Perturbation Methods for Markov-Switching DSGE Models by ** Andrew Foerster^*, Juan Rubio-Ramirez Dan Waggoner^*** and Tao Zha^**** Documento de Trabajo 2013-22

December 2013

* Federal Reserve Bank of Kansas City ** Duke University, Federal Reserve Bank of Atlanta, CEPR, FEDEA, and BBVA Research *** Federal Reserve Bank of Atlanta. **** Federal Reserve Bank of Atlanta, Emory University, and NBER.

Los Documentos de Trabajo se distribuyen gratuitamente a las Universidades e Instituciones de Investigación que lo solicitan. No obstante están disponibles en texto completo a través de Internet: http://www.fedea.es. These Working Paper are distributed free of charge to University Department and other Research Centres. They are also available through Internet: http://www.fedea.es. ISSN:1696-750

Perturbation Methods for Markov-Switching DSGE Models

Andrew Foerstery Juan Rubio-Ramirezz Dan Waggonerx Tao Zha{

October 28, 2013

Abstract

This paper develops a general perturbation methodology for constructing high-order approximations to the solutions of Markov-switching DSGE models. We introduce an important and practical idea of partitioning the Markov-switching parameter space so that a steady state is well deÖned. With this deÖnition, we show that the problem of Önding an approximation of any order can be reduced to solving a system of quadratic equations. We propose using the theory of Grˆbner bases in searching all the solutions to the quadratic system. This approach allows us to obtain all the approximations and ascertain how many of them are stable. Our methodology is applied to three models to illustrate its feasibility and practicality.

The views expressed herein are solely those of the authors and do not necessarily reáect the views of the Federal Reserve Banks of Atlanta and Kansas City, the Federal Reserve System, or the National Bureau of Economic Research. We thank Leonardo Melosi, Seonghoon Cho, Rhys Bidder, seminar participants at Duke, the Federal Reserve Bank of St. Louis, the 2010 Society of Economic Dynamics meetings, the 2011 Federal Reserve System Committee on Business and Financial Analysis Conference, the 2012 Annual Meeting of the American Economic Association, the 8th Dynare Conference, and the 2012 NBER Workshop on Methods and Applications for DSGE Models for helpful comments. This research is supported in part by the National Science Foundation Grants SES-1127665 and SES-1227397.

yFederal Reserve Bank of Kansas City

zDuke University, Federal Reserve Bank of Atlanta, CEPR, FEDEA, and BBVA Research

xFederal Reserve Bank of Atlanta

{Federal Reserve Bank of Atlanta, Emory University, and NBER

1 Introduction

In this paper we show how to use perturbation methods as described in Judd (1998) and Schmitt-Grohe and Uribe (2004) to solve Markov-switching dynamic stochastic general equilibrium (MSDSGE) models. Our contribution advances the current literature in two signiÖcant respects. First, we develop a general methodology for approximating the solution of a larger class of Markov-switching models than currently possible and, second, we show the feasibility and practicality of implementing our methodology when we consider high-order approximations to the model solution. Current methods only allow for Örst-order approximations.

The literature on Markov-switching linear rational expectations (MSLRE) models has been an active Öeld in empirical macroeconomics (Leeper and Zha (2003), Blake and Zampolli (2006), Svensson and Williams (2007), Davig and Leeper (2007), and Farmer et al. (2009)). Building on standard linear rational expectations models, the MSLRE approach allows model parameters to change over time according to discrete Markov chain processes. This nonlinearity has proven to be important in explaining changes in monetary policy and macroeconomic time series (Schorfheide (2005), Davig and Doh (2008), Liu et al. (2011), and Bianchi (2010)) and in modeling the expected e§ects of future Öscal policy changes (Davig et al. (2010), Davig et al. (2011), Bi and Traum (2012)). In particular, Markov-switching models provide a tractable way to study how agents form expectations over possible discrete changes in the economy, such as those in technology and policy.

There are, however, two main shortcomings with the MSLRE approach. First, that approach begins with a system of standard linear rational expectations equations that have been obtained from linearizing equilibrium conditions as though the parameters were constant over time. Discrete Markov chain processes are then annexed to certain parameters. As a consequence, the resultant MSLRE model may be incompatible with the optimizing behavior of agents in an original economic model with Markov-switching parameters. Second, because it builds on linear rational expectations models, the MSLRE approach does not take into account higher-order coe¢ cients in the approximation. Higher-order approximations improve the approximation accuracy and may be potentially important for certain economic questions.

This paper develops a general perturbation methodology for constructing Örst-order and second-order approximations to the solutions of MSDSGE models in which certain parameters vary according to discrete Markov chain processes. Our method can be easily expanded to higher-order approximations. The key is to Önd the approximations using the equilibrium conditions implied by the original economic model when Markov-switching parameters are present. Thus, our method overcomes the aforementioned shortcomings. By working with the original MSDSGE model directly rather than taking a system of linear rational expectations equations with constant parameters as a short-cut, we maintain congruity between the original economic model with Markov-switching parameters and the resultant approximations to the model solution. Such congruity is necessary for researchers to derive both Örst-order and second-order approximations.

Unlike the case with a standard model with constant parameters, one conceptual di¢ culty while working with MSDSGE models is that certain Markov-switching parameters, such as the mean growth rate of technology, complicate the steady-state deÖnition. The deÖnition of the steady-state for MSDSGE models should be independent of the realization of the discrete Markov chain process for any changing parameter. To this end, we introduce a new concept of steadystate. We partition the parameter space such that the subset of Markov-switching parameters that would ináuence the steady-state in the constant parameter case is now a function of the perturbation parameter, while the complementary subset is not. This new concept renders a key to deriving Örst-order and second-order approximations.

Several important results are as follows. First, we Önd that Örst-order approximations to the solutions of MSDSGE models are, in general, not certainty equivalent when a subset of Markov-switching parameters needs to be perturbed. Second, we identify the task of Önding all the solutions to a system of quadratic equations as the only bottleneck in obtaining Örst-order and second-order approximations to the solutions of MSDSGE models.

Third, we propose to remove the bottleneck by applying Grˆbner bases, a key insight of our approach. Grˆbner bases have not only a sound mathematical theory but also many computational applications. For example, Grˆbner bases have been successfully applied by Kubler and Schmedders (2010a) and Kubler and Schmedders (2010b) to Önd multiple equilibria in general equilibrium models; Datta (2010) uses such bases to Önd all the Nash equilibria. For our purpose, Grˆbner bases provide a computationally feasible way to obtain all the solutions to a system of polynomial equations. Once the bottleneck of solving a system of quadratic equations is resolved, the remaining task to obtain Örst-order and second-order approximations involves solving only systems of linear equations even for higher-order approximations.

Fourth, one may use a numerical algorithm to search for a solution to the quadratic system, but there is no guarantee that such an algorithm is capable of Önding all solutions. By employing Grˆbner bases to solve the quadratic system, we can Örst obtain all its solutions and then determine how many approximations are stable, where the stability concept follows the earlier work of Costa et al. (2005), Farmer et al. (2009), Farmer et al. (2011), and Cho (2011). This procedure enables researchers to ascertain both the existence and the uniqueness of a stable approximation.

The rest of the paper is organized as follows. Section 2 presents a general class of MSDSGE models, outlines our methodology, and introduces a the concept of steady-state. Section 3 derives Örst-order approximations and discusses the feature of no certainty equivalence. Section 4 shows how to reduce the problem of Önding Örst-order approximations to that of Önding all the solutions to a system of quadratic equations. Grˆbner bases are proposed to tackle this problem and the concept of stability is introduced in this section. Section 5 derives a secondorder approximation and shows that it involves solving only linear systems once a Örst-order approximation is obtained. Section 6 illustrates how to apply our methodology to three di§erent MSDSGE models. Concluding remarks are o§ered in Section 7.

2 The General Framework

Our general framework is laid out as follows. First, we discuss a general class of MSDSGE models with a simple example to guide the reader through our new notation. Second, we shows the need to deÖne a concept of steady-state that is regime independent. Third, we introduce the steady-state to be used in the paper and discuss a new idea of partitioning the Markov-switching parameter space that will help us to Önd such a steady-state. Finally, we state the goal of this paper.

2.1 The Model

We study a general class of MSDSGE models in which some of the parameters follow a discrete Markov chain process indexed by with the transition matrix . The element represents the probability that given for , where is the number of regimes. When , the model is said to be in regime s at time t. We denote the vector of all Markov-switching parameters by This vector takes values in the following set .²

Given , the equilibrium conditions for MSDSGE models have the general form

\[\mathbb {E} _ {t} f \left(\mathbf {y} _ {t + 1}, \mathbf {y} _ {t}, \mathbf {x} _ {t}, \mathbf {x} _ {t - 1}, \chi \pmb {\varepsilon} _ {t + 1}, \pmb {\varepsilon} _ {t}, \pmb {\theta} _ {t + 1}, \pmb {\theta} _ {t}\right) = \mathbf {0} _ {n _ {x} + n _ {y}},\tag{1}\]

where denotes the mathematical expectation operator conditional on information available at time t, is an vector of zeros, is the vector of (endogenous and exogenous) predetermined variables, is the vector of non-predetermined (control) variables, is the vector of i.i.d. innovations to the exogenous predetermined variables with and , and is the perturbation parameter. Where denotes the identity matrix of size and | indicates transpose. Since the total number of equations in (1) is , the function maps into

¹ Unless otherwise stated, all vectors in this paper are column vectors.

²Note that the vector does not include other parameters that are constant over time. We call those parameters ìconstant parameters.

Finding Örst-order and second-order approximations to the solutions of MSDSGE models using perturbation methods requires us to introduce new notation and lengthy algebraic work. To aid the reader through our new notation and derivations in the paper, we use a simple real business cycle (RBC) model as a clear demonstration of how our proposed methodology works.

The RBC model is an economy with the representative household whose preferences over a stochastic sequence of consumption goods, ; are represented by the utility function

\[\max \mathbb {E} _ {0} \sum_ {t = 0} ^ {\infty} \beta^ {t} \frac {c _ {t} ^ {v}}{v}\]

where denotes the discount factor, and controls the degree of risk aversion. The resource constraint is

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

where is a stock of physical capital and represents a technological change that is a random walk in log with a Markov-switching drift as

\[\log z _ {t} = \mu_ {t} + \log z _ {t - 1} + \sigma_ {t} \varepsilon_ {t},\]

where the drift and the standard deviation take two discrete values dictated by the Markov chain process represented by , and

The three equations characterizing the equilibrium are

\[c _ {t} ^ {\upsilon - 1} = \beta \mathbb {E} _ {t} c _ {t + 1} ^ {\upsilon - 1} \left(\alpha z _ {t + 1} ^ {1 - \alpha} k _ {t} ^ {\alpha - 1} + (1 - \delta)\right),\]

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

\[\mathrm{and} \log z _ {t} = \mu_ {t} + \log z _ {t - 1} + \sigma_ {t} \varepsilon_ {t}.\]

Because log has a unit root, the economy is non-stationary. To derive a stationary equilibrium, deÖne . The transformed (re-scaled) equilibrium conditions become

\[\tilde {c} _ {t} ^ {\upsilon - 1} = \beta \tilde {z} _ {t} ^ {\upsilon - 1} \mathbb {E} _ {t} \tilde {c} _ {t + 1} ^ {\upsilon - 1} \left(\alpha \tilde {z} _ {t + 1} ^ {1 - \alpha} \tilde {k} _ {t} ^ {\alpha - 1} + 1 - \delta\right),\]

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

\[\text { and } \log \tilde {z} _ {t} = \mu (s _ {t}) + \sigma (s _ {t}) \varepsilon_ {t}.\]

Substituting out leads to the following two equilibrium conditions

\[\tilde {c} _ {t} ^ {\upsilon - 1} = \beta \exp \left(\mu (s _ {t}) + \sigma (s _ {t}) \varepsilon_ {t}\right) ^ {\upsilon - 1} \mathbb {E} _ {t} \tilde {c} _ {t + 1} ^ {\upsilon - 1} \left(\alpha (\exp (\mu (s _ {t + 1}) + \sigma (s _ {t + 1}) \chi \varepsilon_ {t + 1})) ^ {1 - \alpha} \tilde {k} _ {t} ^ {\alpha - 1} + 1 - \delta\right),\]

\[\mathrm{and} \tilde {c} _ {t} + \tilde {k} _ {t} \exp (\mu (s _ {t}) + \sigma (s _ {t}) \varepsilon_ {t}) = \exp (\mu (s _ {t}) + \sigma (s _ {t}) \varepsilon_ {t}) ^ {1 - \alpha} \tilde {k} _ {t - 1} ^ {\alpha} + (1 - \delta) \tilde {k} _ {t - 1},\]

Using the notation in this section, we have , and . The equilibrium condition (1) can be speciÖcally expressed as

\[\begin{array}{c} \mathbb {E} _ {t} f \left(\mathbf {y} _ {t + 1}, \mathbf {y} _ {t}, \mathbf {x} _ {t}, \mathbf {x} _ {t - 1}, \chi \pmb {\varepsilon} _ {t + 1}, \pmb {\varepsilon} _ {t}, \pmb {\theta} _ {t + 1}, \pmb {\theta} _ {t}\right) = \\ \mathbb {E} _ {t} \left[ \begin{array}{c} \tilde {c} _ {t} ^ {v - 1} - \beta \exp \left(\mu \left(s _ {t}\right) + \sigma \left(s _ {t}\right) \varepsilon_ {t}\right) ^ {v - 1} \tilde {c} _ {t + 1} ^ {v - 1} \left(\alpha \left(\exp \left(\mu \left(s _ {t + 1}\right) + \sigma \left(s _ {t + 1}\right) \chi \varepsilon_ {t + 1}\right)\right) ^ {1 - \alpha} \tilde {k} _ {t} ^ {\alpha - 1} + 1 - \delta\right) \\ \tilde {c} _ {t} + \tilde {k} _ {t} \tilde {z} _ {t} - \exp \left(\mu \left(s _ {t}\right) + \sigma \left(s _ {t}\right) \varepsilon_ {t}\right) ^ {1 - \alpha} \tilde {k} _ {t - 1} ^ {\alpha} - (1 - \delta) \tilde {k} _ {t - 1} \end{array} \right] \end{array}\tag{2}\]

2.2 Approximating the Solution and the Approximation Point

Consider model solutions of the form

\[\mathbf {y} _ {t} = g \left(\mathbf {x} _ {t - 1}, \pmb {\varepsilon} _ {t}, \chi , s _ {t}\right), \mathbf {y} _ {t + 1} = g \left(\mathbf {x} _ {t}, \chi \pmb {\varepsilon} _ {t + 1}, \chi , s _ {t + 1}\right), \mathrm{and} \mathbf {x} _ {t} = h \left(\mathbf {x} _ {t - 1}, \pmb {\varepsilon} _ {t}, \chi , s _ {t}\right)\tag{3}\]

where maps into and h maps into

In general, we do not know the explicit functional forms of and thus, we need to approximate them. In this paper we propose using Taylor expansions for the approximations.⁴ When taking this road it is important to choose the point of approximation. In constant parameters DSGE models, the Taylor expansions are usually done around the steady-state that can be found when uncertainty related to the innovations to the exogenous predetermined variables is eliminated by setting and . In MSDSGE models, the Markov-switching parameters are an additional source of uncertainty that we need to deal with when choosing the point of approximation.

³In theory, one could allow and to depend on the entire history of the regimes. In practice, this is not tractable. The assumption that and h depend only on the current regime corresponds to the notion of a minimal state variable (MSV) solution in Farmer et al. (2011).

⁴We assume that the solutions g and h are unique.

One natural possibility is to approximate the policy functions around a set of regime-speciÖc steady-states. Under this choice, the policy functions would be approximated around di§erent points depending on the realization of the Markov chain process. The set of regime-speciÖc steady-states can be obtained by perturbing the transition matrix to eliminate regime change when the model is perturbed. In order to accomplish this version of perturbation, we need to rewrite the transition matrix as , where if and 0 otherwise. This set up implies that the transition matrix _s when , and when

Using this new transition matrix, with the assumption that innovations to the exogenous predetermined variables, , are independent of the discrete Markov chain process, , using (3), we can rewrite (1) as

\[\begin{array}{c} \mathbb {F} \left(\mathbf {x} _ {t - 1}, \boldsymbol {\varepsilon} _ {t}, \chi , s _ {t}\right) = \\ \sum_ {s ^ {\prime} = 1} ^ {n _ {s}} \binom{\mathbf {1} _ {\{s _ {t} \}} \left(s ^ {\prime}\right) +}{\chi \left(p _ {s, s ^ {\prime}} - \mathbf {1} _ {\{s \}} \left(s ^ {\prime}\right)\right)} \int f \binom{g \left(h \left(\mathbf {x} _ {t - 1}, \boldsymbol {\varepsilon} _ {t}, \chi , s _ {t}\right), \chi \boldsymbol {\varepsilon} ^ {\prime}, \chi , s ^ {\prime}\right), g \left(\mathbf {x} _ {t - 1}, \boldsymbol {\varepsilon} _ {t}, \chi , s _ {t}\right),}{h \left(\mathbf {x} _ {t - 1}, \boldsymbol {\varepsilon} _ {t}, \chi , s _ {t}\right), \mathbf {x} _ {t - 1}, \chi \boldsymbol {\varepsilon} ^ {\prime}, \boldsymbol {\varepsilon} _ {t}, \boldsymbol {\theta} \left(s ^ {\prime}\right), \boldsymbol {\theta} \left(s _ {t}\right)} \mu \left(\boldsymbol {\varepsilon} ^ {\prime}\right) d \boldsymbol {\varepsilon} ^ {\prime} = \mathbf {0} _ {n _ {y} + n _ {x}} \end{array}\]

for all ; and , where is the density function of the innovations. The function maps into . Using , we can now deÖne the set of regime-speciÖc steady-states.

DeÖnition 1 Let . The vectors and are the regimespeciÖc steady state for regime if they solve the following system of equations

\[\mathbb {F} \left(\mathbf {x} _ {s s} (s _ {t}), \mathbf {0} _ {n _ {\varepsilon}}, 0, s _ {t}\right) = \mathbf {0} _ {n _ {y} + n _ {x}}\]

where and

The next step is to show that there are a set of regime-speciÖc steady states. In order to show that, we need to assume the following about function

Assumption 2 For any , there exists a pair of vectors and that solve the following system of equations

\[f \left(\mathbf {y}, \mathbf {y}, \mathbf {x}, \mathbf {x}, \mathbf {0} _ {n _ {\varepsilon}}, \mathbf {0} _ {n _ {\varepsilon}}, \boldsymbol {\theta}, \boldsymbol {\theta}\right) = \mathbf {0} _ {n _ {y} + n _ {x}}.\]

With this assumption, we can now show that there exists an set of regime-speciÖc steadystates.

Theorem 3 Let . Under Assumption there exist a pair of vectors and that are a regime-speciÖc steady state for regime

Proof. Let . It is easy to see that, when evaluated at , and the function F equals

\[\begin{array}{c}\mathbb {F} \left(\mathbf {x} _ {s s} (s _ {t}), \mathbf {0} _ {n _ {\varepsilon}}, 0, s _ {t}\right) = f \left(\begin{array}{c}g \left(h \left(\mathbf {x} _ {s s} (s _ {t}), \mathbf {0} _ {n _ {\varepsilon}}, 0, s _ {t}\right), 0 \boldsymbol {\varepsilon} ^ {\prime}, 0, s _ {t}\right), g \left(\mathbf {x} _ {s s} (s _ {t}), \mathbf {0} _ {n _ {\varepsilon}}, 0, s _ {t}\right),\\h \left(\mathbf {x} _ {s s} (s _ {t}), \mathbf {0} _ {n _ {\varepsilon}}, 0, s _ {t}\right), \mathbf {x} _ {s s} (s _ {t}), 0 \boldsymbol {\varepsilon} ^ {\prime}, \mathbf {0} _ {n _ {\varepsilon}}, \boldsymbol {\theta} \left(s _ {t}\right), \boldsymbol {\theta} \left(s _ {t}\right)\end{array}\right) =\\f \left(\begin{array}{c}g \left(h \left(\mathbf {x} _ {s s} (s _ {t}), \mathbf {0} _ {n _ {\varepsilon}}, 0, s _ {t}\right), \mathbf {0} _ {n _ {\varepsilon}}, 0, s _ {t}\right), g \left(\mathbf {x} _ {s s} (s _ {t}), \mathbf {0} _ {n _ {\varepsilon}}, 0, s _ {t}\right),\\h \left( \right.\mathbf {x} _ {s s} (s _ {t}), \mathbf {0} _ {n _ {\varepsilon}}, 0, s _ {t}\big) , \mathbf {x} _ {s s} (s _ {t}), \mathbf {0} _ {n _ {\varepsilon}}, \mathbf {0} _ {n _ {\varepsilon}}, \boldsymbol {\theta} \left(s _ {t}\right), \boldsymbol {\theta} \left(s _ {t}\right)\end{array}\right) = \mathbf {0} _ {n _ {y} + n _ {x}}.\end{array}\]

Since we have that and , the result follows.

Clearly, DeÖnition 1 and Theorem 3 imply that the vectors and are the regime-speciÖc steady state for regime if

\[f \left(\mathbf {y} _ {s s} (s _ {t}), \mathbf {y} _ {s s} (s _ {t}), \mathbf {x} _ {s s} (s _ {t}), \mathbf {x} _ {s s} (s _ {t}), \mathbf {0} _ {n _ {\varepsilon}}, \mathbf {0} _ {n _ {\varepsilon}}, \boldsymbol {\theta} (s _ {t}), \boldsymbol {\theta} (s _ {t})\right) = \mathbf {0} _ {n _ {x} + n _ {y}}.\]

Let now try to use this set of regime-speciÖc steady states to approximate the policy functions. First we need to introduce some notation. Let

\[\begin{array}{r l} \mathcal {D} f _ {s s} (s _ {t}) & = \\ & \left[ \mathcal {D} _ {j} f ^ {i} (\mathbf {y} _ {s s} (s _ {t}), \mathbf {y} _ {s s} (s _ {t}), \mathbf {x} _ {s s} (s _ {t}), \mathbf {x} _ {s s} (s _ {t}), \mathbf {0} _ {n _ {\varepsilon}}, \mathbf {0} _ {n _ {\varepsilon}}, \boldsymbol {\theta} (s _ {t}), \boldsymbol {\theta} (s _ {t})) \right] _ {1 \leq i \leq n _ {y} + n _ {x}, 1 \leq j \leq 2 (n _ {y} + n _ {x} + n _ {\varepsilon})} \end{array}\]

denote the matrix of Örst partial derivatives of with respect to , and evaluated at and for all To simplify notation, deÖne

\[\mathcal {D} _ {n, m} f _ {s s} (s _ {t}) = [ \mathcal {D} _ {n} f _ {s s} (s _ {t}) \dots \mathcal {D} _ {m} f _ {s s} (s _ {t}) ]\]

for all and 5

Let and denote the matrix of Örst partial derivatives of and h with respect to and denote the matrix of Örst partial derivatives of and with respect to , and and denote the matrix of Örst partial derivatives of and h with respect to all evaluated at , and for all .

Taking the derivative of with respect to and evaluating at , and for all produces

\[\begin{array}{r} \mathcal {D} _ {1, n _ {y}} f _ {s s} (s _ {t}) g _ {x} (s _ {t}) h _ {x} (s _ {t}) + \mathcal {D} _ {n _ {y} + 1, 2 n _ {y}} f _ {s s} (s _ {t}) g _ {x} (s _ {t}) + \\ \mathcal {D} _ {2 n _ {y} + 1, 2 n _ {y} + n _ {x}} f _ {s s} (s _ {t}) h _ {x} (s _ {t}) + \mathcal {D} _ {2 n _ {y} + n _ {x} + 1, 2 (n _ {y} + n _ {x})} f _ {s s} (s _ {t}) = 0, \end{array}\]

taking the derivative of with respect to and evaluating at , and for all produces

\[\mathcal {D} _ {1, n _ {y}} f _ {s s} (s _ {t}) g _ {x} (s _ {t}) h _ {\varepsilon} (s _ {t}) + \mathcal {D} _ {n _ {y} + 1, 2 n _ {y}} f _ {s s} (s _ {t}) g _ {\varepsilon} (s _ {t}) +\]

\[\mathcal {D} _ {2 n _ {y} + 1, 2 n _ {y} + n _ {x}} h _ {\varepsilon} (s _ {t}) + \mathcal {D} _ {2 (n _ {y} + n _ {x}) + 1, 2 (n _ {y} + n _ {x}) + n _ {\varepsilon}} f _ {s s} (s _ {t}) = 0,\]

and taking the derivative of with respect to and evaluating at , and for all produces

\[\begin{array}{r l r} & & {\mathcal {D} _ {1, n _ {y}} f _ {s s} (s _ {t}) (g _ {\chi} (s _ {t}) + g _ {x} (s _ {t}) h _ {\chi} (s _ {t})) + \mathcal {D} _ {n _ {y} + 1, 2 n _ {y}} f _ {s s} (s _ {t}) g _ {\chi} (s _ {t}) + \mathcal {D} _ {2 n _ {y} + 1, 2 n _ {y} + n _ {x}} f _ {s s} (s _ {t}) h _ {\chi} (s _ {t}) +} \\ & & {\sum_ {s ^ {\prime} = 1} ^ {n _ {s}} (p _ {s _ {t}, s ^ {\prime}} - \mathbf {1} _ {\{s _ {t} \}} (s ^ {\prime})) f (g (\mathbf {x} _ {s s} (s _ {t}), \mathbf {0} _ {n _ {\varepsilon}}, 0; s ^ {\prime}), \mathbf {y} _ {s s} (s _ {t}), \mathbf {x} _ {s s} (s _ {t}), \mathbf {x} _ {s s} (s _ {t}), \mathbf {0} _ {n _ {\varepsilon}}, \mathbf {0} _ {n _ {\varepsilon}}, \pmb {\theta} (s ^ {\prime}), \pmb {\theta} (s _ {t})) = 0.} \end{array}\]

Note that the Örst two derivatives, with respect to and , have expressions identical to the constant parameter case where for all t. Consequently, the solutions for ， , and will not reáect the fact that parameters switch. The third derivative, with respect to requires evaluating for all and . Generally, this expression is only known in the case of . This result implies that it is not possible to approximate the policy functions around a set regime-speciÖc steady-states.

for all .

2.3 A Partition and the Steady-State

As shown above, for MSDSGE models the deÖnition of the steady-state should be independent of the realization of the discrete Markov chain process. To achieve this objective, our key idea is to partition the vector of Markov-switching parameters into two sub-vectors. The Örst sub-vector of parameters we would perturb. The second sub-vector we would not perturb. We denote the Örst sub-vector by and the second sub-vector by . In order to classify which subset of parameters goes into each sub-vector, we would take the derivative of the equilibrium conditions with respect to and and evaluate it at and . Let

\[\mathcal {D} f _ {\theta} \left(\mathbf {y} _ {t + 1}, \mathbf {y} _ {t}, \mathbf {x} _ {t}, \mathbf {x} _ {t - 1}, \boldsymbol {\theta} _ {t + 1}, \boldsymbol {\theta} _ {t}\right) =\]

\[\left[ \mathcal {D} _ {j} f ^ {i} \left(\mathbf {y} _ {t + 1}, \mathbf {y} _ {t}, \mathbf {x} _ {t}, \mathbf {x} _ {t - 1}, 0 \boldsymbol {\varepsilon} _ {t + 1}, \mathbf {0} _ {n _ {\varepsilon}}, \boldsymbol {\theta} _ {t + 1}, \boldsymbol {\theta} _ {t}\right) \right] _ {1 \leq i \leq n _ {y} + n _ {x}, 2 (n _ {y} + n _ {x} + n _ {\varepsilon}) + 1 \leq j \leq 2 (n _ {y} + n _ {x} + n _ {\varepsilon} + n _ {\theta})}\]

denote the matrix of Örst partial derivatives of with respect to and evaluated at and for all , and . Let be the vector of Markov-switching parameters, then we have the following deÖnition for sub-vectors and

DeÖnition 4 Let . We say that if either

\[\mathcal {D} _ {2 (n _ {y} + n _ {x} + n _ {\varepsilon}) + i} f _ {\theta} \left(\mathbf {y} _ {t + 1}, \mathbf {y} _ {t}, \mathbf {x} _ {t}, \mathbf {x} _ {t - 1}, \boldsymbol {\theta} _ {t + 1}, \boldsymbol {\theta} _ {t}\right) \neq \mathbf {0} _ {n _ {y} + n _ {x}} o r\]

\[\mathcal {D} _ {2 (n _ {y} + n _ {x} + n _ {\varepsilon}) + n _ {\theta} + i} f _ {\theta} \left(\mathbf {y} _ {t + 1}, \mathbf {y} _ {t}, \mathbf {x} _ {t}, \mathbf {x} _ {t - 1}, \pmb {\theta} _ {t + 1}, \pmb {\theta} _ {t}\right) \neq \mathbf {0} _ {n _ {y} + n _ {x}}\]

hold and we say that otherwise.

Without loss of generality, we can sort the Markov-switching parameters such that the Örst of them belong to the sub-vector and the rest to the sub-vector . Hence, and have the dimensions and respectively with and we can rewrite equation 1 as

\[\mathbb {E} _ {t} f \left(\mathbf {y} _ {t + 1}, \mathbf {y} _ {t}, \mathbf {x} _ {t}, \mathbf {x} _ {t - 1}, \chi \pmb {\varepsilon} _ {t + 1}, \pmb {\varepsilon} _ {t}, \pmb {\theta} _ {1 t + 1}, \pmb {\theta} _ {2 t + 1}, \pmb {\theta} _ {1 t}, \pmb {\theta} _ {2 t}\right) = \mathbf {0} _ {n _ {x} + n _ {y}}.\tag{4}\]

The motivation for this partition is as follows. We need to set and in order to perturb away any uncertainty related to the innovations to the exogenous predetermined variables. Take an element of the sub-vector . Both derivatives in DeÖnition 4 are zero. This means that by setting and we are already eliminating the e§ects of the elements in the sub-vector in the equilibrium conditions when evaluated at and and no additional perturbation of the elements in this sub-vector is needed. On the other hand, by setting and we do not eliminate the e§ects of the elements in the sub-vector from the equilibrium conditions, because at least one of the derivatives in DeÖnition 4 is not zero, and we do need to perturb the elements of this sub-vector.

To perturb , rewrite the vector as , where maps into . We have

\[\boldsymbol {\theta} _ {t} = \left[ \begin{array}{c c} \boldsymbol {\theta} _ {1 t} ^ {\intercal} & \boldsymbol {\theta} _ {2 t} ^ {\intercal} \end{array} \right] ^ {\intercal} = \left[ \begin{array}{c c} \theta_ {1} (\chi , s _ {t}) ^ {\intercal} & \theta_ {2} (\chi , s _ {t}) ^ {\intercal} \end{array} \right] ^ {\intercal},\tag{5}\]

where and have functional forms

\[\theta_ {1} (\chi , s _ {t}) = \overline {{\boldsymbol {\theta}}} _ {1} + \chi \widehat {\boldsymbol {\theta}} _ {1} (s _ {t}) \mathrm{and} \theta_ {2} (\chi , s _ {t}) = \widehat {\boldsymbol {\theta}} _ {2} (s _ {t})\tag{6}\]

for all where and 1 , and and correspond to elements of the partition and of

The speciÖc functional form (6) is chosen for tractable derivations in the rest of the paper.⁶ We will discuss, below, how to choose the value of . We apply the same partition to the vector , where we simply replace the subscript t with

Two important features stand out from (6). First, is a deviation of from in regime . Second, is not a function of the perturbation parameter . Thus, the perturbation parameter, , a§ects only a subset of Markov-switching parameters, . Since the steady-state will depend on only, a natural choice for this point is the mean of the ergodic distribution across

⁶ Any other functional form, so long as holds for all will be valid.

Using Equations (1) and (3) with the assumption that innovations to the exogenous prede termined variables, are independent of the discrete Markov chain process, we write (1) as

\[\mathbb {G} \left(\mathbf {x} _ {t - 1}, \boldsymbol {\varepsilon} _ {t}, \chi , s _ {t}\right) =\tag{7}\]

\[\sum_ {s ^ {\prime} = 1} ^ {n _ {s}} p _ {s _ {t}, s ^ {\prime}} \int f \left(g (h (\mathbf {x} _ {t - 1}, \pmb {\varepsilon} _ {t}, \chi , s _ {t}), \chi \pmb {\varepsilon} ^ {\prime}, \chi , s ^ {\prime}), g (\mathbf {x} _ {t - 1}, \pmb {\varepsilon} _ {t}, \chi , s _ {t}), h (\mathbf {x} _ {t - 1}, \pmb {\varepsilon} _ {t}, \chi , s _ {t}), \mathbf {x} _ {t - 1}, \chi \pmb {\varepsilon} ^ {\prime}, \pmb {\varepsilon} _ {t}, \theta (\chi , s ^ {\prime}), \theta (\chi , s _ {t})\right) \mu (\pmb {\varepsilon} ^ {\prime}) d \pmb {\varepsilon} ^ {\prime} = \mathbf {0} _ {n _ {y} + n _ {x}}\]

for all ; and . The function G maps into . Using , we are ready to deÖne the steady-state for the case of MSDSGE models.

DeÖnition 5 The vector and the vector are a steady-state if they solve the following system of equations

\[\mathbb {G} \left(\mathbf {x} _ {s s}, \mathbf {0} _ {n _ {\varepsilon}}, 0, s _ {t}\right) = \mathbf {0} _ {n _ {y} + n _ {x}}\]

where and for all .

The next step is to show that the vectors and are a steady state. We now need to make the following assumption about .

Assumption 6 For any , there exists a pair of vectors and that solve the following system of equations

\[f \left(\mathbf {y}, \mathbf {y}, \mathbf {x}, \mathbf {x}, \mathbf {0} _ {n _ {\varepsilon}}, \mathbf {0} _ {n _ {\varepsilon}}, \boldsymbol {\theta} _ {1}, \boldsymbol {\theta} _ {2} ^ {\prime}, \boldsymbol {\theta} _ {1}, \boldsymbol {\theta} _ {2}\right) = \mathbf {0} _ {n _ {y} + n _ {x}}\]

for any vectors

Assumption 6 is less restrictive than Assumption 2 because it is valid for any and . We can now use DeÖnition 5 to show that there is a steady-state.

Theorem 7 Under assumption 6, there exists a pair of steady-state vectors and

Proof. It is easy to see that, when evaluate at , function G equals

\[\begin{array}{c} \mathbb {G} \left(\mathbf {x} _ {s s}, \mathbf {0} _ {n _ {\varepsilon}}, 0, s _ {t}\right) = f \binom{g \left(h \left(\mathbf {x} _ {s s}, \mathbf {0} _ {n _ {\varepsilon}}, 0, s _ {t}\right), 0 \boldsymbol {\varepsilon} ^ {\prime}, 0, s _ {t}\right), g \left(\mathbf {x} _ {s s}, \mathbf {0} _ {n _ {\varepsilon}}, 0, s _ {t}\right),}{h \left(\mathbf {x} _ {s s}, \mathbf {0} _ {n _ {\varepsilon}}, 0, s _ {t}\right), \mathbf {x} _ {s s}, 0 \boldsymbol {\varepsilon} ^ {\prime}, \mathbf {0} _ {n _ {\varepsilon}}, \overline {{\boldsymbol {\theta}}} _ {1}, \widehat {\boldsymbol {\theta}} _ {2} \left(s ^ {\prime}\right), \overline {{\boldsymbol {\theta}}} _ {1}, \widehat {\boldsymbol {\theta}} _ {2} \left(s _ {t}\right)} = \\ f \binom{g \left(h \left(\mathbf {x} _ {s s}, \mathbf {0} _ {n _ {\varepsilon}}, 0, s _ {t}\right), \mathbf {0} _ {n _ {\varepsilon}}, 0, s _ {t}\right), g \left(\mathbf {x} _ {s s}, \mathbf {0} _ {n _ {\varepsilon}}, 0, s _ {t}\right),}{h \left(\mathbf {x} _ {s s}, \mathbf {0} _ {n _ {\varepsilon}}, 0, s _ {t}\right), \mathbf {x} _ {s s}; \mathbf {0} _ {n _ {\varepsilon}}, \mathbf {0} _ {n _ {\varepsilon}}, \overline {{\boldsymbol {\theta}}} _ {1}, \widehat {\boldsymbol {\theta}} _ {2} \left(s ^ {\prime}\right), \overline {{\boldsymbol {\theta}}} _ {1}, \widehat {\boldsymbol {\theta}} _ {2} \left(s _ {t}\right)} = \mathbf {0} _ {n _ {y} + n _ {x}} \end{array}\]

for all and . Since and , the results follow.

