{h}} _ {\mathrm{x}}) (\boldsymbol {\mathrm{d}} _ {\mathrm{t+1} + i} + (\boldsymbol {\mathrm{d}} _ {\mathrm{t+1} + i} + (\boldsymbol {\mathrm{d}} _ {\mathrm{t+1} + i} + (\boldsymbol {\mathrm{d}} _ {\mathrm{t+1} + i} + (\boldsymbol {\mathrm{d}} _ {\mathrm{t+1} + i} + (\boldsymbol {\mathrm{d}} _ {\mathrm{t+1} + i} - (\boldsymbol {\mathrm{d}} _ {\mathrm{t+1} + i} - (\boldsymbol {\mathrm{d}} _ {\mathrm{t+1} + i} - (\boldsymbol {\mathrm{d}} _ {\mathrm{t+1} + i} - (\boldsymbol {\mathrm{d}} _ {\mathrm{t+1} + i} - (\boldsymbol {\mathrm{d}} _ {\mathrm{t+1} + i} + (\boldsymbol {\mathrm{d}} _ {\mathrm{t+1} + i} - (\boldsymbol {\mathrm{d}} _ {\mathrm{t+1} + i} - (\boldsymbol {\mathrm{d}} _ {\mathrm{t+1} + i} - (\boldsymbol {\mathrm{d}} _ {\mathrm{t+1} + i} + (\boldsymbol {\mathrm{d}} _ {\mathrm{t+1} + i} + (\boldsymbol {\mathrm{d}} _ {\mathrm{t+1} + i} - (\boldsymbol {\mathrm{d}} _ {\mathrm{t+1} + i} - (\boldsymbol {\mathrm{d}} _ {\mathrm{t+1} + i} + (\boldsymbol {\mathrm{d}} _ {\mathrm{t+1} + i} - (\boldsymbol {\mathrm{d}} _ {\mathrm{t+1} + i} + (\boldsymbol {\mathrm{d}} _ {\mathrm{t+1} + i} - (\boldsymbol {\mathrm{d}} _ {\mathrm{t+1} + i} - (\boldsymbol {\mathrm{d}} _ {\mathrm{t+1} + i} + (\boldsymbol {\mathrm{d}} _ {\mathrm{t+1} + i} + (\boldsymbol {\mathrm{d}} _ {\mathrm{t+1} + i} + (\boldsymbol {\mathrm{d}} _ {\mathrm{t+1} + i} - (\boldsymbol {\mathrm{d}} _ {\mathrm{t+1} + i} + (\boldsymbol {\mathrm{d}} _ {\mathrm{t+1} + i} - (\boldsymbol {\mathrm{d}} _ {\mathrm{t+1} + i} + (\boldsymbol {\mathrm{d}} _ {\mathrm{t+1} + i} + (\boldsymbol {\mathrm{d}} _ {\mathrm{t+1} + i} - (\boldsymbol {\mathrm{d}} _ {\mathrm{t+1} + i} + (\boldsymbol {\mathrm{d}} _ {\mathrm{t+1} + i} + (\boldsymbol {\mathrm{d}} _ {\mathrm{t+1} + i} + (\boldsymbol {\mathrm{d}} _ {\mathrm{t+1} + i} + (\boldsymbol {\mathrm{d}} _ {\mathrm{t+1} + i} , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . e n d e r a g e n d e r a g e n d e r a g e n d e r a g e n d e r a g e n d e r a g e n d e r a g e n d e r a g e n d e r a g e n d e r a g e n d e r a g e n d e r a g e n d e r a g e n d e r a g e n d e r a g e m a n d e r a g e n d e r a g e n d e r a g e n d e r a g e n d e r a g e n d e r a g e n d e r a g e n d e r a g e n d e r a g e n d e r a g e n d e r a g e n d e r a g e n d e r a g e n d e r a g e n c o n d e r a g e n c o n d e r a g e n c o n d e r a g e n c o n d e r a g e n c o n d e r a g e n c o n d e r a g e n c o n d e r a g e n c o n d e r a g e n c o n d e r a g e n c o n d e r a g e n c c o n d e r a g e n c o n d e r a g e n c o n d e r a g e n c o n d e r a g e n c o n d e r a g e n c o n d e r a g e n c o n d e r a g e n c o n d e r a g e n c o n d e r a g e n c o n d e r a g e k o u t h o u t h o u t h o u t h o u t h o u t h o u t h o u t h o u t h o u t h o u t h o u t h o u t h o u t h o u t h o u t h o u t h o u t h o u t h o u t h o u t h o u t h o u t h o u t h o u t h o u t w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v vv] \\ = E _ { t }\left[ \sum_ {i = 0} ^ {} ( b ^ {- 1 - i}) ( b ^ {- 1 - i}) ( b ^ {- 1 - i}) ( b ^ {- 1 - i}) ( b ^ {- 1 - i}) ( b ^ {- 1 - i}) ( b ^ {- 1 - i}) ( b ^ {- 1 - i}) ( b ^ {- 1 - i}) ( b ^ {- 1 - i}) ( b ^ {- 1 - i}) ( b ^ {- 1 - i}) \( ) \\ + S U T A L I N G M A L I N G M A L I N G M A L I N G M A L I N G M A L I N G M A L I N G M A L I N G M A L I N G M A L I N G M A L I N G M A L I N G M A L I N G M A L I N G M A L I N G M A L I N G M A L I N G M A L I I N G M A L I N G M A L I N G M A L I N G M A L I N G M A L I N G M A L I N G M A L I N G M A L I N G M A L I N G M A L I N G M A L I N G M A L I N G M A L I N G M A L I N G M A L I N G M A L I N G m a l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l / ] \\ = E [ S U T ] [ S U T ] [ S U T ] [ S U T ] [ S U T ] [ S U T ] [ S U T ] [ S U T ] [ S U T ] [ S U T ] [ S U T ] [ S U T ] [ S U T ] [ S U T ] [ S U T ] [ S U T ] [ S U T ] [ S U T ] [ S U T ] [ S U T ] [ S U T ]. \\ = E [ S U T ] [ S U T ] [ S U T ] [ S U T ] [ S U T ] [ S U T ] [ S U T ] [ S U T ] [ S U T ] [ S U T ] [ S U T ] [ S U T ] [ S U T ] [ S U T ] [ S U T ] [ S U T ] [ S U T ] [ S U T ] [ S U T ] [S U T ] [ S U T ] [ S U T ] [ S U T ] [ S U T ] [ S U T ] [ S U T ] [ S U T ] [ S U T ] [ S U T ] [ S U T ] [ S U T ] [ S U T ] [ S U T ] [ S U T ] [ S U T ] [ S U T ] [ S U T ] [ S U T ] [ S U T ] [S U T] [ S U T] [ S U T] [ S U T] [ S U T] [ S U T] [ S U T] [ S U T] [ S U T] [ S U T] [ S U T] [ S U T] [ S U T] [ S U T] [ S U T] [ S U T] [ S U T] [ S U T] [ S U T] [ S U T] [ S U T] [S U T] [S U T] [S U T] [S U T] [S U T] [S U T] [S U T] [S U T] [S U T] [S U T] [S U T] [S U T] [S U T] [S U T] [S U T] [S U T] [S U T] [S U T] [S U T] [S U T] [N V W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W & p q : q : q : q : q : q : q : q : q : q : q : q : q : q : q : q : q : q : q : q : q : q : q : q : q : q : q : q : q : q : q : q : q : q : q : q : q : q : q : q : q : q : q : q : q : q : q : q : q : q : q: p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p: p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p : p :p < 5>5>5>5>5>5>5>5>5>5>5>5>5>5>5>5>5>5>5>5>5>5>5>5>5>5>5>5>5>5>5>5>5>5>5>5>5>5>5>5>5>5>5>5>5>5>5>5>5>5>5< |content_end|>\]

\[\begin{array}{l} + \sum_ {i = 1} ^ {l - 1} \left(\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}\right) ^ {l - 1 - i} \left(\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}\right) (\mathbf {0} - \mathbf {0}) \\ + \left(\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}\right) ^ {l - 1} \left(\sigma \boldsymbol {\eta} \otimes \frac {1}{2} \mathbf {H} _ {\mathbf {x x}}\right) \left((\boldsymbol {\nu} + 0) \otimes \tilde {\mathbf {x}} _ {t} ^ {f} \otimes \tilde {\mathbf {x}} _ {t} ^ {f} - \mathbf {0}\right) \\ + \sum_ {i = 1} ^ {l - 1} \left(\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}\right) ^ {l - 1 - i} \left(\sigma \boldsymbol {\eta} \otimes \frac {1}{2} \mathbf {H} _ {\mathbf {x x}}\right) (\mathbf {0} - \mathbf {0}) ] \end{array}\]

using is a function of which is a function of . The zero-mean iid innovations therefore implies that, and

The same argument implies that

and

\[\begin{array}{r l} & {= E _ {t} [ \sum_ {i = 1} ^ {l - 1} (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {l - 1 - i} (\mathbf {h} _ {\mathbf {x}} \otimes \frac {1}{2} \mathbf {H} _ {\mathbf {x x}}) (\tilde {\mathbf {x}} _ {t + i} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + i} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + i} ^ {f} - \mathbf {x} _ {t + i} ^ {f} \otimes \mathbf {x} _ {t + i} ^ {f} \otimes \mathbf {x} _ {t + i} ^ {f})} \\ & {+ \sum_ {i = 1} ^ {l - 1} (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {l - 1 - i} (\mathbf {h} _ {\mathbf {x}} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}) (\tilde {\mathbf {x}} _ {t + i} ^ {f} - \mathbf {x} _ {t + i} ^ {f})} \\ & {+ (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {l - 1} (\sigma \pmb {\eta} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}) \pmb {\nu}} \\ & {+ (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {l - 1} (\sigma \pmb {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) (\pmb {\nu} \otimes \mathbf {x} _ {t} ^ {s})} \\ & {+ (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) ^ {l - 1} (\sigma \pmb {\eta} \otimes \frac {1}{2} \mathbf {H} _ {\mathbf {x x}}) (\pmb {\nu} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) ]} \end{array}\]

because the shock hits in period , meaning that and similar for

\[\begin{array}{r l} & {= \sum_ {i = 1} ^ {l - 1} \left(\mathbf {h _ {x}} \otimes \mathbf {h _ {x}}\right) ^ {l - 1 - i} \left(\mathbf {h _ {x}} \otimes \frac {1}{2} \mathbf {H _ {x x}}\right) E _ {t} \left[ \left(\tilde {\mathbf {x}} _ {t + i} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + i} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + i} ^ {f} - \mathbf {x} _ {t + i} ^ {f} \otimes \mathbf {x} _ {t + i} ^ {f} \otimes \mathbf {x} _ {t + i} ^ {f}\right) \right]} \\ & {+ \sum_ {i = 1} ^ {l - 1} \left(\mathbf {h _ {x}} \otimes \mathbf {h _ {x}}\right) ^ {l - 1 - i} \left(\mathbf {h _ {x}} \otimes \frac {1}{2} \mathbf {h _ {\sigma \sigma}} \sigma^ {2}\right) E _ {t} \left[ \left(\tilde {\mathbf {x}} _ {t + i} ^ {f} - \mathbf {x} _ {t + i} ^ {f}\right) \right]} \\ & {+ (\mathbf {h _ {x}} \otimes \mathbf {h _ {x}}) ^ {l - 1} \left\{\left(\sigma \pmb {\eta} \otimes \frac {1}{2} \mathbf {h _ {\sigma \sigma}} \sigma^ {2}\right) \pmb {\nu} + (\sigma \pmb {\eta} \otimes \mathbf {h _ {x}}) (\pmb {\nu} \otimes \mathbf {x} _ {t} ^ {s}) + (\sigma \pmb {\eta} \otimes \frac {1}{2} \mathbf {H _ {x x}}) (\pmb {\nu} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) \right\}} \\ & \\ & {= \sum_ {i = 1} ^ {l - 1} (\mathbf {h _ {x}} \otimes \mathbf {h _ {x}}) ^ {l - 1 - i} (\mathbf {h _ {x}} \otimes \frac {1}{2} \mathbf {H _ {x x}}) E _ {t} \left[ \left(\tilde {\mathbf {x}} _ {t + i} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + i} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + i} ^ {f} - \mathbf {x} _ {t + i} ^ {f} \otimes \mathbf {x _ {\tau t + i}} ^ {f} \otimes \mathbf {x _ {\tau t + i}} ^ {f}\right) \right]} \\ & {+ \sum_ {i = 1} ^ {l - 1} (\mathbf {h _ {x}} \otimes \mathbf {h _ {x}}) ^ {l - 1 - i} (\mathbf {h _ {x}} \otimes \frac {1}{2} \mathbf {h _ {\sigma \sigma}} \sigma^ {2}) E _ {t} \left[ (\tilde {\mathbf {x}} _ {t + i} ^ {f} - \mathbf {x} _ {t + i} ^ {f}) \right]} \\ & {+ (\mathbf {h _ {x}} \otimes \mathbf {h _ {x}}) ^ {l - 1} \left\{\sigma \pmb {\eta} \pmb {\nu} \otimes \frac {1}{2} \mathbf {h _ {\sigma \sigma}} \sigma^ {2} + \sigma \pmb {\eta} \pmb {\nu} \otimes \mathbf {h _ {x}} \mathbf {x} _ {t} ^ {s} + \sigma \pmb {\eta} \pmb {\nu} \otimes \frac {1}{2} \mathbf {H _ {\alpha x}} (\mathbf {x} _ {\tau t} ^ {f} \otimes \mathbf {x} _ {\tau t} ^ {f}) \right\}} \\ & {\mathrm{using(A⊗B)(C⊗D)=AC⊗BDifACandBDaredefined}} \end{array}\]

\[\begin{array}{r l} & {= \sum_ {i = 1} ^ {l - 1} \left(\mathbf {h _ {x}} \otimes \mathbf {h _ {x}}\right) ^ {l - 1 - i} \left(\mathbf {h _ {x}} \otimes \frac {1}{2} \mathbf {H _ {x x}}\right) E _ {t} \left[ \left(\tilde {\mathbf {x}} _ {t + i} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + i} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + i} ^ {f} - \mathbf {x} _ {t + i} ^ {f} \otimes \mathbf {x} _ {t + i} ^ {f} \otimes \mathbf {x} _ {t + i} ^ {f}\right) \right]} \\ & {+ \sum_ {i = 1} ^ {l - 1} \left(\mathbf {h _ {x}} \otimes \mathbf {h _ {x}}\right) ^ {l - 1 - i} \left(\mathbf {h _ {x}} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}\right) E _ {t} \left[ \left(\tilde {\mathbf {x}} _ {t + i} ^ {f} - \mathbf {x} _ {t + i} ^ {f}\right) \right]} \\ & {+ \left(\mathbf {h _ {x}} \otimes \mathbf {h _ {x}}\right) ^ {l - 1} \left\{\sigma \eta \pmb {\nu} \otimes \left(\mathbf {h _ {x}} \mathbf {x} _ {t} ^ {s} + \frac {1}{2} \mathbf {H _ {x x}} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) + \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}\right) \right\}} \end{array}\]

To derive a recursive version for this sum, we let

\[\begin{array}{r l} & {X _ {l} = \sum_ {i = 1} ^ {l - 1} \left(\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}\right) ^ {l - 1 - i} \left(\mathbf {h} _ {\mathbf {x}} \otimes \frac {1}{2} \mathbf {H} _ {\mathbf {x x}}\right) \left(\tilde {\mathbf {x}} _ {t + i} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + i} ^ {f} \otimes \tilde {\mathbf {x}} _ {t + i} ^ {f} - \mathbf {x} _ {t + i} ^ {f} \otimes \mathbf {x} _ {t + i} ^ {f} \otimes \mathbf {x} _ {t + i} ^ {f}\right)} \\ & {\qquad + \sum_ {i = 1} ^ {l - 1} \left(\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}\right) ^ {l - 1 - i} \left(\mathbf {h} _ {\mathbf {x}} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}\right) \left(\tilde {\mathbf {x}} _ {t + i} ^ {f} - \mathbf {x} _ {t + i} ^ {f}\right)} \end{array}\]

\[\begin{array} { r l } & { + ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { l - 1 } \left( \sigma \eta \nu \otimes \left( \mathbf { h } _ { \mathbf { x } } \mathbf { x } _ { t } ^ { s } + \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } \left( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \right) + \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } \right) \right) } \\ & \text {so} \\ & { X _ { 1 } = \sigma \eta \nu \otimes \left( \mathbf { h } _ { \mathbf { x } } \mathbf { x } _ { t } ^ { s } + \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } \left( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ f \right) + \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } \right) } \\ & { X _ { 2 } = \sum _ { i = 1 } ^ { 1 } ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { 1 - i } ( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } ) ( \tilde { \mathbf { x } } _ { t + i } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + i } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + i } ^ { f } - \mathbf { x } _ { t + i } ^ { f } \otimes \mathbf { x } _ { t + i } ^ { f } \otimes \mathbf { x } _ { t + i } ^ { f } ) } \\ & { + \sum _ { i = 1 } ^ { 1 } ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ^ { 1 - i } ( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } ) ( \tilde { \mathbf { x } } _ { t + i } ^ { f } - \mathbf { x } _ { t + i } ^ { f } ) } \\ & { + ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ( \sigma \eta \nu \otimes ( \mathbf { h } _ { \mathbf { x } } \mathbf { x } _ { t } ^ { s } + \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ f ) + \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2} ) ) } \\ & { = ( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x} } ) ( \tilde { \mathbf { x } } _ { t + 1 } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + 1 } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + 1 } ^ { f } - \mathbf { x} _ { t + 1 } ^ { f } \otimes \mathbf { x} _ { t + 1 } ^ { f } \otimes \mathbf { x} _ { t + 1 } ^ { f } ) } \\ & { + ( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2} ) ( \tilde { \mathbf { x } } _ { t + 1 } ^ { f } - \mathbf { x} _ { t + 1 } ^ { f } ) } \\ & + ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ \mathbf {\Delta x} ) ) ( \sigma \eta \nu \otimes ( [ | | | | | | | | | | | | | | | | | | | | | | | | | ) ) = ( [ | | | | | | | | | | | | | | | | | | ) , \\ & {\quad + ( [ | | | | | | | | | | | | | | | | | | | ) .} \\ & {\quad = ( [ | | | | | | | | | | | | | | | | | ) X _ { 1 }} \\ & {\quad + ( [ | | | | | | | | | | | | | | | ) X _ { 2 }} \\ & \quad + ( [ | | ] X _ {\mathrm{的}} X _ {\mathrm{的}} X _ {\mathrm{的}} X _ {\mathrm{的}} X _ {\mathrm{的}} X _ {\mathrm{的}} X _ {\mathrm{的}} X _ {\mathrm{的}} X _ {\mathrm{的}} X _ {\mathrm{的}} X _ {\mathrm{的}} X _ {\mathrm{的}} X _ {\mathrm{的}} X _ {\mathrm{的}} X _ {\mathrm{的}} X_{ t + 1} X_{ t + 1} X_{ t + 1} X_{ t + 1} X_{ t + 1} X_{ t + 1} X_{ t + 1} X_{ t + 1} X_{ t + 1} X_{ t + 1} X_{ t + 1} X_{ t + 1} X_{ t + 1} X_{ t + 1} X_{ t + 1} X_ {\mathrm{的}} X_{ t + 1} X_{ t + 1} X_{ t + 1} X_{ t + 1} X_{ t + 1} X_{ t + 1} X_{ t + 1} X_{ t + 1} X_{ t + 1} X_{ t + 1} X_{ t + 1} X_{ t + 1} X_{ t + 1} X_{ t - 1} X_{ t - 1} X_{ t - 1} X_{ t - 1} X_{ t - 1} X_{ t - 1} X_{ t - 1} X_{ t - 1} X_{ t - 1} X_{ t - 1} X_{ t - 1} X_{ t - 1} X_{ t - 1} X_{ t - 1} X_{ t - 2} X_{ t - 2} X_{ t - 2} X_{ t - 2} X_{ t - 2} X_{ t - 2} X_{ t - 2} X_{ t - 2} X_{ t - 2} X_{ t - 2} X_{ t - 2} X_{ t - 2} X_{ t - 2} X_{ t - 2} X_{ t - 2}X_{ t - 3}X_{ t - 3}X_{ t - 3}X_{ t - 3}X_{ t - 3}X_{ t - 3}X_{ t - 3}X_{ t - 3}X_{ t - 3}X_{ t - 3}X_{ t - 3}X_{ t - 3}X_{ t - 3}X_{ t - 3}X_{ t - 3}Y_{t - k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k, k , K , K , K , K , K , K , K , K , K , K , K , K , K , K , K , K , K , K , K , K , K , K , K , K , K , K , K , K , K , K , K , K , K , K , K , K , K , K , K , K , K , K , K , K , K , K , K , K , K , K ,K , K , K , K , K , K , K , K , K , K , K , K , K , K , K , K , K , K , K , K , K , K , K , K , K , K , K , K , K , K , K , K , K ,K ,K ,K ,K ,K ,K ,K ,K ,K ,K ,K ,K ,K ,K ,K ,K ,K ,K ,K ,K ,K ,K ,K ,K ,K ,K ,K ,K ,K ,K ,K ,K ,K ,K ,K ,K ,K ,K ,K ,K ,K ,K ,K ,K ,K ,K ,K ,K ,K ,K ,M,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,K,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,k,lk,lk,lk,lk,lk,lk,lk,lk,lk,lk,lk,lk,lk,lk,lk,lk,lk,lk,lk,lk,lk,lk,lk,lk,lk,lk,lk,lk,lk,lk,lk,lk,lk,lk,lk,lk,lk,lk,lk,lk,lk,lk,lk,lk,lk,lk,lk,lk,lk,lk,lr,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,n,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,p,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P,P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P S p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p p q s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s s / n . m a n d e r e d e n d e r e d e n d e r e d e n d e r e d e n d e r e d e n d e r e d e n d e r e d e n d e r e d e n d e r e d e n d e r e d e n d e r e d e n d e r e d e n d e r e d e n d e r e d e n d e r e d e n c o n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i m a l l u a l l u a l l u a l l u a l l u a l l u a l l u a l l u a l l u a l l u a l l u a l l u a l l u a l l u a l l u a l l u a l l u a l l u a l l u a l l u a l l u a l l u a l l u a l l u a l l u a l l w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W WWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWwWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwhwwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhwhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhWhite< fcel>$X_0 = (\lambda_0^* * N^* * N^* * N^* * N^* * N^* * N^* * N^* * N^* * N^* * N^* * N^* * N^* * N^* * N^* * N^* * N^* * N^* * N^* * N^* * N^* * N^* * N^* * N^* * N^* * N^* *N^* * N^* * N^* * N^* * N^* * N^* * N^* * N^* * N^* * N^* * N^* * N^* * N^* * N^* * N^* * N^* * N^* * N^* * N^* * N^* * N^* * N^* * N^* * N^* * N^* * F(N^*F(N^*F(N^*F(N^*F(N^*F(N^*F(N^*F(N^*F(N^*F(N^*F(N^*F(N^*F(N^*F(N^*F(N^*F(N^*F(N^*F(N^*F(N^*F(N^*F(N^*F(N^*F(N^*F(N^*F(N^*F(N^* F(N^*F(N^*F(N^*F(N^*F(N^*F(N^*F(N^*F(N^*F(N^*F(N^*F(N^*F(N^*F(N^*F(N^*F(N^*F(N^*F(N^*F(N^*F(N^*F(N^*F(N^*F(N^*F(N^*F(N^*F(N^*\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%}\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)\%)(x,y,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,z,Z,x,y,z,y,z,y,z,y,z,y,z,y,z,y,z,y,z,y,z,y,z,y,z,y,z,y,z,y,z,y,z,y,z,y,z,y,z,y,z,y,z,y,z,y,z,y,z,y,z,y,z,y,z,y,z,y,z,y,z,y,z,y,z,y,z,y,z,y,z,y,z,y,z,y,z,y,z,y,z,y,z,y,z,y,Z,x,y,Z,x,Z,x,Z,x,Z,x,Z,x,Z,x,Z,x,Z,x,Z,x,Z,x,Z,x,Z,x,Z,x,Z,x,Z,x,Z,x,Z,x,Z,x,Z,x,Z,x,Z,x,Z,x,Z,x,Z,x,Z,x,Z,x,Z,x,Z,x,Z,x,Z,x,Z,x,Z,x,Z,x,Z,x,Z,x,Z,x,Z,x,Z,x,Z,x,Z,x,Z,x,Z,x,Z,x,Z,x,Z,x,Z,x,Z,x,Z,x,Z,x,Z,x,Z;x,Y,X,Y,X,Y,X,Y,X,Y,X,Y,X,Y,X,Y,X,Y,X,Y,X,Y,X,Y,X,Y,X,Y,X,Y,X,Y,X,Y,X,Y,X,Y,X,Y,X,Y,X,Y,X,Y,X,Y,X,Y,X,Y,X,Y,X,Y,X,Y,X,Y,X,Y,X,Y,X,Y,X,Y,X,Y,X,Y,X,Y,X,Y,X,Y,X,Y,X,Y,X,Y,X,Y,X,Y,X,Y,X,Y,X,Y,X,Y,X,Y,X,Y,X,Y,T,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,M,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m;m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,m,mm;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m;m; $Hence$, $in$, $general$\]

12.5.3 Summarizing

At third order, the total effect on the state variables is:

\[E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} - \mathbf {x} _ {t + l} \right] = E _ {t} \left| \tilde {\mathbf {x}} _ {t + l} ^ {f} - \mathbf {x} _ {t + l} ^ {f} \right| + E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} ^ {s} - \mathbf {x} _ {t + l} ^ {s} \right] + E _ {t} \left[ \tilde {\mathbf {x}} _ {t + l} ^ {r d} - \mathbf {x} _ {t + l} ^ {r d} \right]\]

\[\begin{array} { r l } & { \text {For the control variables:} } \\ & { \mathbf { y } _ { t + l } ^ { r d } = \mathbf { g } _ { \mathbf { x } } \left( \mathbf { x } _ { t + l } ^ { f } + \mathbf { x } _ { t + l } ^ { s } + \mathbf { x } _ { t + l } ^ { r d } \right) + \frac { 1 } { 2 } \mathbf { G } _ { \mathbf { x x } } \left( \left( \mathbf { x } _ { t + l } ^ { f } \otimes \mathbf { x } _ { t + l } ^ { f } \right) + 2 \left( \mathbf { x } _ { t + l } ^ { f } \otimes \mathbf { x } _ { t + l } ^ { s } \right) \right) } \\ & { \quad + \frac { 1 } { 6 } \mathbf { G } _ { \mathbf { x x x } } \left( \mathbf { x } _ { t + l } ^ { f } \otimes \mathbf { x } _ { t + l } ^ { f } \otimes \mathbf { x } _ { t + l } ^ { f } \right) + \frac { 1 } { 2 } \mathbf { g } _ { \sigma \sigma } \sigma ^ { 2 } + \frac { 3 } { 6 } \mathbf { g } _ { \sigma \sigma \mathbf { x } } \sigma ^ { 2 } \mathbf { x } _ { t + l } ^ { f } + \frac { 1 } { 6 } \mathbf { g } _ { \sigma \sigma \sigma } \sigma ^ { 3 } } \\ & { \tilde { \mathbf { y } } _ { t + l } ^ { r d } = \mathbf { g } _ { \mathbf { x } } \left( \tilde { \mathbf { x } } _ { t + l } ^ { f } + \tilde { \mathbf { x } } _ { t + l } ^ { s } + \tilde { \mathbf { x } } _ { t + l } ^ { r d } \right) + \frac { 1 } { 2 } \mathbf { G } _ { \mathbf { x x } } \left( \left( \tilde { \mathbf { x } } _ { t + l } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + l } ^ { f } \right) + 2 \left( \tilde { \mathbf { x } } _ { t + l } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + l } ^ { s } \right) \right) } \\ & { \quad + \frac { 1 } { 6 } \mathbf { G } _ { \mathbf { x x x } } \left( \tilde { \mathbf { x } } _ { t + l } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + l } ^ { f } \otimes \tilde { \mathbf { x } } _ { t + l } ^ { f } \right) + \frac { 1 } { 2 } \mathbf { g } _ { \sigma \sigma } \sigma ^ { 2 } + \frac { 3 } { 6 } \mathbf { g } _ { \sigma \sigma \mathbf { x } } \sigma ^ { 2 } \tilde { \mathbf { x } } _ { t + l } ^ { f } + \frac { 1 } { 6 } \mathbf { g } _ { \sigma \sigma \sigma } \sigma ^ { 3 } } \\ & S o : \\ & E _ { t } [ \tilde { \mathbf { y } } _ { t + l } ^ { r d} - \mathbf { y} _ { t + l } ^ { r d} ] \\ & = \mathbf { g _ { x } } ( E _ { t } [ \tilde { \mathbf { x } } _ { t + l } ^ { f } - \mathbf { x} _ t + l ] ] + E _ { t } [ \tilde { \mathbf { x } } _ { t + l } ^ { s} - \mathbf { x} _ t + l ] ] + E _ { t } [ \tilde { \mathbf { x } } _ t + l ] d a n d i s c ] ) \\ & + \frac 1 2 G _ { x x } ( E _ { t } [ \tilde { \mathbf {\Delta x }} _ t + l ] d a n d i s c ] - X _ \tau , j , k , m , n , p , q , r , s , u , v , w , y , z , w , u , v , v , w , u , v , v , w , v , v , v , v , v , v , v , v , v , v , v , v , v , v , v , v , v , v , v , v , v , v , v , v , v , v , v , v , v , v , v , v , v , v , v , v , v , v , v , v , v , v , v , v , v , v , v , v , v , v , . ) \\ & + \frac 1 6 G _ { x x x} E _ { t } [ \tilde {\mathbf {\Delta x}} _ \tau , j , k , m , n , p , q , r , s , u , v , w , u , v , u , v , u , v , u , v , u , v , u , v , u , v , u , v , u , v , u , v , u , v , u , v , u , v , u , v , u , u , u , u , u , u , u , u , u , u , u , u , u , u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .\]

13 Alternative notation with in the state vector

When deriving the perturbation approximation, the perturbation parameter is treated as a variable. It may therefore be natural to consider as a part of the state vector when constructing the state space system for the approximated model.

We therefore define , , and .

At first-order:

\[\mathbf {y} _ {t} ^ {f} = \left[ \begin{array}{l l} \mathbf {g} _ {\mathbf {x}} & 0 \end{array} \right] \left[ \begin{array}{l} \mathbf {x} _ {t} ^ {f} \\ \sigma \end{array} \right]\]

\[\begin{array}{c} \mathbf {y} _ {t} ^ {f} = \tilde {\mathbf {g}} _ {\tilde {\mathbf {x}}} \tilde {\mathbf {x}} _ {t} ^ {f} \\ \left[ \begin{array}{c} \mathbf {x} _ {t + 1} ^ {f} \\ \sigma \end{array} \right] = \left[ \begin{array}{c c} \mathbf {h} _ {\mathbf {x}} & 0 \\ 0 & 1 \end{array} \right] \left[ \begin{array}{c} \mathbf {x} _ {t} ^ {f} \\ \sigma \end{array} \right] + \left[ \begin{array}{c} \sigma \boldsymbol {\eta} \\ \mathbf {0} \end{array} \right] \boldsymbol {\epsilon} _ {t + 1} \end{array}\]

↓

\[\tilde {\mathbf {x}} _ {t + 1} ^ {f} = \tilde {\mathbf {h}} _ {\tilde {\mathbf {x}}} \tilde {\mathbf {x}} _ {t} ^ {f} + \left[ \begin{array}{c} \sigma \pmb {\eta} \\ \mathbf {0} \end{array} \right] \pmb {\epsilon} _ {t + 1}\]

At second order:

\[y _ {t} ^ {s} \left(i\right) = \left[ \begin{array}{c c} \mathbf {g} _ {\mathbf {x}} \left(i,: \right) & 0 \end{array} \right] \left(\left[ \begin{array}{c} \mathbf {x} _ {t} ^ {f} \\ \sigma \end{array} \right] + \left[ \begin{array}{c} \mathbf {x} _ {t} ^ {s} \\ \sigma \end{array} \right]\right) + \frac {1}{2} \left[ \begin{array}{c c} \left(\mathbf {x} _ {t} ^ {f}\right) ^ {\prime} & \sigma \end{array} \right] \left[ \begin{array}{c c} \mathbf {g} _ {\mathbf {x x}} \left(i,:,:) \right. & 0 \\ \mathbf {0} & g _ {\sigma \sigma} \left(i, 1\right) \end{array} \right] \left[ \begin{array}{c} \mathbf {x} _ {t} ^ {f} \\ \sigma \end{array} \right]\]

↓

\[y _ {t} ^ {s} (i) = \tilde {\mathbf {g}} _ {\tilde {\mathbf {x}}} (i,:) (\tilde {\mathbf {x}} _ {t} ^ {f} + \tilde {\mathbf {x}} _ {t} ^ {s}) + \frac {1}{2} (\tilde {\mathbf {x}} _ {t} ^ {f}) ^ {\prime} \tilde {\mathbf {g}} _ {\tilde {\mathbf {x}} \tilde {\mathbf {x}}} (i,:,:) \tilde {\mathbf {x}} _ {t} ^ {f}\]

\[x _ {t} ^ {s} (i) = \left[ \begin{array}{l l} \mathbf {h} _ {\mathbf {x}} (i,:) & 0 \end{array} \right] \left[ \begin{array}{l} \mathbf {x} _ {t} ^ {s} \\ \sigma \end{array} \right] + \frac {1}{2} \left[ \begin{array}{l l} (\mathbf {x} _ {t} ^ {f}) ^ {\prime} & \sigma \end{array} \right] \left[ \begin{array}{c c} \mathbf {h} _ {\mathbf {x x}} (i,:,:) & 0 \\ \mathbf {0} & h _ {\sigma \sigma} (i, 1) \end{array} \right] \left[ \begin{array}{l} \mathbf {x} _ {t} ^ {f} \\ \sigma \end{array} \right]\]

↓

\[x _ {t} ^ {s} (i) = \tilde {\mathbf {h}} _ {\tilde {\mathbf {x}}} (i,:) \tilde {\mathbf {x}} _ {t} ^ {s} + \frac {1}{2} (\tilde {\mathbf {x}} _ {t} ^ {f}) ^ {\prime} \tilde {\mathbf {h}} _ {\tilde {\mathbf {x}} \tilde {\mathbf {x}}} (i,:,:) \tilde {\mathbf {x}} _ {t} ^ {f}\]

for

At third order:

\[\begin{array}{r c l} y _ {t} ^ {r d} (i) & = & \left[ \begin{array}{c c} \mathbf {g} _ {\mathbf {x}} (i,:) & 0 \end{array} \right] \left(\left[ \begin{array}{c} \mathbf {x} _ {t} ^ {f} \\ \sigma \end{array} \right] + \left[ \begin{array}{c} \mathbf {x} _ {t} ^ {s} \\ \sigma \end{array} \right] + \left[ \begin{array}{c} \mathbf {x} _ {t} ^ {r d} \\ \sigma \end{array} \right]\right) \\ & & + \frac {1}{2} \left[ \begin{array}{c c} (\mathbf {x} _ {t} ^ {f}) ^ {\prime} & \sigma \end{array} \right] \left[ \begin{array}{c c} \mathbf {g} _ {\mathbf {x x}} (i,:,:) & 0 \\ \mathbf {0} & g _ {\sigma \sigma} \end{array} \right] \left(\left[ \begin{array}{c} \mathbf {x} _ {t} ^ {f} \\ \sigma \end{array} \right] + 2 \left[ \begin{array}{c} \mathbf {x} _ {t} ^ {s} \\ \sigma \end{array} \right]\right) \\ & & + \frac {1}{6} \left[ \begin{array}{c c} (\mathbf {x} _ {t} ^ {f}) ^ {\prime} & \sigma \end{array} \right] \left[ \begin{array}{c c} [ (\mathbf {x} _ {t} ^ {f}) ^ {\prime} & \sigma ] [ (\mathbf {g} _ {\mathbf {x x x}} (i, 1,:,:) & 0 \\ \mathbf {0} & 3 g (i, 1) _ {\sigma \sigma \mathbf {x}} ] [ (\mathbf {x} _ {t} ^ {f}) ^ {\prime} & 0 \\ [ (\mathbf {x} _ {t} ^ {f}) ^ {\prime} & \sigma ] [ (\mathbf {g} _ {\mathbf {x x x}} (i, 2,:,:) & 0 \\ \mathbf {0} & 3 g (i, 2) _ {\sigma \sigma \mathbf {x}} ] [ (\mathbf {x} _ {t} ^ {f}) ^ {\prime} & 0 \\ ... & 0 \\ [ (\mathbf {x} _ {t} ^ {f}) ^ {\prime} & \sigma ] [ (\mathbf {g} _ {\mathbf {x x x}} (i, n _ {x},:,:) & 0 \\ \mathbf {0} & 3 g (i, n _ {x}) _ {\sigma \sigma \mathbf {x}} ] [ (\mathbf {x} _ {t} ^ {f}) ^ {\prime} & 0 \\ [ (\mathbf {x} _ {t} ^ {f}) ^ {\prime} & \sigma ] [ (\mathbf {0} & g (i, 1) _ {\sigma \sigma \sigma} ] [ (\mathbf {x} _ {t} ^ {f}) ^ {\prime} & 0 \\ [ (\mathbf {x} _ {t} ^ {f}) ^ {\prime} & g (i, 1) _ {\sigma \sigma \sigma} ] [ (\mathbf {x} _ {t} ^ {f}) ^ {\prime} & 0 \\ [ (\mathbf {x} _ {t} ^ {f}) ^ {\prime} & g (i, 1) _ {\sigma \sigma \sigma} ] [ (\mathbf {x} _ {t} ^ {f}) ^ {\prime} & 0. \end{array} \right]. \end{array}\]

\[\begin{array}{r c l} y _ {t} ^ {r d} (i) & = & \tilde {\mathbf {g}} _ {\tilde {\mathbf {x}}} (i,:) (\tilde {\mathbf {x}} _ {t} ^ {f} + \tilde {\mathbf {x}} _ {t} ^ {s} + \tilde {\mathbf {x}} _ {t} ^ {r d}) \\ & & + \frac {1}{2} (\tilde {\mathbf {x}} _ {t} ^ {f}) ^ {\prime} \left[ \begin{array}{c c} \mathbf {g} _ {\mathbf {x x}} (i,:,:) & 0 \\ \mathbf {0} & g _ {\sigma \sigma} \end{array} \right] (\tilde {\mathbf {x}} _ {t} ^ {f} + 2 \tilde {\mathbf {x}} _ {t} ^ {s}) \\ & & + \frac {1}{6} (\tilde {\mathbf {x}} _ {t} ^ {f}) ^ {\prime} \left[ \begin{array}{c c} (\tilde {\mathbf {x}} _ {t} ^ {f}) ^ {\prime} \left[ \begin{array}{c c} \mathbf {g} _ {\mathbf {x x x}} (i, 1,:,:) & 0 \\ \mathbf {0} & 3 g (i, 1) _ {\sigma \sigma \mathbf {x}} \end{array} \right] \tilde {\mathbf {x}} _ {t} ^ {f} \\ (\tilde {\mathbf {x}} _ {t} ^ {f}) ^ {\prime} \left[ \begin{array}{c c} \mathbf {g} _ {\mathbf {x x x}} (i, 2,:,:) & 0 \\ \mathbf {0} & 3 g (i, 2) _ {\sigma \sigma \mathbf {x}} \end{array} \right] \tilde {\mathbf {x}} _ {t} ^ {f} \\ & & \ldots \\ (\tilde {\mathbf {x}} _ {t} ^ {f}) ^ {\prime} \left[ \begin{array}{c c} \mathbf {g} _ {\mathbf {x x x}} (i, n _ {x},:,::) & 0 \\ \mathbf {0} & 3 g (i, n _ {x}) _ {\sigma \sigma \mathbf {x}} \end{array} \right] \tilde {\mathbf {x}} _ {t} ^ {f} \\ (\tilde {\mathbf {x}} _ {t} ^ {f}) ^ {\prime} \left[ \begin{array}{c c} \mathbf {0} & 0 \\ \mathbf {0} & g (i, 1) _ {\sigma \sigma \sigma} \end{array} \right] \tilde {\mathbf {x}} _ {t} ^ {f} \end{array}\]

Notice that

\[\left[ \begin{array}{cc} \left(\mathbf {x} _ {t} ^ {f}\right) ^ {\prime} & \sigma \\ \end{array} \right]\left[ \begin{array}{cc} \left(\mathbf {x} _ {t} ^ {f}\right) ^ {\prime} & \sigma \\ \end{array} \right]\left[ \begin{array}{cc} \mathbf {g} _ {\mathbf {x x x}} (i, 1,:,:) & 0 \\ \mathbf {0} & 3 g (i, 1) _ {\sigma \sigma \mathbf {x}} \\ \left(\mathbf {x} _ {t} ^ {f}\right) ^ {\prime} & \sigma \\ \end{array} \right]\left[ \begin{array}{c} \mathbf {x} _ {t} ^ {f} \\ \sigma \\ \mathbf {x} _ {t} ^ {f} \\ \sigma \\ \end{array} \right] \\ \cdots \\ \left[ \begin{array}{c c} \left(\mathbf {x} _ {t} ^ {f}\right) ^ {\prime} & \sigma \\ \end{array} \right]\left[ \begin{array}{cc} \mathbf {g} _ {\mathbf {x x x}} (i, n _ {x},,:,:) & 0 \\ \mathbf {0} & 3 g (i, n _ {x}) _ {\sigma \sigma \mathbf {x}} \\ \end{array} \right]\left[ \begin{array}{c} \mathbf {x} _ {t} ^ {f} \\ \sigma \\ \end{array} \right] \\ \left[ \begin{array}{c c} \left(\mathbf {x} _ {t} ^ {f}\right) ^ {\prime} & \sigma \\ \end{array} \right]\left[ \begin{array}{ccc} 0 & & 0 \\ \mathbf {0} & & g (i, 1) _ {\sigma \sigma \sigma} \\ \end{array} \right]\left[ \begin{array}{c} \mathbf {x} _ {t} ^ {f} \\ \sigma \\ \end{array} \right] \\ \end{array} \right]\]

\[= \left[ \begin{array}{c c} {\left(\mathbf {x} _ {t} ^ {f}\right) ^ {\prime}} & {\sigma} \end{array} \right] \left[ \begin{array}{c} {\left(\mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \mathbf {g} _ {\mathbf {x x x}} (i, 1,:,:) \mathbf {x} _ {t} ^ {f} + 3 g (i, 1) _ {\sigma \sigma \mathbf {x}} \sigma^ {2}} \\ {\left(\mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \mathbf {g} _ {\mathbf {x x x}} (i, 2,:,:) \mathbf {x} _ {t} ^ {f} + 3 g (i, 2) _ {\sigma \sigma \mathbf {x}} \sigma^ {2}} \\ \ldots \\ {\left(\mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \mathbf {g} _ {\mathbf {x x x}} (i, n _ {x},,:,:) \mathbf {x} _ {t} ^ {f} + 3 g (i, n _ {x}) _ {\sigma \sigma \mathbf {x}} \sigma^ {2}} \\ g (i, 1) _ {\sigma \sigma \sigma} \sigma^ {2} \end{array} \right]\]

as desired.

\[= \sum_ {k = 1} ^ {n _ {x}} x _ {t} ^ {f} (k) \left(\left(\mathbf {x} _ {t} ^ {f}\right) ^ {\prime} \mathbf {g} _ {\mathbf {x x x}} (i, k,:,:) \mathbf {x} _ {t} ^ {f} + 3 g (i, k) _ {\sigma \sigma \mathbf {x}} \sigma^ {2}\right) + g (i, 1) _ {\sigma \sigma \sigma} \sigma^ {3}\]

\[\begin{array}{r c l} x _ {t} ^ {r d} (i) & = & \left[ \begin{array}{c c} \mathbf {h} _ {\mathbf {x}} (i,:) & 0 \end{array} \right] \left[ \begin{array}{c} \mathbf {x} _ {t} ^ {r d} \\ \sigma \end{array} \right] \\ & & + \left[ \begin{array}{c c} (\mathbf {x} _ {t} ^ {f}) ^ {\prime} & \sigma \end{array} \right] \left[ \begin{array}{c c} \mathbf {h} _ {\mathbf {x x}} (i,:,:) & 0 \\ \mathbf {0} & h _ {\sigma \sigma} \end{array} \right] \left[ \begin{array}{c} \mathbf {x} _ {t} ^ {s} \\ \sigma \end{array} \right] \\ & & + \frac {1}{6} \left[ \begin{array}{c c} (\mathbf {x} _ {t} ^ {f}) ^ {\prime} & \sigma \end{array} \right] \left[ \begin{array}{c c} \left[ \begin{array}{c c} (\mathbf {x} _ {t} ^ {f}) ^ {\prime} & \sigma \end{array} \right] \left[ \begin{array}{c c} \mathbf {h} _ {\mathbf {x x x}} (i, 1,:,:) & 0 \\ \mathbf {0} & 3 h (i, 1) _ {\sigma \sigma \mathbf {x}} \end{array} \right] \left[ \begin{array}{c} \mathbf {x} _ {t} ^ {f} \\ \sigma \end{array} \right] \\ & & \left[ \begin{array}{c c} (\mathbf {x} _ {t} ^ {f}) ^ {\prime} & \sigma \end{array} \right] \left[ \begin{array}{c c} \mathbf {h} _ {\mathbf {x x x}} (i, 2,:,:) & 0 \\ \mathbf {0} & 3 h (i, 2) _ {\sigma \sigma \mathbf {x}} \end{array} \right] \left[ \begin{array}{c} \mathbf {x} _ {t} ^ {f} \\ \sigma \end{array} \right] \\ & & ... \\ & & \left[ \begin{array}{c c} (\mathbf {x} _ {t} ^ {f}) ^ {\prime} & \sigma \end{array} \right] \left[ \begin{array}{c c} \mathbf {h} _ {\mathbf {x x x}} (i, n _ {x},:,:) & 0 \\ \mathbf {0} & 3 h (i, n _ {x}) _ {\sigma \sigma \mathbf {x}} \end{array} \right] \left[ \begin{array}{c} \mathbf {x} _ {t} ^ {f} \\ \sigma \end{array} \right] \\ & & \left[ \begin{array}{c c} (\mathbf {x} _ {t} ^ {f}) ^ {\prime} & \sigma \end{array} \right] \left\lbrack \begin{array}{c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c } 0 \\ 0 & h (i, 1) _ {\sigma \sigma \sigma} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & n _ {t - 1}, n _ {t - 1}, n _ {t - 1}, n _ {t - 1}, n _ {t - 1}, n _ {t - 1}, n _ {t - 1}, n _ {t - 1}, n _ {t - 1}, n _ {t - 1}, n _ {t - 1}, n _ {t - 1}, n _ {t - 1}, n _ {T}, n _ {T}, n _ {T}, n _ {T}, n _ {T}, n _ {T}, n _ {T}, n _ {T}, n _ {T}, n _ {T}, n _ {T}, n _ {T}, n _ {T}, n _ {T}, n _ {T}, n _ {T}, n _ {T}, n _ {T}, n _ {T}, n _ {T}, n _ {S}, n _ {S}, n _ {S}, n _ {S}, n _ {S}, n _ {S}, n _ {S}, n _ {S}, n _ {S}, n _ {S}, n _ {S}, n _ {S}, n _ {S}, n _ {S}, n _ {S}, n _ {S}, n _ {S}, n _ {S}, n _ {S}, n _ {S}, n _ {T}, n _ {T}, n _ {T}, n _ {T}, n _ {T}, n _ {T}, n _ {T}, n _ {T}, n _ {T}, n _ {T}, n _ {T}, n _ {T}, n _ {T}, n _ {T}, n _ {T}, n _ {T}, n _ {T}, n _ {T}, n _ {T}, n _ {N}, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N, N; \\ [ i ] = [ i ] = [ i ] = [ i ] = [ i ] = [ i ] = [ i ] = [ i ] = [ i ] = [ i ] = [ i ] = [ i ] = [ i ] = [ i ] = [ i ] = [ i ] = [ i ] = [ i ] = [ i ] = [ i ] = [ i ] = [ i ] = [ i ] = [ i ] = [ i ] = [ i ] := [ i ] := [ i ] := [ i ] := [ i ] := [ i ] := [ i ] := [ i ] := [ i ] := [ i ] := [ i ] := [ i ] := [ i ] := [ i ] := [ i ] := [ i ] := [ i ] := [ i ] := [ i ] := [ i ] := [ i ] := [ i ] := [ i ] := [ i ] := [ i ] := [ i ] : [ i ] := [ i ] := [ i ] := [ i ] := [ i ] := [ i ] := [ i ] := [ i ] := [ i ] := [ i ] := [ i ] := [ i ] := [ i ] := [ i ] := [ i ] := [ i ] := [ i ] := [ i ] := [ i ] := [ i ] := [ i ] := [ i ] := [ i ] := [ i ] := [ i ]:: [ i ], \\ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. , \\ .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. | \\ .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. | \\ .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. , .. | \\ .. , .. , .. | \\ .. | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | x_{t^{r d}}(i) & = & - x_{t^{r d}}(i) \\ x_{t^{r d}}(i) & = & - x_{t^{r d}}(i) \\ x_{t^{r d}}(i) & = & - x_{t^{r d}}(i) \\ x_{t^{r d}}(i) & = & - x_{t^{r d}}(i) \\ x_{t^{r d}}(i) & = & - x_{t^{r d}}(i) / (x_{t^{r d}}) / (x_{t^{r d}}) / (x_{t^{r d}}) / (x_{t^{r d}}) / (x_{t^{r d}}) / (x_{t^{r d}}) / (x_{t^{r d}}) / (x_{t^{r d}}) / (x_{t^{r d}}) / (x_{t^{r d}}) / (x_{t^{r d}}, x_{t^{r d}}, x_{t^{r d}}, x_{t^{r d}}, x_{t^{r d}}, x_{t^{r d}}, x_{t^{r d}}, x_{t^{r d}}, x_{t^{r d}}, x_{t^{r d}}, x_{t^{r d}}, x_{t^{r d}}, x_{t^{r d}}, x_{t^{r d}}, x_{t^{r d}}, x_{\tau^{\prime}}, x_{\tau^{\prime}}, x_{\tau^{\prime}}, x_{\tau^{\prime}}, x_{\tau^{\prime}}, x_{\tau^{\prime}}, x_{\tau^{\prime}}, x_{\tau^{\prime}}, x_{\tau^{\prime}}, x_{\tau^{\prime}}, x_{\tau^{\prime}}, x_{\tau^{\prime}}, x_{\tau^{\prime}}, x_{-1}\end{array}\]

\[\begin{array}{r c l} x _ {t} ^ {r d} (i) & = & \left[ \begin{array}{l l} \mathbf {h} _ {\mathbf {x}} (i,:) & 0 \end{array} \right] \tilde {\mathbf {x}} _ {t} ^ {r d} \\ & & + (\tilde {\mathbf {x}} _ {t} ^ {f}) ^ {\prime} \left[ \begin{array}{c c} \mathbf {h} _ {\mathbf {x x}} (i,:,: & 0 \\ \mathbf {0} & h _ {\sigma \sigma} \end{array} \right] \tilde {\mathbf {x}} _ {t} ^ {f} \\ & & + \frac {1}{6} (\tilde {\mathbf {x}} _ {t} ^ {f}) ^ {\prime} \left[ \begin{array}{c c} (\tilde {\mathbf {x}} _ {t} ^ {f}) ^ {\prime} \left[ \begin{array}{c c} \mathbf {h} _ {\mathbf {x x x}} (i, 1,:,: & 0 \\ \mathbf {0} & 3 h (i, 1) _ {\sigma \sigma \mathbf {x}} \end{array} \right] \tilde {\mathbf {x}} _ {t} ^ {f} \\ (\tilde {\mathbf {x}} _ {t} ^ {f}) ^ {\prime} \left[ \begin{array}{c c} \mathbf {h} _ {\mathbf {x x x}} (i, 2,:,: & 0 \\ \mathbf {0} & 3 h (i, 2) _ {\sigma \sigma \mathbf {x}} \end{array} \right] \tilde {\mathbf {x}} _ {t} ^ {f} \\ & & \dots \\ (\tilde {\mathbf {x}} _ {t} ^ {f}) ^ {\prime} \left[ \begin{array}{c c} \mathbf {h} _ {\mathbf {x x x}} (i, n _ {x},,:,: & 0 \\ \mathbf {0} & 3 h (i, n _ {x}) _ {\sigma \sigma \mathbf {x}} \end{array} \right] \tilde {\mathbf {x}} _ {t} ^ {f} \\ (\tilde {\mathbf {x}} _ {t} ^ {f}) ^ {\prime} \left[ \begin{array}{c c} \mathbf {0} & 0 \\ \mathbf {0} & h (i, 1) _ {\sigma \sigma \sigma} \end{array} \right] \tilde {\mathbf {x}} _ {t} ^ {f} \end{array} \right] \end{array}\]

14 Accuracy of the pruned state-space system with

This section shows that the errors induced by the pruned state-space system at second order are for , and that the errors induced by the state space system at third order are for . This corresponds to showing that the errors implied by the unpruned and pruned approximations are of the same order for , provided that the unpruned sample path is stable. The procedure for showing these results follow the one applied in the Appendix to Haan & Wind (2012).

