Estimating Dynamic Equilibrium Models with Stochastic Volatility
by * Jesús Fernández-Villaverde Pablo Guerrón-Quintana** *** Juan F. Rubio-Ramírez
Documento de Trabajo 2014-11
October 2014
* University of Pennsylvania, NBER, and CEPR. ** Federal Reserve Bank of Philadelphia. *** Duke University, Federal Reserve Bank of Atlanta, BBVA Research and FEDEA
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NON‐TECHNICAL SUMMARY
En este trabajo analizamos las causas de la gran moderación que la economía de Estados Unidos sufrió hasta la gran recesión. Para ellos escribimos un modelo keynesiano donde tanto los parámetros que controlan la política monetaria como las varianzas de las innovaciones (volatilidad estocástica) se mueven a lo largo del tiempo y lo confrontamos con los datos. Nuestros principales hallazgos empíricos son los siguientes. En primer lugar, la distribución posterior de los parámetros pone la mayor parte de su masa en áreas que denotan una buena cantidad de volatilidad estocástica. En segundo lugar, un ejercicio de comparación de modelo indica que, incluso después de controlar por volatilidad estocástica, los datos prefieren una especificación donde hay cambios en los parámetros de política monetaria. Este hallazgo no debe interpretarse, sin embargo, en el sentido de que los cambios de volatilidad no jugaron un papel en la gran moderación. Significa, en cambio, que un modelo exitoso de la economía de Estados Unidos requiere la presencia de los cambios tanto en volatilidad como en el resto de parámetros. Este resultado que desafía los resultados de Sims y Zha (2006). Por último, se documenta la evolución de la volatilidad estocástica y de los parámetros de la política monetaria. En el apéndice, construimos historias contrafactuales de los datos de Estados Unidos mediante el cierre de la volatilidad estocástica o la imposición de políticas monetarias alternativas.
Jes˙s Fern·ndez-Villaverdey
Pablo GuerrÛn-Quintanaz
Juan F. Rubio-RamÌrezx
September 8, 2014
Abstract
This paper develops a particle Öltering algorithm to estimate dynamic equilibrium models with stochastic volatility using a likelihood-based approach. The algorithm, which exploits the structure and profusion of shocks in stochastic volatility models, is versatile and computationally tractable even in large-scale models. As an application, we use our algorithm and Bayesian methods to estimate a business cycle model of the U.S. economy with both stochastic volatility and parameter drifting in monetary policy. Our application shows the importance of stochastic volatility in accounting for the dynamics of the data.
Keywords: Dynamic equilibrium models, Stochastic volatility, Parameter drifting, Bayesian methods.
JEL classiÖcation numbers: E10, E30, C11.
We thank AndrÈ Kurmann, Jim Nason, Frank Schorfheide, Neil Shephard, Tao Zha, the editor Yacine AÔt-Sahalia, two referees, and participants at several seminars for useful comments, and BÈla SzemÈly for invaluable research assistance. This paper borrows material from our manuscript ìFortune or Virtue: Time-Variant Volatilities Versus Parameter Drifting in U.S. Data.î Beyond the usual disclaimer, we must note that any views expressed herein are those of the authors and not necessarily those of the Federal Reserve Bank of Atlanta, the Federa Reserve Bank of Philadelphia, or the Federal Reserve System. Juan F. Rubio-RamÌrez also thanks the Institute for Economic Analysis (IAE) and the ìPrograma de Excelencia en EducaciÛn e Investigacionî of the Bank of Spain, and the spanish ministry of science and technology Ref. ECO2011-30323-c03-01 for support. Finally, we also thank the NSF for Önancial support.
yUniversity of Pennsylvania, NBER, and CEPR, <jesusfv@econ.upenn.edu>.
zFederal Reserve Bank of Philadelphia, <pablo.guerron@phil.frb.org>.
xDuke University, Federal Reserve Bank of Atlanta, BBVA Research and FEDEA, <juan.rubioramirez@duke.edu>.
1. Introduction
This paper develops a particle Öltering algorithm to estimate dynamic equilibrium models with stochastic volatility using a likelihood-based approach. The novelty of our algorithm is that it does not require the presence of linear measurement errors to evaluate the likelihood function of the model. In order to do that, we characterize the properties of the solution of these models when approximated with the second-order expansion. As an application of our procedure, we estimate a medium-size business cycle economy.
Our results are useful because, motivated by the Öndings of Stock and Watson (2002) and Sims and Zha (2006), many recent papers have built dynamic equilibrium models with volatility shocks (also known as uncertainty shocks). Among them, we can highlight Fern·ndez-Villaverde and Rubio-RamÌrez (2007), Justiniano and Primiceri (2008), Bloom (2009), and Fern·ndez-Villaverde et al. (2010c). In these models, and in the tradition of stochastic volatility (Shephard, 2008), there are two types of shocks: structural shocks (shock to productivity, to preferences, etc.) and volatility shocks (shocks to the standard deviation of the innovations to the structural shocks).
To fulÖll the promise in this literature, we need tools to estimate this class of models. However, the task is complicated by the inherent non-linearity that stochastic volatility generates. Linearization is ill-equipped to handle time-varying volatility because it yields certainty-equivalent policy functions. That is, volatility ináuences neither the agentsídecision rules nor the laws of motion of the aggregate variables. Hence, to consider how stochastic volatility a§ects those factors, it is imperative to employ at least the second-order approximation to the equilibrium dynamics of the economy and to use simulation-based estimators of the likelihood.
To accomplish that last task, one could, in principle, rely on the baseline particle Ölter presented in Fern·ndez-Villaverde and Rubio-RamÌrez (2007). Unfortunately, that version of the particle Ölter requires, when estimating models with stochastic volatility, the presence of linear measurement errors in observables. Otherwise, we would be forced to solve a large quadratic system of equations with multiple solutions, an endeavor for which there are no suitable algorithms. Although measurement errors are plausible, they complicate identiÖcation in small samples and entangle the interpretation of the results.
To get around this problem, we show how to write an alternative particle Ölter that exploits the structure of the second-order approximation to the equilibrium dynamics of an economy with stochastic volatility without the need of linear measurement errors. Second-order approximations accurately capture important implications of stochastic volatility and are convenient because they are not computationally expensive.
We proceed in two steps. First, we characterize the second-order approximation to the decision rules of a dynamic equilibrium model with stochastic volatility. Second, we demonstrate how to use this characterization to write the alternative particle Ölter. The key is to show how the quadratic problem associated with the evaluation of the approximated measurement density is reduced to a much simpler linear problem that only involves a matrix inversion. After we have evaluated the likelihood, we can combine it with a prior and a Markov chain Monte Carlo (McMc) algorithm to draw from the posterior distribution.
Our characterization of the second-order approximation to the decision rules is also of interest in itself. Among other things, it is useful to analyze the equilibrium of the model, to explore the shape of its impulse response functions, or to calibrate it. More concretely, we prove that:
1. The Örst-order approximation to the decision rules of the agents (or any other equilibrium object of interest) does not depend on volatility shocks and they are certainty equivalent.
2. The second-order approximation to the decision rules of the agents only depends on volatility shocks on terms where volatility is multiplied by the innovation to its own structural shock. For instance, if we have a productivity shock and a volatility shock to it, the only non-zero term where the volatility shock to productivity appears is the one where the volatility shock multiplies the innovation to the productivity shock. Thus, only a few of the terms in the second-order approximation are non-zero.
3. The perturbation parameter will only appear in a non-zero term where it is raised to a square. This term is a constant that corrects for precautionary behavior induced by risk.
As an application, we estimate a business cycle model of the U.S. economy. The model incorporates stochastic volatility in the shocks that drive its dynamics and parameter drifting in the parameters that control monetary policy. In that way, we include two of the main mechanisms that researchers have highlighted to account for the time-varying volatility of U.S. time series - heteroscedastic shocks and parameter drifting- and let the likelihood decide which of them better accounts for the data. Last, we have a model that is as rich as many of the models employed in modern quantitative macroeconomics. While estimating such a large model is a computational challenge, we wanted to demonstrate that our procedure is of practical use and to make our application a blueprint for the estimation of other dynamic equilibrium models.
Our main empirical Öndings are as follows. First, the posterior distribution of the parameters puts most of its mass in areas that denote a fair amount of stochastic volatility. Second, a model comparison exercise indicates that, even after controlling for stochastic volatility, the data prefer a speciÖcation where monetary policy changes over time. This Önding should not be interpreted, though, as implying that volatility shocks did not play a role. It means, instead, that a successful model of the U.S. economy requires the presence of both stochastic volatility and parameter drifting, a result that challenges the results of Sims and Zha (2006). Finally, we document the evolution of the structural shocks, of stochastic volatility, and the parameters of monetary policy. We emphasize the conáuence, during the 1970s, of times of high volatility and weak responses to ináation, and, during the 1990s, of positive structural shocks and low volatility even if monetary policy was weaker than often argued. In the appendix, we construct counterfactual histories of the U.S. data by varying some aspect of the model such as shutting down time-varying volatility or imposing alternative monetary policies.
An alternative to our stochastic volatility framework would be to work with Markov regimeswitching models such as those of Bianchi (2009) or Farmer et al. (2009). These models provide a promising extra degree of áexibility in modelling aggregate dynamics. In fact, some of the fast changes in policy parameters that we document in our empirical section suggest that discrete jumps could be a good representation of the data. We hope to undertake in the future a more careful assessment of the advantages and disadvantages of stochastic volatility versus Markov regime-switching models.
Finally, even if the motivation for our approach and the application belong to macroeconomics, the tools we present are not speciÖc to that Öeld. One can think about the importance of estimating dynamic equilibrium models with stochastic volatility in many other Öelds such as Önance (Bansal and Yaron, 2004) or international economics (Fern·ndez-Villaverde et al., 2010c).
The rest of the paper is organized as follows. Section 2 introduces a generic dynamic equilibrium model with stochastic volatility to Öx notation and discuss how to solve it. Section 3 explains the evaluation of the likelihood of the model. Section 4 compares our approach with continuous-time methods. Section 5 presents our application. Section 6 concludes. An extensive technical appendix includes additional material.
2. Dynamic Equilibrium Models with Stochastic Volatility
2.1. The Model
The set of equilibrium conditions of a wide class of dynamic equilibrium models can be written as
\[\mathbb {E} _ {t} f \left(\mathcal {Y} _ {t + 1}, \mathcal {Y} _ {t}, \mathcal {S} _ {t + 1}, \mathcal {S} _ {t}, \mathcal {Z} _ {t + 1}, \mathcal {Z} _ {t}; \gamma\right) = 0,\tag{1}\]
where is the conditional expectation operator at time is the vector of observables at time is the vector of endogenous states at time t, is the vector of structural shocks at time t, f maps into , and is the vector of parameters that describe preferences and technology. In this paper, is also the vector of parameters to be estimated.
We will consider models where follow a stochastic volatility process of the form
\[\mathcal {Z} _ {i t + 1} = \rho_ {i} \mathcal {Z} _ {i t} + \Lambda \sigma_ {i} \sigma_ {i t + 1} \varepsilon_ {i t + 1}\tag{2}\]
for all , where is a perturbation parameter, is the mean volatility, and log the percentage deviation of the standard deviation of the innovations to the structural shocks with respect to its mean, evolves as
\[\log \sigma_ {i t + 1} = \vartheta_ {i} \log \sigma_ {i t} + \Lambda \left(1 - \vartheta_ {i} ^ {2}\right) ^ {\frac {1}{2}} \eta_ {i} u _ {i t + 1}\tag{3}\]
for all . The combination of levels in (2) and logs in (3) ensures a positive We multiply the innovation in (3) by to normalize its size by the persistence of . It will be clear momentarily why we specify (2) and (3) in terms of the perturbation parameter .
It is also convenient to write, for all , the laws of motions for and log
\[\mathcal {Z} _ {i t} = \rho_ {i} \mathcal {Z} _ {i t - 1} + \sigma_ {i} \sigma_ {i t} \varepsilon_ {i t}\tag{4}\]
and
\[\log \sigma_ {i t} = \vartheta_ {i} \log \sigma_ {i t - 1} + (1 - \vartheta_ {i} ^ {2}) ^ {\frac {1}{2}} \eta_ {i} u _ {i t}\tag{5}\]
Note that appears only in equations (2) and (3) but not in equations (4) and (5). This is because this parameter is used to eliminate, when we later determine the point around which we approximate the equilibrium dynamics of the model, uncertainty about the future. Given the information set in equation (1), there is uncertainty about both and log for all , but there is no uncertainty about either or log for any
Let be the vector of volatility shocks, the vector of innovations to the structural shocks, and the vector of innovations to the volatility shocks. We assume that and , where 0 is an vector of zeros and I is an identity matrix. To ease notation, we assume that all structural shocks face volatility shocks, that the volatility shocks are uncorrelated, and that and are normally distributed. It is straightforward, yet cumbersome, to generalize the notation to other cases. In particular, as we will see below, to implement our particle Ölter, we only need to be able to simulate and evaluate the density of
2.2. The Solution
Given equations (1)-(5), the solution to the model -if one exists- that embodies equilibrium dynamics (that is, agentsíoptimization and market clearing conditions) is characterized by a policy function describing the evolution of the endogenous state variables
\[\mathcal {S} _ {t + 1} = h \left(\mathcal {S} _ {t}, \mathcal {Z} _ {t - 1}, \Sigma_ {t - 1}, \mathcal {E} _ {t}, \mathcal {U} _ {t}, \Lambda ; \gamma\right),\tag{6}\]
and two policy functions describing the law of motion of the observables
\[\mathcal {Y} _ {t} = g \left(\mathcal {S} _ {t}, \mathcal {Z} _ {t - 1}, \Sigma_ {t - 1}, \mathcal {E} _ {t}, \mathcal {U} _ {t}, \Lambda ; \gamma\right)\tag{7}\]
and
\[\mathcal {Y} _ {t + 1} = g \left(\mathcal {S} _ {t + 1}, \mathcal {Z} _ {t}, \Sigma_ {t}, \Lambda \mathcal {E} _ {t + 1}, \Lambda \mathcal {U} _ {t + 1}, \Lambda ; \gamma\right),\tag{8}\]
together with equations (4) and (5) describing the laws of motion of the structural and volatility shocks. The policy functions h and g map into and ; respectively, and are indexed by .