Clearly, DeÖnition 5 and Theorem 7 imply that the vectors and are the steady state if

\[f \left(\mathbf {y} _ {s s}, \mathbf {y} _ {s s}, \mathbf {x} _ {s s}, \mathbf {x} _ {s s}, \mathbf {0} _ {n _ {\varepsilon}}, \mathbf {0} _ {n _ {\varepsilon}}, \overline {{\boldsymbol {\theta}}} _ {1}, \widehat {\boldsymbol {\theta}} _ {2} (s _ {t + 1}), \overline {{\boldsymbol {\theta}}} _ {1}, \widehat {\boldsymbol {\theta}} _ {2} (s _ {t})\right) = \mathbf {0} _ {n _ {x} + n _ {y}}.\]

Note that because of the partition deÖned in DeÖnition 4, this is true for any and .

In our simple RBC model, , so matrices of Örst partial derivatives of f with respect to and evaluated at and is given by

\[\begin{array}{c} \mathcal {D} _ {1, 2} f _ {\theta} \left(\mathbf {y} _ {t + 1}, \mathbf {y} _ {t}, \mathbf {x} _ {t}, \mathbf {x} _ {t - 1}, \boldsymbol {\theta} _ {t + 1}, \boldsymbol {\theta} _ {t}\right) = \\ \left[ \begin{array}{c c} \beta \alpha \left(\alpha - 1\right) \tilde {c} _ {t + 1} ^ {\upsilon - 1} \exp \left(\mu_ {t + 1}\right) ^ {\upsilon - 1} \exp \left(\mu_ {t + 1}\right) ^ {1 - \alpha} \tilde {k} _ {t} ^ {\alpha - 1} & 0 \\ 0 & 0 \end{array} \right], \end{array}\]

and

\[\begin{array}{c} \mathcal {D} _ {3, 4} f _ {\theta} \left(\mathbf {y} _ {t + 1}, \mathbf {y} _ {t}, \mathbf {x} _ {t}, \mathbf {x} _ {t - 1}, \boldsymbol {\theta} _ {t + 1}, \boldsymbol {\theta} _ {t}\right) = \\ \left[ \begin{array}{c c} (1 - v) \tilde {c} _ {t + 1} ^ {v - 1} \exp \left(\mu_ {t}\right) ^ {v - 1} \beta \left(\exp \left(\mu_ {t + 1}\right) ^ {1 - \alpha} \tilde {k} _ {t} ^ {\alpha - 1} + 1 - \delta\right) & 0 \\ \exp \left(\mu_ {t}\right) k _ {t} - (1 - \alpha) \exp \left(\mu_ {t}\right) ^ {1 - \alpha} \tilde {k} _ {t - 1} ^ {\alpha} & 0 \end{array} \right]. \end{array}\]

Given that the Örst colums of both of these matrices are nonzero and the second columns of both matrices are zero, we partition into and . Consequently, we set the functional forms

\[\mu_ {t} = \mu (\chi , s _ {t}) = \overline {{\mu}} + \chi \widehat {\mu} (s _ {t}),\]

and

\[\sigma_ {t} = \sigma (\chi , s _ {t}) = \widehat {\sigma} (s _ {t}).\]

We set both and . Thus, and the steady-state of the RBC model consists of and such that

\[\left[ \begin{array}{c} \tilde {c} _ {s s} ^ {v - 1} - \beta \exp (\bar {\mu}) ^ {v - 1} \tilde {c} _ {s s} ^ {v - 1} (\alpha (\exp (\bar {\mu})) ^ {1 - \alpha} \tilde {k} _ {s s} ^ {\alpha - 1} + 1 - \delta) \\ \tilde {c} _ {s s} + \tilde {k} _ {s s} \tilde {z} _ {s s} - \exp (\bar {\mu}) ^ {1 - \alpha} \tilde {k} _ {s s} ^ {\alpha} - (1 - \delta) \tilde {k} _ {s s} \end{array} \right] = \mathbf {0} _ {2},\tag{8}\]

which produces the steady-state values

\[\begin{array}{r l} & {\tilde {k} _ {s s} = \left(\frac {1}{\alpha \exp (\bar {\mu}) ^ {1 - \alpha}} \left(\frac {1}{\beta \exp (\bar {\mu}) ^ {v - 1}} - 1 + \delta\right)\right) ^ {\frac {1}{\alpha - 1}},} \\ & {\mathrm{and} \tilde {c} _ {s s} = \exp (\bar {\mu}) ^ {1 - \alpha} \tilde {k} _ {s s} ^ {\alpha} + (1 - \delta - \exp (\bar {\mu})) \tilde {k} _ {s s}.} \end{array}\]

Once the concept of the steady-state to be used in the approximation and the partition of Markov-switching parameters are clearly deÖned, we can now proceed to approximate the solution of a MSDSGE model. In this paper, we present the results up to second-order Taylor expansions, but our procedure can be easily expanded to higher orders. For the rest of the paper, Örst-order Taylor expansions are referred to as Örst-order approximations and secondorder Taylor expansions as second-order approximations.

This paper has two goals. First, we show how to derive Örst-order and second-order approximations to the functions g and h around the steady-state. Second, we show that obtaining these approximations requires solving a quadratic system. We propose a new methodology to Önd all the solutions to such a system. Each of the solutions to the system corresponds to a di§erent set of Örst-order and second-order approximations. Finding all the solutions is a di¢ cult task but is crucial to determine how many approximations are stable.⁷ Sections 3-5, below, are devoted to accomplishing these two objectives.

⁷ The stability deÖnition is discussed in Section 4.2.3.

3 First-Order Approximations

This section gives a detailed description of how to derive Örst-order approximations to the model solution represented by (3) when approximated around the steady-state deÖned in 2.3. We proceed in several steps. We Örst lay out the notation in Section 3.1 and then provide detailed derivations needed to Önd the approximations in Section 3.2. With the notation and derivations in hand, we show in Section 3.3 how to obtain Örst-order approximations to the model solution. In a Önal subsection, Section 3.4, we discuss one of the most important features of MSDSGE models: the result of no certainty equivalence of Örst-order approximations.

3.1 Notation

We now introduce the notation that will be needed in the rest of the section. We will begin by the derivatives of then the derivatives of , and we will Önish with the derivatives of g and h.

3.1.1 Partial Derivatives of f

We begin with the notation for the derivatives of f evaluated at the steady-state. Let

\[\begin{array}{r l} \mathcal {D} f _ {s s} (s _ {t + 1}, s _ {t}) & = \\ & \left[ \mathcal {D} _ {j} f ^ {i} (\mathbf {y} _ {s s}, \mathbf {y} _ {s s}, \mathbf {x} _ {s s}, \mathbf {x} _ {s s}, \mathbf {0} _ {n _ {\varepsilon}}, \mathbf {0} _ {n _ {\varepsilon}}, \overline {{\boldsymbol {\theta}}} _ {1}, \widehat {\boldsymbol {\theta}} _ {2} (s _ {t + 1}), \overline {{\boldsymbol {\theta}}} _ {1}, \widehat {\boldsymbol {\theta}} _ {2} (s _ {t})) \right] _ {1 \leq i \leq n _ {y} + n _ {x}, 1 \leq j \leq 2 (n _ {y} + n _ {x} + n _ {\varepsilon} + n _ {\theta})} \end{array}\]

denote the matrix of Örst partial derivatives of f with respect to all its variables evaluated at , and for all and . To simplify notation, deÖne

\[\mathcal {D} _ {n, m} f _ {s s} (s _ {t + 1}, s _ {t}) = \left[ \mathcal {D} _ {n} f _ {s s} (s _ {t + 1}, s _ {t}) \dots \mathcal {D} _ {m} f _ {s s} (s _ {t + 1}, s _ {t}) \right]\]

for all and and 8

for all and .

Appendix A shows that our derivations depend on the following matrices:

\[\mathcal {D} _ {1, n _ {y}} f _ {s s} (s _ {t + 1}, s _ {t}), \mathcal {D} _ {n _ {y} + 1, 2 n _ {y}} f _ {s s} (s _ {t + 1}, s _ {t}),\]

\[\mathcal {D} _ {2 n _ {y} + 1, 2 n _ {y} + n _ {x}} f _ {s s} (s _ {t + 1}, s _ {t}), \mathcal {D} _ {2 n _ {y} + n _ {x} + 1, 2 (n _ {y} + n _ {x})} f _ {s s} (s _ {t + 1}, s _ {t}),\]

\[\mathcal {D} _ {2 (n _ {y} + n _ {x}) + 1, 2 (n _ {y} + n _ {x}) + n _ {\varepsilon}} f _ {s s} (s _ {t + 1}, s _ {t}), \mathcal {D} _ {2 (n _ {y} + n _ {x}) + n _ {\varepsilon} + 1, 2 (n _ {y} + n _ {x} + n _ {\varepsilon})} f _ {s s} (s _ {t + 1}, s _ {t}),\]

\[\mathcal {D} _ {2 (n _ {y} + n _ {x} + n _ {\varepsilon}) + 1, 2 (n _ {y} + n _ {x} + n _ {\varepsilon}) + n _ {\theta}} f _ {s s} \left(s _ {t + 1}, s _ {t}\right), \mathrm{and} \mathcal {D} _ {2 (n _ {y} + n _ {x} + n _ {\varepsilon}) + n _ {\theta} + 1, 2 (n _ {y} + n _ {x} + n _ {\varepsilon} + n _ {\theta})} f _ {s s} \left(s _ {t + 1}, s _ {t}\right)\]

for all and

Given the importance of these derivatives, we return to the RBC example for concrete illustration. It follows from (2) that

\[\mathcal {D} _ {1, 1} f _ {s s} \left(s _ {t + 1}, s _ {t}\right) = \left[ \begin{array}{c} \left(1 - v\right) \tilde {c} _ {s s} ^ {v - 2} \\ 0 \end{array} \right], \mathcal {D} _ {2, 2} f _ {s s} \left(s _ {t + 1}, s _ {t}\right) = \left[ \begin{array}{c} \left(v - 1\right) \tilde {c} _ {s s} ^ {v - 2} \\ 1 \end{array} \right],\]

\[\mathcal {D} _ {3, 3} f _ {s s} \left(s _ {t + 1}, s _ {t}\right) = \left[ \begin{array}{c} \beta \alpha \left(1 - \alpha\right) \tilde {c} _ {s s} ^ {v - 1} \exp \left(\bar {\mu}\right) ^ {v - \alpha} \tilde {k} _ {s s} ^ {\alpha - 2} \\ \exp \left(\bar {\mu}\right) \end{array} \right], \mathcal {D} _ {4, 4} f _ {s s} \left(s _ {t + 1}, s _ {t}\right) = \left[ \begin{array}{c} 0 \\ \frac {- 1}{\beta \exp (\bar {\mu}) ^ {v - 1}} \end{array} \right],\]

\[\mathcal {D} _ {5, 5} f _ {s s} \left(s _ {t + 1}, s _ {t}\right) = \left[ \begin{array}{l} 0 \\ 0 \end{array} \right], \mathcal {D} _ {6, 6} f _ {s s} \left(s _ {t + 1}, s _ {t}\right) = \left[ \begin{array}{c} \left(1 - v\right) \tilde {c} _ {s s} ^ {v - 1} \sigma \left(s _ {t}\right) \\ \left(\exp \left(\bar {\mu}\right) k _ {s s} - (1 - \alpha) \exp \left(\bar {\mu}\right) ^ {1 - \alpha} k _ {s s} ^ {\alpha}\right) \sigma \left(s _ {t}\right) \end{array} \right]\]

\[\mathcal {D} _ {7, 8} f _ {s s} \left(s _ {t + 1}, s _ {t}\right) = \left[ \begin{array}{c c} \beta \alpha \left(\alpha - 1\right) \tilde {c} _ {s s} ^ {v - 1} \exp \left(\bar {\mu}\right) ^ {v - \alpha} \tilde {k} _ {s s} ^ {\alpha - 1} & 0 \\ 0 & 0 \end{array} \right],\]

\[\text {and} \mathcal {D} _ {9, 1 0} f _ {s s} (s _ {t + 1}, s _ {t}) = \left[ \begin{array}{c c} (1 - v) \tilde {c} _ {s s} ^ {v - 1} & 0 \\ \exp (\bar {\mu}) k _ {s s} - (1 - \alpha) \exp (\bar {\mu}) ^ {1 - \alpha} \tilde {k} _ {s s} ^ {\alpha} & 0 \end{array} \right]\]

for all and

3.1.2 Partial Derivatives of G

We now introduce the notation for the derivatives of G. Let

\[\mathcal {D} \mathbb {G} (\mathbf {x} _ {t - 1}, \pmb {\varepsilon} _ {t}, \chi , s _ {t}) = [ \mathcal {D} _ {j} \mathbb {G} ^ {i} (\mathbf {x} _ {t - 1}, \pmb {\varepsilon} _ {t}, \chi , s _ {t}) ] _ {1 \leq i \leq n _ {y} + n _ {x}, 1 \leq j \leq n _ {x} + n _ {\varepsilon} + 1}\]

refer to the matrix of Örst partial derivatives of G with respect to for all ; and . Note that there are no derivatives with respect to since it is a discrete variable. Similarly, we let

\[\mathcal {D} \mathbb {G} (\mathbf {x} _ {s s}, \mathbf {0} _ {n _ {\varepsilon}}, 0, s _ {t}) = [ \mathcal {D} _ {j} \mathbb {G} ^ {i} (\mathbf {x} _ {s s}, \mathbf {0} _ {n _ {\varepsilon}}, 0, s _ {t}) ] _ {1 \leq i \leq n _ {y} + n _ {x}, 1 \leq j \leq n _ {x} + n _ {\varepsilon} + 1}\]

refer to the matrix of Örst partial derivatives of G with respect to evaluated at , and for all . To simplify notation, we deÖne

\[\mathcal {D} \mathbb {G} _ {s s} (s _ {t}) = \mathcal {D} \mathbb {G} (\mathbf {x} _ {s s}, \mathbf {0} _ {n _ {\varepsilon}}, 0, s _ {t}) \text {and} \mathcal {D} _ {j} \mathbb {G} _ {s s} ^ {i} (s _ {t}) = \mathcal {D} _ {j} \mathbb {G} ^ {i} (\mathbf {x} _ {s s}, \mathbf {0} _ {n _ {\varepsilon}}, 0, s _ {t})\]

for all and and . Thus,

\[\mathcal {D} \mathbb {G} _ {s s} (s _ {t}) = [ \mathcal {D} _ {j} \mathbb {G} _ {s s} ^ {i} (s _ {t}) ] _ {1 \leq i \leq n _ {y} + n _ {x}, 1 \leq j \leq n _ {x} + n _ {\varepsilon} + 1}\]

for all . Let denote the column vector of for all and . It follows that the Örst partial derivatives of G with respect to evaluated at , and for all can be expressed as

\[\left[ \mathcal {D} _ {1} \mathbb {G} _ {s s} (s _ {t}) \dots \mathcal {D} _ {n _ {x}} \mathbb {G} _ {s s} (s _ {t}) \right],\]

the Örst partial derivatives of with respect to evaluated at , and for all can be expressed as

\[\left[ \mathcal {D} _ {n _ {x} + 1} \mathbb {G} _ {s s} (s _ {t}) \dots \mathcal {D} _ {n _ {x} + n _ {\varepsilon}} \mathbb {G} _ {s s} (s _ {t}) \right],\]

and the Örst partial derivative of G with respect to evaluated at , and for all is

\[\mathcal {D} _ {n _ {x} + n _ {\varepsilon} + 1} \mathbb {G} _ {s s} (s _ {t}).\]

In contrast to one single Örst derivative in the constant parameter case, there are Örst derivatives of , one for each possible value of .

To simply notation further, we deÖne

\[\mathcal {D} _ {n, m} \mathbb {G} _ {s s} (s _ {t}) = [ \mathcal {D} _ {n} \mathbb {G} _ {s s} (s _ {t}) \dots \mathcal {D} _ {m} \mathbb {G} _ {s s} (s _ {t}) ]\]

for all and Therefore, represents the Örst partial derivatives of with respect to represents the Örst partial derivatives of G with respect to , and is the Örst partial derivative of with respect to all of them evaluated at , and for all .

3.1.3 Partial Derivatives of g and h

We are now ready to introduce the notation for the derivatives of g and h evaluated at the steady-state. Denote

\[\mathcal {D} g \left(\mathbf {x} _ {s s}, \mathbf {0} _ {n _ {\varepsilon}}, 0, s _ {t}\right) = \left[ \mathcal {D} _ {j} g ^ {i} \left(\mathbf {x} _ {s s}, \mathbf {0} _ {n _ {\varepsilon}}, 0, s _ {t}\right) \right] _ {1 \leq i \leq n _ {y}, 1 \leq j \leq n _ {x} + n _ {\varepsilon} + 1}\]

as the matrix of Örst partial derivatives of g with respect to evaluated at , and for all and

\[\mathcal {D} h \left(\mathbf {x} _ {s s}, \mathbf {0} _ {n _ {\varepsilon}}, 0, s _ {t}\right) = \left[ \mathcal {D} _ {j} h ^ {i} \left(\mathbf {x} _ {s s}, \mathbf {0} _ {n _ {\varepsilon}}, 0, s _ {t}\right) \right] _ {1 \leq i \leq n _ {x}, 1 \leq j \leq n _ {x} + n _ {\varepsilon} + 1}\]

as the matrix of Örst partial derivatives of h with respect to evaluated at , and for all .

Simplifying the notation further, we deÖne

\[\mathcal {D} g _ {s s} (s _ {t}) = \mathcal {D} g (\mathbf {x} _ {s s}, \mathbf {0} _ {n _ {\varepsilon}}, 0, s _ {t}) \mathrm{and} \mathcal {D} _ {j} g _ {s s} ^ {i} (s _ {t}) = \mathcal {D} _ {j} g ^ {i} (\mathbf {x} _ {s s}, \mathbf {0} _ {n _ {\varepsilon}}, 0, s _ {t})\]

for all , and

\[\mathcal {D} h _ {s s} (s _ {t}) = \mathcal {D} h (\mathbf {x} _ {s s}, \mathbf {0} _ {n _ {\varepsilon}}, 0, s _ {t}) \text {and} \mathcal {D} _ {j} h _ {s s} ^ {i} (s _ {t}) = \mathcal {D} _ {j} h ^ {i} (\mathbf {x} _ {s s}, \mathbf {0} _ {n _ {\varepsilon}}, 0, s _ {t})\]

for all , and . Thus,

\[\mathcal {D} g _ {s s} (s _ {t}) = [ \mathcal {D} _ {j} g _ {s s} ^ {i} (s _ {t}) ] _ {1 \leq i \leq n _ {y}, 1 \leq j \leq n _ {x} + n _ {\varepsilon} + 1} \mathrm{and} \mathcal {D} h _ {s s} (s _ {t}) = [ \mathcal {D} _ {j} h _ {s s} ^ {i} (s _ {t}) ] _ {1 \leq i \leq n _ {x}, 1 \leq j \leq n _ {x} + n _ {\varepsilon} + 1}\]

for all . Let be the column vector of and be the column vector of for all and . We then deÖne

\[\mathcal {D} _ {n, m} g _ {s s} (s _ {t}) = [ \mathcal {D} _ {n} g _ {s s} (s _ {t}) \dots \mathcal {D} _ {m} g _ {s s} (s _ {t}) ] \mathrm{and} \mathcal {D} _ {n, m} h _ {s s} (s _ {t}) = [ \mathcal {D} _ {n} h _ {s s} (s _ {t}) \dots \mathcal {D} _ {m} h _ {s s} (s _ {t}) ]\]

⁹When ; then for all

for all and

As will be discussed in Section 3.3, the following matrices of Örst partial derivatives are our ultimate computational objects to obtain Örst-order approximations to the model solution

\[\begin{array}{r l} \left\{\mathcal {D} _ {1, n _ {x}} g _ {s s} (s _ {t}), \mathcal {D} _ {1, n _ {x}} h _ {s s} (s _ {t}) \right\} _ {s _ {t} = 1} ^ {n _ {s}}, & \left\{\mathcal {D} _ {n _ {x} + 1, n _ {x} + n _ {\varepsilon}} g _ {s s} (s _ {t}), \mathcal {D} _ {n _ {x} + 1, n _ {x} + n _ {\varepsilon}} h _ {s s} (s _ {t}) \right\} _ {s _ {t} = 1} ^ {n _ {s}}, \\ & \mathrm{and} \left\{\mathcal {D} _ {n _ {x} + n _ {\varepsilon} + 1} g _ {s s} (s _ {t}), \mathcal {D} _ {n _ {x} + n _ {\varepsilon} + 1} h _ {s s} (s _ {t}) \right\} _ {s _ {t} = 1} ^ {n _ {s}}, \end{array}\]

where represents the Örst partial derivatives of g with respect to represents the Örst partial derivatives of with respect to represents the Örst partial derivatives of g with respect to represents the Örst partial derivatives of h with respect to is the Örst partial derivative of with respect to , and is the Örst partial derivative of h with respect to all evaluated at , and for all .

Note that

\[\begin{array}{r c l} \mathcal {D} _ {1, n _ {x}} g _ {s s} (s _ {t}) & \in & \mathbb {C} ^ {n _ {y} \times n _ {x}}, \mathcal {D} _ {1, n _ {x}} h _ {s s} (s _ {t}) \in \mathbb {C} ^ {n _ {x} \times n _ {x}}, \\ \mathcal {D} _ {n _ {x} + 1, n _ {x} + n _ {\varepsilon}} g _ {s s} (s _ {t}) & \in & \mathbb {C} ^ {n _ {y} \times n _ {\varepsilon}}, \mathcal {D} _ {n _ {x} + 1, n _ {x} + n _ {\varepsilon}} h _ {s s} (s _ {t}) \in \mathbb {C} ^ {n _ {x} \times n _ {\varepsilon}}, \\ \mathcal {D} _ {n _ {x} + n _ {\varepsilon} + 1} g _ {s s} (s _ {t}) & \in & \mathbb {C} ^ {n _ {y} \times 1}, \mathrm{and} \mathcal {D} _ {n _ {x} + n _ {\varepsilon} + 1} h _ {s s} (s _ {t}) \in \mathbb {C} ^ {n _ {x} \times 1} \end{array}\]

for all . The symbol denotes an matrix over the complex space. These computational objects can be found by solving systems of equations obtained through the chain rule using the Örst partial derivatives of G. The next subsection shows the derivations needed to obtain such systems of equations.

3.2 Derivations

We now show how to use the chain rule using the Örst partial derivatives of to obtain systems of equations that are needed to solve for the matrices

¹⁰ When n = m; then Dn;ng_ss (s_t) = Dng_ss (s_t) and Dn;nh_ss (s_t) = Dnh_ss (s_t) for all .

\[\begin{array}{r l} \{\mathcal {D} _ {1, n _ {x}} g _ {s s} (s _ {t}), \mathcal {D} _ {1, n _ {x}} h _ {s s} (s _ {t}) \} _ {s _ {t} = 1} ^ {n _ {s}}, & \{\mathcal {D} _ {n _ {x} + 1, n _ {x} + n _ {\varepsilon}} g _ {s s} (s _ {t}), \mathcal {D} _ {n _ {x} + 1, n _ {x} + n _ {\varepsilon}} h _ {s s} (s _ {t}) \} _ {s _ {t} = 1} ^ {n _ {s}}, \\ & \mathrm{and} \{\mathcal {D} _ {n _ {x} + n _ {\varepsilon} + 1} g _ {s s} (s _ {t}), \mathcal {D} _ {n _ {x} + n _ {\varepsilon} + 1} h _ {s s} (s _ {t}) \} _ {s _ {t} = 1} ^ {n _ {s}}. \end{array}\]

Denote an matrix of zeros by . Since

\[\mathbb {G} \left(\mathbf {x} _ {t - 1}, \pmb {\varepsilon} _ {t}, \chi , s _ {t}\right) = \mathbf {0} _ {n _ {y} + n _ {x}}\]

for all ; and it follows that

\[\mathcal {D} \mathbb {G} \left(\mathbf {x} _ {t - 1}, \pmb {\varepsilon} _ {t}, \chi , s _ {t}\right) = \mathbf {0} _ {(n _ {y} + n _ {x}) \times (n _ {x} + n _ {\varepsilon} + 1)}\]

for all ; and . In particular,

\[\mathcal {D} \mathbb {G} _ {s s} (s _ {t}) = \mathbf {0} _ {(n _ {y} + n _ {x}) \times (n _ {x} + n _ {\varepsilon} + 1)}\]

for all so that

\[\mathcal {D} _ {1, n _ {x}} \mathbb {G} _ {s s} (s _ {t}) = \mathbf {0} _ {(n _ {y} + n _ {x}) \times n _ {x}}, \mathcal {D} _ {n _ {x} + 1, n _ {x} + n _ {\varepsilon}} \mathbb {G} _ {s s} (s _ {t}) = \mathbf {0} _ {(n _ {y} + n _ {x}) \times n _ {\varepsilon}}, \text {and} \mathcal {D} _ {n _ {x} + n _ {\varepsilon} + 1} \mathbb {G} _ {s s} (s _ {t}) = \mathbf {0} _ {n _ {y} + n _ {x}}\]

for all . Note that in contrast to one single derivative in the constant parameter case, there are Örst partial derivatives of G, one for each possible value of .

The three expressions in (??) imply three systems of equations that will be used to Önd the needed matrices

\[\begin{array}{r l} \left\{\mathcal {D} _ {1, n _ {x}} g _ {s s} (s _ {t}), \mathcal {D} _ {1, n _ {x}} h _ {s s} (s _ {t}) \right\} _ {s _ {t} = 1} ^ {n _ {s}}, & \left\{\mathcal {D} _ {n _ {x} + 1, n _ {x} + n _ {\varepsilon}} g _ {s s} (s _ {t}), \mathcal {D} _ {n _ {x} + 1, n _ {x} + n _ {\varepsilon}} h _ {s s} (s _ {t}) \right\} _ {s _ {t} = 1} ^ {n _ {s}}, \\ & \mathrm{and} \left\{\mathcal {D} _ {n _ {x} + n _ {\varepsilon} + 1} g _ {s s} (s _ {t}), \mathcal {D} _ {n _ {x} + n _ {\varepsilon} + 1} h _ {s s} (s _ {t}) \right\} _ {s _ {t} = 1} ^ {n _ {s}}. \end{array}\]

respectively. In what follows, we describe how to use these three systems of equations in (??) to obtain the needed matrices. We use the following steps to highlight the dependence between these systems.

Step 1 The condition implies a system of quadratic equations that determines

Step 2 The conditions and imply two linear systems of equations that determine and as functions of

Appendix A provides a detailed description of the derivations in Steps 1 and 2. In particular, it derives the Örst partial derivatives of G needed to build the three systems. It shows that can be determined by solving systems of quadratic equations of the form

\[A \left(s _ {t}\right) \left[ \begin{array}{c} \mathbf {I} _ {n _ {x}} \\ \mathcal {D} _ {1, n _ {x}} g _ {s s} \left(1\right) \\ \vdots \\ \mathcal {D} _ {1, n _ {x}} g _ {s s} \left(n _ {s}\right) \end{array} \right] \mathcal {D} _ {1, n _ {x}} h _ {s s} \left(s _ {t}\right) = B \left(s _ {t}\right) \left[ \begin{array}{c} \mathbf {I} _ {n _ {x}} \\ \mathcal {D} _ {1, n _ {x}} g _ {s s} \left(s _ {t}\right) \end{array} \right]\tag{9}\]

for all and and are functions of the Örst partial derivatives of f evaluated at the steady-state as

\[A \left(s _ {t}\right) = \left[ \begin{array}{l l l l} \sum_ {s ^ {\prime} = 1} ^ {n _ {s}} p _ {s _ {t}, s ^ {\prime}} \mathcal {D} _ {2 n _ {y} + 1, 2 n _ {y} + n _ {x}} f _ {s s} \left(s ^ {\prime}, s _ {t}\right) & p _ {s _ {t}, 1} \mathcal {D} _ {1, n _ {y}} f _ {s s} \left(1, s _ {t}\right) & \dots & p _ {s _ {t}, n _ {s}} \mathcal {D} _ {1, n _ {y}} f _ {s s} \left(n _ {s}, s _ {t}\right) \end{array} \right]\]

and

\[B \left(s _ {t}\right) = - \sum_ {s ^ {\prime} = 1} ^ {n _ {s}} p _ {s _ {t}, s ^ {\prime}} \left[ \begin{array}{c c} \mathcal {D} _ {2 n _ {y} + n _ {x} + 1, 2 (n _ {y} + n _ {x})} f _ {s s} \left(s ^ {\prime}, s _ {t}\right) & \mathcal {D} _ {n _ {y} + 1, 2 n _ {y}} f _ {s s} \left(s ^ {\prime}, s _ {t}\right) \end{array} \right]\]

for all .

The fact that we need to solve a quadratic system to determine is one of the key discoveries in the paper. The quadratic system has, in general, many solutions. Each solution corresponds to a di§erent Örst-order approximation. Finding all the solutions to this quadratic system is a di¢ cult task but is crucial to ascertain how many of them imply stable approximations. In Section 4 we propose a new method to solve this quadratic system for all its solutions.

Appendix A shows how, after Önding a solution to (9), Step 2 implies that and can be obtained by simply solving the following two systems of linear equations

\[\left[ \begin{array}{c} \mathcal {D} _ {n _ {x} + 1, n _ {x} + n _ {\varepsilon}} g _ {s s} (1) \\ \vdots \\ \mathcal {D} _ {n _ {x} + 1, n _ {x} + n _ {\varepsilon}} g _ {s s} (n _ {s}) \\ \mathcal {D} _ {n _ {x} + 1, n _ {x} + n _ {\varepsilon}} h _ {s s} (1) \\ \vdots \\ \mathcal {D} _ {n _ {x} + 1, n _ {x} + n _ {\varepsilon}} h _ {s s} (n _ {s}) \end{array} \right] = \left[ \begin{array}{c c} \Theta_ {\varepsilon} & \Phi_ {\varepsilon} \end{array} \right] ^ {- 1} \Psi_ {\varepsilon} \text {and} \left[ \begin{array}{c} \mathcal {D} _ {n _ {x} + n _ {\varepsilon} + 1} g _ {s s} (1) \\ \vdots \\ \mathcal {D} _ {n _ {x} + n _ {\varepsilon} + 1} g _ {s s} (n _ {s}) \\ \mathcal {D} _ {n _ {x} + n _ {\varepsilon} + 1} h _ {s s} (1) \\ \vdots \\ \mathcal {D} _ {n _ {x} + n _ {\varepsilon} + 1} h _ {s s} (n _ {s}) \end{array} \right] = \left[ \begin{array}{c c} \Theta_ {\chi} & \Phi_ {\chi} \end{array} \right] ^ {- 1} \Psi_ {\chi},\tag{10}\]

where

\[\Theta_ {\varepsilon} = \sum_ {s ^ {\prime} = 1} ^ {n _ {s}} \left[ \begin{array}{c c c} p _ {1, s ^ {\prime}} \mathcal {D} _ {n _ {y} + 1, 2 n _ {y}} f _ {s s} (s ^ {\prime}, 1) & \dots & 0 _ {(n _ {x} + n _ {y}) \times n _ {y}} \\ \vdots & \ddots & \vdots \\ 0 _ {(n _ {x} + n _ {y}) \times n _ {y}} & \dots & p _ {n _ {s}, s ^ {\prime}} \mathcal {D} _ {n _ {y} + 1, 2 n _ {y}} f _ {s s} (s ^ {\prime}, n _ {s}) \end{array} \right],\]

\[\begin{array}{r l} & {\Phi_ {\varepsilon} = \sum_ {s ^ {\prime} = 1} ^ {n _ {s}} \left[ \begin{array}{c c c} p _ {1, s ^ {\prime}} \mathcal {D} _ {1, n _ {y}} f _ {s s} (s ^ {\prime}, 1) \mathcal {D} _ {1, n _ {x}} g _ {s s} (s ^ {\prime}) & \dots & 0 _ {(n _ {x} + n _ {y}) \times n _ {x}} \\ \vdots & \ddots & \vdots \\ 0 _ {(n _ {x} + n _ {y}) \times n _ {x}} & \dots & p _ {n _ {s}, s ^ {\prime}} \mathcal {D} _ {1, n _ {y}} f _ {s s} (s ^ {\prime}, n _ {s}) \mathcal {D} _ {1, n _ {x}} g _ {s s} (s ^ {\prime}) \end{array} \right]} \\ & {\qquad + \sum_ {s ^ {\prime} = 1} ^ {n _ {s}} \left[ \begin{array}{c c c} p _ {1, s ^ {\prime}} \mathcal {D} _ {2 n _ {y} + 1, 2 n _ {y} + n _ {x}} f _ {s s} (s ^ {\prime}, 1) & \dots & 0 _ {(n _ {x} + n _ {y}) \times n _ {x}} \\ \vdots & \ddots & \vdots \\ 0 _ {(n _ {x} + n _ {y}) \times n _ {x}} & \dots & p _ {n _ {s}, s ^ {\prime}} \mathcal {D} _ {2 n _ {y} + 1, 2 n _ {y} + n _ {x}} f _ {s s} (s ^ {\prime}, n _ {s}) \end{array} \right],} \end{array}\]

\[\Psi_ {\varepsilon} = - \sum_ {s ^ {\prime} = 1} ^ {n _ {s}} \left[ \begin{array}{c} p _ {1, s ^ {\prime}} \mathcal {D} _ {2 (n _ {y} + n _ {x}) + n _ {\varepsilon} + 1, 2 (n _ {y} + n _ {x} + n _ {\varepsilon})} f _ {s s} (s ^ {\prime}, 1) \\ \vdots \\ p _ {n _ {s}, s ^ {\prime}} \mathcal {D} _ {2 (n _ {y} + n _ {x}) + n _ {\varepsilon} + 1, 2 (n _ {y} + n _ {x} + n _ {\varepsilon})} f _ {s s} (s ^ {\prime}, n _ {s}) \end{array} \right],\]

\[\begin{array}{r l} \Theta_ {\chi} & = \sum_ {s ^ {\prime} = 1} ^ {n _ {s}} \left[ \begin{array}{c c c} p _ {1, s ^ {\prime}} \mathcal {D} _ {n _ {y} + 1, 2 n _ {y}} f _ {s s} (s ^ {\prime}, 1) & \dots & 0 _ {(n _ {x} + n _ {y}) \times n _ {y}} \\ \vdots & \ddots & \vdots \\ 0 _ {(n _ {x} + n _ {y}) \times n _ {y}} & \dots & p _ {n _ {s}, s ^ {\prime}} \mathcal {D} _ {n _ {y} + 1, 2 n _ {y}} f _ {s s} (s ^ {\prime}, n _ {s}) \end{array} \right] \\ & + \left[ \begin{array}{c c c} p _ {1, 1} \mathcal {D} _ {1, n _ {y}} f _ {s s} (1, 1) & \dots & p _ {1, n _ {s}} \mathcal {D} _ {1, n _ {y}} f _ {s s} (n _ {s}, 1) \\ \vdots & \ddots & \vdots \\ p _ {n _ {s}, 1} \mathcal {D} _ {1, n _ {y}} f _ {s s} (1, n _ {s}) & \dots & p _ {n _ {s}, n _ {s}} \mathcal {D} _ {1, n _ {y}} f _ {s s} (n _ {s}, n _ {s}) \end{array} \right], \end{array}\]

\[\begin{array}{r l} \Phi_ {\chi} & = \sum_ {s ^ {\prime} = 1} ^ {n _ {s}} \left[ \begin{array}{c c c} p _ {1, s ^ {\prime}} \mathcal {D} _ {1, n _ {y}} f _ {s s} (s ^ {\prime}, 1) \mathcal {D} _ {1, n _ {x}} g _ {s s} (s ^ {\prime}) & \dots & 0 _ {(n _ {x} + n _ {y}) \times n _ {x}} \\ \vdots & \ddots & \vdots \\ 0 _ {(n _ {x} + n _ {y}) \times n _ {x}} & \dots & p _ {n _ {s}, s ^ {\prime}} \mathcal {D} _ {1, n _ {y}} f _ {s s} (s ^ {\prime}, n _ {s}) \mathcal {D} _ {1, n _ {x}} g _ {s s} (s ^ {\prime}) \end{array} \right] \\ & + \sum_ {s ^ {\prime} = 1} ^ {n _ {s}} \left[ \begin{array}{c c c} p _ {1, s ^ {\prime}} \mathcal {D} _ {2 n _ {y} + 1, 2 n _ {y} + n _ {x}} f _ {s s} (s ^ {\prime}, 1) & \dots & 0 _ {(n _ {x} + n _ {y}) \times n _ {x}} \\ \vdots & \ddots & \vdots \\ 0 _ {(n _ {x} + n _ {y}) \times n _ {x}} & \dots & p _ {n _ {s}, s ^ {\prime}} \mathcal {D} _ {2 n _ {y} + 1, 2 n _ {y} + n _ {x}} f _ {s s} (s ^ {\prime}, n _ {s}) \end{array} \right], \end{array}\]

\[\text {and} \Psi_ {\chi} = - \sum_ {s ^ {\prime} = 1} ^ {n _ {s}} \left[ \begin{array}{c} p _ {1, s ^ {\prime}} \binom{\mathcal {D} _ {2 (n _ {y} + n _ {x} + n _ {\varepsilon}) + 1, 2 (n _ {y} + n _ {x} + n _ {\varepsilon}) + n _ {\theta}} f _ {s s} (s ^ {\prime}, 1) \mathcal {D} \theta_ {s s} (s ^ {\prime}) + \ldots}{\mathcal {D} _ {2 (n _ {y} + n _ {x} + n _ {\varepsilon}) + n _ {\theta} + 1, 2 (n _ {y} + n _ {x} + n _ {\varepsilon} + n _ {\theta})} f _ {s s} (s ^ {\prime}, 1) \mathcal {D} \theta_ {s s} (1)} \\ \vdots \\ p _ {n _ {s}, s ^ {\prime}} \binom{\mathcal {D} _ {2 (n _ {y} + n _ {x} + n _ {\varepsilon}) + 1, 2 (n _ {y} + n _ {x} + n _ {\varepsilon}) + n _ {\theta}} f _ {s s} (s ^ {\prime}, n _ {s}) \mathcal {D} \theta_ {s s} (s ^ {\prime}) + \ldots}{\mathcal {D} _ {2 (n _ {y} + n _ {x} + n _ {\varepsilon}) + n _ {\theta} + 1, 2 (n _ {y} + n _ {x} + n _ {\varepsilon} + n _ {\theta})} f _ {s s} (S ^ {\prime}, n _ {s}) \mathcal {D} \theta_ {s s} (n _ {s})} \end{array} \right],\]

where denotes the derivative of with respect to evaluated at for all . That is,

\[\mathcal {D} \theta_ {s s} (s _ {t}) = \mathcal {D} \theta (0, s _ {t}) = [ \mathcal {D} _ {j} \theta^ {i} (0, s _ {t}) ] _ {1 \leq i \leq n _ {\theta}, j = 1}\]

for all

Since and depend on , we must solve (9) prior to solving (10). Since (10) is just a linear problem, we have that, Örst, there is a di§erent solution to (10) for each solution to (9) and, second, the bottleneck to Önding Örst-order approximations is obtaining by solving the quadratic system (9). The bottleneck will be fully discussed in Section 4.