14.1 Some auxiliary expressions

Recall that the unpruned third-order approximation of the state equation is given by

\[\begin{array}{r c l} \mathbf {x} _ {t + 1} ^ {(3)} & = & \left(\mathbf {h} _ {\mathbf {x}} + \frac {3}{6} \mathbf {h} _ {\sigma \sigma \mathbf {x}} \sigma^ {2}\right) \mathbf {x} _ {t} ^ {(3)} + \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t} ^ {(3)} \otimes \mathbf {x} _ {t} ^ {(3)}\right) \\ & & + \frac {1}{6} \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t} ^ {(3)} \otimes \mathbf {x} _ {t} ^ {(3)} \otimes \mathbf {x} _ {t} ^ {(3)}\right) + \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2} + \frac {1}{6} \mathbf {h} _ {\sigma \sigma \sigma} \sigma^ {3} + \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1} + O (\sigma^ {4}), \end{array}\]

where denotes the unpruned third-order approximation. We know that the errors induced by this approximation is of fourth order, i.e. . For the derivations below, we need some auxiliary expressions. We therefore first note that

\[\begin{array} { r l } & { \mathbf { x } _ { t + 1 } ^ { ( 3 ) } \otimes \mathbf { x } _ { t + 1 } ^ { ( 3 ) } = } \\ & { \left( \left( \mathbf { h } _ { \mathbf { x } } + \frac { 3 } { 6 } \mathbf { h } _ { \sigma \sigma \mathbf { x } } \sigma ^ { 2 } \right) \mathbf { x } _ { t } ^ { ( 3 ) } + \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } \left( \mathbf { x } _ { t } ^ { ( 3 ) } \otimes \mathbf { x } _ { t } ^ { ( 3 ) } \right) + \frac { 1 } { 6 } \mathbf { H } _ { \mathbf { x x } } \left( \mathbf { x } _ { t } ^ { ( 3 ) } \otimes \mathbf { x } _ { t } ^ { ( 3 ) } \otimes \mathbf { x } _ { t } ^ { ( 3 ) } \right) + \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } + \frac { 1 } { 6 } \mathbf { h } _ { \sigma \sigma \sigma } \sigma ^ { 3 } + \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + 1 } \right) \otimes } \\ & { \left( \left( \mathbf { h } _ { \mathbf { x } } + \frac { 3 } { 6 } \mathbf { h } _ { \sigma \sigma \mathbf { x } } \sigma ^ { 2 } \right) \mathbf { x } _ { t } ^ { ( 3 ) } + \frac { 1 } { 2 } \mathbf { H } ^ { ( 3 ) } \left( \mathbf { x } _ { t } ^ { ( 3 ) } \otimes \mathbf { x } _ { t } ^ { ( 3 ) } \right) + \frac { 1 } { 6 } \mathbf { H } _ { \mathbf { x x } } \left( \mathbf { x } _ { t } ^ { ( 3 ) } \otimes \mathbf { x} _ { t } ^ { ( 3 ) } \otimes \mathbf { x } _ { t } ^ { ( 3 ) } \right) + \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } + \frac { 1 } { 6 } \mathbf { h } _ { \sigma \sigma \sigma } \sigma ^ { 3 } + σ \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + 1 } \right) } \\ & { = \left( \left( \mathbf { h } _ { \mathbf { x } } + \frac { 3 } { 6 } \mathbf { h } _ { \sigma \sigma \mathbf { x } } \sigma ^ { 2 } \right) \mathbf { x } _ { t } ^ { ( 3 ) } \right) \otimes } \\ & { \left( \left( \mathbf { h } _ { \mathbf { x } } + \frac { 3 } { 6 } \mathbf { h } _ { \sigma \sigma \mathbf { x } } \sigma ^ { 2 } \right) \mathbf { x } _ { t } ^ { ( 3 ) } + \frac { 1 } { 2} \mathbf { H } ^ { ( 3 ) } \left( \mathbf { x } _ { t } ^ { ( 3 ) } \otimes \mathbf { x} _ { t } ^ { ( 3 ) } \right) + \frac { 1 } { 6 } \mathbf { H } ^ { ( 3 ) } \left( \mathbf { x} _ { t } ^ { ( 3 ) } \otimes \mathbf { x} _ { t } ^ { ( 3 ) } \otimes \mathbf { x} _ { t } ^ { ( 3 ) } \right) + \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } + \frac { 1 } { 6 } \mathbf { h } _ { \sigma \sigma \sigma } \sigma ^ { 3 } + σ {\boldsymbol {\eta}} {\boldsymbol {\epsilon}} _ { t + 1} \right) } \\ & \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & + \\ & +; \\ & - \\ & - \\ & - \\ & - \\ & - \\ & - \\ & - \\ & - \\ & - \\ & - \\ & - \\ & - \\ & - \\ & - \\ & - \\ & - \\ & - \\ & - \\ & - \\ & - \\ & - \\ & - \\ & - \\ & - \\ & - \\ & - \\ & - \\ & - \\ & - \\ & - \\ & - \\ & - \\ & - \\ & -; \\ & -; \\ & -; \\ & -; \\ & -; \\ & -; \\ & -; \\ & -; \\ & -; \\ & -; \\ & -; \\ & -; \\ & -; \\ & -; \\ & -; \\ & -; \\ & -; \\ & -; \\ & -; \\ & -; \\ & -; \\ & -; \\ & -; \\ & -; \\ & -; \\ & - ; \\ & -; \\ & -; \\ & -; \\ & -; \\ & -; \\ & -; \\ & -; \\ & -; \\ & -; \\ & -; \\ & -; \\ & -; \\ & -; \\ & -; \\ & -; \\ & -; \\ & -; \\ & -; \\ & -; \\ & -; \\ & -; \\ & -; \\ & -; \\ & -; \\ & -! ; \\ & -! ; \\ & -! ; \\ & -! ; \\ & -! ; \\ & -! ; \\ & -! ; \\ & -! ; \\ & -! ; \\ & -! ; \\ & -! ; \\ & -! ; \\ & -! ; \\ & -! ; \\ & -! ; \\ & -! ; \\ & -! ; \\ & -! ; \\ & -! ; \\ & -! ; \\ & -! ! ; \\ & -! ; \\ & -! ; \\ & -! ; \\ & -! ; \\ & -! ; \\ & -! ; | |, | |, | |, | |, | |, | |, | |, | |, | |, | |, | |, | |, | |, | |, | |, | |, | |, | |, | |, | |, | |, | |, | |, | |, | |, | |, | |, | |, | |, | |, | |, | |, | |, | |, . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , . . . , i n a d i n e f o r e s e n c o n s e n c o n s e n c o n s e n c o n s e n c o n s e n c o n s e n c o n s e n c o n s e n c o n s e n c o n s e n c o n s e n c o n s e n c o n s e n c o n s e n c o n s e n c o n s e n c o n s e n e n c o n s e n c o n s e n c o n s e n c o n s e n c o n s e n c o n s e n c o n s e n c o n s e n c o n s e n c o n s e n c o n s e n c o n s e n c o n s e n c o n s e n c o n s e n c o n s e n c e n c o n s e n c o n s e n c o n s e n c o n s e n c o n s e n c o n s e n c o n s e n c o n s e n c o n s e n c o n s e n c o n s e n c o n s e n c o n s e n c o n s e n c o n s e n c o n s e n c a l l i m i a l l i m i a l l i m i a l l i m i a l l i m i a l l i m i a l l i m i a l l i m i a l l i m i a l l i m i a l l i m i a l l i m i a l l i m i a l l i m i a l l i m i a l l i m i a l l i m i a l l l i m i a l l i m i a l l i m i a l l i m i a l l i m i a l l i m i a l l i m i a l l i m i a l l i m i a l l i m i a l l i m i a l l i m i a l l i m i a l l i m i a l l i m i a l l i m i a l l i u p p h a r b y f r a r b y f r a r b y f r a r b y f r a r b y f r a r b y f r a r b y f r a r b y f r a r b y f r a r b y f r a r b y f r a r b y f r a r b y f r a r b y f r a r b y f r a r b y f r a r b y f r a r b u r g h a r b y f r a r b y f r a r b y f r a r b y f r a r b y f r a r b y f r a r b y f r a r b y f r a r b y f r a r b y f r a r b y f r a r b y f r a r b y f r a r b y f r a r b y f r a r b y f r a r k g h a r b y f r a r k g h a r k g h a r k g h a r k g h a r k g h a r k g h a r k g h a r k g h a r k g h a r k g h a r k g h a r k g h a r k g h a r k g h a r k g h a r k g h a r k g h a r k g h a r k g h a r k g h a r k g g h a r k g h a r k g h a r k g h a r k g h a r k g h a r k g h a r k g h a r k g h a r k g h a r k g h a r k g h a r k g h a r k g h a r k g h a r k g h a r k g h a r k g h a r k g h a r k g h a r j k g h a r j k g h a r j k g h a r j k g h a r j k g h a r j k g h a r j k g h a r j k g h a r j k g h a r j k g h a r j k g h a r j k g h a r j k g h a r j k g h a r j k g h a r j k g h a r j k g h h a r j k g h a r j k g h a r j k g h a r j k g h a r j k g h a r j k g h a r j k g h a r j k g h a r j k g h a r j k g h a r j k g h a r j k g h a r j k g h a r j k g h a r j k g h a r j k g h a rj k g h a r j k g h a r j k g h a r j k g h a r j k g h a r j k g h a r j k g h a r j k g h a r j k g h a r j k g h a r j k g h a r j k g h a r j k g h a r j k g h a r j k g h a r j k g h a r j k g h q u v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v v w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w ~w~w~w~w~w~w~w~w~w~w~w~w~w~w~w~w~w~w~w~w~w~w~w~w~w~w~w~w~w~w~w~w~w~w~w ~w~w~w~w~w~w~w~w ~w ~w ~w ~w ~w ~w ~w ~w ~w ~w ~w ~w ~w ~w ~w ~w ~w ~w ~w ~w ~w ~w ~w ~w ~w ~w ~w ~w ~w ~w ~w ~w ~w ~w ~w ~w ~w ~w ~w ~w ~w ~w ~w ~w ~w ~w ~w ~w ~w ~w % * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / : p > [ ] < [ ] < [ ] < [ ] < [ ] < [ ] < [ ] < [ ] < [ ] < [ ] < [ ] < [ ] < [ ] < [ ] < [ ] < [ ] < [ ] < [ ] < [ ] < [ ] < [ ] < [ ] < [ ] < [ ] < [ ] < [ ] < < nl>\]

\[\begin{array} { l } + \frac { 1 } { 6 } \mathbf { H } _ { \mathbf { x x } } \left( \mathbf { x } _ { t } ^ { ( 3 ) } \otimes \mathbf { x } _ { t } ^ { ( 3 ) } \otimes \mathbf { x } _ { t } ^ { ( 3 ) } \right) \otimes \left( \mathbf { h } _ { \mathbf { x } } + \frac { 3 } { 6 } \mathbf { h } _ { \sigma \sigma \mathbf { x } } \sigma ^ { 2 } \right) \mathbf { x } _ { t } ^ { ( 3 ) } + \frac { 1 } { 6 } \mathbf { H } _ { \mathbf { x x } } \left( \mathbf { x } _ { t } ^ { ( 3 ) } \otimes \mathbf { x } _ { t } ^ { ( 3 ) } \otimes \mathbf { x } _ { t } ^ { ( 3 ) } \right) \otimes \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } \left( \mathbf { x } _ { t } ^ { ( 3 ) } \otimes \mathbf { x } _ { t } ^ { ( 3 ) } \right) \\ + \frac { 1 } { 6 } \mathbf { H } _ { \mathbf { x x } } \left( \mathbf { x } _ { t } ^ { ( 3 ) } \otimes \mathbf { x } _ { t } ^ { ( 3 ) } \otimes \mathbf { x } _ { t } ^ { ( 3 ) } \right) \otimes \frac { 1 } 6 \mathbf { H } _ { \mathbf { x x } } \left( \mathbf { x } _ { t } ^ { ( 3 ) } \otimes \mathbf { x } _ { t } ^ { ( 3 ) } \otimes \mathbf { x } _ { t } ^ { ( 3 ) } \right) + \frac { 1 } 6 \mathbf { H } _ { \mathbf { x x } } \left( \mathbf { x } _ { t } ^ { ( 3 ) } \otimes \mathbf { x } _ { t } ^ { ( 3 ) } \otimes \mathbf { x } _ { t } ^ { ( 3 ) } \right) \otimes \left( \frac { 1 } 2 \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } + \frac { 1 } 6 \mathbf { h } _ { \sigma \sigma \sigma } \sigma ^ { 3 } \right) \\ + \frac { 1 } 6 \mathbf { H } _ { \mathbf { x x } } \left( \mathbf { x } _ { t } ^ { ( 3 ) } \otimes \mathbf { x } _ { t } ^ { ( 3 ) } \otimes \mathbf { x } _ { t } ^ { ( 3 ) } \right) \otimes \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ { t + 1} \\ + \left( \frac { 1 } 2 \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } + \frac { 1 } 6 \mathbf { h } _ { \sigma \sigma \sigma } \sigma ^ { 3} \right) \otimes \left( \mathbf { h } _ { \mathbf { x } } + \frac { 3 } 6 \mathbf { h } _ { \sigma \sigma \mathbf { x }} \sigma ^ { 2} \right) \mathbf { x } _ { t } ^ { ( 3 ) } + \left( \frac { 1 } 2 \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } + \frac { 1 } 6 \mathbf { h } _ { \sigma \sigma \sigma } \sigma ^ { 3} \right) \otimes \frac { 1 } 2 \mathbf { H } _ { \mathbf { x x } } \left( \mathbf { x } _ { t } ^ { ( 3 ) } \otimes \mathbf { x } _ { t } ^ { ( 3 ) } \right) \\ + \left( \frac { 1 } 2 \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } + \frac { 1 } 6 \mathbf { h } _ { \sigma \sigma \sigma } \sigma ^ { 3} \right) \otimes \frac { 1 } 6 \mathbf { H } _ { \mathbf { x x} } \left( \mathbf { x } _ { t } ^ { ( 3 ) } \otimes \mathbf { x } _ { t } ^ { ( 3 ) } \otimes \mathbf { x } _ { t } ^ { ( 3 )} \right) + \left( \frac { 1 } 2 \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } + \frac { 1 } 6 \mathbf { h } _ { \sigma \sigma \sigma } \sigma ^ { 3} \right) \\ + - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + - - - - - - - - - - - - - - - - - - - - - + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + . \\ =\]

Preserving terms up to third order we have

\[\begin{array} { r l } & { \mathbf { x } _ { t + 1 } ^ { ( 3 ) } \otimes \mathbf { x } _ { t + 1 } ^ { ( 3 ) } = \mathbf { h } _ { \mathbf { x } } \mathbf { x } _ { t } ^ { ( 3 ) } \otimes \mathbf { h } _ { \mathbf { x } } \mathbf { x } _ { t } ^ { ( 3 ) } + \mathbf { h } _ { \mathbf { x } } \mathbf { x } _ { t } ^ { ( 3 ) } \otimes \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } \left( \mathbf { x } _ { t } ^ { ( 3 ) } \otimes \mathbf { x } _ { t } ^ { ( 3 ) } \right) + \mathbf { h } _ { \mathbf { x } } \mathbf { x } _ { t } ^ { ( 3 ) } \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } } \\ & { + \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } \left( \mathbf { x } _ { t } ^ { ( 3 ) } \otimes \mathbf { x } _ { t } ^ { ( 3 ) } \right) \otimes \mathbf { h } _ { \mathbf { x } } \mathbf { x } _ { t } ^ { ( 3 ) } + \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } \otimes \mathbf { h } _ { \mathbf { x } } \mathbf { x } _ { t } ^ { ( 3 ) } } \\ & { + \mathbf { h } _ { \mathbf { x } } \mathbf { x } _ { t } ^ { ( 3 ) } \otimes \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + 1 } + \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } \left( \mathbf { x } _ { t } ^ { ( 3 ) } \otimes \mathbf { x } _ { t } ^ { ( 3 ) } \right) \otimes \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + 1 } + \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } \otimes \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + 1 } } \\ & { + \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { h } _ { \mathbf { x } } \mathbf { x } _ { t } ^ { ( 3 ) } + \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + 1 } \otimes \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } \left( \mathbf { x } _ { t } ^ { ( 3 ) } \otimes \mathbf { x } _ { t } ^ { ( 3 ) } \right) + \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + 1 } \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } + \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + 1 } \otimes \sigma \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + 1 } } \\ & \\ & = ( \mathbf { h _ { x }} \otimes \mathbf { h _ { x } ) } ( \mathbf { x } _ { t } ^ { ( 3 ) } \otimes \mathbf { x } _ { t } ^ { ( 3 ) } ) + ( \mathbf { h _ { x }} \otimes \frac { 1 } { 2 } \mathbf { H _ { x x} ) } ( \mathbf { x } _ { t } ^ { ( 3 ) } \otimes \mathbf { x } _ { t } ^ { ( 3 ) } \otimes \mathbf { x} _ { t } ^ { ( 3 ) } ) + ( \mathbf { h _ { x }} \otimes \frac { 1 } { 2 } \mathbf { h _ { s o r} ) } ( \mathbf { x} _ { t } ^ { ( 3 ) } \otimes \sigma ^ { 2} ) \\ & + ( \frac { 1 } { 2 } \mathbf { H _ { x x} ) }\otimes\]

Preserving terms up to second order we have

\[\begin{array}{r c l} \mathbf {x} _ {t + 1} ^ {(2)} \otimes \mathbf {x} _ {t + 1} ^ {(2)} & = & (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) \left(\mathbf {x} _ {t} ^ {(2)} \otimes \mathbf {x} _ {t} ^ {(2)}\right) + \left(\mathbf {h} _ {\mathbf {x}} \otimes \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1}\right) \left(\mathbf {x} _ {t} ^ {(2)} \otimes \sigma\right) \\ & & + \left(\boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {h} _ {\mathbf {x}}\right) \left(\sigma \otimes \mathbf {x} _ {t} ^ {(2)}\right) + \left(\boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1}\right) \sigma^ {2} \end{array}\]

where denotes the unpruned second-order approximation.

Moreover, preserving terms up to third order in clearly gives

\[\begin{array}{r l} & {\mathbf {x} _ {t + 1} ^ {(3)} \otimes \mathbf {x} _ {t + 1} ^ {(3)} \otimes \mathbf {x} _ {t + 1} ^ {(3)} = (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) (\mathbf {x} _ {t} ^ {(3)} \otimes \mathbf {x} _ {t} ^ {(3)} \otimes \mathbf {x} _ {t} ^ {(3)}) + (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}) (\mathbf {x} _ {t} ^ {(3)} \otimes \mathbf {x} _ {t} ^ {(3)} \otimes \boldsymbol {\epsilon} _ {t + 1})} \\ & {\quad + (\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) (\mathbf {x} _ {t} ^ {(3)} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {(3)}) + (\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}) (\mathbf {x} _ {t} ^ {(3)} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1})} \\ & {\quad + (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) (\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {(3)} \otimes \mathbf {x} _ {t} ^ {(3)}) + (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}) (\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {(3)} \otimes \boldsymbol {\epsilon} _ {t + 1})} \end{array}\]

\[+ \left(\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}\right) \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {(3)}\right) + \left(\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}\right) \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) + O \left(\sigma^ {4}\right)\]

Next, recall that the pruned approximation at third order reads

\[\mathbf {x} _ {t + 1} ^ {f} = \mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t} ^ {f} + \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1}\]

\[\mathbf {x} _ {t + 1} ^ {s} = \mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t} ^ {s} + \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) + \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}\]

\[\mathbf {x} _ {t + 1} ^ {r d} = \mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t} ^ {r d} + \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {s}\right) + \frac {1}{6} \mathbf {H} _ {\mathbf {x x x}} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) + \frac {3}{6} \mathbf {h} _ {\sigma \sigma \mathbf {x}} \sigma^ {2} \mathbf {x} _ {t} ^ {f} + \frac {1}{6} \mathbf {h} _ {\sigma \sigma \sigma} \sigma^ {3}\]

Adding up these terms we have

\[\begin{array}{r c l} \mathbf {x} _ {t + 1} ^ {f} + \mathbf {x} _ {t + 1} ^ {s} + \mathbf {x} _ {t + 1} ^ {r d} & = & \mathbf {h} _ {\mathbf {x}} \left(\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d}\right) + \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \left(\left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) + 2 \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {s}\right)\right) \\ & & + \frac {1}{6} \mathbf {H} _ {\mathbf {x x x}} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) + \frac {3}{6} \mathbf {h} _ {\sigma \sigma \mathbf {x}} \sigma^ {2} \mathbf {x} _ {t} ^ {f} + \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2} + \frac {1}{6} \mathbf {h} _ {\sigma \sigma \sigma} \sigma^ {3} + \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1}. \end{array}\]

For the controls we have without pruning that

\[\mathbf {y} _ {t} ^ {(2)} = \mathbf {g} _ {\mathbf {x}} \mathbf {x} _ {t} ^ {(2)} + \frac {1}{2} \mathbf {G} _ {\mathbf {x x}} \left(\mathbf {x} _ {t} ^ {(2)} \otimes \mathbf {x} _ {t} ^ {(2)}\right) + \frac {1}{2} \mathbf {g} _ {\sigma \sigma} \sigma^ {2}\]

and

\[\begin{array}{r c l} \mathbf {y} _ {t} ^ {(3)} & = & \mathbf {g} _ {\mathbf {x}} \mathbf {x} _ {t} ^ {(3)} + \frac {1}{2} \mathbf {G} _ {\mathbf {x x}} \left(\mathbf {x} _ {t} ^ {(3)} \otimes \mathbf {x} _ {t} ^ {(3)}\right) + \frac {1}{6} \mathbf {G} _ {\mathbf {x x x}} \left(\mathbf {x} _ {t} ^ {(3)} \otimes \mathbf {x} _ {t} ^ {(3)} \otimes \mathbf {x} _ {t} ^ {(3)}\right) \\ & & + \frac {3}{6} \mathbf {g} _ {\sigma \sigma \mathbf {x}} \sigma^ {2} \mathbf {x} _ {t} ^ {(3)} + \frac {1}{2} \mathbf {g} _ {\sigma \sigma} \sigma^ {2} + \frac {1}{6} \mathbf {g} _ {\sigma \sigma \sigma} \sigma^ {3} + O (\sigma^ {4}) \end{array}\]

in an unpruned second- and third-order approximation, respectively

14.2 Proof for a second order approximation

14.2.1 First order terms

At first order we trivially have

\[\begin{array}{r l} & {\mathbf {x} _ {t + 1} ^ {f} - \mathbf {x} _ {t + 1} ^ {(2)} = \mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t} ^ {f} + \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1}} \\ & {\qquad - \left\{\mathbf {h} _ {\mathbf {x}} ^ {(2)} \mathbf {x} _ {t} ^ {(2)} + \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t} ^ {(2)} \otimes \mathbf {x} _ {t} ^ {(2)}\right) + \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2} + \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1} + O (\sigma^ {3}) \right\}} \\ & {\qquad = \mathbf {h} _ {\mathbf {x}} \left(\mathbf {x} _ {t} ^ {f} - \mathbf {x} _ {t} ^ {(2)}\right) + O (\sigma^ {2})} \end{array}\]

We therefore see that errors in are of second order, i.e. . Note, that we do not need to require that the initial errors are of second order, i.e. , provided that t is sufficiently far away from zero. This is because any error committed at time 0 will die out because all eigenvalues of are less than one.

For the controls, the pruned expression is , so . Given that , this clearly also holds for the controls, i.e. .

14.2.2 Second order terms

At second order the pruned approximation of the states is given by

\[\mathbf {x} _ {t + 1} ^ {f} + \mathbf {x} _ {t + 1} ^ {s} = \mathbf {h} _ {\mathbf {x}} \left(\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s}\right) + \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) + \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2} + \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1}\]

Comparing this approximation to a second-order expansion without pruning, we obtain

\[\begin{array}{r l} & {\mathbf {x} _ {t + 1} ^ {f} + \mathbf {x} _ {t + 1} ^ {s} - \mathbf {x} _ {t + 1} ^ {(2)} = \mathbf {h} _ {\mathbf {x}} \left(\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s}\right) + \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) + \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2} + \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1}} \\ & {- \left\{\mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t} ^ {(2)} + \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t} ^ {(2)} \otimes \mathbf {x} _ {t} ^ {(2)}\right) + \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2} + \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} + O (\sigma^ {3}) \right\}} \\ & {\Updownarrow \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad} \\ & {\qquad \mathbf {x} _ {t + 1} ^ {f} + \mathbf {x} _ {t + 1} ^ {s} - \mathbf {x} _ {t + 1} ^ {(2)} = \mathbf {h} _ {\mathbf {x}} \left(\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} - \mathbf {x} _ {t} ^ {(2)}\right) + \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} - \mathbf {x} _ {t} ^ {(2)} \otimes \mathbf {x} _ {t} ^ {(2)}\right) + O (\sigma^ {3})} \end{array}\]

We next want to show that and . To do so, we recall that

\[\begin{array}{r c l} \mathbf {x} _ {t + 1} ^ {f} \otimes \mathbf {x} _ {t + 1} ^ {f} & = & (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) + (\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}) \left(\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) \\ & & + (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) + (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}) \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) \end{array}\]

Hence, we have

\[\begin{array}{l} \mathbf {x} _ {t + 1} ^ {f} \otimes \mathbf {x} _ {t + 1} ^ {f} - \mathbf {x} _ {t + 1} ^ {(2)} \otimes \mathbf {x} _ {t + 1} ^ {(2)} = (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) + (\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}) \left(\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) \\ \quad + (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) + (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}) \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) \\ - \bigl \{(\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) \left(\mathbf {x} _ {t} ^ {(2)} \otimes \mathbf {x} _ {t} ^ {(2)}\right) + \\ \quad + (\mathbf {h} _ {\mathbf {x}} \otimes \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1}) \left(\mathbf {x} _ {t} ^ {(2)} \otimes \sigma\right) + \bigl (\frac {1}{2} \mathbf {h} _ {\sigma \sigma} \otimes \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} \bigr) \sigma^ {3} \\ \quad + (\boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {h} _ {\mathbf {x}}) \left(\sigma \otimes \mathbf {x} _ {t} ^ {(2)}\right) + (\boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1}) \sigma^ {2} + O (\sigma^ {3}) \bigr \} \\ = (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} - \mathbf {x} _ {t} ^ {(2)} \otimes \mathbf {x} _ {t} ^ {(2)}\right) + (\mathbf {h} _ {\mathbf {x}} \otimes \boldsymbol {\eta}) \left(\mathbf {x} _ {t} ^ {f} \otimes \sigma \boldsymbol {\epsilon} _ {t + 1} - \mathbf {x} _ {t} ^ {(2)} \otimes \sigma \boldsymbol {\epsilon} _ {t + 1}\right) \\ + (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} - \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {(2)}\right) + O (\sigma^ {3}) \\ = (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} - \mathbf {x} _ {t} ^ {(2)} \otimes \mathbf {x} _ {t} ^ {(2)}\right). \\ + (\boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) \left(\sigma \boldsymbol {\epsilon} _ {t + 1} \otimes (\mathbf {x} _ {t} ^ {f} - \mathbf {x} _ {t} ^ {(2)})\right) + O (\sigma^ {3}) \end{array}\]

We know that . Now recall the meaning of our notation. We say that a function is

\[f (\sigma) = O \left(\sigma^ {m}\right)\]

\[\lim _ {\sigma \longrightarrow 0} \frac {f (\sigma)}{\sigma^ {m}} < M < \infty\]

This clearly implies that

\[\frac {\sigma f (\sigma)}{\sigma^ {m + 1}} = O (\sigma^ {m + 1}),\]

or . So in our case we have , and therefore . Hence, . This in turn means that as desired.

For the controls we have in the pruned approximation

\[\mathbf {y} _ {t} ^ {s} = \mathbf {g} _ {\mathbf {x}} \left(\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s}\right) + \frac {1}{2} \mathbf {G} _ {\mathbf {x x}} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) + \frac {1}{2} \mathbf {g} _ {\sigma \sigma} \sigma^ {2}\]

So we have

\[\mathbf {y} _ {t} ^ {s} - \mathbf {y} _ {t} ^ {(2)} = \mathbf {g} _ {\mathbf {x}} \left(\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} - \mathbf {x} _ {t} ^ {(2)}\right) + \frac {1}{2} \mathbf {G} _ {\mathbf {x x}} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} - \mathbf {x} _ {t} ^ {(2)} \otimes \mathbf {x} _ {t} ^ {(2)}\right) + O \left(\sigma^ {3}\right)\]

Given that and , we clearly have as desired.

14.3 Proof for a third order approximation

14.3.1 First order terms

At first order we trivially have

\[\begin{array}{r l} & {\mathbf {x} _ {t + 1} ^ {f} - \mathbf {x} _ {t + 1} ^ {(3)} = \mathbf {h} _ {\mathbf {x}} \mathbf {x} _ {t} ^ {f} + \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1}} \\ & {- \big \{\left(\mathbf {h} _ {\mathbf {x}} + \frac {3}{6} \mathbf {h} _ {\sigma \sigma \mathbf {x}} \sigma^ {2}\right) \mathbf {x} _ {t} ^ {(3)} + \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t} ^ {(3)} \otimes \mathbf {x} _ {t} ^ {(3)}\right) + \frac {1}{6} \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t} ^ {(3)} \otimes \mathbf {x} _ {t} ^ {(3)} \otimes \mathbf {x} _ {t} ^ {(3)}\right)} \\ & {\quad + \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2} + \frac {1}{6} \mathbf {h} _ {\sigma \sigma \sigma} \sigma^ {3 \prime} + \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1} + O (\sigma^ {4}) \big \}} \\ & {= \mathbf {h} _ {\mathbf {x}} (\mathbf {x} _ {t} ^ {f} - \mathbf {x} _ {t} ^ {(3)}) + O (\sigma^ {2})} \end{array}\]

We therefore see that errors in are of second order, i.e. . For the controls, the pruned expression is , so . Given that , this clearly also holds for the controls, i.e. .

14.3.2 Second order terms

Comparing this approximation to a third-order expansion without pruning, we obtain

\[\begin{array}{r l} & {\mathbf {x} _ {t + 1} ^ {f} + \mathbf {x} _ {t + 1} ^ {s} - \mathbf {x} _ {t + 1} ^ {(3)} = \mathbf {h} _ {\mathbf {x}} \left(\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s}\right) + \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) + \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2} + \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1}} \\ & {- \bigl \{\left(\mathbf {h} _ {\mathbf {x}} + \frac {3}{6} \mathbf {h} _ {\sigma \sigma \mathbf {x}} \sigma^ {2}\right) \mathbf {x} _ {t} ^ {(3)} + \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t} ^ {(3)} \otimes \mathbf {x} _ {t} ^ {(3)}\right) + \frac {1}{6} \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t} ^ {(3)} \otimes \mathbf {x} _ {t} ^ {(3)} \otimes \mathbf {x} _ {t} ^ {(3)}\right)} \\ & {\quad + \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2} + \frac {1}{6} \mathbf {h} _ {\sigma \sigma \sigma} \sigma^ {3 \prime} + \sigma \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} + O (\sigma^ {4}) \bigr \}} \\ & {\Updownarrow \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad} \\ & {\qquad \mathbf {x} _ {t + 1} ^ {f} + \mathbf {x} _ {t + 1} ^ {s} - \mathbf {x} _ {t + 1} ^ {(3)} = \mathbf {h} _ {\mathbf {x}} (\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} - \mathbf {x} _ {t} ^ {(3)}) + \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} - \mathbf {x} _ {t} ^ {(3)} \otimes \mathbf {x} _ {t} ^ {(3)}) + O (\sigma^ {3})} \end{array}\]

We next want to show that and . To do so, we recall that

\[\begin{array}{r c l} \mathbf {x} _ {t + 1} ^ {f} \otimes \mathbf {x} _ {t + 1} ^ {f} & = & (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) + (\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}) \left(\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) \\ & & + (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}\right) + (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}) \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) \end{array}\]

Hence, we have

\[\begin{array}{l} \mathbf {x} _ {t + 1} ^ {f} \otimes \mathbf {x} _ {t + 1} ^ {f} - \mathbf {x} _ {t + 1} ^ {(3)} \otimes \mathbf {x} _ {t + 1} ^ {(3)} = (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) + (\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}) (\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}) \\ \quad + (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) (\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}) + (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}) (\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}) \\ \quad - \bigl \{(\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) (\mathbf {x} _ {t} ^ {(3)} \otimes \mathbf {x} _ {t} ^ {(3)}) + (\mathbf {h} _ {\mathbf {x}} \otimes \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} + \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \otimes \mathbf {h} _ {\mathbf {x}}) (\mathbf {x} _ {t} ^ {(3)} \otimes \mathbf {x} _ {t} ^ {(3)} \otimes \mathbf {x} _ {t} ^ {(3)}) + \\ \quad + (\mathbf {h} _ {\mathbf {x}} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} + \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \otimes \mathbf {h} _ {\mathbf {x}}) (\mathbf {x} _ {t} ^ {(3)} \otimes \sigma^ {2}) \end{array}\]

\[\begin{array}{l} + \left(\mathbf {h} _ {\mathbf {x}} \otimes \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1}\right) \left(\mathbf {x} _ {t} ^ {(3)} \otimes \sigma\right) + \left(\frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \otimes \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1}\right) \left(\mathbf {x} _ {t} ^ {(3)} \otimes \mathbf {x} _ {t} ^ {(3)} \otimes \sigma\right) + \left(\frac {1}{2} \mathbf {h} _ {\sigma \sigma} \otimes \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1}\right) \sigma^ {3} \\ + \left(\boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {h} _ {\mathbf {x}}\right) \left(\sigma \otimes \mathbf {x} _ {t} ^ {(3)}\right) + \left(\boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} \otimes \frac {1}{2} \mathbf {H} _ {\mathbf {x x}}\right) \left(\sigma \otimes \mathbf {x} _ {t} ^ {(3)} \otimes \mathbf {x} _ {t} ^ {(3)}\right) + \left(\boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma}\right) \sigma^ {3} + \left(\boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\eta} \boldsymbol {\epsilon} _ {t + 1}\right) \sigma^ {2} + O \left(\sigma^ {4}\right) \} \\ = (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} - \mathbf {x} _ {t} ^ {(3)} \otimes \mathbf {x} _ {t} ^ {(3)}\right) + (\mathbf {h} _ {\mathbf {x}} \otimes \boldsymbol {\eta}) \left(\mathbf {x} _ {t} ^ {f} \otimes \sigma \boldsymbol {\epsilon} _ {t + 1} - \mathbf {x} _ {t} ^ {(3)} \otimes \sigma \boldsymbol {\epsilon} _ {t + 1}\right) \\ + (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) \left(\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} - \boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {(3)}\right) + O \left(\sigma^ {3}\right) \\ = (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} - \mathbf {x} _ {t} ^ {(3)} \otimes \mathbf {x} _ {t} ^ {(3)}\right). \\ + (\boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) \left(\sigma \boldsymbol {\epsilon} _ {t + 1} \otimes (\mathbf {x} _ {t} ^ {f} - \mathbf {x} _ {t} ^ {(3)})\right) + O \left(\sigma^ {3}\right) \end{array}\]

We know that . Now recall the meaning of our notation. We say that a function is

\[f (\sigma) = O \left(\sigma^ {m}\right)\]

\[\lim _ {\sigma \longrightarrow 0} \frac {f (\sigma)}{\sigma^ {m}} < M < \infty\]

This clearly implies that

\[\frac {\sigma f (\sigma)}{\sigma^ {m + 1}} = O \left(\sigma^ {m + 1}\right),\]

or . So in our case we have , and therefore . Hence, . This in turn means that as desired.

For the controls we have in the pruned approximation

\[\mathbf {y} _ {t} ^ {s} = \mathbf {g} _ {\mathbf {x}} \left(\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s}\right) + \frac {1}{2} \mathbf {G} _ {\mathbf {x x}} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) + \frac {1}{2} \mathbf {g} _ {\sigma \sigma} \sigma^ {2}\]

So we have

\[\mathbf {y} _ {t} ^ {s} - \mathbf {y} _ {t} ^ {(3)} = \mathbf {g} _ {\mathbf {x}} \left(\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} - \mathbf {x} _ {t} ^ {(3)}\right) + \frac {1}{2} \mathbf {G} _ {\mathbf {x x}} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} - \mathbf {x} _ {t} ^ {(3)} \otimes \mathbf {x} _ {t} ^ {(3)}\right) + O \left(\sigma^ {3}\right)\]

Given that and , we clearly have as desired.

14.3.3 Third order terms

Comparing this approximation to a third-order expansion without pruning, we obtain at third order

\[\begin{array}{r l} & {\mathbf {x} _ {t + 1} ^ {f} + \mathbf {x} _ {t + 1} ^ {s} + \mathbf {x} _ {t + 1} ^ {r d} - \mathbf {x} _ {t + 1} ^ {(3)} = \mathbf {h} _ {\mathbf {x}} \left(\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d}\right) + \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} + \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {s}\right) + \left(\mathbf {x} _ {t} ^ {s} \otimes \mathbf {x} _ {t} ^ {f}\right)\right)} \\ & {\quad + \frac {1}{6} \mathbf {H} _ {\mathbf {x x x}} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) + \frac {3}{6} \mathbf {h} _ {\sigma \sigma \mathbf {x}} \mathbf {x} _ {t} ^ {f} \sigma^ {2} + \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2} + \frac {1}{6} \mathbf {h} _ {\sigma \sigma \sigma} \sigma^ {3} + \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1}} \\ & {\quad - \left\{\left(\mathbf {h} _ {\mathbf {x}} + \frac {3}{6} \mathbf {h} _ {\sigma \sigma \mathbf {x}} \sigma^ {2}\right) \mathbf {x} _ {t} ^ {(3)} + \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t} ^ {(3)} \otimes \mathbf {x} _ {t} ^ {(3)}\right) + \frac {1}{6} \mathbf {H} _ {\mathbf {x x}} \left(\mathbf {x} _ {t} ^ {(3)} \otimes \mathbf {x} _ {t} ^ {(3)} \otimes \mathbf {x} _ {t} ^ {(3)}\right) + \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2} + \frac {1}{6} \mathbf {h} _ {\sigma \sigma \sigma} \sigma^ {3 \prime} + \sigma \pmb {\eta} \pmb {\epsilon} _ {t + 1} \right\}} \\ & {\quad = \mathbf {h} _ {\mathbf {x}} \left(\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d} - \mathbf {x} _ {t} ^ {(3)}\right) + \frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \left(\left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) + \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {s}\right) + \left(\mathbf {x} _ {t} ^ {s} \otimes \mathbf {x} _ {t} ^ {f}\right) - \mathbf {x} _ {t} ^ {(3)} \otimes \mathbf {x} _ {t} ^ {(3)}\right)} \\ & {\quad + \frac {1}{6} \mathbf {H} _ {\mathbf {x x x}} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} - \mathbf {x} _ {t} ^ {(3)} \otimes \mathbf {x} _ {t} ^ {(3)} \otimes \mathbf {x} _ {t} ^ {(3)}\right) + \frac {3}{6} \mathbf {h} _ {\sigma \sigma \mathbf {x}} \sigma^ {2} \left(\mathbf {x} _ {t} ^ {f} - \mathbf {x} _ {t} ^ {(3)}\right)} \end{array}\]

We then need to show that each of the terms are accurate up to . We start by considering the last term. Here, , and therefore , as desired. We clearly also have . To consider the single remaining term, recall from above that

\[\begin{array}{r c l} \mathbf {x} _ {t + 1} ^ {f} \otimes \mathbf {x} _ {t + 1} ^ {f} & = & (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) (\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) + (\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}) (\mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}) \\ & & + (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) (\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f}) + (\sigma \boldsymbol {\eta} \otimes \sigma \boldsymbol {\eta}) (\boldsymbol {\epsilon} _ {t + 1} \otimes \boldsymbol {\epsilon} _ {t + 1}) \end{array}\]

and

\[\begin{array}{r c l} \mathbf {x} _ {t + 1} ^ {f} \otimes \mathbf {x} _ {t + 1} ^ {s} & = & (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {s}\right) + \left(\mathbf {h} _ {\mathbf {x}} \otimes \frac {1}{2} \mathbf {H} _ {\mathbf {x x}}\right) \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) + \left(\mathbf {h} _ {\mathbf {x}} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}\right) \mathbf {x} _ {t} ^ {f} \\ & & + (\sigma \boldsymbol {\eta} \otimes \mathbf {h} _ {\mathbf {x}}) (\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {s}) + \left(\sigma \boldsymbol {\eta} \otimes \frac {1}{2} \mathbf {H} _ {\mathbf {x x}}\right) (\boldsymbol {\epsilon} _ {t + 1} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}) + \left(\sigma \boldsymbol {\eta} \otimes \frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2}\right) \boldsymbol {\epsilon} _ {t + 1} \end{array}\]

and

\[\begin{array}{r c l} \mathbf {x} _ {t + 1} ^ {s} \otimes \mathbf {x} _ {t + 1} ^ {f} & = & (\mathbf {h} _ {\mathbf {x}} \otimes \mathbf {h} _ {\mathbf {x}}) \left(\mathbf {x} _ {t} ^ {s} \otimes \mathbf {x} _ {t} ^ {f}\right) + \left(\frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \otimes \mathbf {h} _ {\mathbf {x}}\right) \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) + \left(\frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2} \otimes \mathbf {h} _ {\mathbf {x}}\right) \mathbf {x} _ {t} ^ {f} \\ & & + (\mathbf {h} _ {\mathbf {x}} \otimes \sigma \boldsymbol {\eta}) \left(\mathbf {x} _ {t} ^ {s} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) + \left(\frac {1}{2} \mathbf {H} _ {\mathbf {x x}} \otimes \sigma \boldsymbol {\eta}\right) \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \boldsymbol {\epsilon} _ {t + 1}\right) + \left(\frac {1}{2} \mathbf {h} _ {\sigma \sigma} \sigma^ {2} \otimes \sigma \boldsymbol {\eta}\right) \boldsymbol {\epsilon} _ {t + 1}. \end{array}\]

\[\begin{array} { r l } & { \mathrm{Thus} } \\ & { \left( \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { f } \right) + \left( \mathbf { x } _ { t + 1 } ^ { f } \otimes \mathbf { x } _ { t + 1 } ^ { s } \right) + \left( \mathbf { x } _ { t + 1 } ^ { s } \otimes \mathbf { x } _ { t + 1 } ^ { f } \right) - \left( \mathbf { x } _ { t + 1 } ^ { ( 3 ) } \otimes \mathbf { x } _ { t + 1 } ^ { ( 3 ) } \right) } \\ & { = ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) \left( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \right) + ( \mathbf { h } _ { \mathbf { x } } \otimes \sigma \boldsymbol { \eta } ) \left( \mathbf { x } _ { t } ^ { f } \otimes \boldsymbol { \epsilon } _ { t + 1 } \right) + ( \sigma \boldsymbol { \eta } \otimes \mathbf { h } _ { \mathbf { x } } ) \left( \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { x } _ { t } ^ { f } \right) + ( \sigma \boldsymbol { \eta } \otimes \sigma \boldsymbol { \eta } ) \left( \boldsymbol { \epsilon } _ { t + 1 } \otimes \boldsymbol { \epsilon } _ { t + 1 } \right) } \\ & { + ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) \left( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { s } \right) + ( \mathbf { h } _ { \mathbf { x } } \otimes \frac 12 H _ { \mathbf { x x } } ) \left( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } \right) + ( \mathbf { h } _ { \mathbf { x } } \otimes \frac 12 h _ { \sigma \sigma } \sigma ^ { 2 } ) x _ { t } ^ { f } } \\ & { + ( \sigma \boldsymbol { \eta } \otimes h _ { \mathbf { x } } ) ( \boldsymbol { \epsilon } _ { t + 1 } \otimes x _ { t } ^ { s } ) + ( \sigma \boldsymbol { \eta } \otimes \frac 12 H _ { \mathbf { x x} } ) ( \boldsymbol { \epsilon } _ { t + 1 } \otimes x _ { t } ^ { f } \otimes x _ { t } ^ { f } ) + ( \sigma \boldsymbol { \eta } \otimes \frac 12 h _ { \sigma \sigma } \sigma ^ { 2 } ) e _ { t + 1 } } \\ & { + ( h _ { \mathbf { x } } \otimes h _ { \mathbf { x } } ) ( x _ { t } ^ { s } \otimes x _ { t } ^ { f } ) + ( \frac 12 H _ { x x } \otimes h _ { x } ) ( x _ { t } ^ { f } \otimes x _ { t } ^ { f } \otimes x _ { t } ^ { f } ) + ( \frac 12 h _ { \sigma \sigma } \sigma ^ { 2 } \otimes h _ { x} ) x _ { t } ^ { f } } \\ & { + ( h _ { x } \otimes \sigma \boldsymbol { \eta} ) ( x _ { t } ^ { s } \otimes e _ { t + 1 } ) + ( \frac 12 H _ { x x } \otimes \sigma \boldsymbol { \eta} ) ( x _ { t } ^ { f } \otimes x _ { t } ^ { f } \otimes e _ { t + 1 } ) + ( \frac 12 h _ { \sigma \sigma } \sigma ^ { 2 } \otimes \sigma \boldsymbol { \eta} ) e _ { t + 1 } } \\ & \\ & - ( ( h _ { x } \otimes h _ { x} ) ( x _ { t } ^ { ( 3 ) } \otimes x _ { t } ^ { ( 3 )} ) + ( h _ { x } \otimes \frac 12 H _ { x x} + \frac 12 H _ { x x} \otimes h _ { x} ) ( x _ { t } ^ { ( 3 ) } \otimes x _ { t } ^ { ( 3 ) } \otimes x _ { t } ^ {( 3 )} ) + \\ & + ( h _ { x } \otimes \frac 12 h _ { \sigma \sigma} + \frac 12 h _ { \sigma \sigma} \otimes h _ { x} ) ( x _ { t } ^ {( 3 )} \otimes {\sigma^ { 2 }} ) \\ & + ( h _ { x } \otimes n e r c o m b i a l y i n g r a d i o n g r a d i o n g r a d i o n g r a d i o n g r a d i o n g r a d i o n g r a d i o n g r a d i o n g r a d i o n g r a d i o n g r a d i o n g r a d i o n g r a d i o n g r a d i o n g r a d i o n , \\ & + ( n e r c o m b i a l y i n g r a d i o n g r a d i o n g r a d i o n g r a d i o n g r a d i o n g r a d i o n g r a d i o n g r a d i o n g r a d i o n g r a d i o n g r a d i o n , \\ & = ( h _ { x } \otimes h _ { x} ) ( x _ { t } ^ { f } \otimes x _ { t } ^ { f } - x _ { t } ^ {( 3 )} \otimes x _ { t } ^ {( 3 )} ) \\ & + ( h _ { x } \otimes s e r g e l y i n g r a d i o n g r a d i o n g r a d i o n g r a d i o n g r a d i o n g r a d i o n g r a d i o n g r a d i o n g r a d i o n g r a d i o n , \\ & + ( s e r g e l y i n g r a d i o n g r a d i o n g r a d i o n g r a d i o n g r a d i o n g r a d i o n g r a d i o n g r a d i o n , \\ & + ( h _ { x } \otimes h _ { x} ) ( x _ { t } ^ { f } \otimes x _ { t } ^ { s} + x _ { t } ^ { s} ) * e q u a l y i n g r a d i o n g r a d i o n g r a d i o n g r a d i o n , \\ & + ( h _ { x } \otimes s e r e l y i n g r a d i o n g r a d i o n g r a d i o n g r a d i o n , \\ & + ( h _ { x } \otimes s e r e l y i n g r a d i o n g r a d i o n , \\ & + ( h _ {\sigma} s e r e l y i n g r a d i o n , \\ & + ( h _ {\sigma} s e r e l y i n g r a d i o n , \\ & + ( h _ {\sigma} s e r e l y i n g r a d i o n , \\ & + ( h _ {\sigma} s e r e l y i n g r a d i o n , \\ & + ( h _ {\sigma} s e r e l y j e l y , \\ & + ( h _ {\sigma} s e r e l y j e l y , \\ & + ( h _ {\sigma} s e r e l y j e l y , \\ & + ( h _ {\sigma} s e r e l y j e l y , \\ & + ( h _ {\sigma} s e r e l y j e l y , \\ & + ( h _ {\sigma} s e r e l y j e l l y , \\ & + ( h _ {\sigma} s e r e l y j e l l y , \\ & + ( h _ {\sigma} s e r e l y j e l l y , \\ & + ( h _ {\sigma} s e r e l y j e l l y , \\ & + ( h _ {\sigma} s e r e l y j e l l y , \\ & + ( h _ {\sigma} s e r e l y j e l l y , \\ & + ( h _ {\sigma} s e r e l y j e l l y , \\ & + ( h _ {\sigma} s e r e l y j e l l y , \\ & + ( h _ {\sigma} s e r e l y j e l l y , \\ & + ( h _ {\sigma} s e r e l y j e l l y , \\ & - ( h _ {\sigma} s e r e l y j e l l y , \\ & - ( h _ {\sigma} s e r e l y j e l l y , \\ & - ( h _ {\sigma} s e r e l y j e l l y , \\ & - ( h _ {\sigma} s e r e l y j e l l y , \\ & - ( h _ {\sigma} s e r el y j e l l y , \\ & - ( h _ {\sigma} s e r e l y j e l l y , \\ & - ( h _ {\sigma} s e r e l y j e l l y , \\ & - ( h _ {\sigma} s e r e l y j e l l y , \\ & - ( h _ {\sigma} s e r e l y j e l l y , \\ & - (h u v a c k u m p u a c k u m p u a c k u m p u a c k u m p u a c k u m p u a c k u m p u a c k u m p u a c k u m p u a c k u m p u a c k u m p u a c k u m p u a c k u m p u a c k u m p u a c k u m p u a c k u m p u a . \\ & - ( h u v a c k u m p u a c k u m p u a c k u m p u a c k u m p u a c k u m p u a c k u m p u a c k u m p u a c k u m p u a c k u m p u a c k u m p u a c k u m p u a c k u m p u a c k u m p u a c k u m p w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w w W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W W WWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWW< fcel>Thus, The first value is the sum of all possible values. However, the second value is the sum of all possible values. However, the third value is the sum of all possible values. However, the fourth value is the sum of all possible values. However, the fifth value is the sum of all possible values. However, the sixth value is the sum of all possible values. However, the seventh value is the sum of all possible values. However, the eight value is the sum of all possible values. However, the nine value is the sum of all possible values. However, the ten number of non-zero numbers in each bin is not included in the data. However, the nine number of non-zero numbers in each bin is not included in the data. However, the nine number of non-zero numbers in each bin is not included in the data. However, the nine number of non-zero numbers in each bin is not included in the data. However, the nine number of non-zero numbers in each bin is not included in the data. However, the nine number of non-zero numbers in each bin is not included in the data. However, the nine number of non-zero number in each bin is not included in the data. However, the nine number of non-zero numbers in each bin is not included in the data. However, the nine number of non-zero numbers in each bin is not included in the data. However, the nine number of non-zero numbers in each bin is not included in the data. However, the nine number of non-zero numbers in each bin is not included in the data. However, the nine number of non-zero numbers in each bin isnot included in the data. However, the nine number of non-zero numbers in each bin is not included in the data. However, the nine number of non-zero numbers in each bin is not included in the data. However, the nine number of non-zero numbers in each bin is not included in the data. However, the nine number of non-zero numbers in each bin is not included in the data. However, the nine number of non-zero numbers in each bin is not included in the data; however, there are no other values that can be considered as “f” or “g” for some segments. However, there are no other values that can be considered as “e” for some segments. However, there are no other values that can be considered as “f” for some segments. However, there are no other values that can be considered as “g” for some segments. However, there are no other values that can be considered as “e” for some segments. However, there are no other values that can be considered as “f” for some segments. However, there are no other values that can be considered as “g” for some segments. However, there are no other values that can be considered as “e” for some segments. However, there are no other values that can be considered as “f” for one segment; however, there are no other values that can be considered as “e” for one segment; however, there are no other values that can be considered as “g” for one segment; however, there are no other values that can be considered as “e” for one segment; however, there are no other values that can be considered as “f” for one segment; however, there are no other values that can be considered as “g” for one segment; however, there are no other values that can be considered as “e” for one segment; however, there are no other values that can be considered as “f” for two segments; however, there are no other values that can be considered as “e” for one segment; however, there are no other values that can be considered as “g” for one segment; however, there are no other values that can be considered as “e” for one segment; however, there are no other values that can be considered as “f” for two segments; however, there are no other values that can be considered as “e” for one segment; however, there are no other values that can be considered as “g” for two segments; however, there are no other values that can be considered as “e” for two segments; however, there are no other values that can be considered as “f” for two segments; however, there are no other values that can be considered as “g” for two segments; however, there are no other values that can be considered as “e” for two segments; however, there are no other values that can be considered as “f” for two segments; however, there are no other values that can be considered as “g” for two segments; however, there are no other values that can be considered as “e” for two segments; however, there is not included in the data.< nl>\]

\[\begin{array} { l } + \left( \sigma \boldsymbol { \eta } \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } + \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \sigma ^ { 2 } \otimes \sigma \boldsymbol { \eta } \right) \boldsymbol { \epsilon } _ { t + 1 } \\ - \{ ( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } + \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ( \mathbf { x } _ { t } ^ { ( 3 ) } \otimes \mathbf { x } _ { t } ^ { ( 3 ) } \otimes \mathbf { x } _ { t } ^ { ( 3 ) } ) + \\ + ( \mathbf { h } _ { \mathbf { x } } \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } + \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \otimes \mathbf { h } _ { \mathbf { x } } ) ( \mathbf { x } _ { t } ^ { ( 3 ) } \otimes \sigma ^ { 2 } ) \\ + ( \mathbf { h } _ { \mathbf { x } } \otimes \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + 1 } ) ( \mathbf { x } _ { t } ^ { ( 3 ) } \otimes \sigma ) + ( \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } \otimes \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + 1 } ) ( \mathbf { x } _ { t } ^ { ( 3 ) } \otimes \mathbf { x } _ { t } ^ { ( 3 ) } \otimes \sigma ) + ( \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } \otimes \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + 1 } ) \sigma ^ { 3 } \\ + ( \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + 1 } \otimes \mathbf { h } _ { \mathbf { x } } ) ( \mathbf { x } _ { t } ^ { ( 3 ) } \otimes \sigma ) + ( \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + 1 } \otimes \frac { 1 } { 2 } \mathbf { H } _ { \mathbf { x x } } ) ( \mathbf { x } _ { t } ^ { ( 3 ) } \otimes \mathbf { x } _ { t } ^ { ( 3 ) } \otimes \sigma ) + ( \boldsymbol { \eta } \boldsymbol { \epsilon } _ { t + 1 } \otimes \frac { 1 } { 2 } \mathbf { h } _ { \sigma \sigma } ) \sigma ^ { 3 } \} \\ = ( \mathbf { h } _ { \mathbf { x } } \otimes \mathbf { h } _ { \mathbf { x } } ) ( \mathbf { x } _ { t } ^ { f } \otimes \mathbf { x } _ { t } ^ { f } - \mathbf { x } _ { t } ^ { ( 3 ) } \otimes \mathbf { x } _ { t } ^ { ( 3 ) } ) \\ + ( \mathbf { h } _ { \mathbf { x } } \otimes \sigma \boldsymbol { \eta} ) ( ( [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ ] ] ] ] ] ] ] ] ] ] ] ] ] ) + ( [ [ [ [ [ [ [ [ [ [ [ ] ] ] ] ] ] ] ] ] ] ) + ( [ [ [ [ [ [ [ [ [ ] ] ] ] ] ] ] ] ) + ( [ [ [ [ [ [ [ [ ] ] ] ] ] ] ] ) + ( [ [ [ [ [ [ [ ] ] ] ] ] ] ) + ( [ [ [ [ [ [ [ ] ] ] ] ] ] ) + ( [ [ [ [ [ [ I n s i g e a l l o d i s t e r m a l l o d i s t e r m a l l o d i s t e r m a l l o d i s t e r m a l l o d i s t e r m a l l o d i s t e r m a l l o d i s t e r m a l l o d i s t e r m a l l o d i s t e r m a l l o d i s t e r m o u s s o r d i s t e r m a l l o d i s t e r m o u s s o r d i s t e r m o u s s o r d i s t e r m o u s s o r d i s t e r m o u s s o r d i s t e r m o u s s o r d i s t e r m o u s s o r d i s t e r m o u s s o r d i s t e r m o u s s o r d i s t e r m o w i n g \\ = ( | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | + ( {\cal O} (\tau , t)) + ( {\cal O} (\tau , t)) + ( {\cal O} (\tau , t)) + ( {\cal O} (\tau , t)) + ( {\cal O} (\tau , t)) + ( {\cal O} (\tau , t)) + ( {\cal O} (\tau , t)) + ( {\cal O} (\tau , t)) + ( {\cal O} (\tau , t)) + ( {\cal O} (\tau , t)) + (\tau , t) + (\tau , t) - (\tau , t) - (\tau , t) - (\tau , t) - (\tau , t) - (\tau , t) - (\tau , t) - (\tau , t) - (\tau , t) - (\tau , t) - (\tau , t) - (\tau , t) - (\tau , t) - (\tau , t) - (\tau , t) - (\tau , t) - (\tau , t) - (\tau ,t) - (\tau , t) - (\tau , t) - (\tau , t) - (\tau , t) - (\tau , t) - (\tau , t) - (\tau , t) - (\tau , t) - (\tau , t) - (\tau , t) - (\tau , t) - (\tau , t) - (\tau , t) - (\tau , t) - (\tau , t) - (\tau , t) - (\textit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . - = ( {\cal H} _ {\mathrm{x}}) ( {\cal X} _ {\mathrm{x}}) ( {\cal X} _ {\mathrm{x}}) ( {\cal X} _ {\mathrm{x}}) ( {\cal X} _ {\mathrm{x}}) ( {\cal X} _ {\mathrm{x}}) ( {\cal X} _ {\mathrm{x}}) ( {\cal X} _ {\mathrm{x}}) ( {\cal X} _ {\mathrm{x}}) ( {\cal X} _ {\mathrm{x}}) ( {\cal X}_ {\mathrm{x}}) ( {\cal X}_ {\mathrm{x}}) ( {\cal X}_ {\mathrm{x}}) ( {\cal X}_ {\mathrm{x}}) ( {\cal X}_ {\mathrm{x}}) ( {\cal X}_ {\mathrm{x}}) ( {\cal X}_ {\mathrm{x}}) ( {\cal X}_ {\mathrm{x}}) ( {\cal X}_ {\mathrm{x}}) ( {\cal X}_ {\mathrm{x}}) ( {\cal X}_{\mathrm{x}}) ( {\cal X}_ {\mathrm{x}}) ( {\cal X}_ {\mathrm{x}}) ( {\cal X}_ {\mathrm{x}}) ( {\cal X}_ {\mathrm{x}}) ( {\cal X}_ {\mathrm{x}}) ( {\cal X}_ {\mathrm{x}}) ( {\cal X}_ {\mathrm{x}}) ( {\cal X}_ {\mathrm{x}}) ( {\cal X}_ {\mathrm{x}}) ( {\cal X}_{\mathrm{x}}) ( {\cal X}_{\mathrm{x}}) ( {\cal X}_{\mathrm{x}}) ( {\cal X}_{\mathrm{x}}) ( {\cal X}_{\mathrm{x}}) ( {\cal X}_{\mathrm{x}}) ( {\cal X}_{\mathrm{x}}) ( {\cal X}_{\mathrm{x}}) ( {\cal X}_{\mathrm{x}}) ( {\cal X}_{\mathrm{x}}) ( {\cal X}_\mathrm{x}) ( {\cal X}_\mathrm{x}) ( {\cal X}_\mathrm{x}) ( {\cal X}_\mathrm{x}) ( {\cal X}_\mathrm{x}) ( {\cal X}_\mathrm{x}) ( {\cal X}_\mathrm{x}) ( {\cal X}_\mathrm{x}) ( {\cal X}_\mathrm{x}) ( {\cal X}_\mathrm{x}) ( {\cal X}_\mathrm{x}) ( {\cal X}_\mathrm{x}) & = \\ = ({\bf H} _ {{\bf x}}) ({\bf H} _ {{\bf x}}) ({\bf X} _ {{\bf x}}) ({\bf X} _ {{\bf x}}) ({\bf X} _ {{\bf x}}) ({\bf X} _ {{\bf x}}) ({\bf X} _ {{\bf x}}) ({\bf X} _ {{\bf x}}) ({\bf X} _ {{\bf x}}) ({\bf X} _ {{\bf x}}) ({\bf X} _ b y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y y< fcel>+ (-50.074678797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797979797978000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000016666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666 8888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888 1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111< nl>\]

We know that , so . We clearly also have that . Similarly, , so . Finally,

, so . This means that . In conclusion we have as desired.

For the controls we have

\[\begin{array}{r c l} \mathbf {y} _ {t} ^ {r d} & = & \mathbf {g} _ {\mathbf {x}} \left(\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d}\right) + \frac {1}{2} \mathbf {G} _ {\mathbf {x x}} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {s} \otimes \mathbf {x} _ {t} ^ {f}\right) + \frac {1}{2} \mathbf {g} _ {\sigma \sigma} \sigma^ {2} + \frac {1}{6} \mathbf {g} _ {\sigma \sigma \sigma} \sigma^ {3} \\ & & + \frac {1}{6} \mathbf {G} _ {\mathbf {x x x}} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f}\right) + \frac {3}{6} \mathbf {g} _ {\sigma \sigma \mathbf {x}} \sigma^ {2} \mathbf {x} _ {t} ^ {f} \end{array}\]

Hence,

\[\begin{array}{r c l} \mathbf {y} _ {t} ^ {r d} - \mathbf {y} _ {t} ^ {(3)} & = & \mathbf {g} _ {\mathbf {x}} \left(\mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {r d} - \mathbf {x} _ {t} ^ {(3)}\right) + \frac {1}{2} \mathbf {G} _ {\mathbf {x x}} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} + \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {s} + \mathbf {x} _ {t} ^ {s} \otimes \mathbf {x} _ {t} ^ {f} - \mathbf {x} _ {t} ^ {(3)} \otimes \mathbf {x} _ {t} ^ {(3)}\right) \\ & & + \frac {1}{6} \mathbf {G} _ {\mathbf {x x x}} \left(\mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} \otimes \mathbf {x} _ {t} ^ {f} - \mathbf {x} _ {t} ^ {(3)} \otimes \mathbf {x} _ {t} ^ {(3)} \otimes \mathbf {x} _ {t} ^ {(3)}\right). \end{array}\]

Given that , and , we clearly have as desired.

15 The pruning schemes in Den Haan and De Wind (2012)

In the paper Haan & Wind (2012) suggest two pruning schemes for perturbation approximations beyond a second order approximation (see page 1490 in their paper). We shortly present each of these suggestions with special focus devoted to a third order approximation.

Throughout this section we adopt the notation in Haan & Wind (2012). That is all endogenous variables in the DSGE model are in and all exogenous shocks are in . The policy function is denoted by f (). Furthermore, we use the notation that is the j'th order term in a n'th order Taylor series expansion. Finally, let denote the stochastic steady state, i.e. where is the n'th order Taylor series expansion

15.1 First proposal

Let denote the stochastic steady state, i.e. where is the n'th order Taylor series expansion. The pruning scheme is as follows

\[\mathbf {x} _ {t} ^ {(j)} - \mathbf {x} _ {s t o c h} = \sum_ {i = 1} ^ {j} \mathbf {f} _ {n ^ {t h}} ^ {(i)} \left(\mathbf {x} _ {t - 1} ^ {(j - i + 1)} - \mathbf {x} _ {s t o c h}, \mathbf {z} _ {t} - \mathbf {z}, \sigma\right)\]

for .

Thus for a third order approximation, we have

\[\mathbf {x} _ {t} ^ {(1)} - \mathbf {x} _ {s t o c h} = \mathbf {f} _ {3 ^ {r d}} ^ {(1)} \left(\mathbf {x} _ {t - 1} ^ {(1)} - \mathbf {x} _ {s t o c h}, \mathbf {z} _ {t} - \mathbf {z}, \sigma\right)\]

\[\begin{array}{r c l} \mathbf {x} _ {t} ^ {(2)} - \mathbf {x} _ {s t o c h} & = & \mathbf {f} _ {3 ^ {r d}} ^ {(1)} \left(\mathbf {x} _ {t - 1} ^ {(2)} - \mathbf {x} _ {s t o c h}, \mathbf {z} _ {t} - \mathbf {z}, \sigma\right) \\ & & + \mathbf {f} _ {3 ^ {r d}} ^ {(2)} \left(\mathbf {x} _ {t - 1} ^ {(1)} - \mathbf {x} _ {s t o c h}, \mathbf {z} _ {t} - \mathbf {z}, \sigma\right) \end{array}\]

\[\begin{array}{r c l} \mathbf {x} _ {t} ^ {(3)} - \mathbf {x} _ {s t o c h} & = & \mathbf {f} _ {3 ^ {r d}} ^ {(1)} \left(\mathbf {x} _ {t - 1} ^ {(3)} - \mathbf {x} _ {s t o c h}, \mathbf {z} _ {t} - \mathbf {z}, \sigma\right) \\ & & + \mathbf {f} _ {3 ^ {r d}} ^ {(2)} \left(\mathbf {x} _ {t - 1} ^ {(2)} - \mathbf {x} _ {s t o c h}, \mathbf {z} _ {t} - \mathbf {z}, \sigma\right) \\ & & + \mathbf {f} _ {3 ^ {r d}} ^ {(3)} \left(\mathbf {x} _ {t - 1} ^ {(1)} - \mathbf {x} _ {s t o c h}, \mathbf {z} _ {t} - \mathbf {z}, \sigma\right) \end{array}\]

Thus, we see that in the expression for , we are squaring some second-order terms in , which therefore is of fourth order. This holds even when is considered as a constant.

15.2 Second proposal

Let denote the stochastic steady state, i.e. where is the n'th order Taylor series expansion. The pruning scheme is as follows:

\[\mathbf {x} _ {t} ^ {(1)} - \mathbf {x} _ {s t o c h} = \mathbf {f} _ {2 ^ {n d}} ^ {(1)} \left(\mathbf {x} _ {t - 1} ^ {(1)} - \mathbf {x} _ {s t o c h}, \mathbf {z} _ {t} - \mathbf {z}, \sigma\right)\]

\[\mathbf {x} _ {t} ^ {(2)} - \mathbf {x} _ {s t o c h} = \mathbf {f} _ {2 ^ {n d}} ^ {(1)} \left(\mathbf {x} _ {t - 1} ^ {(2)} - \mathbf {x} _ {s t o c h}, \mathbf {z} _ {t} - \mathbf {z}, \sigma\right) + \mathbf {f} _ {2 ^ {n d}} ^ {(2)} \left(\mathbf {x} _ {t - 1} ^ {(1)} - \mathbf {x} _ {s t o c h}, \mathbf {z} _ {t} - \mathbf {z}, \sigma\right)\]

\[\mathbf {x} _ {t} ^ {(j)} - \mathbf {x} _ {s t o c h} = \mathbf {f} _ {n ^ {t h}} ^ {(1)} \left(\mathbf {x} _ {t - 1} ^ {(j)} - \mathbf {x} _ {s t o c h}, \mathbf {z} _ {t} - \mathbf {z}, \sigma\right) + \sum_ {i = 2} ^ {j} \mathbf {f} _ {n ^ {t h}} ^ {(i)} \left(\mathbf {x} _ {t - 1} ^ {(j - 1)} - \mathbf {x} _ {s t o c h}, \mathbf {z} _ {t} - \mathbf {z}, \sigma\right)\]

for . Thus for a third order approximation, we have

\[\mathbf {x} _ {t} ^ {(1)} - \mathbf {x} _ {s t o c h} = \mathbf {f} _ {3 ^ {r d}} ^ {(1)} \left(\mathbf {x} _ {t - 1} ^ {(1)} - \mathbf {x} _ {s t o c h}, \mathbf {z} _ {t} - \mathbf {z}, \sigma\right)\]

\[\begin{array}{r c l} \mathbf {x} _ {t} ^ {(2)} - \mathbf {x} _ {s t o c h} & = & \mathbf {f} _ {3 ^ {r d}} ^ {(1)} \left(\mathbf {x} _ {t - 1} ^ {(2)} - \mathbf {x} _ {s t o c h}, \mathbf {z} _ {t} - \mathbf {z}, \sigma\right) \\ & & + \mathbf {f} _ {3 ^ {r d}} ^ {(2)} \left(\mathbf {x} _ {t - 1} ^ {(1)} - \mathbf {x} _ {s t o c h}, \mathbf {z} _ {t} - \mathbf {z}, \sigma\right) \end{array}\]

\[\begin{array}{r c l} \mathbf {x} _ {t} ^ {(3)} - \mathbf {x} _ {s t o c h} & = & \mathbf {f} _ {3 ^ {r d}} ^ {(1)} \left(\mathbf {x} _ {t - 1} ^ {(3)} - \mathbf {x} _ {s t o c h}, \mathbf {z} _ {t} - \mathbf {z}, \sigma\right) \\ & & + \mathbf {f} _ {3 ^ {r d}} ^ {(2)} \left(\mathbf {x} _ {t - 1} ^ {(2)} - \mathbf {x} _ {s t o c h}, \mathbf {z} _ {t} - \mathbf {z}, \sigma\right) \\ & & + \mathbf {f} _ {3 ^ {r d}} ^ {(3)} \left(\mathbf {x} _ {t - 1} ^ {(2)} - \mathbf {x} _ {s t o c h}, \mathbf {z} _ {t} - \mathbf {z}, \sigma\right) \end{array}\]

Thus, in the last term , we are taking the third power of all the second order effects, thus including terms up to sixth order. Again, this holds even when is considered as a constant.