For our purposes, it is important to deÖne the steady state of the model. Our assumptions about the stochastic processes imply that, in the steady state, and . Given and equations (1)-(8), the steady state of the model is vector of observables and an vector of endogenous states such that
\[f \left(g \left(h \left(\mathcal {S}, \mathbf {0}, \mathbf {0}, \mathbf {0}, \mathbf {0}, 0; \gamma\right), \mathbf {0}, \mathbf {0}, \mathbf {0}, \mathbf {0}, 0; \gamma\right), g \left(\mathcal {S}, \mathbf {0}, \mathbf {0}, \mathbf {0}, \mathbf {0}, 0; \gamma\right), h \left(\mathcal {S}, \mathbf {0}, \mathbf {0}, \mathbf {0}, \mathbf {0}, 0; \gamma\right), \mathcal {S}, \mathbf {0}, \mathbf {0}; \gamma\right) = 0.\tag{9}\]
In addition, in the steady state, the following two relationships hold
\[\mathcal {S} = h \left(\mathcal {S}, \mathbf {0}, \mathbf {0}, \mathbf {0}, \mathbf {0}, 0; \gamma\right),\tag{10}\]
and
\[\mathcal {Y} = g \left(\mathcal {S}, \mathbf {0}, \mathbf {0}, \mathbf {0}, \mathbf {0}, 0; \gamma\right).\tag{11}\]
Note that, if ; the model is in the steady state, since we eliminate any uncertainty about the future. If ; the model is not in the steady state and conditions (9)-(11) do not hold. For instance, in general because of the precautionary behavior of agents.
2.3. The State-Space Representation
Given the solution to the model -the policy functions (6)-(8) together with equations (4) and (5)- we can characterize the equilibrium dynamics of the model by its state-space representation written in terms of the transition and the observation equations.
We stack equations (4) to (6) in a transition equation
\[\mathbb {S} _ {t + 1} = \widetilde {h} (\mathbb {S} _ {t}, \Lambda ; \gamma) + \Xi \mathbb {W} _ {t + 1}\tag{12}\]
that describe the evolution of the states (endogenous states, structural shocks, volatility shocks, and their innovations) as a function of lag states, the perturbation parameter, and the vector of parameters, where is the vector of the states and maps into . Also, is a vector of random variables, and are vectors with distributions and and is a matrix with the the top n + 2m rows equal to zero and the bottom matrix equal to the identity matrix. and share distributions with and respectively. If we were to change distributions for either or we would need to do the same with and . Let us deÖne . The linearity of the second term of the right-hand side is a consequence of the autoregressive speciÖcation of the structural shocks and the evolution of their volatilities. However, through the function those shocks and their volatilities can a§ect non-linearly.
The measurement equation uses the policy function (7) to describe the relationship of the observables with the states, the perturbation parameter, and the vector of parameters
\[\mathcal {Y} _ {t} = g \left(\mathbb {S} _ {t}, \Lambda ; \gamma\right).\tag{13}\]
2.4. Approximating the State-Space Representation
In general, when we deal with the class of dynamic equilibrium models with stochastic volatility, the policy functions h and g cannot be found explicitly. Thus, we cannot build the state-space representation described by (12) and (13). Instead, we will approximate numerically the solution of the model and use the result to generate an approximated state-space representation.
There are many di§erent solution algorithms for dynamic equilibrium models. Among them, the perturbation method has emerged as a popular way (see Judd and Guu, 1997 and Aruoba et al., 2006) to obtain higher-order Taylor series approximations to the policy functions (6)-(7) together with equations (4) and (5) around the steady state. We also get the Taylor expansion of (4) because it is a non-linear function. Beyond being extremely fast for systems with a large number of state variables, perturbations are highly accurate even far away from the perturbation point (Aruoba, et al., 2006, and, for cases with stochastic volatility, Caldara et al., 2012).
The perturbation method allows the researcher to approximate the solution to the model up to any order. Since we want to analyze models with volatility shocks, we must go beyond a Örst-order approximation. First-order approximations are certainty equivalent, and hence, they are silent about stochastic volatility. As we will see in section 3.1, volatility shocks only appear starting in the second-order approximation. But since we want to evaluate the likelihood function, we stop at a second-order approximation. Because of dimensionality issues, a higher-order approximation would make the evaluation of the likelihood function exceedingly challenging for current computers for models with a reasonable number of state variables.
Given a second-order approximation to the policy functions (6)-(7) together with equations (4) and (5) around the steady state, the approximated state-space representation can be written in terms of two equations: the approximated transition equation and the approximated measurement equation. The approximated transition equation is
\[\widehat {\mathbb {S}} _ {t + 1} = \left( \begin{array}{c} \Psi_ {s, 1} ^ {1} \widehat {\mathbb {S}} _ {t} \\ \vdots \\ \Psi_ {s, n _ {s}} ^ {1} \widehat {\mathbb {S}} _ {t} \end{array} \right) + \frac {1}{2} \left( \begin{array}{c} \widehat {\mathbb {S}} _ {t} ^ {\prime} \Psi_ {s, 1} ^ {2} \widehat {\mathbb {S}} _ {t} \\ \vdots \\ \widehat {\mathbb {S}} _ {t} ^ {\prime} \Psi_ {s, n _ {s}} ^ {2} \widehat {\mathbb {S}} _ {t} \end{array} \right) + \frac {1}{2} \left( \begin{array}{c} \Psi_ {s, 1} ^ {\Lambda} \\ \vdots \\ \Psi_ {s, n _ {s}} ^ {\Lambda} \end{array} \right) + \Xi \mathbb {W} _ {t + 1}\tag{14}\]
where is the vector of deviations of the states from their steady-state value and . The approximated measurement equation is
\[\mathcal {Y} _ {t} - \mathcal {Y} = \left( \begin{array}{c} \Psi_ {y, 1} ^ {1} \widehat {\mathbb {S}} _ {t} \\ \vdots \\ \Psi_ {y, k} ^ {1} \widehat {\mathbb {S}} _ {t} \end{array} \right) + \frac {1}{2} \left( \begin{array}{c} \widehat {\mathbb {S}} _ {t} ^ {\prime} \Psi_ {y, 1} ^ {2} \widehat {\mathbb {S}} _ {t} \\ \vdots \\ \widehat {\mathbb {S}} _ {t} ^ {\prime} \Psi_ {y, k} ^ {2} \widehat {\mathbb {S}} _ {t} \end{array} \right) + \frac {1}{2} \left( \begin{array}{c} \Psi_ {y, 1} ^ {\Lambda} \\ \vdots \\ \Psi_ {y, k} ^ {\Lambda} \end{array} \right).\tag{15}\]
where is the steady-state value of .
In these equations, is a vector and is an matrix for . The Örst term is the linear approximation to the transition equation for the states, while the second term is the quadratic component of the second-order approximation. Similarly, is a vector and an matrix for . The interpretation of each term is the same as before, but for the measurement equations. The term is the constant that appears in the second-order perturbation that corrects for risk in the evolution of state . Similarly, the term is the constant correction for risk of observable . All these vectors and matrices are non-linear functions of . It is important to emphasize that we are not assuming the presence of any measurement error. We will return to this point in a few paragraphs.
3. Stochastic Volatility and Evaluation of the Likelihood
In this section, we explain how to evaluate the likelihood function of our class of dynamic equilibrium models with volatility shocks in equation (1) using the approximated transition and the measurement equations (14) and (15). If we allow to be the data counterpart of our observables , and (with to be their history up to time given , we can write the likelihood of as
\[\prod_ {t = 1} ^ {T} p \left(\mathcal {Y} _ {t} = \mathbb {Y} _ {t} | \mathbb {Y} ^ {t - 1}; \gamma\right),\]
where
\[\begin{array}{c} p \left(\mathcal {Y} _ {t} = \mathbb {Y} _ {t} | \mathbb {Y} ^ {t - 1}; \gamma\right) = \\ \int \int \int \int p \left(\mathcal {Y} _ {t} = \mathbb {Y} _ {t} | \mathcal {S} _ {t}, \mathcal {Z} _ {t - 1}, \Sigma_ {t - 1}, \mathcal {E} _ {t}; \gamma\right) p \left(\mathcal {S} _ {t}, \mathcal {Z} _ {t - 1}, \Sigma_ {t - 1}, \mathcal {E} _ {t} | \mathbb {Y} ^ {t - 1}; \gamma\right) d \mathcal {S} _ {t} d \mathcal {Z} _ {t - 1} d \Sigma_ {t - 1} d \mathcal {E} _ {t} \end{array}\tag{16}\]
for all and
\[\begin{array}{c} p \left(\mathcal {Y} _ {1} = \mathbb {Y} _ {1}; \gamma\right) \\ = \int \int \int \int p \left(\mathcal {Y} _ {1} = \mathbb {Y} _ {1} | \mathcal {S} _ {1}, \mathcal {Z} _ {0}, \Sigma_ {0}, \mathcal {E} _ {1}; \gamma\right) p \left(\mathcal {S} _ {1}, \mathcal {Z} _ {0}, \Sigma_ {0}, \mathcal {E} _ {1}; \gamma\right) d \mathcal {S} _ {1} d \mathcal {Z} _ {0} d \Sigma_ {0} d \mathcal {E} _ {1}. \end{array}\tag{17}\]
Note that the do not show up in these two expressions. It will be momentarily clear why this comes directly from our procedure below to evaluate the likelihood.
Computing this likelihood is di¢ cult. Since we do not have analytic forms for the terms inside the integral, it cannot be evaluated exactly and deterministic numerical integration algorithms are too slow for practical use (we have four integrals per period over large dimensions). As a feasible alternative, we will show how to use a simple particle Ölter to obtain a simulation-based estimate of (16). K¸nsch (2005) proves, under weak conditions, that the particle Ölter delivers a consistent estimator of the likelihood function and that a central limit theorem applies. A particle Ölter is a sequential simulation device for Öltering of non-linear and/or non-Gaussian space models (Pitt and Shephard, 1999, and Doucet et al., 2001). In economics the particle Ölter was introduced by Fern·ndez-Villaverde and Rubio-RamÌrez (2007).
As mentioned before, the particle Ölter has minimal requirements: the ability to evaluate the approximated measurement density associated with the approximated measurement equation, to simulate from the approximated dynamics of the state using the approximated transition equation, and to draw from the unconditional density of the states implied by the approximated transition equation. Usually, the Örst requirement is the hardest. These three requirements are formally described in the following assumption.
Assumption 1. To implement the particle Ölter, we assume that:
1. We can evaluate the approximated measurement density
\[p \left(\mathcal {Y} _ {t} = \mathbb {Y} _ {t} | \mathcal {S} _ {t}, \mathcal {Z} _ {t - 1}, \Sigma_ {t - 1}, \mathcal {E} _ {t}; \gamma\right).\]
2. We can simulate from the approximated transition equation
\[\left(\mathcal {S} _ {t + 1} ^ {\prime}, \mathcal {Z} _ {t} ^ {\prime}, \Sigma_ {t} ^ {\prime}, \mathcal {E} _ {t + 1} ^ {\prime}\right) ^ {\prime} | \left(\mathcal {S} _ {t} ^ {\prime}, \mathcal {Z} _ {t - 1} ^ {\prime}, \Sigma_ {t - 1} ^ {\prime}, \mathcal {E} _ {t} ^ {\prime}\right) ^ {\prime}, \mathcal {F} \left(\mathbb {Y} ^ {t}\right); \gamma\]
for all , where is the Öltration of .
3. We can draw from the unconditional distribution implied by the approximated transition equation
\[p \left(\mathcal {S} _ {t}, \mathcal {Z} _ {t - 1}, \Sigma_ {t - 1}, \mathcal {E} _ {t}; \gamma\right).\]
The second requirement asks for the Öltration of , because we will need to evaluate the volatility shocks. The last requirement can be easily implemented using the results in Santos and Peralta-Alva (2005). Given our assumption about and the quadratic form of the approximated transition equation (14), the second requirement easily holds. A key novelty of this paper is that we show how the class of dynamic equilibrium models with volatility shocks considered here also satisÖes the Örst requirement.
For our class of models, conditional on having N draws of from the density , each integral (16) can be consistently approximated by
\[p \left(\mathcal {Y} _ {t} = \mathbb {Y} _ {t} | \mathbb {Y} ^ {t - 1}; \gamma\right) \simeq \frac {1}{N} \sum_ {i = 1} ^ {N} p \left(\mathcal {Y} _ {t} = \mathbb {Y} _ {t} | s _ {t} ^ {i}, z _ {t - 1} ^ {i}, \sigma_ {t - 1} ^ {i}, \varepsilon_ {t} ^ {i}; \gamma\right)\tag{18}\]
for all . For , we need N draws from the density , so that the integral (17) can be consistently approximated by
\[p \left(\mathcal {Y} _ {1} = \mathbb {Y} _ {1}; \gamma\right) \simeq \frac {1}{N} \sum_ {i = 1} ^ {N} p \left(\mathcal {Y} _ {1} = \mathbb {Y} _ {1} | s _ {1} ^ {i}, z _ {0} ^ {i}, \sigma_ {0} ^ {i}, \varepsilon_ {1} ^ {i}; \gamma\right).\tag{19}\]
We know that we can make the draws because requirement 3 in assumption 1 holds.
In our framework, checking whether requirement 1 in assumption 1 holds means checking whether we can evaluate, for each draw in , the approximated measurement density
\[p \left(\mathcal {Y} _ {t} = \mathbb {Y} _ {t} | s _ {t} ^ {i}, z _ {t - 1} ^ {i}, \sigma_ {t - 1} ^ {i}, \varepsilon_ {t} ^ {i}; \gamma\right)\tag{20}\]
for all . This evaluation step is crucial, not only because it is required to compute (18) and (19) but also because, as we will explain momentarily, our particle Ölter needs to evaluate (20) to resample from and get draws from to obtain for all recursively.