3.3 Approximations

Given a solution to (9) and (10), it is straightforward to build a Örst-order approximation to the solutions of the MSDSGE model represented by Equation (1). Let a and be Örst-order approximations to g and h around the point . It follows that

\[\begin{array}{r c l} g ^ {\mathrm{first}} (\mathbf {x} _ {t - 1}, \pmb {\varepsilon} _ {t}, \chi , s _ {t}) - \mathbf {y} _ {s s} & = & [ \mathcal {D} _ {1} g _ {s s} (s _ {t}) \dots \mathcal {D} _ {n _ {x}} g _ {s s} (s _ {t}) ] (\mathbf {x} _ {t - 1} - \mathbf {x} _ {s s}) \\ & & + [ \mathcal {D} _ {n _ {x} + 1} g _ {s s} (s _ {t}) \dots \mathcal {D} _ {n _ {x} + n _ {\varepsilon}} g _ {s s} (s _ {t}) ] \pmb {\varepsilon} _ {t} + \mathcal {D} _ {n _ {x} + n _ {\varepsilon} + 1} g _ {s s} (s _ {t}) \chi_ {s s}. \end{array}\]

and

\[\begin{array}{r c l} {h ^ {\mathrm{first}} \left(\mathbf {x} _ {t - 1}, \pmb {\varepsilon} _ {t}, \chi , s _ {t}\right) - \mathbf {x} _ {s s}} & = & {\left[ \mathcal {D} _ {1} h _ {s s} \left(s _ {t}\right) \dots \mathcal {D} _ {n _ {x}} h _ {s s} \left(s _ {t}\right) \right] \left(\mathbf {x} _ {t - 1} - \mathbf {x} _ {s s}\right)} \\ & & {+ \left[ \mathcal {D} _ {n _ {x} + 1} h _ {s s} \left(s _ {t}\right) \dots \mathcal {D} _ {n _ {x} + n _ {\varepsilon}} h _ {s s} \left(s _ {t}\right) \right] \pmb {\varepsilon} _ {t} + \mathcal {D} _ {n _ {x} + n _ {\varepsilon} + 1} h _ {s s} \left(s _ {t}\right) \chi_ {s s}} \end{array}\]

for all ; and . Hence, Örst-order approximations can be rewritten as

\[g ^ {\mathrm{first}} \left(\mathbf {x} _ {t - 1}, \pmb {\varepsilon} _ {t}, \chi , s _ {t}\right) - \mathbf {y} _ {s s} = \mathcal {D} _ {1, n _ {x}} g _ {s s} (s _ {t}) (\mathbf {x} _ {t - 1} - \mathbf {x} _ {s s}) + \mathcal {D} _ {n _ {x} + 1, n _ {x} + n _ {\varepsilon}} g _ {s s} (s _ {t}) \pmb {\varepsilon} _ {t} + \mathcal {D} _ {n _ {x} + n _ {\varepsilon} + 1} g _ {s s} (s _ {t}) \chi_ {s s}.\]

and

\[h ^ {\mathrm{first}} \left(\mathbf {x} _ {t - 1}, \pmb {\varepsilon} _ {t}, \chi , s _ {t}\right) - \mathbf {x} _ {s s} = \mathcal {D} _ {1, n _ {x}} h _ {s s} (s _ {t}) (\mathbf {x} _ {t - 1} - \mathbf {x} _ {s s}) + \mathcal {D} _ {n _ {x} + 1, n _ {x} + n _ {\varepsilon}} h _ {s s} (s _ {t}) \pmb {\varepsilon} _ {t} + \mathcal {D} _ {n _ {x} + n _ {\varepsilon} + 1} h _ {s s} (s _ {t}) \chi\]

for all ; and . To express them in compact form, we have

\[g ^ {\mathrm{first}} (\mathbf {x} _ {t - 1}, \pmb {\varepsilon} _ {t}, \chi , s _ {t}) - \mathbf {y} _ {s s} = \mathcal {D} g _ {s s} (s _ {t}) \mathcal {S} _ {t} \mathrm{and} h ^ {\mathrm{first}} (\mathbf {x} _ {t - 1}, \pmb {\varepsilon} _ {t}, \chi , s _ {t}) - \mathbf {x} _ {s s} = \mathcal {D} h _ {s s} (s _ {t}) \mathcal {S} _ {t}\]

for all and where

We summarize what we have developed and what our next task is. For Örst-order approximations, Section 3.2 lays out a procedure for obtaining the matrices and for all . The bottleneck is to obtain by solving the quadratic system (9). Once the bottleneck is removed, the task of obtaining Örst-order approximations only involves solving linear systems. As mentioned, the quadratic system has, in general, many solutions. Each solution corresponds to a di§erent Örst-order approximation. Finding all the solutions to a quadratic system is central to ascertaining how many approximations are stable. Section 4 is devoted to dealing with this bottleneck by Önding all the solutions to the quadratic system (9).

3.4 Feature of No Certainty Equivalence

As pointed out by Schmitt-Grohe and Uribe (2004), the certainty equivalence of Örst-order approximations is a main result for a constant parameter model. This result implies that Örstorder approximations to constant parameter models are inadequate for analyzing interesting behavior such as economic agentsíresponses to risk. For example, van Binsbergen et al. (2008) and Rudebusch and Swanson (2008) argue that, when using constant parameter models, at least second-order approximations are needed to analyze the e§ects of volatility on agentsídecisions.

While second-order approximations nullify the result of certainty equivalence, they result in a substantially higher degree of computational di¢ culty in performing likelihood-based estimation, as documented by Fern·ndez-Villaverde and Rubio-Ramirez (2007). We show in this section, however, that Örst-order approximations to the solutions of MSDSGE models are not necessarily certainty equivalent. This salient feature opens the door to analyzing risk-related behaviors using Örst-order approximations.

To see how certainty equivalence arises in Örst-order approximations in constant parameter models, consider Equation (10) with only one regime so that . In this degenerate case, we have 1

\[\left[ \begin{array}{c c} \Theta_ {\chi} & \Phi_ {\chi} \end{array} \right] \left[ \begin{array}{c} \mathcal {D} _ {n _ {x} + n _ {\varepsilon} + 1} g _ {s s} (1) \\ \mathcal {D} _ {n _ {x} + n _ {\varepsilon} + 1} h _ {s s} (1) \end{array} \right] = \Psi_ {\chi},\tag{11}\]

where

\[\begin{array}{c} \left[ \begin{array}{c c} \Theta_ {\chi} & \Phi_ {\chi} \end{array} \right] = \\ \left[ \begin{array}{c c} \mathcal {D} _ {n _ {y} + 1, 2 n _ {y}} f _ {s s} (1, 1) + \mathcal {D} _ {1, n _ {y}} f _ {s s} (1, 1) & \mathcal {D} _ {1, n _ {y}} f _ {s s} (1, 1) \mathcal {D} _ {1, n _ {x}} g _ {s s} (1) + \mathcal {D} _ {2 n _ {y} + 1, 2 n _ {y} + n _ {x}} f _ {s s} (1, 1) \end{array} \right] \\ \text {and} \Psi_ {\chi} = - \left[ \begin{array}{c} \left( \begin{array}{c} \mathcal {D} _ {2 (n _ {y} + n _ {x} + n _ {\varepsilon}) + 1, 2 (n _ {y} + n _ {x} + n _ {\varepsilon}) + n _ {\theta}} f _ {s s} (1, 1) \mathcal {D} \theta_ {s s} (1) + \ldots \\ \mathcal {D} _ {2 (n _ {y} + n _ {x} + n _ {\varepsilon}) + n _ {\theta} + 1, 2 (n _ {y} + n _ {x} + n _ {\varepsilon} + n _ {\theta})} f _ {s s} (1, 1) \mathcal {D} \theta_ {s s} (1) \end{array} \right) \end{array} \right]. \end{array}\]

Because in any constant parameter model, we have , implying that . Therefore the linear system (11) is homogeneous. If a unique solution exists, it is given by

\[\mathcal {D} _ {n _ {x} + n _ {\varepsilon} + 1} g _ {s s} (1) = 0 _ {n _ {y}} \text { and } \mathcal {D} _ {n _ {x} + n _ {\varepsilon} + 1} h _ {s s} (1) = 0 _ {n _ {x}}.\tag{12}\]

For the constant parameter case, Örst-order approximations to policy rules are

\[g ^ {\mathrm{first}} \left(\mathbf {x} _ {t - 1}, \varepsilon_ {t}, \chi , 1\right) - \mathbf {y} _ {s s} = \mathcal {D} _ {1, n _ {x}} g _ {s s} (1) (\mathbf {x} _ {t - 1} - \mathbf {x} _ {s s}) + \mathcal {D} _ {n _ {x} + 1, n _ {x} + n _ {\varepsilon}} g _ {s s} (1) \varepsilon_ {t} + \mathcal {D} _ {n _ {x} + n _ {\varepsilon} + 1} g _ {s s} (1) \chi_ {t - 1}.\]

and

\[h ^ {\mathrm{first}} \left(\mathbf {x} _ {t - 1}, \varepsilon_ {t}, \chi , 1\right) - \mathbf {x} _ {s s} = \mathcal {D} _ {1, n _ {x}} h _ {s s} (1) (\mathbf {x} _ {t - 1} - \mathbf {x} _ {s s}) + \mathcal {D} _ {n _ {x} + 1, n _ {x} + n _ {\varepsilon}} h _ {s s} (1) \varepsilon_ {t} + \mathcal {D} _ {n _ {x} + n _ {\varepsilon} + 1} h _ {s s} (1) \chi\]

for all and . Using (12) and these policy rules evaluated at and , we have

\[\begin{array}{r c l} {g ^ {\mathrm{first}} (\mathbf {x} _ {s s}, 0 _ {n _ {\varepsilon}}, 1, 1) - \mathbf {y} _ {s s}} & = & {0 _ {n _ {y}},} \\ {h ^ {\mathrm{first}} (\mathbf {x} _ {s s}, 0 _ {n _ {\varepsilon}}, 1, 1) - \mathbf {x} _ {s s}} & = & {0 _ {n _ {x}}.} \end{array}\]

That is, Örst-order approximations to the solutions of the constant parameter model are certainty equivalent.

With this insight we turn to the MSDSGE case. It is clear from Equation (10) that a necessary condition for Örst-order approximations not to display certainty equivalence is . Let us analyze the circumstance under which the condition is true. For visual convenience, we rewrite the expression for

\[\Psi_ {\chi} = - \sum_ {s ^ {\prime} = 1} ^ {n _ {s}} \left[ \begin{array}{c} p _ {1, s ^ {\prime}} \binom{\mathcal {D} _ {2 (n _ {y} + n _ {x} + n _ {\varepsilon}) + 1, 2 (n _ {y} + n _ {x} + n _ {\varepsilon}) + n _ {\theta}} f _ {s s} \left(s ^ {\prime}, 1\right) \mathcal {D} \theta_ {s s} \left(s ^ {\prime}\right) + \ldots}{\mathcal {D} _ {2 (n _ {y} + n _ {x} + n _ {\varepsilon}) + n _ {\theta} + 1, 2 (n _ {y} + n _ {x} + n _ {\varepsilon} + n _ {\theta})} f _ {s s} \left(s ^ {\prime}, 1\right) \mathcal {D} \theta_ {s s} \left(1\right)} \\ \vdots \\ p _ {n _ {s}, s ^ {\prime}} \binom{\mathcal {D} _ {2 (n _ {y} + n _ {x} + n _ {\varepsilon}) + 1, 2 (n _ {y} + n _ {x} + n _ {\varepsilon}) + n _ {\theta}} f _ {s s} \left(s ^ {\prime}, n _ {s}\right) \mathcal {D} \theta_ {s s} \left(s ^ {\prime}\right) + \ldots}{\mathcal {D} _ {2 (n _ {y} + n _ {x} + n _ {\varepsilon}) + n _ {\theta} + 1, 2 (n _ {y} + n _ {x} + n _ {\varepsilon} + n _ {\theta})} f _ {{s s}} \left(s ^ {\prime}, n _ {s}\right) \mathcal {D} \theta_ {{s s}} \left(n _ {s}\right)} \end{array} \right].\]

Clearly, if for all , then . Thus, a necessary condition for to hold is for some . Given the partition of described in (6), we have

\[\mathcal {D} \theta_ {s s} (s _ {t}) = \left[ \begin{array}{c c} \widehat {\theta} _ {1} (s _ {t}) ^ {\intercal} & 0 _ {n _ {\theta_ {2}}} ^ {\intercal} \end{array} \right] ^ {\intercal}.\]

It follows that for some if and only if for some . In sum, a necessary condition for to hold is for some

The condition for some , however, is insu¢ cient for to be true. A su¢ cient condition is

\[\sum_ {s ^ {\prime} = 1} ^ {n _ {s}} p _ {s t, s ^ {\prime}} \binom{\mathcal {D} _ {2 (n _ {y} + n _ {x} + n _ {\varepsilon}) + 1, 2 (n _ {y} + n _ {x} + n _ {\varepsilon}) + n _ {\theta}} f _ {s s} (s ^ {\prime}, s _ {t}) \mathcal {D} \theta_ {s s} (s ^ {\prime}) +}{\mathcal {D} _ {2 (n _ {y} + n _ {x} + n _ {\varepsilon}) + n _ {\theta} + 1, 2 (n _ {y} + n _ {x} + n _ {\varepsilon} + n _ {\theta})} f _ {s s} (s ^ {\prime}, s _ {t}) \mathcal {D} \theta_ {s s} (s _ {t})} \neq 0 _ {n _ {x} + n _ {y}}\]

for some . This additional condition holds if does not enter the equilibrium conditions multiplicatively with a variable to which the Örst partial derivative of f is zero when evaluated at the steady-state. The following proposition summarizes our Öndings.

Proposition 8 First-order approximations to the solution of an MSDSGE model are not certainty equivalent if and only if both and

\[\sum_ {s ^ {\prime} = 1} ^ {n _ {s}} p _ {s _ {t}, s ^ {\prime}} \binom{\mathcal {D} _ {2 (n _ {y} + n _ {x} + n _ {\varepsilon}) + 1, 2 (n _ {y} + n _ {x} + n _ {\varepsilon}) + n _ {\theta}} f _ {s s} (s ^ {\prime}, s _ {t}) \mathcal {D} \theta_ {s s} (s ^ {\prime}) +}{\mathcal {D} _ {2 (n _ {y} + n _ {x} + n _ {\varepsilon}) + n _ {\theta} + 1, 2 (n _ {y} + n _ {x} + n _ {\varepsilon} + n _ {\theta})} f _ {s s} (s ^ {\prime}, s _ {t}) \mathcal {D} \theta_ {s s} (s _ {t})} \neq 0 _ {n _ {x} + n _ {y}}\]

for some .

Proof. The part of the proof has been provided in the analysis prior to the stated proposition.

For the ìonly part, we prove it by contradiction. Note that the only way for

\[\mathcal {D} _ {n _ {x} + n _ {\varepsilon} + 1} g _ {s s} (s _ {t}) = 0 _ {n _ {y}} \text {and} \mathcal {D} _ {n _ {x} + n _ {\varepsilon} + 1} h _ {s s} (s _ {t}) = 0 _ {n _ {x}}\]

is for . Suppose for all . Then and

\[\mathcal {D} _ {n _ {x} + n _ {\varepsilon} + 1} g _ {s s} (s _ {t}) = 0 _ {n _ {y}} \text {and} \mathcal {D} _ {n _ {x} + n _ {\varepsilon} + 1} h _ {s s} (s _ {t}) = 0 _ {n _ {x}}\]

for all . It follows that Örst-order approximations are certainty equivalent.

Now suppose for some but

\[\sum_ {s ^ {\prime} = 1} ^ {n _ {s}} p _ {s t, s ^ {\prime}} \binom{\mathcal {D} _ {2 (n _ {y} + n _ {x} + n _ {\varepsilon}) + 1, 2 (n _ {y} + n _ {x} + n _ {\varepsilon}) + n _ {\theta}} f _ {s s} (s ^ {\prime}, s _ {t}) \mathcal {D} \theta_ {s s} (s ^ {\prime}) +}{\mathcal {D} _ {2 (n _ {y} + n _ {x} + n _ {\varepsilon}) + n _ {\theta} + 1, 2 (n _ {y} + n _ {x} + n _ {\varepsilon} + n _ {\theta})} f _ {s s} (s ^ {\prime}, s _ {t}) \mathcal {D} \theta_ {s s} (s _ {t})} = 0 _ {n _ {x} + n _ {y}}\]

for all . In this case, it is straightforward to see that and

\[\mathcal {D} _ {n _ {x} + n _ {\varepsilon} + 1} g _ {s s} (s _ {t}) = 0 _ {n _ {y}} \text {and} \mathcal {D} _ {n _ {x} + n _ {\varepsilon} + 1} h _ {s s} (s _ {t}) = 0 _ {n _ {x}}\]

for all . It follows that Örst-order approximations are certainty equivalent.

These contradictions establish the proof of the ìonly portion.

Proposition 8 implies that if Örst-order approximations are not certainty equivalent, approximated policy rules evaluated at and are either

\[g ^ {\mathrm{first}} \left(\mathbf {x} _ {s s}, 0 _ {n _ {\varepsilon}}, 1, s _ {t}\right) - \mathbf {y} _ {s s} = \mathcal {D} _ {n _ {x} + n _ {\varepsilon} + 1} g _ {s s} \left(s _ {t}\right) \neq 0 _ {n _ {y}}\]

or

\[h ^ {\mathrm{first}} \left(\mathbf {x} _ {s s}, 0 _ {n _ {\varepsilon}}, 1, s _ {t}\right) - \mathbf {x} _ {s s} = \mathcal {D} _ {n _ {x} + n _ {\varepsilon} + 1} h _ {s s} (s _ {t}) \neq 0 _ {n _ {x}}\]

for some

4 Generalized Quadratic System and New Solution Method

One main discovery in this paper is that we identify solving a system of quadratic equations as the sole bottleneck in obtaining Örst-order approximations to the solutions of MSDSGE models. Once this bottleneck is removed, we need solve only two systems of linear equations to obtain Örst-order approximations. In this section we show how to Önd all the solutions to this quadratic system by introducing a new method. Finding all the solutions is essential to determining how many Örst-order approximations are stable. Section 4.1 characterizes the quadratic system. Section 4.2 proposes a new solution method by applying the theory of Grˆbner bases. We show that the number of solutions to the quadratic system is Önite in most cases and deÖne both the concept and the condition of stability to be used in the paper. Section 4.3 uses our RBC example to demonstrate how to apply our methodology.

4.1 Generalized Quadratic System

To understand the di¢ culty of solving a system of quadratic equations represented by (9), we begin with the constant parameter case in which . In this case, it is well understood that one can map this special system of quadratic equations to the generalized Schur decomposition problem related to the matrices A(1) and B(1). Thus, solving this special system of quadratic equations is equivalent to using the standard matrix algebra procedure to Önd all the solutions.

In the Markov-switching case when , there are systems of quadratic equations represented by (9). Stacking these systems all together expands to a generalized system of quadratic equations with unknowns . Because appear in every system of the systems, the generalized quadratic system can no longer be mapped to the generalized Schur decomposition problem.¹¹

The quadratic system represented by (9) has multiple solutions in general.¹² Each solution implies a di§erent Örst-order approximation. It is crucial that we Önd all the solutions and then determine how many imply stable approximations. For this purpose we develop, in the next section, a new method by applying Grˆbner bases to our problem.

4.2 New Solution Method

4.2.1 Grˆbner Bases

The quadratic system discussed in Section 4.1 is simply a system of quadratic polynomials. Grˆbner bases provide a computationally practical means to obtain all the solutions to a system of polynomial equations. A more detailed description of Grˆbner bases is provided in Appendix B. In this section we provide an intuitive explanation of how to apply Grˆbner bases to solving a system of multivariate polynomials.

Suppose one wishes to Önd all the solutions to a system of n polynomial equations in n unknowns

\[f _ {1} (x _ {1}, \ldots , x _ {n}) = 0, \ldots , f _ {n} (x _ {1}, \ldots , x _ {n}) = 0.\]

There exist a number of computationally e¢ cient routines that can transform the original system of n polynomial equations to an alternative system of n polynomial equations with the same set of solutions. The following proposition, known as the Shape Lemma, describes a useful transformation into a reduced Grˆbner basis.

¹¹The intuition is that the generalized Schur decomposition cannot deal with the matrices and when

¹²For the remainder of the paper, we refer to the generalized quadratic system (9) as the quadratic system.

Proposition 9 Let

\[f _ {1} (x _ {1}, \ldots , x _ {n}) = 0, \ldots , f _ {n} (x _ {1}, \ldots , x _ {n}) = 0\]

be a system of n polynomials in n unknowns. Under certain regularity conditions, there exists the following set of n polynomials in n unknowns with the same set of roots:

\[x _ {1} - q _ {1} \left(x _ {n}\right) = 0, \dots , x _ {n - 1} - q _ {n - 1} \left(x _ {n}\right) = 0, q _ {n} \left(x _ {n}\right) = 0,\]

where each is a univariate polynomial.

Proof. See Appendix B for details.

There are several important aspects to Proposition 9. First, the alternative set of polynomials in Proposition 9 is known as a Shape basis, a special kind of reduced Grˆbner basis. Second, the regularity conditions, referred to in Proposition 9 and detailed in Appendix B, are satisÖed by the quadratic system in most economic problems. Third, the roots of the univariate polynomial can be found by any standard root Önding algorithm. Fourth, once a root of is found, one can easily compute to obtain

A large strand of the literature deals with the computation of reduced Grˆbner bases. Buchberger (1998)ís algorithm is the original technique. Subsequently, many more e¢ cient variants have been proposed. We refer the interested reader to Cox et al. (1997). In this paper we use Mathematica to Önd a Shape basis.

To illustrate how powerful Proposition 9 is, consider the following example featuring a system of quadratic polynomials in four unknown variables

\[x _ {1} x _ {2} + x _ {3} x _ {4} + 2 = 0,\]

\[x _ {1} x _ {2} + x _ {2} x _ {3} + 3 = 0,\]

\[x _ {1} x _ {3} + x _ {4} x _ {1} + x _ {4} x _ {2} + 6 = 0,\]

\[x _ {1} x _ {3} + 2 x _ {1} x _ {2} + 3 = 0.\]

A Shape basis is

\[x _ {1} - \frac {1}{2 8} (9 x _ {4} ^ {5} + 6 x _ {4} ^ {3} - 1 5 x _ {4}) = 0,\]

\[x _ {2} - \frac {1}{2 8} (- 9 x _ {4} ^ {5} - 6 x _ {4} ^ {3} + 9 9 x _ {4}) = 0,\]

\[x _ {3} - \frac {1}{1 4} (- 3 x _ {4} ^ {5} - 9 x _ {4} ^ {3} - 2 x _ {4}) = 0,\]

\[3 x _ {4} ^ {6} + 9 x _ {4} ^ {4} - 1 9 x _ {4} ^ {2} - 4 9 = 0.\]

The last polynomial is univariate of degree six in . The six roots of this polynomial are

\[\left\{1. 5 5 4 6 1, - 1. 5 5 4 6 1, 1. 3 9 5 9 2 i, - 1. 3 9 5 9 2 i, 1. 8 6 2 3 2 i, - 1. 8 6 2 3 2 i \right\}.\]

Each of these roots can be substituted into the Örst three equations to obtain the following six solutions:

\[\left\{x _ {1} = 2. 8 9 1 0 4, x _ {2} = 1. 7 7 2 8, x _ {3} = - 4. 5 8 3 2 8, x _ {4} = 1. 5 5 4 6 1 \right\},\]

\[\left\{x _ {1} = - 2. 8 9 1 0 4, x _ {2} = - 1. 7 7 2 8, x _ {3} = 4. 5 8 3 2 8, x _ {4} = - 1. 5 5 4 6 1 \right\},\]

\[\left\{x _ {1} = 0. 3 7 2 9 9 7 i, x _ {2} = 3. 8 1 4 7 7 i, x _ {3} = 0. 4 1 3 4 2 i, x _ {4} = 1. 3 9 5 9 2 i \right\},\]

\[\left\{x _ {1} = - 0. 3 7 2 9 9 7 i, x _ {2} = - 3. 8 1 4 7 7 i, x _ {3} = - 0. 4 1 3 4 2 i, x _ {4} = - 1. 3 9 5 9 2 i \right\},\]

\[\left\{x _ {1} = 4. 8 1 8 6 1 i, x _ {2} = 0. 7 6 8 3 4 2 i, x _ {3} = - 0. 9 1 4 0 9 7 i, x _ {4} = 1. 8 6 2 3 2 i \right\},\]

\[\left\{x _ {1} = = - 4. 8 1 8 6 1 i, x _ {2} = - 0. 7 6 8 3 4 2 i, x _ {3} = 0. 9 1 4 0 9 7 i, x _ {4} = - 1. 8 6 2 3 2 i \right\}.\]

It is straightforward to show that these roots solve the original system of quadratic equations.

4.2.2 A Finite Number of Solutions

One of the regularity conditions for Önding reduced Grˆbner bases is that the quadratic system has Önitely many solutions. This condition is met in most economic problems. To see why this result is true, let us Örst consider the constant parameter case when . If the solution of the model has a unique stable Örst-order approximation, the usual practice, as in Schmitt-Grohe and Uribe (2004), involves constructing this stable approximation by ordering the generalized eigenvalues of the two matrices A (1) and B (1) in a particular way. In general, however, the quadratic system has multiple solutions. Each solution implies a di§erent Örst-order approximation. Some of these solutions may imply unstable approximations. For most economic problems, the full set of approximations (stable and unstable) can be found by changing the order of generalized eigenvalues. Hence, the number of approximations is related to the number of possible orderings of eigenvalues. Therefore, the number of solutions to the quadratic system, and therefore the number of approximations (stable and unstable), is bounded by

\[\binom{r a n k (A (1)) - n _ {e x o}}{n _ {x} - n _ {e x o}} = \frac {(r a n k (A (1)) - n _ {e x o}) !}{(n _ {x} - n _ {e x o}) ! (r a n k (A (1)) - n _ {x}) !},\tag{13}\]

where is the number of exogenous predetermined variables so that , rank (A (1)) stands for the rank of the matrix This result is familiar to most readers.

Now consider the Markov-switching case when . We have the quadratic system or quadratic systems of the form (9). If were Öxed for all t, the number of solutions to the quadratic system, and therefore the number of approximations (stable and unstable), would be bounded as in (13). Since we have regimes, the total number of solutions to the quadratic system for most parameter conÖgurations is bounded by

\[\max _ {s _ {t}} \binom{r a n k (A (s _ {t})) - n _ {e x o}}{n _ {x} - n _ {e x o}} ^ {n _ {s}} = \max _ {s _ {t}} \left(\frac {(r a n k (A (s _ {t})) - n _ {e x o}) !}{(n _ {x} - n _ {e x o}) ! (r a n k (A (s _ {t})) - n _ {x}) !}\right) ^ {n _ {s}}\]

where rank stands for the rank of the matrix for all

¹³Note that the matrix A (1) may not be of full rank when there are redundant variables that are linearly dependent upon others and consequently can be eliminated from the quadratic system. A simple example is a leisureless RBC model with three variables (capital, consumption, and output) and three equations (the Euler condition, the resource constraint, and the output deÖnition). If we eliminate the output deÖnition equation and the output variable, the newly formed matrix A(1) will be of full rank.

¹⁴For rare conÖgurations of parameters, there may exist inÖnitely many solutions or the algorithm used by Mathematica to Önd reduced Grˆbner bases ceases to converge. In such a case, numerical procedures can be used to solve the generalized quadratic system (9) to Önd one or possibly more solutions, although none of those proecdures is guaranteed to Önd all solutions. Appendix C describes one such numerical proce dure. Alternative numerical methods, such as Newtonís algorithm used in the RISE toolbox of Maih (2013) (https://github.com/jmaih/RISE_toolbox), may be used to Önd solutions, but again with no guarantee of Önding all solutions.

Once we know that the number of solutions to the quadratic system is Önite, we can use Mathematica to obtain reduced Grˆbner bases for Önding all solutions to the quadratic system and, hence, all Örst-order approximations. The next question is whether any of the approximations are stable and if so, how many. We address this question in the following section.

4.2.3 Mean Square Stability

In the constant parameter case, whether a Örst-order approximation is stable or not can be determined by verifying whether its largest absolute generalized eigenvalue is greater than or equal to one, a condition that holds for most concepts of stability. In the Markov-switching case, the problem is subtle and complicated, and there are alternative concepts of stability. Given a Örst-order approximation, we use the concept of mean square stability (MSS) as deÖned in Costa et al. (2005). Farmer et al. (2009) discuss several advantages of using the MSS concept over alternative ones such as the bounded stability. First, under MSS, even if the largest generalized eigenvalues associated with one particular regime is greater than one, the system, as a whole, can nonetheless be stable. Second, MSS permits applications in which the system has unbounded errors and hence unbounded state variables. This unbounded feature holds in our RBC example with the normally distributed shocks to TFP. Third, in the case of Markov-switching, necessary and su¢ cient conditions for the MSS are easily veriÖable, whereas other stability concepts do not have such conditions. SpeciÖcally, the MSS requires verifying whether the following matrix has all its eigenvalues inside the unit circle

\[T = \left(P ^ {\intercal} \otimes I _ {n _ {x} ^ {2}}\right) \Upsilon ,\tag{14}\]

where

\[\Upsilon = \left[ \begin{array}{c c c} \mathcal {D} _ {1, n _ {x}} h _ {s s} (1) \otimes \mathcal {D} _ {1, n _ {x}} h _ {s s} (1) & \dots & 0 _ {n _ {x} ^ {2} \times n _ {x} ^ {2}} \\ 0 _ {n _ {x} ^ {2} \times n _ {x} ^ {2}} & \dots & 0 _ {n _ {x} ^ {2} \times n _ {x} ^ {2}} \\ \vdots & \ddots & \vdots \\ 0 _ {n _ {x} ^ {2} \times n _ {x} ^ {2}} & \dots & \mathcal {D} _ {1, n _ {x}} h _ {s s} (n _ {s}) \otimes \mathcal {D} _ {1, n _ {x}} h _ {s s} (n _ {s}) \end{array} \right].\]

In the Markov-switching case, after we use a reduced Grˆbner basis to obtain all the solutions for , we verify, for each solution, whether the matrix has all its eigenvalues less than one. If only one solution satisÖes the MSS criterion, the model has a unique stable Örst-order approximation. If there is more than one solution that satisÖes the MSS criterion, the model has multiple stable Örst-order approximations. If none of the solutions satisÖes the MSS criterion or if there is no solution to the reduced Grˆbner basis, the model does not have any stable Örst-order approximations. At this point, it is worth emphasizing that the stability of a second-order approximation (to be deÖned below) depends only on the stability of its Örst-order approximation component. In other words, stability depends only on the eigenvalues of the matrix . The same is true for higher-order approximations.

4.3 The RBC model

At this junction we view it as instructive to use the previous RBC example to illustrate our methodology developed thus far. Consider the following parameterization:

The growth rates and correspond to regimes where annual output growth rates are 3 percent and 1 percent respectively, corresponds to a risk free annual rate of 3 percent in the steady-state, and is set to match the total volatility of TFP growth as estimated in Fern·ndez-Villaverde and Rubio-Ramirez (2007). The standard calibration of implies a capital share of one third, and the value of implies an annual depreciation rate of approximately 10 percent, both in the steady-state. Note that regimes are symmetric in the sense that

Given this parameterization, the steady-state values of capital and consumption are 32:0986 and . We compute the Örst partial derivatives of with respect to all the variables evaluated at the steady-state:

\[\mathcal {D} _ {1, 1} f _ {s s} \left(s _ {t + 1}, s _ {t}\right) = \left[ \begin{array}{c} 0. 2 0 8 6 1 \\ 0 \end{array} \right], \mathcal {D} _ {2, 2} f _ {s s} \left(s _ {t + 1}, s _ {t}\right) = \left[ \begin{array}{c} - 0. 2 0 8 6 \\ 1 \end{array} \right], \mathcal {D} _ {3, 3} f _ {s s} \left(s _ {t + 1}, s _ {t}\right) = \left[ \begin{array}{c} 0. 0 0 0 3 1 \\ 1. 0 0 4 9 9 \end{array} \right]\]

\[\mathcal {D} _ {4, 4} f _ {s s} \left(s _ {t + 1}, s _ {t}\right) = \left[ \begin{array}{c} 0 \\ - 1. 0 0 7 4 \end{array} \right], \mathcal {D} _ {5, 5} f _ {s s} \left(s _ {t + 1}, s _ {t}\right) = \left[ \begin{array}{c} 0 \\ 0 \end{array} \right], \mathcal {D} _ {6, 6} f _ {s s} \left(s _ {t + 1}, s _ {t}\right) = \left[ \begin{array}{c} 0. 0 0 0 1 4 \\ 0. 0 0 9 0 0 \end{array} \right]\]

\[\mathcal {D} _ {7, 7} f _ {s s} \left(s _ {t + 1}, s _ {t}\right) = \left[ \begin{array}{c} - 0. 0 1 4 7 \\ 0 \end{array} \right], \text {and} \mathcal {D} _ {8, 8} f _ {s s} \left(s _ {t + 1}, s _ {t}\right) = \left[ \begin{array}{c} 0. 6 8 1 6 9 \\ 4 4. 9 9 5 3 \end{array} \right]\]

for all and .