16 A New Keynesian Model

16.1 Households

The dynamic optimization problem faced by the representative household is of the form:

\[\underset {c _ {t}, b _ {t}, h _ {t}, k _ {t + 1}, i _ {t} \forall t \geq 0} {M a x} V _ {t} = u \left(\frac {c _ {t} - b c _ {t - 1}}{(z _ {t} ^ {*}) ^ {\phi_ {4}}}, 1 - h _ {t}\right) - \beta \left(E _ {t} \left[ (- V _ {t + 1}) ^ {1 - \phi_ {3}} \right]\right) ^ {\frac {1}{1 - \phi_ {3}}}\]

\[\mathrm{St.} k _ {t + 1} = (1 - \delta) k _ {t} + i _ {t} - i _ {t} \frac {\kappa_ {1}}{2} \left(\frac {i _ {t}}{\Upsilon_ {t} z _ {t} ^ {*} I _ {S S}} - 1\right) ^ {2} - k _ {t} \frac {\kappa_ {2}}{2} \left(\frac {i _ {t}}{k _ {t}} - \frac {I _ {s s}}{K _ {s s}} \mu_ {\Upsilon , s s} \mu_ {z ^ {*}, s s}\right) ^ {2}\]

\[b _ {t} + c _ {t} + \frac {i _ {t}}{\Upsilon_ {t}} = \frac {b _ {t - 1} \exp \left\{r _ {t - 1} ^ {b} \right\}}{\pi_ {t}} + h _ {t} w _ {t} + r _ {t} ^ {k} k _ {t} + d i v _ {t} ^ {h}\]

\[c _ {t}, h _ {t}, k _ {t + 1}, i _ {t} \geq 0 \forall t \geq 0\]

As for the notation:

- consumption

- hours

- the deterministic trend in technology and hence consumption

- the value function

- the capital stock

- investments

- the nominal stochastic discount factor

- deposit in the financial intermediary in time period

- the continuously compounded net rate on deposits offered by the financial intermediary

- deterministic trend in the relative price of investments when measured in terms of consumption good units

- gross inflation of consumption good prices

- the wage level measured in consumption good units

- the rental rate for capital services sold to firms as measured in consumption good units

-

We follow Altig, Christiano, Eichenbaum & Linde (2011) and define , meaning that . The process for is assumed to evolve according to a deterministic tend and so does . That is

\[z _ {t + 1} \equiv z _ {t} \mu_ {z, t + 1}\]

where , and

\[\Upsilon_ {t + 1} \equiv \Upsilon_ {t} \mu_ {\Upsilon , t + 1}\]

where .

We allow for habit formation in consumption via . In the derivation, we allow this habit formation to be external or internal via the indicator function , where

\[1 _ {[ h a \_ i n ]} = \left\{ \begin{array}{l l} 1 & \text { for internal habits } \\ 0 & \text { for external habits } \end{array} \right..\]

To simplify the presentation we follow Rudebusch & Swanson (2012) and consider a finite number of states in each period. As in Rudebusch & Swanson (2012), the optimization problem is formulated as a Lagrange problem maximizing . The Lagranian is therefore given by:

\[\begin{array}{r l} & {\mathcal {L} = V _ {t} + \sum_ {l = 0} ^ {\infty} E _ {t} \beta^ {l} m _ {t + l} \left[ u \left(\frac {c _ {t + l} - b c _ {t - 1 + l}}{(z _ {t + l} ^ {*}) ^ {\phi_ {4}}}, 1 - h _ {t + l}\right) - \beta \left(E _ {t} \left[ (- V _ {t + l + 1}) ^ {1 - \phi_ {3}} \right]\right) ^ {\frac {1}{1 - \phi_ {3}}} - V _ {t + l} \right]} \\ & {\quad + E _ {t} \sum_ {l = 0} ^ {\infty} \beta^ {l} \lambda_ {t + l} [ \frac {b _ {t - 1 + l} \exp \{r _ {t - 1 + l} ^ {b} \}}{\pi_ {t + l}} + h _ {t + l} w _ {t + l} + r _ {t + l} ^ {k} k _ {t + l} + d i v _ {t + l} - b _ {t + l} - c _ {t + l} - \frac {i _ {t + l}}{\Upsilon_ {t + l}} ]} \\ & {\quad + E _ {t} \sum_ {l = 0} ^ {\infty} \beta^ {l} q _ {t + l} \lambda_ {t + l} \left[ (1 - \delta) k _ {t + l} + i _ {t + l} - i _ {t + l} \frac {\kappa_ {1}}{2} \left(\frac {i _ {t + l}}{\Upsilon_ {t + l} z _ {t + l} ^ {*} I _ {S S}} - 1\right) ^ {2} - k _ {t + l} \frac {\kappa_ {2}}{2} \left(\frac {i _ {t + l}}{k _ {t + l}} - \frac {I _ {s s}}{K _ {s s}} \mu_ {\Upsilon , s s} \mu_ {z ^ {*}, s s}\right) ^ {2} - k _ {t + 1 + l} \right]} \end{array}\]

That is, we introduce three lagrange multipliers:

- for the constraint on the utility function

- for the budget constraint

- for the capital accumulation equation

FOC (Kuhn-Tucker conditions)

We look for a solution in the interior, i.e.

\[\begin{array}{l} \text {1) Consumption, c_{t} :} \\ \frac {\partial \mathcal {L}}{\partial c _ {t}} = m _ {t} u _ {c} \left(\frac {c _ {t} - b c _ {t - 1}}{(z _ {t} ^ {*}) ^ {\phi_ {4}}}, 1 - h _ {t}\right) \frac {1}{(z _ {t} ^ {*}) ^ {\phi_ {4}}} - 1 _ {[ h a \_ i n ]} b \beta E _ {t} m _ {t + 1} u _ {c} \left(\frac {c _ {t + 1} - b c _ {t}}{(z _ {t + 1} ^ {*}) ^ {\phi_ {4}}}, 1 - h _ {t + 1}\right) \frac {1}{(z _ {t + 1} ^ {*}) ^ {\phi_ {4}}} - \lambda_ {t} = 0 \\ \Updownarrow \\ \lambda_ {t} = m _ {t} u _ {c} \left(\frac {c _ {t} - b c _ {t - 1}}{(z _ {t} ^ {*}) ^ {\phi_ {4}}}, 1 - h _ {t}\right) \frac {1}{(z _ {t} ^ {*}) ^ {\phi_ {4}}} - 1 _ {[ h a \_ i n ]} b \beta E _ {t m _ {t + 1}} u _ {c} \left(\frac {c _ {t + 1} - b c _ {t}}{(z _ {t + 1} ^ {*}) ^ {\phi_ {4}}}, 1 - h _ {t + 1}\right) \frac {1}{(z _ {t + 1} ^ {*}) ^ {\phi_ {4}}} \end{array}\]

2) Deposits, :

\[\begin{array}{l} \frac {\partial \mathcal {L}}{\partial b _ {t}} = E _ {t} \left[ \beta \lambda_ {t + 1} \frac {\exp \left\{r _ {t} ^ {b} \right\}}{\pi_ {t + 1}} - \lambda_ {t} \right] = 0 \\ \Downarrow \end{array}\]

\[\exp \left\{r _ {t} ^ {b} \right\} E _ {t} \left[ \beta \frac {\lambda_ {t + 1}}{\lambda_ {t}} \frac {1}{\pi_ {t + 1}} \right] = 1\]

3) The labor supply,

\[\frac {\partial \mathcal {L}}{\partial h _ {t}} = - m _ {t} u _ {1 - h} \left(\frac {c _ {t} - b c _ {t - 1}}{(z _ {t} ^ {*}) ^ {\phi_ {4}}}, 1 - h _ {t}\right) + \lambda_ {t} w _ {t} = 0\]

\[m _ {t} u _ {1 - h} \left(\frac {c _ {t} - b c _ {t - 1}}{(z _ {t} ^ {*}) ^ {\phi_ {4}}}, 1 - h _ {t}\right) = \lambda_ {t} w _ {t}\]

4) The physical capital stock, :

\[\frac {\partial \mathcal {L}}{\partial k _ {t + 1}} = \lambda_ {t} q _ {t} (- 1) + E _ {t} \beta \lambda_ {t + 1} [ r _ {t + 1} ^ {k} + q _ {t + 1} (1 - \delta)\]

This assumption is without loss of generality as shown by Epstein & Zin (1989).

\[\begin{array}{r l} & {- q _ {t + 1} \frac {\kappa_ {2}}{2} \left(\frac {i _ {t + 1}}{k _ {t + 1}} - \frac {I _ {s s}}{k _ {s s}} \mu_ {\Upsilon , s s} \mu_ {z ^ {*}, s s}\right) ^ {2} + q _ {t + 1} \kappa_ {2} \left(\frac {i _ {t + 1}}{k _ {t + 1}} - \frac {I _ {s s}}{k _ {s s}} \mu_ {\Upsilon , s s} \mu_ {z ^ {*}, s s}\right) \frac {i _ {i + 1}}{k _ {t + 1} ^ {2}} k _ {t + 1} ] = 0} \\ & \Updownarrow \\ & q _ {t} \lambda_ {t} = E _ {t} \beta \lambda_ {t + 1} \Bigg [ r _ {t + 1} ^ {k} + q _ {t + 1} (1 - \delta) - q _ {t + 1} \frac {\kappa_ {2}}{2} \left(\frac {i _ {t + 1}}{k _ {t + 1}} - \frac {I _ {s s}}{k _ {s s}} \mu_ {\Upsilon , s s} \mu_ {z ^ {*}, s s}\right) ^ {2} + q _ {t + 1} \kappa_ {\mathrm{2}} \left(\frac {i _ {t + 1}}{k _ {t + 1}} - \frac {I _ {s s}}{k _ {s s}} \mu_ {\Upsilon , s s} \mu_ {z ^ {*}, s s}\right) \frac {i _ {i + 1}}{k _ {t + 1}} \Bigg ] \end{array}\]

\[\begin{array}{l} \text {5) Investments, i_{t} :} \\ \frac {\partial \mathcal {L}}{\partial i _ {t}} = - \Upsilon_ {t} ^ {- 1} \lambda_ {t} + q _ {t} \lambda_ {t} \left(1 - \frac {\kappa_ {1}}{2} \left(\frac {i _ {t}}{\Upsilon_ {t} z _ {t} ^ {*} I _ {S S}} - 1\right) ^ {2} - \frac {i _ {t}}{\Upsilon_ {t} z _ {t} ^ {*} i _ {S S}} \kappa_ {1} \left(\frac {i _ {t}}{\Upsilon_ {t} z _ {t} ^ {*} I _ {S S}} - 1\right) - \kappa_ {2} \left(\frac {i _ {t}}{k _ {t}} - \frac {I _ {s s}}{k _ {s s}} \mu_ {\Upsilon , s s} \mu_ {z ^ {*}, s s}\right)\right) = 0 \\ \Updownarrow \\ 1 = q _ {t} \Upsilon_ {t} \left(1 - \frac {\kappa_ {1}}{2} \left(\frac {i _ {t}}{\Upsilon_ {t} z _ {t} ^ {*} I _ {S S}} - 1\right) ^ {2} - \frac {i _ {t}}{\Upsilon_ {t} z _ {t} ^ {*} I _ {S S}} \kappa_ {1} \left(\frac {i _ {t}}{\Upsilon_ {t} z _ {t} ^ {*} I _ {S S}} - 1\right) - \kappa_ {2} \left(\frac {i _ {t}}{k _ {t}} - \frac {I _ {s s}}{k _ {s s}} \mu_ {\Upsilon , s s} {\mu_ {z ^ {*}, s s}}\right)\right) \end{array}\]

6) The value function

\[\begin{array}{r l} & {\frac {\partial \mathcal {L}}{\partial V _ {t + 1} (s)} = m _ {t} (s) \left(- \beta \frac {1}{1 - \phi_ {3}} \left(E _ {t} \left[ (- V _ {t + 1}) ^ {1 - \phi_ {3}} \right]\right) ^ {\frac {1}{1 - \phi_ {3}} - \frac {1 - \phi_ {3}}{1 - \phi_ {3}}} (1 - \phi_ {3}) (- V _ {t + 1} (s)) ^ {- \phi_ {3}} (- 1) p r o b _ {t} (s)\right)} \\ & {\qquad - m _ {t + 1} (s) p r o b _ {t} (s) \beta = 0} \\ & {\Updownarrow} \end{array}\]

\[m _ {t + 1} (s) = m _ {t} (s) \left(E _ {t} \left[ (- V _ {t + 1}) ^ {1 - \phi_ {3}} \right]\right) ^ {\frac {\phi_ {3}}{1 - \phi_ {3}}} (- V _ {t + 1} (s)) ^ {- \phi_ {3}} \quad \mathrm{forallstates}\]

16.2 Firms

16.2.1 Final Good producers

The representative competitive consumption good producer chooses for to solve:

\[\max _ {y _ {t, i}} P _ {t} y _ {t} - \int_ {0} ^ {1} P _ {i, t} y _ {i, t} d i s. t.\]

\[y _ {t} = \left(\int_ {0} ^ {1} \left(y _ {i, t}\right) ^ {\frac {\eta - 1}{\eta}} d i\right) ^ {\frac {\eta}{\eta - 1}}.\]

As for the notation:

- denotes the output from firm at time

- the price of .

- output from the final good producers

-

The first order condition to the problem is given by

\[y _ {i, t} = \left(\frac {P _ {i , t}}{P _ {t}}\right) ^ {- \eta} y _ {t}.\]

To find the expression for the aggregate price level we use the zero-profit condition, i.e.

\[P _ {t} y _ {t} = \int_ {0} ^ {1} P _ {i, t} y _ {i, t} d i\]

\[= \int_ {0} ^ {1} P _ {i, t} \left(\frac {P _ {i , t}}{P _ {t}}\right) ^ {- \eta} y _ {t} d i\]

\[\begin{array}{r l} & = y _ {t} P _ {t} ^ {\eta} \int_ {0} ^ {1} (P _ {i, t}) ^ {1 - \eta} d i \\ \Updownarrow \end{array}\]

\[\begin{array}{l} P _ {t} ^ {1 - \eta} = \int_ {0} ^ {1} (P _ {i, t}) ^ {1 - \eta} d i \\ \Updownarrow \end{array}\]

\[P _ {t} = \left[ \int_ {0} ^ {1} (P _ {i, t}) ^ {1 - \eta} d i \right] ^ {\frac {1}{1 - \eta}}\]

16.2.2 Intermediate Good Producer

This section derives the first-order-conditions for the th firm's optimization problem. We start by deriving the equation for the real dividend payments :

\[d i v _ {i, t} \equiv \left[ \left(\frac {P _ {i , t}}{P _ {t}}\right) y _ {i, t} - r _ {t} ^ {k} k _ {i, t} - w _ {t} h _ {i, t} \right].\]

So the problem is

\[\underset {h _ {i, t}, k _ {i, t}, P _ {i, t}} {\text {Max}} \forall t \geq 0 E _ {t} \sum_ {l = 0} ^ {\infty} D _ {t, t + l} P _ {t + l} d i v _ {i, t + l}\]

\[\mathrm{St.} \quad \phi_ {i, t} \equiv \left(\frac {P _ {i , t}}{P _ {t}}\right) ^ {1 - \eta} y _ {t} - r _ {t} ^ {k} k _ {i, t} - w _ {t} h _ {i, t}\]

\[a _ {t} k _ {i, t} ^ {\theta} \left(z _ {t} h _ {i, t}\right) ^ {1 - \theta} = \left(\frac {P _ {i , t}}{P _ {t}}\right) ^ {- \eta} y _ {t}\]

a no-Ponzi-game condition

Here, we use the following notation:

- the rental rate for capital services sold to firms as measured in consumption good units

- the capital stock used by firm at time

- the wage level measured in consumption good units

- =hours used by firm i at time t

- stationary technology shocks

- non-stationary technology shocks

- the nominal stochastic discount factor, i.e.

The law of motion for is given by

\[\log a _ {t + 1} = \rho_ {a} \log a _ {t} + \sigma_ {a} \epsilon_ {a, t + 1}\]

where

\[\begin{array}{r l} & {\mathrm{Thelagrangianis}} \\ & {\mathcal {L} = E _ {t} \sum_ {l = 0} ^ {\infty} D _ {t, t + l} P _ {t + l} [ (\frac {P _ {i , t + l}}{P _ {t + l}}) ^ {1 - \eta} y _ {t + l} - r _ {t + l} ^ {k} k _ {i, t + l} - w _ {t + l} h _ {i, t + l} ]} \\ & {\qquad + E _ {t} \sum_ {l = 0} ^ {\infty} D _ {t, t + l} P _ {t + l} m c _ {i, t + l} [ a _ {t + l} k _ {i, t + l} ^ {\theta} (z _ {t + l} h _ {i, t + l}) ^ {1 - \theta} - (\frac {P _ {i , t + l}}{P _ {t + l}}) ^ {- \eta} y _ {t + l} ]} \end{array}\]

FOC (Kuhn-Tucker conditions)

We look for a solution in the interior, i.e. .

1) Demand for labor, :

\[\frac {\partial \mathcal {L}}{\partial h _ {i , t}} = P _ {t} \left(- w _ {t} + m c _ {i, t} z _ {t} a _ {t} (1 - \theta) k _ {i, t} ^ {\theta} (z _ {t} h _ {i, t}) ^ {- \theta}\right)\]

Since we get

\[m c _ {i, t} z _ {t} a _ {t} (1 - \theta) k _ {i, t} ^ {\theta} (z _ {t} h _ {i, t}) ^ {- \theta} = w _ {t}\]

2) Demand for capital, :

\[\frac {\partial \mathcal {L}}{\partial k _ {i , t}} = P _ {t} \left(- r _ {t} ^ {k} + m c _ {i, t} \theta a _ {t} k _ {i, t} ^ {\theta - 1} (z _ {t} h _ {i, t}) ^ {1 - \theta}\right)\]

Since we get

\[\begin{array}{r l} & {\mathrm{Since} F _ {t} > 0 \mathrm{weget}} \\ & {r _ {t} ^ {k} = \theta a _ {t} k _ {i, t} ^ {\theta - 1} (z _ {t} h _ {i, t}) ^ {1 - \theta}} \end{array}\]

3) The optimal price, :

We assume Calvo-pricing determined by , giving the probability of a firm not being allowed to change its price in a given period. Notice, that all the re-optimizing firms face the same problem, hence they all set the same price. We denote this price by . The non-optimizing firms let . That is, we have

\[\frac {P _ {i , t + l}}{P _ {t + l}} = \frac {\tilde {P} _ {t} \prod_ {i = 1} ^ {l} \pi_ {t + i - 1} ^ {x}}{P _ {t} \frac {P _ {t + l}}{P _ {t}}} = \frac {\tilde {P} _ {t} \prod_ {i = 1} ^ {l} \frac {\pi_ {t + i - 1} ^ {x}}{\pi_ {t + i}}}{P _ {t}}, \mathrm{where} \pi_ {t + l} \equiv \frac {P _ {t + l}}{P _ {t + l - 1}}.\]

If we only write the probability that the new price last forever, the Lagrangian reads

\[\begin{array}{r l} & {\mathcal {L} = E _ {t} \sum_ {l = 0} ^ {\infty} D _ {t, t + l} P _ {t + l} \alpha^ {l} [ (\frac {\tilde {P} _ {t}}{P _ {t}}) ^ {1 - \eta} \prod_ {i = 1} ^ {l} (\frac {\pi_ {t + i - 1} ^ {x}}{\pi_ {t + i}}) ^ {1 - \eta} y _ {t + l} - r _ {t + l} ^ {k} k _ {i, t + l} - w _ {t + l} h _ {i, t + l} ]} \\ & {\qquad + E _ {t} \sum_ {l = 0} ^ {\infty} D _ {t, t + l} P _ {t + l} \alpha^ {l} m c _ {i, t + l} [ a _ {t + l} k _ {i, t + l} ^ {\theta} (z _ {t + l} h _ {i, t + l}) ^ {1 - \theta} - (\frac {\tilde {P} _ {t}}{P _ {t}}) ^ {- \eta} \prod_ {i = 1} ^ {l} (\frac {\pi_ {t + i - 1} ^ {x}}{\pi_ {t + i}}) ^ {- \eta} y _ {t + l} ]} \end{array}\]

The first-order-condition is:

\[\begin{array}{r l} & {\frac {\partial \mathcal {L}}{\partial \tilde {P} _ {t}} = E _ {t} \sum_ {l = 0} ^ {\infty} D _ {t, t + l} P _ {t + l} \alpha^ {l} [ (1 - \eta) \tilde {P} _ {t} ^ {- \eta} P _ {t} ^ {\eta - 1} \prod_ {i = 1} ^ {l} \left(\frac {\pi_ {t + i - 1} ^ {\chi}}{\pi_ {t + i}}\right) ^ {1 - \eta} y _ {t + l}} \\ & {\quad + m c _ {i, t + l} \eta \tilde {P} _ {t} ^ {- \eta - 1} P _ {t} ^ {\eta} \prod_ {i = 1} ^ {l} \left(\frac {\pi_ {t + i - 1} ^ {\chi}}{\pi_ {t + i}}\right) ^ {- \eta} y _ {t + l} ]} \\ & {\quad = E _ {t} \sum_ {l = 0} ^ {\infty} D _ {t, t + l} P _ {t + l} \alpha^ {l} \left(\frac {\tilde {P} _ {t}}{P _ {t}}\right) ^ {- \eta} \prod_ {i = 1} ^ {l} \left(\frac {\pi_ {t + i - 1} ^ {\chi}}{\pi_ {t + i}}\right) ^ {- \eta} y _ {t + l} \left[ (1 - \eta) P _ {t} ^ {- 1} \prod_ {i = 1} ^ {l} \left(\frac {\pi_ {t + i - 1} ^ {\chi}}{\pi_ {t + i}}\right) + \eta \frac {m c _ {i , t + l}}{\tilde {P} _ {t}} \right]} \end{array}\]

\[= E _ {t} \sum_ {l = 0} ^ {\infty} D _ {t, t + l} P _ {t + l} \alpha^ {l} \left(\frac {\tilde {P} _ {t}}{P _ {t}}\right) ^ {- \eta} \prod_ {i = 1} ^ {l} \left(\frac {\pi_ {t + i - 1} ^ {\chi}}{\pi_ {t + i}}\right) ^ {- \eta} y _ {t + l} \left[ \frac {(\eta - 1)}{\eta} \frac {\tilde {P} _ {t}}{P _ {t}} \prod_ {i = 1} ^ {l} \left(\frac {\pi_ {t + i - 1} ^ {\chi}}{\pi_ {t + i}}\right) - m c _ {i, t + l} \right] \frac {- \eta}{\tilde {P} _ {t}}\]

since and , implies

\[E _ {t} \sum_ {l = 0} ^ {\infty} D _ {t, t + l} P _ {t + l} \alpha^ {l} \left(\frac {\tilde {P} _ {t}}{P _ {t}}\right) ^ {- \eta} \prod_ {i = 1} ^ {l} \left(\frac {\pi_ {t + i - 1} ^ {\chi}}{\pi_ {t + i}}\right) ^ {- \eta} y _ {t + l} \left[ \frac {(\eta - 1)}{\eta} \frac {\tilde {P} _ {t}}{P _ {t}} \prod_ {i = 1} ^ {l} \left(\frac {\pi_ {t + i - 1} ^ {\chi}}{\pi_ {t + i}}\right) - m c _ {i, t + l} \right] = 0\]

16.2.3 Marginal costs

We next show that marginal costs are identical across all firms, i.e. for all i. Note from the first order conditions for and that

\[\frac {m c _ {i , t} a _ {t} z _ {t} (1 - \theta) k _ {i , t} ^ {\theta} (z _ {t} h _ {i , t}) ^ {- \theta}}{m c _ {i , t} a _ {t} \theta k _ {i , t} ^ {\theta - 1} (z _ {t} h _ {i , t}) ^ {1 - \theta}} = \frac {w _ {t}}{r _ {t} ^ {k}}\]

↓

\[\frac {z _ {t} (1 - \theta)}{\theta k _ {i , t} ^ {- 1} (z _ {t} h _ {i , t})} = \frac {w _ {t}}{r _ {t} ^ {k}}\]

\[\frac {1 - \theta}{\theta} \frac {z _ {t} k _ {i , t}}{h _ {i , t}} = \frac {w _ {t}}{r _ {t} ^ {k}}\]

implying that must be constant with respect to . I.e.

\[\frac {k _ {i , t}}{h _ {i , t}} = c o n s _ {t}\]

\[k _ {i, t} = h _ {i, t} c o n s _ {t}\]

↓

\[\begin{array}{l} \int_ {0} ^ {1} k _ {i, t} d i = \int_ {0} ^ {1} h _ {i, t} c o n s _ {t} d i \\ \Updownarrow \end{array}\]

\[k _ {t} = h _ {t} c o n s _ {t}\]

\[c o n s _ {t} = k _ {t} / h _ {t}\]

Hence,

\[\begin{array}{l} m c _ {i, t} a _ {t} \theta k _ {i, t} ^ {\theta - 1} \left(z _ {t} h _ {i, t}\right) ^ {1 - \theta} = r _ {t} ^ {k} \\ \Updownarrow \end{array}\]

\[m c _ {i, t} a _ {t} \theta \left(\frac {k _ {i , t}}{h _ {i , t}}\right) ^ {\theta - 1} (z _ {t}) ^ {1 - \theta} = r _ {t} ^ {k}\]

\[m c _ {i, t} a _ {t} \theta \left(\frac {k _ {t}}{h _ {t}}\right) ^ {\theta - 1} (z _ {t}) ^ {1 - \theta} = r _ {t} ^ {k}\]

This shows that .

Hence, we can wright the first order condition for and as

\[z _ {t} a _ {t} (1 - \theta) \left(z _ {t} \frac {h _ {t}}{k _ {t}}\right) ^ {- \theta} m c _ {t} = w _ {t}\]

\[r _ {t} ^ {k} = \theta a _ {t} \left(z _ {t} \frac {h _ {t}}{k _ {t}}\right) ^ {1 - \theta}\]

16.2.4 The recursive representation of the price relation

We start by defining

\[\begin{array}{r l} & {x _ {t} ^ {1} \equiv E _ {t} \sum_ {l = 0} ^ {\infty} D _ {t, t + l} \alpha^ {l} y _ {t + l} m c _ {t + l} \left(\frac {\tilde {P} _ {t}}{P _ {t}}\right) ^ {- \eta - 1} \prod_ {i = 1} ^ {l} \left(\frac {\pi_ {t + i - 1} ^ {\chi}}{\pi_ {t + i} ^ {(1 + \eta) / \eta}}\right) ^ {- \eta}} \\ & {x _ {t} ^ {2} \equiv E _ {t} \sum_ {l = 0} ^ {\infty} D _ {t, t + l} \alpha^ {l} y _ {t + l} \left(\frac {\tilde {P} _ {t}}{P _ {t}}\right) ^ {- \eta} \prod_ {i = 1} ^ {l} \left(\frac {\pi_ {t + i - 1} ^ {\chi}}{\pi_ {t + i} ^ {\eta / (\eta - 1)}}\right) ^ {1 - \eta}} \end{array}\]

This implies that the first-order-condition for the price relation can be expressed as

\[\begin{array}{l} E _ {t} \sum_ {l = 0} ^ {\infty} D _ {t, t + l} P _ {t + l} \alpha^ {l} \left(\frac {\tilde {P} _ {t}}{P _ {t}}\right) ^ {- \eta} \prod_ {i = 1} ^ {l} \left(\frac {\pi_ {t + i - 1} ^ {\chi}}{\pi_ {t + i}}\right) ^ {- \eta} y _ {t + l} \left[ \frac {\eta - 1}{\eta} \frac {\tilde {P} _ {t}}{P _ {t}} \prod_ {i = 1} ^ {l} \left(\frac {\pi_ {t + i - 1} ^ {\chi}}{\pi_ {t + i}}\right) - m c _ {t + l} \right] = 0 \\ \Updownarrow \end{array}\]

\[\begin{array}{r l} & {{E _ {t} \sum_ {l = 0} ^ {\infty} D _ {t, t + l} P _ {t + l} \alpha^ {l} \left(\frac {\tilde {P} _ {t}}{P _ {t}}\right) ^ {- \eta} \prod_ {i = 1} ^ {l} \left(\frac {\pi_ {t + i - 1} ^ {x}}{\pi_ {t + i}}\right) ^ {- \eta} y _ {t + l} \frac {(\eta - 1)}{\eta} \frac {\tilde {P} _ {t}}{P _ {t}} \prod_ {i = 1} ^ {l} \left(\frac {\pi_ {t + i - 1} ^ {x}}{\pi_ {t + i}}\right)}} \\ & {{- E _ {t} \sum_ {l = 0} ^ {\infty} D _ {t, t + l} P _ {t + l} \alpha^ {l} \left(\frac {\tilde {P} _ {t}}{P _ {t}}\right) ^ {- \eta} \prod_ {i = 1} ^ {l} \left(\frac {\pi_ {t + i - 1} ^ {x}}{{\pi_ {t + i}}}\right) ^ {- \eta} y _ {t + l} m c _ {t + l} = 0}} \\ & {{\mathrm{个}}} \end{array}\]

\[\begin{array}{r l} & E _ {t} \sum_ {l = 0} ^ {\infty} D _ {t, t + l} \alpha^ {l} \tilde {P _ {t}} \left(\frac {\tilde {P _ {t}}}{P _ {t}}\right) ^ {- \eta} \prod_ {i = 1} ^ {l} \left(\frac {\pi_ {t + i - 1} ^ {\chi}}{\pi_ {t + i} ^ {\eta / (\eta - 1)}}\right) ^ {1 - \eta} y _ {t + l} \frac {(\eta - 1)}{\eta} \\ & - E _ {t} \sum_ {l = 0} ^ {\infty} D _ {t, t + l} \alpha^ {l} \tilde {P _ {t}} \left(\frac {\tilde {P _ {t}}}{P _ {t}}\right) ^ {- \eta - 1} \prod_ {i = 1} ^ {l} \left(\frac {\pi_ {t + i - 1} ^ {\chi}}{\pi_ {t + i} ^ {(1 + \eta) / \eta}}\right) ^ {- \eta} y _ {t + l} m c _ {t + l} = 0 \quad \mathrm{seebelow} \end{array}\]

since

\[\underbrace {E _ {t} \sum_ {l = 0} ^ {\infty} D _ {t , t + l} \alpha^ {l} y _ {t + l} \left(\frac {\tilde {P} _ {t}}{P _ {t}}\right) ^ {- \eta} \prod_ {i = 1} ^ {l} \left(\frac {\pi_ {t + i - 1} ^ {\chi}}{\pi_ {t + i} ^ {\eta / (\eta - 1)}}\right) ^ {1 - \eta}} _ {x _ {t} ^ {2}} \frac {(\eta - 1)}{\eta}\]

\[- \underbrace {E _ {t} \sum_ {l = 0} ^ {\infty} D _ {t , t + l} \alpha^ {l} y _ {t + l} m c _ {t + l} \left(\frac {\tilde {P} _ {t}}{P _ {t}}\right) ^ {- \eta - 1} \prod_ {i = 1} ^ {l} \left(\frac {\pi_ {t + i - 1} ^ {\chi}}{\pi_ {t + i} ^ {(1 + \eta) / \eta}}\right) ^ {- \eta}} _ {x _ {t} ^ {1}} = 0\]

↓

\[\begin{array}{l} x _ {t} ^ {2} \frac {(\eta - 1)}{\eta} - x _ {t} ^ {1} = 0 \\ \Updownarrow \end{array}\]

\[\eta x _ {t} ^ {1} + (1 - \eta) x _ {t} ^ {2} = 0\]

We used the fact that

\[P _ {t + l} \left(\frac {\tilde {P} _ {t}}{P _ {t}}\right) ^ {1 - \eta} \prod_ {i = 1} ^ {l} \left(\frac {\pi_ {t + i - 1} ^ {\chi}}{\pi_ {t + i}}\right) ^ {1 - \eta} = \tilde {P} _ {t} ^ {1 - \eta} \prod_ {i = 1} ^ {l} \left(\pi_ {t + i - 1} ^ {\chi}\right) ^ {1 - \eta} P _ {t + l} ^ {\eta} = \tilde {P} _ {t} \left(\frac {\tilde {P} _ {t}}{P _ {t}}\right) ^ {- \eta} \prod_ {i = 1} ^ {l} \left(\frac {\pi_ {t + i - 1} ^ {\chi}}{\pi_ {t + i} ^ {\eta / (\eta - 1)}}\right) ^ {1 - \eta}\]

\[P _ {t + l} \left(\frac {\tilde {P} _ {t}}{P _ {t}}\right) ^ {- \eta} \prod_ {i = 1} ^ {l} \left(\frac {\pi_ {t + i - 1} ^ {\chi}}{\pi_ {t + i}}\right) ^ {- \eta} = \tilde {P} _ {t} ^ {- \eta} \prod_ {i = 1} ^ {l} \left(\pi_ {t + i - 1} ^ {\chi}\right) ^ {- \eta} P _ {t + l} ^ {1 + \eta} = \tilde {P} _ {t} \left(\frac {\tilde {P} _ {t}}{P _ {t}}\right) ^ {- \eta - 1} \prod_ {i = 1} ^ {l} \left(\frac {\pi_ {t + i - 1} ^ {\chi}}{\pi_ {t + i} ^ {(1 + \eta) / \eta}}\right) ^ {- \eta}\]

The dynamic process for

We define . Therefore

\[\begin{array}{r l} {x _ {t} ^ {1}} & {= E _ {t} \sum_ {l = 0} ^ {\infty} D _ {t, t + l} \alpha^ {l} y _ {t + l} m c _ {t + l} \left(\frac {\tilde {P} _ {t}}{\tilde {P} _ {t}}\right) ^ {- \eta - 1} \prod_ {i = 1} ^ {l} \left(\frac {\pi_ {t + i - 1} ^ {\chi}}{\pi_ {t + i} ^ {(1 + \eta) / \eta}}\right) ^ {- \eta}} \\ & {= y _ {t} m c _ {i, t} \tilde {p} _ {t} ^ {- \eta - 1} + E _ {t} \sum_ {l = 1} ^ {\infty} D _ {t, t + l} \alpha^ {l} y _ {t + l} m c _ {t + l} \left(\frac {\tilde {P} _ {t}}{\tilde {P} _ {t}}\right) ^ {- \eta - 1} \prod_ {i = 1} ^ {l} \left(\frac {\pi_ {t + i - 1} \chi}{\pi_ {t + i} ^ {(1 + \eta) / \eta}}\right) ^ {- \eta}} \\ & {= y _ {t} m c _ {t} \tilde {p} _ {t} ^ {- \eta - 1} + \tilde {p} _ {t} ^ {- \eta - 1} E _ {t} \left[ \alpha \beta \frac {\lambda_ {t + 1}}{\lambda_ {t}} \tilde {p} _ {t + 1} ^ {\eta + 1} \left(\frac {\pi_ {t} ^ {\chi}}{\pi_ {t + 1}}\right) ^ {- \eta} x _ {t + 1} ^ {1} \right], \mathrm{seebelow}} \\ & {= y _ {t} m c _ {t} \tilde {p} _ {t} ^ {- \eta - 1} + E _ {t} \left[ \alpha \beta \frac {\lambda_ {t + 1}}{\lambda_ {t}} \left(\frac {\tilde {p} _ {t}}{\tilde {p} _ {t + 1}}\right) ^ {- \eta - 1} \left(\frac {\pi_ {t} ^ {\chi}}{\pi_ {t + 1}}\right) ^ {- \eta} x _ {t + 1} ^ {1} \right]} \end{array}\]

At the third equatility sign we use the fact that

\[x _ {t + 1} ^ {1} = E _ {t + 1} \sum_ {l = 0} ^ {\infty} D _ {t + 1, t + 1 + l} \alpha^ {l} y _ {t + 1 + l} m c _ {t + 1 + l} \left(\frac {\tilde {P} _ {t + 1}}{P _ {t + 1}}\right) ^ {- \eta - 1} \prod_ {i = 1} ^ {l} \left(\frac {\pi_ {t + i} ^ {\chi}}{\pi_ {t + 1 + i} ^ {(1 + \eta) / \eta}}\right) ^ {- \eta}\]

change of index,

\[x _ {t + 1} ^ {1} = \tilde {p} _ {t + 1} ^ {- \eta - 1} E _ {t + 1} \sum_ {j = 1} ^ {\infty} D _ {t + 1, t + j} \alpha^ {j - 1} y _ {t + j} m c _ {t + j} \prod_ {i = 1} ^ {j - 1} \left(\frac {\pi_ {t + i} ^ {\chi}}{\pi_ {t + 1 + i} ^ {(1 + \eta) / \eta}}\right) ^ {- \eta}\]

\[x _ {t + 1} ^ {1} \tilde {p} _ {t + 1} ^ {1 + \eta} \alpha D _ {t, t + 1} = E _ {t + 1} \sum_ {j = 1} ^ {\infty} D _ {t, t + j} \alpha^ {j} y _ {t + j} m c _ {t + j} \prod_ {i = 2} ^ {j} \left(\frac {\pi_ {t + i - 1} ^ {\chi}}{\pi_ {t + i} ^ {(1 + \eta) / \eta}}\right) ^ {- \eta}\]

↓

\[x _ {t + 1} ^ {1} \hat {p} _ {t + 1} ^ {1 + \eta} \alpha D _ {t, t + 1} \left(\frac {\pi_ {t} ^ {\chi}}{\pi_ {t + 1} ^ {(1 + \eta) / \eta}}\right) ^ {- \eta} = E _ {t + 1} \sum_ {j = 1} ^ {\infty} D _ {t, t + j} \alpha^ {j} y _ {t + j} m c _ {t + j} \prod_ {i = 1} ^ {j} \left(\frac {\pi_ {t + i - 1} ^ {\chi}}{\pi_ {t + i} ^ {(1 + \eta) / \eta}}\right) ^ {- \eta}\]

\[\begin{array}{l} x _ {t + 1} ^ {1} \hat {p} _ {t + 1} ^ {1 + \eta} \alpha \beta \frac {\lambda_ {t + 1}}{\lambda_ {t}} \frac {1}{\pi_ {t + 1}} \left(\frac {\pi_ {t} ^ {\chi}}{\pi_ {t + 1} ^ {(1 + \eta) / \eta}}\right) ^ {- \eta} = E _ {t + 1} \sum_ {j = 1} ^ {\infty} D _ {t, t + j} \alpha^ {j} y _ {t + j} m c _ {t + j} \prod_ {i = 1} ^ {j} \left(\frac {\pi_ {t + i - 1} ^ {\chi}}{\pi_ {t + i} ^ {(1 + \eta) / \eta}}\right) ^ {- \eta} \\ \mathrm{since} D _ {t, t + 1} = \beta \frac {\lambda_ {t + 1}}{\lambda_ {t}} \frac {1}{\pi_ {t + 1}} \\ \Updownarrow \end{array}\]

\[\begin{array}{l} x _ {t + 1} ^ {1} \tilde {p} _ {t + 1} ^ {1 + \eta} \alpha \beta \frac {\lambda_ {t + 1}}{\lambda_ {t}} \left(\frac {\pi_ {t} ^ {\chi}}{\pi_ {t + 1}}\right) ^ {- \eta} = E _ {t + 1} \sum_ {j = 1} ^ {\infty} D _ {t, t + j} \alpha^ {j} y _ {t + j} m c _ {t + j} \prod_ {i = 1} ^ {j} \left(\frac {\pi_ {t + i - 1} ^ {\chi}}{\pi_ {t + i} ^ {(1 + \eta) / \eta}}\right) ^ {- \eta} \\ \Downarrow \end{array}\]

\[\begin{array}{l} E _ {t} \left[ x _ {t + 1} ^ {1} \tilde {p} _ {t + 1} ^ {1 + \eta} \alpha \beta \frac {\lambda_ {t + 1}}{\lambda_ {t}} \left(\frac {\pi_ {t} ^ {\chi}}{\pi_ {t + 1}}\right) ^ {- \eta} \right] = E _ {t} \sum_ {j = 1} ^ {\infty} D _ {t, t + j} \alpha^ {j} y _ {t + j} m c _ {t + j} \prod_ {i = 1} ^ {j} \left(\frac {\pi_ {t + i - 1} ^ {\chi}}{\pi_ {t + i} ^ {(1 + \eta) / \eta}}\right) ^ {- \eta} \\ \text { due to the law of iterated expectations. } F _ {t} F _ {t + 1} [ \cdot ] = F _ {t} [ \cdot ]. \end{array}\]

due to the law of iterated expectations, .

\[\begin{array}{r l r} & {\mathrm{Thedynamicprocessfor} x _ {t} ^ {2}} \\ & {x _ {t} ^ {2} = E _ {t} \sum_ {l = 0} ^ {\infty} D _ {t, t + l} \alpha^ {l} y _ {t + l} \left(\frac {\tilde {P} _ {t}}{P _ {t}}\right) ^ {- \eta} \prod_ {i = 1} ^ {l} \left(\frac {\pi_ {t + i - 1} ^ {\chi}}{\pi_ {t + i} ^ {\eta / (\eta - 1)}}\right) ^ {1 - \eta}} \\ & {\quad = y _ {t} \tilde {p} _ {t} ^ {- \eta} + \tilde {p} _ {t} ^ {- \eta} E _ {t} \sum_ {l = 1} ^ {\infty} D _ {t, t + l} \alpha^ {l} y _ {t + l} \prod_ {i = 1} ^ {l} \left(\frac {\pi_ {t + i - 1} ^ {\chi}}{\pi_ {t + i} ^ {\eta / (\eta - 1)}}\right) ^ {1 - \eta}} & {\mathrm{since} \tilde {p} _ {t} = \frac {\tilde {P} _ {t}}{P _ {t}}} \\ & {\quad = y _ {t} \tilde {p} _ {t} ^ {- \eta} + \tilde {p} _ {t} ^ {- \eta} E _ {t} \left[ \alpha \beta \frac {\lambda_ {t + 1}}{\lambda_ {t}} \tilde {p} _ {t + 1} ^ {\eta} \left(\frac {\pi_ {t} ^ {\chi}}{\pi_ {t + 1}}\right) ^ {1 - \eta} x _ {t + 1} ^ {2} \right]} & {\mathrm{,seebelow}} \\ & {\quad = y _ {t} \tilde {p} _ {t} ^ {- \eta} + E _ {t} \left[ \alpha \beta \frac {\lambda_ {t + 1}}{\lambda_ {t}} \left(\frac {\tilde {p} _ {t}}{\tilde {p} _ {t + 1}}\right) ^ {- \eta} \left(\frac {\pi_ {t} ^ {\chi}}{\pi_ {t + 1}}\right) ^ {1 - \eta} x _ {t + 1} ^ {2} \right]} \end{array}\]

At the third equatility sign we use the fact that

\[\begin{array}{l} x _ {t + 1} ^ {2} = E _ {t + 1} \sum_ {l = 0} ^ {\infty} D _ {t + 1, t + 1 + l} \alpha^ {l} y _ {t + 1 + l} \left(\frac {\tilde {P} _ {t + 1}}{P _ {t + 1}}\right) ^ {- \eta} \prod_ {i = 1} ^ {l} \left(\frac {\pi_ {t + i} ^ {\chi}}{\pi_ {t + 1 + i} ^ {\eta / (\eta - 1)}}\right) ^ {1 - \eta} \\ \Updownarrow \text {change of index} j = 1 + l \end{array}\]

\[\begin{array}{r l} {{x _ {t + 1} ^ {2}}} & {{= \tilde {p} _ {t + 1} ^ {- \eta} E _ {t + 1} \sum_ {j = 1} ^ {\infty} D _ {t + 1, t + j} \alpha^ {j - 1} y _ {t + j} \prod_ {i = 1} ^ {j - 1} \left(\frac {\pi_ {t + i} ^ {\chi}}{\pi_ {t + 1 + i} ^ {\eta / (\eta - 1)}}\right) ^ {1 - \eta}}} \\ {{\mathrm{个}}} \end{array}\]

\[x _ {t + 1} ^ {2} \alpha D _ {t, t + 1} \tilde {p} _ {t + 1} ^ {\eta} = E _ {t + 1} \sum_ {j = 1} ^ {\infty} D _ {t, t + j} \alpha^ {j} y _ {t + j} \prod_ {i = 2} ^ {j} \left(\frac {\pi_ {t + i - 1} ^ {\chi}}{\pi_ {t + i} ^ {\eta / (\eta - 1)}}\right) ^ {1 - \eta}\]

\[x _ {t + 1} ^ {2} \alpha D _ {t, t + 1} \tilde {p} _ {t + 1} ^ {\eta} \left(\frac {\pi_ {t} ^ {\chi}}{\pi_ {t + 1} ^ {\eta / (\eta - 1)}}\right) ^ {1 - \eta} = E _ {t + 1} \sum_ {j = 1} ^ {\infty} D _ {t, t + j} \alpha^ {j} y _ {t + j} \prod_ {i = 1} ^ {j} \left(\frac {\pi_ {t + i - 1} ^ {\chi}}{\pi_ {t + i} ^ {\eta / (\eta - 1)}}\right) ^ {1 - \eta}\]

↑↓

\[\begin{array}{l} x _ {t + 1} ^ {2} \alpha \beta \frac {\lambda_ {t + 1}}{\lambda_ {t}} \frac {1}{\pi_ {t + 1}} \tilde {p} _ {t + 1} ^ {\eta} \left(\frac {\pi_ {t} ^ {x}}{\pi_ {t + 1} ^ {\eta / (\eta - 1)}}\right) ^ {1 - \eta} = E _ {t + 1} \sum_ {j = 1} ^ {\infty} D _ {t, t + j} \alpha^ {j} y _ {t + j} \prod_ {i = 1} ^ {j} \left(\frac {\pi_ {t + i - 1} ^ {x}}{\pi_ {t + i} ^ {\eta / (\eta - 1)}}\right) ^ {1 - \eta} \\ \mathrm{since} D _ {t, t + 1} = \beta \frac {\lambda_ {t + 1}}{\lambda_ {t}} \frac {1}{\pi_ {t + 1}} \\ \Updownarrow \end{array}\]

\[\begin{array}{l} x _ {t + 1} ^ {2} \alpha \beta \frac {\lambda_ {t + 1}}{\lambda_ {t}} \tilde {p} _ {t + 1} ^ {\eta} \left(\frac {\pi_ {t} ^ {\chi}}{\pi_ {t + 1}}\right) ^ {1 - \eta} = E _ {t + 1} \sum_ {j = 1} ^ {\infty} D _ {t, t + j} \alpha^ {j} y _ {t + j} \prod_ {i = 1} ^ {j} \left(\frac {\pi_ {t + i - 1} ^ {\chi}}{\pi_ {t + i} ^ {\eta / (\eta - 1)}}\right) ^ {1 - \eta} \\ \Downarrow \end{array}\]

\[E _ {t} \left[ x _ {t + 1} ^ {2} \alpha \beta \frac {\lambda_ {t + 1}}{\lambda_ {t}} \tilde {p} _ {t + 1} ^ {\eta} \left(\frac {\pi_ {t} ^ {x}}{\pi_ {t + 1}}\right) ^ {1 - \eta} \right] = E _ {t} \sum_ {j = 1} ^ {\infty} D _ {t, t + j} \alpha^ {j} y _ {t + j} \prod_ {i = 1} ^ {j} \left(\frac {\pi_ {t + i - 1} ^ {x}}{\pi_ {t + i} ^ {\eta / (\eta - 1)}}\right) ^ {1 - \eta}\]

due to the law of iterated expectations, .

16.3 Financial intermediary

The financial intermediary is owned by the household. Its is assumed that the financial intermediary invests deposits from households in short- and long-term government bonds and offers the household a non-stochastic return on its deposits of . Hence, the financial intermediary carries all the risk from investing in the bond market. The short-term bond is for simplicity assumed to be the one-period bond, whereas the maturity of the long-term bond is denoted by L > 1. The net-worth (measured in nominal terms) in period t of the financial intermediary is given by

\[\begin{array}{r c l} n _ {t} & = & n _ {t - 1} + (1 - \omega) (1 - P _ {t, 1}) b _ {t, 1} + \omega (1 - P _ {t, L}) b _ {t, L} \\ & & + (1 - \omega) (1 - P _ {t, 1 - 1}) b _ {t - 1, 1} + \omega (1 - P _ {t, L - 1}) b _ {t - 1, L} \\ & & - \exp \left\{r _ {t - 1} ^ {b} \right\} b _ {t - 1} + T _ {t}. \end{array}\]

where and denotes the fraction chosen by the financial intermediary invested in the long-term government bond. The components are:

- : net worth from previous time period

- : the amount of deposits and the amount spent on buying bonds today

- : income from selling bonds bought in time period .

- : paying out returns and repaying deposits from time period

- : net lump-sum transfers from the household (in nominal terms)

In relation to the law of motion for we note that in equilibrium, bonds are in zero net supply, i.e. . This implies that the law of motion for net worth reduces to

\[n _ {t} = n _ {t - 1} + T _ {t},\]

and we therefore do not need to include this equation when solving the model. Note that this is exactly the same as in the standard New Keynesian model where households only invest in the central bank account at the rate .

The behavior of the financial intermediary is solely determined by the deposit rate . To state its expression, let the ex ante holding period return on the kth bond be

\[h r _ {t, k} \equiv \mathbb {E} _ {t} \left[ \log P _ {t + 1, k - 1} - \log P _ {t, k} \right],\]

where is the nominal price in period t of a zero-coupon bond maturing in period . The excess holding period return is therefore . We then assume that the deposit rate is equal the ex ante holding period return on the invested bond portfolio, i.e.

\[\begin{array}{r c l} {r _ {t} ^ {b}} & \equiv & {(1 - \omega) \times h r _ {t, 1} + \omega \times h r _ {t, L}} \\ & = & {r _ {t} + \omega \times x h r _ {t, L}} \end{array}\]

because . Here, denotes the fraction chosen by the financial intermediary invested in the long-term government bond. In other words, the financial intermediary is simply a mutual fund trading government bonds.

We only need to describe how the financial intermediary prices government bonds. Given that the financial intermediary is owned by the household, we simply use their stochastic discount factor. That is

\[P _ {t, 1} = \frac {1}{\exp \{r _ {t} \}}\]

and

\[P _ {t, k} = E _ {t} \left[ \beta \frac {\lambda_ {t + 1}}{\lambda_ {t}} \frac {1}{\pi_ {t + 1}} P _ {t + 1, k - 1} \right]\]

for . Note then that the yield curve is then given by

\[r _ {t, k} = - \frac {1}{k} \log P _ {t, k}\]

Note finally that we do not explicitly model a portfolio problem for the financial intermediary but simply assume that it buys these bonds. Although, the considered deposit rate and the policy rate are both risk-free, the deposit rate will on average be higher than the policy rate (because the yield curve is upward sloping) and should therefore be preferred by the households to simply investing in the central bank.

16.4 The central bank

We assume a standard Taylor rule of the form

\[\begin{array}{r c l} \log \left(\frac {\exp \left\{r _ {t} \right\}}{\exp \left\{r _ {s s} \right\}}\right) & = & \rho_ {r} \log \left(\frac {\exp \left\{r _ {t - 1} \right\}}{\exp \left\{r _ {s s} \right\}}\right) \\ & & + (1 - \rho_ {r}) \left(\beta_ {\pi} \log \left(\frac {\pi_ {t}}{\pi_ {s s}}\right) + \beta_ {y} \log \left(\frac {y _ {t}}{z _ {t} ^ {*} Y _ {s s}}\right) + \beta_ {x h r} (x h r _ {t, L} - E [ x h r _ {t, L} ])\right) \end{array}\]

↓

\[\begin{array}{r c l} r _ {t} & = & r _ {s s} \left(1 - \rho_ {r}\right) + \rho_ {r} r _ {t - 1} + \left(1 - \rho_ {r}\right) \left(\beta_ {\pi} \log \left(\frac {\pi_ {t}}{\pi_ {s s}}\right) + \beta_ {y} \log \left(\frac {y _ {t}}{z _ {t} ^ {*} Y _ {s s}}\right)\right) \\ & & + \left(1 - \rho_ {r}\right) \beta_ {x h r} \left(x h r _ {t, L} - E \left[ x h r _ {t, L} \right]\right) \end{array}\]

where is the net one-period risk-free rate. Note that we remove the mean in by subtracting . When solving the model by the perturbation method, we approximate by

\[E \left[ x h r _ {t, L} \right] \approx E _ {t} \left[ (1 - \gamma) \sum_ {l = 0} ^ {\infty} \gamma^ {l} x h r _ {t + l, L} \right]\]

where . This is convenient because we have

\[\begin{array}{r c l} X _ {t, L} & \equiv & E _ {t} \left[ (1 - \gamma) \sum_ {l = 0} ^ {\infty} \gamma^ {l} x h r _ {t + l, L} \right] = (1 - \gamma) x h r _ {t, L} + E _ {t} \left[ (1 - \gamma) \sum_ {l = 1} ^ {\infty} \gamma^ {l} x h r _ {t + l, L} \right] \\ & = & (1 - \gamma) x h r _ {t, L} + \gamma E _ {t} [ X _ {t + 1, L} ] \end{array}\]

as

\[X _ {t + 1, L} = E _ {t + 1} \left[ (1 - \gamma) \sum_ {l = 0} ^ {\infty} \gamma^ {l} x h r _ {t + 1 + l, L} \right] = E _ {t + 1} \left[ (1 - \gamma) \sum_ {l = 1} ^ {\infty} \gamma^ {l - 1} x h r _ {t + l, L} \right]\]

↓

\[\gamma E _ {t + 1} \left[ X _ {t + 1, L} \right] = E _ {t + 1} \left[ (1 - \gamma) \sum_ {l = 1} ^ {\infty} \gamma^ {l} x h r _ {t + l, L} \right]\]

↓

\[\gamma E _ {t} \left[ E _ {t + 1} \left[ X _ {t + 1, L} \right] \right] = E _ {t} \left[ E _ {t + 1} \left[ (1 - \gamma) \sum_ {l = 1} ^ {\infty} \gamma^ {l} x h r _ {t + l, L} \right] \right]\]

↓

\[\gamma E _ {t} \left[ X _ {t + 1, L} \right] = E _ {t} \left[ (1 - \gamma) \sum_ {l = 1} ^ {\infty} \gamma^ {l} x h r _ {t + l, L} \right]\]

16.5 Aggregation

16.5.1 The goods market: final good producer

From the final good producer we have

\[y _ {i, t} = \left(\frac {P _ {i , t}}{P _ {t}}\right) ^ {- \eta} y _ {t}\]

\[a _ {t} k _ {i, t} ^ {\theta} \left(z _ {t} h _ {i, t}\right) ^ {1 - \theta} = \left(\frac {P _ {i , t}}{P _ {t}}\right) ^ {- \eta} y _ {t}\]

We next notice that:

1.

2.

3.