To check how we can evaluate the approximated measurement density (20), we rewrite (15) in terms of draws and , instead of and . Thus, the approximated measurement equation (15) becomes
\[\mathbb {Y} _ {t} - \mathcal {Y} = \left( \begin{array}{c} \Psi_ {y, 1} ^ {1} \widehat {\mathbb {S}} _ {t} ^ {i} \\ \vdots \\ \Psi_ {y, k} ^ {1} \widehat {\mathbb {S}} _ {t} ^ {i} \end{array} \right) + \frac {1}{2} \left( \begin{array}{c} \widehat {\mathbb {S}} _ {t} ^ {i \prime} \Psi_ {y, 1} ^ {2} \widehat {\mathbb {S}} _ {t} ^ {i} \\ \vdots \\ \widehat {\mathbb {S}} _ {t} ^ {i \prime} \Psi_ {y, k} ^ {2} \widehat {\mathbb {S}} _ {t} ^ {i} \end{array} \right) + \frac {1}{2} \left( \begin{array}{c} \Psi_ {y, 1} ^ {\Lambda} \\ \vdots \\ \Psi_ {y, k} ^ {\Lambda} \end{array} \right)\tag{21}\]
where and . The new approximated measurement equation (21) implies that evaluating the approximated measurement density (20) involves solving the system of quadratic equations
\[\mathbb {Y} _ {t} - \mathcal {Y} - \left( \begin{array}{c} \Psi_ {y, 1} ^ {1} \widehat {\mathbb {S}} _ {t} ^ {i} \\ \vdots \\ \Psi_ {y, k} ^ {1} \widehat {\mathbb {S}} _ {t} ^ {i} \end{array} \right) - \frac {1}{2} \left( \begin{array}{c} \widehat {\mathbb {S}} _ {t} ^ {i \prime} \Psi_ {y, 1} ^ {2} \widehat {\mathbb {S}} _ {t} ^ {i} \\ \vdots \\ \widehat {\mathbb {S}} _ {t} ^ {i \prime} \Psi_ {y, k} ^ {2} \widehat {\mathbb {S}} _ {t} ^ {i} \end{array} \right) - \frac {1}{2} \left( \begin{array}{c} \Psi_ {y, 1} ^ {\Lambda} \\ \vdots \\ \Psi_ {y, k} ^ {\Lambda} \end{array} \right) = 0\tag{22}\]
for Ut given ; and .
Solving this system is non-trivial. Since the system is quadratic, we may have either no solution or several di§erent ones. In fact, it is even hard to know how many solutions there are. But even if we knew the number of solutions, we are not aware of any accurate and e¢ cient method to solve quadratic problems that Önds all the solutions. This di¢ culty seemingly prevents us from achieving our goal of evaluating the likelihood function.
A solution would be to introduce, as Fern·ndez-Villaverde and Rubio-RamÌrez (2007) did, a vector of linear measurement errors and solve for those instead of . In this case the system would have a unique, easy-to-Önd solution. However, there are three reasons, in increasing order of importance, why this solution is not satisfactory:
1. Although measurement errors are plausible, their presence complicates the interpretation of any empirical results. In particular, we are interested in measuring how heteroscedastic structural shocks help in accounting for the data and measurement errors can confound us.
2. The absence of measurement errors will help us to illustrate below how dynamic equilibrium models with volatility shocks have a profusion of shocks that we can exploit in our estimation.
3. As we will also show below, volatility shocks would enter linearly (conditional on the draw) in the system equations. Since, by deÖnition, linear measurement errors would also enter in the same fashion, it would be hard to identify one apart from the others.
Our alternative in this paper is to realize that considering stochastic volatility converts the above-described quadratic system into a linear one. Hence, if a rank condition is satisÖed, the system (22) has only one solution and the solution can be found by inverting a matrix. Thus, requirement 1 in assumption 1 holds and we can use the particle Ölter. The core of the argument is to note that, when volatility shocks are considered, the policy functions share a peculiar pattern that we can exploit.
3.1. Structure of the Solution
Our Örst step will characterize the Örst- and second-order derivatives of the policy functions h and g evaluated at the steady state. Then, we will describe an interesting pattern in these derivatives. The second step will take advantage of the pattern to show that, when the number of volatility shocks equals the number of observables, our quadratic system becomes a linear one. This characterization is important both for estimation and, more generally, for the analysis of perturbation solutions to dynamic equilibrium models with stochastic volatility.
3.1.1. First- and Second-order Derivatives of the Policy Functions
Let us begin with the characterization of the Örst- and second-order derivatives of the policy functions. The following theorem shows that the Örst-order derivatives of h and g with respect to any component of and evaluated at the steady state are zero; that is, volatility shocks and their innovations do not a§ect the linear component of the optimal decision rule of the agents. The same occurs with the perturbation parameter . A similar result has been established by Schmitt-GrohÈ and Uribe (2004) for the homoscedastic shocks case.
Theorem 2. Let us denote as the derivative of the element of generic function with respect to the element of generic variable ! evaluated at the non-stochastic steady state (where we drop this index if ! is unidimensional). Then, for the dynamic equilibrium model speciÖed in equation (1), we have that for , and
Proof. See appendix 2.1.
The second theorem shows, among other things, that the second partial derivatives of h and g with respect to either log and any other variable but are also zero for any .
Theorem 3. Furthermore, if we denote as the derivative of the element of generic function with respect to the element of generic variable ! and the element of generic variable evaluated at the non-stochastic steady state (where again we drop the index for unidimensional variables), we have that 2 and
\[\left[ h _ {\Lambda , \mathcal {Z} _ {t - 1}} \right] _ {j} ^ {i _ {1}} = \left[ g _ {\Lambda , \mathcal {Z} _ {t - 1}} \right] _ {j} ^ {i _ {2}} = \left[ h _ {\Lambda , \Sigma_ {t - 1}} \right] _ {j} ^ {i _ {1}} = \left[ g _ {\Lambda , \Sigma_ {t - 1}} \right] _ {j} ^ {i _ {2}} = [ h _ {\Lambda , \mathcal {E} _ {t}} ] _ {j} ^ {i _ {1}} = [ g _ {\Lambda , \mathcal {E} _ {t}} ] _ {j} ^ {i _ {2}} = [ h _ {\Lambda , \mathcal {U} _ {t}} ] _ {j} ^ {i _ {1}} = [ g _ {\Lambda , \mathcal {U} _ {t}} ] _ {j} ^ {i _ {2}} = 0\]
for i1 1; : : : ; n , i2 1; : : : ; k , and j 1; : : : ; m ,
\[\left[ h _ {\mathcal {S} _ {t}, \Sigma_ {t - 1}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {1}} = \left[ g _ {\mathcal {S} _ {t}, \Sigma_ {t - 1}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {2}} = \left[ h _ {\mathcal {S} _ {t}, \mathcal {U} _ {t}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {1}} = \left[ g _ {\mathcal {S} _ {t}, \mathcal {U} _ {t}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {2}} = 0\]
\[f o r i _ {1} \in \{1, \dots , n \}, i _ {2} \in \{1, \dots , k \}, j _ {1} \in \{1, \dots , n \}, a n d j _ {2} \in \{1, \dots , m \},\]
\[\left[ h _ {\mathcal {Z} _ {t - 1}, \Sigma_ {t - 1}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {1}} = \left[ g _ {\mathcal {Z} _ {t - 1}, \Sigma_ {t - 1}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {2}} = \left[ h _ {\Sigma_ {t - 1}, \Sigma_ {t - 1}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {1}} = \left[ g _ {\Sigma_ {t - 1}, \Sigma_ {t - 1}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {2}} = 0\]
and
\[\left[ h _ {\mathcal {Z} _ {t - 1}, \mathcal {U} _ {t}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {1}} = \left[ g _ {\mathcal {Z} _ {t - 1}, \mathcal {U} _ {t}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {2}} = \left[ h _ {\Sigma_ {t - 1}, \mathcal {U} _ {t}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {1}} = \left[ g _ {\Sigma_ {t - 1}, \mathcal {U} _ {t}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {2}} = [ h _ {\mathcal {U} _ {t}, \mathcal {U} _ {t}} ] _ {j _ {1}, j _ {2}} ^ {i _ {1}} = [ g _ {\mathcal {U} _ {t}, \mathcal {U} _ {t}} ] _ {j _ {1}, j _ {2}} ^ {i _ {2}} = 0\]
for , and
\[\left[ h _ {\mathcal {E} _ {t}, \Sigma_ {t - 1}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {1}} = \left[ g _ {\mathcal {E} _ {t}, \Sigma_ {t - 1}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {2}} = \left[ h _ {\mathcal {E} _ {t}, \mathcal {U} _ {t}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {1}} = \left[ g _ {\mathcal {E} _ {t}, \mathcal {U} _ {t}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {2}} = 0\]
for , and
Proof. See appendix 2.2.
We clarify the statement of theorem 3 with table 3.1, in which we characterize the second derivatives of the functions h and g with respect to the di§erent variables _ This pattern is both interesting and useful. The way to read table 3.1 is as follows. Take an arbitrary entry, for instance, entry (1,2), . In this entry, we state that the crossderivatives of h and g with respect to and are di§erent from zero (the table is upper triangular because, given the symmetry of second derivatives, we do not need to report those entries). Similarly, entry (3,5), tells us that the cross-derivatives of h and g with respect to and are all zero. Entries (3,4) and (4,5) have a ì*î to denote that the only cross-derivatives of those entries that are di§erent from zero are those that correspond to the same index j (that is, the cross-derivatives of each innovation to the structural shocks with respect to its own volatility shock and the cross-derivatives of the innovation to the structural shocks to the innovation to its own volatility shock). The lower triangular part of the table is empty because of the symmetry of second derivatives.
Table 3.1 tells us that, of the 21 possible sets of second derivatives, 12 are zero and 9 are not. The implications for the decision rules of agents and for the equilibrium function are striking. The perturbation parameter, , will only have a coe¢ cient di§erent from zero in the term where it appears in a square by itself. This term is a constant that corrects for precautionary behavior induced by risk. Volatility shocks, , appear with coe¢ cients di§erent from zero only in the term where they are multiplied by the innovation to its own structural shock . Finally, innovations to the volatility shocks, , also appear with coe¢ cients di§erent from zero when they show up with the innovation to their own structural shock . Hence, of the terms that complicate the evaluation of the approximated measurement density, only the ones associated with and are non-zero.
1 For ease of exposition, in table 3.1 we are not being explicit about the dimensions of the matrices: it is a qualitative description of the relevant derivatives.
3.1.2. Evaluating the Likelihood Using the Particle Filter
The second step is to use theorems 2 and 3 to show that the system (22) is linear and has only one solution. Corollary 4 shows that the pattern described in table 3.1. has an important implication for the structure of the approximated measurement equation (21).
Corollary 4. The approximated measurement equation (21) can be written as
\[\mathbb {Y} _ {t} - \mathcal {Y} = \left( \begin{array}{c} \widetilde {\Psi} _ {y, 1} ^ {1} \widetilde {\widehat {\mathbb {S}}} _ {t} ^ {i} \\ \vdots \\ \widetilde {\Psi} _ {y, k} ^ {1} \widetilde {\widehat {\mathbb {S}}} _ {t} ^ {i} \end{array} \right) + \frac {1}{2} \left( \begin{array}{c} \widetilde {\widehat {\mathbb {S}}} _ {t} ^ {i \prime} \widetilde {\Psi} _ {y, 1} ^ {2, 1} \widetilde {\widehat {\mathbb {S}}} _ {t} ^ {i} \\ \vdots \\ \widetilde {\widehat {\mathbb {S}}} _ {t} ^ {i \prime} \widetilde {\Psi} _ {y, k} ^ {2, 1} \widetilde {\widehat {\mathbb {S}}} _ {t} ^ {i} \end{array} \right) + \frac {1}{2} \left( \begin{array}{c} \Psi_ {y, 1} ^ {\Lambda} \\ \vdots \\ \Psi_ {y, k} ^ {\Lambda} \end{array} \right) + \left( \begin{array}{c} \varepsilon_ {t} ^ {i \prime} \widetilde {\Psi} _ {y, 1} ^ {2, 2} \\ \vdots \\ \varepsilon_ {t} ^ {i \prime} \widetilde {\Psi} _ {y, k} ^ {2, 2} \end{array} \right) \sigma_ {t - 1} ^ {i} + \left( \begin{array}{c} \varepsilon_ {t} ^ {i \prime} \widetilde {\Psi} _ {y, 1} ^ {2, 3} \\ \vdots \\ \varepsilon_ {t} ^ {i \prime} \widetilde {\Psi} _ {y, k} ^ {2, 3} \end{array} \right) \mathcal {U} _ {t}\]
where is the vector of the simulated states without the stochastic volatility components (that is, endogenous states, structural shocks, and their innovations), is its steady state, and is its deviation from the steady state. Let us deÖne . The matrix is vector that represents the Örst-order component of the second-order approximation to the law of motion for the measurement equation as a function of , where and , for . The matrix is an matrix that represents the second-order component of the second-order approximation to the law of motion for the measurement equation as a function of for . The matrix is an matrix that represents the second-order component of the second-order approximation to the law of motion for the measurement equation as a function of and for The matrix is an matrix that represents the second-order term of the approximated law of motion for the measurement equation as a function of and for
We are now ready to show that the system (22) is linear, and if a rank condition is satisÖed, it has only one solution.