Using these results, one can build the quadratic system as in (9) and solve for . Remember that in this example, , and The new method leads to the following four solutions

Mathematica Önds the four solutions in less than a hundredth of a second. Using the results in Section 4.2.3, one can verify that only solution (1) implies a stable Örst-order approximation under the MSS criterion. Normally, when you have a unique stable Örst-order approximation, you call it the Örst-order approximation. Thus, if we let , and , the Örst-order approximation to the solution of the model is

\[{\left[ \begin{array}{l} {\hat {c} _ {t}} \\ {\hat {k} _ {t}} \end{array} \right]} = {\left[ \begin{array}{l l l} 0. 0 3 8 9 6 & 0. 0 0 0 2 8 & 0. 0 0 9 7 2 \\ 0. 9 6 3 6 4 & - 0. 0 0 9 2 & - 0. 0 8 4 3 \end{array} \right]} S _ {t}\]

if , and

\[\left[ \begin{array}{l} \hat {c} _ {t} \\ \hat {k} _ {t} \end{array} \right] = \left[ \begin{array}{l l l} 0. 0 3 8 9 6 & 0. 0 0 0 2 8 & - 0. 0 0 9 7 2 \\ 0. 9 6 3 6 4 & - 0. 0 0 9 2 & 0. 0 8 4 3 \end{array} \right] \mathcal {S} _ {t}\]

if , where the rest of the derivatives used to form the Örst-order approximation can be obtained by solving the linear system deÖned in (10).

The approximation highlights Proposition 8. First-order approximations are, in general, not certainty equivalent. Since for all and

\[\sum_ {s ^ {\prime} = 1} ^ {n _ {s}} p _ {s _ {t}, s ^ {\prime}} \left(\mathcal {D} _ {7, 7} f _ {s s} \left(s ^ {\prime}, s _ {t}\right) \mathcal {D} \theta_ {s s} \left(s ^ {\prime}\right) + \mathcal {D} _ {8, 8} f _ {s s} \left(s ^ {\prime}, s _ {t}\right) \mathcal {D} \theta_ {s s} \left(s _ {t}\right)\right) \neq 0 _ {2}\]

for all , Örst partial derivatives of and h with respect to will be nonzero. That is, and

5 Second-Order Approximations

Having shown how to construct Örst-order approximations to the solutions of the MSDSGE model, we now show how to construct second-order approximations. In Section 5.1, we introduce additional notation. In Section 5.2, we use the notation to derive second-order approximations to the model solution. We show that given Örst-order approximations, second-order approximations can be obtained by simply solving linear systems. This fact emphasizes that the bottleneck to Önd both Örst-order and second-order approximations is solving the quadratic system deÖned by (9). Section 5.3 returns to the RBC model for illustration.

5.1 Additional Notation

5.1.1 Second Partial Derivatives of f

We begin with additional notation in regard to second partial derivatives of . Denote

\[\mathcal {H} f _ {s s} ^ {i} \left(s _ {t + 1}, s _ {t}\right) =\]

\[\left[ \mathcal {D} _ {k} \mathcal {D} _ {j} f ^ {i} \left(\mathbf {y} _ {s s}, \mathbf {y} _ {s s}, \mathbf {x} _ {s s}, \mathbf {x} _ {s s}, 0 _ {n _ {\varepsilon}}, 0 _ {n _ {\varepsilon}}, \overline {{\theta}} _ {1}, \widehat {\theta} _ {2} (s _ {t + 1}), \overline {{\theta}} _ {1}, \widehat {\theta} _ {2} (s _ {t})\right) \right] _ {1 \leq j, k \leq 2 (n _ {y} + n _ {x} + n _ {\varepsilon} + n _ {\theta})}\]

for all and as the matrix of second partial derivatives of for with respect to all its variables evaluated at

\[\left(\mathbf {y} _ {s s}, \mathbf {y} _ {s s}, \mathbf {x} _ {s s}, \mathbf {x} _ {s s}, 0 _ {n _ {\varepsilon}}, 0 _ {n _ {\varepsilon}}, \overline {{\theta}} _ {1}, \widehat {\theta} _ {2} (s _ {t + 1}), \overline {{\theta}} _ {1}, \widehat {\theta} _ {2} (s _ {t})\right).\]

To conserve space, we do not represent the second partial derivatives of f for our RBC example (they are available upon request).

5.1.2 Second Partial Derivatives of G

Let

\[\mathcal {H} \mathbb {G} ^ {i} (\mathbf {x} _ {t - 1}, \varepsilon_ {t}, \chi , s _ {t}) = [ \mathcal {D} _ {k} \mathcal {D} _ {j} \mathbb {G} ^ {i} (\mathbf {x} _ {t - 1}, \varepsilon_ {t}, \chi , s _ {t}) ] _ {1 \leq j, k \leq n _ {x} + n _ {\varepsilon} + 1}\]

be the matrix of second partial derivatives of with respect to for all and and . It follows that

\[\mathcal {H} \mathbb {G} ^ {i} \left(\mathbf {x} _ {s s}, 0 _ {n _ {\varepsilon}}, 0, s _ {t}\right) = \left[ \mathcal {D} _ {k} \mathcal {D} _ {j} \mathbb {G} ^ {i} \left(\mathbf {x} _ {s s}, 0 _ {n _ {\varepsilon}}, 0, s _ {t}\right) \right] _ {1 \leq j, k \leq n _ {x} + n _ {\varepsilon} + 1}\]

is the matrix of second partial derivatives of with respect to evaluated at for all and

To simplify notation we deÖne

\[\mathcal {H} \mathbb {G} _ {s s} ^ {i} (s _ {t}) = \mathcal {H} \mathbb {G} ^ {i} (\mathbf {x} _ {s s}, 0 _ {n _ {\varepsilon}}, 0, s _ {t}) \text {and} \mathcal {D} _ {k} \mathcal {D} _ {j} \mathbb {G} _ {s s} ^ {i} (s _ {t}) = \mathcal {D} _ {k} \mathcal {D} _ {j} \mathbb {G} ^ {i} (\mathbf {x} _ {s s}, 0 _ {n _ {\varepsilon}}, 0, s _ {t})\]

for all and and . Thus,

\[\mathcal {H} \mathbb {G} _ {s s} ^ {i} (s _ {t}) = [ \mathcal {D} _ {k} \mathcal {D} _ {j} \mathbb {G} _ {s s} ^ {i} (s _ {t}) ] _ {1 \leq j, k \leq n _ {x} + n _ {\varepsilon} + 1}\]

for all and

5.1.3 Second Partial Derivatives of g and h

We now introduce the second partial derivatives of and h. Let

\[\mathcal {H} g ^ {i} \left(\mathbf {x} _ {t - 1}, \varepsilon_ {t}, \chi , s _ {t}\right) = \left[ \mathcal {D} _ {k} \mathcal {D} _ {j} g ^ {i} \left(\mathbf {x} _ {t - 1}, \varepsilon_ {t}, \chi , s _ {t}\right) \right] _ {1 \leq j, k \leq n _ {x} + n _ {\varepsilon} + 1}\]

be the matrix of second partial derivatives of with respect to for all and and . It follows that

\[\mathcal {H} g ^ {i} \left(\mathbf {x} _ {s s}, 0 _ {n _ {\varepsilon}}, 0, s _ {t}\right) = \left[ \mathcal {D} _ {k} \mathcal {D} _ {j} g ^ {i} \left(\mathbf {x} _ {s s}, 0 _ {n _ {\varepsilon}}, 0, s _ {t}\right) \right] _ {1 \leq j, k \leq n _ {x} + n _ {\varepsilon} + 1}\]

is the matrix of second partial derivatives of with respect to evaluated at for all and

To put in compact notation we deÖne

\[\mathcal {H} g _ {s s} ^ {i} (s _ {t}) = \mathcal {H} g _ {s s} ^ {i} (\mathbf {x} _ {s s}, 0 _ {n _ {\varepsilon}}, 0, s _ {t}) \mathrm{and} \mathcal {D} _ {k} \mathcal {D} _ {j} g _ {s s} ^ {i} (s _ {t}) = \mathcal {D} _ {k} \mathcal {D} _ {j} g ^ {i} (\mathbf {x} _ {s s}, 0 _ {n _ {\varepsilon}}, 0, s _ {t})\]

for all and and . Hence,

\[\mathcal {H} g _ {s s} ^ {i} (s _ {t}) = \left[ \mathcal {D} _ {k} \mathcal {D} _ {j} g ^ {i} (\mathbf {x} _ {s s}, 0 _ {n _ {\varepsilon}}, 0, s _ {t}) \right] _ {1 \leq j, k \leq n _ {x} + n _ {\varepsilon} + 1}\]

for all and

Let

\[\mathcal {H} h ^ {i} \left(\mathbf {x} _ {t - 1}, \varepsilon_ {t}, \chi , s _ {t}\right) = \left[ \mathcal {D} _ {k} \mathcal {D} _ {j} h ^ {i} \left(\mathbf {x} _ {t - 1}, \varepsilon_ {t}, \chi , s _ {t}\right) \right] _ {1 \leq j, k \leq n _ {x} + n _ {\varepsilon} + 1}\]

be the matrix of second partial derivatives of with respect to for all ; and and . It follows that

\[\mathcal {H} h ^ {i} \left(\mathbf {x} _ {s s}, 0 _ {n _ {\varepsilon}}, 0, s _ {t}\right) = \left[ \mathcal {D} _ {k} \mathcal {D} _ {j} h ^ {i} \left(\mathbf {x} _ {s s}, 0 _ {n _ {\varepsilon}}, 0, s _ {t}\right) \right] _ {1 \leq j, k \leq n _ {x} + n _ {\varepsilon} + 1}\]

is the matrix of second partial derivatives of with respect to evaluated at for all and

If we deÖne

\[\mathcal {H} h _ {s s} ^ {i} (s _ {t}) = \mathcal {H} h _ {s s} ^ {i} (\mathbf {x} _ {s s}, 0 _ {n _ {\varepsilon}}, 0, s _ {t}) \mathrm{and} \mathcal {D} _ {k} \mathcal {D} _ {j} h _ {s s} ^ {i} (s _ {t}) = \mathcal {D} _ {k} \mathcal {D} _ {j} h ^ {i} (\mathbf {x} _ {s s}, 0 _ {n _ {\varepsilon}}, 0, s _ {t})\]

for all , and , then we have

\[\mathcal {H} h _ {s s} ^ {i} (s _ {t}) = \left[ \mathcal {D} _ {k} \mathcal {D} _ {j} h ^ {i} (\mathbf {x} _ {s s}, 0 _ {n _ {\varepsilon}}, 0, s _ {t}) \right] _ {1 \leq j, k \leq n _ {x} + n _ {\varepsilon} + 1}\]

for all and

With these deÖnitions, the second partial derivatives of g and h evaluated at the steady-state can be expressed as

\[\mathcal {H} g _ {s s} \left(s _ {t}\right) = \left[ \begin{array}{c} v e c \left(\mathcal {H} g _ {s s} ^ {1} \left(s _ {t}\right)\right) ^ {\intercal} \\ \vdots \\ v e c \left(\mathcal {H} g _ {s s} ^ {n _ {y}} \left(s _ {t}\right)\right) ^ {\intercal} \end{array} \right]\]

and

\[\mathcal {H} h _ {s s} \left(s _ {t}\right) = \left[ \begin{array}{c} v e c \left(\mathcal {H} h _ {s s} ^ {1} \left(s _ {t}\right)\right) ^ {\intercal} \\ \vdots \\ v e c \left(\mathcal {H} h _ {s s} ^ {n _ {x}} \left(s _ {t}\right)\right) ^ {\intercal} \end{array} \right]\]

for all .

5.2 Approximations

Let and be a second-order approximation to g and h around the point With the additional notation introduced in Section 5.1, we have

\[g ^ {\mathrm{second}} \left(\mathbf {x} _ {t - 1}, \varepsilon_ {t}, \chi , s _ {t}\right) - \mathbf {y} _ {s s} = \mathcal {D} g _ {s s} \left(s _ {t}\right) \mathcal {S} _ {t} + \frac {1}{2} \mathcal {H} g _ {s s} \left(s _ {t}\right) \left(\mathcal {S} _ {t} \otimes \mathcal {S} _ {t}\right),\]

where is an vector. Similarly, we have

\[h ^ {\text { second }} \left(\mathbf {x} _ {t - 1}, \varepsilon_ {t}, \chi , s _ {t}\right) - \mathbf {x} _ {s s} = \mathcal {D} h _ {s s} \left(s _ {t}\right) \mathcal {S} _ {t} + \frac {1}{2} \mathcal {H} h _ {s s} \left(s _ {t}\right) \left(\mathcal {S} _ {t} \otimes \mathcal {S} _ {t}\right).\]

Similar to our analysis in Section 3.2, given the Örst-order approximations, the remaining task is to derive the matrices

\[\left\{\left\{\mathcal {H} g _ {s s} ^ {i} (s _ {t}) \right\} _ {i = 1} ^ {n _ {y}}, \left\{\mathcal {H} h _ {s s} ^ {i} (s _ {t}) \right\} _ {i = 1} ^ {n _ {x}} \right\} _ {s _ {t} = 1} ^ {n _ {s}}\tag{15}\]

for all , where for all and and for all and and

As in the case of Örst-order approximations, we use the chain rule and the second partial derivatives of to obtain the systems of equations that can be used to solve for the Hessian matrices expressed in (15). Since

\[\mathbb {G} \left(\mathbf {x} _ {t - 1}, \varepsilon_ {t}, \chi , s _ {t}\right) = 0 _ {n _ {y} + n _ {x}}\]

for all ; and it must be the case that

\[\mathcal {H} \mathbb {G} ^ {i} \left(\mathbf {x} _ {t - 1}, \varepsilon_ {t}, \chi , s _ {t}\right) = 0 _ {\left(n _ {x} + n _ {\varepsilon} + 1\right) \times \left(n _ {x} + n _ {\varepsilon} + 1\right)}\]

for all and , and in particular

\[\mathcal {H} \mathbb {G} _ {s s} ^ {i} (s _ {t}) = 0 _ {(n _ {x} + n _ {\varepsilon} + 1) \times (n _ {x} + n _ {\varepsilon} + 1)}\]

for all and It follows that

\[\begin{array}{r c l} \mathcal {H} _ {1, n _ {x}; 1, n _ {x}} \mathbb {G} _ {s s} ^ {i} (s _ {t}) & = & 0 _ {n _ {x} \times n _ {x}}, \mathcal {H} _ {1, n _ {x}; n _ {x} + 1, n _ {x} + n _ {\varepsilon}} \mathbb {G} _ {s s} ^ {i} (s _ {t}) = 0 _ {n _ {x} \times n _ {\varepsilon}}, \\ \mathcal {H} _ {1, n _ {x}; n _ {x} + n _ {\varepsilon} + 1} \mathbb {G} _ {s s} ^ {i} (s _ {t}) & = & 0 _ {n _ {x}}, \mathcal {H} _ {n _ {x} + 1, n _ {x} + n _ {\varepsilon}; n _ {x} + 1, n _ {x} + n _ {\varepsilon}} \mathbb {G} _ {s s} ^ {i} (s _ {t}) = 0 _ {n _ {\varepsilon} \times n _ {\varepsilon}}, \\ \mathcal {H} _ {n _ {x} + 1, n _ {x} + n _ {\varepsilon}; n _ {x} + n _ {\varepsilon} + 1} \mathbb {G} _ {s s} ^ {i} (s _ {t}) & = & 0 _ {n _ {\varepsilon}}, \text {and} \mathcal {H} _ {n _ {x} + n _ {\varepsilon} + 1; n _ {x} + n _ {\varepsilon} + 1} \mathbb {G} _ {s s} ^ {i} (s _ {t}) = 0 \end{array}\tag{16}\]

for all and , where we have used the following deÖnition

\[\mathcal {H} _ {n _ {1}, n _ {2}; m _ {1}, m _ {2}} \mathbb {G} _ {s s} ^ {i} \left(s _ {t}\right) = \left[ \begin{array}{c c c} \mathcal {D} _ {n _ {1}} \mathcal {D} _ {m _ {1}} \mathbb {G} _ {s s} ^ {i} \left(s _ {t}\right) & \ldots & \mathcal {D} _ {n _ {1}} \mathcal {D} _ {m _ {2}} \mathbb {G} _ {s s} ^ {i} \left(s _ {t}\right) \\ \vdots & \ddots & \vdots \\ \mathcal {D} _ {n _ {1}} \mathcal {D} _ {m _ {1}} \mathbb {G} _ {s s} ^ {i} \left(s _ {t}\right) & \ldots & \mathcal {D} _ {n _ {2}} \mathcal {D} _ {m _ {2}} \mathbb {G} _ {s s} ^ {i} \left(s _ {t}\right) \end{array} \right]\]

for all , and

All the equations in the systems represented in (16) depend on

\[\begin{array}{r l} \{\mathcal {D} _ {1, n _ {x}} g _ {s s} (s _ {t}), \mathcal {D} _ {1, n _ {x}} h _ {s s} (s _ {t}) \} _ {s _ {t} = 1} ^ {n _ {s}}, & \{\mathcal {D} _ {n _ {x} + 1, n _ {x} + n _ {\varepsilon}} g _ {s s} (s _ {t}), \mathcal {D} _ {n _ {x} + 1, n _ {x} + n _ {\varepsilon}} h _ {s s} (s _ {t}) \} _ {s _ {t} = 1} ^ {n _ {s}}, \\ & \mathrm{and} \{\mathcal {D} _ {n _ {x} + n _ {\varepsilon} + 1} g _ {s s} (s _ {t}), \mathcal {D} _ {n _ {x} + n _ {\varepsilon} + 1} h _ {s s} (s _ {t}) \} _ {s _ {t} = 1} ^ {n _ {s}}. \end{array}\]

Thus, one must obtain Örst-order approximations prior to obtaining second-order approximations. The following steps describe how to use the systems in (16) to obtain the Hessian matrices expressed in (15) and highlight the dependence between these systems.

\[\mathcal {H} _ {n _ {1}, n _ {2}; m _ {1}} \mathbb {G} _ {s s} ^ {i} (s _ {t}) = \left[ \begin{array}{c} \mathcal {D} _ {n _ {1}} \mathcal {D} _ {m _ {1}} \mathbb {G} _ {s s} ^ {i} (s _ {t}) \\ \vdots \\ \mathcal {D} _ {n _ {1}} \mathcal {D} _ {m _ {1}} \mathbb {G} _ {s s} ^ {i} (s _ {t}) \end{array} \right]\]

for all , and . When we have

\[\mathcal {H} _ {n _ {1}; m _ {1}, m _ {2}} \mathbb {G} _ {s s} ^ {i} (s _ {t}) = \left[ \begin{array}{l l l} \mathcal {D} _ {n _ {1}} \mathcal {D} _ {m _ {1}} \mathbb {G} _ {s s} ^ {i} (s _ {t}) & \ldots & \mathcal {D} _ {n _ {1}} \mathcal {D} _ {m _ {2}} \mathbb {G} _ {s s} ^ {i} (s _ {t}) \end{array} \right]\]

for all , and . When and we have

\[\mathcal {H} _ {n _ {1}; m _ {1}} \mathbb {G} _ {s s} ^ {i} (s _ {t}) = \mathcal {D} _ {n _ {1}} \mathcal {D} _ {m _ {1}} \mathbb {G} _ {s s} ^ {i} (s _ {t})\]

¹⁵By Youngís Theorem, all the Hessian matrices in (15) and for all are symmetric. In the following, we exploit this fact whenever it is applicable and focus on only the relevant portion of the Hessian matrices.

¹⁶When we have

for a

Step 1 The condition for implies a linear system of equations that determines the solution for

Step 2 Given the solution from Step 1, the condition for implies a linear system of equations that determines the solution for

Step 3 Given the solution from Step 2, the condition implies a linear system of equations that determines the solution for

Step 4 Given the solution from Step 1, the condition for implies a linear system of equations that determines the solution for

Step 5 Given the solutions from Steps 1 and the condition for implies a linear system of equations that determines the solution

Step 6 Given the solutions from Steps 1, 3 and 4, the condition for implies a linear system of equations that determines the solution for

In the steps described above, we have used the following deÖnitions:

\[\mathcal {H} _ {n _ {1}, n _ {2}; m _ {1}, m _ {2}} g _ {s s} ^ {i} \left(s _ {t}\right) = \left[ \begin{array}{c c c} \mathcal {D} _ {n _ {1}} \mathcal {D} _ {m _ {1}} g _ {s s} ^ {i} \left(s _ {t}\right) & \ldots & \mathcal {D} _ {n _ {1}} \mathcal {D} _ {m _ {2}} g _ {s s} ^ {i} \left(s _ {t}\right) \\ \vdots & \ddots & \vdots \\ \mathcal {D} _ {n _ {1}} \mathcal {D} _ {m _ {1}} g _ {s s} ^ {i} \left(s _ {t}\right) & \ldots & \mathcal {D} _ {n _ {2}} \mathcal {D} _ {m _ {2}} g _ {s s} ^ {i} \left(s _ {t}\right) \end{array} \right]\]

for all , and ;

\[\mathcal {H} _ {n _ {1}, n _ {2}; m _ {1}, m _ {2}} h _ {s s} ^ {i} \left(s _ {t}\right) = \left[ \begin{array}{c c c} \mathcal {D} _ {n _ {1}} \mathcal {D} _ {m _ {1}} h _ {s s} ^ {i} \left(s _ {t}\right) & \ldots & \mathcal {D} _ {n _ {1}} \mathcal {D} _ {m _ {2}} h _ {s s} ^ {i} \left(s _ {t}\right) \\ \vdots & \ddots & \vdots \\ \mathcal {D} _ {n _ {1}} \mathcal {D} _ {m _ {1}} h _ {s s} ^ {i} \left(s _ {t}\right) & \ldots & \mathcal {D} _ {n _ {2}} \mathcal {D} _ {m _ {2}} h _ {s s} ^ {i} \left(s _ {t}\right) \end{array} \right]\]

for all

Appendix D provides a detailed description of the derivations for Steps 1 to 6. In particular, it derives the second partial derivatives of G needed to construct the six systems. It shows that once we obtain Örst-order approximations to the model solution, each of the equations in Steps 1 to 6 becomes linear. As a result, for each Örst-order approximation, a second-order approximation can be easily derived from the set of linear systems described above.

In addition, Appendix D shows that if Örst-order approximations are certainty equivalent, the systems of equations represented by (cross partial derivatives between and and (cross partial derivatives between and for ¹⁷When , we have

\[\mathcal {H} _ {n _ {1}, n _ {2}; m _ {1}} g _ {s s} ^ {i} (s _ {t}) = \left[ \begin{array}{c} \mathcal {D} _ {n _ {1}} \mathcal {D} _ {m _ {1}} g _ {s s} ^ {i} (s _ {t}) \\ \vdots \\ \mathcal {D} _ {n _ {1}} \mathcal {D} _ {m _ {1}} g _ {s s} ^ {i} (s _ {t}) \end{array} \right]\]

for all , and ;

\[\mathcal {H} _ {n _ {1}, n _ {2}; m _ {1}} h _ {s s} ^ {i} (s _ {t}) = \left[ \begin{array}{c} \mathcal {D} _ {n _ {1}} \mathcal {D} _ {m _ {1}} h _ {s s} ^ {i} (s _ {t}) \\ \vdots \\ \mathcal {D} _ {n _ {1}} \mathcal {D} _ {m _ {1}} h _ {s s} ^ {i} (s _ {t}) \end{array} \right]\]

for all , and . When , we have

\[\mathcal {H} _ {n _ {1}; m _ {1}, m _ {2}} g _ {s s} ^ {i} (s _ {t}) = \left[ \begin{array}{l l l} \mathcal {D} _ {n _ {1}} \mathcal {D} _ {m _ {1}} g _ {s s} ^ {i} (s _ {t}) & \dots & \mathcal {D} _ {n _ {1}} \mathcal {D} _ {m _ {2}} g _ {s s} ^ {i} (s _ {t}) \end{array} \right]\]

for all , and

\[\mathcal {H} _ {n _ {1}; m _ {1}, m _ {2}} h _ {s s} ^ {i} (s _ {t}) = \left[ \begin{array}{l l l} \mathcal {D} _ {n _ {1}} \mathcal {D} _ {m _ {1}} h _ {s s} ^ {i} (s _ {t}) & \ldots & \mathcal {D} _ {n _ {1}} \mathcal {D} _ {m _ {2}} h _ {s s} ^ {i} (s _ {t}) \end{array} \right]\]

for all , and . When and , we have

\[\mathcal {H} _ {n _ {1}; m _ {1}} g _ {s s} ^ {i} (s _ {t}) = \mathcal {D} _ {n _ {1}} \mathcal {D} _ {m _ {1}} g _ {s s} ^ {i} (s _ {t})\]

for all , and

\[\mathcal {H} _ {n _ {1}; m _ {1}} h _ {s s} ^ {i} (s _ {t}) = \mathcal {D} _ {n _ {1}} \mathcal {D} _ {m _ {1}} h _ {s s} ^ {i} (s _ {t})\]

for all

all and are homogeneous in the unknown variables

\[\begin{array}{r l} & {\mathcal {H} _ {1, n _ {x}; n _ {x} + n _ {\varepsilon} + 1} g _ {s s} ^ {i} (s _ {t}), \mathcal {H} _ {1, n _ {x}; n _ {x} + n _ {\varepsilon} + 1} h _ {s s} ^ {j} (s _ {t}),} \\ & {\mathcal {H} _ {n _ {x} + 1, n _ {x} + n _ {\varepsilon}; n _ {x} + n _ {\varepsilon} + 1} g _ {s s} ^ {i} (s _ {t}), \mathrm{and} \mathcal {H} _ {n _ {x} + 1, n _ {x} + n _ {\varepsilon}; n _ {x} + n _ {\varepsilon} + 1} h _ {s s} ^ {j} (s _ {t})} \end{array}\]

for all , and . This property is formally stated in the following proposition.

Proposition 10 If Örst-order approximations are certainty equivalent, then the systems of equations

\[\mathcal {H} _ {1, n _ {x}; n _ {x} + n _ {\varepsilon} + 1} \mathbb {G} _ {s s} ^ {i} (s _ {t}) = 0 _ {n _ {x}} a n d \mathcal {H} _ {n _ {x} + 1, n _ {x} + n _ {\varepsilon}; n _ {x} + n _ {\varepsilon} + 1} \mathbb {G} _ {s s} ^ {i} (s _ {t}) = 0 _ {n _ {x}}\]

for all and are homogeneous in their unknown variables and hence

\[\begin{array}{r c l} \mathcal {H} _ {1, n _ {x}; n _ {x} + n _ {\varepsilon} + 1} g _ {s s} ^ {i} (s _ {t}) & = & 0 _ {n _ {x}}, \mathcal {H} _ {1, n _ {x}; n _ {x} + n _ {\varepsilon} + 1} h _ {s s} ^ {j} (s _ {t}) = 0 _ {n _ {x}}, \\ \mathcal {H} _ {n _ {x} + 1, n _ {x} + n _ {\varepsilon}; n _ {x} + n _ {\varepsilon} + 1} g _ {s s} ^ {i} (s _ {t}) & = & 0 _ {n _ {\varepsilon}} a n d \mathcal {H} _ {n _ {x} + 1, n _ {x} + n _ {\varepsilon}; n _ {x} + n _ {\varepsilon} + 1} h _ {s s} ^ {j} (s _ {t}) = 0 _ {n _ {\varepsilon}}. \end{array}\]

for all , and

Proof. If Örst-order approximations are certainty equivalent, one can see from Proposition 8 that

\[\mathcal {D} _ {n _ {x} + n _ {\varepsilon} + 1} g _ {s s} (s _ {t}) = 0 _ {n _ {y}} \text {and} \mathcal {D} _ {n _ {x} + n _ {\varepsilon} + 1} h _ {s s} (s _ {t}) = 0 _ {n _ {x}}\]

for all . The proof follows directly from the above results.

In the constant parameter case, the cross partial derivatives with respect to and other state variables are always zero because . In the Markov-switching case, Proposition 10 implies that if for some then it may be the case that for some , some , and some . These non-trivial extra terms may result in more accurate second-order approximations.

5.3 Returning to the RBC Model

Let us now return to the RBC example. As shown above, given our parameterization, there is only one stable Örst-order approximation. Thus, there is only one stable second-order approximation. When that is the case, we call it the second-order approximation. We can Önd the second-order approximation by solving a set of linear systems described in Steps 1 to 6. We have

\[\begin{array}{r l r} & & {\left[ \begin{array}{l} \hat {c} _ {t} \\ \hat {k} _ {t} \end{array} \right] = \left[ \begin{array}{l l l} 0. 0 3 8 9 6 & 0. 0 0 0 2 8 & 0. 0 0 9 7 2 \\ 0. 9 6 3 6 4 & - 0. 0 0 9 2 & - 0. 0 8 4 3 \end{array} \right] \mathcal {S} _ {t}} \\ & & {+ \frac {1}{2} \left[ \begin{array}{l l l l l l l l} - 0. 0 0 0 4 & 4 \times 1 0 ^ {- 6} & 0. 0 0 0 1 6 & 4 \times 1 0 ^ {- 6} & 4 \times 1 0 ^ {- 8} & 1 \times 1 0 ^ {- 6} & 0. 0 0 0 1 6 & 1 \times 1 0 ^ {- 6} & - 0. 0 0 0 3 \\ - 0. 0 0 0 2 & - 0. 0 0 0 3 & - 0. 0 0 2 5 & - 0. 0 0 0 3 & 2. 7 \times 1 0 ^ {- 6} & 2 \times 1 0 ^ {- 5} & - 0. 0 0 2 5 & 0. 0 0 0 0 2 & 0. 0 0 0 5 7 \end{array} \right] (\mathcal {S} _ {t} \otimes \mathcal {S} _ {t})} \\ & & {\mathrm{if} s _ {t} = 1 \mathrm{or}} \\ & & {\left[ \begin{array}{l} \hat {c} _ {t} \\ \hat {k} _ {t} \end{array} \right] = \left[ \begin{array}{l l l} 0. 0 3 8 9 6 & 0. 0 0 0 2 8 & - 0. 0 0 9 7 2 \\ 0. 9 6 3 6 4 & - 0. 0 0 9 2 & 0. 0 8 4 3 \end{array} \right] \mathcal {S} _ {t}} \\ & & {+ \frac {1}{2} \left[ \begin{array}{l l l l l l l l} - 0. 0 0 0 4 & 4 \times 1 0 ^ {- 6} & - 0. 0 0 0 2 & 4 \times 1 0 ^ {- 6} & 4 \times 1 0 ^ {- 8} & - 1 \times 1 0 ^ {- 6} & - 0. 0 0 0 2 & - 1 \times 1 0 ^ {- 6} & - 0. 0 0 0 3 \\ - 0. 0 0 0 2 & - 0. 0 0 0 2 & 0. 0 0 2 5 _ {1} & - 0. 0 0 _ {3} & {3 \times {1}} _ {1} ^ {- {6}} & - {2 \times {1}} _ {5} ^ {- {5}} & {0.} {0} {2} _ {5} ^ {1} & - {2 \times {1}} _ {5} ^ {- {5}} & {0.} {0} {0} _ {5} ^ {7} \end{array} \right] (\mathcal {S} _ {t} \otimes \mathcal {S} _ {t})} \end{array}\]

if

Since the Örst-order approximation is not certainty equivalent in this example, the secondorder approximation highlights the result in Proposition 10. The cross derivatives of and h with respect to and other state variables evaluated at the steady-state are

\[\mathcal {H} _ {i, 3} g _ {s s} (s _ {t}) = \mathcal {H} _ {3, i} g _ {s s} (s _ {t}) \neq 0, \mathcal {H} _ {i, 3} h _ {s s} (s _ {t}) = \mathcal {H} _ {3, i} h _ {s s} (s _ {t}) \neq 0\]

for all and . Moreover, since regimes are symmetric, the following terms are symmetric across regimes

\[\begin{array}{r c l} \mathcal {D} _ {3} g _ {s s} (1) & = & - \mathcal {D} _ {3} g _ {s s} (2), \mathcal {D} _ {3} h _ {s s} (1) = \mathcal {D} _ {3} h _ {s s} (2), \\ \mathcal {H} _ {i, 3} g _ {s s} (1) & = & - \mathcal {H} _ {i, 3} g _ {s s} (2), \mathcal {H} _ {i, 3} g _ {s s} (1) = - \mathcal {H} _ {i, 3} g _ {s s} (2), \\ \mathcal {H} _ {i, 3} h _ {s s} (1) & = & - \mathcal {H} _ {i, 3} h _ {s s} (2), \text {and} \mathcal {H} _ {i, 3} h _ {s s} (1) = - \mathcal {H} _ {i, 3} h _ {s s} (2) \end{array}\]

for

6 Applications

Since our theoretical results are new, the purpose of this section is to guide the reader by illustrating how to apply our methodology in practice. For this purpose we Örst continue the previous RBC model with new parameterization and then study two versions of the New-Keynesian model.

6.1 Continuing the RBC Model

Stochastic neoclassical growth models, such as the one discussed in previous sections, are the foundation of modern macroeconomics. Understanding how to solve MSDSGE models of this prototype would enable us to work on richer models such as those commonly used for policy analysis. For pedagogical reasons we consider the shock standard deviation, , to be constant across regimes in all the examples studied in the paper. But our approach can be easily extended, without much computational burden, to cases allowing for Markov-switching volatilities.

The previous RBC example restricts regimes to be symmetric such that . As a result, the cross partial derivatives of g and h evaluated at the steady-state are symmetric across regimes. In this section, we consider an asymmetric-regime case in which . As will be shown, the cross partial derivatives of g and h evaluated at the steady-state are asymmetric across regimes in this case.

Consider the same parameter conÖguration as in Section 4.3, except 2 , and . In this case, Regime 1 has a shorter expected duration, and Regime 2 occurs more frequently in the ergodic distribution. With this alternative parameter conÖguration, the steady-state is represented by , and and the Örst partial derivatives of f evaluated at this steady-state are di§erent from those in the symmetricregime case.¹⁸ Using these results we can solve the quadratic system (9).