Doing the summation with respect to i we get

\[\begin{array}{l} \int_ {0} ^ {1} a _ {t} k _ {i, t} ^ {\theta} (z _ {t} h _ {i, t}) ^ {1 - \theta} d i = y _ {t} \underbrace {\int_ {0} ^ {1} \left(\frac {P _ {i , t}}{P _ {t}}\right) ^ {- \eta} d i} _ {s _ {t + 1}} \\ \Updownarrow \end{array}\]

\[\begin{array}{l} \int_ {0} ^ {1} a _ {t} h _ {i, t} \left(\frac {k _ {i , t}}{h _ {i , t}}\right) ^ {\theta} (z _ {t}) ^ {1 - \theta} d i = y _ {t} s _ {t + 1} \\ \Updownarrow \end{array}\]

\[\begin{array}{l} \int_ {0} ^ {1} a _ {t} h _ {i, t} \left(\text { constant }\right) ^ {\theta} \left(z _ {t}\right) ^ {1 - \theta} d i = y _ {t} s _ {t + 1} \\ \Updownarrow \end{array}\]

\[\begin{array}{l} a _ {t} \left(\text { constant }\right) ^ {\theta} \left(z _ {t}\right) ^ {1 - \theta} \int_ {0} ^ {1} h _ {i, t} d i = y _ {t} s _ {t + 1} \\ \Updownarrow \end{array}\]

\[\begin{array}{l} a _ {t} \left(\frac {k _ {t}}{h _ {t}}\right) ^ {\theta} (z _ {t}) ^ {1 - \theta} h _ {t} = y _ {t} s _ {t + 1} \\ \Updownarrow \end{array}\]

\[\begin{array}{l} a _ {t} \left(k _ {t}\right) ^ {\theta} \left(z _ {t}\right) ^ {1 - \theta} h _ {t} ^ {1 - \theta} = y _ {t} s _ {t + 1} \\ \Updownarrow \end{array}\]

\[a _ {t} (k _ {t}) ^ {\theta} (z _ {t} h _ {t}) ^ {1 - \theta} = y _ {t} s _ {t + 1}\]

\[\begin{array}{l} \text {and} \\ s _ {t + 1} \equiv \int_ {0} ^ {1} \left(\frac {P _ {i , t}}{P _ {t}}\right) ^ {- \eta} d i \\ = \underbrace {(1 - \alpha) \left(\frac {\tilde {P} _ {t}}{P _ {t}}\right) ^ {- \eta}} _ {\text {opt. in period t}} + \underbrace {(1 - \alpha) \alpha \left(\frac {\tilde {P} _ {t - 1} \pi_ {t - 1} ^ {\chi}}{P _ {t}}\right) ^ {- \eta}} _ {\text {opt. in period t - 1}} + \underbrace {(1 - \alpha) \alpha^ {2} \left(\frac {\tilde {P} _ {t - 2} \pi_ {t - 1} ^ {\chi} \pi_ {t - 2} ^ {\chi}}{P _ {t}}\right) ^ {- \eta}} _ {\text {opt. in period t - 2}} +... \\ (0 \text {indexation}) \end{array}\]

\[= (1 - \alpha) \sum_ {j = 0} ^ {\infty} \alpha^ {j} \left[ \frac {\tilde {P} _ {t - j}}{P _ {t}} \prod_ {s = 1} ^ {j} \pi_ {t - j - 1 + s} ^ {\chi} \right] ^ {- \eta}\]

\[(1 - \alpha) \tilde {p} _ {t} ^ {- \eta} + P _ {t} ^ {\eta} \sum_ {j = 1} ^ {\infty} \alpha^ {j} \left[ \tilde {P} _ {t - j} \prod_ {s = 1} ^ {j} \pi_ {t - j - 1 + s} ^ {\chi} \right] ^ {- \eta} \quad \mathrm{where} \tilde {p} _ {t - j} \equiv \frac {\tilde {P} _ {t - j}}{P _ {t}}\]

\[\begin{array}{r l r} & {= (1 - \alpha) \tilde {p} _ {t} ^ {- \eta} + P _ {t} ^ {\eta} \left(s _ {t - 1} \alpha P _ {t - 1} ^ {- \eta} \left(\pi_ {t - 1} ^ {\chi}\right) ^ {- \eta}\right)} & {\mathrm{,seebelow}} \\ & {= (1 - \alpha) \tilde {p} _ {t} ^ {- \eta} + \alpha \left(\frac {P _ {t} / P _ {t - 1}}{\pi_ {t - 1} ^ {\chi}}\right) ^ {\eta} s _ {t}} \end{array}\]

↓

\[s _ {t + 1} = (1 - \alpha) \tilde {p} _ {t} ^ {- \eta} + \alpha \left(\frac {\pi_ {t}}{\pi_ {t - 1} ^ {\chi}}\right) ^ {\eta} s _ {t}\]

\[\begin{array}{l} \text {We used that} \\ s _ {t} = (1 - \alpha) \sum_ {j = 0} ^ {\infty} \alpha^ {j} \left[ \frac {\tilde {P} _ {t - j - 1}}{P _ {t - 1}} \prod_ {s = 1} ^ {j} \pi_ {t - j - 2 + s} ^ {\chi} \right] ^ {- \eta} \\ \Updownarrow \text {change of index:} j + 1 = l \end{array}\]

\[\begin{array}{l} s _ {t} P _ {t - 1} ^ {- \eta} = (1 - \alpha) \sum_ {l = 1} ^ {\infty} \alpha^ {l - 1} \left[ \tilde {P} _ {t - l} \prod_ {s = 1} ^ {l - 1} \pi_ {t - l - 1 + s} ^ {\chi} \right] ^ {- \eta} \\ \Updownarrow \end{array}\]

\[s _ {t} P _ {t - 1} ^ {- \eta} \alpha (\pi_ {t - 1} ^ {\chi}) ^ {- \eta} = (1 - \alpha) \sum_ {l = 1} ^ {\infty} \alpha^ {l} [ \tilde {P} _ {t - l} \prod_ {s = 1} ^ {l} \pi_ {t - l - 1 + s} ^ {\chi} ] ^ {- \eta}\]

So the ressource constraint in the goods market is:

1)

\[2) s _ {t + 1} = (1 - \alpha) \tilde {p} _ {t} ^ {- \eta} + \alpha \left(\frac {\pi_ {t}}{\pi_ {t - 1} ^ {\chi}}\right) ^ {\eta} s _ {t}\]

16.6 The goods market: The relation between the optimale price and the price index

We start by noticing that the number of firms is by construct very large, so there is a fraction of firms reoptimize their prices and the remaining fraction using the indexation rule. This implies

\[\begin{array}{l} P _ {t} \equiv \left[ \int_ {0} ^ {1} P _ {i, t} ^ {1 - \eta} d i \right] ^ {\frac {1}{1 - \eta}} \\ \Updownarrow \\ P _ {t} ^ {1 - \eta} = \int_ {0} ^ {1} \left\{(1 - \alpha) \tilde {P} _ {t} ^ {1 - \eta} + \alpha \left(P _ {i, t - 1} \pi_ {t - 1} ^ {\chi}\right) ^ {1 - \eta} \right\} d i \\ \qquad = (1 - \alpha) \tilde {P} _ {t} ^ {1 - \eta} + \alpha \int_ {0} ^ {1} \left(P _ {i, t - 1} \pi_ {t - 1} ^ {\chi}\right) ^ {1 - \eta} d i \\ \qquad = (1 - \alpha) \tilde {P} _ {t} ^ {1 - \eta} + \alpha \left(\pi_ {t - 1} ^ {\chi}\right) ^ {1 - \eta} \underbrace {\int_ {0} ^ {1} P _ {i , t - 1} ^ {1 - \eta} d i} _ {P _ {t - 1} ^ {1 - \eta}} \\ \Updownarrow \\ P _ {t} ^ {1 - \eta} = (1 - \alpha) \tilde {P} _ {t} ^ {1 - \eta} + \alpha \left(P _ {t - 1} \pi_ {t - 1} ^ {\chi}\right) ^ {1 - \eta} \end{array}\]

\[1 = (1 - \alpha) \left(\frac {\tilde {P} _ {t}}{P _ {t}}\right) ^ {1 - \eta} + \alpha \left(\frac {P _ {t - 1}}{P _ {t}} \pi_ {t - 1} ^ {\chi}\right) ^ {1 - \eta}\]

\[1 = (1 - \alpha) \tilde {p} _ {t} ^ {1 - \eta} + \alpha \left(\frac {\pi_ {t - 1} ^ {\chi}}{\pi_ {t}}\right) ^ {1 - \eta}, \mathrm{since} \tilde {p} _ {t} \equiv \frac {\tilde {P} _ {t}}{P _ {t}}\]

16.7 The resource constraint

Summing the dividend payments from firms we have

\[\begin{array}{r c l} d i v _ {t} & = & \int d i v _ {i, t} d i \\ & = & \int \left[ \left(\frac {P _ {i , t}}{P _ {t}}\right) y _ {i, t} - r _ {t} ^ {k} k _ {i, t} - w _ {t} h _ {i, t} \right] d i \\ & = & \int \left(\frac {P _ {i , t}}{P _ {t}}\right) ^ {1 - \eta} y _ {t} d i - r _ {t} ^ {k} \int k _ {i, t} d i - w _ {t} \int h _ {i, t} d i \\ & = & \frac {1}{P _ {t} ^ {1 - \eta}} y _ {t} \int (P _ {i, t}) ^ {1 - \eta} d i - r _ {t} ^ {k} k _ {t} - w _ {t} h _ {t} \\ & = & y _ {t} - r _ {t} ^ {k} k _ {t} - w _ {t} h _ {t} \end{array}\]

because and . To get the dividends transferred to household, we need to subtract real government consumption equal . That is,

\[\begin{array}{r c l} {d i v _ {t} ^ {h}} & = & {d i v _ {t} - z _ {t} ^ {*} G _ {t}} \\ & = & {y _ {t} - z _ {t} ^ {*} G _ {t} - r _ {t} ^ {k} k _ {t} - w _ {t} h _ {t}.} \end{array}\]

Inserting this expression in the budget constraint for the households, we get

\[b _ {t} + c _ {t} + \frac {i _ {t}}{\Upsilon_ {t}} = \frac {b _ {t - 1} \exp \left\{r _ {t - 1} ^ {b} \right\}}{\pi_ {t}} + h _ {t} w _ {t} + r _ {t} ^ {k} k _ {t} + d i v _ {t} ^ {h}\]

↓

\[b _ {t} + c _ {t} + \frac {i _ {t}}{\Upsilon_ {t}} = \frac {b _ {t - 1} \exp \left\{r _ {t - 1} ^ {b} \right\}}{\pi_ {t}} + y _ {t} - z _ {t} ^ {*} G _ {t}\]

Focusing on the case where deposits are zero in equilibrium, we have

\[c _ {t} + \frac {i _ {t}}{\Upsilon_ {t}} + z _ {t} ^ {*} G _ {t} = y _ {t},\]

which constitutes our resource constraint for the economy. We finally assume that is exogenous given and evolves according to

\[\log \left(\frac {G _ {t + 1}}{G _ {s s}}\right) = \rho_ {G} \log \left(\frac {G _ {t}}{G _ {s s}}\right) + \sigma_ {G} \epsilon_ {G, t + 1}\]

with

16.8 The periodic utility function of the representative household

We assume that

\[u \left(\frac {c _ {t} - b c _ {t - 1}}{z _ {t} ^ {*}}, 1 - h _ {t}\right) = \frac {d _ {t}}{1 - \phi_ {2}} \left(\left(\frac {c _ {t} - b c _ {t - 1}}{(z _ {t} ^ {*}) ^ {\phi_ {4}}}\right) ^ {1 - \phi_ {2}} - (z _ {t} ^ {*}) ^ {(1 - \phi_ {4}) (1 - \phi_ {2})}\right) + d _ {t} \left(z _ {t} ^ {*}\right) ^ {(1 - \phi_ {4}) (1 - \phi_ {2})} \phi_ {0} \frac {(1 - h _ {t}) ^ {1 - \phi_ {1}}}{1 - \phi_ {1}}\]

\[u _ {c} \left(\frac {c _ {t} - b c _ {t - 1}}{z _ {t} ^ {*}}, 1 - h _ {t}\right) = d _ {t} \left[ \left(\frac {c _ {t} - b c _ {t - 1}}{(z _ {t} ^ {*}) ^ {\phi_ {4}}}\right) \right] ^ {- \phi_ {2}} \frac {1}{(z _ {t} ^ {*}) ^ {\phi_ {4}}}\]

\[u _ {1 - h} \left(\frac {c _ {t} - b c _ {t - 1}}{z _ {t} ^ {*}}, 1 - h _ {t}\right) = d _ {t} \left(z _ {t} ^ {*}\right) ^ {(1 - \phi_ {4}) (1 - \phi_ {2})} \phi_ {0} \left(1 - h _ {t}\right) ^ {- \phi_ {1}}\]

Note that we introduce a preference shock , having the following law of motion

\[\log d _ {t + 1} = \rho_ {d} \log d _ {t} + \sigma_ {d} \epsilon_ {d, t + 1}\]

with

16.9 Summarizing

The Households \( V_t = \frac{d_t}{1-\phi_2} \left( \left( \frac{c_t - bc_{t-1}}{(z_t^*)^{\phi_4}} \right)^{1-\phi_2} - (z_t^*)(^{1-\phi_4})(1-\phi_2) \right) + (z_t^*)(^{1-\phi_4})(1-\phi_2) d_t \phi_0 \frac{(1-h_t)^{1-\phi_1}}{1-\phi_1} - \beta \left( E_t \left[ (-V_{t+1})^{1-\phi_3} \right] \right)^{\frac{1}{1-\phi_3}} \) \( \lambda_t = m_t d_t \left( \frac{c_t - bc_{t-1}}{(z_t^*)^{\phi_4}} \right)^{-\phi_2} \frac{1}{(z_t^*)^{\phi_4}} - 1_{[ha\_in]} b \beta E_t m_{t+1} d_{t+1} \left( \frac{c_{t+1} - bc_t}{(z_{t+1}^*)^{\phi_4}} \right)^{-\phi_2} \frac{1}{(z_{t+1}^*)^{\phi_4}} \) \( q_t \lambda_t = E_t \beta \lambda_{t+1}[r_{t+1}^k + q_{t+1}(1-\delta) - q_{t+1} \frac{\kappa_2}{2} \left( \frac{i_{t+1}}{k_{t+1}} - \frac{I_{ss}}{k_{ss}} \mu_{\Upsilon,ss} \mu_{z*,ss} \right)^2 + q_{t+1} \kappa_2 \left( \frac{i_{t+1}}{k_{t+1}} - \frac{I_{ss}}{k_{ss}} \mu_{\Upsilon,ss} \mu_{z*,ss} \right) \frac{i_{t+1}}{k_{t+1}}] \) \( m_t (z_t^*)(^{1-\phi_4})(^{1-\phi_2}) d_t \phi_0 (1-h_t)^{-\phi_1} = \lambda_t w_t \) \( 1 = q_t \Upsilon_t \left( 1 - \frac{\kappa_1}{2} \left( \frac{i_i}{\Upsilon_t z_t^* I_{SS}} - 1 \right)^2 - \frac{i_t}{\Upsilon_t z_t^* i_{SS}} \kappa_1 \left( \frac{i_i}{\Upsilon_t z_t^* I_{SS}} - 1 \right) - \kappa_2 \left( \frac{i_i}{k_t} - \frac{I_{ss}}{k_{ss}} \mu_{\Upsilon,ss} \mu_{z*,ss} \right) \right) \) \( \lambda_t = \beta \exp\{ r_t^b \} E_t [ \frac{\lambda_{t+1}}{\pi_{t+1}} ] \) The Firms \( mc_t a_t z_t (1-\theta) \left( \frac{z_t h_t}{k_t} \right)^{-\theta} = w_t \) \( a_t mc_t \theta \left( \frac{z_t h_t}{k_t} \right)^{1-\theta} = r_t^k \) \( \frac{(\eta-1)x_t^2}{\eta} = y_t mc_t p̃_t^{-\eta-1} + E_t [\alpha\beta\frac{\lambda_{t+1}}{\lambda_t} (\frac{\tilde{p}_t}{p_{t+1}})^{-\eta-1} (\frac{\pi_t^\chi}{\pi_{t+1}})^{-\eta} \frac{(\eta-1)x_{t+1}^2}{\eta}] ] \) \( x_t^2 = y_t p̃_t^{-\eta} + E_t [\alpha\beta\frac{\lambda_{t+1}}{\lambda_t} (\frac{\tilde{p}_t}{p_{t+1}})^{-\eta} (\frac{\pi_t^\chi}{\pi_{t+1}})^{1-\eta} x_{t+1}^2] ] \) \( 1 = (1-\alpha) p̃_t^{1-\eta} + \alpha (\frac{\pi_{t-1}^\chi}{\pi_t})^{1-\eta} \) The Financial Intermediary \( r_t + \omega × xhr_{t,L}, where xhr_{t,L} \equiv E_t [\log(P_{t+1,L-1}/P_{t,L})] - r_t \) \( P_{t,1} = \frac{1}{\exp\{r_t\}} \) \( P_{t,k} = E_t [\beta\frac{\lambda_{t+1}}{\lambda_t} \frac{1}{\pi_{t+1}} P_{t+1,k-1}] for k = 2,3,...,K \) The Central Bank \( r_t = r_{ss}(1-\rho_r) + \rho_r r_{t-1} + (1-\rho_r) (\beta_\pi \log(\frac{\pi_t}{\pi_{ss}}) + \beta_y \log(\frac{y_t}{z_t^* Y_{ss}})) + (1-\rho_r) \beta_{xhr}(xhr_{t,L}-X_{t,L}) X_{t,L} = (1-\gamma) xhr_{t,L} + \gamma E_t [X_{t+1,L}] \) Other relations \( a_t k_t^\theta(z_t h_t)^{1-\theta} = y_t s_{t+1} s_{t+1} = (1-\alpha) p̃_t^{-\eta} + \alpha (\frac{\pi_t}{\pi_{t-1}^\chi})^\eta s_t k_{t+1} = (1-\delta) k_t + i_t - i_t \frac{\kappa_1}{2} (\frac{i_i}{\Upsilon_t z_t^* I_{SS}} - 1)^2 - \frac{\kappa_2}{2} (\frac{i_i}{k_t} - \frac{I_{ss}}{k_{ss}})\mu_{\Upsilon,ss}\mu_{z*,ss})^2 k_t y_t = c_t + Y_t^{-1}i_t + z_t^* G_t z_t^* \equiv Y_t^{-\theta/(-\theta)} z_t and μ_{z*,t} \equiv μ_Y,t^{\theta/(1-\theta)} μ_z,t \) Exogenous processes \( log(\mu_z,t) = log(\mu_z,ss) and z_{t+1} \equiv z_t μ_z,t+1 (i.e. a deterministic trend) \\ \( log(\mu_Y,t) = logμ_Y,ss and Y_{t+1} \equiv Y_t μ_Y,t+1 (i.e. a deterministic trend) \\ \( loga_{t+1} = ρ_a loga_t + σ_a ε_a,t+1 \\ \( log(\frac{G_{t+1}}{G_{ss}}) = ρ_G log(\frac{G_t}{G_{ss}}) + σ_G ε_G,t+1 \\ \( logd_{t+1} = ρ_d logd_t + σ_d ε_d,t+1 \)

16.10 A transformation of the DSGE model

Here we seek a transformation of the economy in such a way that the transformed economy is stationary in the sense that it only contains stationary variables.

We propose and verify the following transformation, where capital letters in general are used to denote the transformed variable:

\[\begin{array} { r l } & C _ { t } \equiv \frac { c _ { t } } { z _ { t } ^ { * } } \\ & R _ { t } ^ { k } \equiv \Upsilon _ { t } r _ { t } ^ { k } \\ & Q _ { t } \equiv \Upsilon _ { t } q _ { t } \\ & I _ { t } \equiv \frac { i _ { t } } { \Upsilon _ { t } z _ { t } ^ { * } } \quad \Longrightarrow \quad i _ { t } = I _ { t } \Upsilon _ { t } \left( \Upsilon _ { t } ^ { \frac { \theta } { 1 - \theta } } z _ { t } \right) = I _ { t } \Upsilon _ { t } ^ { \frac { 1 - \theta + \theta } { 1 - \theta } } z _ { t } = I _ { t } \Upsilon _ { t } ^ { \frac { 1 } { 1 - \theta } } z _ { t } \\ & \mathrm{So,~} \mu _ { i , t } = \mu _ { \Upsilon , t } \mu _ { z ^ { * } , t } = \mu _ { \Upsilon , t } ^ { \frac { 1 } { 1 - \theta } } \mu _ { z , t } \\ & W _ { t } \equiv \frac { w _ { t } } { z _ { t } ^ { * } } \\ & Y _ { t } \equiv \frac { y _ { t } } { z _ { t } ^ { * } } \quad \Longrightarrow \quad \mu _ { y , s s } = \mu _ { z ^ { * } , s s } \\ & K _ { t + 1 } \equiv \frac { k _ { t + 1 } } { \Upsilon _ { t } ^ { \frac { 1 } { 1 - \theta } } z _ { t } } = \frac { k _ { t + 1 } } { \Upsilon _ { t } ^ { \frac { 1 } { 1 - \theta } } \Upsilon _ { t } ^ { \frac { - \theta } { 1 - \theta } } z _ { t } ^ { * } } = \frac { k _ { t + 1 } } { \Upsilon _ { t } z _ { t } ^ { * } } \qquad \mathrm{since~} z _ { t } ^ { * } \equiv \Upsilon _ { t } ^ { \frac { \theta } { 1 - \theta } } z _ { t } \\ & \widetilde V _ { t } \equiv \frac { V _ { t } } { ( z _ { t } ^ { * }) ^ { ( 1 - \phi _ { 4 } ) ( 1 - \phi _ { 2 } ) } } \\ & \Lambda _ { t } \equiv \frac { \lambda _ { t } } { m _ { t } ( z _ { t } ^ { *}) ^ { - \phi _ { 2 } ( 1 - \phi _ { 4 } ) - \phi _ { 4 } } } \\ & \Downarrow \\ & \mu _ { \lambda , t + 1 } \equiv \frac { \lambda _ { t + 1 } } { \lambda _ { t } } \\ & = \frac { \Lambda _ { t + 1 } m _ { t + 1 } ( z _ { t + 1 } ^ { * }) ^ { - \phi _ { 2 } ( 1 - \phi _ { 4 } ) - \phi _ { 4 } } } { \Lambda _ { t } m _ { t } ( z _ { t } ^ { *}) ^ { - \phi _ { 2 } ( 1 - \phi _ { 4 } ) - \phi _ { 4 } } } \\ & \Updownarrow \\ & \mu _ { \lambda , t + 1 } = \frac { \Lambda _ { t + 1 } } { \Lambda _ { t } } \mu _ { z ^ { * }, t + 1 } ^ { - \phi _ { 2 } ( 1 - \phi _ { 4 } ) - \phi _ { 4 } } ( E _ { t } [ ( - V _ { t + 1 } ) ^ { 1 - \phi _ { 3 } } ] ) ^ {\frac {\phi _ { 3 }}{ 1 - \phi _ { 3} }} ( - V _ { t + 1 } ) ^ { - \phi _ { 3 } } . \end{array}\]

\[X _ {t} ^ {2} = \frac {x _ {t} ^ {2}}{z _ {t} ^ {*}}\]

All remaining variables are stationary and do not need to be transformed. We now verify that given these variables we can transform our DSGE model from a non-stationary to a stationary economy.

\[\begin{array}{r l}&{\mathrm{EQ1}}\\&{V _ {t} = \frac {d _ {t}}{1 - \phi_ {2}} \left(\left(\frac {c _ {t} - b c _ {t - 1}}{(z _ {t} ^ {*}) ^ {\phi_ {4}}}\right) ^ {1 - \phi_ {2}} - (z _ {t} ^ {*}) ^ {(1 - \phi_ {4}) (1 - \phi_ {2})}\right) + (z _ {t} ^ {*}) ^ {(1 - \phi_ {4}) (1 - \phi_ {2})} d _ {t} \phi_ {0} \frac {(1 - h _ {t}) ^ {1 - \phi_ {1}}}{1 - \phi_ {1}} - \beta \left(E _ {t} \left[ (- V _ {t + 1}) ^ {1 - \phi_ {3}} \right]\right) ^ {\frac {1}{1 - \phi_ {3}}}}\\&{\Downarrow}\\&{\frac {V _ {t}}{(z _ {t} ^ {*}) ^ {(1 - \phi_ {4}) (1 - \phi_ {2})}} = \frac {d _ {t}}{1 - \phi_ {2}} \left(\left(\frac {c _ {t} - b c _ {t - 1}}{(z _ {t} ^ {*}) ^ {\phi_ {4}} (z _ {t} ^ {*}) ^ {(1 - \phi_ {4})}}\right) ^ {1 - \phi_ {2}} - 1\right) + d _ {t} \phi_ {0} \frac {(1 - h _ {t}) ^ {1 - \phi_ {1}}}{1 - \phi_ {1}} - \beta \left( \right.E _ {t} \left[ \right. (- \frac {V _ {t + 1}}{(z _ {t} ^ {*}) ^ {(1 - \phi_ {4}) (1 - \phi_ {2})}} \frac {(z _ {t + 1} ^ {*}) ^ {(1 - \phi_ {4}) (1 - \phi_ {2})}}{(z _ {t + 1} ^ {*}) ^ {(1 - \phi_ {4}) (1 - \phi_ {2})}}\left. \right) ^ {1 - \phi_ {3}} \left. \right]\left. \right) ^ {\frac {1}{1 - \phi_ {3}}}}\\&{\Updownarrow}\end{array}\]

\[\begin{array}{l}\widetilde {V} _ {t} = \frac {d _ {t}}{1 - \phi_ {2}} \left(\left(\frac {c _ {t} - b c _ {t - 1}}{(z _ {t} ^ {*}) ^ {\phi_ {4}} (z _ {t} ^ {*}) ^ {(1 - \phi_ {4})}}\right) ^ {1 - \phi_ {2}} - 1\right) + d _ {t} \phi_ {0} \frac {(1 - h _ {t}) ^ {1 - \phi_ {1}}}{1 - \phi_ {1}} - \beta \left( \right.E _ {t} \left[ \right. (- \widetilde {V _ {t + 1}} \mu_ {z ^ {*}, t + 1} ^ {(1 - \phi_ {4}) (1 - \phi_ {2})}\left. \right) ^ {1 - \phi_ {3}} \left. \right]\left. \right) ^ {\frac {1}{1 - \phi_ {3}}}\\\Updownarrow\end{array}\]

\[\begin{array}{l}\widetilde {V} _ {t} = \left[ \frac {d _ {t}}{1 - \phi_ {2}} \left(\left(\frac {c _ {t} - b c _ {t - 1}}{z _ {t} ^ {*}}\right) ^ {1 - \phi_ {2}} - 1\right) + d _ {t} \phi_ {0} \frac {(1 - h _ {t}) ^ {1 - \phi_ {1}}}{1 - \phi_ {1}} \right] - \beta \left( \right.E _ {t} \left[ \right. (- \widetilde {V _ {t + 1}} \mu_ {z ^ {*}, t + 1} ^ {(1 - \phi_ {4}) (1 - \phi_ {2})}\left. \right) ^ {1 - \phi_ {3}} \left. \right]\left. \right) ^ {\frac {1}{1 - \phi_ {3}}}\\\Updownarrow\end{array}\]

\[\widetilde {V} _ {t} = \left[ \frac {d _ {t}}{1 - \phi_ {2}} \left(\left(C _ {t} - b C _ {t - 1} \mu_ {z ^ {*}, t} ^ {- 1}\right) ^ {1 - \phi_ {2}} - 1\right) + d _ {t} \phi_ {0} \frac {(1 - h _ {t}) ^ {1 - \phi_ {1}}}{1 - \phi_ {1}} \right] - \beta \left(E _ {t} \left[ \left(- \widetilde {V _ {t + 1}} \mu_ {z ^ {*}, t + 1} ^ {(1 - \phi_ {4}) (1 - \phi_ {2})}\right) ^ {1 - \phi_ {3}} \right]\right) ^ {\frac {1}{1 - \phi_ {3}}}\]

We note because only has a deterministic trend, then we can simplify the above to:

\[\widetilde {V} _ {t} = \left[ \frac {d _ {t}}{1 - \phi_ {2}} \left(\left(C _ {t} - b C _ {t - 1} \mu_ {z ^ {*}, t} ^ {- 1}\right) ^ {1 - \phi_ {2}} - 1\right) + d _ {t} \phi_ {0} \frac {(1 - h _ {t}) ^ {1 - \phi_ {1}}}{1 - \phi_ {1}} \right] - \beta \mu_ {z ^ {*}, s s} ^ {(1 - \phi_ {4}) (1 - \phi_ {2})} \left(E _ {t} \left[ \left(- \widetilde {V _ {t + 1}}\right) ^ {1 - \phi_ {3}} \right]\right) ^ {\frac {1}{1 - \phi_ {3}}}\]

EQ 2

\[\lambda_ {t} = d _ {t} m _ {t} \left(\frac {c _ {t} - b c _ {t - 1}}{(z _ {t} ^ {*}) ^ {\phi_ {4}}}\right) ^ {- \phi_ {2}} \frac {1}{(z _ {t} ^ {*}) ^ {\phi_ {4}}} - 1 _ {[ h a \_ i n ]} b \beta E _ {t} \left[ m _ {t + 1} d _ {t + 1} \left(\frac {c _ {t + 1} - b c _ {t}}{(z _ {t + 1} ^ {*}) ^ {\phi_ {4}}}\right) ^ {- \phi_ {2}} \frac {1}{(z _ {t + 1} ^ {*}) ^ {\phi_ {4}}} \right]\]

↓

\[\begin{array}{r l} & {\frac {\lambda_ {t}}{m _ {t} (z _ {t} ^ {*}) ^ {- \phi_ {2} (1 - \phi_ {4}) - \phi_ {4}}} = d _ {t} \left(\frac {c _ {t} - b c _ {t - 1}}{(z _ {t} ^ {*}) ^ {\phi_ {4}}}\right) ^ {- \phi_ {2}} \frac {1}{(z _ {t} ^ {*}) ^ {\phi_ {4}}} \frac {1}{(z _ {t} ^ {*}) ^ {- \phi_ {2} (1 - \phi_ {4}) - \phi_ {4}}} - 1 _ {[ h a \_ i n ]} b \beta E _ {t} \left[ \frac {m _ {t + 1}}{m _ {t}} d _ {t + 1} \left(\frac {c _ {t + 1} - b c _ {t}}{(z _ {t + 1} ^ {*}) ^ {\phi_ {4}}}\right) ^ {- \phi_ {2}} \frac {1}{(z _ {t + 1} ^ {*}) ^ {\phi_ {4}}} \frac {1}{(z _ {t} ^ {*}) ^ {- \phi_ {2} (1 - \phi_ {4}) - \phi_ {4}}} \right]} \\ & {\Updownarrow} \end{array}\]

\[\begin{array}{r l} & {\Lambda_ {t} = d _ {t} \left(\frac {c _ {t} - b c _ {t - 1}}{(z _ {t} ^ {*}) ^ {\phi_ {4}} (z _ {t} ^ {*}) ^ {(1 - \phi_ {4})}}\right) ^ {- \phi_ {2}} \frac {1}{(z _ {t} ^ {*}) ^ {\phi_ {4}}} \frac {1}{(z _ {t} ^ {*}) ^ {- \phi_ {4}}} - 1 _ {[ h a _ {-} i n ]} b \beta E _ {t} [ \frac {m _ {t + 1}}{m _ {t}} d _ {t + 1} \left(\frac {c _ {t + 1} - b c _ {t}}{(z _ {t + 1} ^ {*}) ^ {\phi_ {4}}}\right) ^ {- \phi_ {2}} \frac {1}{(z _ {t + 1} ^ {*}) ^ {\phi_ {4}}}} \\ & {\qquad \frac {(z _ {t + 1} ^ {*}) ^ {- \phi_ {2} (1 - \phi_ {4}) - \phi_ {4}}}{(z _ {t + 1} ^ {*}) ^ {- \phi_ {2} (1 - \phi_ {4}) - \phi_ {4}}} \frac {1}{(z _ {t} ^ {*}) ^ {- \phi_ {2} (1 - \phi_ {4}) - \phi_ {4}}} ]} \end{array}\]

\[\Lambda_ {t} = d _ {t} \left(C _ {t} - b C _ {t - 1} \mu_ {z ^ {*}, t} ^ {- 1}\right) ^ {- \phi_ {2}} - 1 _ {[ h a \_ i n ]} b \beta E _ {t} \left[ \frac {m _ {t + 1}}{m _ {t}} d _ {t + 1} \left(\frac {c _ {t + 1} - b c _ {t}}{\left(z _ {t + 1} ^ {*}\right) ^ {\phi_ {4}}}\right) ^ {- \phi_ {2}} \left(\mu_ {z ^ {*}, t + 1}\right) ^ {- \phi_ {2} (1 - \phi_ {4}) - \phi_ {4}} \frac {1}{\left(z _ {t + 1} ^ {*}\right) ^ {- \phi_ {2} (1 - \phi_ {4})}} \right]\]

↓

\[\Lambda_ {t} = d _ {t} \left(C _ {t} - b C _ {t - 1} \mu_ {z ^ {*}, t} ^ {- 1}\right) ^ {- \phi_ {2}} - 1 _ {[ h a \_ i n ]} b \beta E _ {t} \left[ \frac {m _ {t + 1}}{m _ {t}} d _ {t + 1} \left(\frac {c _ {t + 1} - b c _ {t}}{\left(z _ {t + 1} ^ {*}\right) ^ {\phi_ {4}} \left(z _ {t + 1} ^ {*}\right) ^ {(1 - \phi_ {4})}}\right) ^ {- \phi_ {2}} \left(\mu_ {z ^ {*}, t + 1}\right) ^ {- \phi_ {2} (1 - \phi_ {4}) - \phi_ {4}} \right]\]

↓

\[\Lambda_ {t} = d _ {t} \left(C _ {t} - b C _ {t - 1} \mu_ {z ^ {*}, t} ^ {- 1}\right) ^ {- \phi_ {2}} - 1 _ {[ h a \_ i n ]} b \beta E _ {t} \left[ \frac {m _ {t + 1}}{m _ {t}} d _ {t + 1} \left(C _ {t + 1} - b C _ {t} \mu_ {z ^ {*}, t + 1} ^ {- 1}\right) ^ {- \phi_ {2}} \left(\mu_ {z ^ {*}, t + 1}\right) ^ {- \phi_ {2} (1 - \phi_ {4}) - \phi_ {4}} \right]\]

\[\begin{array}{l} \text { and using } m _ {t + 1} = m _ {t} \left(E _ {t} \left[ (- V _ {t + 1}) ^ {1 - \phi_ {3}} \right]\right) ^ {\frac {\phi_ {3}}{1 - \phi_ {3}}} (- V _ {t + 1} (s)) ^ {- \phi_ {3}} \\ \Updownarrow \end{array}\]

\[\begin{array}{r} \Lambda_ {t} = d _ {t} \left(C _ {t} - b C _ {t - 1} \mu_ {z ^ {*}, t} ^ {- 1}\right) ^ {- \phi_ {2}} - 1 _ {[ h a _ {-} i n ]} b \beta E _ {t} \{\left[ \left(E _ {t} \left[ (- V _ {t + 1}) ^ {1 - \phi_ {3}} \right]\right) ^ {\frac {\phi_ {3}}{1 - \phi_ {3}}} (- V _ {t + 1} (s)) ^ {- \phi_ {3}} \right] \\ d _ {t + 1} \left(C _ {t + 1} - b C _ {t} \mu_ {z ^ {*}, t + 1} ^ {- 1}\right) ^ {- \phi_ {2}} (\mu_ {z ^ {*}, t + 1}) ^ {- \phi_ {2} (1 - \phi_ {4}) - \phi_ {4}} \} \end{array}\]

↓

\[\begin{array}{r} \Lambda_ {t} = d _ {t} \left(C _ {t} - b C _ {t - 1} \mu_ {z ^ {*}, t} ^ {- 1}\right) ^ {- \phi_ {2}} - 1 _ {[ h a \_ i n ]} b \beta E _ {t} \{\left[ \left(\frac {\left[ E _ {t} \left[ (- V _ {t + 1}) ^ {1 - \phi_ {3}} \right] \right] ^ {\frac {1}{1 - \phi_ {3}}}}{- V _ {t + 1} (s)}\right) ^ {\phi_ {3}} \right] \\ d _ {t + 1} \left(C _ {t + 1} - b C _ {t} \mu_ {z ^ {*}, t + 1} ^ {- 1}\right) ^ {- \phi_ {2}} \left(\mu_ {z ^ {*}, t + 1}\right) ^ {- \phi_ {2} (1 - \phi_ {4}) - \phi_ {4}} \} \end{array}\]

\[\begin{array}{r} \Lambda_ {t} = d _ {t} \left(C _ {t} - b C _ {t - 1} \mu_ {z ^ {*}, t} ^ {- 1}\right) ^ {- \phi_ {2}} - 1 _ {[ h a _ {-} i n ]} b \beta E _ {t} \{\left[ \left(\frac {\left[ E _ {t} \left[ (- V _ {t + 1}) ^ {1 - \phi_ {3}} \right] \right] ^ {\frac {1}{1 - \phi_ {3}}}}{- V _ {t + 1} (s)} \frac {\left(z _ {t + 1} ^ {*}\right) ^ {(1 - \phi_ {4}) (1 - \phi_ {2})}}{\left(z _ {t + 1} ^ {*}\right) ^ {(1 - \phi_ {4}) (1 - \phi_ {2})}}\right) ^ {\phi_ {3}} \right] \\ d _ {t + 1} \left(C _ {t + 1} - b C _ {t} \mu_ {z ^ {*}, t + 1} ^ {- 1}\right) ^ {- \phi_ {2}} \left(\mu_ {z ^ {*}, t + 1}\right) ^ {- \phi_ {2} (1 - \phi_ {4}) - \phi_ {4}} \} \end{array}\]

\[\begin{array}{r} \Lambda_ {t} = d _ {t} \left(C _ {t} - b C _ {t - 1} \mu_ {z ^ {*}, t} ^ {- 1}\right) ^ {- \phi_ {2}} - 1 _ {[ h a _ {-} i n ]} b \beta E _ {t} \{\left[ \left(\frac {\left[ E _ {t} \left[ \left(- \widetilde {V _ {t + 1}}\right) ^ {1 - \phi_ {3}} \right] \right] ^ {\frac {1}{1 - \phi_ {3}}}}{- \widetilde {V _ {t + 1}} (s)}\right) ^ {\phi_ {3}} \right] \\ d _ {t + 1} \left(C _ {t + 1} - b C _ {t} \mu_ {z ^ {*}, t + 1} ^ {- 1}\right) ^ {- \phi_ {2}} \left(\mu_ {z ^ {*}, t + 1}\right) ^ {- \phi_ {2} (1 - \phi_ {4}) - \phi_ {4}} \} \end{array}\]

because is deterministic

\[\begin{array} { r l } & { \mathbf { E Q 3 } } \\ & { q _ { t } \lambda _ { t } = E _ { t } \beta \lambda _ { t + 1 } [ r _ { t + 1 } ^ { k } + q _ { t + 1 } ( 1 - \delta ) } \\ & { \qquad - q _ { t + 1 } \frac { \kappa _ { 2 } } { 2 } \left( \frac { i _ { t + 1 } } { k _ { t + 1 } } - \frac { I _ { s s } } { k _ { s s } } \mu _ { \Upsilon , s s } \mu _ { z ^ { * } , s s } \right) ^ { 2 } + q _ { t + 1 } \kappa _ { 2 } \left( \frac { i _ { t + 1 } } { k _ { t + 1 } } - \frac { I _ { s s } } { k _ { s s } } \mu _ { \Upsilon , s s } \mu _ { z ^ { * } , s s } \right) \frac { i _ { t + 1 } } { k _ { t + 1 } } ] } \\ & \Updownarrow \\ & { \Upsilon _ { t } q _ { t } = E _ { t } \beta \mu _ { \lambda , t + 1 } [ \Upsilon _ { t } r _ { t + 1 } ^ { k } + \Upsilon _ { t } q _ { t + 1 } ( 1 - \delta ) } \\ & { \qquad - \Upsilon _ { t } q _ { t + 1 } \frac { \kappa _ { 2 } } { 2 } \left( \frac { i _ { t + 1 } } { k _ { t + 1 } } - \frac { I _ { s s } } { k _ { s s } } \mu _ { \Upsilon , s s } \mu _ { z ^ { * } , s s } \right) ^ { 2} + \Upsilon _ { t } q _ { t + 1 } \kappa _ { 2 } \left( \frac { i _ { t + 1 } } { k _ { t + 1 } } - \frac { I _ { s s } } { k _ { s s } } \mu _ { \Upsilon , s s } \mu _ { z ^ { * } , s s } \right) \frac { i _ { t + 1 } }{ k _ { t + 1 } } ] } \\ & \Updownarrow \\ & { Q _ { t } = E _ { t } \beta \mu _ { \lambda , t + 1 } [ \Upsilon _ { t } \frac { \Upsilon _ { t + 1 } } { \Upsilon _ { t + 1 } } r _ { t + 1 } ^ { k } + \Upsilon _ { t } \frac { \Upsilon _ { t + 1 } } { \Upsilon _ { t + 1 } } q _ { t + 1 } ( 1 - \delta ) } \\ & \qquad - \Upsilon _ { t } \frac { \Upsilon _ { t + 1 } } { \Upsilon _ { t + 1 } } q _ { t + 1 } \frac { \kappa _ { 2 } } { 2 } \left( \frac { i _ { t + 1 } } { k _ { t + 1 } } - \frac { I _ { s s } } { k _ { s s } } \mu _ { \Upsilon , s s } \mu _ { z ^ { * } , s s } \right) ^ { 3 } + \Upsilon _ { t } \frac { \Upsilon _ { t + 1 } } { \Upsilon _ { t + 1 } } q _ { t + 1 } \kappa _ { 2 } \left( \frac { i _ { t + 1 } } { k _ { t + 1 } } - \frac { I _ { s s } } { k _ { s s } } \mu _ { \Upsilon , s s } \mu _ { z ^ { * } , s s } \right) \frac { i _ { t + 1 } to k _ { t + 1} ] } \\ & \Updownarrow \\ & { Q _ { t } = E _ { t } \beta \mu _ { \lambda , t + 1 } [ \frac { \Upsilon _ { t } } { \Upsilon _ { t + 1 } } R _ { t + 1 } ^ { k } + \frac { \Upsilon _ { t } } { \Upsilon _ { t + 1 } } Q _ { t + 1 } ( 1 - \delta ) } \\ & \qquad - \frac { \Upsilon _ { t } } { \Upsilon _ { t + 1 } } Q _ { t + 1 } \frac { \kappa _ { 2 } } { 2 } \left( \frac { i _ { t + 1 }}{ k _ { t + 1 }} - \frac { I _ { s s }}{ k _ { s s }} \mu _ { \Upsilon , s s } \mu _ { z ^ { * } , s s } \right) ^ { 2 } + \frac { \Upsilon _ { t } } { \Upsilon _ { t + 1 } } Q _ { t + 1 } \kappa _ { 2 } \left( \frac { i _ { t + 1 }}{ k _ { t + 1 }} - \frac { I _ { s s }}{ k _ { s s }} \mu _ { \Upsilon , s s } \mu _ { z ^ {*} , s s } \right) \frac { i _ { t + 1 }}{ k _ { t + 1} ] } \\ & {\Updownarrow \\ & Q _ { t } = E _ { t } \beta \mu _ { \lambda , t + 1 } [ \mu _ { \Upsilon , t + 1 } ^ { - 1 } R _ { t + 1 } ^ { k } + \mu _ { \Upsilon , t + 1 } ^ { - 1} Q _ { t + 1 } ( 1 - \delta ) . } \end{array}\]

\[\begin{array}{r l} & {- \mu_ {\Upsilon , t + 1} ^ {- 1} Q _ {t + 1} \frac {\kappa_ {2}}{2} \left(\frac {i _ {t + 1}}{k _ {t + 1}} - \frac {I _ {s s}}{k _ {s s}} \mu_ {\Upsilon , s s} \mu_ {z ^ {*}, s s}\right) ^ {2} + \mu_ {\Upsilon , t + 1} ^ {- 1} Q _ {t + 1} \kappa_ {2} \left(\frac {i _ {t + 1}}{k _ {t + 1}} - \frac {I _ {s s}}{k _ {s s}} \mu_ {\Upsilon , s s} \mu_ {z ^ {*}, s s}\right) \frac {i _ {t + 1}}{k _ {t + 1}} ]} \\ & {\mathrm{Nownotethat}} \\ & {\frac {i _ {t + 1}}{k _ {t + 1}} = \frac {i _ {t + 1}}{k _ {t + 1}} \frac {\Upsilon_ {t} z _ {t} ^ {*}}{\Upsilon_ {t} z _ {t} ^ {*}} \frac {\Upsilon_ {t + 1} z _ {t + 1} ^ {*}}{\Upsilon_ {t + 1} z _ {t + 1} ^ {*}} = \frac {i _ {t} / (\Upsilon_ {t + 1} z _ {t + 1} ^ {*})}{k _ {t + 1} / (\Upsilon_ {t} z _ {t} ^ {*})} \frac {1}{\Upsilon_ {t} z _ {t} ^ {*}} \frac {\Upsilon_ {t + 1} z _ {t + 1} ^ {*}}{1} = \frac {I _ {t + 1}}{K _ {t + 1}} \mu_ {\Upsilon , t + 1} \mu_ {z ^ {*}, t + 1}} \\ & {\Updownarrow} \end{array}\]

\[\begin{array}{r l} & Q _ {t} = E _ {t} \frac {\beta \mu_ {\lambda , t + 1}}{\mu_ {\Upsilon , t + 1}} [ R _ {t + 1} ^ {k} + Q _ {t + 1} (1 - \delta) \\ & \qquad - Q _ {t + 1} \frac {\kappa_ {2}}{2} \left(\frac {I _ {t + 1}}{K _ {t + 1}} \mu_ {\Upsilon , t + 1} \mu_ {z ^ {*}, t + 1} - \frac {I _ {s s}}{k _ {s s}} \mu_ {\Upsilon , s s} \mu_ {z ^ {*}, s s}\right) ^ {2} + Q _ {t + 1} \kappa_ {2} \left(\frac {I _ {t + 1}}{K _ {t + 1}} \mu_ {\Upsilon , t + 1} \mu_ {z ^ {*}, t + 1} - \frac {I _ {s s}}{k _ {s s}} \mu_ {\Upsilon , s s} \mu_ {z ^ {*}, s s}\right) \frac {I _ {t + 1}}{K _ {t + 1}} \mu_ {\Upsilon , t + 1} \mu_ {z ^ {*}, t + 1} ] \end{array}\]

\[\begin{array}{l} \mathbf {E Q 4} \\ \gamma_ {t} (z _ {t} ^ {*}) ^ {(1 - \phi_ {4}) (1 - \phi_ {2})} d _ {t} \phi_ {0} (1 - h _ {t}) ^ {- \phi_ {1}} = \lambda_ {t} w _ {t} \\ \Updownarrow \end{array}\]

\[\begin{array}{l} (z _ {t} ^ {*}) ^ {(1 - \phi_ {4}) (1 - \phi_ {2})} d _ {t} \phi_ {0} (1 - h _ {t}) ^ {- \phi_ {1}} = \frac {\lambda_ {t}}{\gamma_ {t} (z _ {t} ^ {*}) ^ {- 1}} \frac {w _ {t}}{z _ {t} ^ {*}} \\ \Updownarrow \end{array}\]

\[\begin{array}{l} d _ {t} \phi_ {0} \left(1 - h _ {t}\right) ^ {- \phi_ {1}} = \frac {\lambda_ {t}}{\left(z _ {t} ^ {*}\right) ^ {(1 - \phi_ {4}) (1 - \phi_ {2}) - 1} \gamma_ {t}} \frac {w _ {t}}{z _ {t} ^ {*}} \\ \Updownarrow \end{array}\]

\[\begin{array}{l} d _ {t} \phi_ {0} \left(1 - h _ {t}\right) ^ {- \phi_ {1}} = \frac {\lambda_ {t}}{\left(z _ {t} ^ {*}\right) ^ {1 - \phi_ {4} - \phi_ {2} + \phi_ {2} \phi_ {4} - 1} \gamma_ {t}} \frac {w _ {t}}{z _ {t} ^ {*}} \\ \Updownarrow \end{array}\]

\[\begin{array}{l} d _ {t} \phi_ {0} (1 - h _ {t}) ^ {- \phi_ {1}} = \frac {\lambda_ {t}}{(z _ {t} ^ {*}) ^ {- \phi_ {2} (1 - \phi_ {4}) - \phi_ {4}} \gamma_ {t}} \frac {w _ {t}}{z _ {t} ^ {*}} \\ \Updownarrow \end{array}\]

\[d _ {t} \phi_ {0} (1 - h _ {t}) ^ {- \phi_ {1}} = \Lambda_ {t} W _ {t}\]

\[\begin{array}{r l} & {\mathbf {E Q 5}} \\ & {1 = q _ {t} \Upsilon_ {t} \left(1 - \frac {\kappa_ {1}}{2} \left(\frac {i _ {t}}{\Upsilon_ {t} z _ {t} ^ {*} i _ {S S}} - 1\right) ^ {2} - \frac {i _ {t}}{\Upsilon_ {t} z _ {t} ^ {*} i _ {S S}} \kappa_ {1} \left(\frac {i _ {t}}{\Upsilon_ {t} z _ {t} ^ {*} i _ {S S}} - 1\right) - \kappa_ {2} \left(\frac {i _ {t}}{k _ {t}} - \frac {I _ {s s}}{K _ {s s}} \mu_ {\Upsilon , s s} \mu_ {z ^ {*}, s s}\right)\right)} \\ & {\Updownarrow} \\ & {1 = Q _ {t} \left(1 - \frac {\kappa_ {1}}{2} \left(\frac {I _ {t}}{I _ {S S}} - 1\right) ^ {2} - \frac {I _ {t}}{I _ {S S}} \kappa_ {1} \left(\frac {I _ {t}}{I _ {S S}} - 1\right) - \kappa_ {2} \left(\frac {i _ {t}}{k _ {t}} - \frac {I _ {s s}}{K _ {s s}} \mu_ {\Upsilon , s s} \mu_ {z ^ {*}, s s}\right)\right)} \\ & {\mathrm{since} I _ {t} \equiv \frac {i _ {t}}{\Upsilon_ {t} z _ {t} ^ {*}}} \\ & {\Updownarrow} \\ & {1 = Q _ {t} \left(1 - \frac {\kappa_ {1}}{2} \left(\frac {I _ {t}}{I _ {S S}} - 1\right) ^ {2} - \frac {I _ {t}}{I _ {S S}} \kappa_ {1} \left(\frac {I _ {t}}{I _ {S E}} - 1\right) - \kappa_ {2} \left(\frac {i _ {t}}{k _ {t}} \frac {\Upsilon_ {t} z _ {t} ^ {*}}{\Upsilon_ {t} z _ {t} ^ {*}} \frac {\Upsilon_ {t - 1} z _ {t - 1} ^ {*}}{\Upsilon_ {t - 1} z _ {t - 1} ^ {*}} - \frac {I _ {s s}}{K _ {s s}} \mu_ {\Upsilon , s s} \mu_ {z ^ {*}, s s}\right)\right)} \\ & {\Updownarrow} \\ & {1 = Q _ {t} \left(1 - \frac {\kappa_ {1}}{2} \left(\frac {I _ {t}}{I _ {S S}} - 1\right. ^ {2} - \frac {I _ {t}}{I _ {S S}} \kappa_ {1} \left(\frac {I _ {t}}{I _ {S S}} - 1\right) - \kappa_ {2} \left(\frac {I _ {t}}{k _ {t}} \frac {\Upsilon_ {t} z _ {t} ^ {*}}{1} \frac {1}{\Upsilon_ {t - 1} z _ {t - 1} ^ {*}} - \frac {I _ {s s}}{K _ {s s}} \mu_ {\Upsilon , s s} \mu_ {z ^ {*}, s s}\right)\right)} \\ & {\mathrm{recall} K _ {t + 1} \equiv \frac {k _ {t + 1}}{\Upsilon_ {t} z _ {t} ^ {*}}} \end{array}\]

\[\begin{array}{l} \Updownarrow \\ 1 = Q _ {t} \left(1 - \frac {\kappa_ {1}}{2} \left(\frac {I _ {t}}{I _ {S S}} - 1\right) ^ {2} - \frac {I _ {t}}{I _ {S S}} \kappa_ {1} \left(\frac {I _ {t}}{I _ {S S}} - 1\right) - \kappa_ {2} \left(\frac {I _ {t}}{k _ {t}} \mu_ {\Upsilon , t} \mu_ {z ^ {*}, t} - \frac {I _ {s s}}{K _ {s s}} \mu_ {\Upsilon , s s} \mu_ {z ^ {*}, s s}\right)\right) \end{array}\]

\[\begin{array}{l} \textbf {E Q 6} \\ 1 = E _ {t} \left[ \beta \frac {\lambda_ {t + 1}}{\lambda_ {t}} \frac {\exp \left\{r _ {t} ^ {b} \right\}}{\pi_ {t + 1}} \right] \\ \Updownarrow \end{array}\]

\[1 = E _ {t} \left[ \beta \mu_ {\lambda t + 1} \frac {\exp \{r _ {t} ^ {b} \}}{\pi_ {t + 1}} \right]\]

\[\begin{array}{l} \textbf {E Q 7} \\ m c _ {t} \theta z _ {t} ^ {1 - \theta} a _ {t} k _ {t} ^ {\theta - 1} h _ {t} ^ {1 - \theta} = r _ {t} ^ {k} \\ \Updownarrow \end{array}\]

\[\begin{array}{l} m c _ {t} a _ {t} \theta z _ {t} ^ {1 - \theta} \Upsilon_ {t} \left(h _ {t}\right) ^ {1 - \theta} k _ {t} ^ {\theta - 1} = \Upsilon_ {t} r _ {t} ^ {k} \\ \Updownarrow \end{array}\]

\[m c _ {t} a _ {t} \theta \left(z _ {t}\right) ^ {1 - \theta} \Upsilon_ {t} h _ {t} ^ {1 - \theta} \left(k _ {t} \frac {\Upsilon_ {t - 1} ^ {\frac {1}{1 - \theta}} z _ {t - 1}}{\Upsilon_ {t - 1} ^ {\frac {1}{1 - \theta}} z _ {t - 1}}\right) ^ {\theta - 1} = R _ {t} ^ {k}\]

\[m c _ {t} a _ {t} \theta (z _ {t}) ^ {1 - \theta} \Upsilon_ {t} h _ {t} ^ {1 - \theta} \left(K _ {t} \Upsilon_ {t - 1} ^ {\frac {1}{1 - \theta}} z _ {t - 1}\right) ^ {\theta - 1} = R _ {t} ^ {k}\]

\[\begin{array}{l} m c _ {t} a _ {t} \theta \Upsilon_ {t} \Upsilon_ {t - 1} ^ {- 1} (z _ {t}) ^ {1 - \theta} z _ {t - 1} ^ {\theta - 1} K _ {t} ^ {\theta - 1} h _ {t} ^ {1 - \theta} = R _ {t} ^ {k} \\ \Updownarrow \end{array}\]

\[m c _ {t} a _ {t} \theta \mu_ {\Upsilon , t} \mu_ {z, t} ^ {1 - \theta} K _ {t} ^ {\theta - 1} h _ {t} ^ {1 - \theta} = R _ {t} ^ {k}\]

\[\begin{array}{l} \textbf {E Q 8} \\ m c _ {t} (1 - \theta) z _ {t} ^ {1 - \theta} a _ {t} k _ {t} ^ {\theta} h _ {t} ^ {- \theta} = w _ {t} \\ \Updownarrow \end{array}\]

\[\begin{array}{l} m c _ {t} \left(1 - \theta\right) \frac {a _ {t} z _ {t} ^ {1 - \theta}}{z _ {t} ^ {*}} k _ {t} ^ {\theta} h _ {t} ^ {- \theta} = \frac {w _ {t}}{z _ {t} ^ {*}} \\ \Updownarrow \end{array}\]

\[m c _ {t} \left(1 - \theta\right) \frac {a _ {t} z _ {t} ^ {1 - \theta}}{z _ {t} ^ {*}} \left(k _ {t} \frac {\Upsilon_ {t - 1} ^ {\frac {1}{1 - \theta}} z _ {t - 1}}{\Upsilon_ {t - 1} ^ {\frac {1}{1 - \theta}} z _ {t - 1}}\right) ^ {\theta} h _ {t} ^ {- \theta} = W _ {t}\]

\[\begin{array}{l} m c _ {t} \left(1 - \theta\right) \frac {a _ {t} z _ {t} ^ {1 - \theta}}{z _ {t} ^ {*}} \left(K _ {t} \Upsilon_ {t - 1} ^ {\frac {1}{1 - \theta}} z _ {t - 1}\right) ^ {\theta} h _ {t} ^ {- \theta} = W _ {t} \\ \Updownarrow \end{array}\]

\[m c _ {t} \left(1 - \theta\right) \frac {a _ {t} z _ {t} ^ {1 - \theta}}{\Upsilon_ {t} ^ {\frac {\theta}{1 - \theta}} z _ {t}} \left(K _ {t} \Upsilon_ {t - 1} ^ {\frac {1}{1 - \theta}} z _ {t - 1}\right) ^ {\theta} h _ {t} ^ {- \theta} = W _ {t}\]

\[m c _ {t} \left(1 - \theta\right) \frac {a _ {t} z _ {t} ^ {- \theta}}{\Upsilon_ {t} ^ {\frac {\theta}{1 - \theta}}} \left(\Upsilon_ {t - 1} ^ {\frac {1}{1 - \theta}} z _ {t - 1}\right) ^ {\theta} K _ {t} ^ {\theta} h _ {t} ^ {- \theta} = W _ {t}\]

\[m c _ {t} (1 - \theta) \frac {a _ {t} \Upsilon_ {t - 1} ^ {\frac {\theta}{1 - \theta}}}{\Upsilon_ {t} ^ {\frac {\theta}{1 - \theta}}} \left(\frac {z _ {t - 1}}{z _ {t}}\right) ^ {\theta} K _ {t} ^ {\theta} h _ {t} ^ {- \theta} = W _ {t}\]

↓

\[m c _ {t} (1 - \theta) a _ {t} \mu_ {\Upsilon , t} ^ {\frac {- \theta}{1 - \theta}} \mu_ {z, t} ^ {- \theta} K _ {t} ^ {\theta} h _ {t} ^ {- \theta} = W _ {t}\]

EQ 9

\[\frac {(\eta - 1) x _ {t} ^ {2}}{\eta} = y _ {t} m c _ {t} \tilde {p} _ {t} ^ {- \eta - 1} + E _ {t} \left[ \alpha \beta \frac {\lambda_ {t + 1}}{\lambda_ {t}} \left(\frac {\tilde {p} _ {t}}{\tilde {p} _ {t + 1}}\right) ^ {- \eta - 1} \left(\frac {\pi_ {t} ^ {\chi}}{\pi_ {t + 1}}\right) ^ {- \eta} \frac {(\eta - 1) x _ {t + 1} ^ {2}}{\eta} \right]\]

\[\frac {(\eta - 1)}{\eta} \frac {x _ {t} ^ {2}}{z _ {t} ^ {*}} = \frac {y _ {t}}{z _ {t} ^ {*}} m c _ {t} \tilde {p} _ {t} ^ {- \eta - 1} + E _ {t} \left[ \alpha \beta \mu_ {\lambda , t + 1} \left(\frac {\tilde {p} _ {t}}{\tilde {p} _ {t + 1}}\right) ^ {- \eta - 1} \left(\frac {\pi_ {t} ^ {\chi}}{\pi_ {t + 1}}\right) ^ {- \eta} \frac {(\eta - 1)}{\eta} \frac {x _ {t + 1} ^ {2}}{z _ {t} ^ {*}} \right]\]

↓

\[\frac {(\eta - 1)}{\eta} X _ {t} ^ {2} = Y _ {t} m c _ {t} \tilde {p} _ {t} ^ {- \eta - 1} + E _ {t} \left[ \alpha \beta \mu_ {\lambda , t + 1} \left(\frac {\tilde {p} _ {t}}{\tilde {p} _ {t + 1}}\right) ^ {- \eta - 1} \left(\frac {\pi_ {t} ^ {\chi}}{\pi_ {t + 1}}\right) ^ {- \eta} \frac {(\eta - 1)}{\eta} X _ {t + 1} ^ {2} \mu_ {z ^ {*}, t + 1} \right]\]

EQ 10

\[x _ {t} ^ {2} = y _ {t} \tilde {p} _ {t} ^ {- \eta} + E _ {t} \left[ \alpha \beta \frac {\lambda_ {t + 1}}{\lambda_ {t}} \left(\frac {\tilde {p} _ {t}}{\tilde {p} _ {t + 1}}\right) ^ {- \eta} \left(\frac {\pi_ {t} ^ {\chi}}{\pi_ {t + 1}}\right) ^ {1 - \eta} x _ {t + 1} ^ {2} \right]\]

↓

\[\begin{array}{l} \frac {x _ {t} ^ {2}}{z _ {t} ^ {*}} = \frac {y _ {t}}{z _ {t} ^ {*}} \tilde {p} _ {t} ^ {- \eta} + E _ {t} \left[ \alpha \beta \mu_ {\lambda , t + 1} \left(\frac {\tilde {p} _ {t}}{\bar {p} _ {t + 1}}\right) ^ {- \eta} \left(\frac {\pi_ {t} ^ {\chi}}{\pi_ {t + 1}}\right) ^ {1 - \eta} \frac {x _ {t + 1} ^ {2}}{z _ {t} ^ {*}} \right] \\ \Updownarrow \end{array}\]

\[X _ {t} ^ {2} = Y _ {t} \tilde {p} _ {t} ^ {- \eta} + E _ {t} \left[ \alpha \beta \mu_ {\lambda , t + 1} \left(\frac {\tilde {p} _ {t}}{\tilde {p} _ {t + 1}}\right) ^ {- \eta} \left(\frac {\pi_ {t} ^ {x}}{\pi_ {t + 1}}\right) ^ {1 - \eta} X _ {t + 1} ^ {2} \mu_ {z ^ {*}, t + 1} \right]\]

EQ 11

\[1 = (1 - \alpha) \tilde {p} _ {t} ^ {1 - \eta} + \alpha \left(\frac {\pi_ {t - 1} ^ {\chi}}{\pi_ {t}}\right) ^ {1 - \eta}\]

EQ 12

\[r _ {t} ^ {b} = r _ {t} + \omega \times x h r _ {t, L}\]

EQ 13

\[P _ {t, 1} = \frac {1}{\exp \{r _ {t} \}}\]

\[\begin{array}{l} \textbf {E Q 1 4} \\ P _ {t, k} = E _ {t} \left[ \beta \frac {\lambda_ {t + 1}}{\lambda_ {t}} \frac {1}{\pi_ {t + 1}} P _ {t + 1, k - 1} \right] \\ \Updownarrow \end{array}\]

\[P _ {t, k} = E _ {t} \left[ \beta \mu_ {\lambda , t + 1} \frac {1}{\pi_ {t + 1}} P _ {t + 1, k - 1} \right]\]

EQ 15

\[\begin{array}{l} r _ {t} = r _ {s s} \left(1 - \rho_ {r}\right) + \rho_ {r} r _ {t - 1} + \left(1 - \rho_ {r}\right) \left(\beta_ {\pi} \log \left(\frac {\pi_ {t}}{\pi_ {s s}}\right) + \beta_ {y} \log \left(\frac {y _ {t}}{z _ {t} ^ {*} Y _ {s s}}\right)\right) \\ + \left(1 - \rho_ {r}\right) \beta_ {x h r} \left(x h r _ {t, L} - X _ {t, L}\right) \\ \Updownarrow \end{array}\]

\[\begin{array}{l} {r _ {t} = r _ {s s} \left(1 - \rho_ {r}\right) + \rho_ {r} r _ {t - 1} + \left(1 - \rho_ {r}\right) \left(\beta_ {\pi} \log \left(\frac {\pi_ {t}}{\pi_ {s s}}\right) + \beta_ {y} \log \left(\frac {Y _ {t}}{Y _ {s s}}\right)\right)} \\ {+ \left(1 - \rho_ {r}\right) \beta_ {x h r} (x h r _ {t, L} - X _ {t, L})} \end{array}\]