DeÖne
\[\begin{array}{c} \mathbb {A} \left(\mathbb {Y} _ {t}, s _ {t} ^ {i}, z _ {t - 1} ^ {i}, \sigma_ {t - 1} ^ {i}, \varepsilon_ {t} ^ {i}; \gamma\right) \\ \equiv \mathbb {Y} _ {t} - \mathcal {Y} - \left( \begin{array}{c} \widetilde {\Psi} _ {y, 1} ^ {1} \widetilde {\widehat {\mathbb {S}}} _ {t} ^ {i} \\ \vdots \\ \widetilde {\Psi} _ {y, k} ^ {1} \widetilde {\widehat {\mathbb {S}}} _ {t} ^ {i} \end{array} \right) - \frac {1}{2} \left( \begin{array}{c} \widetilde {\widehat {\mathbb {S}}} _ {t} ^ {i \prime} \widetilde {\Psi} _ {y, 1} ^ {2, 1} \widetilde {\widehat {\mathbb {S}}} _ {t} ^ {i} \\ \vdots \\ \widetilde {\widehat {\mathbb {S}}} _ {t} ^ {i \prime} \widetilde {\Psi} _ {y, k} ^ {2, 1} \widetilde {\widehat {\mathbb {S}}} _ {t} ^ {i} \end{array} \right) - \frac {1}{2} \left( \begin{array}{c} \Psi_ {y, 1} ^ {\Lambda} \\ \vdots \\ \Psi_ {y, k} ^ {\Lambda} \end{array} \right) - \left( \begin{array}{c} \varepsilon_ {t} ^ {i \prime} \widetilde {\Psi} _ {y, 1} ^ {2, 2} \\ \vdots \\ \varepsilon_ {t} ^ {i \prime} \widetilde {\Psi} _ {y, k} ^ {2, 2} \end{array} \right) \sigma_ {t - 1} ^ {i} \end{array}\]
and
\[\mathbb {B} \left(\varepsilon_ {t} ^ {i}, \gamma\right) \equiv \left( \begin{array}{c c c} \left(\varepsilon_ {t} ^ {i \prime} \widetilde {\Psi} _ {y, 1} ^ {2, 3}\right) ^ {\prime} & \ldots & \left(\varepsilon_ {t} ^ {i \prime} \widetilde {\Psi} _ {y, k} ^ {2, 3}\right) ^ {\prime} \end{array} \right) ^ {\prime}.\]
Let , then is a matrix. If B is full rank, the solution to the system (22) can be written as
\[\mathcal {U} _ {t} = \mathbb {B} ^ {- 1} \left(\varepsilon_ {t} ^ {i}; \gamma\right) \mathbb {A} \left(\mathbb {Y} _ {t}, s _ {t} ^ {i}, z _ {t - 1} ^ {i}, \sigma_ {t - 1} ^ {i}, \varepsilon_ {t} ^ {i}; \gamma\right).\]
The next theorem shows how to use this solution to evaluate the approximated measurement density.
Theorem 5. Let , then B is a matrix. is full rank, the approximated measurement density can be written as
\[p \left(\mathcal {Y} _ {t} = \mathbb {Y} _ {t} | s _ {t} ^ {i}, z _ {t - 1} ^ {i}, \sigma_ {t - 1} ^ {i}, \varepsilon_ {t} ^ {i}; \gamma\right) = \det \mathbb {B} ^ {- 1} \left(\varepsilon_ {t} ^ {i}; \gamma\right) p \left(\mathcal {U} _ {t} = \mathbb {B} ^ {- 1} \left(\varepsilon_ {t} ^ {i}; \gamma\right) \mathbb {A} \left(\mathbb {Y} _ {t}, s _ {t} ^ {i}, z _ {t - 1} ^ {i}, \sigma_ {t - 1} ^ {i}, \varepsilon_ {t} ^ {i}; \gamma\right)\right)\tag{23}\]
for each draw in for all , which can be evaluated given that we know , and the distribution of .
Proof. The theorem is a straightforward application of the change of variables theorem.
Theorem 5 shows that requirement 1 in assumption 1 holds and, thus, we can apply our particle Ölter. We are also requiring to be full rank. When can not be full rank? would fail to have full rank when the impact of volatility innovations is identical across several elements of . This would mean that lacks enough information to tell volatility shocks apart and that, to estimate the model, we need a new set of observables.
Note that theorem 5 assumes . Given the notation in section 2.1, this means that the number of structural shocks equals the number of observables. This is not always necessary. What the theorem needs is that the number of volatility shocks equals the number of observables. Since, to simplify notation, we have assumed that all structural shocks face volatility shocks, the number of structural shocks equals the number of observables. As mentioned in section 2.1, we could have structural shocks that do not face volatility shocks (this will be the case in our application to follow). In that case, we could have more structural shocks than observables. From the theorem, it is also clear that if we used rather than to compute the approximated measurement equation, we would have to solve a quadratic system, a challenging task.
An outline of the algorithm in pseudo-code is:
Step 0: Set . Sample N values from
Step 1: Compute
\[p \left(\mathcal {Y} _ {t} = \mathbb {Y} _ {t} | \mathbb {Y} ^ {t - 1}; \gamma\right) \simeq \frac {1}{N} \sum_ {i = 1} ^ {N} \det \mathbb {B} ^ {- 1} \left(\varepsilon_ {t} ^ {i}; \gamma\right) p \left(\mathcal {U} _ {t} = \mathbb {B} ^ {- 1} \left(\varepsilon_ {t} ^ {i}; \gamma\right) \mathbb {A} \left(\mathbb {Y} _ {t}, s _ {t} ^ {i}, z _ {t - 1} ^ {i}, \sigma_ {t - 1} ^ {i}, \varepsilon_ {t} ^ {i}; \gamma\right)\right)\]
using expression (23) and the importance weights for each draw
\[q _ {t} ^ {i} = \frac {\det \mathbb {B} ^ {- 1} \left(\varepsilon_ {t} ^ {i} ; \gamma\right) p \left(\mathcal {U} _ {t} = \mathbb {B} ^ {- 1} \left(\varepsilon_ {t} ^ {i} ; \gamma\right) \mathbb {A} \left(\mathbb {Y} _ {t} , s _ {t} ^ {i} , z _ {t - 1} ^ {i} , \sigma_ {t - 1} ^ {i} , \varepsilon_ {t} ^ {i} ; \gamma\right)\right)}{\sum_ {i = 1} ^ {N} \det \mathbb {B} ^ {- 1} \left(\varepsilon_ {t} ^ {i} ; \gamma\right) p \left(\mathcal {U} _ {t} = \mathbb {B} ^ {- 1} \left(\varepsilon_ {t} ^ {i} ; \gamma\right) \mathbb {A} \left(\mathbb {Y} _ {t} , s _ { t } ^ {i} , z _ { t - 1 } ^ {i} , \sigma_ {t - 1} ^ {i} , \varepsilon_ {t} ^ {i} ; \gamma\right)\right)}.\]
Step 2: Sample N times with replacement from and probability . This delivers
Step 3: Simulate from the approximated transition equation
\[\left(\mathcal {S} _ {t + 1} ^ {\prime}, \mathcal {Z} _ {t} ^ {\prime}, \Sigma_ {t} ^ {\prime}, \mathcal {E} _ {t + 1} ^ {\prime}\right) ^ {\prime} | \left(s _ {t | t} ^ {i \prime}, z _ {t - 1 | t} ^ {i \prime}, \sigma_ {t - 1 | t} ^ {i \prime}, \varepsilon_ {t | t} ^ {i \prime}\right) ^ {\prime}, \mathcal {F} \left(\mathbb {Y} ^ {t}\right); \gamma .\]
Step 4: If ; set and go to step 1. Otherwise stop.
Once we have evaluated the likelihood, we can nest this algorithm either with an McMc to perform Bayesian inference (as done in our application; see Flury and Shephard, 2011, for technical details) or with some optimization algorithm to undertake maximum likelihood estimation (as done, in a model without volatility shocks, by Van Binsbergen et al., 2012). In this last case, care must be taken to use an optimization algorithm that does not rely on derivatives, as the particle Ölter implies an evaluation of the likelihood function that is not di§erentiable.
4. Comparison with Continuous-Time Models
As is common in the business-cycle literature, we wrote our generic dynamic equilibrium model in discrete time. However, much research in Önance and, increasingly, in macroeconomics is using continuous-time models. Thus, it is useful to sketch how we could adapt our framework to continuous time:
1. We would write the equilibrium conditions of the continuous-time model as in equation (1). The main di§erence with discrete time is that, often, it is not possible to eliminate from those conditions the value functions (and their partial derivatives) of the agents of the model. This is not particularly problematic beyond increasing the number of equilibrium conditions (as we will need to have the concentrated Hamilton-Bellman-Jacobi equations deÖning those value functions in a recursive way).
2. We would solve for the policy functions of the agents using the continuous-time version of the perturbation method outlined in section 2.4 of this paper. Judd (1998, chapters 13 and 14) discusses how to apply perturbation methods to continuous-time dynamic equilibrium models. Parra-£lvarez (2013) applies the method to the solution of business cycle models. The solution of the model would give us a multivariate di§usion process for the evolution of the states (the equivalent of our transition equation 12) and a density for the observables (the equivalent of our transition equation 7).2
3. We would then use the di§usion process and the density to build a continuous-time statespace representation analogous to equations (14) and (15). This is done, for example, in Chib, Pitt, and Shephard (2010) for several Önance models.
2Another possibility would be solving the Hamilton-Bellman-Jacobi equation of the agents using a projection method as in Fern·ndez-Villaverde, Rubio-RamÌrez, and Posch (2013).
4. We would then use the methods proposed by AÔt-Sahalia (2002 and 2008), and AÔt-Sahalia and Kimmel (2007) to Önd closed-form expansions for the log-likelihood function of the model. As documented in those papers, since the coe¢ cients of the expansion are calculated explicitly by exploiting the special structure a§orded by the di§usion model, the computations are fast and e¢ cient. Then, we can either maximize the log-likelihood or nest it inside an McMc algorithm (Stramer, Bognar, and Schneider, 2010).
While this approach is promising, it has not been applied in macroeconomics and it remains to assess its performance in real-life applications. In comparison, the discrete-time approach to the solution and estimation of dynamic equilibrium models has been tested in dozens of empirical applications. Therefore (and in addition to the fact that discrete time is still more popular in macroeconomics), we prefer to explore how to handle dynamic equilibrium models with stochastic volatility Örst in a discrete-time framework and leave the analysis of the continuous-time case for ongoing research.
5. An Application: A Business Cycle Model with Stochastic Volatility
As an illustration of our procedure, we estimate a business cycle model of the U.S. economy with nominal rigidities and stochastic volatility. We will show: 1) how we can characterize posterior distributions of the parameters of interest and 2) how we can recover and analyze the smoothed structural and volatility shocks. Those are two of the most relevant exercises in terms of the estimation of dynamic equilibrium models. Once the estimation has been undertaken, there are many potential exercises. For instance, in the appendix, we include several of them such as: 1) Önding the impulse response functions (IRFs) of the model; 2) evaluating the Öt of the model in comparison with some alternatives; and 3) building counterfactuals and running alternative
histories of the evolution of the U.S. economy.
The model we present is a natural example for this paper because it is the base of much applied policy analysis. We will depart only along two dimensions from the standard speciÖcation: we will have stochastic volatility in the shocks that drive the dynamics of the economy and parameter drifting in the monetary policy rule. In that way, the likelihood has the chance of picking between two of the alternatives that the literature has highlighted to account for the well-documented time-varying volatility of the U.S. aggregate time series, either a reduced volatility of the shocks that hit the economy (Sims and Zha, 2006) or a di§erent monetary policy (Clarida et al., 2000), which makes the application of interest in itself.
5.1. Households
The economy is populated by a continuum of households indexed by j and preferences:
\[\mathbb {E} _ {0} \sum_ {t = 0} ^ {\infty} \beta^ {t} d _ {t} \left\{\log \left(c _ {j t} - h c _ {j t - 1}\right) + v \log \left(\frac {m _ {j t}}{p _ {t}}\right) - \varphi_ {t} \psi \frac {l _ {j t} ^ {1 + \vartheta}}{1 + \vartheta} \right\},\]
which are separable in consumption, , real money balances, ; and hours worked, . In our notation, is the conditional expectations operator, is the discount factor, h controls habit persistence, is the inverse of the Frisch labor supply elasticity, is a shifter to intertemporal preference that follows log where ; and is a labor supply shifter that evolves as log where : The two preference shocks are common to all households.
The principal novelty of these preferences is that, for both shifters and the standard deviations, and , of their innovations, and , are indexed by time; that is, they
stochastically move according to:
\[\log \sigma_ {d t} = \rho_ {\sigma_ {d}} \log \sigma_ {d t - 1} + \left(1 - \rho_ {\sigma_ {d}} ^ {2}\right) ^ {\frac {1}{2}} \eta_ {d} u _ {d t} \mathrm{where} u _ {d t} \sim \mathcal {N} (0, 1)\]
and
\[\log \sigma_ {\varphi t} = \rho_ {\sigma_ {\varphi}} \log \sigma_ {\varphi t - 1} + \left(1 - \rho_ {\sigma_ {\varphi}} ^ {2}\right) ^ {\frac {1}{2}} \eta_ {\varphi} u _ {\varphi t} \text {where} u _ {\varphi t} \sim \mathcal {N} (0, 1).\]
This parsimonious speciÖcation introduces only four new parameters, and , while being surprisingly powerful in capturing important features of the data (Shephard, 2008). All the shocks and innovations throughout the model are observed by the agents when they are realized. Agents have, as well, rational expectations about how these shocks (and all the other shocks to the economy) evolve over time. We can interpret the shocks to preferences and to their volatility as reáecting the random evolution of more complicated phenomena, such as changing demographics (see Fern·ndez-Villaverde and Rubio-Ramirez, 2008).
Although we assume complete Önancial markets, to ease notation we drop the Arrow securities implied by that assumption from the budget constraints (they are in net zero supply at the aggregate level). Households also hold government bonds that pay a nominal gross interest rate of . Therefore, the householdís budget constraint is given by:
\[c _ {j t} + x _ {j t} + \frac {m _ {j t}}{p _ {t}} + \frac {b _ {j t + 1}}{p _ {t}} = w _ {j t} l _ {j t} + \left(r _ {t} u _ {j t} - \mu_ {t} ^ {- 1} \Phi [ u _ {j t} ]\right) k _ {j t - 1} + \frac {m _ {j t - 1}}{p _ {t}} + R _ {t - 1} \frac {b _ {j t}}{p _ {t}} + T _ {t} + F _ {t}\]
where is investment, is the real wage, is the real rental price of capital, is the rate of use of capital, is the cost of using capital at rate in terms of the Önal good, ; is an investment-speciÖc technological level, is a lump-sum transfer, and is ÖrmsíproÖts. We specify that , a form that satisÖes that and . This function carries the normalization that in the balanced growth path of the economy. Using the relevant Örst-order conditions, we can Önd where is the (rescaled) steady-state rental price of capital (determined by the other parameters in the model). This leaves us with only one free parameter,
Given a depreciation rate and an investment adjustment cost parameter the capital accumulated by household at the end of period t is given by:
\[k _ {j t} = (1 - \delta) k _ {j t - 1} + \mu_ {t} \left(1 - \frac {\kappa}{2} \left(\frac {x _ {j t}}{x _ {j t - 1}} - \Lambda_ {x}\right) ^ {2}\right) x _ {j t}.\]
This function is written in deviations with respect to the balanced growth rate of investment, The investment-speciÖc technology level , follows a random walk in logs, log with and where is the drift of the process and is the innovation to its growth rate. The standard deviation also evolves as:
\[\log \sigma_ {\mu t} = \rho_ {\sigma_ {\mu}} \log \sigma_ {\mu t - 1} + \left(1 - \rho_ {\sigma_ {\mu}} ^ {2}\right) ^ {\frac {1}{2}} \eta_ {\mu} u _ {\mu t} \text {where} u _ {\mu t} \sim \mathcal {N} (0, 1).\]
Again, we can interpret this stochastic volatility as a stand-in for a more detailed explanation of technological progress in capital production that we do not model explicitly.