There are four solutions to the quadratic system under this alternative parameter conÖgu-

¹⁸ In the interest of space, we omit many matrices from this section. They are available upon request.

ration

As before, Mathematica Önds the four solutions in less than a hundredth of a second. Solution (1) is the only one associated with a stable Örst-order approximation. Remember that the rest of the coe¢ cients necessary to construct the second-order approximation can be obtained by solving linear systems and that, given that the Örst-order approximation is stable, the secondorder approximation is also stable. The second-order approximation to the model solution is

\[\begin{array}{r l r} & & {\left[ \begin{array}{l} \hat {c} _ {t} \\ \hat {k} _ {t} \end{array} \right] = \left[ \begin{array}{l l l} 0. 0 3 7 0 8 & 0. 0 0 0 2 9 & 0. 0 0 6 3 7 \\ 0. 9 6 5 4 5 & - 0. 0 1 0 0 & - 0. 1 4 1 2 \end{array} \right] S _ {t}} \\ & & {+ \frac {1}{2} \left[ \begin{array}{l l l l l l l l} - 0. 0 0 0 4 & 4 \times 1 0 ^ {- 6} & 0. 0 0 0 0 9 & 4 \times 1 0 ^ {- 6} & 5 \times 1 0 ^ {- 8} & 1 \times 1 0 ^ {- 6} & 0. 0 0 0 0 9 & 1 \times 1 0 ^ {- 6} & - 5 \times 1 0 ^ {- 6} \\ - 0. 0 0 0 2 & - 0. 0 0 0 3 & - 0. 0 0 4 0 & - 0. 0 0 0 3 & 3 \times 1 0 ^ {- 6} & 0. 0 0 0 0 4 & - 0. 0 0 4 0 & 0. 0 0 0 0 4 & 0. 0 0 0 6 5 \end{array} \right] (\mathcal {S} _ {t} \otimes \mathcal {S} _ {t})} \end{array}\]

if and

\[\begin{array}{r l r} & & {\left[ \begin{array}{l} \hat {c} _ {t} \\ \hat {k} _ {t} \end{array} \right] = \left[ \begin{array}{l l l} 0. 0 3 7 0 8 & 0. 0 0 0 2 9 & - 0. 0 0 1 3 \\ 0. 9 6 5 4 5 & - 0. 0 1 0 0 & 0. 0 2 8 2 3 \end{array} \right] \mathcal {S} _ {t}} \\ & & {+ \frac {1}{2} \left[ \begin{array}{l l l l l l l l} - 0. 0 0 0 4 & 4 \times 1 0 ^ {- 6} & - 1 \times 1 0 ^ {- 5} & 4 \times 1 0 ^ {- 6} & 5 \times 1 0 ^ {- 8} & - 2 \times 1 0 ^ {- 7} & - 1 \times 1 0 ^ {- 5} & - 2 \times 1 0 ^ {- 7} & - 6 \times 1 0 ^ {- 5} \\ - 0. 0 0 0 2 & - 0. 0 0 0 3 & 0. 0 0 0 8 0 & - 0. 0 0 0 3 & 3 \times 1 0 ^ {- 6} & - 8 \times 1 0 ^ {- 6} & 0. 0 0 0 8 0 & - 8 \times 1 0 ^ {- 6} & 0. 0 0 0 0 9 \end{array} \right] (\mathcal {S} _ {t} \otimes \mathcal {S} _ {t})} \end{array}\]

if

Clearly, the Örst-order approximation can be constructed by only considering the Örst matrix of the right hand side the above expressions.

As in the symmetric regime case, the Örst-order approximation is not certainty equivalent and the cross derivatives of g and h with respect to and other state variables evaluated at the steady-state are non-zero. These results follow directly from Propositions 8 and 10. Because for all , we have that

\[\sum_ {s ^ {\prime} = 1} ^ {n _ {s}} p _ {s _ {t}, s ^ {\prime}} \left(\mathcal {D} _ {7, 7} f _ {s s} \left(s ^ {\prime}, s _ {t}\right) \mathcal {D} \theta_ {s s} \left(s ^ {\prime}\right) + \mathcal {D} _ {8, 8} f _ {s s} \left(s ^ {\prime}, s _ {t}\right) \mathcal {D} \theta_ {s s} \left(s _ {t}\right)\right) \neq 0 _ {2}\]

for all . Consequently, , and

\[\mathcal {H} _ {i, 3} g _ {s s} (s _ {t}) = \mathcal {H} _ {3, i} g _ {s s} (s _ {t}) \neq 0, \mathcal {H} _ {i, 3} h _ {s s} (s _ {t}) = \mathcal {H} _ {3, i} h _ {s s} (s _ {t}) \neq 0\]

for all and

The results show that plays a crucial role in determining whether the Örst-order approximations are certainty equivalent and whether the cross partial derivatives of g and h with respect to and other state variables evaluated at the steady-state are zero. Unlike the symmetric regime case, however, the cross partial derivatives of g and f at the steady-state are not the same across regimes. SpeciÖcally,

\[\begin{array}{r c l} \mathcal {D} _ {3} g _ {s s} (1) & \neq & - \mathcal {D} _ {3} g _ {s s} (2), \mathcal {D} _ {3} h _ {s s} (1) \neq \mathcal {D} _ {3} h _ {s s} (2), \\ \mathcal {H} _ {i, 3} g _ {s s} (1) & \neq & - \mathcal {H} _ {i, 3} g _ {s s} (2), \mathcal {H} _ {3, i} g _ {s s} (1) \neq - \mathcal {H} _ {3, i} g _ {s s} (2), \\ \mathcal {H} _ {i, 3} h _ {s s} (1) & \neq & - \mathcal {H} _ {i, 3} h _ {s s} (2), \text {and} \mathcal {H} _ {3, i} h _ {s s} (1) \neq - \mathcal {H} _ {3, i} h _ {s s} (2) \end{array}\]

for

Given the Örst-order and second-order approximations, we can compute and compare their accuracy using Euler equation errors as suggested in Judd (1998) and Aruoba et al. (2006). The Euler equation error at point for the approximation order Örst, second is

\[\begin{array}{c} E E ^ {o r d e r} \left(\tilde {k} _ {t - 1}, \varepsilon_ {t}, 1; s _ {t}\right) = \\ 1 - \beta \int \sum_ {s ^ {\prime} = 1} ^ {2} p _ {s _ {t}, s ^ {\prime}} \left( \begin{array}{c} \frac {\tilde {c} ^ {o r d e r} \left(\tilde {k} _ {t - 1 , \varepsilon_ {t} , 1 ; s _ {t}}\right)}{\tilde {c} ^ {o r d e r} \left(\tilde {k} ^ {o r d e r} \left(\tilde {k} _ {t - 1 , \varepsilon_ {t} , 1 ; s _ {t}}\right) , \varepsilon^ {\prime} ; 1\right)} \exp \left(\frac {\mu (s _ {t}) + \sigma \varepsilon_ {t}}{\alpha - 1}\right) \\ \times \left(\alpha \exp \left(\mu (s ^ {\prime}) + \sigma \varepsilon^ {\prime}\right) \tilde {k} ^ {o r d e r} \left(\tilde {k} _ {t - 1}, \varepsilon_ {t}, 1; s _ {t}\right) ^ {\alpha - 1} + 1 - \delta\right) \end{array} \right) \mu (\varepsilon^ {\prime}) d \varepsilon^ {\prime} \end{array}\]

where and are the appropriate approximations to the policy functions. The unconditional absolute Euler equation error is

\[E E ^ {o r d e r} = \sum_ {s _ {t}} \int \left| E E ^ {o r d e r} \left(\tilde {k} _ {t - 1}, \varepsilon_ {t}, 1; s _ {t}\right) \right| \mu^ {o r d e r} \left(\tilde {k} _ {t - 1}, \varepsilon_ {t}, 1; s _ {t}\right) d \tilde {k} _ {t - 1} d \varepsilon_ {t}\]

where is the unconditional distribution of the variables and

To approximate this integral numerically, we simulate a long path for the economy, saving , , and along the path. The simulation length is 10,000 periods, with the Örst 1,000 periods discarded as a burn-in. Then, for each state , we draw 10,000 normally distributed ís to compute the expectation over , and use the transition values to compute the expectation over . This entire process takes approximately an hour.

The following table shows the base-10 logarithms of absolute values of Euler equation errors for the Örst-order and second-order approximations in both symmetric and asymmetric cases. Both the Örst-order and second-order approximations produce a high degree of accuracy: a value of 5 implies an error of $1 for each $100,000 of consumption. As can be seen, the second-order approximation achieves an even higher degree of accuracy.

Unconditional Absolute Euler Equation Errors

6.2 A New-Keynesian Model

New-Keynesian models, such as those in Woodford (2003) and Christiano et al. (2005), are often used in policy analysis. This section discusses a version of the New-Keynesian model, illustrates how to partition the vector of Markov-switching parameters and discusses the conditions under which multiple stable approximations exist.

6.2.1 The Model

We consider a New-Keynesian model with quadratic price adjustment costs. The monetary authority follows a Taylor Rule. There are two Markov-switching parameters: the technology drift parameter and the coe¢ cient on ináation in the Taylor rule. Davig and Leeper (2007), Farmer et al. (2011), and Bianchi (2010), among others, argue that Markov-switching in the Taylor rule coe¢ cient on ináation captures a switch in U.S. policy regime since the mid 1980s.

The model features the representative consumer maximizing the expected lifetime utility

over consumption and hours worked

\[\mathbb {E} _ {0} \sum_ {t = 0} ^ {\infty} \beta^ {t} (\log C _ {t} - H _ {t})\]

subject to the budget constraint

\[C _ {t} + \frac {B _ {t}}{P _ {t}} = W _ {t} H _ {t} + R _ {t - 1} \frac {B _ {t - 1}}{P _ {t}} + T _ {t} + D _ {t},\]

where is nominal bonds, is the real wage, is the nominal return on bonds, is lump-sum transfers, and is proÖts from Örms. The Örst-order conditions are

\[\begin{array}{c} 1 = \beta \mathbb {E} _ {t} \frac {C _ {t}}{C _ {t + 1}} \frac {R _ {t}}{\Pi_ {t + 1}} \\ \text {and} C _ {t} = W _ {t}. \end{array}\]

The competitive Önal goods producer combines a continuum of intermediate goods into a Önal good according to the constant elasticity of substitution (CES) aggregation technology

\[Y _ {t} = \left(\int_ {0} ^ {1} Y _ {j, t} ^ {\frac {\eta - 1}{\eta}} d j\right) ^ {\frac {\eta}{\eta - 1}}.\]

Intermediate goods are produced by Örms taking the wage and the demand function

\[Y _ {j, t} = \left(\frac {P _ {j , t}}{P _ {t}}\right) ^ {- \eta} Y _ {t}\]

as given. The price is set and hours are demanded according to

\[Y _ {j, t} = A _ {t} H _ {j, t},\]

where is a technology shock following the law of motion

\[\log A _ {t} = \mu_ {t} + \log A _ {t - 1},\]

where, similar to the RBC model, the drift takes two discrete values dictated by the Markov chain process represented by

These intermediate-goods Örms face quadratic price adjustment costs according to

\[A C _ {j, t} = \frac {\kappa}{2} \left(\frac {P _ {j , t}}{P _ {j , t - 1}} - 1\right) ^ {2} Y _ {t}.\]

The Örms maximize the expected discounted proÖts subject to the demand function, the production function, and adjustment costs. In the symmetric equilibrium, , and for all , and the optimality conditions are

\[W _ {t} = m c _ {t} A _ {t}\]

\[\mathrm{and} \kappa (\Pi_ {t} - 1) \Pi_ {t} = (1 - \eta) + \eta m c _ {t} + \beta \kappa \mathbb {E} _ {t} (\Pi_ {t + 1} - 1) \Pi_ {t + 1} \frac {C _ {t}}{C _ {t + 1}} \frac {Y _ {t + 1}}{Y _ {t}},\]

where is the marginal cost faced by the Örms.

The monetary authority sets the interest rate by the following Taylor rule

\[\frac {R _ {t}}{R _ {s s}} = \left(\frac {R _ {t - 1}}{R _ {s s}}\right) ^ {\rho} \Pi_ {t} ^ {(1 - \rho) \psi_ {t}} \exp (\sigma \varepsilon_ {t})\]

where the coe¢ cient on ináation, , takes two discrete values dictated by the same Markov chain process represented by .

The market clearing condition is

\[Y _ {t} = C _ {t} + \frac {\kappa}{2} \left(\Pi_ {t} - 1\right) ^ {2} Y _ {t}.\]

Substituting out , and leads to the equilibrium conditions

\[1 = \beta \mathbb {E} _ {t} \frac {\left(1 - \frac {\kappa}{2} (\Pi_ {t} - 1) ^ {2}\right)}{\left(1 - \frac {\kappa}{2} (\Pi_ {t + 1} - 1) ^ {2}\right)} \frac {Y _ {t}}{Y _ {t + 1}} \frac {R _ {t}}{\Pi_ {t + 1}},\]

\[\kappa (\Pi_ {t} - 1) \Pi_ {t} = (1 - \eta) + \eta \left(1 - \frac {\kappa}{2} (\Pi_ {t} - 1) ^ {2}\right) \frac {Y _ {t}}{A _ {t}} + \beta \mathbb {E} _ {t} \kappa (\Pi_ {t + 1} - 1) \Pi_ {t + 1} \frac {(1 - \frac {\kappa}{2} (\Pi_ {t} - 1) ^ {2})}{(1 - \frac {\kappa}{2} (\Pi_ {t + 1} - 1) ^ {2})},\]

\[\mathrm{and} \frac {R _ {t}}{R _ {s s}} = \left(\frac {R _ {t - 1}}{R _ {s s}}\right) ^ {\rho} \Pi_ {t} ^ {(1 - \rho) \psi_ {t}} \exp (\sigma \varepsilon_ {t}).\]

Since the technology shock has a unit root, the model is non-stationary. To obtain a stationary equilibrium, we deÖne , which leads to the stationary equilibrium conditions as

\[1 = \beta \mathbb {E} _ {t} \frac {\left(1 - \frac {\kappa}{2} \left(\Pi_ {t} - 1\right) ^ {2}\right) \tilde {Y} _ {t}}{\left(1 - \frac {\kappa}{2} \left(\Pi_ {t + 1} - 1\right) ^ {2}\right) \tilde {Y} _ {t + 1}} \frac {1}{\exp (\mu_ {t + 1})} \frac {R _ {t}}{\Pi_ {t + 1}},\]

\[\kappa (\Pi_ {t} - 1) \Pi_ {t} = (1 - \eta) + \eta \left(1 - \frac {\kappa}{2} (\Pi_ {t} - 1) ^ {2}\right) \tilde {Y} _ {t} + \beta \mathbb {E} _ {t} \kappa \frac {(1 - \frac {\kappa}{2} (\Pi_ {t} - 1) ^ {2})}{(1 - \frac {\kappa}{2} (\Pi_ {t + 1} - 1) ^ {2})} (\Pi_ {t + 1} - 1) \Pi_ {t + 1},\]

\[\text { and } \frac {R _ {t}}{R _ {s s}} = \left(\frac {R _ {t - 1}}{R _ {s s}}\right) ^ {\rho} \Pi_ {t} ^ {(1 - \rho) \psi_ {t}} \exp (\sigma \varepsilon_ {t}).\]

We partition into and . We have

\[\mu_ {t} = \mu (\chi , s _ {t}) = \overline {{\mu}} + \chi \widehat {\mu} (s _ {t})\]

\[\text { and } \psi_ {t} = \psi (\chi , s _ {t}) = \widehat {\psi} (s _ {t}).\]

Hence, the drift parameter depends on the perturbation parameter , while the Taylor-rule coe¢ cient on ináation does not. We choose this partition because would enter the deÖnition of steady-state in a constant parameter model, while would not.

Using the notation in Section 2, and , we express the stationary equilibrium condition as

\[\begin{array}{l l} f \left(\mathbf {y} _ {t + 1}, \mathbf {y} _ {t}, \mathbf {x} _ {t}, \mathbf {x} _ {t - 1}, \chi \varepsilon_ {t + 1}, \varepsilon_ {t}, \theta_ {t + 1}, \theta_ {t}\right) & = \\ & \left[ \begin{array}{c} 1 - \beta \frac {\left(1 - \frac {\kappa}{2} (\Pi_ {t} - 1) ^ {2}\right) \tilde {Y} _ {t}}{\left(1 - \frac {\kappa}{2} (\Pi_ {t + 1} - 1) ^ {2}\right) \tilde {Y} _ {t + 1}} \frac {1}{\exp \left(\mu_ {t + 1}\right)} \frac {R _ {t}}{\Pi_ {t + 1}} \\ (1 - \eta) + \eta \left(1 - \frac {\kappa}{2} (\Pi_ {t} - 1) ^ {2}\right) \tilde {Y} _ {t} + \beta \kappa \frac {\left(1 - \frac {\kappa}{2} (\Pi_ {t} - 1) ^ {2}\right)}{\left(1 - \frac {\kappa}{2} (\Pi_ {t + 1} - 1) ^ {2}\right)} (\Pi_ {t + 1} - 1) \Pi \\ \left(\frac {R _ {t - 1}}{R _ {s s}}\right) ^ {\rho} \Pi_ {t} ^ {(1 - \rho) \psi_ {t}} \exp (\sigma \varepsilon_ {t}) - \frac {R _ {t}}{R _ {s s}} \end{array} \right. \end{array}\]

The model solution takes the form of

\[\begin{array}{r l} & {\left[ \begin{array}{l l} \Pi_ {t} & \tilde {Y} _ {t} \end{array} \right] ^ {\intercal} = g (R _ {t - 1}, \varepsilon_ {t}, \chi , s _ {t}),} \\ & {\left[ \begin{array}{l l} \Pi_ {t + 1} & \tilde {Y} _ {t + 1} \end{array} \right] ^ {\intercal} = g (R _ {t}, \chi \varepsilon_ {t + 1}, \chi , s _ {t + 1}),} \\ & {\mathrm{and} R _ {t} = h (R _ {t - 1}, \varepsilon_ {t}, \chi , s _ {t}).} \end{array}\]

6.2.2 Solving the Model

The following subsections show the sequential steps of obtaining approximations to the solution of the model. First, we Önd the steady-state. Second, we derive the Örst partial derivatives of with respect to all its variables evaluated at steady-state. Third, we construct second-order approximations to the solution. We present the results for two parameter conÖgurations to show how to use reduced Grˆbner bases to obtain approximations and the MSS criterion to ascertain whether we have a unique stable approximation.

Steady-State To obtain the steady-state, we set and , so that 2 , and . Thus the equilibrium conditions in the steady-state become

\[\left[ \begin{array}{c} 1 - \beta \frac {\left(1 - \frac {\kappa}{2} (\Pi_ {s s} - 1) ^ {2}\right) \tilde {Y} _ {s s}}{\left(1 - \frac {\kappa}{2} (\Pi_ {s s} - 1) ^ {2}\right) \tilde {Y} _ {s s}} \frac {1}{\exp (\bar {\mu})} \frac {R _ {s s}}{\Pi_ {s s}} \\ (1 - \eta) + \eta \left(1 - \frac {\kappa}{2} (\Pi_ {s s} - 1) ^ {2}\right) \tilde {Y} _ {s s} + \beta \kappa \frac {\left(1 - \frac {\kappa}{2} (\Pi_ {s s} - 1) ^ {2}\right)}{\left(1 - \frac {\kappa}{2} (\Pi_ {s s} - 1) ^ {2}\right)} (\Pi_ {s s} - 1) \Pi_ {s s} - \kappa (\Pi_ {s s} - 1) \Pi_ {s s} \\ \left(\frac {R _ {s s}}{R _ {s s}}\right) ^ {\rho} \Pi_ {s s} ^ {(1 - \rho) \psi_ {t}} - \frac {R _ {s s}}{R _ {s s}} \end{array} \right] = 0 _ {3 \times 1}.\]

Assuming , we have the steady-state values as

\[R _ {s s} = \frac {\exp (\bar {\mu})}{\beta}, \tilde {Y} _ {s s} = \frac {\eta - 1}{\eta}.\]

This result conÖrms our partition such that a§ects the steady-state, while does not. In principle, we could let so that perturbation would apply to as well. To obtain a numerical solution that is as accurate as possible, however, one should keep the number of perturbed parameters at the minimum. The converse is not true. That is, one cannot let because the steady-state in the constant parameter case would depend on technology drift.

Partial Derivatives of In this example, , and . Thus, the Örst partial derivatives of with respect to all its variables evaluated at the steady-state are

\[\mathcal {D} _ {1, 2} f _ {s s} \left(s _ {t + 1}, s _ {t}\right) = \left[ \begin{array}{c c} \frac {\eta}{\eta - 1} & 1 \\ 0 & \beta \kappa \\ 0 & 0 \end{array} \right], \mathcal {D} _ {3, 4} f _ {s s} \left(s _ {t + 1}, s _ {t}\right) = \left[ \begin{array}{c c} \frac {\eta}{1 - \eta} & 0 \\ \eta & - \kappa \\ 0 & (1 - \rho) \widehat {\psi} \left(s\right) \end{array} \right]\]

\[\mathcal {D} _ {5, 5} f _ {s s} \left(s _ {t + 1}, s _ {t}\right) = \left[ \begin{array}{c} - \beta e ^ {- \bar {\mu}} \\ 0 \\ - \beta e ^ {- \bar {\mu}} \end{array} \right], \mathcal {D} _ {6, 6} f _ {s s} \left(s _ {t + 1}, s _ {t}\right) = \left[ \begin{array}{c} 0 \\ 0 \\ \rho \beta e ^ {- \bar {\mu}} \end{array} \right], \mathcal {D} _ {7, 7} f _ {s s} \left(s _ {t + 1}, s _ {t}\right) = \left[ \begin{array}{c} 0 \\ 0 \\ 0 \end{array} \right],\]

\[\mathcal {D} _ {8, 8} f _ {s s} \left(s _ {t + 1}, s _ {t}\right) = \left[ \begin{array}{l} 0 \\ 0 \\ \sigma \end{array} \right], \mathcal {D} _ {9, 1 0} f _ {s s} \left(s _ {t + 1}, s _ {t}\right) = \left[ \begin{array}{l l} 1 & 0 \\ 0 & 0 \\ 0 & 0 \end{array} \right], \text {and} \mathcal {D} _ {1 1, 1 2} f _ {s s} \left(s _ {t + 1}, s _ {t}\right) = \left[ \begin{array}{l l} 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{array} \right],\]

for all and . Note that enters the matrices because we do not perturb

Unique Stable Approximation Consider the following parameterization

The growth rates and correspond to regimes where the annual growth rate are percent and 1 percent respectively, corresponds to an annual risk-free rate of 3 percent, has the steady-state markup of 11 percent, and and match the estimates in Fernandez-Villaverde et al. (2009). The two monetary policy parameters and are such that would imply a unique stable approximation if for all t and would lead to multiple stable approximations if for all .

Given this parameter conÖguration, we can calculate the steady-state values of the nominal rate and output as and . The resulting Örst partial derivatives of with respect to all its variables evaluated at the steady-state are

\[\mathcal {D} _ {1, 2} f _ {s s} \left(s _ {t + 1}, s _ {t}\right) = \left[ \begin{array}{c c} 1. 1 1 1 1 1 & 1 \\ 0 & 1 6 0. 6 1 4 \\ 0 & 0 \end{array} \right], \mathcal {D} _ {3, 4} f _ {s s} \left(s _ {t + 1}, s _ {t}\right) = \left[ \begin{array}{c c} - 1. 1 1 1 1 1 & 0 \\ 1 0 & - 1 6 1. \\ 0 & 0. 2 \widehat {\psi} \left(s _ {t}\right) \end{array} \right],\]

\[\mathcal {D} _ {5, 5} f _ {s s} \left(s _ {t + 1}, s _ {t}\right) = \left[ \begin{array}{c} - 0. 9 9 2 6 \\ 0 \\ - 0. 9 9 2 6 \end{array} \right], \mathcal {D} _ {6, 6} f _ {s s} \left(s _ {t + 1}, s _ {t}\right) = \left[ \begin{array}{c} 0 \\ 0 \\ 0. 7 7 9 4 1 \end{array} \right], \mathcal {D} _ {7, 7} f _ {s s} \left(s _ {t + 1}, s _ {t}\right) = \left[ \begin{array}{c} 0 \\ 0 \\ 0 \end{array} \right],\]

\[\mathcal {D} _ {8, 8} f _ {s s} \left(s _ {t + 1}, s _ {t}\right) = \left[ \begin{array}{c} 0 \\ 0 \\ 0. 0 0 2 5 \end{array} \right], \mathcal {D} _ {9, 1 0} f _ {s s} \left(s _ {t + 1}, s _ {t}\right) = \left[ \begin{array}{c c} 1 & 0 \\ 0 & 0 \\ 0 & 0 \end{array} \right], \text {and} \mathcal {D} _ {1 1, 1 2} f _ {s s} \left(s _ {t + 1}, s _ {t}\right) = \left[ \begin{array}{c c} 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{array} \right]\]

for all and

In this example, we obtain the following nine solutions for

Mathematica Önds the nine solutions in less than three hundredths of a second. The only solution that produces a stable Örst-order approximation is (1) and hence there is a unique stable approximation. Given solution (1), we can solve the linear systems that allow us to obtain the second-order approximation. Let , and deÖne . The second-order approximation is

\[\begin{array}{r l r} & & {\left[ \begin{array}{c} \widehat {\hat {Y}} _ {t} \\ \hat {\Pi} _ {t} \\ \hat {R} _ {t} \end{array} \right] = \left[ \begin{array}{c c c} - 1. 9 2 8 2 & - 0. 0 0 6 2 & 0. 0 0 4 8 1 \\ - 0. 3 2 7 9 & - 0. 0 0 1 1 & 0. 0 0 0 1 4 \\ 0. 5 9 5 1 7 & 0. 0 0 1 9 1 & 0. 0 0 0 0 8 \end{array} \right] \mathcal {S} _ {t}} \\ & & {+ \frac {1}{2} \left[ \begin{array}{c c c c c c c c} 2 1. 3 7 7 1 & 0. 0 6 2 4 7 & - 0. 0 1 8 8 & 0. 0 6 2 4 7 & 0. 0 0 0 2 0 & - 0. 0 0 0 1 & - 0. 0 1 8 8 & 0. 0 0 0 2 0 & - 0. 0 0 0 4 \\ 0. 4 9 7 9 3 & 0. 0 0 0 5 6 & - 0. 0 0 0 8 & 0. 0 0 0 5 6 & 2 \times 1 0 ^ {- 6} & - 2 \times 1 0 ^ {- 6} & - 0. 0 0 0 8 & 2 \times 1 0 ^ {- 6} & - 0. 0 0 0 3 \\ - 0. 1 9 8 6 & 0. 0 0 1 2 4 & - 0. 0 0 0 4 & 0. 0 0 1 2 4 & 4 \times 1 0 ^ {- 6} & - 1 \times 1 0 ^ {- 6} & - 0. 0 0 0 4 & 4 \times 1 0 ^ {- 6} & - 0. 0 0 0 2 \end{array} \right] (\mathcal {S} _ {t} \otimes \mathcal {S} _ {t})} \end{array}\]

if and

\[\begin{array}{l} \left[ \begin{array}{c} \widehat {\tilde {Y}} _ {t} \\ \hat {\Pi} _ {t} \\ \hat {R} _ {t} \end{array} \right] = \left[ \begin{array}{c c c} - 2. 9 5 4 1 & - 0. 0 0 9 0 & - 0. 0 0 9 4 \\ - 0. 5 5 4 7 & - 0. 0 0 1 7 & - 0. 0 0 2 6 \\ 0. 6 9 9 4 1 & 0. 0 0 2 1 4 & - 0. 0 0 0 6 \end{array} \right] \mathcal {S} _ {t} \\ + \frac {1}{2} \left[ \begin{array}{c c c c c c c c} 5 6. 9 7 3 3 & 0. 1 6 4 8 7 & 0. 2 3 1 7 4 & 0. 1 6 4 8 7 & 0. 0 0 0 5 0 & 0. 0 0 0 7 1 & 0. 2 3 1 7 4 & 0. 0 0 0 5 0 & - 0. 0 0 1 6 \\ 0. 9 9 3 3 3 & 0. 0 0 1 3 6 & 0. 0 0 0 3 3 & 0. 0 0 1 3 6 & 4 \times 1 0 ^ {- 6} & 1 \times 1 0 ^ {- 6} & 0. 0 0 0 3 3 & 4 \times 1 0 ^ {- 6} & - 0. 0 0 1 0 \\ - 0. 1 8 4 2 & 0. 0 0 1 6 0 & - 0. 0 0 0 5 & 0. 0 0 1 6 0 & 5 \times 1 0 ^ {- 6} & - 2 \times 1 0 ^ {- 6} & - 0. 0 0 0 5 & 5 \times 1 0 ^ {- 6} & - 0. 0 0 0 2 \end{array} \right] (\mathcal {S} _ {t} \otimes \mathcal {S} _ {t}) \end{array}\]

if

Again, the Örst-order approximation is easily obtained from the above expressions. As in the RBC model, there is no certainty equivalence and the cross derivatives of g and h with respect to and other state variables evaluated at the steady-state are non-zero. Note that the Örst partial derivatives of g and h with respect to and are di§erent across regimes. This result occurs because a§ects the Örst derivatives of . Perturbing would make these derivatives equal and the approximation less accurate.

Accuracy: Euler Equation Errors The Euler equation error at the point is that for order , second_g,

\[E E ^ {o r d e r} \left(R _ {t - 1}, \varepsilon_ {t}, 1; s _ {t}\right) = 1 - \beta \int \sum_ {s ^ {\prime} = 1} ^ {2} p _ {s _ {t}, s ^ {\prime}} \times\]

\[\left( \begin{array}{c} \frac {1 - \frac {\kappa}{2} \left(\Pi^ {o r d e r} (R _ {t - 1} , \varepsilon_ {t} , 1 ; s _ {t}) - 1\right) ^ {2}}{1 - \frac {\kappa}{2} \left(\Pi^ {o r d e r} \left(R ^ {o r d e r} (R _ {t - 1} , \varepsilon_ {t} , 1 ; s _ {t}) , \varepsilon_ {t + 1} , 1 ; s _ {t + 1}\right) - 1\right) ^ {2}} \times \\ \frac {\tilde {Y} ^ {o r d e r} (R _ {t - 1} , \varepsilon_ {t} , 1 ; s _ {t})}{\tilde {Y} ^ {o r d e r} \left(R ^ {o r d e r} (R _ {t - 1} , \varepsilon_ {t} , 1 ; s _ {t}) , \varepsilon_ {t + 1} , 1 ; s _ {t + 1}\right)} \times \\ \frac {1}{\exp (\mu^ {\prime})} \frac {R ^ {o r d e r} (R _ {t - 1} , \varepsilon_ {t} , 1 ; s _ {t})}{\Pi^ {o r d e r} \left(R ^ {o r d e r} (R _ {t - 1} , \varepsilon_ {t} , 1 ; s _ {t}) , \varepsilon_ {t + 1} , 1 ; s _ {t + 1}\right)} \end{array} \right) \mu \left(\varepsilon^ {\prime}\right) d \varepsilon^ {\prime}\]

where , and are the approximations to the policy functions. The unconditional absolute Euler equation error is

\[E E ^ {o r d e r} = \sum_ {s _ {t}} \int \left| E E ^ {o r d e r} \left(R _ {t - 1}, \varepsilon_ {t}, 1; s _ {t}\right) \right| \mu^ {o r d e r} \left(R _ {t - 1}, \varepsilon_ {t}, 1; s _ {t}\right) d R _ {t - 1} d \varepsilon_ {t}\]

where is the unconditional distribution of the variables R and

Numerical approximation of this integral simulates a 10; 000-period path, with the Örst 1; 000 periods discarded as a burn-in. For each state along the simulated path, we draw 10,000 normally distributed to compute the expectation over , and use the transition values to compute the expectation over . This entire process takes about an hour.

The following table shows the base-10 logarithms of absolute Euler equation errors for the Örst-order and second-order approximations. Both the Örst-order and the second-order approximations produce a high degree of accuracy: a value of 4 implies an error of $1 for each $10,000 of consumption.

Unconditional Absolute Euler Equation Errors

Multiple Stable Approximations As an alternative parameter conÖguration, we consider the same parameters as in the previous section except . That is, the second regime now has a slightly lower response by the monetary authority to ináation. In this case, the steady-state is still the same. As before, Regime 1 would imply a unique stable approximation if considered in isolation, whereas Regime 2 would imply multiple stable approximations.

The Örst partial derivatives of f with respect to all its variables evaluated at the steady-state are the same as those in the previous section except

\[\mathcal {D} _ {3, 4} f _ {s s} \left(s _ {t + 1}, s _ {t}\right) = \left[ \begin{array}{c c} - 1. 1 1 1 1 1 & 0 \\ 1 0 & - 1 6 1. \\ 0 & 0. 2 \widehat {\psi} \left(s _ {t}\right) \end{array} \right]\]

for all and which depends on . Using these results, we obtain the following 9 solutions for

Now we have two solutions ñ(1) and (3) ñthat produce stable approximations. Thus the solution to the model has multiple stable approximations. This example illustrates how we can use reduced Grˆbner bases and the MSS criterion to determine the number of stable approximations.

6.3 A New-Keynesian Model with Habit

Now consider a slight variant of the previously discussed New-Keynesian model, but with the representative household having habit formation. We consider this application because deriving Örst-order and second-order approximations is more involved when habit formation is present and the reader would be interested in knowing how to accomplish this task. The household in this economy maximizes

\[\mathbb {E} _ {0} \sum_ {t = 0} ^ {\infty} \beta^ {t} \left(\log \left(C _ {t} - \varphi C _ {t - 1}\right) - H _ {t}\right)\]

where denotes habit persistence. Letting denote the Lagrange multiplier on the budget constraint, the Örst-order conditions are

\[\begin{array}{r} \lambda_ {t} = \frac {1}{C _ {t} - \varphi C _ {t - 1}} - \beta \mathbb {E} _ {t} \frac {\varphi}{C _ {t + 1} - \varphi C _ {t}}, \\ \lambda_ {t} = \beta \mathbb {E} _ {t} \lambda_ {t + 1} \frac {R _ {t}}{\Pi_ {t + 1}}, \\ \mathrm{and} 1 = \lambda_ {t} W _ {t}. \end{array}\]

Final good and intermediate goods producers face problems identical to the previous example and

\[\log A _ {t} = \mu_ {t} + \log A _ {t - 1}.\]

Thus, the Örmís optimality conditions are

\[\begin{array}{c} W _ {t} = m c _ {t} A _ {t} \\ \text {and} \kappa (\Pi_ {t} - 1) \Pi_ {t} = (1 - \eta) + \eta m c _ {t} + \beta \kappa \mathbb {E} _ {t} (\Pi_ {t + 1} - 1) \Pi_ {t + 1} \frac {\lambda_ {t + 1}}{\lambda_ {t}} \frac {Y _ {t + 1}}{Y _ {t}}. \end{array}\]

The resource constraint is

\[Y _ {t} = C _ {t} + \frac {\kappa}{2} \left(\Pi_ {t} - 1\right) ^ {2} Y _ {t}.\]

For clarity we assume that the monetary authority does not smooth interest rates and follows the rule

\[R _ {t} = R _ {s s} \Pi_ {t} ^ {\psi_ {t}} \exp (\sigma \varepsilon_ {t}).\]

With habits, consumption appears at di§erent dates, as , and , in the equilibrium conditions. Substituting out and , we have the equilibrium conditions as

\[\lambda_ {t} = \frac {1}{C _ {t} - \varphi C _ {t - 1}} - \beta \mathbb {E} _ {t} \frac {\varphi}{C _ {t + 1} - \varphi C _ {t}},\]

\[\lambda_ {t} = \beta \mathbb {E} _ {t} \lambda_ {t + 1} \frac {R _ {s s} \Pi_ {t} ^ {\psi_ {t}}}{\Pi_ {t + 1}},\]

\[\mathrm{and} \kappa (\Pi_ {t} - 1) \Pi_ {t} = (1 - \eta) + \frac {\eta}{A _ {t} \lambda_ {t}} + \beta \kappa \mathbb {E} _ {t} (\Pi_ {t + 1} - 1) \Pi_ {t + 1} \frac {\lambda_ {t + 1}}{\lambda_ {t}} \frac {C _ {t + 1}}{C _ {t}} \frac {1 - \frac {\kappa}{2} (\Pi_ {t} - 1) ^ {2}}{1 - \frac {\kappa}{2} (\Pi_ {t + 1} - 1) ^ {2}}.\]

The economy is nonstationary because of the presence of the unit root. If we deÖne and , we have the stationary equilibrium conditions as

\[\tilde {\lambda} _ {t} = \frac {1}{\tilde {C} _ {t} - \varphi \exp (- \mu_ {t}) \tilde {C} _ {t - 1}} - \beta \mathbb {E} _ {t} \frac {\varphi}{\tilde {C} _ {t + 1} \exp (\mu_ {t + 1}) - \varphi \tilde {C} _ {t}},\]

\[\tilde {\lambda} _ {t} = \beta \mathbb {E} _ {t} \frac {\tilde {\lambda} _ {t + 1}}{\exp (\mu_ {t + 1})} \frac {R _ {s s} \Pi_ {t} ^ {\psi_ {t}} \exp (\sigma \varepsilon_ {t})}{\Pi_ {t + 1}},\]

\[\mathrm{and} \kappa (\Pi_ {t} - 1) \Pi_ {t} = (1 - \eta) + \frac {\eta}{\tilde {\lambda} _ {t}} + \beta \kappa \mathbb {E} _ {t} (\Pi_ {t + 1} - 1) \Pi_ {t + 1} \frac {\tilde {\lambda} _ {t + 1}}{\tilde {\lambda} _ {t}} \frac {\tilde {C} _ {t + 1}}{\tilde {C} _ {t}} \frac {1 - \frac {\kappa}{2} (\Pi_ {t} - 1) ^ {2}}{1 - \frac {\kappa}{2} (\Pi_ {t + 1} - 1) ^ {2}}.\]

To solve the model, we deÖne the two auxiliary variables and . Using the notation in Section 2, we have and the equilibrium conditions are

\[\begin{array}{r l r} {f \left(\mathbf {y} _ {t + 1}, \mathbf {y} _ {t}, \mathbf {x} _ {t}, \mathbf {x} _ {t - 1}, \chi^ {\varepsilon_ {t + 1}}, \varepsilon_ {t}, \theta_ {t + 1}, \theta_ {t}\right)} & = & \\ & & {\frac {1}{\tilde {C} _ {t} - \varphi \exp (- \mu_ {t}) \tilde {C} _ {t - 1}} - \beta \frac {\varphi}{\tilde {X} _ {t + 1} \exp (\mu_ {t + 1}) - \varphi \tilde {C} _ {t}} - \tilde {\lambda} _ {t}} \\ & & {\beta \frac {\tilde {\lambda} _ {t + 1}}{\exp (\mu_ {t + 1})} \frac {R _ {s s} \Pi_ {t} ^ {\psi_ {t}} \exp (\sigma \varepsilon_ {t})}{\Pi_ {t + 1}} - \tilde {\lambda} _ {t}} \\ & & {(1 - \eta) + \frac {\eta}{\tilde {\lambda} _ {t}} + \beta \kappa \left(\Pi_ {t + 1} - 1\right) \Pi_ {t + 1} \frac {\tilde {\lambda} _ {t + 1}}{\tilde {\lambda} _ {t}} \frac {\tilde {X} _ {t + 1}}{\tilde {C} _ {t}} \frac {1 - \frac {\kappa}{2} (\Pi_ {t} - 1) ^ {2}}{1 - \frac {\kappa}{2} (\Pi_ {t + 1} - 1) ^ {2}} - \kappa \left(\Pi_ {t} - 1\right) \Pi_ {t + 1}} \\ & & {\tilde {X} _ {t} - \tilde {C} _ {t}} \end{array}\]

6.3.1 Solving the Model

Similar to the previous examples, the following subsections show how to solve the model: Önd the steady-state, deÖne the matrices of Örst partial derivatives of f with respect to all its variables evaluated at the steady-state, and Önd the second-order approximation to the policy functions. As in the basic New-Keynesian model, we consider two parameter conÖgurations to show how to use reduced Grˆbner bases to Önd all the approximations and how to use the MSS criterion to determine how many of them are stable.