EQ 16

\[X _ {t, L} = (1 - \gamma) x h r _ {t, L} + \gamma E _ {t} [ X _ {t + 1, L} ]\]

\[\begin{array}{l} \mathbf {E Q 1 7} \\ a _ {t} k _ {t} ^ {\theta} (z _ {t} h _ {t}) ^ {1 - \theta} = y _ {t} s _ {t + 1} \\ \Updownarrow \end{array}\]

\[\begin{array}{l} a _ {t} \frac {k _ {t} ^ {\theta}}{z _ {t} ^ {*}} (z _ {t} h _ {t}) ^ {1 - \theta} = \frac {y _ {t}}{z _ {t} ^ {*}} s _ {t + 1} \\ \Updownarrow \end{array}\]

\[\begin{array}{l} a _ {t} \frac {1}{z _ {t} ^ {*}} \left(k _ {t} \frac {\Upsilon_ {t - 1} ^ {\frac {1}{1 - \theta}} z _ {t - 1}}{\Upsilon_ {t - 1} ^ {\frac {1}{1 - \theta}} z _ {t - 1}}\right) ^ {\theta} (z _ {t} h _ {t}) ^ {1 - \theta} = Y _ {t} s _ {t + 1} \\ \Updownarrow \end{array}\]

\[\begin{array}{l} a _ {t} \frac {1}{\Upsilon_ {t} ^ {\frac {\theta}{1 - \theta}} z _ {t}} \left(K _ {t} \Upsilon_ {t - 1} ^ {\frac {1}{1 - \theta}} z _ {t - 1}\right) ^ {\theta} (z _ {t} h _ {t}) ^ {1 - \theta} = Y _ {t} s _ {t + 1} \\ \Updownarrow \end{array}\]

\[\begin{array}{l} a _ {t} \left(K _ {t} \frac {\Upsilon_ {t - 1} ^ {\frac {1}{1 - \theta}}}{\Upsilon_ {t} ^ {\frac {1}{1 - \theta}}} z _ {t - 1}\right) ^ {\theta} z _ {t} ^ {- \theta} h _ {t} ^ {1 - \theta} = Y _ {t} s _ {t + 1} \\ \Updownarrow \end{array}\]

\[\begin{array}{l} a _ {t} \left(K _ {t} \frac {\Upsilon_ {t - 1} ^ {\frac {1}{1 - \theta}}}{\Upsilon_ {t} ^ {\frac {1}{1 - \theta}}} \frac {z _ {t - 1}}{z _ {t}}\right) ^ {\theta} h _ {t} ^ {1 - \theta} = Y _ {t} s _ {t + 1} \\ \Updownarrow \end{array}\]

\[a _ {t} \left(K _ {t} \mu_ {\Upsilon , t} ^ {\frac {- 1}{1 - \theta}} \mu_ {z, t} ^ {- 1}\right) ^ {\theta} h _ {t} ^ {1 - \theta} = Y _ {t} s _ {t + 1}\]

EQ 18

\[s _ {t + 1} = (1 - \alpha) \tilde {p} _ {t} ^ {- \eta} + \alpha \left(\frac {\pi_ {t}}{\pi_ {t - 1} ^ {\chi}}\right) ^ {\eta} s _ {t}\]

EQ 19

\[\begin{array}{l} k _ {t + 1} = (1 - \delta) k _ {t} + i _ {t} - i _ {t} \frac {\kappa_ {1}}{2} \left(\frac {i _ {i}}{\Upsilon_ {t} z _ {t} ^ {*} I _ {S S}} - 1\right) ^ {2} - k _ {t} \frac {\kappa_ {2}}{2} \left(\frac {i _ {i}}{k _ {t}} - \frac {I _ {s s}}{K _ {s s}} \mu_ {\Upsilon , s s} \mu_ {z ^ {*}, s s}\right) ^ {2} \\ \Updownarrow \end{array}\]

\[\begin{array}{l} \frac {k _ {t + 1}}{\Upsilon_ {t} z _ {t} ^ {*}} = (1 - \delta) \frac {k _ {t}}{\Upsilon_ {t} z _ {t} ^ {*}} + \frac {i _ {t}}{\Upsilon_ {t} z _ {t} ^ {*}} - \frac {i _ {t}}{\Upsilon_ {t} z _ {t} ^ {*}} \frac {\kappa_ {1}}{2} \left(\frac {i _ {i}}{\Upsilon_ {t} z _ {t} ^ {*} I _ {S S}} - 1\right) ^ {2} - \frac {k _ {t}}{\Upsilon_ {t} z _ {t} ^ {*}} \frac {\kappa_ {2}}{2} \left(\frac {i _ {i}}{k _ {t}} - \frac {I _ {s s}}{K _ {s s}} \mu_ {\Upsilon , s s} \mu_ {z ^ {*}, s s}\right) ^ {2} \\ \Updownarrow \end{array}\]

\[K _ {t + 1} = (1 - \delta) \frac {k _ {t}}{\Upsilon_ {t} z _ {t} ^ {*}} \frac {\Upsilon_ {t - 1} z _ {t - 1} ^ {*}}{\Upsilon_ {t - 1} z _ {t - 1} ^ {*}} + I _ {t} - I _ {t} \frac {\kappa_ {1}}{2} \left(\frac {I _ {i}}{I _ {S S}} - 1\right) ^ {2} - \frac {k _ {t}}{\Upsilon_ {t} z _ {t} ^ {*}} \frac {\Upsilon_ {t - 1} z _ {t - 1} ^ {*}}{\Upsilon_ {t - 1} z _ {t - 1} ^ {*}} \frac {\kappa_ {2}}{2} \left(\frac {i _ {i}}{k _ {t}} \frac {\Upsilon_ {t} z _ {t} ^ {*}}{\Upsilon_ {t} z _ {t} ^ {*}} \frac {\Upsilon_ {t - 1} z _ {t - 1} ^ {*}}{\Upsilon_ {t - 1} z _ {t - 1} ^ {*}} - \frac {I _ {s s}}{K _ {s s}} \mu_ {\Upsilon , s s} \mu_ {z ^ {*}, s s}\right) ^ {2}\]

because and

\[\begin{array}{l} K _ {t + 1} = (1 - \delta) \frac {K _ {t}}{\Upsilon_ {t} z _ {t} ^ {*}} \frac {\Upsilon_ {t - 1} z _ {t - 1} ^ {*}}{1} + I _ {t} - I _ {t} \frac {\kappa_ {1}}{2} \left(\frac {I _ {i}}{I _ {S S}} - 1\right) ^ {2} - \frac {K _ {t}}{\Upsilon_ {t} z _ {t} ^ {*}} \frac {\Upsilon_ {t - 1} z _ {t - 1} ^ {*}}{1} \frac {\kappa_ {2}}{2} \left(\frac {I _ {i}}{K _ {t}} \frac {\Upsilon_ {t} z _ {t} ^ {*}}{1} \frac {1}{\Upsilon_ {t - 1} z _ {t - 1} ^ {*}} - \frac {I _ {s s}}{K _ {s s}} \mu_ {\Upsilon , s s} \mu_ {z ^ {*}, s s}\right) ^ {2} \\ \Updownarrow \end{array}\]

\[K _ {t + 1} = (1 - \delta) K _ {t} (\mu_ {\Upsilon , t} \mu_ {z ^ {*}, t}) ^ {- 1} + I _ {t} - I _ {t} \frac {\kappa_ {1}}{2} \left(\frac {I _ {i}}{I _ {S S}} - 1\right) ^ {2} - K _ {t} (\mu_ {\Upsilon , t} \mu_ {z ^ {*}, t}) ^ {- 1} \frac {\kappa_ {2}}{2} \left(\frac {I _ {i}}{K _ {t}} \mu_ {\Upsilon , t} \mu_ {z ^ {*}, t} - \frac {I _ {s s}}{K _ {s s}} \mu_ {\Upsilon , s s} \mu_ {z ^ {*}, s s}\right) ^ {2}\]

EQ 20

\[\begin{array}{l} y _ {t} = c _ {t} + \Upsilon_ {t} ^ {- 1} i _ {t} + z _ {t} ^ {*} G _ {t} \\ \Updownarrow \\ \frac {y _ {t}}{z _ {t} ^ {*}} = \frac {\left(c _ {t} + \Upsilon_ {t} ^ {- 1} i _ {t}\right)}{z _ {t} ^ {*}} + G _ {t} \\ \Updownarrow \\ Y _ {t} = C _ {t} + I _ {t} + G _ {t} \end{array}\]

The Households 1 2 3 4 5 6 The Firms 7 8 9 10 11 The Financial Intermediary 12 , where 13 14 for k=2,3,...,K The Central Bank 15 16 Other relations 17 18 19 20 21 Exogenous processes 22 and (i.e. a deterministic trend) 23 and (i.e. a deterministic trend) 24 25 26

In dealing with the term we follow the procedure suggested by Rudebusch & Swanson (2012).

\[\begin{array}{l} \text { that is we define } \\ E V _ {t} = E _ {t} \left[ \left(\frac {- V _ {t + 1}}{A A}\right) ^ {1 - \phi_ {3}} \right] \quad \text { where } E V _ {s s} = \left(\frac {- V _ {s s}}{A A}\right) ^ {1 - \phi_ {3}} > 0 \\ P E V _ {t} = A A \times E V _ {t} ^ {\frac {1}{1 - \phi_ {3}}} \\ = A A \times \left(E _ {t} \left[ \left(\frac {- V _ {t + 1}}{A A}\right) ^ {1 - \phi_ {3}} \right]\right) ^ {\frac {1}{1 - \phi_ {3}}} \\ = A A \times \left(A A ^ {- (1 - \phi_ {3})} E _ {t} \left[ (- V _ {t + 1}) ^ {1 - \phi_ {3}} \right]\right) ^ {\frac {1}{1 - \phi_ {3}}} \\ = A A \times A A ^ {- 1} \left(E _ {t} \left[ (- V _ {t + 1}) ^ {1 - \phi_ {3}} \right]\right) ^ {\frac {1}{1 - \phi_ {3}}} \\ = \left(E _ {t} \left[ (- V _ {t + 1}) ^ {1 - \phi_ {3}} \right]\right) ^ {\frac {1}{1 - \phi_ {3}}} \\ \text { where } A A \text { is a scaling constant } \end{array}\]

16.11 Market completeness

This subsection shows that our model implies market completeness although it is only the financial intermediary that trades bonds. We first note that the deposit rate only enters in the following tree equations within our model:

\[\begin{array}{r l} & 1 = E _ {t} \left[ \beta \mu_ {\lambda t + 1} \frac {\exp \left\{r _ {t} ^ {b} \right\}}{\pi_ {t + 1}} \right] \\ & r _ {t} ^ {b} = r _ {t} + \omega \times x h r _ {t, L} \\ & r _ {t} = (1 - \rho_ {r}) r _ {s s} + \rho_ {r} r _ {t - 1} + (1 - \rho_ {r}) \left(\beta_ {\pi} \log \left(\frac {\pi_ {t}}{\pi_ {s s}}\right) + \beta_ {y} \log \left(\frac {y _ {t}}{z _ {t} ^ {*} Y _ {s s}}\right)\right) + (1 - \rho_ {r}) \beta_ {x h r} (x h r _ {t, L} - X _ {t, L}) \end{array}\]

Next, note that and substituted into the Taylor rule implies

\[\begin{array}{r l} & {r _ {t} ^ {b} - \omega \times x h r _ {t, L} = (1 - \rho_ {r}) r _ {s s} + \rho_ {r} (r _ {t - 1} ^ {b} - \omega \times x h r _ {t - 1, L})} \\ & {\qquad + (1 - \rho_ {r}) (\beta_ {\pi} \log (\frac {\pi_ {t}}{\pi_ {s s}}) + \beta_ {y} \log (\frac {y _ {t}}{z _ {t} ^ {*} Y _ {s s}})) + (1 - \rho_ {r}) \beta_ {x h r} (x h r _ {t, L} - X _ {t, L})} \\ & {\Updownarrow} \\ & {r _ {t} ^ {b} = (1 - \rho_ {r}) r _ {s s} + \rho_ {r} r _ {t - 1} ^ {b} + (1 - \rho_ {r}) (\beta_ {\pi} \log (\frac {\pi_ {t}}{\pi_ {s s}}) + \beta_ {y} \log (\frac {y _ {t}}{z _ {t} ^ {*} Y _ {s s}}))} \\ & {+ \omega \times x h r _ {t, L} - \rho_ {r} \omega \times x h r _ {t - 1, L} + (1 - \rho_ {r}) \beta_ {x h r} (x h r _ {t, L} - X _ {t, L})} \end{array}\]

Hence, our model is equivalent to a standard New Keynesian model with market completeness but with a Taylor rule for that depends on fast and current values of excess holding period return on the long bond.

16.12 The intertemporal elasticity of substitution (IES)

We want to compute expression for the intertemporal elasticity of substitution (IES) under perfect foresight and then evaluate it at the deterministic steady state. We start by considering the case of external habit formation before considering the case of internal habit formation.

16.12.1 External habit formation

We consider the specification

\[U = \sum_ {l = 0} ^ {\infty} \beta^ {l} u (c _ {t + l} - H _ {t + l})\]

where we omit leisure (because it does not affect the IES) and denote the external habit level by . We start by computing the intertemporal marginal rate of substitution (IMRS) which equals in this case

\[d U = u _ {c} \left(c _ {t} - H _ {t}\right) d c _ {t} + \beta u _ {c} \left(c _ {t + 1} - H _ {t + 1}\right) d c _ {t + 1} = 0\]

↓

\[\beta u _ {c} (c _ {t + 1} - H _ {t + 1}) d c _ {t + 1} = - u _ {c} (c _ {t} - H _ {t}) d c _ {t}\]

\[I M R S _ {t}: \frac {d c _ {t + 1}}{d c _ {t}} = - \frac {1}{\beta} \frac {u _ {c} (c _ {t} - H _ {t})}{u _ {c} (c _ {t + 1} - H _ {t + 1})}.\]

The IES is then defined as

\[\begin{array}{r c l} I E S _ {t} & = & \frac {d \left(\frac {c _ {t + 1}}{c _ {t}}\right) / \left(\frac {c _ {t + 1}}{c _ {t}}\right)}{d I M R S _ {t} / I M R S _ {t}} \\ & = & \frac {I M R S _ {t} / \left(\frac {c _ {t + 1}}{c _ {t}}\right)}{d I M R S _ {t} / d \left(\frac {c _ {t + 1}}{c _ {t}}\right)} \end{array}\]

To compute the denominator, we then note that

\[\begin{array}{r c l} I M R S _ {t} & = & - \frac {1}{\beta} \frac {u _ {c} (c _ {t} - H _ {t})}{u _ {c} (c _ {t + 1} - H _ {t + 1})} \\ & = & - \frac {1}{\beta} \frac {u _ {c} (\frac {c _ {t}}{c _ {t}} - \frac {H _ {t}}{c _ {t}})}{u _ {c} (\frac {c _ {t + 1}}{c _ {t}} - \frac {H _ {t + 1}}{c _ {t}})} \\ & = & - \frac {1}{\beta} u _ {c} (1 - \frac {H _ {t}}{c _ {t}}) [ u _ {c} (\frac {c _ {t + 1}}{c _ {t}} - \frac {H _ {t + 1}}{c _ {t}}) ] ^ {- 1} \end{array}\]

provided is homogenous of some order (as assumed for the our considered functional form). Then

\[\frac {d I M R S _ {t}}{d \left(\frac {c _ {t + 1}}{c _ {t}}\right)} = \frac {1}{\beta} u _ {c} \left(1 - \frac {H _ {t}}{c _ {t}}\right) \left[ u _ {c} \left(\frac {c _ {t + 1}}{c _ {t}} - \frac {H _ {t + 1}}{c _ {t}}\right) \right] ^ {- 2} u _ {c c} \left(\frac {c _ {t + 1}}{c _ {t}} - \frac {H _ {t + 1}}{c _ {t}}\right)\]

Thus, we have

\[\begin{array}{r c l} I E S _ {t} & = & \frac {- \frac {1}{\beta} u _ {c} \left(1 - \frac {H _ {t}}{c _ {t}}\right) \left[ u _ {c} \left(\frac {c _ {t + 1}}{c _ {t}} - \frac {H _ {t + 1}}{c _ {t}}\right) \right] ^ {- 1} / \left(\frac {c _ {t + 1}}{c _ {t}}\right)}{\frac {1}{\beta} u _ {c} \left(1 - \frac {H _ {t}}{c _ {t}}\right) \left[ u _ {c} \left(\frac {c _ {t + 1}}{c _ {t}} - \frac {H _ {t + 1}}{c _ {t}}\right) \right] ^ {- 2} u _ {c c} \left(\frac {c _ {t + 1}}{c _ {t}} - \frac {H _ {t + 1}}{c _ {t}}\right)} \\ & = & - \frac {u _ {c} \left(\frac {c _ {t + 1}}{c _ {t}} - \frac {H _ {t + 1}}{c _ {t}}\right) / \left(\frac {c _ {t + 1}}{c _ {t}}\right)}{u _ {c c} \left(\frac {c _ {t + 1}}{c _ {t}} - \frac {H _ {t + 1}}{c _ {t}}\right)} \\ & = & - \frac {u _ {c} \left(\frac {c _ {t + 1}}{c _ {t}} - \frac {H _ {t + 1}}{c _ {t}}\right)}{u _ {c c} \left(\frac {c _ {t + 1}}{c _ {t}} - \frac {H _ {t + 1}}{c _ {t}}\right)} \frac {1}{\frac {c _ {t + 1}}{c _ {t}}}. \end{array}\]

The considered functional form for in our case is , implying

\[u _ {c} \left(c _ {t} - b c _ {t - 1}\right) = \left(\frac {c _ {t} - b c _ {t - 1}}{\left(z _ {t} ^ {*}\right) ^ {\phi_ {4}}}\right) ^ {- \phi_ {2}} \frac {1}{\left(z _ {t} ^ {*}\right) ^ {\phi_ {4}}}\]

\[u _ {c c} \left(c _ {t} - b c _ {t - 1}\right) = - \phi_ {2} \left(\frac {c _ {t} - b c _ {t - 1}}{\left(z _ {t} ^ {*}\right) ^ {\phi_ {4}}}\right) ^ {- \phi_ {2} - 1} \frac {1}{\left(z _ {t} ^ {*}\right) ^ {2 \phi_ {4}}}\]

Thus we get

\[\begin{array}{r c l} I E S _ {t} & = & - \frac {\left(\frac {c _ {t + 1} - b c _ {t}}{\left(z _ {t + 1} ^ {*}\right) ^ {\phi_ {4}} c _ {t}}\right) ^ {- \phi_ {2}} \frac {1}{\left(z _ {t + 1} ^ {*}\right) ^ {\phi_ {4}}}}{\left(- \phi_ {2} \left(\frac {c _ {t + 1} - b c _ {t}}{\left(z _ {t + 1} ^ {*}\right) ^ {\phi_ {4}} c _ {t}}\right) ^ {- \phi_ {2} - 1} \frac {1}{\left(z _ {t + 1} ^ {*}\right) ^ {2 \phi_ {4}}}\right)} \frac {1}{\frac {c _ {t + 1}}{c _ {t}}} \\ & = & \frac {\left(\frac {c _ {t + 1} - b c _ {t}}{\left(z _ {t + 1} ^ {*}\right) ^ {\phi_ {4}} c _ {t}}\right)}{\phi_ {2} \frac {1}{\left(z _ {t + 1} ^ {*}\right) ^ {\phi_ {4}}}} \frac {1}{\frac {c _ {t + 1}}{c _ {t}}} \\ & = & \frac {1}{\phi_ {2}} \left(\frac {c _ {t + 1} - b c _ {t}}{c _ {t}}\right) \frac {c _ {t}}{c _ {t + 1}} \\ & = & \frac {1}{\phi_ {2}} \left(\frac {c _ {t + 1} - b c _ {t}}{c _ {t + 1}}\right) \\ & = & \frac {1}{\phi_ {2}} \left(1 - b \frac {c _ {t}}{c _ {t + 1}} \frac {z _ {t} ^ {*}}{z _ {t} ^ {*}} \frac {z _ {t + 1} ^ {*}}{z _ {t + 1} ^ {*}}\right) \\ & = & \frac {1}{\phi_ {2}} \left(1 - b \frac {C _ {t}}{C _ {t + 1}} \frac {z _ {t} ^ {*}}{1} \frac {1}{z _ {t + 1} ^ {*}}\right) \\ & = & \frac {1}{\phi_ {2}} \left(1 - b \frac {C _ {t}}{C _ {t + 1}} \mu_ {z ^ {*}, t} ^ {- 1}\right) \end{array}\]

So in the steady state

\[I E S _ {s s} = \frac {1}{\phi_ {2}} \left(1 - \frac {b}{\mu_ {z ^ {*} , s s}}\right).\]

Note that this expression corresponds to the one obtained by using the standard formula evaluated at the steady state.

16.12.2 Internal habit formation

We consider the general specification

\[U = \sum_ {l = 0} ^ {\infty} \beta^ {l} u \left(c _ {t + l} - b c _ {t - 1 + l}\right)\]

Note also that we omit leisure (because it does not affect the IES). We start by computing the intertemporal marginal rate of substitution (IMRS) which equals

\[\left[ u _ {c} \left(c _ {t} - b c _ {t - 1}\right) - \beta b u _ {c} \left(c _ {t + 1} - b c _ {t}\right) \right] d c _ {t} + \left[ \beta u _ {c} \left(c _ {t + 1} - b c _ {t}\right) - \beta^ {2} b u _ {c} \left(c _ {t + 2} - b c _ {t + 1}\right) \right] d c _ {t + 1} = 0\]

↓

\[\beta \left[ u _ {c} (c _ {t + 1} - b c _ {t}) - \beta b u _ {c} (c _ {t + 2} - b c _ {t + 1}) \right] d c _ {t + 1} = - \left[ u _ {c} (c _ {t} - b c _ {t - 1}) - \beta b u _ {c} (c _ {t + 1} - b c _ {t}) \right] d c _ {t}\]

↓

\[I M R S _ {t}: \frac {d c _ {t + 1}}{d c _ {t}} = - \frac {1}{\beta} \frac {u _ {c} (c _ {t} - b c _ {t - 1}) - \beta b u _ {c} (c _ {t + 1} - b c _ {t})}{u _ {c} (c _ {t + 1} - b c _ {t}) - \beta b u _ {c} (c _ {t + 2} - b c _ {t + 1})}.\]

The IES is then defined as

\[\begin{array}{r c l} I E S _ {t} & = & \frac {d \left(\frac {c _ {t + 1}}{c _ {t}}\right) / \left(\frac {c _ {t + 1}}{c _ {t}}\right)}{d I M R S _ {t} / I M R S _ {t}} \\ & = & \frac {I M R S _ {t} / \left(\frac {c _ {t + 1}}{c _ {t}}\right)}{d I M R S _ {t} / d \left(\frac {c _ {t + 1}}{c _ {t}}\right)} \end{array}\]

To compute the denominator, we then note that

\[\begin{array}{r c l} I M R S _ {t} & = & - \frac {1}{\beta} \frac {u _ {c} (c _ {t} - b c _ {t - 1}) - \beta b u _ {c} (c _ {t + 1} - b c _ {t})}{u _ {c} (c _ {t + 1} - b c _ {t}) - \beta b u _ {c} (c _ {t + 2} - b c _ {t + 1})} \\ & = & - \frac {1}{\beta} \frac {u _ {c} (1 - b \frac {c _ {t - 1}}{c _ {t}}) - \beta b u _ {c} (\frac {c _ {t + 1}}{c _ {t}} - b)}{u _ {c} (\frac {c _ {t + 1}}{c _ {t}} - b) - \beta b u _ {c} (\frac {c _ {t + 2}}{c _ {t}} - b \frac {c _ {t + 1}}{c _ {t}})} \\ & = & - \frac {1}{\beta} [ u _ {c} (1 - b \frac {c _ {t - 1}}{c _ {t}}) - \beta b u _ {c} (\frac {c _ {t + 1}}{c _ {t}} - b) ] \\ & & \times [ u _ {c} (\frac {c _ {t + 1}}{c _ {t}} - b) - \beta b u _ {c} (\frac {c _ {t + 2}}{c _ {t}} - b \frac {c _ {t + 1}}{c _ {t}}) ] ^ {- 1}. \end{array}\]

provided is homogenous of some order (as assumed for the our considered functional form). Then

\[\begin{array}{r c l} \frac {d I M R S _ {t}}{d \left(\frac {c _ {t + 1}}{c _ {t}}\right)} & = & b u _ {c c} \left(\frac {c _ {t + 1}}{c _ {t}} - b\right) \left[ u _ {c} \left(\frac {c _ {t + 1}}{c _ {t}} - b\right) - \beta b u _ {c} \left(\frac {c _ {t + 2}}{c _ {t}} - b \frac {c _ {t + 1}}{c _ {t}}\right) \right] ^ {- 1} \\ & & + \frac {1}{\beta} \left[ u _ {c} \left(1 - b \frac {c _ {t - 1}}{c _ {t}}\right) - \beta b u _ {c} \left(\frac {c _ {t + 1}}{c _ {t}} - b\right) \right] \\ & & \times \left[ u _ {c} \left(\frac {c _ {t + 1}}{c _ {t}} - b\right) - \beta b u _ {c} \left(\frac {c _ {t + 2}}{c _ {t}} - b \frac {c _ {t + 1}}{c _ {t}}\right) \right] ^ {- 2} \\ & & \times \left[ u _ {c c} \left(\frac {c _ {t + 1}}{c _ {t}} - b\right) + \beta b ^ {2} u _ {c c} \left(\frac {c _ {t + 2}}{c _ {t}} - b \frac {c _ {t + 1}}{c _ {t}}\right) \right] \\ & = & b u _ {c c} \left(\frac {c _ {t + 1}}{c _ {t}} - b\right) \left(- \beta \left[ u _ {c} \left(1 - b \frac {c _ {t - 1}}{c _ {t}}\right) - \beta b u _ {c} \left(\frac {c _ {t + 1}}{c _ {t}} - b\right) \right] ^ {- 1} I M R S _ {t}\right) \\ & & - I M R S _ {t} \left[ u _ {c} \left(\frac {c _ {t + 1}}{c _ {t}} - b\right) - \beta b u _ {c} \left(\frac {c _ {t + 2}}{c _ {t}} - b \frac {c _ {t + 1}}{c _ {t}}\right) \right] ^ {- 1} \\ & & \times \left[ u _ {c c} \left(\frac {c _ {t + 1}}{c _ {t}} - b\right) + \beta b ^ {2} u _ {c c} \left(\frac {c _ {t + 2}}{c _ {t}} - b \frac {c _ {t + 1}}{c _ {t}}\right) \right] \\ & & - I M R S _ {t} \left\{\frac {b \beta u _ {c c} \left(\frac {c _ {t + 1}}{c _ {t}} - b\right)}{u _ {c} \left(1 - b \frac {c _ {t - 1}}{c _ {t}}\right) - \beta b u _ {c} \left(\frac {c _ {t + 1}}{c _ {t}} - b\right)} + \frac {u _ {c c} \left(\frac {c _ {t + 1}}{c _ {t}} - b\right) + \beta b ^ {2} u _ {c c} \left(\frac {c _ {t + 2}}{c _ {t}} - b \frac {c _ {t + 1}}{c _ {t}}\right)}{u _ {c} \left(\frac {c _ {t + 1}}{c _ {t}} - b\right) - \beta b u _ {c} \left(\frac {c _ {t + 2}}{c _ {t}} - b \frac {c _ {t + 1}}{c _ {t}}\right)} \right\} \end{array}\]

Thus, we get

\[I E S _ {t} = \frac {I M R S _ {t} / \left(\frac {c _ {t + 1}}{c _ {t}}\right)}{d I M R S _ {t} / d \left(\frac {c _ {t + 1}}{c _ {t}}\right)}\]

\[\begin{array}{l} = \frac {I M R S _ {t} / \left(\frac {c _ {t + 1}}{c _ {t}}\right)}{- I M R S _ {t} \left\{\frac {b \beta u _ {c c} \left(\frac {c _ {t + 1}}{c _ {t}} - b\right)}{u _ {c} \left(1 - b \frac {c _ {t - 1}}{c _ {t}}\right) - \beta b u _ {c} \left(\frac {c _ {t + 1}}{c _ {t}} - b\right)} + \frac {u _ {c c} \left(\frac {c _ {t + 1}}{c _ {t}} - b\right) + \beta b ^ {2} u _ {c c} \left(\frac {c _ {t + 2}}{c _ {t}} - b \frac {c _ {t + 1}}{c _ {t}}\right)}{u _ {c} \left(\frac {c _ {t + 1}}{c _ {t}} - b\right) - \beta b u _ {c} \left(\frac {c _ {t + 2}}{c _ {t}} - b \frac {c _ {t + 1}}{c _ {t}}\right)} \right\}} \\ = \frac {- 1 / \left(\frac {c _ {t + 1}}{c _ {t}}\right)}{\frac {b \beta u _ {c c} \left(\frac {c _ {t + 1}}{c _ {t}} - b\right)}{u _ {c} \left(1 - b \frac {c _ {t - 1}}{c _ {t}}\right) - \beta b u _ {c} \left(\frac {c _ {t + 1}}{c _ {t}} - b\%\right)} + \frac {u _ {c c} \left(\frac {c _ {t + 1}}{c _ {t}} - b\right) + \beta b ^ {2} u _ {c c} \left(\frac {c _ {t + 2}}{c _ {t}} - b \frac {c _ {t + 1}}{c _ {t}}\right)}{u _ {c} \left({\frac {c _ {t + 1}}{c _ {t}}} - b\right) - \beta b u _ {c} \left(\frac {c _ {t + 2}}{c _ {t}} - b \frac {c _ {t + 1}}{c _ {t}}\right)}} \\ = - \left[ \frac {b \beta u _ {c c} \left(\frac {c _ {t + 1}}{c _ {t}} - b\right)}{u _ {c} \left(1 - b \frac {c _ {t - 1}}{c _ {t}}\right) - \beta b u _ {c} \left(\frac {c _ {t + 1}}{c _ {t}} - b\right.} + \frac {u _ {c c} \left(\frac {c _ {t + 1}}{c _ {t}} - b\right) + \beta b ^ {2} u _ {c c} \left(\frac {c _ {t + 2}}{c _ {t}} - b \frac {c _ {t + 1}}{c _ {t}}\right)}{u _ {c} \left(\frac {\dot {} c t + 1}{c t} - b\right) - \beta b u _ {c} \left(\frac {\dot {} c t + 2}{c t} - b \frac {\dot {} c t + 1}{\dot {} c t}\right)} \right] ^ {- 1} \left[ \frac {c _ {t + 1}}{c t} \right] ^ {- 1} \\ = - \left[ \left(\frac {b \beta u _ {c c} \left(\frac {c _ {t + 1}}{c _ {t}} - b\right)}{u _ {c} \left(1 - b \frac {\dot {} c t - 1}{\dot {} c t}\right) - \beta b u _ {c} \left(\frac {\dot {} c t + 1}{\dot {} c t} - b\right)} + \frac {u _ {c c} \left(\frac {\dot {} c t + 1}{\dot {} c t} - b\right) + \beta b ^ {2} u _ {c c} \left(\frac {\dot {} c t + 2}{\dot {} c t} - b \frac {\dot {} c t + 1}{\dot {} c t}\right)}{u _ {c} \left(\frac {\dot {} c t + 1}{\dot {} c t} - b\right) - \beta b u _ {c} \left(\frac {\dot {} c t + 2}{\dot {} c t} - b \frac {\dot {} c t + 1}{\dot {} c t}\right)}\right) \frac {\dot {} c t + 1}{\dot {} c t} \right] ^ {- 1} \\ = - (a) ^ {- a}, a = - (a, a) ^ {- a}. \end{array}\]

We next express these terms for the transformed economy, i.e.

\[\begin{array}{r l} & {\frac {c _ {t - 1}}{c _ {t}} = \frac {C _ {t - 1} z _ {t - 1} ^ {*}}{C _ {t} z _ {t} ^ {*}} = \frac {C _ {t - 1}}{C _ {t}} \mu_ {z ^ {*}, t} ^ {- 1}} \\ & {\frac {c _ {t + 1}}{c _ {t}} = \frac {C _ {t + 1} z _ {t + 1} ^ {*}}{C _ {t} z _ {t} ^ {*}} = \frac {C _ {t + 1}}{C _ {t}} \mu_ {z ^ {*}, t + 1}} \\ & {\frac {c _ {t + 2}}{c _ {t}} = \frac {C _ {t + 2} z _ {t + 2} ^ {*}}{C _ {t} z _ {t} ^ {*}} \frac {z _ {t + 1} ^ {*}}{z _ {t + 1} ^ {*}} = \frac {C _ {t + 2}}{C _ {t}} \mu_ {z ^ {*}, t + 2} \mu_ {z ^ {*}, t + 1}} \\ & {\mathrm{Thus,}} \\ & {I E S _ {t} = - \left[ \left(\frac {b \beta u _ {c c} \left(\frac {C _ {t + 1}}{C _ {t}} \mu_ {z ^ {*} , t + 1} - b\right)}{u _ {c} \left(1 - b \frac {C _ {t - 1}}{C _ {t}} \mu_ {z ^ {*} , t}\right) - \beta b u _ {c} \left(\frac {C _ {t + 1}}{C _ {t}} \mu_ {z ^ {*} , t + 1} - b\right)} + \frac {u _ {c c} \left(\frac {C _ {t + 1}}{C _ {t}} \mu_ {z ^ {*} , t + 1} - b\right) + \beta b ^ {2} u _ {c c} \left(\frac {C _ {t + 2}}{C _ {t}} \mu_ {z ^ {*} , t + 2} \mu_ {z ^ {*} , t + 1} - b \frac {C _ {t + 1}}{C _ {t}} \mu_ {z ^ {*} , t + 1}\right)}{u _ {c} \left(\frac {C _ {t + 1}}{C _ {t}} \mu_ {z ^ {*} , t + 1} - b\right) - \beta b u _ {c} \left(\frac {C _ {t + 2}}{C _ {t}} \mu_ {z ^ {*} , t + 2} \mu_ {z ^ {*} , t + 1} - b \frac {C _ {t + 1}}{C _ {t}} \mu_ {z ^ {*} , t + 1}\right)}\right) \mu_ {z ^ {*}, t + 1} \right] ^ {- 1}.} \end{array}\]

The considered functional form for in our case is , implying

\[u _ {c} (c _ {t} - b c _ {t - 1}) = (c _ {t} - b c _ {t - 1}) ^ {- \phi_ {2}} (z _ {t} ^ {*}) ^ {(\phi_ {2} - 1) \phi_ {4}}\]

\[u _ {c c} (c _ {t} - b c _ {t - 1}) = - \phi_ {2} (c _ {t} - b c _ {t - 1}) ^ {- \phi_ {2} - 1} (z _ {t} ^ {*}) ^ {(\phi_ {2} - 1) \phi_ {4}}\]

Hence,

\[\begin{array} { r l } I E S _ { t } = - \left[ \left( \frac { b \beta u _ { c c } \left( \frac { C _ { t + 1 } } { C _ { t } } \mu _ { z ^ { * } , t + 1 } - b \right) } { u _ { c } \left( 1 - b \frac { C _ { t - 1 } } { C _ { t } } \mu _ { z ^ { * } , t } ^ { - 1 } \right) - \beta b u _ { c } \left( \frac { C _ { t + 1 } } { C _ { t } } \mu _ { z ^ { * } , t + 1 } - b \right) } + \frac { u _ { c c } \left( \frac { C _ { t + 1 } } { C _ { t } } \mu _ { z ^ { * } , t + 1 } - b \right) + \beta b ^ { 2 } u _ { c c } \left( \frac { C _ { t + 2 } } { C _ { t } } \mu _ { z ^ { * } , t + 2 } \mu _ { z ^ { * } , t + 1 } - b \frac { C _ { t + 1 } } { C _ { t } } \mu _ { z ^ { * } , t + 1 } \right) } { u _ { c } \left( \frac { C _ { t + 1 } } { C _ { t } } \mu _ { z ^ { * } , t + 1 } - b \right) - \beta b u _ { c } \left( \frac { C _ { t + 2 } } { C _ { t } } \mu _ { z ^ { * } , t + 2 } \mu _ { z ^ { * } , t + 1 } - b \frac { C _ { t + 1 } } { C _ { t } } \mu _ { z ^ { * } , t + 1 } \right) } \right) \mu _ { z ^ { * } , t + 1 } \right] ^ { - 1 } \\ & = - \left[ \left( \begin{array} { c } - \phi _ { 2 } b \beta \left( \frac { C _ { t + 1 } } { C _ { t } } \mu _ { z ^ { * } , t + 1 } - b \right) ^ { - \phi _ { 2 } - 1 } \left( z _ { t + 1 } ^ { * } \right) ^ { ( \phi _ { 2 } - 1 ) \phi _ { 4 } } \\ \hline \left( 1 - b \frac { C _ { t - 1 } } { C _ { t } } \mu _ { z ^ { * } , t } ^ { - 1 } \right) ^ { - \phi _ { 2 } } ( z _ { t } ^ { * } ) ^ { ( \phi _ { 2 } - 1 ) \phi _ { 4 } } - \beta b \left( \frac { C _ { t + 1 } } { C _ { t } } \mu _ { z ^ { * } , t + 1 } - b \right) ^ { - \phi _ { 2 } } ( z _ { t + 1 } ^ { * } ) ^ { ( \phi _ { 2 } - 1 ) \phi _ { 4 } } \\ + \frac - \phi _ { 2 } \left( \frac { C _ { t + 1 } } { C _ { t } } \mu _ { z ^ { * } , t + 1 } - b \right) ^ { - \phi _ { 2 } - 1 } ( z _ { t + 1 } ^ { * } ) ^ { ( \phi _ { 2 } - 1 ) \phi _ { 4 } } - \phi _ { 2 } \beta b ^ { 2 } \left( \frac { C _ { t + 2 } } { C _ { t } } \mu _ { z ^ { * } , t + 2 } \mu _ { z ^ { * } , t + 1 } - b \frac { C _ { t + 1 } } { C _ { t } } \mu _ { z ^ { * } , t + 1 } \right) ^ { - \phi _ { 2 } - 1 } ( z _ { t + 2 } ^ { * } ) ^ { ( \phi _ { 2 } - 1 ) \phi _ { 4 } } \\ \hline ( C _ { t + 1 } / C _ { t} / C _ { t + 1 + 2 }) ^ { ( \phi _ { 2 - 1 ) / 4 }} - b b ( C _ { t + 2 } / C _ { t} / C _ { t + 2 + 2 }) ^ ( z _ { t + 1 + 1}) ^ ( ( z _ t + 2 + 2 + 2 + 2 + 3 ) / ( z _ t + 2 + 2 + 3 + 4 ) / ( z _ t + 2 + 2 + 3 + 4 + 5 ) / ( z _ t + 2 + 2 + 3 + 4 + 5 ) / ( z _ t + 2 + 2 + 3 + 4 + 5 ) / ( z _ t + 2 + 2 + 3 + 4 + 5 ) / ( z _ t + 2 + 2 + 3 + 4 + 5 ) / ( z _ t + 2 + n + n + n + n + n + n + n + n + n + n + n + n + n + n + n + n + n + n + n + n + n + n + n + n + n + n + n + n + n + n + n + n + n + n + n + n + n + n + n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . \\ = - [ ( - [ ( A B D ] ) ] ] ^ {- 1} \\ & = - [ ( A B D ) ] ^ {- 1} \\ & = - [ ( A B D ) ] ^ {- 1} \\ & = - [ ( A B D ) ] ^ {- 1} \\ & = - [ ( A B D ) ] ^ {- 1} \\ & = - [ ( A B D ) ] ^ {- 1} \\ & = - [ ( A B D ) ] ^ {- 1} \\ & = - [ ( A B D ) ] ^ {- i k a} \\ & = - [ ( A B D ) ] ^ {- i k a} \\ & = - [ ( A B D ) ] ^ {- i k a} \\ & = - [ ( A B D ) ] ^ {- i k a} \\ & = - [ ( A B D ) ] ^ {- i k a} \\ & = - [ ( A B D ) ] ^ {- i k a} \\ & = - [ ( A B D ) ] ^ {- j k a} \\ & = - [ ( A B D ) ] ^ {- j k a} \\ & = - [ ( A B D ) ] ^ {- j k a} \\ & = - [ ( A B D ) ] ^ {- j k a} \\ & = - [ ( A B D ) ] ^ {- j k a} \\ & = - [ ( A B D ) ] ^ {- j k a} \\ & = - [ ( A B D ) ] e q u a d i s e f o r e x i g h e r m a x e r e f o r e x i g h e r m a x e r e f o r e x i g h e r m a x e r e f o r e x i g h e r m a x e r e f o r e x i g h e r m a x e r e f o r e x i g h e r m a x e r e f o r e x i g h e r m a x e r e f o l o w i v e r e f o l o w i v e r e f o l o w i v e r e f o l o w i v e r e f o l o w i v e r e f o l o w i v e r e f o l o w i v e r e f o l o w i v e r e f o l o w i v e r e f o l o w i v e r e f o l o w i w e r e f o l o w i v e r e f o l o w i v e r e f o l o w i v e r e f o l o w i v e r e f o l o w i v e r e f o l o w i v e r e f o l o w i v e r e f o l o w i v e r e f o l o w i v e r e f o l o w i V i v e r e f o l o w i V i v e r e f o l o w i V i v e r e f o l o w i V i v e r e f o l o w i V i v e r e f o l o w i V i v e r e f o l o w i V i v e r e f o l o w i V i v e r e f o l o w i V i v e v e r e f o l o w i V i v e r e f o l o w i V i v e r e f o l o w i V i v e r e f o l o w i V i v e r e f o l o w i V i v e r e f o l o w i V i v e r e f o l o w i V i v e r e f o l o w i V i v e r F O W I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I ] ^ {- 1} \\ & = - [ ( A B D ) ] ^ {- j k a} \\ & = - [ ( A B D ) ] ^ {- j k a} \\ & = - [ ( A B D ) ] ^ {- j k a} \\ & = - [ ( A B D ) ] ^ {- j k a} \\ & = - [ ( A B D ) ] ^ {- j k a} \\ & = - [ ( A B D ) ] ^ {- k a} \\ & = - [ ( A B D ) ] ^ {- j k a} \\ & = - [ ( A B D ) ] ^ {- j k a} \\ & = - [ ( A B D ) ] ^ {- j k a} \\ & = - [ ( A B D ) ] ^ {- j k a} \\ & = - [ ( A B D ) ] ^ {- j k a} \\ & = - [ ( A B D ) ]^{ j k a} \\ & = - [ ( A B D ) ]^{ j k a} \\ & = - [ ( A B D ) ]^{ j k a} \\ & = - [ ( A B D ) ]^{ j k a} \\ & = - [ ( A B D ) ]^{ j k a} \\ & = - [ ( A B D ) ]^{ j k a} \\ & = - [ ( A B D ) ]^{ j k a}. \\ & = - [ ( A B D ) ]^{ j k a}. \\ & = - [ ( A B D ) ]^{ j k a}. \\ & = - [ ( A B D ) ]^{ j k a}. \\ & = - [ ( A B D ) ]^{ j k a}. \\ & = - [ ( A B D ) ]^{ j k a}. \\ & = - [ ( A B D ) ]^{ j k a}. \\ & =- [ ( A B D ) ] ^ {- j k a}. \\ & =- [ ( A B D ) ] ^ {- j k a}. \\ & =- [ ( A B D ) ] ^ {- j k a}. \\ & =- [ ( A B D ) ] ^ {- j k a}. \\ & =- [ ( A B D ) ] ^ {- j k a}. \\ & =- [ ( A B D ) ] ^ {- j k a}. \\ & =- [ ( A B D ) ]. \\ & =- [ ( A B D ) ]. \\ & =- [ ( A B D ) ]. \\ & =- [ ( A B D ) ]. \\ & =- [ ( A B D ) ]. \\ & =- [ ( A B D ) ]. \\ & =- [ ( A B D ) ]. \\ & =- [ ( A B D ) ]. \\ & =- [ ( A B D ) ]. \\ & =- [ ( A B D ) ]. \\ & =- [ ( A B D ) ]. \\ & =- [ ( A B D ) ]. \\ & =- [ ( A B D ) ]. \\ & =- [ ( A B D ) ]. \\ & =- [ ( A B D ) ]. \\ & =- [ ( A B D ) ]. \\ & =- [ ( A B D ) ]. \\ & =- [ ( A B D ) ]. \\ & =- [ ( A B D ) ]. ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | J E S_{t} = -S_{t+1}, S_{t+2}, S_{t+3}, S_{t+4}, S_{t+5}, S_{t+6}, S_{t+7}, S_{t+8}, S_{t+9}, S_{t+10}, S_{t+11}, S_{t+12}, S_{t+13}, S_{t+14}, S_{t+15}, S_{t+16}, S_{t+17}, S_{t+18}, S_{t+19}, S_{t+20}, S_{t+21}, S_{t+22}, S_{t+23}, S_{t+24}, S_{t+25}, S_{t+26}, S_{t+27}, S_{t+28}, S_{t+29}, S_{t+30}, S_{t+31}, S_{t+32}, S_{t+33}, S_{t+34}, S_{t+35}, S_{t+36}, S_{t+37}, S_{t+38}, S_{t+39}, S_{t+40}, S_{t+41}, S_{t+42}, S_{t+43}, S_{t+44}, S_{t+45}, S_{t+46}, S_{t+47}, S_{t+48}, S_{t+49}, S_{t+50}, S_{t+51}, S_{t+52}, S_{t+53}, S_{t+54}, S_{t+55}, S_{t+56}, S_{t+57}, S_{t+58}, S_{t+59}, S_{t+60}, S_{t+61}, S_{t+62}, S_{t+63}, S_{t+64}, S_{t+65}, S_{t+66}, S_{t+67}, S_{t+68}, S_{t+69}, S_{t+70}, S_{t+71}, S_{t+72}, S_{t+73}, S_{t+74}, S_{t+75}, S_{t+76}, S_{t+77}, S_{t+78}, S_{t+79}, S_{t+80}, S_{t+81}, S_{t+82}, S_{t+83}, S_{t+84}, S_{t+85}, S_{t+86}, S_{t+87}, S_{t+88}, S_{t+89}, S_{t+90}, S.\\ & = - [ (\alpha^{\prime})^{\prime}\alpha^{\prime}\alpha^{\prime}\alpha^{\prime}\alpha^{\prime}\alpha^{\prime}\alpha^{\prime}\alpha^{\prime}\alpha^{\prime}\alpha^{\prime}\alpha^{\prime}\alpha^{\prime}\alpha^{\prime}\alpha^{\prime}\alpha^{\prime}\alpha^{\prime}\alpha^{\prime}\alpha^{\prime}\alpha^{\prime}\alpha^{\prime}\alpha^{\prime}\beta^{\prime}\beta^{\prime}\beta^{\prime}\beta^{\prime}\beta^{\prime}\beta^{\prime}\beta^{\prime}\beta^{\prime}\beta^{\prime}\beta^{\prime}\beta^{\prime}\beta^{\prime}\beta^{\prime}\beta^{\prime}\beta^{\prime}\beta^{\prime}\beta^{\prime}\beta^{\prime}\beta^{\prime}\beta^{\prime}\gamma^{\prime}\gamma^{\prime}\gamma^{\prime}\gamma^{\prime}\gamma^{\prime}\gamma^{\prime}\gamma^{\prime}\gamma^{\prime}\gamma^{\prime}\gamma^{\prime}\gamma^{\prime}\gamma^{\prime}\gamma^{\prime}\gamma^{\prime}\gamma^{\prime}\gamma^{\prime}\gamma^{\prime}\gamma^{\prime}\gamma^{\prime}\gamma^{\prime}\chi_ \chi_ \chi_ \chi_ \chi_ \chi_ \chi_ \chi_ \chi_ \chi_ \chi_ \chi_ \chi_ \chi_ \chi_ \chi_ \chi_ \chi_ \chi_ \chi_ \chi_ \chi_ \chi_ \chi_ \chi_ {\chi_ {\chi_ {\chi_ {\chi_ {\chi_ {\chi_ {\chi_ {\chi_ {\chi_ {\xi}}}}}}}) / (\Delta T) / (\Delta T) / (\Delta T) / (\Delta T) / (\Delta T) / (\Delta T) / (\Delta T) / (\Delta T) / (\Delta T) / (\Delta T) / (\Delta T) / (\Delta T) / (\Delta T) / (\Delta T) / (\Delta T) / (\Delta T) / (\Delta T) / (\Delta T) / (\Delta T) / (\Delta T) / (\Delta T/(\Delta T)/\Delta T/(\Delta T)/\Delta T/(\Delta T)/\Delta T/(\Delta T)/\Delta T/(\Delta T)/\Delta T/(\Delta T)/\Delta T/(\Delta T)/\Delta T/(\Delta T)/\Delta T/(\Delta T)/\Delta T/(\Delta T)/\Delta T/(\Delta T)/\Delta T/(\Delta T)/\Delta T/(\Delta T)/\Delta T/ (\Delta T/(\Delta T)/\Delta T/(\Delta T/(\Delta T)/\Delta T/(\Delta T/(\Delta T)/\Delta T/(\Delta T/(\Delta T)/\Delta T/(\Delta T/(\Delta T)/\Delta T/(\Delta T/(\Delta T)/\Delta T/(\Delta T/(\Delta T)/\Delta T/(\Delta T/(\Delta T)/\Delta T/(\Delta T/(\Delta T)/\Delta T/(\Delta T/ (\Delta T/(\Delta T)/\Delta T/(\Delta T/(\Delta T/(\Delta T/(\Delta T/(\Delta T/(\Delta T/(\Delta T/(\Delta T/(\Delta T/(\Delta T/(\Delta T/(\Delta T/(\Delta T/(\Delta T/(\Delta T/(\Delta T/(\Delta T/(\Delta T/(\Delta T/(\Delta T/(\Delta T/(\Delta T/(\Delta T/(\Delta T/(\Delta T/ (\Delta T/(\Delta T/ (\Delta T/ (\Delta T/ (\Delta T/ (\Delta T/ (\Delta T/ (\Delta T/ (\Delta T/ (\Delta T/ (\Delta T/ (\Delta T/ (\Delta T/ (\Delta T/ (\Delta T/ (\Delta T/ (\Delta T/ (\Delta T/ (\Delta T/ (\Delta T/ (\Delta T/ (\Delta T/ (\Delta T/ (\Delta T/ (\Delta T/ (\Delta T/ (\Delta T/ ({\bf L})})})}{\sqrt {|b|}} \\ & = - [ b ] ^ {- j k a}. \\ & = - [ b ] ^ {- j k a}. \\ & = - [ b ] ^ {- j k a}. \\ & = - [ b ] ^ {- j k a}. \\ & = - [ b ] ^ {- j k a}. \\ & = - [ b ] ^ {- j k a}. \\ & = - [ b ] ^ {- j k a}. \\ & = - [ b ] ^ {- j k a}. \\ & = - [ b ] ^{- j k a}. \\ & = - [ b ] ^{- j k a}. \\ & = - [ b ] ^{- j k a}. \\ & = - [ b ] ^{- j k a}. \\ & = - [ b ] ^{- j k a}. \\ & = - [ b ] ^{- j k a}. \\ & = - [ b ] ^{- j k a}. \\ & = - [ b ] ^{- j k a}. \\ & = - {[ b ]}.\end{array} \right]\]

Evaluated in the steady state, we get

\[\begin{array} { r l } & I E S _ { s s } = - \left[ \left( \begin{array} { c } - \phi _ { 2 } b \beta \big ( \mu _ { z ^ { * } , s s } - b \big ) ^ { - \phi _ { 2 } - 1 } \\ \hline \frac { - \phi _ { 2 } b \beta \big ( \mu _ { z ^ { * } , s s } - b \big ) ^ { - \phi _ { 2 } - 1 } } { \left( 1 - b \mu _ { z ^ { * } , s s } ^ { - 1 } \right) ^ { - \phi _ { 2 } } \left( \mu _ { z ^ { * } , s s } \right) ^ { - ( \phi _ { 2 } - 1 ) \phi _ { 4 } } - \beta b \big ( \mu _ { z ^ { * } , s s } - b \big ) ^ { - \phi _ { 2 } } } \\ + \frac { - \phi _ { 2 } \big ( \mu _ { z ^ { * } , s s } - b \big ) ^ { - \phi _ { 2 } - 1 } - \phi _ { 2 } \beta b ^ { 2 } \big ( \mu _ { z ^ { * } , s s } ^ { 2 } - b \mu _ { z ^ { * } , s s } \big ) ^ { - \phi _ { 2 } - 1 } \big ( \mu _ { z ^ { * } , s s } \big ) ^ { ( \phi _ { 2 } - 1 ) \phi _ { 4 } } } \\ \left( \mu _ { z ^ { * } , s s } - b \right) ^ { - \phi _ { 2 } } - \beta b \big ( \mu _ { z ^ { * } , s s } ^ { 2 } - b \mu _ { z ^ { * } , s s } \big ) ^ { - \phi _ { 2 } } \big ( \mu _ { z ^ { * } , s s } \big ) ^ { ( \phi _ { 2 } - 1 ) \phi _ { 4 } } \end{array} \right) \mu _ { z ^ { * } , s s } \\ & = - \left[ \left( \begin{array} { c } - \phi _ { 2 } b \beta \mu _ { z ^ { * } , s s } ^ { - \phi _ { 2 } - 1 } \left( 1 - \frac { b } { \mu _ { z ^ { * } , s s } } \right) ^ { - \phi _ { 2 } - 1 } \\ \hline \frac { - \phi _ { 2 } b \beta \mu _ { z ^ { * } , s s } ^ { - \phi _ { 2 } - 1 } ( 1 - \frac { b } { \mu _ { z ^ { * } , s s } } ) ^ { - \phi _ { 2 } - 1 } } { ( 1 - \frac { b } { \mu _ { z ^ { * } , s s } } ) ^ { - \phi _ { 2 } } ( \mu _ { z ^ { * } , s s } ) ^ { - ( \phi _ { 2 } - 1 ) \phi _ { 4 } } - \mu _ { z ^ { * } , s s } ^ { - \phi _ { 2 } } \beta b ( 1 - \frac { b } { \mu _ { z ^ { * } , s s } } ) ^ { - \phi _ { 2 } } } \\ + \frac { - \phi _ { 2 } ( \mu _ { z ^ { * } , s s } - b ) ^ { - \phi _ { 2 } - 1 } - \phi _ { 2 } \beta b ^ { 2 } ( \mu _ { z ^ { * } , s s } - b ) ^ { - \phi _ { 2 } - 1 } ( \mu _ { z ^ { * } , s s } ) ^ { ( \phi _ { 2 } - 1 ) \phi _ { 4 } - \phi _ { 2 } - 1 } } \\ ( \mu _ { z ^ { * } , s s } - b ) ^ { - \phi _ { 2 } } - \beta b ( \mu _ { z ^ { * } , s s } - b ) ^ { - \phi _ { 2 } } ( \mu _ { z ^ { * } , s s } ) ^ { ( \phi _ { 2 } - 1 ) \phi _ { 4 } - \phi _ { 2 } } ) \end{array} \right) \mu _ { z ^ { * } , s s } \\ & = - [ [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ ( [ (\mathrm{中})) ] ) ] ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ], \\ & = - [ [ (\frac { - \phi _ { 2 } b \beta \mu _ { z ^ { * } , s s } ^ { - \phi _ { 2 } - 1 } ( 1 - \frac { b } { \mu _ { z ^ { * } , s s } } ) ) ^ { - \phi _ { 2 } - 1 }} [ ( [ ( [ u a d i n e g h t h e r m a x t h e r m a x t h e r m a x t h e r m a x t h e r m a x t h e r m a x t h e r m a x t h e r m a x t h e r m a x t h e r m a x t h e r m a x t h e r m a x t h e r m a x t h e r m a x t h e r m a x t t h e r m a x t h e r m a x t t h e r m a x t t h e r m a x t t h e r m a x t t h e r m a x t t h e r m a x t t h e r m a x t t h e r m a x t t h e r m a x t t h e r m a x t t h e r m a x t t h e r m a x t t h e r m a x t t h e t h e r m a x t t h e r m a x t t h e r m a x t t h e r m a x t t h e r m a x t t h e r m a x t t h e r m a x t t h e r m a x t t h e r m a x t t h e r m a x t t h e r m a x t t h e r m a x t t h e f o r d i n i n g n o w i n g n o w i n g n o w i n g n o w i n g n o w i n g n o w i n g n o w i n g n o w i n g n o w i n g n o w i n g n o w i n g n o w i n g n o w i n g n o w i n g n o w i n g n o w i n g n o w i n g N O W I N G I N G I N G I N G I N G I N G I N G I N G I N G I N G I N G I N G I N G I N G I N G I N G I N G I N G I N G I N G I N G I N G I N G I N G I N G I N G I N G I N G I N G I N G I N G I N G I N G I N G | J | J | J | J | J | J | J | J | J | J | J | J | J | J | J | J | J | J | J | J | J | J | J | J | J | J | J | J | J | J | J | J | J | J | J | J | J | J | J | J | J | J | J | J | J | J | J | J | J | J |J | J | J | J | J | J | J | J | J | J | J | J | J | J | J | J | J | J | J | J | J |J |J |J |J |J |J |J |J |J |J |J |J |J |J |J |J |J |J |J |J |J |J |J |J |J |J |J |J |J |J |J |J |J |J |J |J |J |J |J |J |J |J |J |J |J |J |J |J |J |J |K |K |K |K |K |K |K | | & = + (\frac {- b}{b}) + (\frac {- b}{b}) + (\frac {- b}{b}) + (\frac {- b}{b}) + (\frac {- b}{b}) + (\frac {- b}{b}) + (\frac {- b}{b}) + (\frac {- b}{b}) + (\frac {- b}{b}) + (\frac {- b}{b}) + (\frac {- b}{b}) + (\frac {- b}{b}) + (\frac {- b}{b}) + (\frac {\mathrm{d} B}{B}) + (\frac {\mathrm{d} B}{B}) + (\frac {\mathrm{d} B}{B}) + (\frac {\mathrm{d} B}{B}) + (\frac {\mathrm{d} B}{B}) + (\frac {\mathrm{d} B}{B}) + (\frac {\mathrm{d} B}{B}) + (\frac {\mathrm{d} B}{B}) + (\frac {\mathrm{d} B}{B}) + (\mathrm{d} B / B) + (\frac {\mathrm{d} B}{B}) + (\frac {\mathrm{d} B}{B}) + (\frac {\mathrm{d} B}{B}) + (\frac {\mathrm{d} B}{B}) + (\frac {\mathrm{d} B}{B}) + (\frac {\mathrm{d} B}{B}) + (\frac {\mathrm{d} B}{B}) + (\frac {\mathrm{d} B}{B}) + (\frac{\mathrm{d} B}{B}) + (\frac{\mathrm{d} B}{B}) + (\frac{\mathrm{d} B}{B}) + (\frac{\mathrm{d} B}{B}) + (\frac{\mathrm{d} B}{B}) + (\frac{\mathrm{d} B}{B}) + (\frac{\mathrm{d} B}{B}) + (\frac{\mathrm{d} B}{B}) + (\frac{\mathrm{d}\delta B}{B}) + (\frac{\mathrm{d}\delta B}{B}) + (\frac{\mathrm{d}\delta B}{B}) + (\frac{\mathrm{d}\delta B}{B}) + (\frac{\mathrm{d}\delta B}{B}) + (\frac{\mathrm{d}\delta B}{B}) + (\frac{\mathrm{d}\delta B}{B}) + (\frac{\mathrm{d}\delta B}{B}) + (\frac{\delta B}{B}) + (\frac{\delta B}{B}) + (\delta B / B) + (\delta B / B) + (\delta B / B) + (\delta B / B) + (\delta B / B) + (\delta B / B) + (\delta B / B) + (\delta B / B) + (\delta B / B) + (\delta B / B) + (\delta B / B) + (\delta B / B) + (\delta B / B) + (\delta B / B) + (\delta B / B) +(\delta B / B) + (\delta B / B) + (\delta B / B) + (\delta B / B) + (\delta B / B) + (\delta B / B) + (\delta B / B) + (\delta B / B) + (\delta B / B) + (\delta B / B) + (\delta B / B) + (\delta B / B) + (\delta B / B) + (\delta B / B) + (\delta C / B) + (\delta C / B) + (\delta C / B) + (\delta C / B) + (\delta C / B) + (\delta C / B) + (\delta C / B) + (\delta C / B) + (\delta C / B) + (\delta C / B) + (\delta C / B) + (\delta C / B) + (\delta C / B) + (\delta C / B) + (\delta C / R) + (\delta C / R) + (\delta C / R) + (\delta C / R) + (\delta C / R) + (\delta C / R) + (\delta C / R) + (\delta C / R) + (\delta C / R) + (\delta C / R) + (\delta C / R) + (\delta C / R) + (\delta C / R) + (\delta C / R) + (\delta C / R) . \\ & = [ [ (- b q u v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v i v j u j k l y j u j k l y j u j k l y j u j k l y j u j k l y j u j k l y j u j k l y j u j k l y j u j k l y j u j k l y j u j k l y j u j k l y j u j k l y j u j k l y j u j k l y j u j k l y j u j k l y j u j l y j u j l y j u j l y j u j l y j u j l y j u j l y j u j l y j u j l y j u j l y j u j l y j u j l y j u j l y j u j l y j u j l y j u j l y j u j l y j u j l y j u j l y j u j l y j u j l y j uj l y j u j l y j u j l y j u j l y j u j l y j u j l y j u j l y j u j l y j u j l y j u j l y j u j l y j u j l y j u j l y j u j l y ]. \\ & = [ [ (- b q u v i v i v q u q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q p o p p o p p o p p o p p o p p o p p o p p o p p o p p o p p o p p o p p o p p o p p o p p o p p o p p o p p o p p o p p o p p o p p o p p o p p o p p o p p o p p o p p o p p o p p o p p o p p o p p o p c o n d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d d D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D DD D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D DD A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U AU A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A U A M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M L P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P S T S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S,S T S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S_S T_S T_S T_S T_S T_S T_S T_S T_S T_S T_S T_S T_S T_S T_S T_S T_S T_S T_S T_S T_S T_S T_S T_S T_S T_S T_S T_S T_S T_S T_S T_S T_S T_S T_S T_S T_S T_S T_S T_S T_S T_S T_S T_S T_S T_S T_S T_S T_S T_S T_STTSSTTSSTTSSTTSSTTSSTTSSTTSSTTSSTTSSTTSSTTSSTTSSTTSSTTSSTTSSTTSSTTSSTTSSTTSSTTSSTTSSTTSSTTSSTTSSTTSSTTSSTTSSTTSSTTSSTTSSTTSSTTSSTTSSTTSSTTSSTTSSTTSSTTSSTTSSTTSSTTSSTTSSTTSSTTSSTTSSTTSSTTSSTTSSTTSSTTSSTTS STSSTSSTSSTSSTSSTSSTSSTSSTSSTSSTSSTSSTSSTSSTSSTSSTSSTSSTSSTSSTSSTSSTSSTSSTSSTSSTSSTSSTSSTSSTSSTSSTSSTSSTSSTSSTSSTSSTSSTSSTSSTSSTSSTSSTSSTSSTSSTSSTSSTSSTS STS STS STS STS STS STS STS STS STS STS STS STS STS STS STS STS STS STS STS STS STS STS STS STS STS STS STS STS STS STS STS STS STS STS STS STS STS STS STS STS STS STS STS STS STS STS STS STS STS STS \(T\) .\\ & = [ [ (-bqu vviv i vviv i vviv i vviv i vviv i vviv i vviv i vviv i vviv i vviv i vviv i vviv i vviv i vviv i vviv i vviv i vviv i vviv i vviv i vviv i vviv i vviv i vviv i vviv i vviv i vvivi v vvivi .\\ & = [ [ (-bqu vviv i vviv i vviv i vviv i vviv i vviv i vviv i vviv i vviv i vviv i vviv i vviv i vviv i vviv i vviv i vviv i vvivi .\\ & = [ [ (-bqu vviv i vvivii] .\\ & = [ [ (-bqu wvivii] .\\ & = [ [ (-bqu wvivii] .\\ & = [ [ (-bqu wvivii] .\\ & = [ [ (-bqu wvivii] .\\ & = [ [ (-bqu wvivii] .\\ & = [ [ (-bqu wvivii] .\\ & = [ [ (-bqu wvivii] .\\ & = [[ (-bqu wvivii] .\\ & = [[ (-bqu wvivii] .\\ & = [[ (-bqu wvivii] .\\ & = [[ (-bqu wvivii] .\\ & = [[ (-bqu wvivii] .\\ & = [[ (-bqu wvivii] .\\ & = [[ (-bqu wvivii] ..\\ & = [[ (-bqu wvivii] ..\\ & = [[ (-bqu wvivii] ..\\ & = [[ (-bqu wvivii] ..\\ & = [[ (-bqu wvivii] ..\\ & = [[ (-bqu wvivii] ..\\ & = [[ (-bqu wvivii] ..\\ & = [[ (-bquwviowii] ..\\ & = [[ (-bquwviowii] ..\\ & = [[ (-bquwviowii] ..\\ & = [[ (-bquwviowii] ..\\ & = [[ (-bquwviowii] ..\\ & = [[ (-bquwviowii] ..\\ & = [[ (-bquwviowii] ..\\ & = [[ (-cqutwviowii] ..\\ & = [[ (-bqutwviowii] ..\\ & = [[ (-bqutwviowii] ..\\ & = [[ (-bqutwviowii] ..\\ & = [[ (-bqutwviowii] ..\\ & = [[ (-bqutwviowii] ..\\ & = [[ (-bqutwviowii] ..\\ & = [[ (-bqutwviowii] ..\\& = [[ (-bqutwviowii] ..\\& = [[ (-bqutwviowii] ..\\& = [[ (-bqutwviowii] ..\\& = [[ (-bqutwviowii] ..\\& = [[ (-bqutwviowii] ..\\& = [[ (-bqutwviowii] ..\\& = [[ (-bqutwviovii] ..\\& = [[ (-bqutwviowii] ..\\& = [[ (-bqutwviowii] ..\\& = [[ (-bqutwviowii] ..\\& = [[ (-bqutwviowii] ..\\& = [[ (-bqutwviowii] ..\\& = [[ (-bqutwviowii] ..\\& = [[ (-\Delta K L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O LO L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O L O\]