Each household supplies a di§erent type of labor services that are aggregated by a labor packer into homogeneous labor with the production function that is rented to intermediate good producers at wage . The labor packer is perfectly competitive and it takes all wages as given. Households set their wages with a Calvo pricing mechanism. At the start of every period, a randomly selected fraction of households can reoptimize their wages (where, by a law of large numbers, individual probabilities and aggregate fractions are equal). All other households index their wages given past ináation with an indexation parameter
5.2. Firms
There is one Önal good producer that aggregates a continuum of intermediate goods according to:
\[y _ {t} = \left(\int_ {0} ^ {1} y _ {i t} ^ {\frac {\varepsilon - 1}{\varepsilon}} d i\right) ^ {\frac {\varepsilon}{\varepsilon - 1}}\tag{24}\]
where is the elasticity of substitution. The Önal good producer is perfectly competitive and minimizes its costs subject to the production function (24) and taking as given all intermediate goods prices and the Önal good price
Each intermediate good is produced by a monopolistic competitor with technology ; where is the capital rented by the Örm, is the amount of the ìpackedî labor input rented by the Örm, and is neutral productivity. Productivity evolves as log where is the drift of the process and is the innovation to its growth rate. The time-varying standard deviation of this innovation follows:
\[\log \sigma_ {A t} = \rho_ {\sigma_ {A}} \log \sigma_ {A t - 1} + \left(1 - \rho_ {\sigma_ {A}} ^ {2}\right) ^ {\frac {1}{2}} \eta_ {A} u _ {A t} \text {where} u _ {A t} \sim \mathcal {N} (0, 1).\]
Intermediate good producers meet the quantity demanded by the Önal good producer by renting and at prices and . Given their demand function, these producers set prices to maximize proÖts. However, when they do so, they follow a Calvo pricing scheme. In each period, a fraction of intermediate good producers reoptimize their prices. All other Örms partially index their prices by past ináation with an indexation parameter
5.3. The Monetary Authority
A monetary authority sets the nominal interest rate (as a deviation with respect to the balanced growth path nominal interest rate) by following a Taylor rule:
\[\frac {R _ {t}}{R} = \left(\frac {R _ {t - 1}}{R}\right) ^ {\gamma_ {R}} \left(\left(\frac {\Pi_ {t}}{\Pi}\right) ^ {\gamma_ {\Pi} \gamma_ {\Pi , t}} \left(\frac {y _ {t} ^ {d}}{y _ {t - 1} ^ {d}} / \exp \left(\Lambda_ {y ^ {d}}\right)\right) ^ {\gamma_ {y} \gamma_ {y, t}}\right) ^ {1 - \gamma_ {R}} \xi_ {t}.\tag{25}\]
The Örst term on the right-hand side represents a desire for interest rate smoothing. The second term responds to the deviation of ináation from its balanced growth path level . The third term is a ìgrowth gapî: the ratio between the growth rate of the economy and , the balanced path gross growth rate of , where is aggregate demand (deÖned precisely in appendix 3). The last term is the monetary policy shock, where log with an innovation and a time-varying standard deviation, , that follows an autoregressive process
\[\log \sigma_ {\xi t} = \rho_ {\sigma_ {\xi}} \log \sigma_ {\xi t - 1} + \left(1 - \rho_ {\sigma_ {\xi}} ^ {2}\right) ^ {\frac {1}{2}} \eta_ {\xi} u _ {\xi , t} \text {where} u _ {\xi , t} \sim \mathcal {N} (0, 1).\]
In this policy rule, we have two drifting parameters: the responses of the monetary authority to the ináation gap and the growth gap. The parameters drift over time as:
\[\log \gamma_ {\Pi t} = \rho_ {\gamma_ {\Pi}} \log \gamma_ {\Pi t - 1} + \sigma_ {\pi} \varepsilon_ {\pi t} \text {and} \log \gamma_ {y t} = \rho_ {\gamma_ {u}} \log \gamma_ {y t - 1} + \sigma_ {y} \varepsilon_ {y t} \text {where} \varepsilon_ {\pi t}, \varepsilon_ {y t} \sim \mathcal {N} (0, 1).\]
For simplicity, the volatility of the innovation to this processes is Öxed over time. The agents perfectly observe these changes in monetary policy parameters.
5.4. Equilibrium and Solution
We characterize the equilibrium of the model in appendix 3. The equilibrium conditions are nonstationary because we have two unit roots in the processes for technology. We circumvent this problem by rescaling the model using the variable in the form: , and . In the notation of section 2.1 we have that:
1. The states of the (rescaled) economy are
\[\mathcal {S} _ {t} = \left(\log \widetilde {k} _ {t - 1}, \log \widetilde {c} _ {t - 1}, \log \widetilde {x} _ {t - 1}, \log \widetilde {y} _ {t - 1}, \log v _ {t - 1} ^ {p}, \log v _ {t - 1} ^ {w}, \log \widetilde {w} _ {t - 1}, \log R _ {t - 1}, \log \Pi_ {t - 1}\right) ^ {\prime}\]
2. The structural shocks are . The rameter drifts are handled as structural shocks in the state-space representation.
3. The volatility shocks are ; where the last two zeros correspond to the processes for parameter drifting, which have constant volatilities (see also in vector below). Here, we can see that we do not need as many volatility shocks (5 of them) as structural shocks (7 of them).
4. The innovations to the structural and the volatility shocks are and ; respectively.
We pick as observables the Örst di§erence of the log of the relative price of investment, the log federal funds rate, log ináation, the Örst di§erence of log output, and the Örst di§erence of log real wages, in our notation : We select these variables because they bring us information about aggregate behavior (output), the stance of monetary policy (the interest rate and ináation), and the di§erent shocks (the relative price of investment about investment-speciÖc technological change, the other four variables about technology and preference shocks) that we are concerned about. Note that we have the same number of observables as we do of volatility shocks, as required by theorem 5.
Also, note that a second-order approximation is even more relevant in our application than in the standard case of dynamic equilibrium models with stochastic volatility because a linearization would also imply that the parameter drift in the Taylor rule would disappear as well from the equilibrium dynamics (see appendix 4 for more details).
5.5. A Userís Guide to Computing the Results
In this subsection, we outline a userís guide to the computation of our result. The main steps involved are:
1. We rescale the equilibrium conditions of the model to make them stationary and write them in Mathematica.
2. We ask Mathematica to take all the analytic derivatives required to solve for the second-order approximation of the model. We take advantage of the symbolic computation capabilities of Mathematica to express them as functions of parameters of the model. In that way, we do not need to recompute the derivatives, the most time-intensive step, for each set of parameter values in our estimation.
3. Once we have all the relevant derivatives, we export them into Fortran Öles. At this stage, we have a set of Fortran Öles that solves the second-order approximation of the dynamics of the model as a function of the parameters (steps 2 and 3 take about 3 hours).
4. Then, we compile the resulting Öles with the Intel Fortran Compiler version 10:1:025 with IMSL. Compilation takes about 18 hours. The project has 1798 Öles and occupies 2:33 Gbytes of memory.
5. Given some parameter values, we use the derivatives from step 3 to solve for the second-order approximation of the model. For this task, Fortran takes around 5 seconds (remember that we have 9 states, 7 structural shocks, and 5 volatility shocks).
6. We build, in Fortran, the state-space representation associated with the second-order approximation.
7. We approximate the likelihood with the particle Ölter using 10; 000 particles. This number delivered a good compromise between accuracy and time to compute the likelihood. Hence, the solution of the model plus the evaluation of one likelihood requires 22 seconds on a Dell server with 8 processors.
8. We use steps 2 to 7 within a random-walk Metropolis-Hastings algorithm to draw from the posterior of the parameters. Drawing 5,000 times from the posterior takes around 38 hours. In order to initialize the chain, we extensively search on a grid parameter for high values of the likelihood.
The Mathematica and Fortran codes were highly optimized in order to 1) keep the size of the project within reasonable dimensions (otherwise, the compiler cannot parse the Öles) and 2) provide a rapid solution of the model and a fast computation of the likelihood. Perhaps the most important task in that optimization was the parallelization of the Fortran code using OPENMP as well as the compilation options: OG (global optimizations) and Loop Unroll. Without the parallelization, the solution of the model and evaluation of its likelihood take about 70 seconds.
Also, note that implementing corollary 1 needed in step 7 requires the solution of a linear system of equations and the computation of a Jacobian. For our application, we found that the following sequence of LAPACK operations delivered the fastest solution:
1. DGESV (computes the solution to a real system of linear equations A X = B).
2. DGETRI (computes the inverse of a matrix using the LU factorization from the previous line).
3. DGETRF (helps to compute the determinant of the inverse from the previous line).
With respect to the random-walk Metropolis-Hastings, we performed an intensive process of Öne-tuning of the chain, both in terms of initial conditions as well as in terms of getting the right acceptance level. While the tuning was time-intensive, it did not involve any non-standard step. The only other important remark is to remember to keep the random numbers used for resampling in the particle Ölter constant across draws of the Markov chain. This is required to reduce the numerical variance of the procedure, which was a serious concern for us given the complexity of our problem.
5.6. Data and Estimation
We estimate our model using the Öve time series for the U.S. economy described above. Our sample covers 1959.Q1 to 2007.Q1, with 192 observations. We stop at 2007 to avoid having to deal with the Önancial crisis, which would make it di¢ cult to appreciate the points we want to illustrate about how to econometrically deal with stochastic volatility. This could be Öxed at the cost of a lengthier discussion. Appendix 5 explains how we construct the series.
Once we have evaluated the likelihood, we combine it with a prior. We pick áat priors on a bounded support for all the parameters. The bounds are either natural economic restrictions (for instance, the Calvo and indexation parameters lie between 0 and 1) or are so wide that the likelihood assigns (numerically) zero probability to values outside them. Bounded áat priors induce a proper posterior, a convenient feature for our exercises below. We resort to áat priors for two reasons. First, to reduce the impact of presample information and show that our results arise mainly from the shape of the likelihood and not from the prior (although, of course, áat priors are not invariant to reparameterization). Thus, we could interpret our posterior modes as maximum likelihood point estimates. Second, as we learned in Fern·ndez-Villaverde et al. (2010c), eliciting priors for stochastic volatility is di¢ cult, since we deal with unfamiliar units, such as the variance of volatility shocks, about which we do not have clear beliefs. Flat priors come, though, at a price: before proceeding to the estimation, we have to Öx several parameters to reduce the dimensionality of the problem.
Table 5.1 lists the Öxed parameters. Our guiding criterion in selecting them was to pick conventional values. The discount factor, , is a default choice, habit persistence, ， matches the observed sluggish response of consumption to shocks, the parameter controlling the level of labor supply, ; captures the average amount of hours in the data, and the depreciation rate, , induces the appropriate capital-output ratio. The elasticities of substitution, , deliver average mark-ups of around 10 percent, a common value in these models. We set the cost of capital utilization, , to a small number to introduce some curvature in this decision. Three parameter values are borrowed from Fern·ndez-Villaverde et al. (2009). The Örst is the inverse of the Frisch labor elasticity, . This aggregate elasticity is compatible with the micro data, once we allow for intensive and extensive margins on labor supply. The second is the coe¢ cient of the intermediate goods production function, . This value is lower than the common calibration in real business cycle models because, in our environment, we have positive proÖts that appear as capital income in the National Income and Product Accounts. Finally, the adjustment cost, , is in line with other estimates from similar models ( would be particularly hard to identify, since investment is not one of our observables).
The autoregressive parameter of the evolution of the response to ináation, , is set to 0.95. In preliminary estimations, we discovered that the likelihood pushed this parameter to 1. When this happened, the simulations became numerically unstable: after a series of positive innovations to log , the reaction of nominal interest rates to ináation could be too tepid for too long. The 0.95 value seems to be the highest value of such that the problem does not appear. The last two parameters, and ; are equal to zero because, also in exploratory estimations, the likelihood favored values of : Thus, we decided to forget about them and make
To Önd the posterior, we proceed as follows. First, we deÖne a grid of parameter values and check for the regions of high posterior density by evaluating the likelihood function in each point of the grid. This is a time-consuming procedure, but it ensures that we are searching in the right zone of the parameter space. Once we have identiÖed the global maximum in the grid, we initialize a random-walk Metropolis-Hastings algorithm from this point. After an extensive Öne-tuning of the algorithm, we use 10,000 draws from the chain to compute posterior moments.
5.7. Results I: Parameter Estimates
Our Örst empirical result is the parameter estimates. To ease the discussion, we group them in di§erent tables, one for each set of parameters dealing with related aspects of the model. In all cases, we report the mode of the posterior and the standard deviation in parenthesis below (in the interest of space, we do not include the whole histograms of the posterior).
Table 5.2 presents the results for the nominal rigidities and the stochastic processes for the structural shock parameters. Our estimates indicate an economy with substantial rigidities in prices, which are reoptimized roughly once every Öve quarters, and in wages, which are reoptimized approximately every three quarters. Moreover, since the standard deviations are small, there is enough information in the data about this result. At the same time, there is a fair amount of indexation, between 0.62-0.63, which brings a strong persistence of ináation. While it is tempting to compare our estimates with the micro evidence on the individual duration of prices, in our model all prices and wages change every quarter. That is why, to a naive observer, our economy would look like one displaying tremendous price áexibility.
We estimate a low persistence of the intertemporal preference shock and a high persistence of the intratemporal one. The low estimate of produces the quick variations in marginal utilities of consumption that match output growth and ináation áuctuations. The intratemporal shock is persistent to account for long-lived movements in hours worked. We estimate mean growth rates of technology of 0.0034 (neutral) and 0.0028 (investment-speciÖc). Those numbers give us an average growth of the economy of 0.44 percent per quarter (0.46 in the data). Technology shocks, in our model, are deviations with respect to these drifts. Thus, we estimate that falls in only 8 of the 192 quarters in our sample (which roughly corresponds to the percentage of quarters where measured productivity falls in the data), even if we estimate negative innovations to neutral technology in 103 quarters.