Steady-State Calculating the steady-state involves setting and . Therefore, we have and . The equilibrium conditions in the steady-state are

\[\left[ \begin{array}{c} \frac {1}{\tilde {C} _ {s s} - \varphi \exp (- \bar {\mu}) \tilde {C} _ {s s}} - \beta \frac {\varphi}{\tilde {X} _ {s s} \exp (\bar {\mu}) - \varphi \tilde {C} _ {s s}} - \tilde {\lambda} _ {s s} \\ \beta \frac {\tilde {\lambda} _ {s s}}{\exp (\bar {\mu})} \frac {R _ {s s} \Pi_ {s s} ^ {\psi_ {t}}}{\Pi_ {s s}} - \tilde {\lambda} _ {s s} \\ (1 - \eta) + \frac {\eta}{\tilde {\lambda} _ {s s}} + \beta \kappa (\Pi_ {s s} - 1) \Pi_ {s s} \frac {\tilde {\lambda} _ {s s}}{\tilde {\lambda} _ {s s}} \frac {\tilde {X} _ {s s}}{\tilde {C} _ {s s}} \frac {1 - \frac {\kappa}{2} (\Pi_ {s s} - 1) ^ {2}}{1 - \frac {\kappa}{2} (\Pi_ {s s} - 1) ^ {2}} - \kappa (\Pi_ {s s} - 1) \Pi_ {s s} \\ \tilde {X} _ {s s} - \tilde {C} _ {s s} \\ \end{array} \right] = 0 _ {4 \times 1}.\]

Assuming , the steady-state satisÖes

\[\begin{array}{c} {R _ {s s} = \frac {\exp (\bar {\mu})}{\beta},} \\ {\tilde {\lambda} _ {s s} = \frac {\eta}{\eta - 1},} \\ {\mathrm{and} \tilde {C} _ {s s} = \tilde {X} _ {s s} = \frac {\exp (\bar {\mu}) - \beta \varphi}{\exp (\bar {\mu}) - \varphi} \frac {\eta - 1}{\eta}.} \end{array}\]

Partial Derivatives of In this example, , and . Thus, the Örst partial derivatives of with respect to all its variables evaluated at the steady-state are

\[\mathcal {D} _ {1, 3} f _ {s s} \left(s _ {t + 1}, s _ {t}\right) = \left[ \begin{array}{c c c} 0 & \frac {\varphi \beta \exp (\bar {\mu})}{\bar {C} _ {s s} ^ {2} (\exp (\bar {\mu}) - \varphi) ^ {2}} & 0 \\ - \lambda_ {s s} & 0 & 1 \\ \beta \kappa & 0 & 0 \\ 0 & 0 & 0 \end{array} \right], \mathcal {D} _ {4, 6} f _ {s s} \left(s _ {t + 1}, s _ {t}\right) = \left[ \begin{array}{c c c} 0 & 0 & - 1 \\ \lambda_ {s s} \psi \left(s _ {t}\right) & 0 & - 1 \\ - \kappa & 0 & - \frac {\eta}{\tilde {\lambda} _ {s s} ^ {2}} \\ 0 & 1 & 0 \end{array} \right]\]

\[\mathcal {D} _ {7, 7} f _ {s s} \left(s _ {t + 1}, s _ {t}\right) = \left[ \begin{array}{c} - \frac {\exp (2 \bar {\mu}) + \beta \varphi^ {2}}{\tilde {C} _ {s s} ^ {2} (\exp (\bar {\mu}) - \varphi) ^ {2}} \\ 0 \\ 0 \\ - 1 \end{array} \right], \mathcal {D} _ {8, 8} f _ {s s} \left(s _ {t + 1}, s _ {t}\right) = \left[ \begin{array}{c} \frac {\exp (\bar {\mu}) \varphi}{\tilde {C} _ {s s} ^ {2} (\exp (\bar {\mu}) - \varphi) ^ {2}} \\ 0 \\ 0 \\ 0 \end{array} \right], \mathcal {D} _ {9, 9} f _ {s s} \left(s _ {t + 1}, s _ {t}\right) = \left[ \begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \end{array} \right]\]

\[\mathcal {D} _ {1 0, 1 0} f _ {s s} \left(s _ {t + 1}, s _ {t}\right) = \left[ \begin{array}{c} 0 \\ \lambda_ {s s} \sigma \\ 0 \\ 0 \end{array} \right], \mathcal {D} _ {1 1, 1 2} f _ {s s} \left(s _ {t + 1}, s _ {t}\right) = \left[ \begin{array}{c c} \frac {\exp (\bar {\mu}) \beta \varphi}{\tilde {C} _ {s s} (\exp (\bar {\mu}) - \varphi) ^ {2}} & 0 \\ - \lambda_ {s s} & 0 \\ 0 & 0 \\ 0 & 0 \end{array} \right],\]

\[\mathrm{and} \mathcal {D} _ {1 3, 1 4} f _ {s s} = \left[ \begin{array}{c c} - \frac {\exp (\bar {\mu}) \varphi}{\tilde {C} _ {s s} (\exp (\bar {\mu}) - \varphi) ^ {2}} & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{array} \right]\]

for all and .

Unique Stable Approximation Consider the Örst parameter conÖguration

All parameters other than (the degree of habit formation) are similar to those in the previous New-Keynesian example. The steady-state values are and Consequently the numerical values of the matrices of Örst partial derivatives of with respect to all its variables evaluated at the steady-state are

\[\mathcal {D} _ {1, 3} f _ {s s} \left(s _ {t + 1}, s _ {t}\right) = \left[ \begin{array}{c c c} 0 & 9. 2 1 1 5 9 & 0 \\ - 1. 1 1 1 1 & 0 & 1 \\ 1 6 0. 6 1 4 & 0 & 0 \\ 0 & 0 & 0 \end{array} \right], \mathcal {D} _ {4, 6} f _ {s s} \left(s _ {t + 1}, s _ {t}\right) = \left[ \begin{array}{c c c} 0 & 0 & - 1 \\ 1. 1 1 1 1 1 \psi \left(s _ {t}\right) & 0 & - 1 \\ - 1 6 1. & 0 & - 8. 1 \\ 0 & 1 & 0 \end{array} \right],\]

\[\mathcal {D} _ {7, 7} f _ {s s} \left(s _ {t + 1}, s _ {t}\right) = \left[ \begin{array}{c} - 1 9. 6 7 3 1 \\ 0 \\ 0 \\ - 1 \end{array} \right], \mathcal {D} _ {8, 8} f _ {s s} \left(s _ {t + 1}, s _ {t}\right) = \left[ \begin{array}{c} 9. 2 3 3 7 5 \\ 0 \\ 0 \\ 0 \end{array} \right], \mathcal {D} _ {9, 9} f _ {s s} \left(s _ {t + 1}, s _ {t}\right) = \left[ \begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \end{array} \right],\]

\[\mathcal {D} _ {1 0, 1 0} f _ {s s} \left(s _ {t + 1}, s _ {t}\right) = \left[ \begin{array}{c} 0 \\ 0. 0 0 2 7 7 7 7 8 \\ 0 \\ 0 \end{array} \right], \mathcal {D} _ {1 1, 1 2} f _ {s s} \left(s _ {t + 1}, s _ {t}\right) = \left[ \begin{array}{c c} 8. 3 3 6 0 9 & 0 \\ - 1. 1 1 1 1 1 & 0 \\ 0 & 0 \\ 0 & 0 \end{array} \right],\]

\[\mathrm{and} \mathcal {D} _ {1 3, 1 4} f _ {s s} (s _ {t + 1}, s _ {t}) = \left[ \begin{array}{c c} - 8. 3 5 6 1 5 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \end{array} \right]\]

for all and .

Using these matrices, we can solve the quadratic system for In this example there are a total of 16 solutions. To conserve space, we report only:

Mathematica Önds 16 solutions in approximately 5 seconds. The only solution that produces a stable approximation is (1). Given Solution (1), we calculate the rest of the matrices needed to obtain the second-order approximation through solving systems of linear equations. Let , and deÖne . The

second-order approximation is

\[+ \left[ \begin{array}{c c c c c c c c} 0 & 0 & - 0. 0 0 2 0 & 0 & - 2 \times 1 0 ^ {- 7} & - 1 \times 1 0 ^ {- 6} & - 0. 0 0 2 0 & - 1 \times 1 0 ^ {- 6} & 5 \times 1 0 ^ {- 5} \\ 0 & 0 & 0. 0 0 0 2 0 & 0 & 2 \times 1 0 ^ {- 7} & - 4 \times 1 0 ^ {- 7} & 0. 0 0 0 2 0 & - 4 \times 1 0 ^ {- 7} & - 6 \times 1 0 ^ {- 5} \\ 0 & 0 & - 0. 0 0 2 0 & 0 & - 2 \times 1 0 ^ {- 7} & - 1 \times 1 0 ^ {- 6} & - 0. 0 0 2 0 & - 1 \times 1 0 ^ {- 6} & 5 \times 1 0 ^ {- 5} \\ 0 & 0. & 0. 0 0 1 4 3 & 0 & 6 \times 1 0 ^ {- 6} & - 1 \times 1 0 ^ {- 5} & 0. 0 0 1 4 3 & - 1 \times 1 0 ^ {- 5} & - 0. 0 0 0 3 \end{array} \right] (\mathcal {S} _ {t} \otimes \mathcal {S} _ {t})\]

for and

\[\left[ \begin{array}{c} \widehat {\tilde {C}} _ {t} \\ \hat {\Pi} _ {t} \\ \widehat {\tilde {X}} _ {t} \\ \widehat {\tilde {\lambda}} _ {t} \end{array} \right] = \left[ \begin{array}{c c c} 0. 6 9 6 5 1 & - 0. 0 0 0 2 & - 0. 0 0 0 5 \\ 0 & - 0. 0 0 0 1 & - 0. 0 0 3 3 \\ 0. 6 9 6 5 1 & - 0. 0 0 0 2 & - 0. 0 0 0 5 \\ 0 & 0. 0 0 2 6 1 & 0. 0 0 7 0 7 \end{array} \right] \mathcal {S} _ {t}\]

\[+ \left[ \begin{array}{c c c c c c c c} 0 & 0 & 0. 0 0 1 5 1 & 0 & - 2 \times 1 0 ^ {- 7} & 1 \times 1 0 ^ {- 6} & 0. 0 0 1 5 1 & 1 \times 1 0 ^ {- 6} & - 6 \times 1 0 ^ {- 5} \\ 0 & 0 & 0. 0 0 1 6 5 & 0 & 3 \times 1 0 ^ {- 7} & 6 \times 1 0 ^ {- 6} & 0. 0 0 1 6 5 & 6 \times 1 0 ^ {- 6} & - 0. 0 0 0 2 \\ 0 & 0 & 0. 0 0 1 5 1 & 0 & - 2 \times 1 0 ^ {- 7} & 1 \times 1 0 ^ {- 6} & 0. 0 0 1 5 1 & 1 \times 1 0 ^ {- 6} & - 6 \times 1 0 ^ {- 5} \\ 0 & 1 & 0. 0 0 1 5 6 & 0 & 7 \times 1 0 ^ {- 6} & 2 \times 1 0 ^ {- 5} & 0. 0 0 1 5 6 & 2 \times 1 0 ^ {- 5} & 0. 0 0 0 {3} {6} \end{array} \right] (\mathcal {S} _ {t} \otimes \mathcal {S} _ {t}).\]

for . Again, the Örst-order approximation can be easily obtained from the above expressions.

Accuracy: Euler Equation Errors in the previous examples, we compare the accuracies of the Örst-order and second-order approximations. The Euler equation error at point for an approximation order is

\[E E ^ {o r d e r} \left(\tilde {C} _ {t - 1}, \varepsilon_ {t}, 1; s _ {t}\right) = 1 - \beta \int \sum_ {s ^ {\prime} = 1} ^ {2} p _ {s _ {t}, s ^ {\prime}} \times\]

\[\left( \begin{array}{c} \beta \frac {\tilde {\lambda} ^ {o r d e r} (\tilde {C} ^ {o r d e r} (\tilde {C} _ {t - 1} , \varepsilon_ {t} , 1 ; s _ {t}) , \chi \varepsilon_ {t + 1} , 1 ; s _ {t + 1})}{\Pi^ {o r d e r} (\tilde {C} ^ {o r d e r} (\tilde {C} _ {t - 1} , \varepsilon_ {t} , 1 ; s _ {t}) , \chi \varepsilon_ {t + 1} , 1 ; s _ {t + 1})} \times \\ \frac {R _ {s s} \Pi^ {o r d e r} (\tilde {C} _ {t - 1} , \varepsilon_ {t} , 1 ; s _ {t}) ^ {\psi_ {t}} \exp (\sigma \varepsilon_ {t})}{\exp (\mu (s ^ {\prime})) \tilde {\lambda} ^ {o r d e r} (\tilde {C} _ {t - 1} , \varepsilon_ {t} , 1 ; s _ {t})} \end{array} \right) \mu (\varepsilon^ {\prime}) d \varepsilon^ {\prime}\]

where , and are the approximations to the policy functions. The unconditional absolute Euler equation error is

\[E E ^ {o r d e r} = \int \left| E E ^ {o r d e r} \left(\tilde {C} _ {t - 1}, \varepsilon_ {t}, \chi ; s _ {t}\right) \right| \mu^ {o r d e r} \left(\tilde {C} _ {t - 1}, \varepsilon_ {t}, \chi ; s _ {t}\right) d \tilde {C} _ {t - 1} d \varepsilon_ {t} d s _ {t}\]

where is the unconditional distribution of the variables implied by the approximated solution.

Numerical approximation of this integral simulates a 10; 000-period path, with the Örst 1; 000 periods discarded as a burn-in. For each state along the simulated path, we draw 10,000 normally distributed to compute the expectation over , and use the transition values to compute the expectation over . The entire process takes approximately an hour.

The table below reports the base-10 logarithms of absolute Euler equation errors for the Örst-order and the second-order approximations. Both the Örst-order and the second-order approximations produce a high degree of accuracy: a value of 3 implies an error of $1 for each $1,000.

Unconditional Absolute Euler Equation Errors

Multiple Stable Approximations As an alternative parameterization, consider the same parameters as above except that The second regime now has a slightly lower response by the monetary authority to ináation. The numerical values of the matrices of partial derivatives of with respect to all its variables evaluated at steady-state are unchanged except

\[\mathcal {D} _ {4, 6} f _ {s s} \left(s _ {t + 1}, s _ {t}\right) = \left[ \begin{array}{c c c} 0 & 0 & - 1 \\ 1. 1 1 1 1 1 \psi \left(s _ {t}\right) & 0 & - 1 \\ - 1 6 1. & 0 & - 8. 1 \\ 0 & 1 & 0 \end{array} \right]\]

for all and , which depends on . Using this new matrix as well as the other matrices, we solve the quadratic system for . There are 16 solutions to the quadratic system, and to conserve space we present only:

There are two solutions ñ(1) and (5) ñthat produce stable approximations. The examples in the previous New-Keynesian model and the current variant with habit might suggest that the

only parameter a§ecting the uniqueness of a stable approximation is , which governs the monetary authorityís response to ináation. This conclusion is incorrect.

Consider a higher degree of habit persistence, while keeping and . Given these parameter values, the steady-state is . There are 16 solutions to the quadratic system. The following table presents only:

\[D _ {1, 1} h _ {s s} (1)\]

\[D _ {1, 1} h _ {s s} (2)\]

There is only one solution, which is (1), that produces stable approximations. Remember that there are two stable approximations when we set as in the previous example.

7 Conclusion

Markov switching has been introduced as an essential ingredient to a large class of models usable for analyses of structural breaks and regime changes in policy, ranging from backward-looking models (Hamilton (1989) and Sims and Zha (2006)) to forward-looking rational expectations models (Clarida et al. (2000), Lubik and Schorfheide (2004), Davig and Leeper (2007), Farmer et al. (2011)). This paper expands this literature by developing a general methodology for constructing Örst-order and second-order approximations to the solutions of MSDSGE models. We resolve conceptual issues related to the steady-state and certainty equivalence of Örst-order approximations; we reduce the potentially very complicated problem to a task of solving a system of quadratic equations; we propose using Grˆbner bases to solve such a system; and we apply the MSS criterion to verify the existence and uniqueness of a stable approximation to the solution.

The contribution of this paper is not only theoretical but also practical. We show that after the quadratic system is successfully dealt with, obtaining up to second-order approximations is straightforward, as the remaining task involves Önding a solution to only a system of linear equations. Our application to three MSDSGE models illustrates the practical value of our methodology. It is our hope that the advance made in this paper enables applied researchers to estimate MSDSGE models by focusing on improving the e¢ ciency of our methods.

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8 Appendix A:

This appendix derives, in detail, the two steps described in Section 3.2.

8.1 Step 1: Obtaining the derivatives of

Taking Örst partial derivatives of G with respect to in (7) produces the expression for

\[\mathcal {D} _ {1, n _ {x}} \mathbb {G} _ {s s} (s _ {t}) = \sum_ {s ^ {\prime} = 1} ^ {n _ {s}} p _ {s _ {t}, s ^ {\prime}} \int \left( \begin{array}{c} \mathcal {D} _ {1, n _ {y}} f _ {s s} (s ^ {\prime}, s _ {t}) \mathcal {D} _ {1, n _ {x}} g _ {s s} (s ^ {\prime}) \mathcal {D} _ {1, n _ {x}} h _ {s s} (s _ {t}) \\ + \mathcal {D} _ {n _ {y} + 1, 2 n _ {y}} f _ {s s} (s ^ {\prime}, s _ {t}) \mathcal {D} _ {1, n _ {x}} g _ {s s} (s _ {t}) \\ + \mathcal {D} _ {2 n _ {y} + 1, 2 n _ {y} + n _ {x}} f _ {s s} (s ^ {\prime}, s _ {t}) \mathcal {D} _ {1, n _ {x}} h _ {s s} (s _ {t}) \\ + \mathcal {D} _ {2 n _ {y} + n _ {x} + 1, 2 (n _ {y} + n _ {x})} f _ {s s} (s ^ {\prime}, s _ {t}) \end{array} \right) \mu (\varepsilon^ {\prime}) d \varepsilon^ {\prime}\]

for all . Taking into account, one can simplify the above expression to

\[\mathcal {D} _ {1, n _ {x}} \mathbb {G} _ {s s} \left(s _ {t}\right) = \sum_ {s ^ {\prime} = 1} ^ {n _ {s}} p _ {s t, s ^ {\prime}} \left( \begin{array}{c} \mathcal {D} _ {1, n _ {y}} f _ {s s} \left(s ^ {\prime}, s _ {t}\right) \mathcal {D} _ {1, n _ {x}} g _ {s s} \left(s ^ {\prime}\right) \mathcal {D} _ {1, n _ {x}} h _ {s s} \left(s _ {t}\right) \\ + \mathcal {D} _ {n _ {y} + 1, 2 n _ {y}} f _ {s s} \left(s ^ {\prime}, s _ {t}\right) \mathcal {D} _ {1, n _ {x}} g _ {s s} \left(s _ {t}\right) \\ + \mathcal {D} _ {2 n _ {y} + 1, 2 n _ {y} + n _ {x}} f _ {s s} \left(s ^ {\prime}, s _ {t}\right) \mathcal {D} _ {1, n _ {x}} h _ {s s} \left(s _ {t}\right) \\ + \mathcal {D} _ {2 n _ {y} + n _ {x} + 1, 2 (n _ {y} + n _ {x})} f _ {s s} \left(s ^ {\prime}, s _ {t}\right) \end{array} \right)\]

for all . Rearranging the above expression for each leads to

\[\begin{array}{c} \mathcal {D} _ {1, n _ {x}} \mathbb {G} _ {s s} \left(s _ {t}\right) = \\ \sum_ {s ^ {\prime} = 1} ^ {n _ {s}} p _ {s _ {t}, s ^ {\prime}} \left( \begin{array}{c} \left(\mathcal {D} _ {1, n _ {y}} f _ {s s} \left(s ^ {\prime}, s _ {t}\right) \mathcal {D} _ {1, n _ {x}} g _ {s s} \left(s ^ {\prime}\right) + \mathcal {D} _ {2 n _ {y} + 1, 2 n _ {y} + n _ {x}} f _ {s s} \left(s ^ {\prime}, s _ {t}\right)\right) \mathcal {D} _ {1, n _ {x}} h _ {s s} \left(s _ {t}\right) \\ + \mathcal {D} _ {n _ {y} + 1, 2 n _ {y}} f _ {s s} \left(s ^ {\prime}, s _ {t}\right) \mathcal {D} _ {1, n _ {x}} g _ {s s} \left(s _ {t}\right) + \mathcal {D} _ {2 n _ {y} + n _ {x} + 1, 2 (n _ {y} + n _ {x})} f _ {s s} \left(s ^ {\prime}, s _ {t}\right) \end{array} \right). \end{array}\tag{17}\]

Putting together the versions of (17), one for each value of and equating them to zero as implied by (??) yields a quadratic system of equations. The number of equations is exactly the same as the number of unknowns . Writing (17) in matrix form leads to a system of quadratic equations expressed in (9).

8.2 Step 2: Obtaining the derivatives of and

8.2.1 Obtaining the derivatives of

By taking Örst partial derivatives with respect to in (7) we obtain the expression for

\[\sum_ {s ^ {\prime} = 1} ^ {n _ {s}} p _ {s _ {t}, s ^ {\prime}} \int \left( \begin{array}{c} \mathcal {D} _ {n _ {x} + 1, n _ {x} + n _ {\varepsilon}} \mathbb {G} _ {s s} \left(s _ {t}\right) = \\ \mathcal {D} _ {1, n _ {y}} f _ {s s} \left(s ^ {\prime}, s _ {t}\right) \mathcal {D} _ {1, n _ {x}} g _ {s s} \left(s ^ {\prime}\right) \mathcal {D} _ {n _ {x} + 1, n _ {x} + n _ {\varepsilon}} h _ {s s} \left(s _ {t}\right) + \\ \mathcal {D} _ {n _ {y} + 1, 2 n _ {y}} f _ {s s} \left(s ^ {\prime}, s _ {t}\right) \mathcal {D} _ {n _ {x} + 1, n _ {x} + n _ {\varepsilon}} g _ {s s} \left(s _ {t}\right) + \\ \mathcal {D} _ {2 n _ {y} + 1, 2 n _ {y} + n _ {x}} f _ {s s} \left(s ^ {\prime}, s _ {t}\right) \mathcal {D} _ {n _ {x} + 1, n _ {x} + n _ {\varepsilon}} h _ {s s} \left(s _ {t}\right) + \\ \mathcal {D} _ {2 (n _ {y} + n _ {x}) + n _ {\varepsilon} + 1, 2 (n _ {y} + n _ {x} + n _ {\varepsilon})} f _ {s s} \left(s ^ {\prime}, s _ {t}\right) \end{array} \right) \mu \left(\varepsilon^ {\prime}\right) d \varepsilon^ {\prime}\]

for all . With , this expression simpliÖes to

\[\sum_ {s ^ {\prime} = 1} ^ {n _ {s}} p _ {s _ {t}, s ^ {\prime}} \left( \begin{array}{c} \mathcal {D} _ {n _ {x} + 1, n _ {x} + n _ {\varepsilon}} \mathbb {G} _ {s s} \left(s _ {t}\right) = \\ \left(\mathcal {D} _ {1, n _ {y}} f _ {s s} \left(s ^ {\prime}, s _ {t}\right) \mathcal {D} _ {1, n _ {x}} g _ {s s} \left(s ^ {\prime}\right) + \mathcal {D} _ {2 n _ {y} + 1, 2 n _ {y} + n _ {x}} f _ {s s} \left(s ^ {\prime}, s _ {t}\right)\right) \mathcal {D} _ {n _ {x} + 1, n _ {x} + n _ {\varepsilon}} h _ {s s} \left(s _ {t}\right) \\ \mathcal {D} _ {n _ {y} + 1, 2 n _ {y}} f _ {s s} \left(s ^ {\prime}, s _ {t}\right) \mathcal {D} _ {n _ {x} + 1, n _ {x} + n _ {\varepsilon}} g _ {s s} \left(s _ {t}\right) \\ \mathcal {D} _ {2 (n _ {y} + n _ {x}) + n _ {\varepsilon} + 1, 2 (n _ {y} + n _ {x} + n _ {\varepsilon})} f _ {s s} \left(s ^ {\prime}, s _ {t}\right) \end{array} \right)\tag{18}\]

for all .

Putting together the versions of (18), one for each value of , and equating them to zero as implied by (??) yields a system of equations. The number of equations is the same as that of unknowns . The system is linear and can be written in matrix form as

\[\left[ \begin{array}{c c} \Theta_ {\varepsilon} & \Phi_ {\varepsilon} \end{array} \right] \left[ \begin{array}{c} \mathcal {D} _ {n _ {x} + 1, n _ {x} + n _ {\varepsilon}} g _ {s s} (1) \\ \vdots \\ \mathcal {D} _ {n _ {x} + 1, n _ {x} + n _ {\varepsilon}} g _ {s s} (n _ {s}) \\ \mathcal {D} _ {n _ {x} + 1, n _ {x} + n _ {\varepsilon}} h _ {s s} (1) \\ \vdots \\ \mathcal {D} _ {n _ {x} + 1, n _ {x} + n _ {\varepsilon}} h _ {s s} (n _ {s}) \end{array} \right] = \Psi_ {\varepsilon}.\tag{19}\]

The solution to this system is given in (10).

8.2.2 Obtaining the derivatives of

We obtain the expression for by taking Örst partial derivatives with respect to in (7)

\[\sum_ {s ^ {\prime} = 1} ^ {n _ {s}} p _ {s _ {t}, s ^ {\prime}} \int \left( \begin{array}{c} \mathcal {D} _ {n _ {x} + n _ {\varepsilon} + 1} \mathbb {G} _ {s s} \left(s _ {t}\right) = \\ \mathcal {D} _ {1, n _ {y}} f _ {s s} \left(s ^ {\prime}, s _ {t}\right) \left[ \begin{array}{c} \mathcal {D} _ {1, n _ {x}} g _ {s s} \left(s ^ {\prime}\right) \mathcal {D} _ {n _ {x} + n _ {\varepsilon} + 1} h _ {s s} \left(s _ {t}\right) \\ + \mathcal {D} _ {n _ {x} + 1, n _ {x} + n _ {\varepsilon}} g _ {s s} \left(s ^ {\prime}\right) \varepsilon^ {\prime} + \mathcal {D} _ {n _ {x} + n _ {\varepsilon} + 1} g _ {s s} \left(s ^ {\prime}\right) \end{array} \right] + \\ \mathcal {D} _ {n _ {y} + 1, 2 n _ {y}} f _ {s s} \left(s ^ {\prime}, s _ {t}\right) \mathcal {D} _ {n _ {x} + n _ {\varepsilon} + 1} g _ {s s} \left(s _ {t}\right) + \\ \mathcal {D} _ {2 n _ {y} + 1, 2 n _ {y} + n _ {x}} f _ {s s} \left(s ^ {\prime}, s _ {t}\right) \mathcal {D} _ {n _ {x} + n _ {\varepsilon} + 1} h _ {s s} \left(s _ {t}\right) + \\ \mathcal {D} _ {2 n _ {y} + n _ {x} + 1, 2 n _ {y} + n _ {x} + n _ {\varepsilon}} f _ {s s} \left(s ^ {\prime}, s _ {t}\right) \varepsilon^ {\prime} + \\ \mathcal {D} _ {2 (n _ {y} + n _ {x} + n _ {\varepsilon}) + 1, 2 (n _ {y} + n _ {x} + n _ {\varepsilon}) + n _ {\theta}} f _ {s s} \left(s ^ {\prime}, s _ {t}\right) \mathcal {D} \theta_ {s s} \left(s ^ {\prime}\right) + \\ \mathcal {D} _ {2 (n _ {y} + n _ {x} + n _ {\varepsilon}) + n _ {\theta} + 1, 2 (n _ {y} + n _ {x} + n _ {\varepsilon} + n _ {\theta})} f _ {s s} \left(s ^ {\prime}, s _ {t}\right) \mathcal {D} \theta_ {s s} \left(s _ {t}\right) \end{array} \right) \mu \left(\varepsilon^ {\prime}\right) d \varepsilon^ {\prime}\]

for all . Note that is the derivative of with respect to evaluated at That is,

\[\mathcal {D} \theta_ {s s} (s _ {t}) = \mathcal {D} \theta (0, s _ {t}) = [ \mathcal {D} _ {j} \theta^ {i} (0, s _ {t}) ] _ {1 \leq i \leq n _ {\theta}, j = 1}\]

for all .

Using the two equalities and , one can simplify the expression for as

\[\sum_ {s ^ {\prime} = 1} ^ {n _ {s}} p _ {s _ {t}, s ^ {\prime}} \left( \begin{array}{c} \mathcal {D} _ {n _ {x} + n _ {\varepsilon} + 1} \mathbb {G} _ {s s} (s _ {t}) = \\ \mathcal {D} _ {1, n _ {y}} f _ {s s} (s ^ {\prime}, s _ {t}) \left\{\mathcal {D} _ {1, n _ {x}} g _ {s s} (s ^ {\prime}) \mathcal {D} _ {n _ {x} + n _ {\varepsilon} + 1} h _ {s s} (s _ {t}) + \mathcal {D} _ {n _ {x} + n _ {\varepsilon} + 1} g _ {s s} (s ^ {\prime}) \right\} + \\ \mathcal {D} _ {n _ {y} + 1, 2 n _ {y}} f _ {s s} (s ^ {\prime}, s _ {t}) \mathcal {D} _ {n _ {x} + n _ {\varepsilon} + 1} g _ {s s} (s _ {t}) + \mathcal {D} _ {2 n _ {y} + 1, 2 n _ {y} + n _ {x}} f _ {s s} (s ^ {\prime}, s _ {t}) \mathcal {D} _ {n _ {x} + n _ {\varepsilon} + 1} h _ {s s} (s _ {t}) \\ + \mathcal {D} _ {2 (n _ {y} + n _ {x} + n _ {\varepsilon}) + 1, 2 (n _ {y} + n _ {x} + n _ {\varepsilon}) + n _ {\theta}} f _ {s s} (s ^ {\prime}, s _ {t}) \mathcal {D} \theta_ {s s} (s ^ {\prime}) + \\ \mathcal {D} _ {2 (n _ {y} + n _ {x} + n _ {\varepsilon}) + n _ {\theta} + 1, 2 (n _ {y} + n _ {x} + n _ {\varepsilon} + n _ {\theta})} f _ {s s} (s ^ {\prime}, s _ {t}) \mathcal {D} \theta_ {s s} (s _ {t}) \end{array} \right)\tag{20}\]

for all .

Putting together the versions of (20), one for each value of and equating them to zero as implied by (??), yields a system of linear equations. The number of these equations is the same as that of unknowns . The linear

system can be written in matrix form as

\[\left[ \begin{array}{c c} \Theta_ {\chi} & \Phi_ {\chi} \end{array} \right] \left[ \begin{array}{c} \mathcal {D} _ {n _ {x} + n _ {\varepsilon} + 1} g _ {s s} (1) \\ \vdots \\ \mathcal {D} _ {n _ {x} + n _ {\varepsilon} + 1} g _ {s s} (n _ {s}) \\ \mathcal {D} _ {n _ {x} + n _ {\varepsilon} + 1} h _ {s s} (1) \\ \vdots \\ \mathcal {D} _ {n _ {x} + n _ {\varepsilon} + 1} h _ {s s} (n _ {s}) \end{array} \right] = \Psi_ {\chi}.\tag{21}\]

The solution to (21) is given in (10).

9 Appendix B: Grˆbner bases

In this appendix we give an overview of Grˆbner bases and describe how they can be applied to our problem. See Becker et al. (1998) for a more detailed description and other applications.

We wish to Önd all the solutions of a system of n polynomials in n variables. Let the polynomial system under study be

\[\begin{array}{r c l} f _ {1} (x _ {1}, \ldots , x _ {n}) & = & 0, \\ & \vdots \\ f _ {n} (x _ {1}, \ldots , x _ {n}) & = & 0. \end{array}\]

Each equation in this system deÖnes a manifold of dimension in and the set of solutions of the system is the intersection of these manifolds.

When all the are linear, the solution set consists of a linear subspace of . It is well known that there are three possible outcomes: a unique solution, no solutions, or inÖnitely many solutions. More importantly, the set of linear systems with no solution or inÖnitely many solutions is of measure zero in the set of all linear systems. When there is a unique solution, it can be easily found.

When the are higher-order polynomials, the solution set is more complicated, but the intuition from the linear case still holds. For almost all polynomial systems of n equations in n variables, there are only Önitely many solutions and these solutions can be easily found.

To describe how solutions are computed for polynomial systems, we need to develop the concept of an ideal and its Grˆbner basis. Given a set of polynomials in n variables, 2 the ideal generated by is the set of all polynomials of the form

\[g _ {1} (x _ {1}, \dots , x _ {n}) f _ {1} (x _ {1}, \dots , x _ {n}) + \dots + g _ {m} (x _ {1}, \dots , x _ {n}) f _ {m} (x _ {1}, \dots , x _ {n})\tag{22}\]

where vary over all polynomials in n variables. We denote this ideal by For our purpose we focus on one important feature of an ideal. The point is a zero of the polynomials if and only if it is a zero of every polynomial in the ideal . This feature implies that if two di§erent sets of polynomials generate the same ideal, then they have the same zeros. The goal is to Önd a generating set for which it is easy to compute zeros.