Suppose , then

\[\begin{array}{l} I E S _ {s s} = \left[ \phi_ {2} \left(1 - \frac {b}{\mu_ {z ^ {*} , s s}}\right) ^ {- 1} \left(\frac {b \beta}{\mu_ {z ^ {*} , s s} - \beta b} + \frac {\left(1 + \beta b ^ {2} (\mu_ {z ^ {*} , s s}) ^ {- 2}\right)}{1 - \beta b (\mu_ {z ^ {*} , s s}) ^ {- 1}}\right) \right] ^ {- 1} \\ = \frac {1}{\phi_ {2}} \left[ \frac {1 - \frac {b}{\mu_ {z ^ {*} , s s}}}{\frac {b \beta}{\mu_ {z ^ {*} , s s} - \beta b} + \frac {1 + \beta b ^ {2} (\mu_ {z ^ {*} , s s}) ^ {- 2}}{1 - \beta b (\mu_ {z ^ {*} , s s}) ^ {- 1}}} \right] \\ = \frac {1}{\phi_ {2}} \left[ \frac {1 - \frac {b}{\mu_ {z ^ {*} , s s}}}{\frac {b \beta}{\mu_ {z ^ {*} , s s} - \beta b} + \frac {\mu_ {z ^ {*} , s s} + \beta b ^ {2} (\mu_ {z ^ {*} , s s}) ^ {- 1}}{\mu_ {z ^ {*} , s s} - \beta b}} \right] \\ = \frac {1}{\phi_ {2}} \left[ \frac {\left(1 - \frac {b}{\mu_ {z ^ {*} , s s}}\right) (\mu_ {z ^ {*} , s s} - \beta b)}{\mu_ {z ^ {*} , s s} + b \beta + \beta b ^ {2} (\mu_ {z ^ {*} , s s}) ^ {- 1}} \right] \\ \text {Consider the case} \mu_ {z ^ {*}, s s} = 1 \text {and} \beta = 1, \text {which implies} \end{array}\]

\[I E S _ {s s} = \frac {1}{\phi_ {2}} \left[ \frac {(1 - b) (1 - b)}{1 + b + b ^ {2}} \right] = \frac {1}{\phi_ {2}} \frac {(1 - b) ^ {2}}{1 + b + b ^ {2}}\]

Suppose that then

\[\begin{array}{l} I E S _ {t} = \left[ \phi_ {2} \left(1 - \frac {b}{\mu_ {z ^ {*} , s s}}\right) ^ {- 1} \left(\frac {b \beta}{\left(\mu_ {z ^ {*} , s s}\right) ^ {\phi_ {2}} - \beta b} + \frac {\left(1 + \beta b ^ {2} \left(\mu_ {z ^ {*} , s s}\right) ^ {- \phi_ {2} - 1}\right)}{1 - \beta b \left(\mu_ {z ^ {*} , s s}\right) ^ {- \phi_ {2}}}\right) \right] ^ {- 1} \\ = \left[ \phi_ {2} \left(1 - \frac {b}{\mu_ {z ^ {*} , s s}}\right) ^ {- 1} \left(\frac {b \beta \left(\mu_ {z ^ {*} , s s}\right) ^ {- \phi_ {2}}}{1 - \beta b \left(\mu_ {z ^ {*} , s s}\right) ^ {- \phi_ {2}}} + \frac {\left(1 + \beta b ^ {2} \left(\mu_ {z ^ {*} , s s}\right) ^ {- \phi_ {2} - 1}\right)}{1 - \beta b \left(\mu_ {z ^ {*} , s s}\right) ^ {- \phi_ {2}}}\right) \right] ^ {- 1} \\ = \frac {1}{\phi_ {2}} \left[ \frac {1 - \frac {b}{\mu_ {z ^ {*} , s s}}}{\frac {b \beta \left(\mu_ {z ^ {*} , s s}\right) ^ {- \phi_ {2}}}{1 - \beta b \left(\mu_ {z ^ {*} , s s}\right) ^ {- \phi_ {2}}} + \frac {1 + \beta b ^ {2} \left(\mu_ {z ^ {*} , s s}\right) ^ {- \phi_ {2} - 1}}{1 - \beta b \left(\mu_ {z ^ {*} , s s}\right) ^ {- \phi_ {2}}}} \right] \\ = \frac {1}{\phi_ {2}} \left[ \frac {\frac {1}{\mu_ {z ^ {*} , s s}} \left(\mu_ {z ^ {*} , s s} - b\right) \left(1 - \beta b \left(\mu_ {z ^ {*} , s s}\right) ^ {- \phi_ {2}}\right)}{b \beta \left(\mu_ {z ^ {*} , s s}\right) ^ {- \phi_ {2}} + 1 + \beta b ^ {2} \left(\mu_ {z ^ {*} , s s}\right) ^ {- \phi_ {2} - 1}} \right] \\ = \frac {1}{\phi_ {2}} \left[ \frac {\left(\mu_ {z ^ {*} , s s} - b\right) \left(1 - \beta b \left(\mu_ {z ^ {*} , s s}\right) ^ {- \phi_ {2}}\right)}{b \beta \left(\mu_ {z ^ {*} , s s}\right) ^ {- \phi_ {2} + 1} + \mu_ {z ^ {*} , s s} + \beta b ^ {2} \left(\mu_ {z ^ {*} , s s}\right) ^ {- \phi_ {2}}} \right] \end{array}\]

Consider the case and , which implies

\[I E S _ {s s} = \frac {1}{\phi_ {2}} \left[ \frac {(1 - b) (1 - b)}{b + 1 + b ^ {2}} \right] = \frac {1}{\phi_ {2}} \frac {(1 - b) ^ {2}}{1 + b + b ^ {2}}\]

Digression

Another approach for the case of to directly obtain IES at the steady state is to evaluate at this point to get

\[I E S _ {s s} = - \left[ \left(\frac {b \beta u _ {c c} (\mu_ {z ^ {*} , s s} - b)}{u _ {c} (1 - \frac {b}{\mu_ {z ^ {*} , s s}}) - \beta b u _ {c} (\mu_ {z ^ {*} , s s} - b)} + \frac {u _ {c c} (\mu_ {z ^ {*} , s s} - b) + \beta b ^ {2} u _ {c c} (\mu_ {z ^ {*} , s s} ^ {2} - b \mu_ {z ^ {*} , s s})}{u _ {c} (\mu_ {z ^ {*} , s s} - b) - \beta b u _ {c} (\mu_ {z ^ {*} , s s} ^ {2} - b \mu_ {z ^ {*} , s s})}\right) \mu_ {z ^ {*}, s s} \right] ^ {- 1}.\]

Next let , then and so

\[\begin{array} { r l } I E S _ { s s } & = - \left[ \left( \frac { - b \beta \phi _ { 2 } \left( \mu _ { z ^ { * } , s s } - b \right) ^ { - \phi _ { 2 } - 1 } } { \left( 1 - \frac { b } { \mu _ { z ^ { * } , s s } } \right) ^ { - \phi _ { 2 } } - \beta b \left( \mu _ { z ^ { * } , s s } - b \right) ^ { - \phi _ { 2 } } } + \frac { - \phi _ { 2 } \left( \mu _ { z ^ { * } , s s } - b \right) ^ { - \phi _ { 2 } - 1 } - \beta b ^ { 2 } \phi _ { 2 } \left( \mu _ { z ^ { * } , s s } ^ { 2 } - b \mu _ { z ^ { * } , s s } \right) ^ { - \phi _ { 2 } - 1 } } { \left( \mu _ { z ^ { * } , s s } - b \right) ^ { - \phi _ { 2 } } - \beta b \left( \mu _ { z ^ { * } , s s } ^ { 2 } - b \mu _ { z ^ { * } , s s } \right) ^ { - \phi _ { 2 } } } \right) \mu _ { z ^ { * } , s s } \right] ^ { - 1 } \\ & = - \left[ \left( \frac { - b \beta \phi _ { 2 } \left( \mu _ { z ^ { * } , s s } - b \right) ^ { - \phi _ { 2 } - 1 } } { \left( 1 - \frac { b } { \mu _ { z ^ { * } , s s } } \right) ^ { - \phi _ { 2 }} - \beta b \mu _ { z ^ { * } , s s } ^ { - \phi _ { 2 } } \left( 1 - \frac { b } { \mu _ { z ^ { * } , s s } } \right) ^ { - \phi _ { 2 } } } + \frac { - \phi _ { 2 } \left( \mu _ { z ^ { * } , s s } - b \right) ^ { - \phi _ { 2 } - 1 } - \beta b ^ { 2 } \mu _ { z ^ { * } , s s } ^ { - \phi _ { 2 } - 1 } \phi _ { 2 } \left( \mu _ { z ^ { * } , s s } - b \right) ^ { - \phi _ { 2 } - 1 } } { \left( \mu _ { z ^ { * } , s s } - b \right) ^ { - \phi _ { 2 } } - \beta b \mu _ { z ^ { * } , s s } ^ { - \phi _ { 2 } } \left( \mu _ { z ^ { * } , s s } - b \right) ^ { - \phi _ { 2 } } } \right) \mu _ { z ^ { * } , s s } \\ & = - \left[ \left( \frac { - b \beta \phi _ { 2 } \mu _ { z ^ { * } , s s } ^ { - \phi _ { 2 } - 1 } \left( 1 - \frac { b } { \mu _ { z ^ { * } , s s } } \right) ^ { - \phi _ { 2 } - 1 } } { \left( 1 - \frac { b } { \mu _ { z ^ { * } , s s } } \right) ^ { - \phi _ { 2 } } [ 1 - \beta b \mu _ { z ^ { * } , s s } ^ { - \phi _ { 2 } } ] } + \frac { - \phi _ { 2 } ( \mu _ { z ^ { * } , s s } - b ) ^ { - \phi _ { 2 } - 1 } [ 1 + \beta b ^ { 2 } \mu _ { z ^ { * } , s s } ^ { - \phi _ { 2 } - 1 } ] } { ( \mu _ { z ^ { * } , s s } - b ) ^ { - \phi _ { 2 } } [ 1 - \beta b \mu _ { z ^ { * } , s s } ^ { - \phi _ { 2 } } ] } \right) \mu _ { z ^ { * } , s s } \right] ^ { - 1 } \\ & = - \left[ \left( \frac { - b \beta \phi _ { 2 } \mu _ { z ^ { * } , s s } ^ { - \phi _ { 2 } - 1 } ( 1 - \frac { b } { \mu _ { z ^ { * } , s s } } ) ^ { - 1 } } [ 1 - \beta b \mu _ { z ^ { * } , s s } ^ { - \phi _ { 2 } } ] + \frac { - \phi _ { 2 } ( \mu _ { z ^ {*} , s s} - b ) ^ { - 1 } [ 1 + \beta b ^ { 2 } \mu _ { z ^ {*} , s s } ^ { - \phi _ { 2 } - 1 } ] } [ 1 - \beta b \mu _ { z ^ {*} , s s } ^ { - \phi _ { 2 } } ] \right) \mu _ { z ^ {*} , s s } \right] ^ { - 1 } \\ & = \frac { 1 - \beta b u _ { z ^ {*} , s s} ^ { - \phi _ { 2} }} ( e f f ( e f f ( e f f ( e f f ( e f f ( e f f ( e f f ( e f f ( e f f ( e f f ( e f f ( e f f ( e f f ( e f f ( e f f ( e f f ( e f f ( e f f ( e f f ( e f f ( e f f ( e f f ( e f f ( e f f ( e f f ( e f i n t h a d o n t h a d o n t h a d o n t h a d o n t h a d o n t h a d o n t h a d o n t h a d o n t h a d o n t h a d o n t h a d o n t h a d o n t h a d o n t h a d o n t h a d o n t h a d o n t h a d o n t h o u p h i g h y m a x i v e r c o n t i o n g w i r k l y . \\ & = \frac \left( e f f ( e f f ( e f f ( e f f ( e f f ( e f f ( e f f ( e f f ( e f f ( e f f ( e f f ( e f f ( e f f ( e f f ( e f f ( e f f ( e f f ( e f f ( e f f ( e f f ( e f f ( e f i n t h a d o n t h a d o non t h a d o n t h a d o n t h a d o n t h a d o n t h a d o n t h a d o n t h a d o n t h a d o n t h a d o n t h a d o n t h a d o n t h a d o n t h a d o n t h a d o n t h a d o n t h a d o n t h a n d i v e r c o n t i o n g w i r k l y . \\ & = \frac \left( e f f ( e f i n t h a d o n t h a d o n t h a d o n t h a d o n t h a d o n t h a d o n t h a d o n t h a d o n t h a d o n t h a d o n t h a d o n t h a d o n t h a d o n t h a d o n t h a d o n t H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W HW H W W H W W H W W H W H W W H W W H W W H W W H W W H W W H W W H W W H W W H W W H W W H W W H W W H W W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W H W N O U P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P L I N O U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N OI N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O IN O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I N O I NOI N OIINOIINOIINOIINOIINOIINOIINOIINOIINOIINOIINOIINOIINOIINOIINOIINOIINOIINOIINOIINOIINOIINOIINOIINOIINOIINOIINOIINOIINOIINOIINOIINOIINOIINOiINOiINOiINOiINOiINOiINOiINOiINOiINOiINOiINOiINOiINOiINOiINOiINOiINOiINOiINOiINOiINOiINOiINOiINOiInoioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioioiosususiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiusiausiausiausiausiausiausiausiausiausiausiausiausiausiausiausiausiausiausiausiausiausiausiausiausiausiausiausiausiausiausiausiausiausiausiausiausiausiausiausiausiausiausiausiausiausiausiausiausiasuasuausiausiausiausiausiausiausiausiausiausiausiausiausiausiausiausiausiausiausiausiausiausiausiausiausiausiausiausiausiausiausiausiasuaesuaesuaesuaesuaesuaesuaesuaesuaesuaesuaesuaesuaesuaesuaesuaesuaesuaesuaesuaesuaesuaesuaesuaesuaesuaesuaesuaesuaesuaesuaesuaesuaesuaesuaesuaesuaesuaesuaesuaesuaesuaesuaesuaesuaesuaesuaesuaesuaesuaesuaeueauu u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u u / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & >\\& = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = \\& = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = & = \\& = & = & = & = & = & = & = & = & = & = \\& = & = & = & = & = \\& = & = & = & = \\& = & = & = & = \\& = & = & = \\& = & = & = \\& = & = & = \\& = & = & = \\& = & = & = \\& = & = & = \\& = & = & = \\& = & = & = \\& = & = & = \\& = & = & = \\& = & = & = \\& = & < v> < v> < v> < v> < v> < v> < v> < v> < v> < v> < v> < v> < v> < v> < v> < v> < v> < v> < v> < v> < v> < v> < v> < v> < v> < v > < v> < v> < v> < v> < v> < v> < v> < v> < v> < v> < v> < v> < v> < v> < v> < v> < v> < v> < v> < v> < v> < v> < v> < v> < v> < v >< |content_end|>\]

Note that for , we get as desired.

16.12.3 Comparing internal and external habits

Suppose and , then we get

\[I E S _ {s s} ^ {i n t e r n a l} = \frac {(1 - b) ^ {2}}{\phi_ {2} (1 + b ^ {2} + b)}\]

whereas we for comparison have because

\[\begin{array}{l} \frac {(1 - b) ^ {2}}{(1 + b ^ {2} + b)} < (1 - b) \\ \Updownarrow \end{array}\]

\[\frac {1 - b}{(1 + b ^ {2} + b)} < 1\]

\[1 - b < 1 + b ^ {2} + b\]

\[\dot {0} < b ^ {2} + 2 b\]

which is always true.

Note finally that if we have and b = 0.7 then we get

\[\begin{array}{l} {I E S _ {s s} ^ {i n t e r n a l} = \frac {(1 - 0 . 7) ^ {2}}{0 . 5 (1 + 0 . 7 ^ {2} + 0 . 7)} = 8. 2 1 9 2 \times 1 0 ^ {- 2}} \\ {I E S _ {s s} ^ {e x t e r n a l} = \frac {(1 - 0 . 7)}{0 . 5} = 0. 6} \end{array}\]

16.13 The Frisch labor supply elasticity

Recall that this elasticity is given by

\[e l a s _ {F} = \frac {u _ {h}}{h \left(u _ {h h} - \frac {u _ {c h}}{u _ {c c}}\right)}\]

\[\begin{array}{l} \text {In our case} \\ u _ {h} = - (z _ {t} ^ {*}) ^ {(1 - \phi_ {4}) (1 - \phi_ {2})} \phi_ {0} (1 - h _ {t}) ^ {- \phi_ {1}} \\ u _ {h h} = - \phi_ {1} (z _ {t} ^ {*}) ^ {(1 - \phi_ {4}) (1 - \phi_ {2})} \phi_ {0} (1 - h _ {t}) ^ {- \phi_ {1} - 1} \\ u _ {c h} = 0 \end{array}\]

So, in the steady state we have

\[\begin{array}{r l} & e l a s _ {F} = \frac {u _ {h}}{h _ {s s} \left(u _ {h h} - \frac {u _ {c h}}{u _ {c c}}\right)} \\ & \quad = \frac {- (z _ {s s} ^ {*}) ^ {(1 - \phi_ {4}) (1 - \phi_ {2})} \phi_ {0} (1 - h _ {s s}) ^ {- \phi_ {1}}}{h _ {s s} \left(- \phi_ {1} (z _ {s s} ^ {*}) ^ {(1 - \phi_ {4}) (1 - \phi_ {2})} \phi_ {0} (1 - h _ {s s}) ^ {- \phi_ {1} - 1}\right)} \\ & \quad = \frac {(1 - h _ {s s})}{\phi_ {1} h _ {s s}} \\ & \quad = \frac {1}{\phi_ {1}} \left(\frac {1}{h _ {s s}} - 1\right) \end{array}\]

16.14 Measures of relative risk aversion

We follow Swanson (2012) and compute two measures of relative risk aversion in our model. With recursive Epstein-Zin preferences controlled by , there are two measures of relative risk aversion. The first measure applies when there is no upper bound for labor and therefore total household wealth equals the present discounted value of consumption. The other measure applies when the upper bound for the household's time endowment is well-specified, meaning that total household wealth equals the present discounted value of leisure plus consumption.

Throughout this section we use the notational convention in Swanson (2012) where a variable in the steady state is denoted without a subscript. For instance c is the steady state value of .

16.14.1 External Habits

The general formulas are (see Swanson (2012), page 24, eq 53 and eq 54)

\[R R A ^ {c l} = \frac {- u _ {1 1} + \lambda u _ {1 2}}{u _ {1}} \frac {c + w (1 - h)}{1 + w \lambda} + \phi_ {3} \frac {(c + w (1 - h)) u _ {1}}{u}\]

\[R R A ^ {c} = c \left(\frac {- u _ {1 1} + \lambda u _ {1 2}}{u _ {1}} \frac {1}{1 + w \lambda} + \phi_ {3} \frac {u _ {1}}{u}\right)\]

where

\[w = - \frac {u _ {2}}{u _ {1}}\]

\[\lambda = \frac {w u _ {1 1} + u _ {1 2}}{u _ {2 2} + w u _ {1 2}}\]

Note that

\[R R A ^ {c l} = \frac {- u _ {1 1} + \lambda u _ {1 2}}{u _ {1}} \frac {c + w (1 - h)}{1 + w \lambda} + \phi_ {3} \frac {(c + w (1 - h)) u _ {1}}{u}\]

\[= \frac {c + w (1 - h)}{c} c \left[ \frac {- u _ {1 1} + \lambda u _ {1 2}}{u _ {1}} \frac {1}{1 + w \lambda} + \phi_ {3} \frac {u _ {1}}{u} \right]\]

\[R R A ^ {c l} = \left(1 + \frac {w}{c} (1 - h)\right) R R A ^ {c}\]

Here, we use the same notation that

- the utility index

- the partial derivative of with respect to consumption

- the partial derivative of with respect to hours worked

- the steady state wage level

- the steady state consumption level

- hours worked

Note that the way Swanson (2012) defines is different from the way we define in our model (i.e. the lagrange multiplier on households' budget restriction). Importantly, Swanson (2012) shows that the above formulas also hold with balanced growth (see Swanson (2012) page 48). Recall that our utility function reads (ignoring as )

\[\begin{array}{r l} & u \left(\frac {c _ {t} - b c _ {t - 1}}{z _ {t} ^ {*}}, 1 - h _ {t}\right) = \frac {1}{1 - \phi_ {2}} \left(\left(\frac {c _ {t} - b c _ {t - 1}}{(z _ {t} ^ {*}) ^ {\phi_ {4}}}\right) ^ {1 - \phi_ {2}} - (z _ {t} ^ {*}) ^ {(1 - \phi_ {4}) (1 - \phi_ {2})}\right) + (z _ {t} ^ {*}) ^ {(1 - \phi_ {4}) (1 - \phi_ {2})} \phi_ {0} \frac {(1 - h _ {t}) ^ {1 - \phi_ {1}}}{1 - \phi_ {1}} \\ & \qquad = \frac {1}{1 - \phi_ {2}} (z _ {t} ^ {*}) ^ {- \phi_ {4} (1 - \phi_ {2})} ((c _ {t} - b c _ {t - 1}) ^ {1 - \phi_ {2}} - (z _ {t} ^ {*}) ^ {(1 - \phi_ {2})}) + (z _ {t} ^ {*}) ^ {(1 - \phi_ {4}) (1 - \phi_ {2})} \phi_ {0} \frac {(1 - h _ {t}) ^ {1 - \phi_ {1}}}{1 - \phi_ {1}} \\ & \qquad = (z _ {t} ^ {*}) ^ {- \phi_ {4} (1 - \phi_ {2})} \left\{\frac {1}{1 - \phi_ {2}} (c _ {t} - b c _ {t - 1}) ^ {1 - \phi_ {2}} - \frac {1}{1 - \phi_ {2}} (z _ {t} ^ {*}) ^ {(1 - \phi_ {2})} + (z _ {t} ^ {*}) ^ {(1 - \phi_ {2})} \phi_ {0} \frac {(1 - h _ {t}) ^ {1 - \phi_ {1}}}{1 - \phi_ {1}} \right\} \\ & \qquad = (z _ {t} ^ {*}) ^ {- \phi_ {4} (1 - \phi_ {2})} \left\{\frac {1}{1 - \phi_ {2}} (c _ {t} - b c _ {t - 1}) ^ {1 - \phi_ {2}} - \frac {1}{1 - \phi_ {2}} (z _ {t} ^ {*}) ^ {\left(1 - \phi_ {2}\right)} + (z _ {t} ^ {*}) ^ {\left(1 - \phi_ {2}\right)} \phi_ {0} \frac {(1 - h _ {t}) ^ {1 - \phi_ {1}}}{1 - \phi_ {1}} \right\} \end{array}\]

From the formulas of risk-aversion we see that scaling u by a constant does not affect the measure of relative risk aversion. Hence, we can without loss of generality consider

\[u \left(\frac {c _ {t} - b c _ {t - 1}}{z _ {t} ^ {*}}, 1 - h _ {t}\right) = \frac {1}{1 - \phi_ {2}} \left(c _ {t} - b c _ {t - 1}\right) ^ {1 - \phi_ {2}} - \frac {1}{1 - \phi_ {2}} \left(z _ {t} ^ {*}\right) ^ {(1 - \phi_ {2})} + \left(z _ {t} ^ {*}\right) ^ {(1 - \phi_ {2})} \phi_ {0} \frac {(1 - h _ {t}) ^ {1 - \phi_ {1}}}{1 - \phi_ {1}}\]

Hence, we have

\[\begin{array}{l} \text {Hence, we have} \\ u _ {1} = (c _ {t} - b c _ {t - 1}) ^ {- \phi_ {2}} \\ u _ {1 1} = - \phi_ {2} (c _ {t} - b c _ {t - 1}) ^ {- \phi_ {2} - 1} \\ u _ {2} = - (z _ {t} ^ {*}) ^ {(1 - \phi_ {2})} \phi_ {0} (1 - h _ {t}) ^ {- \phi_ {1}} \\ u _ {2 2} = - (- \phi_ {1}) (z _ {t} ^ {*}) ^ {(1 - \phi_ {2})} \phi_ {0} (1 - h _ {t}) ^ {- \phi_ {1} - 1} (- 1) = - \phi_ {1} (z _ {t} ^ {*}) ^ {(1 - \phi_ {2})} \phi_ {0} (1 - h _ {t}) ^ {- \phi_ {1} - 1} \\ u _ {1 2} = 0 \end{array}\]

Note that in the steady state . Hence

\[u _ {1} = \left(c - b c \mu_ {z ^ {*}} ^ {- 1}\right) ^ {- \phi_ {2}}\]

\[u _ {1 1} = - \phi_ {2} \left(c - b c \mu_ {z ^ {*}} ^ {- 1}\right) ^ {- \phi_ {2} - 1}\]

\[u _ {2} = - \left(z ^ {*}\right) ^ {\left(1 - \phi_ {2}\right)} \phi_ {0} (1 - h) ^ {- \phi_ {1}}\]

\[\begin{array}{l} u _ {2 2} = - \phi_ {1} \left(z ^ {*}\right) ^ {(1 - \phi_ {2})} \phi_ {0} \left(1 - h\right) ^ {- \phi_ {1} - 1} \\ u _ {1 2} = 0 \\ \text {Thus} \\ w = - \frac {u _ {2}}{u _ {1}} = - \frac {- (z ^ {*}) ^ {(1 - \phi_ {2})} \phi_ {0} (1 - h) ^ {- \phi_ {1}}}{\left(c - b c \mu_ {z ^ {*}} ^ {- 1}\right) ^ {- \phi_ {2}}} = \frac {(z ^ {*}) ^ {(1 - \phi_ {2})} \phi_ {0} (1 - h) ^ {- \phi_ {1}}}{\left(c - b c \mu_ {z ^ {*}} ^ {- 1}\right) ^ {- \phi_ {2}}} \\ \Updownarrow \\ \phi_ {0} = \frac {w \big (c - b c \mu_ {z ^ {*}} ^ {- 1} \big) ^ {- \phi_ {2}}}{(z ^ {*}) ^ {(1 - \phi_ {2})} (1 - h) ^ {- \phi_ {1}}} \end{array}\]

And

\[\begin{array}{l} \text {And} \\ \lambda = \frac {w u _ {1 1} + u _ {1 2}}{u _ {2 2} + w u _ {1 2}} = \frac {\frac {(z ^ {*}) ^ {(1 - \phi_ {2})} \phi_ {0} (1 - h) ^ {- \phi_ {1}}}{(c - b c \mu_ {z} ^ {- 1}) ^ {- \phi_ {2}}} \left[ - \phi_ {2} (c - b c \mu_ {z ^ {*}} ^ {- 1}) ^ {- \phi_ {2} - 1} \right]}{- \phi_ {1} (z ^ {*}) ^ {(1 - \phi_ {2})} \phi_ {0} (1 - h) ^ {- \phi_ {1} - 1}} \\ = \frac {\frac {1}{(c - b c \mu_ {z} ^ {- 1}) ^ {- \phi_ {2}}} \phi_ {2} (c - b c \mu_ {z ^ {*}} ^ {- 1}) ^ {- \phi_ {2} - 1}}{\phi_ {1} (1 - h) ^ {- 1}} \\ = \frac {\phi_ {2} (1 - h)}{\phi_ {1} (c - b c \mu_ {z ^ {*}} ^ {- 1})} \\ \text {Note that} \\ w \lambda = \frac {(z ^ {*}) ^ {(1 - \phi_ {2})} \phi_ {0} (1 - h) ^ {- \phi_ {1}}}{(c - b c \mu_ {z ^ {*}} ^ {- 1}) ^ {- \phi_ {2}}} \frac {\phi_ {2} (1 - h)}{\phi_ {1} (c - b c \mu_ {z ^ {*}} ^ {- 1})} = \frac {\phi_ {2}}{\phi_ {1}} \frac {(z ^ {*}) ^ {(1 - \phi_ {2})} \phi_ {0} (1 - h) ^ {1 - \phi_ {1}}}{(c - b c \mu_ {z ^ {*}} ^ {- 1}) ^ {1 - \phi_ {2}}} \end{array}\]

Hence, the first measure of relative risk-aversion is:

\[\begin{array} { r l } & { R R A ^ { c } = c \left( \frac { u _ { 1 1 } } { u _ { 1 } } \frac { 1 } { 1 + w \lambda } + \phi _ { 3 } \frac { u _ { 1 } } { u } \right) } \\ & = c \frac { \phi _ { 2 } ( c - b c \mu _ { z ^ { * } } ^ { - 1 } ) ^ { - \phi _ { 2 } - 1 } } { ( c - b c \mu _ { z ^ { * } } ^ { - 1 } ) ^ { - \phi _ { 2 } } } \frac { 1 } { 1 + \frac { \phi _ { 2 } } { \phi _ { 1 } } \frac { ( z ^ { * } ) ^ { ( 1 - \phi _ { 2 } ) } \phi _ { 0 } ( 1 - h ) ^ { 1 - \phi _ { 1 } } } { ( c - b c \mu _ { z ^ { * } } ^ { - 1 } ) ^ { 1 - \phi _ { 2 } } } } \\ & + c \phi _ { 3 } \frac { ( c - b c \mu _ { z ^ { * } } ^ { - 1 } ) ^ { - \phi _ { 2 } } } { \frac { 1 } { 1 - \phi _ { 2 } } ( c - b c \mu _ { z ^ { * } } ^ { - 1 } ) ^ { 1 - \phi _ { 2 } } - \frac { 1 } { 1 - \phi _ { 2 } } ( z _ { t } ^ { * } ) ^ { ( 1 - \phi _ { 2 } ) } + ( z ^ { * } ) ^ { ( 1 - \phi _ { 2 } ) } \phi _ { 0 } \frac { ( 1 - h ) ^ { 1 - \phi _ { 1 } } } { 1 - \phi _ { 1 } } } \\ & = c \frac { \phi _ { 2 } ( c - b c \mu _ { z ^ { * } } ^ { - 1 } ) ^ { - 1 } } { 1 } \frac { ( c - b c \mu _ { z ^ { * } } ^ { - 1 } ) ^ { 1 - \phi _ { 2 } } } { ( c - b c \mu _ { z ^ { * } } ^ { - 1 } ) ^ { 1 - \phi _ { 2 } } + \frac { \phi _ { 2 } } { \phi _ { 1 } } ( z ^ { * } ) ^ { ( 1 - \phi _ { 2 } ) } \phi _ { 0 } ( 1 - h ) ^ { 1 - \phi _ { 1 } } } \\ & { + c \phi _ { 3 } \frac { ( c - b c \mu _ { z ^ { * } } ^ { - 1 } ) ^ { - \phi _ { 2 } } } { \left\{ \frac { 1 } { 1 - \phi _ { 2 } } ( c - b c \mu _ { z ^ { * } } ^ { - 1 } ) ^ { 1 - \phi _ { 2 } } - \frac { 1 } { 1 - \phi _ { 2 } } ( z _ { t } ^ { * } ) ^ { ( 1 - \phi _ { 2 } ) } + [ ( z ^ { * } ) ^ { ( 1 - \phi _ { 2 } ) } \phi _ { 0 } \frac { ( 1 - h ) ^ { 1 - \phi _ { 1 } } } { 1 - \phi _ { 1 } } \right\} }} \\ & = c \phi _ { 2 } \frac { ( c - b c \mu _ { z ^ { * } } ^ { - 1 } ) ^ { - \phi _ { 2 } } } { ( c - b c \mu _ { z ^ { * } } ^ { - 1 } ) ^ { 1 - \phi _ { 2 } } + \frac { \phi _ { 2 } } { \phi _ { 1 } } ( z ^ { * } ) ^ { ( 1 - \phi _ { 2 } ) } \phi _ { 0 } ( 1 {- h}) ^ { 1 {- \phi _ { 1} } } } \\ & + c \phi _ { 3 } \frac { ( c - b c \mu _ { z ^ { * } } ^ { - 1 } ) ^ { - \phi _ { 2} } } { \left\{ \frac { 1 } { 1 - \phi _ { 2 } } ( c - b c \mu _ { z ^ { * } } ^ { - 1 } ) ^ { 1 {- \phi _ { 2} } } - \frac { 1 } { 1 - \phi _ { 2 } } ( z _ { t } ^ { * ) ( 1 {- \phi _ { 2} )} + ( z ^ {*} ) ^ {( 1 {- \phi _ { 2} })} \phi _ { 0} \frac {( 1 {- h}) ^ { 1 {- \phi _ { 1} } }}{ 1 {- \phi _ { 1} } } \right\} }} \\ & = c \phi _ { 2 } \frac { 1 } ( c - b c \mu _ { z ^ {* *} } ^ { - 1 } ) + \frac { \phi _ { 2 } } { \phi _ { 1 } } ( z ^ {*} ) ^ { ( 1 {- \phi _ { 2} )} }\phi _ { 0 } ( 1 {- h}) ^ { 1 {- \phi _ { 1} }} ( c - b c \mu _ z ^ {*} * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . + c \phi _ { 3 } \frac { 1 } \left\{ \frac { 1}{ 1 - \phi _ { 2 }} ( c - b c \mu _ { z ^ {*} *} ^ {- 1} ) - \frac {( c - b c \mu _ { z ^ {*} *} ^ {- 1} ) ^ {\phi _ { 2} }}{ 1 {- \phi _ { 2} }} ( z _ {\tau} ^ {*}) ^ {( 1 {- \phi _ { 2} })} + ( z ^ {*} ) ^ {( 1 {- \phi _ { 2} })} \phi _ { 0 }\frac {( 1 {- h}) ^{ 1 {- \phi _ { 1} }}}\mathrm~l~o~l~o~l~o~l~o~l~o~l~o~l~o~l~o~l~o~l~o~l~o~l~o~l~o~l~o~l~o~l~o~l~o~l~o~l~o~l~o~l~o~l~o~l~o~l~o~l~o~l~a ~ l ~ o ~ l ~ o ~ l ~ o ~ l ~ o ~ l ~ o ~ l ~ o ~ l ~ o ~ l ~ o ~ l ~ o ~ l ~ o ~ l ~ o ~ l ~ o ~ l ~ o ~ l ~ o ~ l ~ o ~ l ~ o ~ l ~ o ~ l ~ o ~ l ~ o ~ l ~ o ~ l ~ o ~ l ~ o ~ l ~ o ~ l ~ o ~ l ~ o ~ l ~ a ~ l ~ o ~ l ~ o ~ l ~ o ~ l ~ o ~ l ~ o ~ l ~ o ~ l ~ o ~ l ~ o ~ l ~ o ~ l ~ o ~ l ~ o ~ l ~ o ~ l ~ o ~ l ~ o ~ l ~ o ~ l ~ o ~ l ~ o ~ l ~ o ~ l ~ o ~ l ~ o ~ l ~ o ~ l ~ o ~ l ~ o ~ l ~ o ~ l ~ e ~ i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d i n d e f f e r e s t e r m a x e r e s t e r m a x e r e s t e r m a x e r e s t e r m a x e r e s t e r m a x e r e s t e r m a x e r e s t e r m a x e r e s t e r m a x e r e s t e r m a x e r e s t e r m a x e r e s t e r m a X e S E M a X E S E M a X E S E M a X E S E M a X E S E M a X E S E M a X E S E M a X E S E M a X E S E M a X E S E M a X E S E M a X E S E M a X E S E M a X E S E M a X E S E M a X E S E M a X E S E M a X E S E M A X E S E M A X E S E M A X E S E M A X E S E M A X E S E M A X E S E M A X E S E M A X E S E M A X E S E M A X E S E M A X E S E M A X E S E M A X E S E M A X E S E M A X E S E M A X E S E M A X E S S E M A X E S S E M A X E S S E M A X E S S E M A X E S S E M A X E S S E M A X E S S E M A X E S S E M A X E S S E M A X E S S E M A X E S S E M A X E S S E M A X E S S E M A X E S S E M A X E S S S E M A X E S S S E M A X E S S S E M A X E S S S E M A X E S S S E M A X E S S S E M A X E S S S E M A X E S S S S E M A X E S S S S E M A X E S S S S E M A X E S S S S E M A X E S S S S E M A X E S S S S S F e r e s t e r m a x e r m a x e r m a x e r m a x e r m a x e r m a x e r m a x e r m a x e r m a x e r m a x e r m a x e r m a x e r m a x e r m a x e r m a x e r m a x e r m a x e r m a x e r m a x e r m a x e r m a , v i v e r s t e r m a x e r m a x e r m a x e r m a x e r m a x e r m a x e r m a x e r m a x e r m a x e r m a x e r m a x e r m a x e r m a x e r m a x e r m a x e r m a x e r m a x e r m a x e r m a x e r s t e r m a x e r m a x e r m a x e r m a x e r m a x e r m a x e r m a x e r m a x e r m a x e r m a x e r m a x e r m a x e r m a x e r m a x e r m a x e r m a x e r m a x e r m a x e r m a x & = c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c & = C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C C & = B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B W & = D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D D I & = N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N NNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNInNInNInNInNInNInNInNInNInNInNInNInNInNInNInNInNInNInNInNInNInNInNInNInNInNInNInNInNInNInNInNInNInNInNInNInNInNInNInNInNInNInNInNInNInNInNInNInNInNInN InIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIinIainiainiainiainiainiainiainiainiainiainiainiainiainiainiainiainiainiainiainiainiainiainiainiainiainiainiainiainiainiainiainiainiainiainiainiainiainiainiainiainiainiainiainiainiainiainiainiainiainiainiainkai inkai inkai inkai inkai inkai inkai inkai inkai inkai inkai inkai inkai inkai inkai inkai inkai inkai inkai inkai inkai inkai inkai inkai inkai inkai inkai inkai inkai inkai inkai inkai inkai inkai in kai inkai inkai inkai inkai inkai inkai inkai inkai inkai inkai inkai inkai inkai inkai inkai inkai inkai inkai inkai inkai inkai inkai inkai inkai inkai inkai inkai inkai inkai inkai inkai inkai inkau in kau in kau in kau in kau in kau in kau in kau in kau in kau in kau in kau in kau in kau in kau in kau in kau in kau in kau in kau in kau in kau in kau in kau in kau in kau in kau in kau in kau in kau in kau in kau in kau in kau in kou and kau. & = K U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U I & = K I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I II & = K J K J K J K J K J K J K J K J K J K J K J K J K J K J K J K J K J K J K J K J K J K J K J K J K J K J K J K J K J K J K J K J K J K J K J K J K J K J K J K J K J K J K J K J K J K J K J K J K J K J KJKJKJKJKJKJKJKJKJKJKJKJKJKJKJKJKJKJKJKJKJKJKJKJKJKJKJKJKJKJKJKJKJKJKJKJKJKJKJKJKJKJKJKJKJKJKJKJKJKJKJLIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII, II III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III III II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II II-II-II-II-II-II-II-II-II-II-II-II-II-II-II-II-II-II-II-II-II-II-II-II-II-II-II-II-II-II-II-II-II-II-II-II-II-II-II-II-II-II-II-II-II-II-II-II-II-II-II-II-II-II-II-II-II-II-II-II-II-II-II-III, IV, V, V, V, V, V, V, V, V, V, V, V, V, V, V, V, V, V, V, V, V, V, V, V, V, V, V, V, V, V, V, V, V, V, V, V, V, V, V, V, V, V, V, V, V, V, V, V, V, V, V, and viiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii. \\ & = Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z Z ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZzZZzZZzZZzZZzZZzZZzZZzZZzZZzZZzZZzZZzZZzZZzZZzZZzZZzZZzZZzZZzZZzZZzZZzZZzZZzZZzZZzZZzZZzZZzZZzZZzZZzZZzZZzZZzZZzZZzZZzZZzZZzZZzZZzZZzZZzZZzZZzZZzZZzZZzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzggggggggggggggggggggggggggggggggggggggggggggggggggggggggggggggggggggggggggggggggggggggggggggggggggggggggggggggggggggggggggggggggggggggggggggggggggggggggttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttlllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllllcccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcctcactccctccctccctccctccctccctccctccctccctccctccctccctccctccctccctccctccctccctccctccctccctccctccctccctccctccctccctccctccctccctccctccctccctccctccctccctccctccctccctccctccctccaaccaaccaaccaaccaaccaaccaaccaaccaaccaaccaaccaaccaaccaaccaaccaaccaaccaaccaaccaaccaaccaaccaaccaaccaaccaaccaaccaaccaaccaaccaaccaaccaaccaaccaaccaaccaaccaaccaaccaaccaaccaaccaaccaaccaaccaaccaaccaaccaaccaaccaacaaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaacaatacaatacaatacaatacaatacaatacaatacaatacaatacaatacaatacaatacaatacaatacaatacaatacaatacaatacaatacaatacaatacaatacaatacaatacaatacaatacaatacaatacaatacaatacaatacaatacaatacaatacaatacaatacaatacaatacaatacaatacaatacaatacaatacaatacaatacaatacaatacaatacaatacaatacaataccaataccaataccaataccaataccaataccaataccaataccaataccaataccaataccaataccaataccaataccaataccaataccaataccaataccaataccaataccaataccaataccaataccaataccaataccaataccaataccaataccaataccaataccaataccaataccaataccaataccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatcca atccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatccaatCCAATCCAATCCAATCCAATCCAATCCAATCCAATCCAATCCAATCCAATCCAATCCAATCCAATCCAATCCAATCCAATCCAATCCAATCCAATCCAATCCAATCCAATCCAATCCAATCCAATCCAATCCAATCCAATCCAATCCAATCCAATCCAATCCAATCCAATCCAATCCAATCCAATCCAATCCAATCCAATCCAATCCAATCCAATCCAATCCAATCCAATCCAATCCAATCCAATCCAATCCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATT CAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATTCAATT CAATTCAATTCAATT CAATT CAATT CAATT CAATT CAATT CAATT CAATT CAATT CAATT CAATT CAATT CAATT CAATT CAATT CAATT CAATT CAATT CAATT CAATT CAATT CAATT CAATT CAATT CAATT CAATT CAATT CAATT CAATT CAATT CAATT CAATT CAATT CAATT CAATT CAATT CAATT CAATT CAATT CAINT CAINT CAINT CAINT CAINT CAINT CAINT CAINT CAINT CAINT CAINT\]

\[\begin{array}{l} + c \phi_ {3} \frac {1}{\left\{\frac {1}{1 - \phi_ {2}} \left(c - b c \mu_ {z ^ {*}} ^ {- 1}\right) - \frac {\left(c - b c \mu_ {z ^ {*}} ^ {- 1}\right) ^ {\phi_ {2}}}{1 - \phi_ {2}} (z _ {t} ^ {*}) ^ {(1 - \phi_ {2})} + (z ^ {*}) ^ {(1 - \phi_ {2})} \frac {w \left(c - b c \mu_ {z ^ {*}} ^ {- 1}\right) ^ {- \phi_ {2}}}{(z ^ {*}) ^ {(1 - \phi_ {2})} (1 - h) ^ {- \phi_ {1}}} \frac {(1 - h) ^ {1 - \phi_ {1}}}{1 - \phi_ {1}} \left(c - b c \mu_ {z ^ {*}} ^ {- 1}\right) ^ {\phi_ {2}} \right\}} \\ \text {using} \phi_ {0} = \frac {w \left(c - b c \mu_ {z ^ {*}} ^ {- 1}\right) ^ {- \phi_ {2}}}{(z ^ {*}) ^ {(1 - \phi_ {2})} (1 - h) ^ {- \phi_ {1}}} \\ = c \left(\phi_ {2} \frac {1}{\left(c - b c \mu_ {z ^ {*}} ^ {- 1}\right) + \frac {\phi_ {2}}{\phi_ {1}} \frac {w (1 - h)}{1}} + \phi_ {3} \frac {1}{\left\{\frac {1}{1 - \phi_ {2}} \left(c - b c \mu_ {z ^ {*}} ^ {- 1}\right) - \frac {\left(c - b c \mu_ {z ^ {*}} ^ {- 1}\right) ^ {\phi_ {2}}}{1 - \phi_ {2}} (z _ {t} ^ {*}) ^ {( 1 - \phi_ {2})} + \frac {w}{1} \frac {(1 - h)}{1 - \phi_ {1}} \right\}}\right) \\ = \frac {\phi_ {2}}{1 - b \mu_ {z ^ {*}} ^ {- 1} + \frac {\phi_ {2}}{\phi_ {1}} \frac {w (1 - h)}{c}} + \phi_ {3} \frac {1 - \phi_ {2}}{1 - b \mu_ {z ^ {*}} ^ {- 1} - \left(c - b c \mu_ {z ^ {*}} ^ {- 1}\right) ^ {\phi_ {2}} \frac {(z _ {t} ^ {*}) ^ {(1 - \phi_ {2})}}{c} + \frac {w (1 - h)}{c} \frac {1 - \phi_ {2}}{1 - \phi_ {1}}} \end{array}\]

We will now express this expression in terms of variables for the transformed economy:

\[\begin{array}{r l} & {R R A ^ {c} = \frac {\phi_ {2}}{1 - b \mu_ {z ^ {*}} ^ {- 1} + \frac {\phi_ {2}}{\phi_ {1}} \frac {w (1 - h)}{c} \frac {z ^ {*}}{z ^ {*}}} + \phi_ {3} \frac {1 - \phi_ {2}}{1 - b \mu_ {z ^ {*}} ^ {- 1} - (c - b c \mu_ {z ^ {*}} ^ {- 1}) ^ {\phi_ {2}} \frac {(z _ {t} ^ {*}) ^ {(1 - \phi_ {2})}}{c} + \frac {w (1 - h)}{c} \frac {1 - \phi_ {2}}{1 - \phi_ {1}} \frac {z ^ {*}}{z ^ {*}}}} \\ & {\qquad = \frac {\phi_ {2}}{1 - b \mu_ {z ^ {*}} ^ {- 1} + \frac {\phi_ {2}}{\phi_ {1}} \frac {w (1 - h)}{c} \frac {z ^ {*}}{z ^ {*}}} + \phi_ {3} \frac {1 - \phi_ {2}}{1 - b \mu_ {z ^ {*}} ^ {- 1 -} (\frac {c}{z _ {t} ^ {*}} - b \frac {c}{z _ {t} ^ {*}} \mu_ {z ^ {*}} ^ {- 1}) ^ {\phi_ {2}} \frac {1}{c / z _ {t} ^ {*}} + \frac {w (1 - h)}{c} \frac {1 - \phi_ {2}}{1 - \phi_ {1}} \frac {z ^ {*}}{z ^ {*}}}} \\ & {\qquad = \frac {\phi_ {2}}{1 - b \mu_ {z ^ {*}} ^ {- 1} + \frac {\phi_ {2}}{\phi_ {1}} \frac {w (1 - H)}{c} \frac {z ^ {*}}{z ^ {*}}} + \phi_ {3} \frac {1 - \phi_ {2}}{1 - b \mu_ {z ^ {*}} ^ {- 1} - (C - b C \mu_ {z ^ {*}} ^ {- 1}) ^ {\phi_ {2}} \frac {1}{C} + \frac {w (1 - h)}{c} \frac {1 - \phi_ {2}}{1 - \phi_ {1}} \frac {z ^ {*}}{z ^ {*}}}} \\ & {\qquad = \frac {\phi_ {2}}{1 - b \mu_ {z ^ {*}} ^ {- 1} + \frac {\phi_ {2}}{\phi_ {1}} \frac {W (1 - H)}{C}} + \phi_ {3} \frac {1 - \phi_ {2}}{1 - b \mu_ {z ^ {*}} ^ {- 1} - (C - b C \mu_ {z ^ {*}} ^ {- 1}) ^ {\phi_ {2}} \frac {1}{C} + \frac {W (1 - h)}{C} \frac {1 - \phi_ {2}}{1 - \phi_ {1}}}} \\ & {\qquad = \frac {\phi_ {2}}{1 - b \mu_ {z ^ {*}} ^ {- 1} + \frac {\phi_ {2}}{\phi_ {1}} \frac {W (1 - H)}{C}} + \phi_ {3} \frac {1 - \phi_ {2}}{1 - b \mu_ {z ^ {*}} ^ {- 1} - (1 - b \mu_ {z ^ {*}} ^ {- 1}) ^ {\phi_ {2}} C \phi_ {2} - 1 + \frac {W (1 - h)}{C} \frac {1 - \phi_ {2}}{1 - \phi_ {1}}}.} \end{array}\]

\[\begin{array}{l} \text {If we follow Swanson (2012) and assume that C = hW, then} \\ R R A ^ {c} = \frac {\phi_ {2}}{1 - b \mu_ {z ^ {*}} ^ {- 1} + \frac {\phi_ {2}}{\phi_ {1}} \frac {W (1 - h)}{h W}} + \phi_ {3} \frac {1 - \phi_ {2}}{1 - b \mu_ {z ^ {*}} ^ {- 1} - \left(C - b C \mu_ {z ^ {*}} ^ {- 1}\right) ^ {\phi_ {2}} \frac {1}{C} + \frac {W (1 - h)}{h W} \frac {1 - \phi_ {2}}{1 - \phi_ {1}}} \\ = \frac {\phi_ {2}}{1 - b \mu_ {z ^ {*}} ^ {- 1} + \frac {\phi_ {2}}{\phi_ {1}} \frac {(1 - h)}{h}} + \phi_ {3} \frac {1 - \phi_ {2}}{1 - b \mu_ {z ^ {*}} ^ {- 1} - \left(C - b C \mu_ {z ^ {*}} ^ {- 1}\right) ^ {\phi_ {2}} \frac {1}{C} + \frac {(1 - h)}{h} \frac {1 - \phi_ {2}}{1 - \phi_ {1}}}. \end{array}\]

\[\begin{array}{l} \text {And finally:} \\ R R A ^ {c l} = \left(1 + \frac {w}{c} (1 - h)\right) R R A ^ {c} = \left(1 + \frac {W}{C} (1 - h)\right) R R A ^ {c} \end{array}\]

16.14.2 Internal habits

The general formulas with internal habits are (see Swanson (2012), page 29, eq 72 and eq 73, where the Epstein-Zin coefficient is and )

\[\begin{array}{r c l} R R A ^ {c l} & = & \frac {- u _ {1 1} + \lambda u _ {1 2}}{u _ {1}} \frac {(1 - \beta b) (c + w (1 - h))}{1 + (1 - \beta b) w \lambda} + \phi_ {3} \frac {(c + w (1 - h)) u _ {1}}{u} (1 - \beta b) \\ & = & (1 - \beta b) \left[ \frac {- u _ {1 1} + \lambda u _ {1 2}}{u _ {1}} \frac {(c + w (1 - h))}{1 + (1 - \beta b) w \lambda} + \phi_ {3} \frac {(c + w (1 - h)) u _ {1}}{u} \right] \end{array}\]

and

\[\begin{array}{r c l} {R R A ^ {c}} & = & {\frac {- u _ {1 1} + \lambda u _ {1 2}}{u _ {1}} \frac {(1 - \beta b) c}{1 + (1 - \beta b) w \lambda} + \phi_ {3} \frac {c u _ {1}}{u} (1 - \beta b)} \\ & = & {(1 - \beta b) c \left[ \frac {- u _ {1 1} + \lambda u _ {1 2}}{u _ {1}} \frac {1}{1 + (1 - \beta b) w \lambda} + \phi_ {3} \frac {u _ {1}}{u} \right]} \end{array}\]

As for external habits, we have

\[R R A ^ {c l} = \left(1 + \frac {w}{c} (1 - h)\right) R R A ^ {c}\]

In the stated expressions, we use the fact that we have , so in the steady state.