The results for the parameters of the stochastic volatility processes appear in table 5.3. In all cases, the ís and the ís are far away from zero: the likelihood strongly favors values where stochastic volatility plays an important role. The standard deviations of the innovations of the intertemporal preference shock and of the monetary policy shock are the most persistent, while the standard deviation of the innovation of the intratemporal preference shock is the least persistent. The standard deviation of the innovations of the volatility shock to the intratemporal preference shock, , is large: the model asks for fast changes in the size of movements in marginal utilities of leisure to reproduce the hours data.
In table 5.4., we have the estimates of the policy parameters. The autoregressive component of the federal funds rate is high, 0:7855, although somewhat smaller than in estimations without parameter drift. The value of (0.24 in levels) is similar to other results in the literature and shows that the likelihood clearly likes parameter drifting, although with mild persistence. The estimated value of plus the correction on equilibrium ináation implied by second-order e§ects of the solution match the average ináation in the data.3 Finally, the estimated value of (1.045 in levels) guarantees local determinacy of the equilibrium even if is temporarily below 1 (see appendix ?? for details).
In appendix 6.2 we plot the impulse response functions of the model implied by our estimates. This exercise allows us to check that the estimates are sensible and that the behavior of the model is consistent with the behavior of related models in the literature. In appendix 6.3 we compare our model against an alternative version without parameter drifting but still with stochastic volatility. That is, we ask whether, once we have included stochastic volatility, it is still important to allow for changes in the monetary policy rule to account for the time-varying volatility of U.S. aggregate data over the last several decades. The results show that, even after controlling for stochastic volatility, the data strongly prefer a speciÖcation where the monetary policy rule has changed over time. In appendix 6.4 we show that this Önding does not imply that volatility shocks did not play an important role in the time-varying volatilities of U.S. aggregate time series. In other words, both stochastic volatility and parameter drifting are key parts of a successful dynamic equilibrium
3Also, these second-order e§ects complicate the introduction of time-variation in . The likelihood wants to match the moments of the ergodic distribution of ináation, not the level of , which is ináation along the balanced growth path. When we have non-linearities, the mean of that ergodic distribution may be far from : Thus, learning about is hard. Learning about a time-varying is even harder.
model of the U.S. economy.
5.8. Results II: Smoothed Shocks
Figure 5.1 reports the log-deviations with respect to their means for the smoothed intertemporal, intratemporal, and monetary shocks and deviations of the growth rate of the investment and technological shocks with respect to their means ( 2 standard deviations). We color di§erent vertical bars to represent each of the periods at the Federal Reserve: the Martin years from the start of our sample in 1959 to the appointment of Burns in February 1970 (white), the Burns-Miller era (light blue), the Volcker interlude from August 1979 to August 1987 (grey), the Greenspan times (orange), and Bernankeís tenure from February 2006 (yellow).
We see in the top left panel of Ögure 5.1 that the intertemporal shock, log ; is particularly high in the 1970s. This increases householdsídesire for current consumption (for instance, because of the entrance of baby boomers into adulthood). A higher aggregate demand triggers, in the model, the higher ináation observed in the data for those years. The shock has a dramatic drop in the second quarter of 1980. This is precisely the quarter in which the Carter administration invoked the Credit Control Act (March 14, 1980). Schreft (1990) documents that this measure caused turmoil in Önancial markets and, most likely, distorted intertemporal choices of households, which is reáected in the large negative innovation to log . The low values of log in the 1990s with respect to the 1970s and 1980s eased the ináationary pressures in the economy.
The shock to the utility of leisure, log grows in the 1970s and falls in the 1980s to stabilize at a very low value in the 1990s. The likelihood wants to track, in this way, the path of average hours worked: low in the 1970s, increasing in the 1980s, and stabilizing in the 1990s. Higher hours also lower the marginal cost of Örms (wages fall relative to the technology level). The reduction in marginal costs also helped to reduce ináation during Greenspanís tenure.
The evolution of the investment-speciÖc technology, log , shows a sharp drop after 1973 (when it is likely that energy-intensive capital goods su§ered the consequences of the oil shocks in the form of economic obsolescence) and large positive realizations in the late 1990s (our model interprets the sustained boom of those years as the consequence of strong improvements in investment technology). These positive realizations were an additional help to contain ináation during the 1990s. In comparison, the neutral-technology shocks, log , have been stable since 1959, with only a few big shocks at the end of the sample.
The evolution of the monetary policy shock, log , reveals large innovations in the early 1980s. This is due both to the fast change in policy brought about by Volcker and to the fact that a Taylor rule might not fully capture the dynamics of monetary policy during a period in which money growth targeting was attempted. Sims and Zha (2006) also Önd that the Volcker period appears to be one with large disturbances to the policy rule and argue that the Taylor rule formalism can be a misleading perspective from which to view policy during that time. Our evidence from the estimated intertemporal, intratemporal, and investment shocks suggests that monetary authorities faced a more di¢ cult environment in the 1970s and early 1980s than in the 1990s.
As a way to gauge the level of uncertainty of our smoothed estimates, we also plot in Ögure 5.1 the same shock (2 standard deviations). In all cases, the data are informative about the history we just narrated.
We plot, in Ögure 5.2, the evolution of the volatility shocks, all of them in log-deviations with respect to their estimated means (plus/minus two standard deviations). We see in this Ögure that the standard deviation of the intertemporal shock was particularly high in the 1970s and only slowly went down during the 1980s and early 1990s. By the end of the sample, the standard deviation of the intertemporal shock was roughly at the level where it started. In comparison, the standard deviation of all the other shocks is relatively stable except, perhaps, for a large drop in the standard deviation of the monetary policy shock in the early 1980s as well as large changes in the standard deviation of the investment shock during the period of the oil price shocks. Hence, the 1970s and the 1980s were more volatile than the 1960s and the 1990s, creating a tougher environment for monetary policy. This result also conÖrms Blanchard and Simonís (2001) and Nason and Smithís (2008) observation that volatility had a downward trend in the 20th century with an abrupt and temporal increase in the 1970s. Also, from the size of the plus/minus two standard deviations, we conclude that the big movements in the di§erent series that we report can be ascertained with a reasonable degree of conÖdence.
Finally, in Ögure 5.3, we plot the evolution of the response of monetary policy to ináation plus/minus a two-standard-deviation interval. In particular, we graph . This graph shows us an intriguing narrative. The parameter started the sample around its estimated mean, slightly over 1, and it grew more or less steadily during the 1960s until reaching a peak in early 1968. After that year, it su§ered a fast collapse that took it below 1 in 1971. To put this evolution in perspective, it is useful to remember that Burns was appointed chairman in February 1970. The parameter stayed below 1 for all of the 1970s. The arrival of Volcker is quickly picked up by our smoothed estimates: it increases to over 2 after a few months and stays high during all the years of Volckerís tenure. Our estimate captures well the observation by Goodfriend and King (2007) that monetary policy tightened in the spring of 1980 as ináation and long-run ináation expectations continued to grow. Its level stayed roughly constant at this high during the remainder of Volckerís tenure. But as quickly as it rose when Volcker arrived, it went down again when he departed. Greenspanís tenure at the Fed meant that, by 1990, the response of the monetary authority to ináation was again below 1. Moreover, our estimates are relatively tight. Fern·ndez-Villaverde et al. (2010a) discuss how the results of the estimation relate to historical evidence.
6. Conclusion
In this paper, we have shown how to estimate dynamic equilibrium models with stochastic volatility. The key to the procedure is to realize that a second-order perturbation to the solution of this class of models has a very particular structure that can be easily exploited to build an e¢ cient particle Ölter. The recent boom in the literature on dynamic equilibrium models with stochastic volatility suggests that this procedure may have many uses. Our characterization of the solution might also be, on many occasions, of interest in itself to understand the dynamic properties of the equilibrium even if the researcher does not want to estimate the model.
As an application to illustrate how the procedure works we have estimated a business cycle model with both stochastic volatility in the structural shocks that drive the economy and parameter drifting in the monetary policy rule. Such a model is motivated by the need to have an empirical framework where we can account for the time-varying volatility of U.S. aggregate time series. In particular, we have explained how you obtain point estimates in such a model and how to recover and analyze the smoothed structural and volatility shocks. Finally, through di§erent comments -even if brief and not exhaustive- we have discussed the di§erent empirical lessons that one can learn from all these steps.
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Table 3.1: Second Derivatives
| $\mathcal{S}_{t}\mathcal{S}_{t} \neq 0$ | $\mathcal{S}_{t}\mathcal{Z}_{t-1} \neq 0$ | $\mathcal{S}_{t}\Sigma_{t-1} = 0$ | $\mathcal{S}_{t}\mathcal{E}_{t} \neq 0$ | $\mathcal{S}_{t}\mathcal{U}_{t} = 0$ | $\mathcal{S}_{t}\Lambda = 0$ |
| $\mathcal{Z}_{t-1}\mathcal{Z}_{t-1} \neq 0$ | $\mathcal{Z}_{t-1}\Sigma_{t-1} = 0$ | $\mathcal{Z}_{t-1}\mathcal{E}_{t} \neq 0$ | $\mathcal{Z}_{t-1}\mathcal{U}_{t} = 0$ | $\mathcal{Z}_{t-1}\Lambda = 0$ | |
| $\Sigma_{t-1}\Sigma_{t-1} = 0$ | $\Sigma_{t-1}\mathcal{E}_{t} \neq 0^{*}$ | $\Sigma_{t-1}\mathcal{U}_{t} = 0$ | $\Sigma_{t-1}\Lambda = 0$ | ||
| $\mathcal{E}_{t}\mathcal{E}_{t} \neq 0$ | $\mathcal{E}_{t}\mathcal{U}_{t} \neq 0^{*}$ | $\mathcal{E}_{t}\Lambda = 0$ | |||
| $\mathcal{U}_{t}\mathcal{U}_{t} = 0$ | $\mathcal{U}_{t}\Lambda = 0$ | ||||
| $\Lambda\Lambda \neq 0$ |
Table 5.1: Fixed Parameters
| $\beta$ | h | $\psi$ | $\vartheta$ | $\delta$ | $\alpha$ | $\kappa$ | $\varepsilon$ | $\eta$ | $\Phi_{2}$ | $\rho_{\gamma_{\Pi}}$ | $\rho_{\gamma_{y}}$ | $\sigma_{y}$ |
| 0.99 | 0.9 | 8 | 1.17 | 0.025 | 0.21 | 9.5 | 10 | 10 | 0.001 | 0.95 | 0 | 0 |
Table 5.2: Posterior, Parameters of Nominal Rigidities and Structural Shocks
| $\theta_p$ | $\chi$ | $\theta_w$ | $\chi_w$ | $\rho_d$ | $\rho_\varphi$ | $\Lambda_\mu$ | $\Lambda_A$ |
| 0.8139(0.0143) | 0.6186(0.024) | 0.6869(0.0432) | 0.6340(0.0074) | 0.1182(0.0049) | 0.9331(0.0425) | 0.0034(6.6e-5) | 0.0028(4.1e-5) |
Table 5.3: Posterior, Parameters of the Stochastic Processes for Volatility Shocks
| $\log \sigma_d$ | $\log \sigma_\varphi$ | $\log \sigma_\mu$ | $\log \sigma_A$ | $\log \sigma_\xi$ |
| -1.9834(0.0726) | -2.4983(0.0917) | -6.0283(0.1278) | -3.9013(0.0745) | -6.000(0.1471) |
| $\rho_{\sigma_d}$ | $\rho_{\sigma_\varphi}$ | $\rho_{\sigma_\mu}$ | $\rho_{\sigma_a}$ | $\rho_{\sigma_\xi}$ |
| 0.9506(0.0298) | 0.1275(0.0032) | 0.7508(0.035) | 0.2411(0.005) | 0.8550(0.0231) |
| $\eta_d$ | $\eta_\varphi$ | $\eta_\mu$ | $\eta_a$ | $\eta_\xi$ |
| 0.1007(0.0083) | 2.8316(0.0669) | 0.3115(0.006) | 0.7720(0.013) | 0.5723(0.0185) |
Inter. Shock +/- 2 Std. Dev.
Table 5.4: Posterior, Policy Parameters
| $\gamma_R$ | $\log \gamma_y$ | $\Pi$ | $\log \gamma_{\Pi}$ | $\sigma_\pi$ |
| 0.7855(0.0162) | -1.4034(0.0498) | 1.0005(0.0043) | 0.0441(0.0005) | 0.145(0.002) |
Figure 5.1: Smoothed intertemporal shock, intratemporal shock, investment-speciÖc shock, technology shock, and monetary policy shock
Std. Dev. Inter. Shock +/- 2 Std. Dev. Std. Dev. Tech. Shock +/- 2 Std. Dev.
Std. Dev. Intra. Shock +/- 2 Std. Dev.
Std. Dev. Invest. Shock +/- 2 Std. Dev.
Std. Dev. Mon. Shock +/- 2 Std. Dev.
Figure 5.2: Smoothed standard deviation shocks to the intertemporal shock, the intratemporal shock, the investment-speciÖc shock, the technology shock, and the monetary policy shock +/- 2 s.d. Drift on Taylor Rule Param. on Inflation +/- 2 Std. Dev. Figure 5.3: Smoothed path for the Taylor rule parameter on ináation standard deviations.
1. Technical Appendix (Not for Publication)
This technical appendix is organized as follows. First, it presents the proofs of theorems 2 and 3. Second, it describes the equilibrium of the model in more detail. Third, it shows how parameter drifting in the monetary policy rule does not appear in the Örst-order approximation. Fourth, it o§ers details on how we built the data. Finally, it includes some additional empirical results.