Before giving the deÖnition of a Grˆbner basis, we must Örst deÖne what we mean by the leading term of a polynomial. A polynomial in is a sum of terms of the form , where is a non-negative integer and c is a non-zero real number. The product is called a monomial in . The degree of a term is the sum of its exponents, . For polynomials in a single variable, the leading term is deÖned to be the one of highest degree. For polynomials of several variables, there may be many terms of the same degree. Thus, one deÖnes the leading term relative to a monomial ordering. For instance, the lexicographical ordering of monomials implies that if and only if there is an i such that and for . In general, a monomial order must satisfy

1. The monomial is the smallest monomial.

2. If X, Y , and Z are monomials with , then

The leading term of a polynomial is the largest term with respect to the monomial ordering. With these notions in hand, we are ready to deÖne a Grˆbner basis. The set is a Grˆbner basis for the ideal if

1.

2. The leading term of any polynomial in is divisible by the leading term of for some i.

Consider the following example. Consider the ideal generated by

\[\{2 x _ {1} x _ {2} - x _ {1}, x _ {2} - x _ {3}, x _ {3} ^ {2} - 1 \}.\]

Note that the Örst term of each polynomial is the leading term with respect to the lexicographical order. Is this generating set a Grˆbner basis? The answer is negative because the leading term in is not divisible by any of the leading terms in the generating set.

As an illustration, we show how to use Buchbergerís Algorithm to construct a Grˆbner basis. There are many other algorithms, most of which are based on Buchbergerís Algorithm, that can also be used to construct Grˆbner bases. The algorithm begins with constructing S-polynomials. The polynomial is called the S-polynomial of and because factors and were chosen so that the leading terms would cancel. After the S-polynomial has been formed, it must be reduced using the elements of the generating set. The reduction step is illustrated by the following example.

Consider the S-polynomial of and , which is . This polynomial is reduced by using the leading terms in the generating set to eliminate terms in the S-polynomial. In this case the reduction proceeds as follows:

\[\begin{array}{r c l} x _ {1} x _ {2} - \frac {1}{2} x _ {1} x _ {3} ^ {2} & \Rightarrow & (x _ {1} x _ {2} - \frac {1}{2} x _ {1} x _ {3} ^ {2}) - \frac {1}{2} (2 x _ {1} x _ {2} - x _ {1}) = - \frac {1}{2} x _ {1} x _ {3} ^ {2} + \frac {1}{2} x _ {1} \\ - \frac {1}{2} x _ {1} x _ {3} ^ {2} + \frac {1}{2} x _ {1} & \Rightarrow & (- \frac {1}{2} x _ {1} x _ {3} ^ {2} + \frac {1}{2} x _ {1}) + \frac {1}{2} x _ {1} (x _ {3} ^ {2} - 1) = 0 \end{array}\]

So the reduction of the S-polynomial of and gives the zero polynomial. Readers should convince themselves that the S-polynomial of and given above cannot be reduced further and that the S-polynomial of and can be reduced to zero. Note that the reduction is in general not unique. It can depend on the order in which the terms are eliminated and on particular elements of the generating set that are used to eliminate the terms. One can always devise an algorithm to reduce any polynomial in Önite steps.

Buchbergerís Algorithm proceeds as follows. Successively form the S-polynomials from pairs of polynomials in the generating set and reduce them. If a reduced non-zero S-polynomial is obtained, add it to the generating set. Continue until all S-polynomials formed from pairs of the enlarged generating set can be reduced to zero. This algorithm is guaranteed to terminate in a Grˆbner basis. See Buchberger (1998) or Becker et al. (1998) for details.

Continuing with our example, the reduced S-polynomial of and is We add it to our generating set to obtain

\[\{2 x _ {1} x _ {2} - x _ {1}, x _ {2} - x _ {3}, x _ {3} ^ {2} - 1, x _ {1} x _ {3} - \frac {1}{2} x _ {1} \}.\]

As discussed above, the S-polynomials of both the pair and and the pair and are zero. Note also that the S-polynomial of and reduces to zero, but the S-polynomial of and is and reduces to

\[\begin{array}{r c l} \frac {1}{2} x _ {1} x _ {2} - x _ {1} x _ {3} ^ {2} & \Rightarrow & (\frac {1}{2} x _ {1} x _ {2} - x _ {1} x _ {3} ^ {2}) - \frac {1}{4} (2 x _ {1} x _ {2} - x _ {1}) = - x _ {1} x _ {3} ^ {2} + \frac {1}{4} x _ {1} \\ - x _ {1} x _ {3} ^ {2} + \frac {1}{4} x _ {1} & \Rightarrow & (- x _ {1} x _ {3} ^ {2} + \frac {1}{4} x _ {1}) + x _ {1} (x _ {3} ^ {2} - 1) = - \frac {3}{4} x _ {1}. \end{array}\]

We add this non-zero polynomial to our generating set to obtain

\[\{2 x _ {1} x _ {2} - x _ {1}, x _ {2} - x _ {3}, x _ {3} ^ {2} - 1, 2 x _ {1} x _ {3} - x _ {1}, - \frac {3}{4} x _ {1} \}.\]

The reader should verify that all S-polynomials of pairs from this generating set will reduce to zero. Thus we have obtained a Grˆbner basis.

Grˆbner bases are not unique, because adding any element from the ideal generated by a Grˆbner basis will result in another Grˆbner basis. To obtain uniqueness, with respect to the monomial ordering, we work with a reduced Grˆbner basis. A Grˆbner basis is said to be reduced if

1. The coe¢ cient of the leading term of each polynomial in the basis is one.

2. Each polynomial in the basis cannot be further reduced with respect to the other polynomials in the basis.

Any Grˆbner basis can be easily transformed to a reduced Grˆbner basis by Örst reducing each polynomial in the basis with respect to the other polynomials in the basis and then dividing the resultant leading term by the leading coe¢ cient. For instance, the Grˆbner basis obtained above is not reduced because both and can be reduced to zero. Thus these polynomials must be eliminated to obtain the reduced Grˆbner basis

\[\{x _ {1}, x _ {2} - x _ {3}, x _ {3} ^ {2} - 1 \}.\]

The reduced basis above is called a Shape basis because it is of the form

\[\{x _ {1} - q _ {1} (x _ {n}), \dots , x _ {n - 1} - q _ {n - 1} (x _ {n}), q _ {n} (x _ {n}) \},\]

where are polynomials in a single variable with the degree of strictly less than the degree of for . Shape bases are particularly useful because it is straightforward to Önd all the zeros from this representation. One Örst Önds the values of that are zeros of and then substitutes each of these values into through to obtain the values of through

Not all reduced Grˆbner bases are Shape bases. The Shape lemma, below, gives the conditions under which the reduced Grˆbner basis is a Shape basis.

Lemma 11 Let be polynomials in . The reduced Grˆbner basis with respect to the lexicographical ordering of the ideal is a Shape basis and only if the following conditions hold.

1. The system has only Önitely many zeros.

2. If and are two distinct zeros, then

3. Each zero is either a simple point or a multiple point of local dimension one.

4. If a zero is a multiple point, then the tangent line at the zero does not contain the hyperplane

The meaning of Conditions 1 and 2 is clear, but Conditions 3 and 4 need further explanation. The point is a zero of the polynomial system if and only if . If there exists an i such that

\[\left\langle f _ {1}, \ldots , f _ {n} \right\rangle \subseteq \left\langle x _ {1} - a _ {1}, \ldots , (x _ {i} - a _ {i}) ^ {2}, \ldots , x _ {n} - a _ {n} \right\rangle \subset \left\langle x _ {1} - a _ {1}, \ldots , x _ {n} - a _ {n} \right\rangle ,\]

then we say the zero is a multiple point; otherwise, the zero is simple. One can verify that the zero is a multiple point if and only if there exists an i such that for all . The tangent space at the zero is the set of all such that

\[\left[ \begin{array}{c c c} \frac {\partial f _ {1}}{\partial x _ {1}} | _ {(a _ {1}, \ldots , a _ {n})} & \dots & \frac {\partial f _ {1}}{\partial x _ {n}} | _ {(a _ {1}, \ldots , a _ {n})} \\ \vdots & \ddots & \vdots \\ \frac {\partial f _ {n}}{\partial x _ {1}} | _ {(a _ {1}, \ldots , a _ {n})} & \dots & \frac {\partial f _ {n}}{\partial x _ {n}} | _ {(a _ {1}, \ldots , a _ {n})} \end{array} \right] \left[ \begin{array}{c} x _ {1} \\ \vdots \\ x _ {n} \end{array} \right] = 0.\]

Note that this matrix of partial derivatives is the Jacobian. If the zero is simple, then this deÖnition of the tangent space corresponds to our usual geometric notion of a tangent space, but the correspondence breaks down if the zero is a multiple point. The local dimension of a zero is the dimension of the tangent space. Note that the local dimension is zero if and only if the Jacobian is of full rank. Thus, if the Jacobian is of full rank, then the zero will be simple. The converse, however, is not necessarily true.

One can verify that if the reduced Grˆbner basis is a Shape basis, then Conditions 1-4 will hold. The converse is also true, but the veriÖcation requires much more work. See Becker et al. (1993) for details. If the Jacobian at each zero is of full rank, then each zero is isolated and there can only be Önitely many zeros. Thus, Conditions 1-4 hold in this case. Since the set of polynomial systems of n equations in n unknowns whose Jacobian is not of full rank is of measure zero in the set of all such systems, Conditions 1-4 hold almost surely.

In summary, for almost all polynomial systems, there are only Önitely many zeros and Buch bergerís Algorithm can be used to Önd them. While Buchbergerís Algorithm is instructive, it can be ine¢ cient for certain problems. Active research continues to develop variants of this algorithm that improve e¢ ciency.

10 Appendix C: Iterative algorithm

This section describes an iterative procedure to Önd solutions of the quadratic system (9). In general, this method will not provide us with all the solutions to the system. Recall the quadratic system to be solved is

\[A \left(s _ {t}\right) \left[ \begin{array}{c} I _ {n _ {x}} \\ \mathcal {D} _ {1, n _ {x}} g _ {s s} (1) \\ \vdots \\ \mathcal {D} _ {1, n _ {x}} g _ {s s} \left(n _ {s}\right) \end{array} \right] \mathcal {D} _ {1, n _ {x}} h _ {s s} \left(s _ {t}\right) = B \left(s _ {t}\right) \left[ \begin{array}{c} I _ {n _ {x}} \\ \mathcal {D} _ {1, n _ {x}} g _ {s s} \left(s _ {t}\right) \end{array} \right]\]

for all , where

\[A \left(s _ {t}\right) = \left[ \begin{array}{l l l l} \sum_ {s ^ {\prime} = 1} ^ {n _ {s}} p _ {s _ {t}, s ^ {\prime}} \mathcal {D} _ {2 n _ {y} + 1, 2 n _ {y} + n _ {x}} f _ {s s} \left(s ^ {\prime}, s _ {t}\right) & p _ {s _ {t}, 1} \mathcal {D} _ {1, n _ {y}} f _ {s s} \left(1, s _ {t}\right) & \dots & p _ {s _ {t}, n _ {s}} \mathcal {D} _ {1, n _ {y}} f _ {s s} \left(n _ {s}, s _ {t}\right) \end{array} \right]\]

and

\[B \left(s _ {t}\right) = - \sum_ {s ^ {\prime} = 1} ^ {n _ {s}} p _ {s _ {t}, s ^ {\prime}} \left[ \begin{array}{c c} \mathcal {D} _ {2 n _ {y} + n _ {x} + 1, 2 (n _ {y} + n _ {x})} f _ {s s} \left(s ^ {\prime}, s _ {t}\right) & \mathcal {D} _ {n _ {y} + 1, 2 n _ {y}} f _ {s s} \left(s ^ {\prime}, s _ {t}\right) \end{array} \right]\]

for all

The idea of the algorithm is to guess a set of policy functions and solve for each regimeís policy functions as in the constant parameter model case using the generalized Schur decomposition. When the solutions are close to the guesses, then a solution has been found.

Algorithm 12 Let denote solution at iteration .

1. Set and initialize and

2. For each , construct the transformed system

\[\begin{array}{r l} & A \left(s _ {t}, \left\{\mathcal {D} _ {1, n _ {x}} g _ {s s} ^ {(j - 1)} (s ^ {\prime}) \right\} _ {s ^ {\prime} = 1, s ^ {\prime} \neq s _ {t}} ^ {n _ {s}}\right) \left[ \begin{array}{c} I _ {n _ {x}} \\ \mathcal {D} _ {1, n _ {x}} g _ {s s} ^ {(j)} (s _ {t}) \end{array} \right] \mathcal {D} _ {1, n _ {x}} h _ {s s} ^ {(j)} (s _ {t}) \\ & \quad = B (s _ {t}) \left[ \begin{array}{c} I _ {n _ {x}} \\ \mathcal {D} _ {1, n _ {x}} g _ {s s} ^ {(j)} (s _ {t}) \end{array} \right] \end{array}\tag{23}\]

where

\[\begin{array}{c} A \left(s _ {t}, \left\{\mathcal {D} _ {1, n _ {x}} g _ {s s} ^ {(j)} \left(s ^ {\prime}\right) \right\} _ {s ^ {\prime} = 1, s ^ {\prime} \neq s _ {t}} ^ {n _ {s}}\right) = \\ \left[ \left( \begin{array}{c} \sum_ {s ^ {\prime} = 1} ^ {n _ {s}} p _ {s _ {t}, s ^ {\prime}} \mathcal {D} _ {2 n _ {y} + 1, 2 n _ {y} + n _ {x}} f _ {s s} \left(s ^ {\prime}, s _ {t}\right) \\ + \sum_ {s ^ {\prime} = 1, s ^ {\prime} \neq s _ {t}} ^ {n _ {s}} p _ {s _ {t}, s ^ {\prime}} \mathcal {D} _ {1, n _ {y}} f _ {s s} \left(s ^ {\prime}, s _ {t}\right) \mathcal {D} _ {1, n _ {x}} g _ {s s} ^ {(0)} \left(s ^ {\prime}\right) \end{array} \right) p _ {s _ {t}, s _ {t}} \mathcal {D} _ {1, n _ {y}} f _ {s s} \left(s _ {t}, s _ {t}\right) \right] \end{array}\]

3. The set of systems (23) is identical to a constant parameter model case. When Önding , use the "most stable" generalized eigenvalues, those with the smallest modulus.

4. Check max If yes, then stop and check for MSS. If no, then set and return to step 2.

11 Appendix D: Second partial derivatives

This appendix derives, in detail, the six steps described in Section 5.2. Let us Örst deÖne

\[\begin{array}{r l r} \mathcal {D} _ {k} \mathcal {D} _ {j} f _ {s s} ^ {i} (s _ {t + 1}, s _ {t}) & = \\ & & \mathcal {D} _ {k} \mathcal {D} _ {j} f ^ {i} (\mathbf {y} _ {s s}, \mathbf {y} _ {s s}, \mathbf {x} _ {s s}, \mathbf {x} _ {s s}, 0 _ {n _ {\varepsilon}}, 0 _ {n _ {\varepsilon}}, \overline {{\theta}} _ {1}, \widehat {\theta} _ {2} (s _ {t + 1}), \overline {{\theta}} _ {1}, \widehat {\theta} _ {2} (s _ {t})) \end{array}\]

for all and and and and

\[\mathcal {H} _ {n _ {1}, n _ {2}; m _ {1}, m _ {2}} f _ {s s} ^ {i} \left(s _ {t + 1}, s _ {t}\right) = \left[ \begin{array}{c c c} \mathcal {D} _ {n _ {1}} \mathcal {D} _ {m _ {1}} f _ {s s} ^ {i} \left(s _ {t + 1}, s _ {t}\right) & \ldots & \mathcal {D} _ {n _ {1}} \mathcal {D} _ {m _ {2}} f _ {s s} ^ {i} \left(s _ {t + 1}, s _ {t}\right) \\ \vdots & \ddots & \vdots \\ \mathcal {D} _ {n _ {2}} \mathcal {D} _ {m _ {1}} f _ {s s} ^ {i} \left(s _ {t + 1}, s _ {t}\right) & \ldots & \mathcal {D} _ {n _ {2}} \mathcal {D} _ {m _ {2}} f _ {s s} ^ {i} \left(s _ {t + 1}, s _ {t}\right) \end{array} \right]\]

for all and and

11.1 Step 1: Obtaining the derivatives of twice

The Örst equation is the derivative with respect to twice.

\[\begin{array} { r l }{ \mathcal { H } _ { 1 , n _ { x } ; 1 , n _ { x } } \mathbb { G } _ { s s } ^ { i } ( s _ { t } ) }& = \sum _ { s ^ { \prime } = 1 } ^ { n _ { s } } p _ { s _ { t } , s ^ { \prime } } \times\\&{ } \left( ( \mathcal { D } _ { 1 , n _ { x } } g _ { s s } ( s ^ { \prime } ) \mathcal { D } _ { 1 , n _ { x } } h _ { s s } ( s _ { t } ) ) ^ { \intercal } \left( \left(\begin{array} { c }\mathcal { H } _ { 1 , n _ { y } ; 1 , n _ { y } } f _ { s s } ^ { i } ( s ^ { \prime } , s _ { t } ) \mathcal { D } _ { 1 , n _ { x } } g _ { s s } ( s ^ { \prime } )\\+ 2 \mathcal { H } _ { 1 , n _ { y } ; 2 n _ { y } + 1 , 2 n _ { y } + n _ { x } } f _ { s s } ^ { i } ( s ^ { \prime } , s _ { t } )\end{array}\right) \mathcal { D } _ { 1 , n _ { x } } h _ { s s } ( s _ { t } ) \right. \right.\\&{ } \left. + 2 \mathcal { H } _ { 1 , n _ { y } ; n _ { y } + 1 , 2 n _ { y } } f _ { s s } ^ { i } ( s ^ { \prime } , s _ { t } ) \mathcal { D } _ { 1 , n _ { x } } g _ { s s } ( s _ { t } ) + 2 \mathcal { H } _ { 1 , n _ { y } ; 2 n _ { y } + n _ { x } + 1 , 2 ( n _ { y } + n _ { x } ) } f _ { s s } ^ { i } ( s ^ { \prime } , s _ { t } ) \right)\\&{ } + \mathcal { D } _ { 1 , n _ { x } } g _ { s s } ( s _ { t } ) ^ { \intercal } \left(\begin{array} { c }\mathcal { H } _ { n _ { y } + 1 , 2 n _ { y } ; n _ { y } + 1 , 2 n _ { y } } f _ { s s } ^ { i } ( s ^ { \prime } , s _ { t } ) \mathcal { D } _ { 1 , n _ { x } } g _ { s s } ( s _ { t } )\\+ 2 \mathcal { H } _ { n _ { y } + 1 , 2 n _ { y } ; 2 n _ { y } + 1 , 2 n _ { y } + n _ { x } } f _ { s s } ^ { i } ( s ^ { \prime } , s _ { t } ) \mathcal { D } _ { 1 , n _ { x } } h _ { s s } ( s _ { t } )\\+ 2 \mathcal { H } _ { n _ { y } + 1 , 2 n _ { y } ; 2 n _ { y } + n _ { x } + 1 , 2 ( n _ { y } + n _ { x } ) } f _ { s s } ^ { i } ( s ^ { \prime } , s _ { t } )\end{array}\right)\\& + \mathcal { D } _ { 1 , n _ { x } } h _ { s s } ( s _ { t } ) ^ { \intercal } \left( \right.\left(\begin{array} { c }\mathcal { H } _ { 2 n _ { y } + 1 , 2 n _ { y } + n _ { x } ; 2 n _ { y } + 1 , 2 n _ { y } + n _ { x } } f _ { s s } ^ { i } ( s ^ { \prime } , s _ { t } )\\+ \mathcal { D } _ { 1 , n _ { y } } f _ { s s } ^ { i } ( s ^ { \prime } , s _ { t} ) \mathcal { H } _ { 1 , n _ { x } ; 1 , n _ { x } } g _ { s s } ( s ^ { \prime} )\\+ 2 \mathcal { H } _ { 2 n _ { y } + 1 , 2 n _ { y } + n _ { x } ; 2 n _ { y } + n _ { x } + 1 , 2 ( n _ { y } + n _ { x} ) } f _ { s s } ^ { i } ( s ^ { \prime } , s _ { t} )\end{array}\right)\\&+ \binom{ \mathcal { D } _ { 1 , n _ { y } } f _ { s s } ^ { i } ( s ^ { \prime } , s _ { t} ) \mathcal { D } _ { 1 , n _ { x } } g _ { s s } ( s ^ { \prime} )}{+ \mathcal { D } _ { 2 n _ { y } + 1 , 2 n _ { y } + n _ { x } } f _ { s s } ^ { i } ( s ^ { \prime} , s _ { t} )} \mathcal { H } _ { 1 , n _ { x } ; 1 , n _ { x } } h _ { s s } ( s _ { t})\\&+ \mathcal { D } _ { n _ { y } + 1 , 2 n _ { y } } f _ { s s } ^ { i } ( s ^ { \prime} , s _ { t} ) \mathcal { H } _ { 1 , n _ { x} ; 1 , n _ { x} } g _ { s s } ( s _ { t} )\\&+ \mathcal { H } _ { 2 n _ { y } + n _ { x } + 1 , 2 ( n _ { y } + n _ { x} ) ; 2 n _ { y } + n _ { x } + 1 , 2 ( n _ { y } + n _ { x} ) } f _ { s s } ^ { i } ( s ^ { \prime} , s _ { t} )\end{array}\]

\[\begin{array}{r l r} & {\mathrm{Thecondition} \mathcal {H} _ {1, n _ {x}; 1, n _ {x}} \mathbb {G} _ {s s} ^ {i} (s _ {t}) = 0 _ {n _ {x} \times n _ {x}} \mathrm{for} 1 \leq i \leq n _ {y} + n _ {x} \mathrm{impliesalinearsystemofequationsthatdeterminesthesolutionfor}} \\ & {\left\{\left\{\mathcal {H} _ {1, n _ {x}; 1, n _ {x}} g _ {s s} ^ {i} (s _ {t}) \right\} _ {i = 1} ^ {n _ {y}}, \left\{\mathcal {H} _ {1, n _ {x}; 1, n _ {x}} h _ {s s} ^ {i} (s _ {t}) \right\} _ {i = 1} ^ {n _ {x}} \right\} _ {s _ {t} = 1} ^ {n _ {s}}.} \end{array}\]

11.2 Step 2: Obtaining the derivatives of and

The second equation is the derivative with respect to and .

\[\begin{array} { r l } \mathcal { H } _ { 1 , n _ { x } ; n _ { x } + 1 , n _ { x } + n _ { \varepsilon } } \mathbb { G } _ { s s } ^ { i } \left( s _ { t } \right) & = \sum _ { s ^ { \prime } = 1 } ^ { n _ { s } } p _ { s _ { t } , s ^ { \prime } } \times \\ & \left( \begin{array} { c } ( \mathcal { D } _ { 1 , n _ { x } } g _ { s s } \left( s ^ { \prime } \right) \mathcal { D } _ { 1 , n _ { x } } h _ { s s } \left( s _ { t } \right) ) ^ { \intercal } \\ \end{array} \right) \left( \begin{array} { c } \left( \begin{array} { c } \mathcal { H } _ { 1 , n _ { y } ; 1 , n _ { y } } f _ { s s } ^ { i } \left( s ^ { \prime } , s _ { t } \right) \mathcal { D } _ { 1 , n _ { x } } g _ { s s } \left( s ^ { \prime } \right) \\ + 2 \mathcal { H } _ { 1 , n _ { y } ; 2 n _ { y } + 1 , 2 n _ { y } + n _ { x } } f _ { s s } ^ { i } \left( s ^ { \prime } , s _ { t } \right) \end{array} \right) \mathcal { D } _ { n _ { x } + 1 , n _ { x } - n _ { \varepsilon } } \\ + \mathcal { H } _ { 1 , n _ { y } ; n _ { y } + 1 , 2 n _ { y } } f _ { s s } ^ { i } \left( s ^ { \prime } , s _ { t } \right) \mathcal { D } _ { n _ { x } + 1 , n _ { x } + n _ { \varepsilon } } g _ { s s } \left( s ^ { \prime } , s _ { t } \right) \\ + \mathcal { H } _ { 1 , n _ { y } ; 2 ( n _ { y } + n _ { x } ) + n _ { \varepsilon } + 1 , 2 ( n _ { x } + n _ { y } + n _ { \varepsilon } ) } f _ { s s } ^ { i } \left( s ^ { \prime } , s _ { t } \right) \\ + \mathcal { D } _ { 1 , n _ { x } } g _ { s s } \left( s _ { t } \right) ^ { \intercal } \\ & + \mathcal { D } _ { 1 , n _ { x } } g _ { s s } \left( s _ { t } \right) ^ { \intercal } \\ & + \mathcal { D } _ { 1 , n _ { x } } h _ { s s } \left( s _ { t } \right) ^ { \intercal } \\ & + \mathcal { D } _ { 1 , n _ { x } } h _ { s s } \left( s _ { t } \right) ^ { \intercal } \\ & + \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 & + \mathcal { H } _ 2 n _ { y } + 1 , 2 n _ { y } + n _ { x } ; 2 n _ { y } + 1 , 2 n _ { y } f _ { s s } ^ { i } ( s ^ { \prime } , s _ { t} ) d t \\ + \mathcal { H } _ 2 n _ { y } + 1 , 2 n _ { y } + n _ { x } ; 2 ( n _ { y } + n _ x ) + n _ { \varepsilon } + 1 , 2 ( n _ { x } + n _ { y } + n _ \varepsilon ) ) f _ { s s } ^ { i } ( s ^ { \prime } , s _ { t} ) d t \\ + \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 + d \\ + d & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & 6 \\ + d & & & & & & 3 & 3 & 3 & 3 & 3 & 3 & 3 & 3 & 3 & 3 & 3 & 3 & 3 & 3 & 3 & 3 & 3 & 3 & 3 & 3 & 3 & 3 & 3 \\ + d & & b & b & b & b & b & b & b & b & b & b & b & b & b & b & b & b & b & b & b & b & b & b & b & b & b & b \\ + d & b & a & a & a & a & a & a & a & a & a & a & a & a & a & a & a & a & a & a & a & a & a & a & a \\ + d & b & e & e & e & e & e & e & e & e & e & e & e & e & e & e & e & e & e & e & e & e & e & e \\ + d & b & f & f & f & f & f & f & f & f & f & f & f & f & f & f & f & f & f & f & f & f & f & f \\ + d & b & g & g & g & g & g & g & g & g & g & g & g & g & g & g & g & g & g & g & g & g \\ + d & b & h & h & h & h & h & h & h & h & h & h & h & h & h & h \\ + d \\ + d \\ + d \\ + d \\ + d \\ + d \\ + d \\ + d \\ + d \\ + d \\ + d \\ + d \\ + d \\ + d \\ + d \\ + d \\ + d \\ + d \\ + d \\ + d \\ + d \\ + d \\ + d \\ + d \\ + d \\ + d \\ + d \\ + d \\ + d \\ + d \\ + d \\ + d \\ + d \\ + d / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / . \\ + d \\ - d \\ - d \\ - d \\ - d \\ - d \\ - d \\ - d \\ - d \\ - d \\ - 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 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | - k|k|k|k|k|k|k|k|k|k|k|k|k|k|k|k|k|k|k|k|k|k|k|k|k|k|k|k|k|k|k|k|k|k|k|k|k|k|k|k|k|k|k|k|k|k|k|k|k|k|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|\ldots\\ k\end{array} \right)\]

Conditional on the solution from Step 1, the condition for implies a linear system of equations that determines the solution

\[\left\{\left\{\mathcal {H} _ {1, n _ {x}; n _ {x} + 1, n _ {x} + n _ {\varepsilon}} g _ {s s} ^ {i} (s _ {t}) \right\} _ {i = 1} ^ {n _ {y}}, \left\{\mathcal {H} _ {1, n _ {x}; n _ {x} + 1, n _ {x} + n _ {\varepsilon}} h _ {s s} ^ {i} (s _ {t}) \right\} _ {i = 1} ^ {n _ {x}} \right\} _ {s _ {t} = 1} ^ {n _ {s}}.\]

11.3 Step 3: Obtaining the derivatives of and

The third equation is the derivative with respect to and

\[\begin{array} { r l } \mathcal { H } _ { 1 , n _ { x } ; n _ { x } + n _ { \varepsilon } + 1 } \mathbb { G } _ { s s } ^ { i } \left( s _ { t } \right) & = \sum _ { s ^ { \prime } = 1 } ^ { n _ { s } } p _ { s _ { t } , s ^ { \prime } } \times \\ & \left( ( \mathcal { D } _ { 1 , n _ { x } } g _ { s s } \left( s ^ { \prime } \right) \mathcal { D } _ { 1 , n _ { x } } h _ { s s } \left( s _ { t } \right) ) ^ { \intercal } \left( \begin{array} { c } \mathcal { H } _ { 1 , n _ { y } ; 1 , n _ { y } } f _ { s s } ^ { i } \left( s ^ { \prime } , s _ { t } \right) \binom { \mathcal { D } _ { n _ { x } + n _ { \varepsilon } + 1 } g _ { s s } \left( s ^ { \prime } \right) } { + \mathcal { D } _ { 1 , n _ { x } } g _ { s s } \left( s ^ { \prime } \right) \mathcal { D } _ { n _ { x } + n _ { \varepsilon } + 1 } h _ { s s } \left( s _ { t } \right) } \\ + \mathcal { H } _ 1 , n _ { y } ; n _ { y } + 1 , 2 n _ { y } f _ { s s } ^ { i } \left( s ^ { \prime } , s _ { t } \right) \mathcal { D } _ { n _ { x } + n _ { \varepsilon } + 1 } g _ { s s } \left( s _ { t } \right) \\ + \mathcal { H } _ 1 , n _ { y } ; 2 n _ { y } + 1 , 2 n _ { y } + n _ { x } f _ { s s } ^ { i } \left( s ^ { \prime } , s _ { t } \right) \mathcal { D } _ { n _ { x } + n _ { \varepsilon } + 1 } h _ { s s } \left( s _ { t } \right) \\ + \mathcal { H } _ 1 , n _ { y } ; 2 ( n _ { x } + n _ { y } + n _ { \varepsilon } ) + 1 , 2 ( n _ { x } + n _ { y } + n _ { \varepsilon } ) + n _ { \theta } f _ { s s } ^ { i } \left( s ^ { \prime } , s _ { t } \right) \mathcal { D } \theta _ { s s } \left( s ^ { \prime } \right) \\ + \mathcal { H } _ 1 , n _ { y } ; 2 ( n _ { x } + n _ { y } + n _ { \varepsilon } + n _ { \theta } ) + 1 , 2 ( n _ { x } + n _ { y } + n _ { \varepsilon } + n _ { \theta } ) f _ { s s } ^ { i } \left( s ^ { \prime } , s _ { t } \right) \mathcal { D } \theta _ { s s } \left( s _ { t } \right) ) \\ & \\ + \mathcal { D } _ { 1 , n _ { x } } h _ { s s } \left( s _ { t } \right) ^ { \intercal } \left( \begin{array} { c } \mathcal { H } _ 2 n _ { y } + 1 , 2 n _ { y } + n _ { x } ; 1 , n _ { y } f _ { s s } ^ { i } \left( s ^ { \prime } , s _ { t } \right) \binom { \mathcal { D } _ { n _ { x } + n _ { \varepsilon } + 1 } g _ { s s } \left( s ^ { \prime } \right) } + \mathcal { D } _ { 1 , n _ { x } } g _ \textit ~s ~ s ~ s ~ ( ~ s ~ ^ { \prime }) ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ - 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 ,~ - 1 ,~ - 1 ,~ - 1 ,~ - 1 ,~ - 1 ,~ - 1 ,~ - 1 ,\\ + \mathcal { H } _ 2 n _ { y } + 1 , 2 n _ { y } + n _ { x}; 2 n _ { y } + 1 , 2 n _ { y } + n _ { x} f _ { s s} ^ { i } \left( s ^ { \prime } , s _ { t } \right) \mathcal { D } _ { n _ { x } + n _ { \varepsilon } + 1 } h _ { s s } \left( s _ { t } \right) \\ + \mathcal { H } _ 2 n _ { y } + 1 , 2 n _ { y } + n _ { x}; 2 n _ { y } + 1 , 2 n _ { y } + n _ { x} f _ { s s} ^ { i} \left( s ^ { \prime } , s _ { t} \right) \mathcal { D } _ { n _ { x } + n _ { \varepsilon } + 1 } h _ { s s } \left( s _ { t} \right) \\ + \mathcal { H } _ 2 n _ { y } + 1 , 2 n _ { y } + n _ { x}; 2 ( n _ { x } + n _ { y } + n _ { \varepsilon} ) + 1 , 2 ( n _ { x } + n _ { y } + n _ { \varepsilon} ) + n _ { \theta } f _ { s s} ^ { i} \left( s ^ { \prime } , s _ { t} \right) \mathcal { D} \theta _ { s s} \left( s ^ { \prime} \right) \\ + \mathcal { H } _ 2 n _ { y } + 1 , 2 n _ { y } + n _ { x}; 2 ( n _ { x } + n _ { y } + n _ { \varepsilon} ) + n _ { \theta } + 1 , 2 ( n _ { x } + n _ { y } + n _ { \varepsilon} + n _ { \theta} ) f _ { s s} ^ { i} \left( s ^ { \prime } , s _ { t} \right) \mathcal { D} \theta _ { s s} \left( s ^ { t} \right) \\ & \\ {\mathcal D}_{1, n_{y}} f_{s_{s}}^{i}\left( (s^{\prime},s_{t})\binom{ {\mathcal H}_{1, n_x;n_x+n_x+1}g_{ss}\left( s^{\prime} \\+ {\mathcal H}_{1, n_x;1, n_x}g_{ss}\left( s^{\prime} \right)\mathcal D_{n_x+n_x+1}h_{ss}\left( s_ t\right) }\end{array} ) \\ & \\ + {\mathcal D}_{1, n_{x}} g_{s s}\left( s_{t}\right) ^{ \intercal }\left( \begin{array}{c}\mathcal H_{n_{y}+ 1,2n_{y};1,n_{y}}f_{s_{s}}^{i}\left( s^{\prime },s_{t}\right)\binom {\mathcal D}_{n_{x}+n_{\varepsilon}+1}g_{s_{s}}\left( s^{\prime} \\+ {\mathcal D}_{1,n_{x}}g_{s_{s}}\left( s^{\prime} \right)\mathcal D_{n_{x}+n_{\varepsilon}+1}h_{s_{s}}\left( s_ t\right)\end{array} )\\ + {\mathcal H}_{n_{y}+ 1,2n_{y};n_{y}+ 1,2n_{y}}f_{s_{s}}^{i}\left( s^{\prime },s_{t}\right)\mathcal D_{n_{x}+n_{\varepsilon}+1}g_{s_{s}}\left( s_ t\right)\\ + {\mathcal H}_n_{y}+ 1,2n_{y};2n_{y}+ 1,2n_{y}+n_{x}f_{s_{s}}^{i}\left( s^{\prime },s_ t\right)\mathcal D_{n_{x}+n_{\varepsilon}+1}h_{s_{s}}\left( s_ t\right)\\ + {\mathcal H}_{n_{y}+ 1,2n_{y};2(n_{x}+n_{y}+n_{\varepsilon})+ 1,2(n_{x}+n_{y}+n_{\varepsilon})+n_{\theta }}f_{s_{s}}^{i}\left( s^{\prime },s_ t\right)\mathcal D_{\theta_{s}s}\left( s^{\prime}\right)\\ + {\mathcal H}_n_{y}+ 1,2n_{y};2(n_{x}+n_{y}+n_{\varepsilon}+n_{\theta})+ 1,2(n_{x}+n_{y}+n_{\varepsilon}+n_{\theta})f_{s_s}^{i}\left( s^{\prime },s_ t\right)\mathcal D_{\theta_{s}s}\left( s_ t\right)\\ + {\mathcal H}_n_{y}+ 1,2n_{y};2(n_{x}+n_{y}+n_{\varepsilon}+n_{\theta})+ 1,2(n_{x}+n_{y}+n_{\varepsilon}+n_{\theta})f_{s_s}^{i}\left( s^{\prime },s_ t\right)\mathcal D_{0.5}(s_ t)\end{array} ) \\ & \\ & + {\mathcal H}_{2n_{{y}}+n_{{x}}+1,2(n_{{x}}+n_{{y}});1,n_{{y}}}f_{s{s}}^{i}\left(s^{\prime },s_ {{t}}\right)\binom{{\mathcal D}_{n_{{x}}+n_{{\varepsilon}}+1}g_{s{s}}\left(s^{\prime}\right)}{+ {\mathcal D}_{1,n_x}g_{s{s}}\left(s^{\prime}\right)\mathcal D_{n_x+n_x+1}h_{s{s}}\left(s_ {{t}}\right)}\\ + {\mathcal H}_{2n_{{y}}+n_{{x}}+1,2(n_{{y}}+n_{{x}});n_{{y}}+1,2n_{{y}}}f_{s{s}}^{i}\left(s^{\prime },s_ {{t}}\right)\mathcal D_{n_x+n_x+1}g_{s{s}}\left(s_ {{t}}\right)\\ + {\mathcal H}_{2n_{{y}}+n_{{x}}+1,2(n_{{y}}+n_{{x}});2n_{{y}}+1,2n_{{y}}+n_{{x}}f_{s{s}}^{i}\left(s^{\prime },s_ {{t}}\right)\mathcal D_{n_x+n_x+1}h_{s{s}}\left(s_ {{t}}\right)\\ + {\mathcal H}_{2n_{{y}}+n_{{x}}+1,2(n_{{x}}+n_{{y}});2(n_{{x}}+n_{{y}}+n_{{\varepsilon}})+1,2(n_{{x}}+n_{{y}}+n_{{\varepsilon}})+n_{{\theta }}}f_{s{s}}^{i}\left(s^{\prime },s_ {{t}}\right)\mathcal D_{0.5}(s^{\prime})\\ & \\ & + {\mathcal H}_{2n_{{y}}+n_{{x}}+1,2(n_{{x}}+n_{{y}});2(n_{{x}}+n_{{y}}+n_{{\varepsilon}})+n_{{\theta }}+1,2(n_{{x}}+n_{{y}}+n_{{\varepsilon}}+n_{{\theta }})}f_{s{s}}^{i}\left(s^{\prime },s_ {{t}}\right)\theta_ {\chi}\left(s\right)\\ & \\ & + {\Bigg ( }\binom{\mathcal D}_{-l_n_y f_s^i}\binom{s^{\prime},s_t}{+\mathcal D}_{-l_n_y f_s^i}\binom{s^{\prime},s_t}{+\mathcal D}_{-l_n_y f_s^i}\binom{s^{\prime},s_t}{+\mathcal D}_{-l_n_y f_s^i}\binom{s^{\prime},s_t}{+\mathcal D}_{-l_n_y f_s^i}\binom{s^{\prime},s_t}{+\mathcal D}_ {-l_n_y f_s^i}\binom{s^{\prime},s_t}{+\mathcal D}_ {-l_n_y f_s^i}\binom{s^{\prime},s_t}{+\mathcal D}_ {-l_n_y f_s^i}\binom{s^{\prime},s_t}{+\mathcal D}_ {-l_n_y f_s^i}\binom{s^{\prime},s_t}{+\mathcal D}_ {-l_n_y F_s^i}\binom{s^{\prime},s_t}{+\mathcal D}_ {-l_n_y F_s^i}\binom{s^{\prime},s_t}{+\mathcal D}_ {-l_n_y F_s^i}\binom{s^{\prime},s_t}{+\mathcal D}_ {-l_n_y F_s^i}\binom{s^{\prime},s_t}{+\mathcal D}_ {-l_n_y F_s^i}\\ & \\ & + {\mathcal D}_{-l_n_y f_s^i}\binom{s^{\prime},s_t}{+\mathcal D}_{-l_n_y f_s^i}\binom{s^{\prime},s_t}{+\mathcal D}_{-l_n_y f_s^i}\binom{s^{\prime},s_t}{+\mathcal D}_{-l_n_y F_s^i}\binom{s^{\prime},s_t}{+\mathcal D}_{-l_n_y F_s^i}\binom{s^{\prime},s_t}{+\mathcal D}_{-l_n_y F_s^i}\binom{s^{\prime},s_t}{+\mathcal D}_{-l_n_y F_s^i}\binom{s^{\prime},s_t}{+\mathcal D}_{-l_n_y F_s^i}\\ & \\ & + {\mathcal D}_{-l_n_y f_s^i}\binom{s^{\prime},s_t}{+\mathcal D}_{-l_n_y f_s^i}\binom{s^{\prime},s_t}{+\mathcal D}_{-l_n_y F_s^i}\binom{s^{\prime},s_t}{+\mathcal D}_{-l_n_y F_s^i}\binom{s^{\prime},s_t}{+\mathcal D}_ {-l_n_y f_s^i}\binom{s^{\prime},s_t}{+\mathcal D}_ {-l_n_y F_s^i}\binom{s^{\prime},s_t}{+\mathcal D}_ {-l_n_y F_s^i}\binom{s^{\prime},s_t}{+\mathcal D}_ {-l_n_y F_s^i}\\ & \\ & + {\mathcal D}_{-l_n_y f_s^i}\binom{s^{\ell},s_t}{+\mathcal D}_{-l_n_y f_s^i}\binom{s^{\ell},s_t}{+\mathcal D}_{-l_n_y F_s^i}\binom{s^{\ell},s_t}{+\mathcal D}_{- l_n_y F_s^i}\binom{s^{\ell},s_t}{+\mathcal D}_{- l_n_y F_s^i}\binom{s^{\ell},s_t}{+\mathcal D}_{- l_n_y F_s^i}\binom{s^{\ell},s_t}{+\mathcal D}_{- l_n_y F_s^i}\binom{s^{\ell},s_t}{+\mathcal D}_{- l_n_y F_S^n}\binom{s^{\ell},s_t}{+\mathcal D}_{- l_n_y F_S^n}\binom{s^{\ell},s_t}{+\mathcal D}_{- l_n_y F_S^n}\binom{s^{\ell},s_t}{+\mathcal D}_{- l_n_y F_S^n}\binom{s^{\ell},s_t}{+\mathcal D}_{- l_n_y F_S^n}\binom{s^{\ell}-s_t}{+\mathcal D}_{- l_n_y F_S^n}\binom{s^{\ell}-s_t}{+\mathcal D}_{- l_n_y F_S^n}\binom{s^{\ell}-s_t}{+\mathcal D}_{- l_n_y F_S^n}\binom{s^{\ell}-s_t}{+\mathcal D}_{- l_n_y F_S^n}\binom{s^{\ell}-s_t}{+\mathcal D}_ {- l_n_y F_S^n}\binom{s^{\ell}-s_t}{+\mathcal D}_ {- l_n_y F_S^n}\binom{s^{\ell}-s_t}{+\mathcal D}_ {- l_n_y F_S^n}\binom{s^{\ell}-s_t}{+\mathcal D}_ {- l_n_y F_S^n}\binom{s^{\ell}-s_t}{+\mathcal D}_ {- l_n_y F_S^n}\sin(\frac{n_m}{m})}\\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & \\ & / .\\\]