We next note that the expressions for and are different with internal habits. In the Appendix in Swanson (2012) (the proof of proposition 15) we have with internal habits that:

\[w = - \frac {u _ {2}}{u _ {1}} \frac {1}{1 - \beta b}\]

\[\lambda = \frac {w (1 - \beta b) u _ {1 1} + u _ {1 2}}{u _ {2 2} + w (1 - \beta b) u _ {1 2}}\]

Recall that we have in the steady state:

\[\begin{array}{l} u _ {1} = \left(c - b c \mu_ {z ^ {*}} ^ {- 1}\right) ^ {- \phi_ {2}} \\ u _ {1 1} = - \phi_ {2} \left(c - b c \mu_ {z ^ {*}} ^ {- 1}\right) ^ {- \phi_ {2} - 1} \\ u _ {2} = - (z ^ {*}) ^ {(1 - \phi_ {2})} \phi_ {0} (1 - h) ^ {- \phi_ {1}} \\ u _ {2 2} = - \phi_ {1} (z ^ {*}) ^ {(1 - \phi_ {2})} \phi_ {0} (1 - h) ^ {- \phi_ {1} - 1} \\ u _ {1 2} = 0 \\ \text {Thus} \\ w = - \frac {u _ {2}}{u _ {1}} \frac {1}{1 - \beta b} = - \frac {- (z ^ {*}) ^ {(1 - \phi_ {2})} \phi_ {0} (1 - h) ^ {- \phi_ {1}}}{(c - b c \mu_ {z ^ {*}} ^ {- 1}) ^ {- \phi_ {2}}} \frac {1}{1 - \beta b} = \frac {(z ^ {*}) ^ {(1 - \phi_ {2})} \phi_ {0} (1 - h) ^ {- \phi_ {1}}}{(c - b c \mu_ {z ^ {*}} ^ {- 1}) ^ {- \phi_ {2}}} \frac {1}{1 - \beta b} \\ \Updownarrow \\ \phi_ {0} = \frac {w (1 - \beta b) (c - b c \mu_ {z ^ {*}} ^ {- 1}) ^ {- \phi_ {2}}}{(1 - h) ^ {- \phi_ {1}} (z ^ {*}) ^ {(1 - \phi_ {2})}} \\ \lambda = \frac {w (1 - \beta b) u _ {1 1}}{u _ {2 2}} = \frac {(z ^ {*}) ^ {(1 - \phi_ {2})} \phi_ {0} (1 - h) ^ {- \phi_ {1}}}{(c - b c \mu_ {z ^ {*}} ^ {- 1}) ^ {- \phi_ {2}}} \frac {(1 - \beta b)}{1 - \beta b} \frac {- \phi_ {2} (c - b c \mu_ {z ^ {*}} ^ {- 1}) ^ {- \phi_ {2} - 1}}{- \phi_ {1} (z ^ {*}) ^ {(1 - \phi_ {2})} \phi_ {0} (1 - h) ^ {- \phi_ {1} - 1}} \\ = \frac {(1 - \beta b)}{1 - \beta b} \frac {\phi_ {2} (c - b c \mu_ {z ^ {*}} ^ {- 1}) ^ {- 1}}{\phi_ {1} (1 - h) ^ {- 1}} \\ = \frac {\phi_ {2} (1 - h)}{\phi_ {1} (c - b c \mu_ {z ^ {*}} ^ {- 1})} \\ \text {Note also that} \\ w \lambda = \frac {(z ^ {*}) ^ {(1 - \phi_ {2})} \phi_ {0} (1 - h) ^ {- \phi_ {1}}}{(c - b c \mu_ {z ^ {*}} ^ {- 1}) ^ {- \phi_ {2}}} \frac {1}{1 - \beta b} \frac {1 - \beta b}{1 - \beta b} \frac {\phi_ {2} (1 - h)}{\phi_ {1} (c - b c \mu_ {z ^ {*}} ^ {- 1})} \\ = \frac {(z ^ {*}) ^ {(1 - \phi_ {2})} \phi_ {0} (1 - h) ^ {1 - \phi_ {1}}}{(c - b c \mu_ {z ^ {*}} ^ {- 1}) ^ {1 - \phi_ {2}}} \frac {1}{1 - \beta b} \frac {\phi_ {2}}{\phi_ {1}} \end{array}\]

Hence,

\[\begin{array}{r l} & {R R A ^ {c} = (1 - \beta b) c \left[ \frac {- u _ {1 1}}{u _ {1}} \frac {1}{1 + (1 - \beta b) w \lambda} + \phi_ {3} \frac {u _ {1}}{u} \right]} \\ & \quad = (1 - \beta b) c \frac {\phi_ {2} (c - b c \mu_ {z ^ {*}} ^ {- 1}) ^ {- \phi_ {2} - 1}}{(c - b c \mu_ {z ^ {*}} ^ {- 1}) ^ {- \phi_ {2}}} \frac {1}{1 + (1 - \beta b) \frac {(z ^ {*}) ^ {(1 - \phi_ {2})} \phi_ {0} (1 - h) ^ {1 - \phi_ {1}}}{(c - b c \mu_ {z ^ {*}} ^ {- 1}) ^ {1 - \phi_ {2}}} \frac {1}{1 - \beta b} \frac {\phi_ {2}}{\phi_ {1}}} \\ & {\quad + (1 - \beta b) c \phi_ {3} \frac {(c - b c \mu_ {z ^ {*}} ^ {- 1}) ^ {- \phi_ {2}}}{\frac {1}{1 - \phi_ {2}} (c - b c \mu_ {z ^ {*}} ^ {- 1}) ^ {1 - \phi_ {2}} - \frac {1}{1 - \phi_ {2}} (z _ {t} ^ {*}) ^ {(1 - \phi_ {2})} + (z ^ {*}) ^ {(1 - \phi_ {2})} \phi_ {0} \frac {(1 - h) ^ {1 - \phi_ {1}}}{1 - \phi_ {1}}}} \end{array}\]

\[\begin{array} { r l } & = ( 1 - \beta b ) c \frac { \phi _ { 2 } \left( c - b c \mu _ { z ^ { * } } ^ { - 1 } \right) ^ { - 1 } } { 1 } \frac { \left( c - b c \mu _ { z ^ { * } } ^ { - 1 } \right) ^ { 1 - \phi _ { 2 } } } { \left( c - b c \mu _ { z ^ { * } } ^ { - 1 } \right) ^ { 1 - \phi _ { 2 } } + ( 1 - \beta b ) ( z ^ { * } ) ^ { ( 1 - \phi _ { 2 } ) } \phi _ { 0 } ( 1 - h ) ^ { 1 - \phi _ { 1 } } \frac { 1 } { 1 - \beta b } \frac { \phi _ { 2 } } { \phi _ { 1 } } } \\ & + ( 1 - \beta b ) c \phi _ { 3 } \frac { \left( c - b c \mu _ { z ^ { * } } ^ { - 1 } \right) ^ { - \phi _ { 2 } } } { \frac { 1 } { 1 - \phi _ { 2 } } \left( c - b c \mu _ { z ^ { * } } ^ { - 1 } \right) ^ { 1 - \phi _ { 2 } } - \frac { 1 } { 1 - \phi _ { 2 } } ( z _ { t } ^ { * } ) ^ { ( 1 - \phi _ { 2 } ) } + ( z ^ { * } ) ^ { ( 1 - \phi _ { 2 } ) } \phi _ { 0 } \frac { ( 1 - h ) ^ { 1 - \phi _ { 1 } } } { 1 - \phi _ { 1 } } } \\ & = ( 1 - \beta b ) c \frac { \phi _ { 2 } } { 1 } \frac { \left( c - b c \mu _ { z ^ { * } } ^ { - 1 } \right) ^ { - \phi _ { 2 } } } \left( c - b c \mu _ { z ^ { * } } ^ { - 1 } \right) ^ { 1 - \phi _ { 2 } } + ( 1 - \beta b ) ( z ^ { * } ) ^ { ( 1 - \phi _ { 2 } ) } \frac { w ( 1 - \beta b ) \left( c - b c \mu _ { z ^ { * } } ^ { - 1 } \right) ^ { - \phi _ { 2 } } } { ( 1 - h ) ^ { - \phi _ { 1 } } ( z ^ { * } ) ^ { ( 1 - \phi _ { 2 } ) } } ( 1 - h ) ^ { 1 - \phi _ { 1 } } \frac { 1 } { 1 - \beta b } \frac { \phi _ { 2 } } { \phi _ { 1 } } + \\ & + ( 1 - \beta b ) c \phi _ { 3 } \frac { \left( c - b c \mu _ { z ^ { * } } ^ { - 1 } \right) ^ { - \phi _ { 2 } } } \frac { 1 } { 1 - \phi _ { 2 } } \left( c - b c \mu _ { z ^ { * } } ^ { -1} \right) ^ { 1 - \phi _ { 2 } } - \frac { 1 } { 1 - \phi _ { 2 } } ( z _ { t } ^ { * } ) ^ { ( 1 - \phi _ { 2 } ) } + ( z ^ { * } ) ^ { ( 1 - \phi _ { 2 } ) } \frac { w ( 1 - \beta b ) \left( c - b c \mu _ { z ^ { * } } ^ { -1} \right) ^ { - \phi _ { 2 } } } { ( 1 - h ) ^ { - \phi _ { 1 } } ( z ^ { * } ) ^ { ( 1 - \phi _ { 2 } ) } } \frac { ( 1 - h ) ^ { 1 - \phi _ { 1 } } } { 1 - \phi _ { 1 } } \\ & u s i n g \phi _ { 0 } = \frac { w ( 1 - \beta b ) \left( c - b c \mu _ { z ^ { * } } ^ { -1} \right) ^ {- \phi _ { 2 } } } { ( 1 - h ) ^ { - \phi _ { 1 } } ( z ^ { * } ) ^ { ( 1 - \phi _ { 2} ) } } \\ & = ( 1 - \beta b ) c [ \frac { \phi _ { 2 } } { 1 } \frac { ( c - b c \mu _ { z ^ { * } } ^ { -1} ) ^ { - \phi _ { 2 } } } ( c - b c \mu _ { z ^ {*} } ^ { -1} ) ^ { 1 - \phi _ { 2 } } + ( 1 - \beta b ) \frac { w ( c - b c \mu _ { z ^ {*} } ^ { -1} ) ^ { - \phi _ { 2} } } { 1 } ( 1 - h ) \frac { \phi _ { 2} } { \phi _ { 1} } + \phi _ { 3 } \frac { ( c - b c \mu _ { z ^ {*} } ^ { -1} ) ^ { - \phi _ { 2} } } { \frac { 1 } { 1 - \phi _ { 2 } } ( c - b c \mu _ { z ^ {*} } ^ { -1} ) ^ { 1 - \phi _ { 2 } } - \frac { 1 } { 1 - \phi _ { 2 } } ( z _ { t } ^ { * } ) ^ { ( 1 - \phi _ { 2} ) } + \frac { w ( 1 - \beta b ) ( c - b c \mu _ { z ^ {*} } ^ { -1} ) ^ { - \phi _ { 2} } } { 1 }} ( \frac 1 - h ) ^ ( p + q + q + q + q + q + q + q + q + q + q + q + q + q + q + q + q + q + q + q + q + q + q + q + q + q + q + q + q + q + q + q + q + q + q + q + q + q + q + q + q + q + q + q + q + q + q + q + q + q + q + k , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . e f f o r e d i n e a l l e m a x e n t i o n e f f o r e d i n e a l l e m a x e n t i o n e f f o r e d i n e a l l e m a x e n t i o n e f f o r e d i n e a l l e m a x e n t i o n e f f o r e d i n e a l l e m a x e n t i o n e f f o r e d e f f o r e d i n e a l l e m a x e n t i o n e f f o r e d e f f o r e d i n e a l l e m a x e n t i o n e f f o r e d e f f o r e d i n e a l l e m a x e n t i o n e f f o r e d e f f o r e d i n i o n e f f o r e d e f f o r e d i n i o n e f f o r e d e f f o r e d i n i o n e f f o r e d e f f o r e d i n i o n e f f o r e d e f f o r e d i n i o n e f f o r e d e f f o r e d i n i o n ef f o r e d e f f o r e d i n i o n e f f o r e d e f f o r e d i n i o n e f f o r e d e f f o r e d i n i o n e f f o r e d e f f o r e d i n i o n e f f o r e d e f f o r e d i n i o n e f f o r e t h a l l u a l l u a l l u a l l u a l l u a l l u a l l u a l l u a l l u a l l u a l l u a l l u a l l u a l l u a l l u a l l u a l l u a l l u a l l u a l l u a l l u a l l u a l l u a l l u a l l u a l l , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | j | j | j | j | j | j | j | j | j | j | j | j | j | j | j | j | j | j | j | j | j | j | j | j | j | j | j | j | j | j | j | j | j | j | j | j | j | j | j | j | j | j | j | j | j | j | j | j | j | j |j | j | j | j | j | j | j | j | j | j | j | j | j | j | j | j | j | j | j | j | j | j | j | j | j | j | j | j | j | j | j | j | j | j | j | j | j | | & = (\mathrm{if} b > (\mathrm{if} b < b < b < b < b < b < b < b < b < b < b < b < b < b < b < b < b < b < b < b < b < b < b < b < b < b < b < b < b < b < b < b < b < b < b < b < b < b < b < b < b < b < b < b < b < b < b < b < b < b < b < b< |content_end|>\]

Expressed in terms of the transformed economy:

\[R R A ^ {c} = \frac {\phi_ {2}}{\frac {1 - b \mu_ {z ^ {*}} ^ {- 1}}{1 - \beta b} + \frac {W (1 - h)}{C} \frac {\phi_ {2}}{\phi_ {1}}} + \phi_ {3} \frac {1 - \phi_ {2}}{\frac {1 - b \mu_ {z ^ {*}} ^ {- 1}}{1 - \beta b} - \frac {(C - b C \mu_ {z ^ {*}} ^ {- 1}) ^ {\phi_ {2}}}{1 - \beta b} \frac {1}{C} + \frac {W (1 - h)}{C} \frac {1 - \phi_ {2}}{1 - \phi_ {1}}}\]

And finally:

\[R R A ^ {c l} = \left(1 + \frac {w}{c} (1 - h)\right) R R A ^ {c} = \left(1 + \frac {W}{C} (1 - h)\right) R R A ^ {c}\]

16.15 The steady state

This section derives the values for the transformed variables in steady state as a function of the structural parameters. We denote variables in steady state by subscript ss. The steady state value of labor, i.e. , is assumed to be given and we then back out the value of . We also assume that is known.

The growth rate in

\[\begin{array}{r l} & {\mu_ {\lambda , t + 1} \equiv \frac {\Lambda_ {t + 1}}{\Lambda_ {t}} \left(\frac {\left[ E _ {t} \left[ \left(- \widetilde {V _ {t + 1}}\right) ^ {1 - \phi_ {3}} \right] \right] ^ {\frac {1}{1 - \phi_ {3}}}}{- \widetilde {V _ {t + 1}}}\right) ^ {\phi_ {3}} \mu_ {z ^ {*}, t + 1} ^ {- \phi_ {2} (1 - \phi_ {4}) - \phi_ {4}}} \\ & {\Downarrow} \\ & {\mu_ {\lambda , s s} \equiv \left(\frac {\left[ E _ {t} \left[ \left(- \widetilde {V _ {s s}}\right) ^ {1 - \phi_ {3}} \right] \right] ^ {\frac {1}{1 - \phi_ {3}}}}{- \widetilde {V _ {s s}}}\right) ^ {\phi_ {3}} \mu_ {z ^ {*}, s s} ^ {- \phi_ {2} (1 - \phi_ {4}) - \phi_ {4}}} \\ & {\Updownarrow} \\ & {\mu_ {\lambda , s s} \equiv \mu_ {z ^ {*}, s s} ^ {- \phi_ {2} (1 - \phi_ {4}) - \phi_ {4}}} \end{array}\]

The optimal relative price, .

From EQ 11

\[1 = (1 - \alpha) \tilde {p} _ {t} ^ {1 - \eta} + \alpha \left(\frac {\pi_ {t - 1} ^ {\chi}}{\pi_ {t}}\right) ^ {1 - \eta}\]

\[\stackrel {\vee} {1 - \alpha \pi_ {s s} ^ {(\chi - 1) (1 - \eta)}} = (1 - \alpha) \tilde {p} _ {s s} ^ {1 - \eta}\]

\[\tilde {p} _ {s s} = \left[ \frac {1 - \alpha \pi_ {s s} ^ {(\chi - 1) (1 - \eta)}}{1 - \alpha} \right] ^ {\frac {1}{1 - \eta}}\]

The state variable for distortion due to price stickyness,

From EQ 18

\[s _ {t + 1} = (1 - \alpha) \tilde {p} _ {t} ^ {- \eta} + \alpha \left(\frac {\pi_ {t}}{\pi_ {t - 1} ^ {\chi}}\right) ^ {\eta} s _ {t}\]

\[\begin{array}{l} s _ {s s} = (1 - \alpha) \tilde {p} _ {s s} ^ {- \eta} + \alpha \pi_ {s s} ^ {(1 - \chi) \eta} s _ {s s} \\ \Updownarrow \end{array}\]

\[s _ {s s} \left(1 - \alpha \pi_ {s s} ^ {(1 - \chi) \eta}\right) = (1 - \alpha) \tilde {p} _ {s s} ^ {- \eta}\]

\[s _ {s s} = \frac {(1 - \alpha) \tilde {p} _ {s s} ^ {- \eta}}{1 - \alpha \pi_ {s s} ^ {(1 - \chi) \eta}}\]

The nominal one period deposit rate,

From EQ 6

\[1 = E _ {t} \left[ \beta \mu_ {\lambda t + 1} \frac {\exp \{r _ {t} ^ {b} \}}{\pi_ {t + 1}} \right]\]

\[r _ {s s} ^ {b} = \log \left(\frac {\pi_ {s s}}{\beta \mu_ {\lambda s s}}\right)\]

and using we get

\[r _ {s s} ^ {b} = \log \left(\frac {\pi_ {s s}}{\beta \mu_ {z ^ {*} , s s} ^ {- \phi_ {2} (1 - \phi_ {4}) - \phi_ {4}}}\right)\]

\[= \log \left(\frac {\pi_ {s s}}{\beta}\right) - \left(- \phi_ {2} \left(1 - \phi_ {4}\right) - \phi_ {4}\right) \log \mu_ {z ^ {*}, s s}\]

Note, if , then , implying that increasing also increases the steady state level of the deposit rate as . However, when , then and increasing no longer increases the steady state level of the deposit rate.

The real price of capital,

From EQ 5

\[1 = Q _ {t} \left(1 - \frac {\kappa_ {1}}{2} \left(\frac {I _ {t}}{I _ {S S}} - 1\right) ^ {2} - \frac {I _ {t}}{I _ {S S}} \kappa_ {1} \left(\frac {I _ {t}}{I _ {S S}} - 1\right) - \kappa_ {2} \left(\frac {I _ {t}}{k _ {t}} \mu_ {\Upsilon , t} \mu_ {z ^ {*}, t} - \frac {I _ {s s}}{K _ {s s}} \mu_ {\Upsilon , s s} \mu_ {z ^ {*}, s s}\right)\right)\]

We immediately get that

\[\begin{array}{r l} & {\mathrm{Therealpriceofcapital,} R _ {s s} ^ {k}} \\ & {\mathrm{FromEQ3}} \\ & {Q _ {t} = E _ {t} \frac {\beta \mu_ {\lambda , t + 1}}{\mu_ {\Upsilon , t + 1}} [ R _ {t + 1} ^ {k} + Q _ {t + 1} (1 - \delta)} \\ & {- Q _ {t + 1} \frac {\kappa_ {2}}{2} \left(\frac {I _ {t + 1}}{K _ {t + 1}} \mu_ {\Upsilon , t + 1} \mu_ {z ^ {*}, t + 1} - \frac {I _ {s s}}{k _ {s s}} \mu_ {\Upsilon , s s} \mu_ {z ^ {*}, s s}\right) ^ {2} + Q _ {t + 1} \kappa_ {2} \left(\frac {I _ {t + 1}}{K _ {t + 1}} \mu_ {\Upsilon , t + 1} \mu_ {z ^ {*}, t + 1} - \frac {I _ {s s}}{k _ {s s}} \mu_ {\Upsilon , s s} \mu_ {z ^ {*}, s s}\right) \frac {I _ {t + 1}}{K _ {t + 1}} \mu_ {\Upsilon , t + 1} \mu_ {z ^ {*}, t + 1} ]} \end{array}\]

↓

\[\stackrel {\vee} {Q} _ {s s} = \beta \frac {\mu_ {\lambda , s s}}{\mu_ {\Upsilon , s s}} \left[ R _ {s s} ^ {k} + Q _ {s s} (1 - \delta) \right]\tag{↓}\]

\[\begin{array}{l} \stackrel {{\vee}} {{\mu_ {\Upsilon , s s}}} Q _ {s s} = \beta \mu_ {\lambda , s s} \left[ R _ {s s} ^ {k} + Q _ {s s} (1 - \delta) \right] \\ \Updownarrow \end{array}\]

\[\frac {\mu_ {\Upsilon , s s} Q _ {s s}}{\beta \mu_ {\lambda , s s}} - Q _ {s s} (1 - \delta) = R _ {s s} ^ {k}\]

\[R _ {s s} ^ {k} = Q _ {s s} \left[ \frac {\mu_ {\Upsilon , s s}}{\beta \mu_ {\lambda , s s}} - (1 - \delta) \right]\]

The marginal costs in the firms,

First from EQ 9

\[\frac {(\eta - 1)}{\eta} X _ {t} ^ {2} = Y _ {t} m c _ {t} \tilde {p} _ {t} ^ {- \eta - 1} + E _ {t} \left[ \alpha \beta \mu_ {\lambda , t + 1} \left(\frac {\tilde {p} _ {t}}{\tilde {p} _ {t + 1}}\right) ^ {- \eta - 1} \left(\frac {\pi_ {t} ^ {\chi}}{\pi_ {t + 1}}\right) ^ {- \eta} \frac {(\eta - 1)}{\eta} X _ {t + 1} ^ {2} \mu_ {z ^ {*}, t + 1} \right]\]

\[\begin{array}{l} \stackrel {\searrow} {\underset {\eta} {\longrightarrow}} ^ {(\eta - 1)} X _ {s s} ^ {2} = Y _ {s s} m c _ {s s} \tilde {p} _ {s s} ^ {- \eta - 1} + \alpha \beta \mu_ {\lambda , s s} \pi_ {s s} ^ {(1 - \chi) \eta} \frac {(\eta - 1)}{\eta} X _ {s s} ^ {2} \mu_ {z ^ {*}, s s} \\ \Downarrow \end{array}\]

\[X _ {s s} ^ {2} \left[ \frac {(\eta - 1)}{\eta} - \alpha \beta \mu_ {\lambda , s s} \pi_ {s s} ^ {(1 - \chi) \eta} \frac {(\eta - 1)}{\eta} \mu_ {z ^ {*}, s s} \right] = Y _ {s s} m c _ {s s} \tilde {p} _ {s s} ^ {- \eta - 1}\]

\[X _ {s s} ^ {2} = \frac {Y _ {s s} m c _ {s s} \tilde {p} _ {s s} ^ {- \eta - 1}}{\frac {(\eta - 1)}{\eta} - \alpha \beta \mu_ {\lambda , s s} \pi_ {s s} ^ {(1 - \chi) \eta} \frac {(\eta - 1)}{\eta} \mu_ {z ^ {*} , s s}}\]

And from EQ 10

\[X _ {t} ^ {2} = Y _ {t} \tilde {p} _ {t} ^ {- \eta} + E _ {t} \left[ \alpha \beta \mu_ {\lambda , t + 1} \left(\frac {\tilde {p} _ {t}}{\tilde {p} _ {t + 1}}\right) ^ {- \eta} \left(\frac {\pi_ {t} ^ {\chi}}{\pi_ {t + 1}}\right) ^ {1 - \eta} X _ {t + 1} ^ {2} \mu_ {z ^ {*}, t + 1} \right]\]

↓

\[X _ {s s} ^ {2} = Y _ {s s} \tilde {p} _ {s s} ^ {- \eta} + \alpha \beta \mu_ {\lambda , s s} \pi_ {s s} ^ {(\chi - 1) (1 - \eta)} X _ {s s} ^ {2} \mu_ {z ^ {*}, s s}\]

\[X _ {s s} ^ {2} \left(1 - \alpha \beta \mu_ {\lambda , s s} \pi_ {s s} ^ {(\chi - 1) (1 - \eta)} \mu_ {z ^ {*}, s s}\right) = Y _ {s s} \tilde {p} _ {s s} ^ {- \eta}\]

\[X _ {s s} ^ {2} = \frac {Y _ {s s} \tilde {p} _ {s s} ^ {- \eta}}{1 - \alpha \beta \mu_ {\lambda , s s} \pi_ {s s} ^ {(\chi - 1) (1 - \eta)} \mu_ {z ^ {*} , s s}}\]

So by setting the two equations equal to one another we get:

\[\begin{array}{r l} & {\frac {Y _ {s s} m c _ {s s} \tilde {p} _ {s s} ^ {- \eta - 1}}{\frac {(\eta - 1)}{\eta} - \alpha \beta \mu_ {\lambda , s s} \pi_ {s s} ^ {(1 - \chi) \eta} \frac {(\eta - 1)}{\eta} \mu_ {z ^ {*} , s s}} = \frac {Y _ {s s} \tilde {p} _ {s s} ^ {- \eta}}{1 - \alpha \beta \mu_ {\lambda , s s} \pi_ {s s} ^ {(\chi - 1) (1 - \eta)} \mu_ {z ^ {*} , s s}}} \\ & {\Updownarrow} \\ & {\frac {m c _ {s s}}{\left(1 - \alpha \beta \mu_ {\lambda , s s} \pi_ {s s} ^ {(1 - \chi) \eta} \mu_ {z ^ {*} , s s}\right) \frac {(\eta - 1)}{\eta} \tilde {p} _ {s s}} = \frac {1}{1 - \alpha \beta \mu_ {\lambda , s s} \pi_ {s s} ^ {(\chi - 1) (1 - \eta)} \mu_ {z ^ {*} , s s}}} \\ & {\Updownarrow} \\ & {m c _ {s s} = \frac {\left(1 - \alpha \beta \mu_ {\lambda , s s} \pi_ {s s} ^ {(1 - \chi) \eta} \mu_ {z ^ {*} , s s}\right) \frac {(\eta - 1)}{\eta} \tilde {p} _ {s s}}{1 - \alpha \beta \mu_ {\lambda , s s} \pi_ {s s} ^ {(\chi - 1) (1 - \eta)} \mu_ {z ^ {*} , s s}}} \end{array}\]

The Capital level,

From EQ 7

\[m c _ {t} a _ {s s} \theta \mu_ {\Upsilon , t} \mu_ {z, t} ^ {1 - \theta} K _ {t} ^ {\theta - 1} h _ {t} ^ {1 - \theta} = R _ {t} ^ {k}\tag{↓}\]

\[m c _ {s s} \theta \mu_ {\Upsilon , s s} \mu_ {z, s s} ^ {1 - \theta} K _ {s s} ^ {\theta - 1} h _ {s s} ^ {1 - \theta} = R _ {s s} ^ {k}\]

↓

since

\[\begin{array}{l} \left(\frac {K _ {s s}}{h _ {s s}}\right) ^ {\theta - 1} = \frac {R _ {s s} ^ {k}}{m c _ {s s} \theta \mu_ {\Upsilon , s s} \mu_ {z , s s} ^ {1 - \theta}} \\ \Updownarrow \end{array}\]

\[K _ {s s} = h _ {s s} \left(\frac {R _ {s s} ^ {k}}{m c _ {s s} \theta \mu_ {\Upsilon , s s} \mu_ {z , s s} ^ {1 - \theta}}\right) ^ {\frac {1}{\theta - 1}}\]

The wage level,

From EQ 8

\[\begin{array}{l} m c _ {t} \left(1 - \theta\right) a _ {t} \mu_ {\Upsilon , t} ^ {\frac {- \theta}{1 - \theta}} \mu_ {z, t} ^ {- \theta} K _ {t} ^ {\theta} h _ {t} ^ {- \theta} = W _ {t} \\ \Downarrow \end{array}\]

\[m c _ {s s} (1 - \theta) \mu_ {\Upsilon , s s} ^ {\frac {- \theta}{1 - \theta}} \mu_ {z, s s} ^ {- \theta} \left(\frac {K _ {s s}}{h _ {s s}}\right) ^ {\theta} = W _ {s s}\]

since

\[W _ {s s} = m c _ {s s} \left(1 - \theta\right) \mu_ {\Upsilon , s s} ^ {\frac {- \theta}{1 - \theta}} \mu_ {z, s s} ^ {- \theta} \left(\frac {K _ {s s}}{h _ {s s}}\right) ^ {\theta}\]

The investment level,

From EQ 19

\[K _ {t + 1} = (1 - \delta) K _ {t} \left(\mu_ {\Upsilon , t} \mu_ {z ^ {*}, t}\right) ^ {- 1} + I _ {t} - I _ {t} \frac {\kappa_ {1}}{2} \left(\frac {I _ {t}}{I _ {S S}} - 1\right) ^ {2} - K _ {t} \left(\mu_ {\Upsilon , t} \mu_ {z ^ {*}, t}\right) ^ {- 1} \frac {\kappa_ {2}}{2} \left(\frac {I _ {t}}{K _ {t}} \mu_ {\Upsilon , t} \mu_ {z ^ {*}, t} - \frac {I _ {s s}}{k _ {s s}} \mu_ {\Upsilon , s s} \mu_ {z ^ {*}, s s}\right) ^ {2}\]

\[\begin{array}{l} \stackrel {{\vee}} {{K}} _ {s s} = (1 - \delta) K _ {t} \left(\mu_ {\Upsilon , s s} \mu_ {z ^ {*}, s s}\right) ^ {- 1} + I _ {s s} \\ \Updownarrow \end{array}\]

\[I _ {s s} = K _ {s s} - (1 - \delta) K _ {s s} \mu_ {\Upsilon , s s} ^ {\frac {- 1}{1 - \theta}} \mu_ {z, s s} ^ {- 1}\]

The Consumption level,

From 17

\[a _ {t} \left(K _ {t} \mu_ {\Upsilon , t} ^ {\frac {- 1}{1 - \theta}} \mu_ {z, t} ^ {- 1}\right) ^ {\theta} h _ {t} ^ {1 - \theta} = Y _ {t} s _ {t + 1}\]

Note then that Eq 20 implies

\[Y _ {s s} = C _ {s s} + I _ {s s} + G _ {s s}\]

\[\stackrel {\vee} {Y} _ {s s} = C _ {s s} + I _ {s s} + \frac {G _ {s s}}{Y _ {s s}} Y _ {s s}\]

\[Y _ {s s} \left(1 - \frac {G _ {s s}}{Y _ {s s}}\right) = C _ {s s} + I _ {s s}\]

\[\stackrel {\vee} {Y} _ {s s} = \frac {C _ {s s} + I _ {s s}}{1 - \frac {G _ {s s}}{Y _ {s s}}}\]

Thus, we have

\[\frac {\left(K _ {s s} \mu_ {\Upsilon , s s} ^ {\frac {- 1}{1 - \theta}} \mu_ {z , s s} ^ {- 1}\right) ^ {\theta} h _ {s s} ^ {1 - \theta}}{s _ {s s}} = \frac {C _ {s s} + I _ {s s}}{1 - \frac {G _ {s s}}{Y _ {s s}}}\]

↓

\[\frac {\left(1 - \frac {g _ {s s}}{Y _ {s s}}\right) \left(K _ {s s} \mu_ {\Upsilon , s s} ^ {\frac {- 1}{1 - \theta}} \mu_ {z , s s} ^ {- 1}\right) ^ {\theta} h _ {s s} ^ {1 - \theta}}{s _ {s s}} - I _ {s s} = C _ {s s}\]

The value of

From EQ 2

\[\begin{array}{r l} & {\Lambda_ {t} = d _ {t} \left(C _ {t} - b C _ {t - 1} \mu_ {z ^ {*}, t} ^ {- 1}\right) ^ {- \phi_ {2}} - 1 _ {[ h a _ {-} i n ]} b \beta E _ {t} \bigg \{\left[ \left(\frac {\left[ E _ {t} \left[ \left(- \widetilde {V _ {t + 1}}\right) ^ {1 - \phi_ {3}} \right] \right] ^ {\frac {1}{1 - \phi_ {3}}}}{- \widetilde {V _ {t + 1}} (s)}\right) ^ {\phi_ {3}} \right]} \\ & {\qquad \times d _ {t + 1} \left(C _ {t + 1} - b C _ {t} \mu_ {z ^ {*}, t + 1} ^ {- 1}\right) ^ {- \phi_ {2}} \left(\mu_ {z ^ {*}, t + 1}\right) ^ {- \phi_ {2} (1 - \phi_ {4}) - \phi_ {4}} \bigg \}} \end{array}\]

↓

\[\Lambda_ {s s} = \left(C _ {s s} - b C _ {s s} \mu_ {z ^ {*}, s s} ^ {- 1}\right) ^ {- \phi_ {2}} - 1 _ {[ h a \_ i n ]} b \beta \left(C _ {s s} - b C _ {s s} \mu_ {z ^ {*}, s s} ^ {- 1}\right) ^ {- \phi_ {2}} \left(\mu_ {z ^ {*}, s s}\right) ^ {- \phi_ {2} (1 - \phi_ {4}) - \phi_ {4}}\]

The value of

From EQ 4

\[d _ {t} \phi_ {0} (1 - h _ {t}) ^ {- \phi_ {1}} = \Lambda_ {t} W _ {t}\]

\[\phi_ {0} = \Lambda_ {s s} W _ {s s} (1 - h _ {s s}) ^ {\phi_ {1}}\]

The level of the value function,

From EQ 1

\[\begin{array}{l} \widetilde {V} _ {t} = \left[ \frac {d _ {t}}{1 - \phi_ {2}} \left(\left(C _ {t} - b C _ {t - 1} \mu_ {z ^ {*}, t} ^ {- 1}\right) ^ {1 - \phi_ {2}} - 1\right) + d _ {t} \phi_ {0} \frac {(1 - h _ {t}) ^ {1 - \phi_ {1}}}{1 - \phi_ {1}} \right] - \beta \mu_ {z ^ {*}, s s} ^ {(1 - \phi_ {4}) (1 - \phi_ {2})} \left(E _ {t} \left[ \left(- \widetilde {V _ {t + 1}}\right) ^ {1 - \phi_ {3}} \right]\right) ^ {\frac {1}{1 - \phi_ {3}}} \\ \Downarrow \end{array}\]

\[\widetilde {V _ {s s}} = \frac {1}{1 - \phi_ {2}} \left(\left(C _ {s s} - b C _ {s s} \mu_ {z ^ {*}, s s} ^ {- 1}\right) ^ {1 - \phi_ {2}} - 1\right) + \phi_ {0} \frac {(1 - h _ {s s}) ^ {1 - \phi_ {1}}}{1 - \phi_ {1}} - \beta \mu_ {z ^ {*}, s s} ^ {(1 - \phi_ {4}) (1 - \phi_ {2})} \left(- \widetilde {V _ {s s}}\right)\]

\[\begin{array}{l} \widetilde {V _ {s s}} \left(1 - \beta \mu_ {z ^ {*}, s s} ^ {(1 - \phi_ {4}) (1 - \phi_ {2})}\right) = \frac {1}{1 - \phi_ {2}} \left(\left(C _ {s s} - b C _ {s s} \mu_ {z ^ {*}, s s} ^ {- 1}\right) ^ {1 - \phi_ {2}} - 1\right) + \phi_ {0} \frac {(1 - h _ {s s}) ^ {1 - \phi_ {1}}}{1 - \phi_ {1}} \Updownarrow \\ \Updownarrow \end{array}\]

\[\begin{array}{l} {{\widetilde {V _ {s s}} = \frac {1}{1 - \beta \mu_ {z ^ {*} , s s} ^ {(1 - \phi_ {4}) (1 - \phi_ {2})}} \left[ \frac {1}{1 - \phi_ {2}} \left(\left(C _ {s s} - b C _ {s s} \mu_ {z ^ {*}, s s} ^ {- 1}\right) ^ {1 - \phi_ {2}} - 1\right) + \phi_ {0} \frac {(1 - h _ {s s}) ^ {1 - \phi_ {1}}}{1 - \phi_ {1}} \right]}} \\ {{\text {企}}} \end{array}\]

\[\widetilde {V _ {s s}} = \frac {1}{1 - \beta \mu_ {z ^ {*} , s s} ^ {(1 - \phi_ {4}) (1 - \phi_ {2})}} \left[ \frac {1}{1 - \phi_ {2}} \left(\left(C _ {s s} - b C _ {s s} \mu_ {z ^ {*}, s s} ^ {- 1}\right) ^ {1 - \phi_ {2}} - 1\right) + \phi_ {0} \frac {(1 - h _ {s s}) ^ {1 - \phi_ {1}}}{1 - \phi_ {1}} \right]\]

If , then we consider

\[- \widetilde {V _ {s s}} = \frac {- 1}{1 - \beta \mu_ {z ^ {*} , s s} ^ {(1 - \phi_ {4}) (1 - \phi_ {2})}} \left[ \frac {1}{1 - \phi_ {2}} \left(\left(C _ {s s} - b C _ {s s} \mu_ {z ^ {*}, s s} ^ {- 1}\right) ^ {1 - \phi_ {2}} - 1\right) + \phi_ {0} \frac {(1 - h _ {s s}) ^ {1 - \phi_ {1}}}{1 - \phi_ {1}} \right]\]

and we do the perturbation for where because .

The value of

\[G _ {s s} = \frac {G _ {s s}}{Y _ {s s}} Y _ {s s}\]

16.16 The observables and their moments

16.16.1 Calculating the observables

This section shows how to calculate the considered observables used in the estimation from the approximation of the DSGE model.

The presence of non-stationary shocks (i.e. and ) imply that variables such as , , and and are non-stationary, and this fact must be taken into account when solving the model. We adopt the standard method to deal with this feature by approximating the model's solution around the economy's balanced growth path. This is done by scaling the non-stationary variables such that they become stationary. For instance, and are scaled by and is scaled by , and this implies that , , and are stationary variables.

Applying a log-transformation, the output from approximating the DSGE model is

\[\hat {\mathbf {y}} _ {t} \equiv \mathbf {g} (\hat {\mathbf {x}} _ {t})\]

where we use the standard notation that a hat denotes deviation from the deterministic steady state, i.e. . The elements in must be transformed to make them comparable to empirical data series. We now show how this transformation is done.

1. Consumption growth

The expressions for real quarterly consumption growth is given by

\[\begin{array}{r l r} \log \mu_ {c, t} ^ {o b s} & \equiv & \log \frac {c _ {t}}{c _ {t - 1}} = \log \frac {C _ {t} z _ {t} ^ {*}}{C _ {t - 1} z _ {t - 1} ^ {*}} = \log \frac {C _ {t}}{C _ {s s}} - \log \frac {C _ {t - 1}}{C _ {s s}} + \log \frac {z _ {t} ^ {*}}{z _ {t - 1} ^ {*}} \\ & = & \hat {C} _ {t} - \hat {C} _ {t - 1} + \log \mu_ {z ^ {*}, s s} \end{array}\]

2. Investment growth

For the quarterly growth rate in investments

\[\begin{array}{r l r} \log \mu_ {i, t} ^ {o b s} & \equiv & \log \frac {i _ {t}}{i _ {t - 1}} = \log \frac {I _ {t} z _ {t} ^ {*} \Upsilon_ {t}}{I _ {t - 1} z _ {t - 1} ^ {*} \Upsilon_ {t - 1}} = \log \frac {I _ {t}}{I _ {s s}} - \log \frac {I _ {t - 1}}{I _ {s s}} + \log \frac {z _ {t} ^ {*}}{z _ {t - 1} ^ {*}} + \log \frac {\Upsilon_ {t}}{\Upsilon_ {t - 1}} \\ & = & \hat {I} _ {t} - \hat {I} _ {t - 1} + \log \mu_ {z ^ {*}, s s} + \log \mu_ {\Upsilon , s s} \end{array}\]

3. Inflation

The quarterly inflation rate is given by

\[\log \pi_ {t} = \log \pi_ {s s} + \hat {\pi} _ {t}.\]

4. One-period nominal interest rate

The quarterly nominal interest rate is

\[r _ {t} = r _ {s s} + \hat {r} _ {t},\]

5. 40-period nominal interest rate

The quarterly rate is

\[r _ {t, 4 0} = r _ {s s, 4 0} + \hat {r} _ {t, 4 0},\]

6. Excess holding-period return for the 40-period bond

We compute the ex post excess holding period return by

\[x h r _ {t, 4 0} = \log \left(\frac {P _ {t , 3 9}}{P _ {t - 1 , 4 0}}\right) - r _ {t - 1}\]

\[= \log (P _ {t, 3 9}) - \log (P _ {t - 1, 4 0}) - r _ {t - 1}\]

\[\begin{array}{r l} & = \log (P _ {t, 3 9}) - \log P _ {s s, 4 0} + \log P _ {s s, 4 0} - \log (P _ {t - 1, 4 0}) - r _ {t - 1} \\ & = \hat {P} _ {t, 3 9} - \log P _ {s s, 1} - \hat {P} _ {t - 1, 4 0} - r _ {t - 1} \\ & = \hat {P} _ {t, 3 9} - \hat {P} _ {t - 1, 4 0} - (r _ {t - 1} + \log P _ {s s, 1}) \\ & = \hat {P} _ {t, 3 9} - \hat {P} _ {t - 1, 4 0} - (r _ {t - 1} - r _ {s s}) \\ & x h r _ {t, 4 0} = (\hat {P} _ {t, 3 9} - \hat {P} _ {t - 1, 4 0}) - \hat {r} _ {t - 1} \end{array}\]

7. Ratio of government spending to output

We have

\[\begin{array}{r l} \log \left(\frac {g}{y}\right) _ {t} ^ {o b s} & \equiv \log \left(\frac {z _ {t} ^ {*} G _ {t}}{z _ {t} ^ {*} Y _ {t}}\right) = \log \left(\frac {G}{Y}\right) _ {t} \\ & = \log \left(\frac {G}{Y}\right) _ {s s} + \log \left(\frac {G}{Y}\right) _ {t} - \log \left(\frac {G}{Y}\right) _ {s s} \\ & = \log \left(\frac {G}{Y}\right) _ {s s} + \widehat {\left(\frac {G}{Y}\right)} _ {t} \end{array}\]

8. log of hours

\[\begin{array}{l} \text {Recall} \\ h _ {t} = h _ {s s} e ^ {\log h _ {t} - \log h _ {s s}} = h _ {s s} e ^ {\hat {h} _ {t}} \\ \Downarrow \end{array}\]

\[\log h _ {t} = \log h _ {s s} + \hat {h} _ {t}\]

Note that we scale the empirical measure for average hours per week by in order to normalize its value to take values between 0 and 1.

16.16.2 Moments of growth rates and excess holding period return

This section discusses how to compute moments including consumption growth and investment growth. For numerical convenience, we prefer to only have as the state variable when solving the DSGE model.

1. Consumption growth

\[\begin{array}{l} \log \mu_ {c, t} ^ {o b s} = \hat {C} _ {t} - \hat {C} _ {t - 1} + \log \mu_ {z ^ {*}, s s} \\ \qquad = \left(\mathbf {C} ^ {(i)} (c _ {t},:) \mathbf {z} _ {t} ^ {(i)} + \mathbf {d} ^ {(i)} (c _ {t}, 1)\right) - \left(\mathbf {C} ^ {(i)} (c _ {t},:) \mathbf {z} _ {t - 1} ^ {(i)} + \mathbf {d} ^ {(i)} (c _ {t}, 1)\right) + \log \mu_ {z ^ {*}, s s} \end{array}\]

for approximation order where we use the compact representation of the pruned state space system Note that is the row of matrix which relates to consumption. A similar notation is used for .

\[\mathbf {C} ^ {(i)} (c _ {t},:) \mathbf {z} _ {t} ^ {(i)} - \mathbf {C} ^ {(i)} (c _ {t},:) \mathbf {z} _ {t - 1} ^ {(i)} + \log \mu_ {z ^ {*}, s s}\]

\[= \log \mu_ {z ^ {*}, s s} + \left[ \begin{array}{l l} \mathbf {C} ^ {(i)} (c _ {t},:) & - \mathbf {C} ^ {(i)} (c _ {t},:) \end{array} \right] \left[ \begin{array}{l} \mathbf {z} _ {t} ^ {(i)} \\ \mathbf {z} _ {t - 1} ^ {(i)} \end{array} \right]\]

2. Investment growth

\[\begin{array}{r l} & {\log \mu_ {i, t} ^ {o b s} = \hat {I} _ {t} - \hat {I} _ {t - 1} + \log \mu_ {z ^ {*}, s s} + \log \mu_ {\Upsilon , s s}} \\ & {\qquad = \log \mu_ {z ^ {*}, s s} + \log \mu_ {\Upsilon , s s} + \hat {I} _ {t} - \hat {I} _ {t - 1}} \\ & {\qquad = \log \mu_ {z ^ {*}, s s} + \log \mu_ {\Upsilon , s s} + \left(\mathbf {C} ^ {(i)} (i _ {t},:) \mathbf {z} _ {t} ^ {(i)} + \mathbf {d} ^ {(i)} (i _ {t}, 1)\right) - \left(\mathbf {C} ^ {(i)} (i _ {t},:) \mathbf {z} _ {t - 1} ^ {(i)} + \mathbf {d} ^ {(i)} (i _ {t}, 1)\right)} \\ & {\qquad = \log \mu_ {z ^ {*}, s s} + \log \mu_ {\Upsilon , s s} + \mathbf {C} ^ {(i)} (i _ {t},:) \mathbf {z} _ {t} ^ {(i)} - \mathbf {C} ^ {(i)} (i _ {t},:) \mathbf {z} _ {t - 1} ^ {(i)}} \\ & {\qquad = \log \mu_ {z ^ {*}, s s} + \log \mu_ {\Upsilon , s s} + \left[ \begin{array}{l l} \mathbf {C} ^ {(i)} (i _ {t},:) & - \mathbf {C} ^ {(i)} (i _ {t},:) \end{array} \right] \left[ \begin{array}{l} \mathbf {z} _ {t} ^ {(i)} \\ \mathbf {z} _ {t - 1} ^ {(i)} \end{array} \right]} \end{array}\]

3. Inflation

\[\log \pi_ {s s} + \left[ \begin{array}{l l} {\bf C} ^ {(i)} (\pi_ {t},:) & {\bf 0} \end{array} \right] \left[ \begin{array}{l} {\bf z} _ {t} ^ {(i)} \\ {\bf z} _ {t - 1} ^ {(i)} \end{array} \right] + {\bf d} ^ {(i)} (\pi_ {t}, 1)\]

4. One-period nominal interest rate

\[r _ {t} = r _ {s s} + \hat {r} _ {t}\]

\[= r _ {s s} + \left[ \begin{array}{l l} \mathbf {C} ^ {(i)} (r _ {t},:) & \mathbf {0} \end{array} \right] \left[ \begin{array}{l} \mathbf {z} _ {t} ^ {(i)} \\ \mathbf {z} _ {t - 1} ^ {(i)} \end{array} \right] + \mathbf {d} ^ {(i)} (r _ {t}, 1)\]

5. 40-period nominal interest rate

\[r _ {t, 4 0} = r _ {s s} + \hat {r} _ {t, 4 0},\]

\[= r _ {s s} + \left[ \begin{array}{l l} \mathbf {C} ^ {(i)} (r _ {t, 4 0},:) & \mathbf {0} \end{array} \right] \left[ \begin{array}{l} \mathbf {z} _ {t} ^ {(i)} \\ \mathbf {z} _ {t - 1} ^ {(i)} \end{array} \right] + \mathbf {d} ^ {(i)} (r _ {t, 4 0}, 1)\]

6. Excess holding-period return for the 40-period bond

We compute the ex post excess holding period return by

\[\begin{array}{r l} & x h r _ {t, 4 0} = \left(\hat {P} _ {t, 3 9} - \hat {P} _ {t - 1, 4 0}\right) - \hat {r} _ {t - 1} \\ & \qquad = \left(\mathbf {C} ^ {(i)} \left(P _ {t, 3 9},:\right) \mathbf {z} _ {t} ^ {(i)} + \mathbf {d} ^ {(i)} \left(P _ {t, 3 9}, 1\right)\right) - \left(\mathbf {C} ^ {(i)} \left(P _ {t, 4 0},:\right) \mathbf {z} _ {t - 1} ^ {(i)} + \mathbf {d} ^ {(i)} \left(P _ {t, 4 0}, 1\right)\right) \\ & \qquad - \left(\mathbf {C} ^ {(i)} \left(r _ {t},:\right) \mathbf {z} _ {t - 1} ^ {(i)} + \mathbf {d} ^ {(i)} \left(r _ {t}, 1\right)\right) \\ & \qquad = \mathbf {C} ^ {(i)} \left(P _ {t, 3 9},:\right) \mathbf {z} _ {t} ^ {(i)} + \mathbf {d} ^ {(i)} \left(P _ {t, 3 9}, 1\right) - \left(\mathbf {C} ^ {(i)} \left(P _ {t, 4 0},:\right) - \mathbf {C} ^ {(i)} \left(r _ {t},:\right)\right) \mathbf {z} _ {t - 1} ^ {(i)} - \mathbf {d} ^ {(i)} \left(P _ {t, 4 0}, 1\right) - \mathbf {d} ^ {(i)} \left(r _ {t}, 1\right) \\ & \qquad = \mathbf {C} ^ {(i)} \left(P _ {t, 3 9},:\right) \mathbf {z} _ {t} ^ {(i)} - \left(\mathbf {C} ^ {(i)} \left(P _ {t, 4 0},:\right) - \mathbf {C} ^ {(i)} \left(r _ {t},:\right)\right) \mathbf {z} _ {t - 1} ^ {(i)} + \mathbf {d} ^ {(i)} \left(P _ {t, 3 9}, 1\right) - \mathbf {d} ^ {(i)} \left(P _ {t, 4 0}, 1\right) - \mathbf {d} ^ {(i)} \left(r _ {t}, 1\right) \end{array}\]

\[\begin{array}{l} 7. \text { Ratio of government spending to output } \\ \log \left(\frac {g}{y}\right) _ {t} ^ {o b s} = \log \left(\frac {G}{Y}\right) _ {s s} + \widehat {\left(\frac {G}{Y}\right)} _ {t} \\ = \log \left(\frac {G}{Y}\right) _ {s s} + \mathbf {C} ^ {(i)} \left(\frac {G}{Y},: \cdot\right) \mathbf {z} _ {t} ^ {(i)} + \mathbf {d} ^ {(i)} \left(\frac {G}{Y}, 1\right) \end{array}\]

\[= \log \left(\frac {G}{Y}\right) _ {s s} + \left[ \begin{array}{c c} \mathbf {C} ^ {(i)} \left(\frac {G}{Y},:\right) & \mathbf {0} \end{array} \right] \left[ \begin{array}{c} \mathbf {z} _ {t} ^ {(i)} \\ \mathbf {z} _ {t - 1} ^ {(i)} \end{array} \right] + \mathbf {d} ^ {(i)} \left(\frac {G}{Y}, 1\right)\]

8. Log of hours

\[\begin{array}{r l} & {\log h _ {t} = \log h _ {s s} + \hat {h} _ {t}} \\ & {\qquad = \log h _ {s s} + \left[ \begin{array}{l l} \mathbf {C} ^ {(i)} (h _ {t},:) & \mathbf {0} \end{array} \right] \left[ \begin{array}{l} \mathbf {z} _ {t} ^ {(i)} \\ \mathbf {z} _ {t - 1} ^ {(i)} \end{array} \right] + \mathbf {d} ^ {(i)} (h _ {t}, 1)} \end{array}\]

To summarize we have

\[\begin{array}{r l} & {\mathbf {y} _ {t} ^ {o b s} \equiv \left[ \begin{array}{c} \log \mu_ {c, t} ^ {o b s} \\ \log \mu_ {i, t} ^ {o b s} \\ \log \pi_ {t} \\ r _ {t} \\ r _ {t, 4 0} \\ x h r _ {t, 4 0} \\ \log \left(\frac {G}{Y}\right) _ {t} ^ {o b s} \\ \log h _ {t} \end{array} \right] = \left[ \begin{array}{c} \log \mu_ {z ^ {*}, s s} \\ \log \mu_ {z ^ {*}, s s} + \log \mu_ {\Upsilon , s s} \\ \log \pi_ {s s} + \mathbf {d} ^ {(i)} (\pi_ {t}, 1) \\ \log R _ {s s} + \mathbf {d} ^ {(i)} (r _ {t}, 1) \\ \log R _ {s s} + \mathbf {d} ^ {(i)} (r _ {t, 4 0}, 1) \\ \mathbf {d} ^ {(i)} (P _ {t, 3 9}, 1) - \mathbf {d} ^ {(i)} (P _ {t, 4 0}, 1) - \mathbf {d} ^ {(i)} (r _ {t}, 1) \\ \log \left(\frac {G}{Y}\right) _ {s s} + \mathbf {d} ^ {(i)} (\frac {G}{Y}, 1) \\ \log h _ {s s} + \mathbf {d} ^ {(i)} (h _ {t}, 1) \end{array} \right]} \\ & {\quad + \left[ \begin{array}{c c} \mathbf {C} ^ {(i)} (c _ {t},:) & - \mathbf {C} ^ {(i)} (c _ {t},:) \\ \mathbf {C} ^ {(i)} (i _ {t},:) & - \mathbf {C} ^ {(i)} (i _ {t},:) \\ \mathbf {C} ^ {(i)} (\pi_ {t},:) & \mathbf {0} \\ \mathbf {C} ^ {(i)} (r _ {t},:) & \mathbf {0} \\ \mathbf {C} ^ {(i)} (r _ {t, 4 0},:) & \mathbf {0} \\ \mathbf {C} ^ {(i)} (P _ {t, 3 9},:) & - \mathbf {C} ^ {(i)} (P _ {t - 1, 4 0},:) - \mathbf {C} ^ {(i)} (r _ {t},:) \\ \mathbf {C} ^ {(i)} (\frac {G}{Y},:) & \mathbf {0} \\ \mathbf {C} ^ {(i)} (h _ {t},:) & \mathbf {0} \end{array} \right] \left[ \begin{array}{c} \mathbf {z} _ {t} ^ {(i)} \\ \mathbf {z} _ {t - 1} ^ {(i)} \end{array} \right]} \end{array}\]

\[= \left[ \begin{array}{c} \log \mu_ {z ^ {*}, s s} \\ \log \mu_ {z ^ {*}, s s} + \log \mu_ {\Upsilon , s s} \\ \log \pi_ {s s} + \mathbf {d} ^ {(i)} (\pi_ {t}, 1) \\ \log R _ {s s} + \mathbf {d} ^ {(i)} (r _ {t}, 1) \\ \log R _ {s s} + \mathbf {d} ^ {(i)} (r _ {t, 4 0}, 1) \\ \mathbf {d} ^ {(i)} (P _ {t, 3 9}, 1) - \mathbf {d} ^ {(i)} (P _ {t, 4 0}, 1) - \mathbf {d} ^ {(i)} (r _ {t}, 1) \\ \log \left(\frac {G}{Y}\right) _ {s s} + \mathbf {d} ^ {(i)} \left(\frac {G}{Y}, 1\right) \\ \log h _ {s s} + \mathbf {d} ^ {(i)} (h _ {t}, 1) \end{array} \right] + \left[ \begin{array}{c c} \tilde {\mathbf {C}} _ {1, 1} ^ {(i)} & \tilde {\mathbf {C}} _ {1, 2} ^ {(i)} \\ \tilde {\mathbf {C}} _ {2, 1} ^ {(i)} & \tilde {\mathbf {C}} _ {2, 2} ^ {(i)} \\ \tilde {\mathbf {C}} _ {3, 1} ^ {(i)} & \tilde {\mathbf {C}} _ {3, 2} ^ {(i)} \\ \tilde {\mathbf {C}} _ {4, 1} ^ {(i)} & \tilde {\mathbf {C}} _ {4, 2} ^ {(i)} \\ \tilde {\mathbf {C}} _ {5, 1} ^ {(i)} & \tilde {\mathbf {C}} _ {5, 2} ^ {(i)} \\ \tilde {\mathbf {C}} _ {6, 1} ^ {(i)} & \tilde {\mathbf {C}} _ {6, 2} ^ {(i)} \\ \tilde {\mathbf {C}} _ {7, 1} ^ {(i)} & \tilde {\mathbf {C}} _ {7, 2} ^ {(i)} \\ \tilde {\mathbf {C}} _ {8, 1} ^ {(i)} & \tilde {\mathbf {C}} _ {8, 2} ^ {(i)} \end{array} \right] \left[ \begin{array}{c} \mathbf {z} _ {t} ^ {(i)} \\ \mathbf {z} _ {t - 1} ^ {(i)} \end{array} \right]\]

Computing all mean values are straightforward. It is more difficult to compute all second moments and their formulas are derived below. Ignoring the constant term we have

\[y _ {j, t} ^ {o b s} = \left\{\mathbf {\tilde {C}} _ {j, 1} ^ {(i)} \mathbf {z} _ {t} ^ {(i)} + \mathbf {\tilde {C}} _ {j, 2} ^ {(i)} \mathbf {z} _ {t - 1} ^ {(i)} \right\} _ {j = 1} ^ {7}\]