2. Proofs
2.1. Theorem 2
We start by proving theorem 2, which characterizes the Örst-order derivatives of the policy functions h and g evaluated at the steady state. We Örst show that the Örst partial derivatives of h and g with respect to any component of , or evaluated at the steady state are zero (in other words, that the Örst-order approximation of the policy functions do not depend on volatility shocks nor their innovations nor on the perturbation parameter). Before proceeding, note that using (2), we can write , in a compact manner, as a function of , and
\[\mathcal {Z} _ {t + 1} = \varsigma \left(\mathcal {Z} _ {t}, \Sigma_ {t}, \Lambda \mathcal {E} _ {t + 1}, \Lambda \mathcal {U} _ {t + 1}; \gamma\right),\tag{26}\]
that using can be expressed as a function of , and
\[\Sigma_ {t + 1} = \vartheta \Sigma_ {t} + \eta \Lambda \mathcal {U} _ {t + 1},\tag{27}\]
that using (4) we can write as a function of ; and
\[\mathcal {Z} _ {t} = \varsigma \left(\mathcal {Z} _ {t - 1}, \Sigma_ {t - 1}, \mathcal {E} _ {t}, \mathcal {U} _ {t}; \gamma\right),\tag{28}\]
and that using (5) can be expressed as
\[\Sigma_ {t} = \vartheta \Sigma_ {t - 1} + \eta \mathcal {U} _ {t}\tag{29}\]
where # and are both m m diagonal matrices with diagonal elements equal to and respectively. If we substitute the policy functions (6)-(8) and (26)-(29) into the
set of equilibrium conditions (1), we get that
\[\begin{array}{c} F \left(\mathcal {S} _ {t}, \mathcal {Z} _ {t - 1}, \Sigma_ {t - 1}, \mathcal {E} _ {t}, \mathcal {U} _ {t}, \Lambda\right) \equiv \\ \mathbb {E} _ {t} f \left( \begin{array}{c} g \left(h \left(\mathcal {S} _ {t}, \mathcal {Z} _ {t - 1}, \Sigma_ {t - 1}, \mathcal {E} _ {t}, \mathcal {U} _ {t}, \Lambda\right), \varsigma \left(\mathcal {Z} _ {t - 1}, \Sigma_ {t - 1}, \mathcal {E} _ {t}, \mathcal {U} _ {t}\right), \vartheta \Sigma_ {t - 1} + \eta \mathcal {U} _ {t}, \Lambda \mathcal {E} _ {t + 1}, \Lambda \mathcal {U} _ {t + 1}, \Lambda\right), \\ g \left(\mathcal {S} _ {t}, \mathcal {Z} _ {t - 1}, \Sigma_ {t - 1}, \mathcal {E} _ {t}, \mathcal {U} _ {t}, \Lambda\right), h \left(\mathcal {S} _ {t}, \mathcal {Z} _ {t - 1}, \Sigma_ {t - 1}, \mathcal {E} _ {t}, \mathcal {U} _ {t}, \Lambda\right), \mathcal {S} _ {t}, \\ \varsigma \left(\varsigma \left(\mathcal {Z} _ {t - 1}, \Sigma_ {t - 1}, \mathcal {E} _ {t}, \mathcal {U} _ {t}\right), \vartheta \Sigma_ {t - 1} + \eta \mathcal {U} _ {t}, \Lambda \mathcal {E} _ {t + 1}, \Lambda \mathcal {U} _ {t+ 1}\right), \varsigma \left(\mathcal {Z} _ {t - 1}, \Sigma_ {t - 1}, \mathcal {E} _ {t}, \mathcal {U} _ {t}\right) \end{array} \right) = 0 \end{array}\]
where, to ease notation, we do not explicitly write that the functions above depend on .
Proof. We want to show that
\[\left[ h _ {\Sigma_ {t - 1}} \right] _ {j} ^ {i _ {1}} = \left[ g _ {\Sigma_ {t - 1}} \right] _ {j} ^ {i _ {2}} = [ h _ {\mathcal {U} _ {t}} ] _ {j} ^ {i _ {1}} = [ g _ {\mathcal {U} _ {t}} ] _ {j} ^ {i _ {2}} = [ h _ {\Lambda} ] ^ {i _ {1}} = [ g _ {\Lambda} ] ^ {i _ {2}} = 0\]
for , and
We show this result in three steps that basically repeat the same argument based on the homogeneity of a system of linear equations:
1. We write the derivative of the element of F with respect to the element of as
\[\left[ F _ {\Sigma_ {t - 1}} \right] _ {j} ^ {i} = \left[ f _ {\mathcal {Y} _ {t + 1}} \right] _ {i _ {2}} ^ {i} \left(\left[ g _ {\mathcal {S} _ {t}} \right] _ {i _ {1}} ^ {i _ {2}} \left[ h _ {\Sigma_ {t - 1}} \right] _ {j} ^ {i _ {1}} + \left[ g _ {\Sigma_ {t - 1}} \right] _ {j} ^ {i _ {2}} \vartheta_ {j}\right) + \left[ f _ {\mathcal {Y} _ {t}} \right] _ {i _ {2}} ^ {i} \left[ g _ {\Sigma_ {t - 1}} \right] _ {j} ^ {i _ {2}} + \left[ f _ {\mathcal {S} _ {t + 1}} \right] _ {i _ {1}} ^ {i} \left[ h _ {\Sigma_ {t - 1}} \right] _ {j} ^ {i _ {1}} = 0\]
for and . This is a homogeneous system on and for , and . Thus
\[\left[ h _ {\Sigma_ {t - 1}} \right] _ {j} ^ {i _ {1}} = \left[ g _ {\Sigma_ {t - 1}} \right] _ {j} ^ {i _ {2}} = 0\]
for
2. We write the derivative of the element of F with respect to the element of as
\[[ F _ {\mathcal {U} _ {t}} ] _ {j} ^ {i} = \left[ f _ {\mathcal {Y} _ {t + 1}} \right] _ {i _ {2}} ^ {i} \left([ g _ {\mathcal {S} _ {t}} ] _ {i _ {1}} ^ {i _ {2}} [ h _ {\mathcal {U} _ {t}} ] _ {j} ^ {i _ {1}} + \left[ g _ {\Sigma_ {t - 1}} \right] _ {j} ^ {i _ {2}} \left(1 - \vartheta_ {j} ^ {2}\right) ^ {\frac {1}{2}} \eta_ {j}\right) + [ f _ {\mathcal {Y} _ {t}} ] _ {i _ {2}} ^ {i} [ g _ {\mathcal {U} _ {t}} ] _ {j} ^ {i _ {2}} + \left[ f _ {\mathcal {S} _ {t + 1}} \right] _ {i _ {1}} ^ {i} [ h _ {\mathcal {U} _ {t}} ] _ {j} ^ {i _ {1}} = 0\]
for and . Since we have already shown that
0 for and , this is a homogeneous system on and for , and . Thus
\[[ h _ {\mathcal {U} _ {t}} ] _ {j} ^ {i _ {1}} = [ g _ {\mathcal {U} _ {t}} ] _ {j} ^ {i _ {2}} = 0\]
for , and
3. Finally, we write the derivative of the element of F with respect to as
\[\left[ F _ {\Lambda} \right] ^ {i} = \left[ f _ {\mathcal {Y} _ {t + 1}} \right] _ {i _ {2}} ^ {i} \left(\left[ g _ {\mathcal {S} _ {t}} \right] _ {i _ {1}} ^ {i _ {2}} \left[ h _ {\Lambda} \right] ^ {i _ {1}} + \left[ g _ {\Lambda} \right] ^ {i _ {2}}\right) + \left[ f _ {\mathcal {Y} _ {t}} \right] _ {i _ {2}} ^ {i} \left[ g _ {\Lambda} \right] ^ {i _ {2}} + \left[ f _ {\mathcal {S} _ {t + 1}} \right] _ {i _ {1}} ^ {i} \left[ h _ {\Lambda} \right] ^ {i _ {1}} = 0\]
for . Since this is a homogeneous system on and for and , we have that
\[\left[ h _ {\Lambda} \right] ^ {i _ {1}} = \left[ g _ {\Lambda} \right] ^ {i _ {2}} = 0\]
for and
2.2. Theorem 3
Let us now prove theorem 3. We show, among other things, that the second partial derivatives of h and g with respect to either log and any other variable but are also zero for any . We divide the proof into three parts.
Proof, part 1. The Örst part of the proof deals with the cross-derivatives of the policy functions h and g with respect to and any of , or and it shows that all of them are equal to zero. In particular, we want to show that
\[[ h _ {\Lambda , \mathcal {S} _ {t}} ] _ {j} ^ {i _ {1}} = [ g _ {\Lambda , \mathcal {S} _ {t}} ] _ {j} ^ {i _ {2}} = 0\]
for , and and
\[\left[ h _ {\Lambda , \mathcal {Z} _ {t - 1}} \right] _ {j} ^ {i _ {1}} = \left[ g _ {\Lambda , \mathcal {Z} _ {t - 1}} \right] _ {j} ^ {i _ {2}} = \left[ h _ {\Lambda , \Sigma_ {t - 1}} \right] _ {j} ^ {i _ {1}} = \left[ g _ {\Lambda , \Sigma_ {t - 1}} \right] _ {j} ^ {i _ {2}} = [ h _ {\Lambda , \mathcal {E} _ {t}} ] _ {j} ^ {i _ {1}} = [ g _ {\Lambda , \mathcal {E} _ {t}} ] _ {j} ^ {i _ {2}} = [ h _ {\Lambda , \mathcal {U} _ {t}} ] _ {j} ^ {i _ {1}} = [ g _ {\Lambda , \mathcal {U} _ {t}} ] _ {j} ^ {i _ {2}} = 0\]
for
We show this result in Öve steps. We again exploit the homogeneity of a system of linear equations.
1. We consider the cross-derivative of the element of F with respect to and the element of
\[[ F _ {\Lambda , \mathcal {S} _ {t}} ] _ {j} ^ {i} = \left[ f _ {\mathcal {Y} _ {t + 1}} \right] _ {i _ {2}} ^ {i} \left([ g _ {\mathcal {S} _ {t}} ] _ {i _ {1}} ^ {i _ {2}} [ h _ {\Lambda , \mathcal {S} _ {t}} ] _ {j} ^ {i _ {1}} + [ g _ {\Lambda , \mathcal {S} _ {t}} ] _ {i _ {1}} ^ {i _ {2}} [ h _ {\mathcal {S} _ {t}} ] _ {j} ^ {i _ {1}}\right) + [ f _ {\mathcal {Y} _ {t}} ] _ {i _ {2}} ^ {i} [ g _ {\Lambda , \mathcal {S} _ {t}} ] _ {j} ^ {i _ {2}} + \left[ f _ {\mathcal {S} _ {t + 1}} \right] _ {i _ {1}} ^ {i} [ h _ {\Lambda , \mathcal {S} _ {t}} ] _ {j} ^ {i _ {1}} = 0\]
for and . This is a homogeneous system on and for , and . Thus
\[\left[ h _ {\Lambda , \mathcal {S} _ {t}} \right] _ {j} ^ {i _ {1}} = \left[ g _ {\Lambda , \mathcal {S} _ {t}} \right] _ {j} ^ {i _ {2}} = 0\]
for
2. We consider the cross-derivative of the element of F with respect to and the element of
\[\begin{array}{c} \left[ F _ {\Lambda , \mathcal {Z} _ {t - 1}} \right] _ {j} ^ {i} = \left[ f _ {\mathcal {Y} _ {t + 1}} \right] _ {i _ {2}} ^ {i} \left([ g _ {\mathcal {S} _ {t}} ] _ {i _ {1}} ^ {i _ {2}} \left[ h _ {\Lambda , \mathcal {Z} _ {t - 1}} \right] _ {j} ^ {i _ {1}} + [ g _ {\Lambda , \mathcal {S} _ {t}} ] _ {i _ {1}} ^ {i _ {2}} \left[ h _ {\mathcal {Z} _ {t - 1}} \right] _ {j} ^ {i _ {1}} + \left[ g _ {\Lambda , \mathcal {Z} _ {t - 1}} \right] _ {j} ^ {i _ {2}} \rho_ {j}\right) \\ + [ f _ {\mathcal {Y} _ {t}} ] _ {i _ {2}} ^ {i} \left[ g _ {\Lambda , \mathcal {Z} _ {t - 1}} \right] _ {j} ^ {i _ {2}} + \left[ f _ {\mathcal {S} _ {t + 1}} \right] _ {i _ {1}} ^ {i} \left[ h _ {\Lambda , \mathcal {Z} _ {t - 1}} \right] _ {j} ^ {i _ {1}} = 0 \end{array}\]
for and . Since for and , this is a homogeneous system on and for ， , and . Hence
\[\left[ h _ {\Lambda , \mathcal {Z} _ {t - 1}} \right] _ {j} ^ {i _ {1}} = \left[ g _ {\Lambda , \mathcal {Z} _ {t - 1}} \right] _ {j} ^ {i _ {2}} = 0\]
for , and
3. We consider the cross-derivative of the element of F with respect to and the
element of
\[\begin{array}{c} \left[ F _ {\Lambda , \Sigma_ {t - 1}} \right] _ {j} ^ {i} = \left[ f y _ {t + 1} \right] _ {i _ {2}} ^ {i} \left(\left[ g _ {\mathcal {S} _ {t}} \right] _ {i _ {1}} ^ {i _ {2}} \left[ h _ {\Lambda , \Sigma_ {t - 1}} \right] _ {j} ^ {i _ {1}} + \left[ g _ {\Lambda , \Sigma_ {t - 1}} \right] _ {j} ^ {i _ {2}} \vartheta_ {j}\right) \\ + \left[ f y _ {t} \right] _ {i _ {2}} ^ {i} \left[ g _ {\Lambda , \Sigma_ {t - 1}} \right] _ {j} ^ {i _ {2}} + \left[ f _ {\mathcal {S} _ {t + 1}} \right] _ {i _ {1}} ^ {i} \left[ h _ {\Lambda , \Sigma_ {t - 1}} \right] _ {j} ^ {i _ {1}} = 0 \end{array}\]
for and . This is a homogeneous system on and for , and . Hence
\[\left[ h _ {\Lambda , \Sigma_ {t - 1}} \right] _ {j} ^ {i _ {1}} = \left[ g _ {\Lambda , \Sigma_ {t - 1}} \right] _ {j} ^ {i _ {2}} = 0\]
for , and
4. We consider the cross-derivative of the element of F with respect to and the element of
\[\begin{array}{r} [ F _ {\Lambda , \mathcal {E} _ {t}} ] _ {j} ^ {i} = [ f _ {\mathcal {Y} _ {t + 1}} ] _ {i _ {2}} ^ {i} \left([ g _ {\mathcal {S} _ {t}} ] _ {i _ {1}} ^ {i _ {2}} [ h _ {\Lambda , \mathcal {E} _ {t}} ] _ {j} ^ {i _ {1}} + [ g _ {\Lambda , \mathcal {S} _ {t}} ] _ {i _ {1}} ^ {i _ {2}} [ h _ {\mathcal {E} _ {t}} ] _ {j} ^ {i _ {1}} + [ g _ {\Lambda , \mathcal {Z} _ {t - 1}} ] _ {j} ^ {i _ {2}} \sigma_ {j} \exp^ {\vartheta_ {j} \log \sigma_ {j, t - 1}}\right) \\ + [ f _ {\mathcal {Y} _ {t}} ] _ {i _ {2}} ^ {i} [ g _ {\Lambda , \mathcal {E} _ {t}} ] _ {j} ^ {i _ {2}} + [ f _ {\mathcal {S} _ {t + 1}} ] _ {i _ {1}} ^ {i} [ h _ {\Lambda , \mathcal {E} _ {t}} ] _ {j} ^ {i _ {1}} = 0 \end{array}\]
for and . Since for and and for and , this is a homogeneous system on and for , and . Thus
\[\left[ h _ {\Lambda , \mathcal {E} _ {t}} \right] _ {j} ^ {i _ {1}} = \left[ g _ {\Lambda , \mathcal {E} _ {t}} \right] _ {j} ^ {i _ {2}} = 0\]
for
5. We consider the cross-derivative of the element of F with respect to and the element of
\[\begin{array}{c} [ F _ {\Lambda , \mathcal {U} _ {t}} ] _ {j} ^ {i} = \left[ f _ {\mathcal {Y} _ {t + 1}} \right] _ {i _ {2}} ^ {i} \left([ g _ {\mathcal {S} _ {t}} ] _ {i _ {1}} ^ {i _ {2}} [ h _ {\Lambda , \mathcal {U} _ {t}} ] _ {j} ^ {i _ {1}} + \left[ g _ {\Lambda , \Sigma_ {t - 1}} \right] _ {j} ^ {i _ {2}} \left(1 - \vartheta_ {j} ^ {2}\right) ^ {\frac {1}{2}} \eta_ {j}\right) \\ + [ f _ {\mathcal {Y} _ {t}} ] _ {i _ {2}} ^ {i} [ g _ {\Lambda , \mathcal {U} _ {t}} ] _ {j} ^ {i _ {2}} + \left[ f _ {\mathcal {S} _ {t + 1}} \right] _ {i _ {1}} ^ {i} [ h _ {\Lambda , \mathcal {U} _ {t}} ] _ {j} ^ {i _ {1}} = 0 \end{array}\]
for and . Since we have shown that for and ; we have that the above system is a homogeneous system on and for , and . Then
\[[ h _ {\Lambda , \mathcal {U} _ {t}} ] _ {j} ^ {i _ {1}} = [ g _ {\Lambda , \mathcal {U} _ {t}} ] _ {j} ^ {i _ {2}} = 0\]
for
Proof, part 2. The second part of the proof deals with the cross-derivatives of the policy functions h and with respect to and any of , or and it shows that all of them are equal to zero with one exception. In particular, we want to show that
\[\left[ h _ {\mathcal {S} _ {t}, \Sigma_ {t - 1}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {1}} = \left[ g _ {\mathcal {S} _ {t}, \Sigma_ {t - 1}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {2}} = 0\]
for , and ，
\[\left[ h _ {\mathcal {Z} _ {t - 1}, \Sigma_ {t - 1}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {1}} = \left[ g _ {\mathcal {Z} _ {t - 1}, \Sigma_ {t - 1}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {2}} = \left[ h _ {\Sigma_ {t - 1}, \Sigma_ {t - 1}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {1}} = \left[ g _ {\Sigma_ {t - 1}, \Sigma_ {t - 1}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {2}} = 0\]
for , and , and
\[\left[ h _ {\mathcal {E} _ {t}, \Sigma_ {t - 1}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {1}} = \left[ g _ {\mathcal {E} _ {t}, \Sigma_ {t - 1}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {2}} = 0\]
for , and
We show this result in four steps (and where we have already taken advantage of the terms that we know to be equal to zero from previous derivations).