Given the solution from Step 2, the condition for implies a linear system of equations that determines the solution

11.4 Step 4: Obtaining the derivatives of twice

The fourth equation is the derivative with respect to twice.

\[\left( \begin{array} { c } \mathcal { H } _ { n _ { x } + 1 , n _ { x } + n _ { \varepsilon } ; n _ { x } + 1 , n _ { x } + n _ { \varepsilon } } \mathbb { G } _ { s s } ^ { i } \left( s _ { t } \right) = \sum _ { s ^ { \prime } = 1 } ^ { n _ { s } } p _ { s _ { t } , s ^ { \prime } } \times \\ [ \mathcal { D } _ { 1 , n _ { x } } g _ { s s } \left( s ^ { \prime } \right) \mathcal { D } _ { n _ { x } + 1 , n _ { x } + n _ { \varepsilon } } h _ { s s } \left( s _ { t } \right) ] ^ { \intercal } \left[ \begin{array} { c } \left[ \begin{array} { c } \mathcal { H } _ { 1 , n _ { y } ; 1 , n _ { y } } f _ { s s } ^ { i } \left( s ^ { \prime } , s _ { t } \right) \mathcal { D } _ { 1 , n _ { x } } g _ { s s } \left( s ^ { \prime } \right) \\ + 2 \mathcal { H } _ { 1 , n _ { y } ; 2 n _ { y } + 1 , 2 n _ { y } + n _ { x } } f _ { s s } ^ { i } \left( s ^ { \prime } , s _ { t } \right) \end{array} \right] \mathcal { D } _ { n _ { x } + 1 , n _ { x } + n _ { \varepsilon } } h _ { s s } \left( s _ { t } \right) \\ + 2 \mathcal { H } _ { 1 , n _ { y } ; n _ { y } + 1 , 2 n _ { y } } f _ { s s } ^ { i } \left( s ^ { \prime } , s _ { t } \right) \mathcal { D } _ { n _ { x } + 1 , n _ { x } + n _ { \varepsilon } } g _ { s s } \left( s _ { t } \right) \\ + 2 \mathcal { H } _ { 1 , n _ { y } ; 2 ( n _ { y } + n _ { x } ) + n _ { \varepsilon } + 1 , 2 ( n _ { x } + n _ { y } + n _ { \varepsilon } ) } f _ { s s } ^ { i } \left( s ^ { \prime } , s _ { t } \right) \\ \end{array} \right) \\ + \mathcal { D } _ { n _ { x } + 1 , n _ { x } + n _ { \varepsilon } } h _ { s s } \left( s _ { t } \right) ^ {\intercal} \left[ \begin{array} { c } \left[ \begin{array} { c } \mathcal { H } _ { 2 n _ { y } + 1 , 2 n _ { y } + n _ { x } ; 2 n _ { y } + 1 , 2 n _ { y } + n _ { x } } f _ { s s } ^ { i } \left( s ^ { \prime } , s _ { t } \right) \\ + \mathcal { D } _ { 1 , n _ { y } } f _ { s s } ^ { i } \left( s ^ { \prime } , s _ { t} \right) \mathcal { H } _ { 1 , n _ { x } ; 1 , n _ { x } } g _ { s s } \left( s ^ { \prime} \right) \\ + 2 \mathcal { H } _ { 2 n _ { y } + 1 , 2 n _ { y } + n _ { x } ; 2 ( n _ { x } + n _ { y } ) + n _ { \varepsilon } + 1 , 2 ( n _ { x } + n _ { y } + n _ { \varepsilon } ) } f _ { s s } ^ { i } \left( s ^ { \prime } , s _ { t} \right) \\ \end{array} \right] \\ + \mathcal { D } _ { n _ { x } + 1 , n _ { x } + n _ { \varepsilon } } g _ { s s } \left( s _ { t } \right) ^ {\intercal} \left[ \begin{array} { c } \mathcal { H } _ { n _ { y } + 1 , 2 n _ { y } ; n _ { y } + 1 , 2 n _ { y } } f _ { s s } ^ { i } \left( s ^ { \prime } , s _ { t} \right) \mathcal { D } _ { n _ { x } + 1 , n _ { x } + n _ { \varepsilon } } g _ { s s } \left( s _ { t} \right) \\ + 2 \mathcal { H } _ { n _ { y } + 1 , 2 n _ { y } ; 2 n _ { y } + 1 , 2 n _ { y } + n _ { x } } f _ { s s } ^ { i } \left( s ^ { \prime } , s _ { t} \right) \mathcal { D } _ { n _ { x } + 1 , n _ { x } + n _ { \varepsilon } } h _ { s s } \left( s _ { t} \right) \\ + 2 \mathcal { H } _ { n _ { y } + 1 , 2 n _ { y } ; 2 ( n _ { y } + n _ { x } ) + n _ { \varepsilon } + 1 , 2 ( n _ { x } + n _ { y } + n _ { \varepsilon} ) } f _ { s s } ^ { i } \left( s ^ { \prime } , s _ { t} \right) \\ \end{array} \right] \\ + \left[ \begin{array} { c } \mathcal { D } _ { 1 , n _ { y }} f _ { s s } ^ { i } \left( s ^ { \prime } , s _ { t} \right) \mathcal { D } _ { 1 , n _ { x }} g _ { s s } \left( s ^ { \prime} \right) \\ + \mathcal { D } _ { 2 n _ { y } + 1 , 2 n _ { y } + n _ { x }} f _ { s s } ^ { i } \left( s ^ { \prime } , s _ { t} \right) \\ \end{array} \right] \\ + \mathcal { D } _ { n _ { y } + 1 , 2 n _ { y }} f _ s s s ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ). \\ + {\mathcal H} _ \mathrm{2(n_{y}+n_{x})+n_{\varepsilon}+1,2(n_{x}+n_{y}+n_{\varepsilon}) ;2(n_{y}+n_{x})+n_{\varepsilon}+1,2(n_{x}+n_{y}+n_{\varepsilon})} f ^ {\mathrm{i}}_ \mathrm{s} ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) . \\ - {\mathcal H} ^ {- 1 / 2} [ d ] & - {\mathcal H} ^ {- 1 / 2} [ d ] \\ - {\mathcal H} ^ {- 1 / 2} [ d ] & - {\mathcal H} ^ {- 1 / 2} [ d ] \\ - {\mathcal H} ^ {- 1 / 2} [ d ] & - {\mathcal H} ^ {- 1 / 2} [ d ] \\ - {\mathcal H} ^ {- 1 / 2} [ d ] & - {\mathcal H} ^ {-\mathrm{的}} [ d ] \\ - {\mathcal H} ^ {-\mathrm{的}} [ d ] & - {\mathcal H} ^ {-\mathrm{的}} [ d ] \\ - {\mathcal H} ^ {-\mathrm{的}} [ d ] & - {\mathcal H} ^ {-\mathrm{的}} [ d ] \\ - {\mathcal H} ^ {-\mathrm{的}} [ d ] & - {\mathcal H} ^ {-\mathrm{的}} [ d ] \\ - {\mathcal H} ^ {-\mathrm{(i)} / 2} [ d ] & - {\mathcal H} ^ {-\mathrm{(i)} / 2} [ d ] \\ - {\mathcal H} ^ {-\mathrm{(i)} / 2} [ d ] & - {\mathcal H} ^ {-\mathrm{(i)} / 2} [ d ] \\ - {\mathcal H} ^ {-\mathrm{(i)} / 2} [ d ] & - {\mathcal H} ^ {-\mathrm{(i)} / 2} [ d ] \\ - {\mathcal H} ^ {-\mathrm{(ii)} / 2} [ d ] & - {\mathcal H} ^ {-\mathrm{(ii)} / 2} [ d ] \\ - {\mathcal H} ^ {-\mathrm{(ii)} / 2} [ d ] & - {\mathcal H} ^ {-\mathrm{(ii)} / 2} [ d ] \\ - {\mathcal H} ^ {-\mathrm{(ii)} / 2} [ d ] & - {\mathcal H} ^ {-\mathrm{(ii)} / 2} [ d ] \\ - {\mathcal H}^ - | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | \end{array}\right)\]

Given the solution from Step 1, the condition for implies a linear system of equations that determines the solution

11.5 Step 5: Obtaining the derivatives of and

The Öfth equation is the derivative with respect to and .

\[\begin{array} { c } \mathcal { H } _ { n _ { x } + 1 , n _ { x } + n _ { \varepsilon } ; n _ { x } + n _ { \varepsilon } + 1 } \mathbb { G } _ { s s } ^ { i } ( s _ { t } ) = \sum _ { s ' = 1 } ^ { n _ { s } } p _ { s _ { t } , s ' } \times \\ \left( \mathcal { D } _ { 1 , n _ { x } } g _ { s s } ( s ^ { \prime } ) \mathcal { D } _ { n _ { x } + 1 , n _ { x } + n _ { \varepsilon } } h _ { s s } ( s _ { t } ) \right) ^ { \intercal } \left( \begin{array} { c } \mathcal { H } _ { 1 , n _ { y } ; 1 , n _ { y } } f _ { s s } ^ { i } ( s ^ { \prime } , s _ { t } ) \binom { \mathcal { D } _ { n _ { x } + n _ { \varepsilon } + 1 } g _ { s s } ( s ^ { \prime } ) } { + \mathcal { D } _ { 1 , n _ { x } } g _ { s s } ( s ^ { \prime } ) \mathcal { D } _ { n _ { x } + n _ { \varepsilon } + 1 } h _ { s s } ( s _ { t } ) } \\ + \mathcal { H } _ { 1 , n _ { y } ; 2 n _ { y } + 1 , 2 n _ { y } + n _ { x } } f _ { s s } ^ { i } ( s ^ { \prime } , s _ { t } ) \mathcal { D } _ { n _ { x } + n _ { \varepsilon } + 1 } g _ { s s } ( s _ { t } ) \\ + \mathcal { H } _ 1 , n _ { y } ; 2 ( n _ { x } + n _ { y } + n _ { \varepsilon } ) + 1 , 2 ( n _ { x } + n _ { y } + n _ { \varepsilon } ) + n _ { \theta } f _ { s s } ^ { i } ( s ^ { \prime } , s _ { t } ) \mathcal { D } \theta _ { s s } ( s ^ { \prime } ) \\ + \mathcal { H } _ 1 , n _ { y } ; 2 ( n _ { x } + n _ { y } + n _ { \varepsilon } + n _ { \theta } ) + 1 , 2 ( n _ { x } + n _ { y } + n _ { \varepsilon } + n _ { \theta } ) f _ { s s } ^ { i } ( s ^ { \prime } , s _ { t } ) \mathcal { D } \theta _ { s s } ( s _ { t } ) \end{array} \right) \\ + \mathcal { D } _ { n _ { x } + 1 , n _ { x } + n _ { \varepsilon } } h _ { s s } ( s _ { t } ) ^ { \intercal } \left( \begin{array} { c } \mathcal { H } _ { 2 n _ { y } + 1 , 2 n _ { y } + n _ { x } ; 1 , n _ { y } } f _ { s s } ^ { i } ( s ^ { \prime } , s _ { t } ) \binom { \mathcal { D } _ { n _ { x } + n _ { \varepsilon } + 1 } g _ { s s } ( s ^ { \prime } ) } { + \mathcal { D } _ { 1 , n _ { x } } f _ { s s } ( s ^ { \prime }) \mathcal { D } _ { n _ { x } + n _ { \varepsilon } + 1 } h _ { s s } ( s _ { t } ) } \\ + \mathcal { H } _ 2 n _ { y } + 1 , 2 n _ { y } + n _ { x } ; 2 n _ { y } + 1 , 2 n _ { y } + n _ { x } f _ { s s } ^ { i } ( s ^ { \prime } , s _ { t }) \mathcal { D } _ { n _ { x } + n _ { \varepsilon } + 1 } g _ { s s } ( s _ { t } ) \\ + \mathcal { H } _ { 2 n _ { y } + 1 , 2 n _ { y } + n _ { x } ; 2 ( n _ { x } + n _ { y } + n _ { \varepsilon } ) + 1 , 2 ( n _ { x } + n _ { y } + n _ { \varepsilon ) } + n _ { \theta } f _ { s s} ^ { i } ( s ^ { \prime }, s _ { t}) \mathcal { D} \theta _ { s s } ( s ^ { \prime }) \\ + \mathcal { H } _ { 2 n _ { y } + 1 , 2 n _ { y } + n _ { x } ; 2 ( n _ { x } + n _ { y } + n _ { \varepsilon ) } + n _ { \theta } + 1 , 2 ( n _ { x } + n _ { y } + n _ { \varepsilon } + n _ { \theta ) } f _ { s s} ^ { i } ( s ^ { \prime }, s _ { t}) \mathcal { D} \theta _ { s s} ( s _ { t} ) \\ \mathcal { D } _ { 1 , n _ { y }} f _ { s s} ^ { i } ( s ^ {\prime}, s _ { t}) \binom{ {\mathcal H} ^ {\intercal}}{+ {\mathcal H} ^ {\intercal}} \\ + {\mathcal D} _ { n _ { x } + 1 , n _ { x } + n _ {\varepsilon }} g _ { s s }( s _ { t }) ^ {\intercal} \\ - {\mathcal H} ^ {\intercal} = - {\mathcal H} ^ {\intercal} = - {\mathcal H} ^ {\intercal} = - {\mathcal H} ^ {\intercal} = - {\mathcal H} ^ {\intercal} = - {\mathcal H} ^ {\intercal} = - {\mathcal H} ^ {\intercal} = - {\mathcal H} ^ {\intercal} = - {\mathcal H} ^ {\intercal} = - {\mathbf A} ^ {\intercal} = - {\mathbf A} ^ {\intercal} = - {\mathbf A} ^ {\intercal} = - {\mathbf A} ^ {\intercal} = - {\mathbf A} ^ {\intercal} = - {\mathbf A} ^ {\intercal} = - {\mathbf A} ^ {\intercal} = - {\mathbf A} ^ {\intercal} = - {\mathbf A} ^ {\cong} = - {\mathbf A} ^ {\intercal} = - {\mathbf A} ^ {\intercal} = - {\mathbf A} ^ {\intercal} = - {\mathbf A} ^ {\intercal} = - {\mathbf A} ^ {\intercal} = - {\mathbf A} ^ {\intercal} = - {\mathbf A} ^ {\intercal} = - {\mathbf A} ^ {\intercal} = - {{\mathbf A}} ^ {\intercal} = - {{\mathbf A}} ^ {\intercal} = - {{\mathbf A}} ^ {\intercal} = - {{\mathbf A}} ^ {\intercal} = - {{\mathbf A}} ^ {\intercal} = - {{\mathbf A}} ^ {\intercal} = - {{\mathbf A}} ^ {\intercal} = - {{\mathbf A}} ^ {\intercal} = - {{0.5}}} \\ - {\mathcal H} ^ {\intercal} = - {\mathcal H} ^ {\intercal} = - {\mathcal H} ^ {\intercal} = - {\mathcal H} ^ {\intercal} = - {\mathbf A} ^ {\intercal} = - {{\mathbf A}} ^ {\intercal} = - {{\mathbf A}} ^ {\intercal} = - {{\mathbf A}} ^ {\intercal} = - {{\mathbf A}} ^ {\intercal} = - {{\mathbf A}} ^ {\intercal} = - {{\mathbf A}} ^ {\intercal} = - {{0.5}}} \\ - {\mathcal H} ^ {\infty_ {-}} = - {\mathcal H} ^ {\infty_ {-}} = - {\mathbf A} ^ - 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 . \\ - \mathfrak C o r c e c t i o m a l l o w h e r m a l l o w h e r m a l l o w h e r m a l l o w h e r m a l l o w h e r m a l l o w h e r m a l l o w h e r m a l l o w h e r m a l l o w h e r m a l l o w h e r m a l l o w h e r m a l l o w h e r l o w h e r l o w h e r l o w h e r l o w h e r l o w h e r l o w h e r l o w h e r l o w h e r l o w h e r l o w h e r l o w h e r l o w h e r l o w h e r l o w h e r l o w h e r l o w h e r l o w l o w h e r l o w h e r l o w h e r l o w h e r l o w h e r l o w h e r l o w h e r l o w h e r l o w h e r l o w h e r l o w h e r l o w h e r l o w h e r l o w h e r l o w h e r l o w h e r l o w v e r d i j k u a d i j k u a d i j k u a d i j k u a d i j k u a d i j k u a d i j k u a d i j k u a d i j k u a d i j k u a d i j k u a d i j k u a d i j k u a d i j k u a d i j k u a d i j k u a d i j k u a d i jk u a d i j k u a d i j k u a d i j k u a d i j k u a d i j k u a d i j k u a d i j k u a d i j k u a d i j k u a d i j k u a d i j k u a d i j k u a d i j k u a d i j k u a d i j k u a d i j k u a d | j u a d | j u a d | j u a d | j u a d | j u a d | j u a d | j u a d | j u a d | j u a d | j u a d | j u a d | j u a d | j u a d | j u a d | j u a d | j u a d | j u a d | j u a d | j u a d | j u a d | J U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U U 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 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 S E X S E X S E X S E X S E X S E X S E X S E X S E X S E X S E X S E X S E X S E X S E X S E X S E X S E X S E X S E X S E X S E X S E X S E X S E X S E X S E X S E X S E X S E X S E X S E X S E X S E X 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 NNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINN INNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNINNISIENIENIENIENIENIENIENIENIENIENIENIENIENIENIENIENIENIENIENIENIENIENIENIENIENIENIENIENIENIENIENIENIENIENIENIENIENIENIENIENIENIENIENIENIENIENIENIENIENIENIEniEniEniEniEniEniEniEniEniEniEniEniEniEniEniEniEniEniEniEniEniEniEniEniEniEniEniEniEniEniEniEniEniEniEniEniEniEniEniEniEniEniEniEniEniEniEniEniEniEniEnnienenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenenen ennienennienniennienniennienniennienniennienniennienniennienniennienniennienniennienniennienniennienniennienniennienniennienniennienniennienniennienniennienniennienniennienniennienniennienniennienniennienniennlennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennennnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnmm< nl>\]

Given the solution from Steps 1 and 3, the condition for implies a linear system of equations that determines the solution

11.6 Step 6: Obtaining the derivatives of twice

The sixth equation is the derivative with respect to twice.

\[\left( \begin{array} { c } \mathcal { H } _ { n _ { x } + n _ { \varepsilon } + 1 ; n _ { x } + n _ { \varepsilon } + 1 } \mathbb { G } _ { s s } ^ { i } ( s _ { t } ) = \sum _ { s ^ { \prime } = 1 } ^ { n _ { s } } p _ { s _ { t } , s ^ { \prime } } \times \\ \left( \begin{array} { c } \mathcal { D } _ { n _ { x } + n _ { \varepsilon } + 1 } g _ { s s } ( s ^ { \prime } ) \\ + \mathcal { D } _ { 1 , n _ { x } } g _ { s s } ( s ^ { \prime } ) \mathcal { D } _ { n _ { x } + n _ { \varepsilon } + 1 } h _ { s s } ( s _ { t } ) \end{array} \right) ^ { \intercal } \mathcal { H } _ { 1 , n _ { y } ; 1 , n _ { y } } f _ { s s } ^ { i } ( s ^ { \prime } , s _ { t } ) \left( \begin{array} { c } \mathcal { D } _ { n _ { x } + n _ { \varepsilon } + 1 } g _ { s s } ( s ^ { \prime } ) \\ + \mathcal { D } _ { 1 , n _ { x } } g _ { s s } ( s ^ { \prime } ) \mathcal { D } _ { n _ { x } + { n } _ { \varepsilon } + 1 } h _ { s s } ( s _ { t } ) \end{array} \right) \\ + 2 \left( \begin{array} { c } \mathcal { D } _ { n _ { x } + n _ { \varepsilon } + 1 } g _ { s s } ( s ^ { \prime } ) \\ + \mathcal { D } _ { 1 , n _ { x } } g _ { s s } ( s ^ { \prime } ) \mathcal { D } _ { n _ { x } + {{ n } _ { \varepsilon } + 1 } } h _ { s s } ( s _ { t } ) \end{array} \right) ^ { \intercal } \left( \begin{array} { c } \mathcal { H } _ { 1 , n _ { y } ; n _ { y } + 1 , 2 n _ { y } } f _ { s s } ^ { i } ( s ^ { \prime } , s _ { t } ) \mathcal { D } _ { n _ { x } + n _ { \varepsilon } + 1 } g _ { s s } ( s _ { t } ) \\ + \mathcal { H } _ { 1 , n _ { y } ; 2 n _ { y } + 1 , 2 n _ { y } + n _ { x } } f _ { s s } ^ { i } ( s ^ { \prime } , s _ { t } ) \mathcal { D } _ { n _ { x } + n _ { \varepsilon } + 1 } h _ { s s } ( s _ { t } ) \\ + \mathcal { H } _ 1 , n _ { y } ; 2 ( n _ { x } + n _ { y } + n _ { \varepsilon } ) + 1 , 2 ( n _ { x } + n _ { y } + n _ { \varepsilon } ) + n _ { \theta } ^ { i } f _ { s s } ^ { i } ( s ^ { \prime } , s _ { t } ) \mathcal { D } \theta _ { s s } ( s ^ { \prime } ) \\ + \mathcal { H } _ 1 , n _ { y } ; 2 ( n _ { x } + n _ { y } + n _ { \varepsilon } + n _ { \theta } ) + 1 , 2 ( n _ { x } + n _ { y } + n _ { \varepsilon } + n _ { \theta } ) f _ { s s } ^ { i } ( s ^ { \prime } , s _ { t } ) \mathcal { D } \theta _ { s s } ( s _ { t } ) \end{array} \right) \\ + \mathcal { D } _ { n _ { x } + n _ { \varepsilon } + 1 } g _ { s s } ( s _ { t }) ^ { \intercal } \left( \begin{array} { c } \mathcal { H } _ { n _ { y } + 1 , 2 n _ { y } ; n _ { y } + 1 , 2 n _ { y }} f _ { s s } ^ { i } ( s ^ { \prime } , s _ { t } ) \mathcal { D } _ { n _ { x } + n _ { \varepsilon } + 1 } g _ { s s } ( s _ { t } ) \\ + 2 \mathcal { H } _ n _ { y } + 1 , 2 n _ { y } ; 2 n _ { y } + 1 , 2 n _ { y } + n _ { x } f _ { s s} ^ { i } ( s ^ { \prime } , s _ { t } ) \mathcal { D } _ { n _ { x } + n _ { \varepsilon } + 1 } h _ { s s } ( s _ { t } ) \\ + 2 \mathcal { H } _ n _ { y } + 1 , 2 n _ { y} ; 2 ( n _ { x } + n _ { y } + n _ { \varepsilon} ) + 1 , 2 ( n _ { x } + n _ { y } + n _ { \varepsilon} ) + n _ { \theta } f _ { s s} ^ { i } ( s ^ { \prime } , s _ { t} ) \mathcal { D } \theta _ { s s } ( s ^ { \prime} ) \\ + 2 \mathcal { H } _ n _ { y } + 1 , 2 n _ { y} ; 2 ( n _ { x } + n _ { y } + n _ { \varepsilon} ) + 1 , 2 ( n _ { x } + n _ { y } + n _ { \varepsilon} + n _ { \theta} ) f _ { s s} ^ { i } ( s ^ {\prime} , s t ) \mathcal { D} \theta _ { s s} ( s t ) \\ + 2 \mathcal { H } _ 2 n y + 1 , 2 n y + n x ; 2 ( n x + n y + n e ) + 1 , 2 ( n x + n y + n e ) + n e f i o f f i o f f i o f f i o f f i o f f i o f f i o f f i o f f i o f f i o f f i o f f i o f f i o f f i o f f i o f f i o f f i o f f i o f f i o f f i o f f i o f f i o f f i o f f i o f f i o f f i o l e d e r m a x e d e r m a x e d e r m a x e d e r m a x e d e r m a x e d e r m a x e d e r m a x e d e r m a x e d e r m a x e d e r m a x e d e r m a x e d e r m a x e d e r m a x e d e r m a x e d e r m a x e l e d e r m a x e d e r m a x e d e r m a x e d e r m a x e d e r m a x e d e r m a x e d e r m a x e d e r m a x e d e r m a x e d e r m a x e d e r m a x e d e r m a x e d e r m a x e d e r m a x u l l u l l u l l u l l u l l u l l u l l u l l u l l u l l u l l u l l u l l u l l u l l u l l u l l u l l u l l u l l u l l u l l u l l u l l u l l u l l u l l u l l u l l u l l u l l u l l u l l u l l v o r b j k . \\ + (\varepsilon ^{ \prime}) ^{ \intercal }\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 & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & & \\ - 2 A B C E F I O U S U P L I N T I O U S U P L I N T I O U S U P L I N T I O U S U P L I N T I O U S U P L I N T I O U S U P L I N T I O U S U P L I N T I O U S U P L I N T I O U S U P L I N T I O U S U P L I N T I O U S U P L I N t h r m a x e d \\ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ [ ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] ] | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | - 3.37564890000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000< fcel>+ 2: F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W FW F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F W F V K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K K 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 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 v k k . \\ - 3.3756489000000000000000000000000000000000000000000000000000000000000000000000000000000000000 ① : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / / \\ - 3.3756489199999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999998:88888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888888886:6666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666< 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>\]

Given the solutions from Steps 1, 3 and 4, the condition for implies a linear system of equations that determines the solution

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1. 2013-11: “Gender quotas and the quality of politicians”, Audinga Baltrunaite, Piera Bello, Alessandra Casarico y Paola Profeta.

References

1. 2013-10: “Brechas de Género en los Resultados de PISA :El Impacto de las Normas Sociales y la Transmisión Intergeneracional de las Actitudes de Género”, Sara de la Rica y Ainara González de San Román.

References

1. 2013-09: “¿Cómo escogen los padres la escuela de sus hijos? Teoría y evidencia para España”, Caterina Calsamiglia, Maia Güell.

References

1. 2013-08: “Evaluación de un programa de educación bilingüe en España: El impacto más allá del aprendizaje del idioma extranjero”, Brindusa Anghel, Antonio Cabrales y Jesús M. Carro.

References

1. 2013-07: “Publicación de los resultados de las pruebas estandarizadas externas: ¿Tiene ello un efecto sobre los resultados escolares?”, Brindusa Anghel, Antonio Cabrales, Jorge Sainz e Ismael Sanz.

References

1. 2013-06: “DYPES: A Microsimulation model for the Spanish retirement pension system”, F. J. Fernández-Díaz, C. Patxot y G. Souto.

References

1. 2013-05: “Vertical differentiation, schedule delay and entry deterrence: Low cost vs. full service airlines”, Jorge Validoa, M. Pilar Socorroa y Francesca Medda.

References

1. 2013-04: “Dropout Trends and Educational Reforms: The Role of the LOGSE in Spain”, Florentino Felgueroso, María Gutiérrez‐Domènech y Sergi Jiménez‐Martín.

References

1. 2013-03: “Understanding Different Migrant Selection Patterns in Rural and Urban Mexico”, Simone Bertoli, Herbert Brücker y Jesús Fernández-Huertas Moraga.

References

1. 2013-02: “Understanding Different Migrant Selection Patterns in Rural and Urban Mexico”, Jesús Fernández-Huertas Moraga.

References

1. 2013-01: “Publicizing the results of standardized external tests: Does it have an effect on school outcomes?, Brindusa Anghel, Antonio Cabrales, Jorge Sainz y Ismael Sanz.

References

1. 2012-12: “Visa Policies, Networks and the Cliff at the Border”, Simone Bertoli, Jesús Fernández-Huertas Moraga.

References

1. 2012-11: “Intergenerational and Socioeconomic Gradients of Child Obesity”, Joan Costa-Fonta y Joan Gil.

References

1. 2012-10: “Subsidies for resident passengers in air transport markets”, Jorge Valido, M. Pilar Socorro, Aday Hernández y Ofelia Betancor.

References

1. 2012-09: “Dual Labour Markets and the Tenure Distribution: Reducing Severance Pay or Introducing a Single Contract?”, J. Ignacio García Pérez y Victoria Osuna.

References

1. 2012-08: “The Influence of BMI, Obesity and Overweight on Medical Costs: A Panel Data Approach”, Toni Mora, Joan Gil y Antoni Sicras-Mainar.

References

1. 2012-07: “Strategic behavior in regressions: an experimental”, Javier Perote, Juan Perote-Peña y Marc Vorsatz.

References

1. 2012-06: “Access pricing, infrastructure investment and intermodal competition”, Ginés de Rus y M. Pilar Socorro.

References

1. 2012-05: “Trade-offs between environmental regulation and market competition: airlines, emission trading systems and entry deterrence”, Cristina Barbot, Ofelia Betancor, M. Pilar Socorro y M. Fernanda Viecens.

References

1. 2012-04: “Labor Income and the Design of Default Portfolios in Mandatory Pension Systems: An Application to Chile”, A. Sánchez Martín, S. Jiménez Martín, D. Robalino y F. Todeschini.

References

1. 2012-03: “Spain 2011 Pension Reform”, J. Ignacio Conde-Ruiz y Clara I. Gonzalez.