So all contemporary co-variances are given by

\[C o v \left(y _ {j, t} ^ {o b s}, y _ {k, t} ^ {o b s}\right) = C o v \left(\tilde {\mathbf {C}} _ {j, 1} ^ {(i)} \mathbf {z} _ {t} ^ {(i)} + \tilde {\mathbf {C}} _ {j, 2} ^ {(i)} \mathbf {z} _ {t - 1} ^ {(i)}, \tilde {\mathbf {C}} _ {k, 1} ^ {(i)} \mathbf {z} _ {t} ^ {(i)} + \tilde {\mathbf {C}} _ {k, 2} ^ {(i)} \mathbf {z} _ {t - 1} ^ {(i)}\right)\]

\[\begin{array}{l} = \tilde {\mathbf {C}} _ {j, 1} ^ {(i)} C o v \left(\mathbf {z} _ {t} ^ {(i)}, \tilde {\mathbf {C}} _ {k, 1} ^ {(i)} \mathbf {z} _ {t} ^ {(i)} + \tilde {\mathbf {C}} _ {k, 2} ^ {(i)} \mathbf {z} _ {t - 1} ^ {(i)}\right) \\ + \tilde {\mathbf {C}} _ {j, 2} ^ {(i)} C o v \left(\mathbf {z} _ {t - 1} ^ {(i)}, \tilde {\mathbf {C}} _ {k, 1} ^ {(i)} \mathbf {z} _ {t} ^ {(i)} + \tilde {\mathbf {C}} _ {k, 2} ^ {(i)} \mathbf {z} _ {t - 1} ^ {(i)}\right) \end{array}\]

\[\begin{array}{l} = \tilde {\mathbf {C}} _ {j, 1} ^ {(i)} \left(C o v \left(\mathbf {z} _ {t} ^ {(i)}, \mathbf {z} _ {t} ^ {(i)}\right) \left(\tilde {\mathbf {C}} _ {k, 1} ^ {(i)}\right) ^ {\prime} + C o v \left(\mathbf {z} _ {t} ^ {(i)}, \mathbf {z} _ {t - 1} ^ {(i)}\right) \left(\tilde {\mathbf {C}} _ {k, 2} ^ {(i)}\right) ^ {\prime}\right) \\ + \tilde {\mathbf {C}} _ {j, 2} ^ {(i)} \left(C o v \left(\mathbf {z} _ {t - 1} ^ {(i)}, \mathbf {z} _ {t} ^ {(i)}\right) \left(\tilde {\mathbf {C}} _ {k, 1} ^ {(i)}\right) ^ {\prime} + C o v \left(\mathbf {z} _ {t - 1} ^ {(i)}, \mathbf {z} _ {t - 1} ^ {(i)}\right) \left(\tilde {\mathbf {C}} _ {k, 2} ^ {(i)}\right) ^ {\prime}\right) \\ = \tilde {\mathbf {C}} _ {j, 1} ^ {(i)} C o v \left(\mathbf {z} _ {t} ^ {(i)}, \mathbf {z} _ {t} ^ {(i)}\right) \left(\tilde {\mathbf {C}} _ {k, 1} ^ {(i)}\right) ^ {\prime} + \tilde {\mathbf {C}} _ {j, 1} ^ {(i)} C o v \left(\mathbf {z} _ {t} ^ {(i)}, \mathbf {z} _ {t - 1} ^ {(i)}\right) \left(\tilde {\mathbf {C}} _ {k, 2} ^ {(i)}\right) ^ {\prime} \\ + \tilde {\mathbf {C}} _ {j, 2} ^ {(i)} C o v \left(\mathbf {z} _ {t - 1} ^ {(i)}, \mathbf {z} _ {t} ^ {(i)}\right) \left(\tilde {\mathbf {C}} _ {k, 1} ^ {(i)}\right) ^ {\prime} + \tilde {\mathbf {C}} _ {j, 2} ^ {(i)} C o v \left(\mathbf {z} _ {t - 1} ^ {(i)}, \mathbf {z} _ {t - 1} ^ {(i)}\right) \left(\tilde {\mathbf {C}} _ {k, 2} ^ {(i)}\right) ^ {\prime} \\ = \tilde {\mathbf {C}} _ {j, 1} ^ {(i)} V a r \left(\mathbf {z} _ {t} ^ {(i)}\right) \left(\tilde {\mathbf {C}} _ {k, 1} ^ {(i)}\right) ^ {\prime} + \tilde {\mathbf {C}} _ {j, 1} ^ {(i)} C o v \left(\mathbf {z} _ {t} ^ {(i)}, \mathbf {z} _ {t - 1} ^{(i)}\right) \left(\tilde {\mathbf {C}} _ {k, 2} ^ {(i)}\right) ^ {\prime} \\ + \tilde {\mathbf {C}} _ {j, 2} ^ {(i)} C o v \left(\mathbf {z} _ {t - 1} ^ {(i)}, \mathbf {z} _ {t} ^ {(i)}\right) \left(\tililde {\mathbf {C}} _ {k, 1} ^ {(i)}\right) ^ {\prime} + \tilde {\mathbf {C}} _ {j, 2} ^ {(i)} V a r \left(\mathbf {z} _ {t} ^ {(i)}\right) \left(\tilde {\mathbf {C}} _ {k, 2} ^ {(i)}\right) ^ {\prime} \\ j, k = 1, 2,..., 7 \end{array}\]

\[\begin{array}{r l} & {\mathrm{Allcovariancesoftheform} C o v \left(y _ {j, t} ^ {o b s}, y _ {k, t - s} ^ {o b s}\right) \mathrm{aregivenby}} \\ & {C o v \left(y _ {j, t} ^ {o b s}, y _ {k, t - s} ^ {o b s}\right) = C o v \left(\tilde {\mathbf {C}} _ {j, 1} ^ {(i)} \mathbf {z} _ {t} ^ {(i)} + \tilde {\mathbf {C}} _ {j, 2} ^ {(i)} \mathbf {z} _ {t - 1} ^ {(i)}, \tilde {\mathbf {C}} _ {k, 1} ^ {(i)} \mathbf {z} _ {t - s} ^ {(i)} + \tilde {\mathbf {C}} _ {k, 2} ^ {(i)} \mathbf {z} _ {t - 1 - s} ^ {(i)}\right)} \\ & {\qquad = C o v \left(\tilde {\mathbf {C}} _ {j, 1} ^ {(i)} \mathbf {z} _ {t} ^ {(i)}, \tilde {\mathbf {C}} _ {k, 1} ^ {(i)} \mathbf {z} _ {t - s} ^ {(i)} + \tilde {\mathbf {C}} _ {k, 2} ^ {(i)} \mathbf {z} _ {t - 1 - s} ^ {(i)}\right)} \\ & {\qquad + C o v \left(\tilde {\mathbf {C}} _ {j, 2} ^ {(i)} \mathbf {z} _ {t - 1} ^ {(i)}, \tilde {\mathbf {C}} _ {k, 1} ^ {(i)} \mathbf {z} _ {t - s} ^ {(i)} + \tilde {\mathbf {C}} _ {k, 2} ^ {(i)} \mathbf {z} ^ {(i)} _ {t - 1 - s}\right)} \\ & {\qquad = C o v \left(\tilde {\mathbf {C}} _ {j, 1} ^ {(i)} \mathbf {z} _ {t} ^ {(i)}, \tilde {\mathbf {C}} _ {k, 1} ^ {(i)} \mathbf {z} _ {t - s} ^ {(i)}\right) + C o v \left(\tilde {\mathbf {C}} _ {j, 1} ^ {(i)} \mathbf {z} _ {t} ^ {(i)}, \tilde {\mathbf {C}} _ {k, 2} ^ {(i)} \mathbf {z} _ {t - 1 - s} ^ {(i)}\right)} \\ & {\qquad + C o v \left(\tilde {\mathbf {C}} _ {j, 2} ^ {(i)} \mathbf {z} _ {t - 1} ^ {(i)}, \tilde {\mathbf {C}} ^ {(i)} _ {k, 1} \mathbf {z} _ {t - s} ^ {(i)}\right) + C o v \left(\tilde {\mathbf {C}} _ {j, 2} ^ {(i)} \mathbf {z} _ {t - 1} ^ {(i)}, \tilde {\mathbf {C}} _ {k, 2} ^ {(i)} \mathbf {z} _ {t - 1 - s} ^ {(i)}\right)} \\ & {\qquad = \tilde {\mathbf {C}} _ {j, 1} ^ {(i)} C o v \left(\mathbf {z} _ {t} ^ {(i)}, \mathbf {z} _ {t - s} ^ {(i)}\right) \left(\tilde {\mathbf {C}} _ {k, 1} ^ {(i)}\right) ^ {\prime} + \tilde {\mathbf {C}} _ {j, 1} ^ {(i)} C o v \left(\mathbf {z} _ {t} ^ {(i)}, \mathbf {z} _ {t - 1 - s} ^ {(i)}\right) \left(\tilde {\mathbf {C}} _ {k, 2} ^ {(i)}\right) ^ {\prime}} \\ & {\qquad + \tilde {\mathbf {C}} _ {j, 2} ^ {(i)} C o v \left(\mathbf {z} _ {t - 1} ^ {(i)}, \mathbf {z} _ {t - s} ^ {(i)}\right) \left(\tilde {\mathbf {C}} _ {k, 1} ^ {(i)}\right) ^ {\prime} + \tilde {\mathbf {C}} _ {j, 2} ^ {(i)} C o v \left(\mathbf {z} _ {t - 1} ^ {(i)}, \mathbf {z} _ {t - 1 - s} ^ {(i)}\right) \left(\tilde {\mathbf {C}} _ {k, 2} ^ {(i)}\right) ^ {\prime}} \\ & {\mathrm{for} j, k = 1, 2,..., 7 \mathrm{andforalls=1,2,...}} \end{array}\]

\[\begin{array} { r l } & { \mathrm{So~for~instance~(1~lag)} : } \\ & { C o v \left( y _ { j , t } ^ { o b s } , y _ { k , t - 1 } ^ { o b s } \right) = \tilde { \mathbf { C } } _ { j , 1 } ^ { ( i ) } C o v \left( \mathbf { z } _ { t } ^ { ( i ) } , \mathbf { z } _ { t - 1 } ^ { ( i ) } \right) \left( \tilde { \mathbf { C } } _ { k , 1 } ^ { ( i ) } \right) ^ { \prime } + \tilde { \mathbf { C } } _ { j , 1 } ^ { ( i ) } C o v \left( \mathbf { z } _ { t } ^ { ( i ) } , \mathbf { z } _ { t - 2 } ^ { ( i ) } \right) \left( \tilde { \mathbf { C } } _ { k , 2 } ^ { ( i ) } \right) ^ { \prime } } \\ & { \qquad + \tilde { \mathbf { C } } _ { j , 2 } ^ { ( i ) } C o v \left( \mathbf { z } _ { t - 1 } ^ { ( i ) } , \mathbf { z } _ { t - 1 } ^ { ( i ) } \right) \left( \tilde { \mathbf { C } } _ { k , 1 } ^ { ( i ) } \right) ^ { \prime } + \tilde { \mathbf { C } } _ { j , 2 } ^ { ( i ) } C o v \left( \mathbf { z } _ { t - 1 } ^ { ( i ) } , \mathbf { z } _ { t - 2 } ^ { ( i ) } \right) \left( \tilde { \mathbf { C } } _ { k , 2 } ^ { ( i ) } \right) ^ { \prime } } \\ & { = \tilde { \mathbf { C } } _ { j , 1 } ^ { ( i ) } C o v \left( \mathbf { z } _ { t } ^ { ( i ) } , \mathbf { z } _ { t - 1 } ^ { ( i ) } \right) \left( \tilde { \mathbf { C } } _ { k , 1 } ^ { ( i ) } \right. ^ { \prime } + \tilde { \mathbf { C } } _ { j , 1 } ^ { ( i ) } C o v \left( \mathbf { z } _ { t } ^ { ( i ) } , \mathbf { z } _ { t - 2 } ^ { ( i ) } \right) \left( \tilde { \mathbf { C } } _ { k , 2 } ^ { (i ) } \right) ^ { \prime } } \\ & \qquad + \tilde { \mathbf { C } } _ { j , 2 } ^ { ( i ) } C o v \left( \mathbf { z } _ { t - 1 } ^ { ( i ) } , \mathbf { z } _ { t - 1 } ^ { ( i ) } \right) \left( \tilde { \mathcal { C } } _ { k , 1 } ^ { ( i ) } \right) ^ { \prime } + \tilde { \mathbf { C } } _ { j , 2 } ^ { ( i ) } C o v \left( \mathbf { z } _ { t - 1 } ^ { ( i ) } , \mathbf { z } _ { t - 2 } ^ { ( i ) } \right) \binom{1}{0} ^ {\prime} \\ & = \tilde { \mathbf { C } } _ { j , 1 } ^ { ( i ) } C o v \left( \mathbf { z } _ { t } ^ { ( i ) } , \mathbf { z } _ { t - 1 } ^ { ( i ) } \right) \left( \tilde { \mathbf { C } } _ { k , 1 } ^ { ( i )} \right) ^ { \prime } + \tilde { \mathbf { C } } _ { j , 1 } ^ { ( i ) } C o v \left( \mathbf { z } _ { t } ^ { ( i ) } , \mathbf { z } _ { t - 2 } ^ { ( i ) } \right) \left( \tilde { \mathbf { C } } _ { k , 2 }^{ ( i )} \right) ^ { \prime } \\ & { + \tilde { \mathbf { C } } _ { j , 2 } ^ { ( i ) } V a r \left( \mathbf { z } _ { t } ^ { ( i )} \right) \left( \tilde { \mathbf { C } } _ { k , 1 } ^ { ( i )} \right) ^ { \prime } + \tilde { \mathbf { C } } _ { j , 2 } ^ { ( i ) } C o v \left( \mathbf { z } _ { t } ^ { ( i ) } , \mathbf { z } _ { t - 1 } ^ { ( i )} \right) \left( \tilde { \mathbf { C } } _ { k , 2 } ^ { ( i )} \right) ^ { \prime }} \\ & ~ + ~ . ~ . ~ . ~ . ~ . ~ . ~ . ~ . ~ . ~ . ~ . ~ . ~ . ~ . ~ . ~ . ~ . ~ . ~ . ~ . ~ . ~ . ~ . ~ . ~ . ~ . ~ . ~ . ~ . ~ . ~ . ~ . ~ . ~ . ~ . ~ . ~ . ~ . ~ . ~ . ~ . ~ . ~ . ~ . ~ . ~ . ~ . ~ . ~ . ~ . ~. ~ . ~ . ~ . ~ . ~ . ~ . ~ . ~ . ~ . ~ . ~ . ~ . ~ . ~ . ~ . ~ . ~. ~. ~. ~. ~. ~. ~. ~. ~. ~. ~. ~. ~. ~. ~. ~. ~. ~. ~. ~. ~. ~. ~. ~. ~. ~. ~. ~. ~. ~.~.~.~.~.~.~.~.~.~.~.~.~.~.~.~.~.~.~.~.~.~.~.~.~.~.~.~.~.~.~.~.~.~.~.~.~.~.~.~.~.~.~.~.~.~.~.~.~.~.~.< nl>\]

\[\begin{array}{l} \text {and with 10 lags (for instance)} \\ C o v \left(y _ {j, t} ^ {o b s}, y _ {k, t - 1 0} ^ {o b s}\right) = \tilde {\mathbf {C}} _ {j, 1} ^ {(i)} C o v \left(\mathbf {z} _ {t} ^ {(i)}, \mathbf {z} _ {t - 1 0} ^ {(i)}\right) \left(\tilde {\mathbf {C}} _ {k, 1} ^ {(i)}\right) ^ {\prime} + \tilde {\mathbf {C}} _ {j, 1} ^ {(i)} C o v \left(\mathbf {z} _ {t} ^ {(i)}, \mathbf {z} _ {t - 1 1} ^ {(i)}\right) \left(\tilde {\mathbf {C}} _ {k, 2} ^ {(i)}\right) ^ {\prime} \\ \qquad + \tilde {\mathbf {C}} _ {j, 2} ^ {(i)} C o v \left(\mathbf {z} _ {t - 1} ^ {(i)}, \mathbf {z} _ {t - 1 0} ^ {(i)}\right) \left(\tilde {\mathbf {C}} _ {k, 1} ^ {(i)}\right) ^ {\prime} + \tilde {\mathbf {C}} _ {j, 2} ^ {(i)} C o v \left(\mathbf {z} _ {t - 1} ^ {(i)}, \mathbf {z} _ {t - 1 1} ^ {(i)}\right) \left(\tilde {\mathbf {C}} _ {k, 2} ^ {(i)}\right) ^ {\prime} \end{array}\]

\[\begin{array}{l} = \tilde {\mathbf {C}} _ {j, 1} ^ {(i)} C o v \left(\mathbf {z} _ {t} ^ {(i)}, \mathbf {z} _ {t - 1 0} ^ {(i)}\right) \left(\tilde {\mathbf {C}} _ {k, 1} ^ {(i)}\right) ^ {\prime} + \tilde {\mathbf {C}} _ {j, 1} ^ {(i)} C o v \left(\mathbf {z} _ {t} ^ {(i)}, \mathbf {z} _ {t - 1 1} ^ {(i)}\right) \left(\tilde {\mathbf {C}} _ {k, 2} ^ {(i)}\right) ^ {\prime} \\ + \tilde {\mathbf {C}} _ {j, 2} ^ {(i)} C o v \left(\mathbf {z} _ {t} ^ {(i)}, \mathbf {z} _ {t - 9} ^ {(i)}\right) \left(\tilde {\mathbf {C}} _ {k, 1} ^ {(i)}\right) ^ {\prime} + \tilde {\mathbf {C}} _ {j, 2} ^ {(i)} C o v \left(\mathbf {z} _ {t} ^ {(i)}, \mathbf {z} _ {t - 1 0} ^ {(i)}\right) \left(\tilde {\mathbf {C}} _ {k, 2} ^ {(i)}\right) ^ {\prime} \end{array}\]

16.17 Understanding the dynamics of the price dispersion index

First note that (for )

\[\begin{array}{l} 1 = (1 - \alpha) \hat {p} _ {t} ^ {1 - \eta} + \alpha \left(\frac {1}{\pi_ {t}}\right) ^ {1 - \eta} \\ \Updownarrow \\ 1 - \alpha \left(\frac {1}{\pi_ {t}}\right) ^ {1 - \eta} = (1 - \alpha) \tilde {p} _ {t} ^ {1 - \eta} \\ \Updownarrow \\ \frac {1 - \alpha \left(\frac {1}{\pi_ {t}}\right) ^ {1 - \eta}}{(1 - \alpha)} = \tilde {p} _ {t} ^ {1 - \eta} \\ \Updownarrow \\ \tilde {p} _ {t} = \left[ \frac {1 - \alpha \left(\frac {1}{\pi_ {t}}\right) ^ {1 - \eta}}{(1 - \alpha)} \right] ^ {\frac {1}{1 - \eta}} \end{array}\]

\[\begin{array}{r l} & {\mathrm{Thenwenotethat}} \\ & {s _ {t + 1} = (1 - \alpha) \tilde {p} _ {t} ^ {- \eta} + \alpha \pi_ {t} ^ {\eta} s _ {t}} \\ & {\quad = (1 - \alpha) \frac {1}{\tilde {p} _ {t} ^ {\eta}} + \alpha \pi_ {t} ^ {\eta} s _ {t}} \\ & {\quad = (1 - \alpha) \frac {1}{\left[ \frac {1 - \alpha \left(\frac {1}{\pi_ {t}}\right) ^ {1 - \eta}}{1 - \alpha} \right] ^ {\frac {\eta}{1 - \eta}}} + \alpha \pi_ {t} ^ {\eta} s _ {t}} \\ & {\quad = (1 - \alpha) \left[ \frac {1 - \alpha \left(\frac {1}{\pi_ {t}}\right) ^ {1 - \eta}}{1 - \alpha} \right] ^ {\frac {\eta}{\eta - 1}} + \alpha \pi_ {t} ^ {\eta} s _ {t}} \\ & {\quad = (1 - \alpha) \left(\frac {1}{1 - \alpha}\right) ^ {\frac {\eta}{\eta - 1}} \left[ 1 - \alpha \left(\frac {1}{\pi_ {t}}\right) ^ {1 - \eta} \right] ^ {\frac {\eta}{\eta - 1}} + \alpha \pi_ {t} ^ {\eta} s _ {t}} \\ & {\quad = (1 - \alpha) ^ {\frac {1 - \eta}{1 - \eta}} (1 - \alpha) ^ {\frac {\eta}{1 - \eta}} \left[ 1 - \alpha \left(\frac {1}{\pi_ {t}}\right) ^ {1 - \eta} \right] ^ {\frac {\eta}{\eta - 1}} + \alpha \pi_ {t} ^ {\eta} s _ {t}} \\ & {\quad = (1 - \alpha) ^ {\frac {1}{1 - \eta}} \left[ 1 - \alpha \pi_ {t} ^ {\eta - 1} \right] ^ {\frac {\eta}{\eta - 1}} + \alpha \pi_ {t} ^ {\eta} s _ {t}} \end{array}\]

Note also that the steady state value is given by

\[\begin{array}{r l} & s _ {s s} = (1 - \alpha) ^ {\frac {1}{1 - \eta}} \left[ 1 - \alpha \pi_ {s s} ^ {\eta - 1} \right] ^ {\frac {\eta}{\eta - 1}} + \alpha \pi_ {s s} ^ {\eta} s _ {s s} \\ & \Updownarrow \\ & s _ {s s} = \frac {(1 - \alpha) ^ {\frac {1}{1 - \eta}} \left[ 1 - \alpha \pi_ {s s} ^ {\eta - 1} \right] ^ {\frac {\eta}{\eta - 1}}}{1 - \alpha \pi_ {s s} ^ {\eta}} \\ & \text {and therefore} \\ & \frac {\partial s _ {s s}}{\partial \pi_ {s s}} = \frac {\frac {\eta}{\eta - 1} (1 - \alpha) ^ {\frac {1}{1 - \eta}} \left[ 1 - \alpha \pi_ {s s} ^ {\eta - 1} \right] ^ {\frac {\eta}{\eta - 1} - 1} (- \alpha (\eta - 1) \pi_ {s s} ^ {\eta - 2}) (1 - \alpha \pi_ {s s} ^ {\eta}) - (1 - \alpha) ^ {\frac {1}{1 - \eta}} \left[ 1 - \alpha \pi_ {s s} ^ {\eta - 1} \right] ^ {\frac {\eta}{\eta - 1}} (- \alpha \eta \pi_ {s s} ^ {\eta - 1})}{(1 - \alpha \pi_ {s s} ^ {\eta}) ^ {2}} \\ & = \frac {\frac {- \eta}{\eta - 1} (1 - \alpha) ^ {\frac {1}{1 - \eta}} \left[ 1 - \alpha \pi_ {s s} ^ {\eta - 1} \right] ^ {\frac {\eta}{\eta - 1} - 1} \alpha (\eta - 1) \pi_ {s s} ^ {\eta - 2} (1 - \alpha \pi_ {s s} ^ {\eta}) + (1 - \alpha) ^ {\frac {1}{1 - \eta}} \left[ 1 - \alpha \pi_ {s s} ^ {\eta - 1} \right] ^ {\frac {\eta}{\eta - 1}} \alpha \eta \pi_ {s s} ^ {\eta - 1}}{(1 - \alpha \pi_ {s s} ^ {\eta}) ^ {2}} \end{array}\]

\[\begin{array}{r l} & = \frac {(1 - \alpha) ^ {\frac {1}{1 - \eta}} \left[ 1 - \alpha \pi_ {s s} ^ {\eta - 1} \right] ^ {\frac {\eta}{\eta - 1}}}{(1 - \alpha \pi_ {s s} ^ {\eta}) ^ {2}} \left[ - \eta \left[ 1 - \alpha \pi_ {s s} ^ {\eta - 1} \right] ^ {- 1} \alpha \pi_ {s s} ^ {\eta - 2} (1 - \alpha \pi_ {s s} ^ {\eta}) + \alpha \eta \pi_ {s s} ^ {\eta - 1} \right] \\ & = \frac {\alpha \eta (1 - \alpha) ^ {\frac {1}{1 - \eta}} \left[ 1 - \alpha \pi_ {s s} ^ {\eta - 1} \right] ^ {\frac {\eta}{\eta - 1}}}{(1 - \alpha \pi_ {s s} ^ {\eta}) ^ {2}} \left[ - \frac {1 - \alpha \pi_ {s s} ^ {\eta}}{1 - \alpha \pi_ {s s} ^ {\eta - 1}} \pi_ {s s} ^ {\eta - 2} + \pi_ {s s} ^ {\eta - 1} \right] \\ & = \frac {\pi_ {s s} ^ {\eta - 2} \alpha \eta (1 - \alpha) ^ {\frac {1}{1 - \eta}} \left[ 1 - \alpha \pi_ {s s} ^ {\eta - 1} \right] ^ {\frac {\eta}{\eta - 1}}}{(1 - \alpha \pi_ {s s} ^ {\eta}) ^ {2}} \left[ - \frac {1 - \alpha \pi_ {s s} ^ {\eta}}{{1 - \alpha \pi_ {s s} ^ {\eta - 1}}} + \pi_ {s s} \right] \\ & = \frac {\pi_ {s s} ^ {\eta - 2} \alpha \eta (1 - \alpha) ^ {\frac {1}{1 - \eta}} \left[ 1 - \alpha \pi_ {s s} ^ {\eta - 1} \right] ^ {\frac {\eta}{\eta - 1}}}{(1 - \alpha \pi_ {s s} ^ {\eta}) ^ {{2}}} \left[ \frac {- 1 + \alpha \pi_ {s s} ^ {\eta}}{{1 - \alpha \pi_ {s s} ^ {\eta - 1}}} + \frac {\pi_ {s s} - \alpha \pi_ {s s} ^ {\eta}}{{1 - \alpha \pi_ {s s} ^ {\eta - 1}}} \right] \\ & = \frac {\pi_ {s s} ^ {\eta - 2} \alpha \eta (1 - \alpha) ^ {\frac {1}{1 - \eta}} [ 1 - \alpha \pi_ {s s} ^ {\eta - 1} ] ^ {\frac {\eta}{\eta - 1}}}{(1 - \alpha \pi_ {s s} ^ {\eta}) ^ {{2}}} \left[ \frac {\pi_ {s s} - 1}{1 - \alpha \pi_ {s s} ^ {\eta - 1}} \right] \geq 0 \\ & = 0 \\ & = 0, \\ & = 0, \\ & = 0, \\ & = 0, \\ & = 0, \\ & = 0, \\ & = 0, \\ & = 0, \\ & = 0, \\ & = 0, \\ & = 0, \\ & = 0, \\ & = 0, \\ & = 0, \\ & = 0, \\ & = 0, \\ & = 0, \\ & = n _ {s s}, \\ & = n _ {s s}, \\ & = n _ {s s}, \\ & = n _ {s s}, \\ & = n _ {s s}, \\ & = n _ {s s}, \\ & = n _ {s s}, \\ & = n _ {s s}, \\ & = n _ {s s}, \\ & = n _ {s s}, \\ & = n _ {s s}, \\ & = n _ {s s}, n _ {s s}. \\ & = n _ {s s}. \\ & = n _ {s s}. \\ & = n _ {s s}. \\ & = n _ {s s}. \\ & = n _ {s s}. \\ & = n _ {s s}. \\ & = n _ {s s}. \\ & = n _ {s s}. \\ & = n _ {s s}. \\ & = n _ {s s}. \\ & = n _ {s s}. \\ & = m _ {s s}. \\ & = m _ {s s}. \\ & = m _ {s s}. \\ & = m _ {s s}. \\ & = m _ {s s}. \\ & = m _ {s s}. \\ & = m _ {s s}. \\ & = m _ {s s}. \\ & = m _ {s s}. \\ & = m _ {s s}. \\ & = m _ {s s}. \\ & = m_{ss}. \\ & = m_{ss}. \\ & = m_{ss}. \\ & = m_{ss}. \\ & = m_{ss}. \\ & = m_{ss}. \\ & = m_{ss}. \\ & = m_{ss}. \\ & = m_{ss}. \\ & = m_{ss}. \\ & = m_{ss}. \\ & = m_{ss}. \\ & = m_{ss}. \\ & = m_{ss}. \\ & = m_{ss}. m_{ss}. \\ & = m_{ss}. m_{ss}. \\ & = m_{ss}. m_{ss}. \\ & = m_{ss}. m_{ss}. \\ & = m_{ss}. m_{ss}. \\ & = m_{ss}. m_{ss}. \\ & = m_{ss}. m_{ss}. \\ & = m_{ss}. m_{ss}. \\ & = m_{ss}. m_{ss}.\end{array}\]

16.18 Understanding term premium in the model

We start by computing an expression for term premium as in Rudebusch & Swanson (2012), after which we look at the excess holding period return

16.18.1 Term premium

We define term premia as , where is the yield-to-maturity on a zero-coupon bond under risk-neutral valuation by the financial intermediary, i.e. . Hence,

\[\begin{array}{r l} & T P _ {t, k} = r _ {t, k} - \tilde {r} _ {t, k} \\ & \qquad = \frac {- 1}{k} \log P _ {t, k} - \left(\frac {- 1}{k} \log \hat {P} _ {t, k}\right) \\ & \qquad = \frac {1}{k} \left(\log \hat {P} _ {t, k} - \log P _ {t, k}\right) \\ & \qquad \approx \frac {1}{k} \left(\log \hat {P} _ {s s, k} + \frac {1}{\hat {P} _ {s s , k}} \left(\hat {P} _ {t, k} - \hat {P} _ {s s, k}\right) - \left(\log P _ {s s, k} + \frac {1}{P _ {s s , k}} (P _ {t, k} - P _ {s s, k})\right)\right) \\ & \qquad \log x _ {t} \approx \log x _ {s s} + \frac {1}{x _ {s s}} (x _ {t} - x _ {s s}) \\ & \qquad = \frac {1}{k} \left(\log P _ {s s, k} + \frac {1}{P _ {s s , k}} \left(\hat {P} _ {t, k} - P _ {s s, k}\right) - \left(\log P _ {s s, k} + \frac {1}{P _ {s s , k}} (P _ {t, k} - P _ {s s, k})\right)\right) \\ & \text {because} P _ {s s, k} = \hat {P} _ {s s, k} \\ & \qquad = \frac {1}{k} \left(\frac {1}{P _ {s s , k}} \hat {P} _ {t, k} - \frac {1}{P _ {s s , k}} P _ {t, k}\right) \\ & \qquad = \frac {- 1}{k P _ {s s , k}} \left(P _ {t, k} - \hat {P} _ {t, k}\right) \\ & \text {Next note as in} \\ & P _ {t, k} - \hat {P} _ {t, k} = E _ {t} [ M _ {t, t + 1} P _ {t + 1, k - 1} ] - e ^ {- r _ {t}} E _ {t} [ \tilde {P} _ {t + 1, k - 1} ] \\ & \qquad = C o v _ {t} (M _ {t, t + 1}, P _ {t + 1, k - 1}) + E _ {t} [ M _ {t, t + 1} ] E _ {t} [ P _ {t + 1, k - 1} ] - e ^ {- r _ {t}} E _ {t} [ \tilde {P} _ {t + 1, k - 1} ] \end{array}\]

\[\begin{array} { l }= C o v _ { t } \left( M _ { t , t + 1 } , P _ { t + 1 , k - 1 } \right) + e ^ { - r _ { t } - \omega \times x h r _ { t , L } } E _ { t } \left[ P _ { t + 1 , k - 1 } \right] - e ^ { - r _ { t } } E _ { t } \left[ \tilde { P } _ { t + 1 , k - 1 } \right]\\\text {because} E _ { t } \left[ M _ { t , t + 1 } \right] = e ^ { - r _ { t } - \omega \times x h r _ { t , L } }\\= C o v _ { t } \left( M _ { t , t + 1 } , P _ { t + 1 , k - 1 } \right) + e ^ { - r _ { t } } \left\{ e ^ { - \omega \times x h r _ { t , L } } E _ { t } \left[ P _ { t + 1 , k - 1 } \right] - E _ { t } \left[ \tilde { P } _ { t + 1 , k - 1 } \right] \right\}\\= C o v _ { t } \left( M _ { t , t + 1 } , P _ { t + 1 , k - 1 } \right) + E _ { t } e ^ { - r _ { t } } \left\{ e ^ { - \omega \times x h r _ { t , L } } P _ { t + 1 , k - 1 } - \tilde { P } _ { t + 1 , k - 1 } \right\}\\= C o v _ { t } \left( M _ { t , t + 1 } , P _ { t + 1 , k - 1 } \right) + E _ { t } e ^ { - r _ { t } } \left\{ e ^ { - \omega \times x h r _ { t , L } } E _ { t + 1 } \left[ M _ { t + 1 , t + 2 } P _ { t + 2 , k - 2 } \right] - e ^ { - r _ { t + 1 } } E _ { t + 1 } \left[ \tilde { P } _ { t + 2 , k - 2 } \right] \right\}\\= C o v _ { t } \left( M _ { t , t + 1 } , P _ { t + 1 , k - 1 } \right)\\+ E _ { t } e ^ { - r _ { t } } \left\{ e ^ { - \omega \times x h r _ { t , L } } C o v _ { t + 1 } \left( M _ { t + 1 , t + 2 } , P _ { t + 2 , k - 2 } \right) + e ^ { - \omega \times x h r _ { t , L } } E _ { t + 1 } \left[ M _ { t + 1 , t + 2 } \right] E _ { t + 1 } \left[ P _ { t + 2 , k - 2 } \right] - e ^ { - r _ { t + 1 } } E _ { t + 1 } \left[ \tilde { P } _ { t + 2 , k - 2 } \right] \right\}\\= C o v _ { t } \left( M _ { t , t + 1 } , P _ { t + 1 , k - i } \right)\\+ E _ { t } e ^ { - r _ { t } } \left\{ \right. e ^ { - \omega \times x h r _ { t , L } } C o v _ { t + 1 } \left( M _ { t + 1 , t + 2 } , P _ { t + 2 , k - 2 } \right) + e ^ { - \omega \times x h r _ {\mathrm{t}, L} } E _ { t + 1 } \left[ M _ { t + 1 , t + 2 } \right] E _ { t + 1 } \left[ P _ { t + 2 , k - 2 } \right] - e ^ { - r _ { t + 1 } } E _ { t + 1 } \left[ \right. \tilde { P } _ { t + 2 , k - 2 } \left[ P _ { t + 2 , k - 2 } \right]\left. \right\}\\b e c a u s e E _ { t + 1 } [ M _ t + 1 , t + 2 ] ] = e ^ { - r _ { t + 1} - \omega \times x h r _ { t + 1 , L } }\\= C o v _ { t } ( M _ { t , t + 1 } , P _ { t + 1 , k - i})\\+ E _ { t } e ^ { - r _ { t } } \left\{ e ^ { - \omega \times x h r _ { t , L } } C o v _ { t + 1 } ( M _ { t + 1 , t + 2 }, P _ { t + 2 , k - 2 }) + e ^ {- r _ { t + 1}} e ^ {- (\omega \times ( x h r _ { t , L } + x h r _ { t + 1 , L} )} E _ { t + 1 } [ P _ { t + 2 , k - 2} ] - E _ { t + 1 } [ \tilde { P } _ { t + 2 , k - 2} ] \right\}\\= C o v _ { t } ( M _ { t , t + 1 } , P _ { t + 1 , k - i})\\+ E _ { t } e ^ { - r _ { t } } \left\{ e ^ { - \omega \times x h r _ { t , L } } C o v _ { t + 1 } ( M _ { t + 1 , t + 2 }, P _ {t + 2 , k - 2}) \right\}\\+ E _ { t } e ^ {- r _ { t }} \left\{ e ^ {- \omega \times x h r _ { t , L }} C o v _ { t + 1 } ( M _ { t + 1 , t + 2 }, P _ {t + 2 , k - 2}) \right\}\\= C o v _ { t } ( M _ { t , t + 1 } , P _ { t + 1 , k - i})\\+ E _ { t } e ^ {- r _ { t }} \left\{ e ^ {- \omega \times x h r _ {\mathrm{t}, L}} C o v _ { t + 1 } ( M _ { t + 1 , t + 2 }, P _ {t + 2 , k - 2}) \right\}\\= C o v _ { t } ( M _ t, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, .\\= C o v _ { t } ( M _ t, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, n, .\\= C o v _ { t } ( M _ t, n, n, n, n, n, n, n, n, n, n, m a x y z | p | q | q | q | q | q | q | q | q | q | q | q | q | q | q | q | q | q | q | q | q | q | q | q | q | q | q | q | q | q | q | q | q | q | q | q | q | q | q | q | q | q | q | q | q | q | q | q | q | q | q | p | p | p | p | p | p | p | p | p | p | p | p | p | p | p | p | p | p | p | p | p | p | p | p | p | p | p | p | p | p | p | p | p | p | p | p | p | p | p | p | p | p | p&= C o v ( M )\\= C o v ( M )\\= C o v ( M )\\= C o v ( M )\\= C o v ( M )\\= C o v ( M )\\= C o v ( M )\\= C o v ( M )\\= C o v ( M )\\= C o v ( M )\\= C o v ( M )\\= C o v ( M )\\= C o v ( M )\\= C o c o v ( M )\\= C o c o v ( M )\\= C o c o v ( M )\\= C o c o v ( M )\\= C o c o v ( M )\\= C o c o v ( M )\\= C o c o v ( M )\\= C o c o v ( M )\\= C o c o v ( M )\\= C o c o v ( M )\\= C o d i s i s i s i s i s i s i s i s i s i s i s i s i s i s i s i s i s i s i s i s i s i s i s i s i s i s i s i s i s i s i s i s i s i s i s i s i s i s i s i s i s i s i s i s i s i s i s i s i s i s i c a l l e f f a l l e f f a l l e f f a l l e f f a l l e f f a l l e f f a l l e f f a l l e f f a l l e f f a l l e f f a l l e f f a l l e f f a l l e f f a l l e f f a l l e f f a l l e f f a l l e f f a l l l e f f a l l l e f f a l l l e f f a l l l e f f a l l l e f f a l l l e f f a l l l e f f a l l l e f f a l l l e f f a l l l e f f a l l l e f f a l l l e f f a l l l e f f a l l l e f f a l l / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / .\\= C o c o v ( M )\\= C o c o v ( M )\\= C o c o v ( M )\\= C o c o v ( M )\\= C o c o v ( M )\\= C o c o v ( M )\\= C o c o v ( M )\\= C o c o v ( M )\\= C o c o v ( M )\\= C o c o v ( M )\end{array}\]

\[\begin{array}{l} + E _ {t} e ^ {- r _ {t}} \left\{e ^ {- \omega \times x h r _ {t, L}} C o v _ {t + 1} \left(M _ {t + 1, t + 2}, P _ {t + 2, k - 2}\right) \right\} \\ + E _ {t} e ^ {- r _ {t} - r _ {t + 1}} \left\{e ^ {- \omega \times (x h r _ {t, L} + x h r _ {t + 1, L})} C o v _ {t + 2} \left(M _ {t + 2, t + 3}, P _ {t + 3, k - 3}\right) \right\} \\ + E _ {t} e ^ {- r _ {t} - r _ {t + 1} - r _ {t + 2}} \left\{e ^ {- \omega \times (x h r _ {t, L} + x h r _ {t + 1, L} + x h r _ {t + 2, L})} P _ {t + 3, k - 3} - \tilde {P} _ {t + 3, k - 3} \right\} \end{array}\]

and hence in general

\[\begin{array}{r c l} P _ {t, k} - \hat {P} _ {t, k} & = & E _ {t} \left[ \sum_ {j = 0} ^ {k - 1} e ^ {- \sum_ {m = 0} ^ {j - 1} r _ {t + m} + \omega \times x h r _ {t + m, L}} C o v _ {t + j} (M _ {t + j, t + j + 1}, P _ {t + j + 1, k - j - 1}) \right] \\ & & + E _ {t} \left[ e ^ {- \sum_ {m = 0} ^ {k - 1} \omega \times x h r _ {t + m, L}} - 1 \right] \end{array}\]

Thus, we get

\[\begin{array}{l} T P _ {t, k} = \frac {- 1}{k P _ {s s , k}} \left(P _ {t, k} - \hat {P} _ {t, k}\right) \\ \Updownarrow \\ T P _ {t, k} = \frac {- 1}{k P _ {s s , k}} \left\{ \begin{array}{c} E _ {t} \left[ \sum_ {j = 0} ^ {k - 1} e ^ {- \sum_ {m = 0} ^ {j - 1} (r _ {t + m} + \omega \times x h r _ {t + m, L})} C o v _ {t + j} \left(M _ {t + j, t + j + 1}, P _ {t + j + 1, k - j - 1}\right) \right] \\ + E _ {t} \left[ e ^ {- \sum_ {m = 0} ^ {k - 1} \omega \times x h r _ {t + m, L}} - 1 \right] \end{array} \right\} \end{array}\]

Note that if , then we get

\[\begin{array}{r c l} T P _ {t, k} & = & \frac {- 1}{k P _ {s s , k}} \left\{E _ {t} \left[ \sum_ {j = 0} ^ {k - 1} e ^ {- \sum_ {m = 0} ^ {j - 1} r _ {t + m}} C o v _ {t + j} \left(M _ {t + j, t + j + 1}, P _ {t + j + 1, k - j - 1}\right) \right] + E _ {t} \left[ e ^ {0} - 1 \right] \right\} \\ & = & \frac {- 1}{k P _ {s s , k}} E _ {t} \left[ \sum_ {j = 0} ^ {k - 1} e ^ {- \sum_ {m = 0} ^ {j - 1} r _ {t + m}} C o v _ {t + j} \left(M _ {t + j, t + j + 1}, P _ {t + j + 1, k - j - 1}\right) \right] \end{array}\]

as in Rudebusch & Swanson (2012), their Eq 35. Given that generally is positive, we see that adds more discounting to but it also makes substantially less than 1 (i.e. increases the final term in absolute value) and this serves to increase term premia. Note that this final term arises because we price the risk-neutral bond by and not by the deposit rate which is only provided to households and not to the bond trading financial intermediary.

16.18.2 The excess holding period return

To understand the dynamic properties of term premium in the model consider the ex ante excess holding period return:

\[\begin{array}{r l} & x h r _ {t, L} \equiv \mathbb {E} _ {t} \left[ \log \left(P _ {t + 1, L - 1} / P _ {t, L}\right) \right] - r _ {t} \\ & \qquad = \mathbb {E} _ {t} \left[ \log \left(P _ {t + 1, L - 1}\right) - \log P _ {t, L} \right] - r _ {t} \\ & \qquad = \mathbb {E} _ {t} \left[ \log \left(P _ {t + 1, L - 1}\right) - \log \left\{\mathbb {E} _ {t} \left[ M _ {t, t + 1} P _ {t + 1, L - 1} \right] \right\} \right] - r _ {t} \\ & \qquad = \mathbb {E} _ {t} \left[ \log \left(P _ {t + 1, L - 1}\right) - \log \left\{\mathbb {E} _ {t} \left[ M _ {t, t + 1} \right] \mathbb {E} _ {t} \left[ P _ {t + 1, L - 1} \right] + C o v _ {t} \left(M _ {t, t + 1}, P _ {t + 1, L - 1}\right) \right\} \right] - r _ {t} \end{array}\]

Now recall that a first-order approximation implies

Applied in our case with and , implying that , we have

\[x h r _ {t, L} \approx \mathbb {E} _ {t} \left[ \log \left(P _ {t + 1, L - 1}\right) - \left\{\log P _ {s s, L} + \frac {1}{P _ {s s , L}} \left(\mathbb {E} _ {t} \left[ M _ {t, t + 1} \right] \mathbb {E} _ {t} \left[ P _ {t + 1, L - 1} \right] - M _ {s s, s s + 1} P _ {s s, L - 1}\right) + \frac {1}{P _ {s s , L}} C o v _ {t} \left(M _ {t, t + 1}, P _ {t + 1, L - 1}\right) \right\} \right]\]

\[\begin{array}{r l} & {- r _ {t}} \\ & {= \mathbb {E} _ {t} \left[ \log \left(P _ {t + 1, L - 1}\right) - \log P _ {s s, L} - \frac {1}{P _ {s s , L}} \mathbb {E} _ {t} \left[ M _ {t, t + 1} \right] \mathbb {E} _ {t} \left[ P _ {t + 1, L - 1} \right] + \frac {M _ {s s , s s + 1} P _ {s s , L - 1}}{P _ {s s , L}} - \frac {1}{P _ {s s , L}} C o v _ {t} \left(M _ {t, t + 1}, P _ {t + 1, L - 1}\right) \right]} \\ & {- r _ {t}} \\ & {= \mathbb {E} _ {t} \left[ \log \left(P _ {t + 1, L - 1}\right) \right] - \log P _ {s s, L} - \frac {1}{P _ {s s , L}} \mathbb {E} _ {t} \left[ M _ {t, t + 1} \right] \mathbb {E} _ {t} \left[ P _ {t + 1, L - 1} \right] + \frac {M _ {s s , s s + 1} P _ {S S , L - 1}}{P _ {s s , L}} - \frac {1}{P _ {s s , L}} C o v _ {t} \left(M _ {t, t + 1}, P _ {t + 1, L - 1}\right)} \\ & {- r _ {t}} \\ & {= \mathbb {E} _ {t} \left[ \log \left(P _ {t + 1, L - 1}\right) \right] - \log P _ {s s, L} - \frac {\mathbb {E} _ {t} \left[ P _ {t + 1 , L - 1} \right]}{P _ {s s , L}} e ^ {- r _ {t} - \omega \times x h r _ {t, L}} + 1 - \frac {C o v _ {t} (M _ {t , t + 1} , P _ {t + 1 , L - 1})}{P _ {s s , L}} - r _ {t}} \\ & {\mathrm{because} \mathbb {E} _ {t} \left[ M _ {t, t + 1} e ^ {r _ {t} + \omega \times x h r _ {t, L}} \right] = 1 \Longleftrightarrow e ^ {- r _ {t} - \omega \times x h r _ {t, L}} = \mathbb {E} _ {t} \left[ M _ {t, t + 1} \right]} \\ & {\mathrm{and} P _ {s s, L} = M _ {s s, s s + 1} P _ {s s, L - 1}} \\ & {= \mathbb {E} _ {t} \left[ \log \left(\frac {P _ {t + 1 , L - 1}}{P _ {s s , L}}\right) \right] - \frac {\mathbb {E} _ {t} [ P _ {t + 1 , L - 1} ]}{P _ {s s , L}} e ^ {- r _ {t} - \omega \times x h r _ {t, L}} - \frac {C o v _ {t} (M _ {t , t + 1} , P _ {t + 1 , L - 1})}{P _ {s s , L}} + 1 - r _ {t}} \end{array}\]

Now consider the first-order approximation round :

\[\begin{array} { l }\text {Now consider the first - order approximation round } x _ { s s } .\\e ^ { x _ { t } } \approx e ^ { x _ { s s } } + e ^ { x _ { s s } } \left( x _ { t } - x _ { s s } \right) = e ^ { x _ { s s } } \left( 1 + x _ { t } - x _ { s s } \right)\\\text {Applied to } x _ { t } = - r _ { t } - \omega \times x h r _ { t , L } \text {and} x _ { s s } = - r _ { s s } , \text {we get}\\e ^ { - r _ { t } - \omega \times x h r _ { t , L } } \approx e ^ { - r _ { s s } } \left( 1 - r _ { t } - \omega \times x h r _ { t , L } + r _ { s s } \right)\\\text {Thus we get}\\x h r _ { t , L } \approx \mathbb { E } _ { t } \left[ \log \left( \frac { P _ { t + 1 , L - 1 } } { P _ { s s , L } } \right)\right] - \frac { \mathbb { E } _ { t } [ P _ { t + 1 , L - 1 } ] } { P _ { s s , L } } \left( \right. e ^ { - r _ { s s } } \left( 1 - r _ { t } - \omega \times x h r _ { t , L } + r _ { s s } ) \right) - \frac { C o v _ { t } ( M _ { t , t + 1 } , P _ { t + 1 , L - 1 } ) } { P _ { s s , L } }\\+ 1 - r _ { t }\\= \mathbb { E } _ { t } \left[ \log \left( \frac { P _ { t + 1 , L - 1 } } { P _ { s s , L } } \right) \right] - \frac { \mathbb { E } _ { t } [ P _ { t + 1 , L - 1 } ] } { P _ { s s , L } } M _ { s s , s s + 1 } \left( 1 - r _ { t } - \omega \times x h r _ { t , L } + r _ { s s } \right) - \frac { C o v _ { t } ( M _ { t , t + 1 } , P _ { t + 1 , L - 1 } ) } { P _ { s s , L } }\\+ 1 - r _ { t }\\= \mathbb { E } _ { t } \left[ \log \left( \frac { P _ { t + 1 , L - 1 } } {P _ { s s , L } } \right) \right] - \frac { \mathbb { E } _ { t } [ P _ { t + 1 , L - 1 } ] } { P _ { s s , L - 1 } } ( 1 - r _ { t } - \omega \times x h r _ { t , L } + r _ { s s } ) - \frac { C o v _ { t } ( M _ { t , t + 1 } , P _ { t + 1 , L - 1 } ) } { P _ { s s , L } } + 1 - r _ { t }\\\text {Remember that} M _ { s s , s s + 1} P _ { s s , L - 1} = P _ { s s , L } \Longleftrightarrow \frac { M _ { s s , s s + 1 }} { P _ { s s , L } } = \frac { 1 } { P _ { s s , L - 1 } }\\\Updownarrow\\x h r _ { t , L } \left( 1 - \omega \times \frac { \mathbb { E } _ { t } [ P _ { t + 1 , L - 1 } ] } { P _ { s s , L - 1 } } \right) = \mathbb { E } _ { t } \left[ \log \left( \frac { P _ { t + 1 , L - 1 } } { P _ { s s , L } } \right) \right] - \frac { \mathbb { E } _ { t } [ P _ { t + 1 , L - 1 } ] } { P _ { s s , L - 1 } } ( 1 - ( r _ { t } - r _ { s s} ) ) - \frac { C o v _ { t } ( M _ { t , t + 1 } , P _ { t + 1 , L - 1 } ) } { P _ { s s , L } } + 1 - r _ { t }\\\Updownarrow\\h r _ { t , L } = \frac { 1 } { 1 - \omega \times \frac { \mathbb { E } _ { t } [ P _ { t + 1 , L - 1 } ] } { P _ { s s , L - 1 } } } \left[ \mathbb { E } _ { t } \left[ \log \left( \frac { P _ { t + 1 , L - 1 } } { P _ { s s , L } } \right) \right] - \frac { \mathbb { E } _ { t } [ P _ { t + 1 , L - 1} ] } { P _ { s s , L - 1 } } ( 1 - ( r _ { t } - r _ { s s} ) ) - \frac { C o v _ { t } ( M _ { t , t + 1 } , P _ { t + 1 , L - 1} ) } { P _ { s s , L } } + 1 - r _ { t } \right]\\\end{array}\]

Note that in the steady state, we have

\[\begin{array}{r l} & x h r _ {s s, L} = \frac {1}{1 - \omega} \left[ \mathbb {E} _ {t} \left[ \log \left(\frac {P _ {s s , L - 1}}{P _ {s s , L}}\right) \right] - (1 - (r _ {s s} - r _ {s s})) + 1 - r _ {t} \right] \\ & \qquad = \frac {1}{1 - \omega} \left[ \log \left(\frac {P _ {s s , L - 1}}{P _ {s s , L}}\right) - r _ {t} \right] \\ & \qquad = \frac {1}{1 - \omega} \left[ \log \left(\frac {P _ {s s , L - 1}}{M _ {s s , s s + 1} P _ {s s , L - 1}}\right) - r _ {s s} \right] \\ & \qquad \mathrm{because} M _ {s s, s s + 1} P _ {s s, L - 1} = P _ {s s, L} \\ & \qquad = \frac {1}{1 - \omega} \left[ \log \left(\frac {1}{M _ {s s , s s + 1}}\right) - r _ {s s} \right] \\ & \qquad = \frac {1}{1 - \omega} \left[ - \log (M _ {s s, s s + 1}) - r _ {s s} \right] \end{array}\]

\[\begin{array} { r l } & = \frac { 1 } { 1 - \omega } \left[ - \log \left( e ^ { - r _ { s s } } \right) - r _ { s s } \right] \\ & \text {using} M _ { s s , s s + 1 } = e ^ { - r _ { s s } } \\ & = \frac { 1 } { 1 - \omega } \left[ r _ { s s } - r _ { s s } \right] \\ & = 0 \\ & \text {as desired.} \\ & \\ & \text {Let us re-write the above expression a bit:} \\ & x h r _ { t , L } = x h r _ { t , L } = \frac { 1 } { 1 - \omega \times \frac { \mathbb { E } _ { t } \left[ P _ { t + 1 , L - 1 } \right] } { P _ { s s , L - 1 } } } \left[ \mathbb { E } _ { t } \left[ \log \left( \frac { P _ { t + 1 , L - 1 } } { P _ { s s , L } } \right) \right] - \frac { \mathbb { E } _ { t } [ P _ { t + 1 , L - 1} ] } { P _ { s s , L - 1 } } ( 1 - ( r _ { t } - r _ { s s } ) ) - \frac { C o v _ { t } ( M _ { t , t + 1 } , P _ { t + 1 , L - 1 } ) } { P _ { s s , L } } + 1 - r _ { t } \right] \\ & = \frac { 1 } { 1 - \omega \times \frac { \mathbb { E } _ { t } [ P _ { t + 1 , L - 1 } ] } { P _ { s s , L - 1 } } } \left[ \mathbb { E } _ { t } \left[ \log \left( \frac { P _ { t + 1 , L - 1 } } { M _ { s s , s s + 1 } P _ { s s , L - 1 } } \right) \right] - \frac { \mathbb { E } _ { t } [ P _ { t + 1 , L - 1} ] } { P _ { s s , L - 1 } } ( 1 - ( r _ { t } - r _ { s s } ) ) - \frac { C o v _ { t } ( M _ { t , t + 1 } , P _ { t + 2 , L - 1 } ) } { M _ { s s , s s + 1 } P _ { s s , L - 1 } } + 1 - r _ { t } \right] \\ & \text {using} M _ { s s , s s + 1 } P _ { s s , L - 1 } = P _ { s s , L } \\ & = \frac { 1 } { 1 - \omega \times \frac { \mathbb { E } _ { t } [ P _ { t + 1 , L - 1 } ] } { P _ { s s , L - 1 } } } \left[ \mathbb { E } _ { t } \log \left( \frac { P _ { t + 1 , L - 1 } } { P _ { s s , L - 1 } } \right) - \log M _ { s s , s s + 1 } - \frac { \mathbb { E } _ { t } [ P _ { t + 1 , L - 1 } ] } { P _ { s s , L - 1 } } ( 1 - ( r _ { t } - r _ { s s } ) ) - C o v _ { t } \left( \frac { M _ { t , t + 1 } } { M _ { s s , s s + 1 } }, \frac { P _ { t + 1 , L - 1 } } { P _ { s s , L - 1 } } \right) + 1 - r _ { t } \right] \\ & = \frac { 1 } { 1 - \omega \times \frac { \mathbb { E } _ { t } [ P _ { t + 1 , L - 1 } ] } { P _ { s s , L - 1 } } } \left[ \mathbb { E } _ { t } \log \left( \frac { P _ { t + 1 , L - 1 } } { P _ {s s , L - 1 } } \right) - \log e ^ {- r _ { s s }} - \frac { \mathbb { E } _ { t } [ P _ { t + 1 , L - 1} ] } { P _ { s s , L - 1 } } ( 1 - ( r _ { t } - r _ { s s} ) ) - C o v _ { t } \left( \frac { M _ { t , t + 1 } } { M _ { s s , s s + 1 } }, \frac { P _ { t + 1 , L - 1 } } { P _ { s s , L - 1 } } \right) + 1 - r _ { t } \right] \\ & \text {using} M _ { s s , s s + 1 } = e ^ {- r _ { s s }} \\ & = \frac { 1 } { 1 - \omega \times \frac { \mathbb { E } _ { t } [ P _ { t + 1 , L - 1} ] } { P _ { s s , L - 1 } } } \left[ \mathbb { E } _ { t } \log \left( \frac { P _ { t + 1 , L - 1 } } { P _ { s s , L - 1 } } \right) + r _ { s s } - \frac { \mathbb { E } _ { t } [ P _ { t + 1 , L - 1} ] } { P _ { s s , L - 1 } } ( 1 - ( r _ { t } - r _ { s s} ) ) - C o v _ { t } \left( \frac { M _ { t , t + 1 } } { M _ { s s , s s + 1} }, \frac { P _ { t + 1 , L - 1 } } { P _ { s s , L - 1} } \right) + 1 - r _ { t } \right] \\ & = \frac { 1 } { 1 - \omega \times \frac { \mathbb { E } _ { t } [ P _ { t + 1 , L - 1} ] } { P _ { s s , L - 1 } } } \left[ \mathbb { E } _ { t } \log \left( \frac { P _ { t + 1 , L - 1} ] } { P _ { s s , L - 1} } \right) - \frac { \mathbb { E } _ { t } [ P _ { t + 1 , L - 1} ] } { P _ { s s , L - 1} } ( 1 - ( r _ { t } - r _ { s s} ) ) - C o v _ { t } \left( \frac { M _ { t , t + 1 } }{ M _ { s s , s s + 1} }, \frac { P _ { t + 1 , L - 1} ] } { P _ { p s , L - 1} }\right) + 1 - ( r _ { t } - r _ { s s} ) \right] \\ & = \frac { 1 } { 1 - \omega \times \frac { \mathbb { E } _ { t } [ P _ { t + 1 , L - 1} ] } { P _ { p s , L - 1} } } \left[ \mathbb { E } _ { t } \log \left( \frac { P _ { t + 1 , L - 1} ] } { P _ {\mathrm{ss,ss+1}} }\right) + ( 1 - \frac E [ P [ P [ T ] ] ] | p | | p | | p | | p | | p | | p | | p | | p | | p | | p | | p | | p | | p | | p | | p | | p | | p | | m | | m | | m | | m | | m | | m | | m | | m | | m | | m | | m | | m | | m | | m | | m | | m | | m | m | m | m | m | m | m | m | m | . \\ & = ( R ^ {\prime}, R ^ {\prime}) ^ {\prime}. \\ & = ( R ^ {\prime}, R ^ {\prime}) ^ {\prime}. \\ & = ( R ^ {\prime}, R ^ {\prime}) ^ {\prime}. \\ & = ( R ^ {\prime}, R ^ {\prime}) ^ {\prime}. \\ & = ( R ^ {\prime}, R ^ {\prime}) ^ {\prime}. \\ & = ( R ^ {\prime}, R ^ {\prime}) ^ {\prime}. \\ & = ( R ^ {\prime}, K ^ {\prime}). \\ & = ( R ^ {\prime}, K ^ {\prime}) ^ {\prime}. \\ & = ( R ^ {\prime}, K ^ {\prime}) ^ {\prime}. \\ & = ( R ^ {\prime}, K ^ {\prime}) ^ {\prime}. \\ & = ( R ^ {\prime}, K ^ {\prime}) ^ {\prime}. \\ & = ( R ^ {\prime}, K ^ {\prime}) ^ {\prime}. \\ & = ( R ^ {\prime}, K ^ {\prime}) ^ {\prime}. \\ & = ( R ^ {\prime}, k ^ {\prime}). \\ & = ( R ^ {\prime}, k ^ {\prime}). \\ & = ( R ^ {\prime}, k ^ {\prime}). \\ & = ( R ^ {\prime}, k ^ {\prime}). \\ & = ( R ^ {\prime}, k ^ {\prime}). \\ & = ( R ^ {\prime}, k ^ {\prime}). \\ & = ( R ^ {\prime}, k ^ {\prime}). \\ & = ( R ^ {\prime}, k ^ {\prime}). \\ and i n d i n g e f i c i o n e n d i n g e f i c i o n e n d i n g e f i c i o n e n d i n g e f i c i o n e n d i n g e f i c i o n e n d i n g e f i c i o n e n d i n g e f i c i o n e n d i n g e f i c i o n e n d i n g e f i n g e f i c i o n e n d i n g e f i c i o n e n d i n g e f i c i o n e n d i n g e f i c i o n e n d i n g e f i c i o n e n d i n g e f i c i o n e n d i n g e f i c i o n e n d i n g e f i c i u n d i n g e f i c i o n e n d i n g e f i c i u n d i n g e f i c i u n d i n g e f i c i u n d i n g e f i c i u n d i n g e f i c i u n d i n g e f i c i u n d i n g e f i c i u n d i n g e f i c i u n d i n g e f i c i u n d i n g e f i c i u u n d i n g e f i c i u u n d i n g e f i c i u u n d i n g e f i c i u u n d i n g e f i c i u u n d i n g e f i c i u u n d i n g e f i c i u u n d i n g e f i c i u u n d i n g e f i c i u u n d j a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y a l l y . \\ & = ( R ^ {\prime}, R ^ {\prime}). \\ & = ( R ^ {\prime}, R ^ {\prime}). \\ & = ( R ^ {\prime}, R ^ {\prime}). \\ & = ( R ^ {\prime}, R ^ {\prime}). \\ & = ( R ^ {\prime}, R ^ {\prime}). \\ & = ( R ^ {\prime}, R ^ {\prime}). \\ & = ( R ^ {\prime}, R ^ {\prime}). \\ & = ( R ^ {\prime}, R ^ {\prime}). \\ * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * . \\ & = ( R ^ {\prime}, R ^ {\prime}). \\ & = ( R ^ {\prime}, R ^ {\prime}). \\ & = ( R ^ {\prime}, R ^ {\prime}). \\ & = ( R ^ {\prime}, R ^ {\prime}). \\ & = ( R ^ {\prime}, R ^ {\prime}). \\ & = ( R ^ {\prime}, R ^ {\prime}). \\ & = ( R ^ {\prime}, R ^ {\prime}). ; \\ & = ( R ^ {\prime}, R ^ {\prime}). \\ & = ( R ^ {\prime}, R ^ {\prime}). \\ & = ( R ^ {\prime}, R ^ {\prime}). \\ & = ( R ^ {\prime}, R ^ {\prime}). \\ & = ( R ^ {\prime}, R ^ {\prime}). \\ & = ( R ^ {\prime}, R ^ {\prime}). \\ & = ( R ^ {\prime}, R ^ {\prime}). \\ # > N, N < j < k, k < j < k, k < j < k, k < j < k, k < j < k, k < j < k, k < j < k, k < j < k, k < j < k, k < j < k, k < j < k, k < j < k, k < j < k, k < j< |content_end|>\]

\[\begin{array}{r l} & {\mathrm{Nowafirstorderapproximationoflog} (x _ {t}) \mathrm{around} x _ {s s} = 1 \mathrm{gives}} \\ & {\log (x _ {t}) \approx \log (1) + \frac {1}{x _ {s s}} (x _ {t} - x _ {s s}) = x _ {t} - 1} \\ & {\mathrm{Appliedinourcaseto} \log \left(\frac {P _ {t + 1 , L - 1}}{P _ {s s , L - 1}}\right) \mathrm{weget}} \\ & {x h r _ {t, L} \approx \frac {1}{1 - \omega \times \frac {\mathbb {E} _ {t} [ P _ {t + 1 , L - 1} ]}{P _ {s s , L - 1}}} \left[ \mathbb {E} _ {t} \left(\frac {P _ {t + 1 , L - 1}}{P _ {s s , L - 1}} - 1\right) + \left(1 - \frac {\mathbb {E} _ {t} [ P _ {t + 1 , L - 1} ]}{P _ {s s , L - 1}}\right) (1 - (r _ {t} - r _ {s s})) - C o v _ {t} \left(\frac {M _ {t , t + 1}}{M _ {s s , s s + 1}}, \frac {P _ {t + 1 , L - 1}}{P _ {s s , L - 1}}\right) \right]} \\ & {\quad = \frac {1}{1 - \omega \times \frac {\mathbb {E} _ {t} [ P _ {t + 1 , L - 1} ]}{P _ {s s , L - 1}}} \left[ \left(\frac {\mathbb {E} _ {t} [ P _ {t + 1 , L - 1} ]}{P _ {s s , L - 1}} - 1\right) + \left(1 - \frac {\mathbb {E} _ {t} [ P _ {t + 1 , L - 1} ]}{P _ {s s , L - 1}}\right) - \left(1 - \frac {\mathbb {E} _ {t} [ P _ {t + 1 , L - 1} ]}{P _ {s s , L - 1}}\right) (r _ {t} - r _ {s s}) - C o v _ {t} \left(\frac {M _ {t , t + 1}}{M _ {s s , s s + 1}}, \frac {P _ {t + 1 , L - 1}}{P _ {s s , L - 1}}\right) \right]} \\ & {\quad = \frac {1}{1 - \omega \times \frac {\mathbb {E} _ {t} [ \mathrm{} P _ {t + 1 , L - 1} ]}{\mathrm{} P _ {s s , L - 1}}} \left[ \left(\frac {\mathbb {E} _ {t} [ P _ {t + 1 , L - 1} ]}{P _ {s s , L - 1}} - 1\right) (r _ {t} - r _ {s s}) - C o v _ {t} \left(\frac {M _ {t , t + 1}}{M _ {s s , s s + 1}}, \frac {P _ {t + 1 , L - 1}}{P _ {s s , L - 1}}\right) \right]} \end{array}\]

That is, up to a first order approximation, we have

\[x h r _ {t, L} \approx \frac {1}{1 - \omega \times \frac {\mathbb {E} _ {t} [ P _ {t + 1 , L - 1} ]}{P _ {s s , L - 1}}} \left[ \left(\frac {\mathbb {E} _ {t} [ P _ {t + 1 , L - 1} ]}{P _ {s s , L - 1}} - 1\right) (r _ {t} - r _ {s s}) - C o v _ {t} \left(\frac {M _ {t , t + 1}}{M _ {s s , s s + 1}}, \frac {P _ {t + 1 , L - 1}}{P _ {s s , L - 1}}\right) \right]\]

Importantly, we have a multiplication effect in our mode if , meaning that the effect of is amplified. This explains why we can settle with lower variation in and hence to explain term premia in our model with feedback effects.

References

1. Altig, D., Christiano, L. J., Eichenbaum, M. & Linde, J. (2011), ‘Firm-specific capital, nominal rigidities and the business cycle’, Review of Economic Dynamics 14, 225–247.
2. Andreasen, M. M. (2012), ‘An estimated DSGE model: Explaining variation in nominal term premia, real term premia, and inflation risk premia’, European Economic Review 56, 1656–1674.
3. Epstein, L. G. & Zin, S. E. (1989), ‘Substitution, risk aversion, and the temporal behavior of consumption and asset returns: A theoretical framework’, Econometrica 57(4), 937–969.
4. Haan, W. J. D. & Wind, J. D. (2012), ‘Nonlinear and stable perturbation-based approximations’, Journal of Economic Dynamic and Control 36, 1477–1497.
5. Rudebusch, G. D. & Swanson, E. T. (2012), ‘The bond premium in a DSGE model with long-run real and nominal risks’, American Economic Journal: Macroeconomics 4(1), 1–43.
6. Schmitt-Grohé, S. & Uribe, M. (2004), ‘Solving dynamic general equilibrium models using a second-order approximation to the policy function’, Journal of Economic Dynamics and Control 28, 755–775.
7. Swanson, E. (2012), ‘Risk aversion and the labor margin in dynamic equilibrium models’, Working Paper version.