1. We consider the cross-derivative of the element of with respect to the element of and the element of
\[\begin{array}{r} \left[ F _ {\mathcal {S} _ {t}, \Sigma_ {t - 1}} \right] _ {j _ {1}, j _ {2}} ^ {i} = \left[ f _ {\mathcal {Y} _ {t + 1}} \right] _ {i _ {2}} ^ {i} \left([ g _ {\mathcal {S} _ {t}} ] _ {i _ {1}} ^ {i _ {2}} \left[ h _ {\mathcal {S} _ {t}, \Sigma_ {t - 1}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {1}} + \left[ g _ {\mathcal {S} _ {t}, \Sigma_ {t - 1}} \right] _ {i _ {1}, j _ {2}} ^ {i _ {2}} [ h _ {\mathcal {S} _ {t}} ] _ {j _ {1}} ^ {i _ {1}} \vartheta_ {j _ {2}}\right) \\ + [ f _ {\mathcal {Y} _ {t}} ] _ {i _ {2}} ^ {i} \left[ g _ {\mathcal {S} _ {t}, \Sigma_ {t - 1}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {2}} + \left[ f _ {\mathcal {S} _ {t + 1}} \right] _ {i _ {1}} ^ {i} \left[ h _ {\mathcal {S} _ {t}, \Sigma_ {t - 1}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {1}} = 0 \end{array}\]
for , and . This is a homogeneous system on and ，
and . Therefore
\[\left[ h _ {\mathcal {S} _ {t}, \Sigma_ {t - 1}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {1}} = \left[ g _ {\mathcal {S} _ {t}, \Sigma_ {t - 1}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {2}} = 0\]
for
2. We consider the cross-derivative of the element of F with respect to the element of and the element of
\[\begin{array}{c} \left[ F _ {\mathcal {Z} _ {t - 1}, \Sigma_ {t - 1}} \right] _ {j _ {1}, j _ {2}} ^ {i} \\ = \left[ f _ {\mathcal {Y} _ {t + 1}} \right] _ {i _ {2}} ^ {i} \left([ g _ {\mathcal {S} _ {t}} ] _ {i _ {1}} ^ {i _ {2}} \left[ h _ {\mathcal {Z} _ {t - 1}, \Sigma_ {t - 1}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {1}} + \left[ g _ {\mathcal {S} _ {t}, \Sigma_ {t - 1}} \right] _ {i _ {1}, j _ {2}} ^ {i _ {2}} \left[ h _ {\mathcal {Z} _ {t - 1}} \right] _ {j _ {1}} ^ {i _ {1}} \vartheta_ {j _ {2}} + \left[ g _ {\mathcal {Z} _ {t - 1}, \Sigma_ {t - 1}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {2}} \rho_ {j _ {1}} \vartheta_ {j _ {2}}\right) \\ + [ f _ {\mathcal {Y} _ {t}} ] _ {i _ {2}} ^ {i} \left[ g _ {\mathcal {Z} _ {t - 1}, \Sigma_ {t - 1}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {2}} + \left[ f _ {\mathcal {S} _ {t + 1}} \right] _ {i _ {1}} ^ {i} \left[ h _ {\mathcal {Z} _ {t - 1}, \Sigma_ {t - 1}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {1}} = 0 \end{array}\]
for , and . Since we just found that 0 for , and , this is a homogeneous system on and for , and . Therefore
\[\left[ h _ {\mathcal {Z} _ {t - 1}, \Sigma_ {t - 1}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {1}} = \left[ g _ {\mathcal {Z} _ {t - 1}, \Sigma_ {t - 1}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {2}} = 0\]
for
3. We consider the cross-derivative of the element of F with respect to the element of and the element of
\[\begin{array}{c} \left[ F _ {\Sigma_ {t - 1}, \Sigma_ {t - 1}} \right] _ {j _ {1}, j _ {2}} ^ {i} = \\ \left[ f _ {\mathcal {Y} _ {t + 1}} \right] _ {i _ {2}} ^ {i} \left(\left[ g _ {\mathcal {S} _ {t}} \right] _ {i _ {1}} ^ {i _ {2}} \left[ h _ {\Sigma_ {t - 1}, \Sigma_ {t - 1}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {1}} + \left[ g _ {\Sigma_ {t - 1}, \Sigma_ {t - 1}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {2}} \vartheta_ {j _ {1}} \vartheta_ {j _ {2}}\right) \\ + \left[ f _ {\mathcal {Y} _ {t}} \right] _ {i _ {2}} ^ {i} \left[ g _ {\Sigma_ {t - 1}, \Sigma_ {t - 1}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {2}} + \left[ f _ {\mathcal {S} _ {t + 1}} \right] _ {i _ {1}} ^ {i} \left[ h _ {\Sigma_ {t - 1}, \Sigma_ {t - 1}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {1}} = 0 \end{array}\]
for and . This is a homogeneous system on and for , and therefore
\[\left[ h _ {\Sigma_ {t - 1}, \Sigma_ {t - 1}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {1}} = \left[ g _ {\Sigma_ {t - 1}, \Sigma_ {t - 1}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {2}} = 0\]
for , and
4. We consider the cross-derivative of the element of with respect to the element of and the element of
\[\begin{array}{c} {\left[ F _ {{\mathcal {E}} _ {t}, \Sigma_ {t - 1}} \right] _ {j _ {1}, j _ {2}} ^ {i} =} \\ {\left[ f _ {{\mathcal {Y}} _ {t + 1}} \right] _ {i _ {2}} ^ {i} \left([ g _ {{\mathcal {S}} _ {t}} ] _ {i _ {1}} ^ {i _ {2}} \left[ h _ {{\mathcal {E}} _ {t}, \Sigma_ {t - 1}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {1}} + \left[ g _ {{\mathcal {S}} _ {t}, \Sigma_ {t - 1}} \right] _ {i _ {1}, j _ {2}} ^ {i _ {2}} [ h _ {{\mathcal {E}} _ {t}} ] _ {j _ {1}} ^ {i _ {1}} \vartheta_ {j _ {2}} + \left[ g _ {{\mathcal {Z}} _ {t - 1}, \Sigma_ {t - 1}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {2}} \sigma_ {j _ {1}} \exp^ {\vartheta_ {j _ {1}} \log \sigma_ {j _ {1}, t - 1}} \vartheta_ {j _ {2}}\right)} \\ + [ f _ {{\mathcal {Y}} _ {t}} ] _ {i _ {2}} ^ {i} \left[ g _ {{\mathcal {E}} _ {t}, \Sigma_ {t - 1}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {2}} + \left[ f _ {{\mathcal {S}} _ {t + 1}} \right] _ {i _ {1}} ^ {i} \left[ h _ {{\mathcal {E}} _ {t}, \Sigma_ {t - 1}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {1}} = 0 \end{array}\]
for and . Since we know that for , and , this is a homogeneous system on and for , and if . Therefore
\[\left[ h _ {\mathcal {E} _ {t}, \Sigma_ {t - 1}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {1}} = \left[ g _ {\mathcal {E} _ {t}, \Sigma_ {t - 1}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {2}} = 0\]
for , and
Note that if , we have that
\[\begin{array}{c} \left[ F _ {\mathcal {E} _ {t}, \Sigma_ {t - 1}} \right] _ {j _ {1}, j _ {1}} ^ {i} = \left[ f _ {\mathcal {Y} _ {t + 1}} \right] _ {i _ {2}} ^ {i} * \\ * \left( \begin{array}{c} [ g _ {\mathcal {S} _ {t}} ] _ {i _ {1}} ^ {i _ {2}} \left[ h _ {\mathcal {E} _ {t}, \Sigma_ {t - 1}} \right] _ {j _ {1}, j _ {1}} ^ {i _ {1}} + \left[ g _ {\mathcal {S} _ {t}, \Sigma_ {t - 1}} \right] _ {i _ {1}, j _ {1}} ^ {i _ {2}} [ h _ {\mathcal {E} _ {t}} ] _ {j _ {1}} ^ {i _ {1}} \vartheta_ {j _ {1}} + \\ \left(\left[ g _ {\mathcal {Z} _ {t - 1}, \Sigma_ {t - 1}} \right] _ {j _ {1}, j _ {1}} ^ {i _ {2}} + \left[ g _ {\mathcal {Z} _ {t - 1}} \right] _ {j _ {1}} ^ {i _ {2}}\right) \sigma_ {j _ {1}} \exp^ {\vartheta_ {j _ {1}} \log \sigma_ {j _ {1}, t - 1}} \vartheta_ {j _ {1}} \\ + [ f _ {\mathcal {Y} _ {t}} ] _ {i _ {2}} ^ {i} \left[ g _ {\mathcal {E} _ {t}, \Sigma_ {t - 1}} \right] _ {j _ {1}, j _ {1}} ^ {i _ {2}} + \left[ f _ {\mathcal {S} _ {t + 1}} \right] _ {i _ {1}} ^ {i} \left[ h _ {\mathcal {E} _ {t}, \Sigma_ {t - 1}} \right] _ {j _ {1}, j _ {1}} ^ {i _ {1}} \\ + \left([ f _ {\mathcal {Z} _ {t}} ] _ {j _ {1}} ^ {i} + \left[ f _ {\mathcal {Z} _ {t + 1}} \right] _ {j _ {1}} ^ {i} \rho_ {j _ {1}}\right) \sigma_ {j _ {1}} \exp^ {\vartheta_ {j _ {1}} \log \sigma_ {j _ {1}, t - 1}} \vartheta_ {j _ {1}} = 0 \end{array} \right. \end{array}\]
and since and are di§erent from zero in general for and ; we have that this system is not homogeneous and
\[\left[ h _ {\mathcal {E} _ {t}, \Sigma_ {t - 1}} \right] _ {j _ {1}, j _ {1}} ^ {i _ {1}} = \left[ g _ {\mathcal {E} _ {t}, \Sigma_ {t - 1}} \right] _ {j _ {1}, j _ {1}} ^ {i _ {2}} \neq 0\]
for
Proof, part 3. The Önal part of the proof deals with the cross-derivatives of the policy functions h and g with respect to and any of , or and it shows that all of them are equal to zero with one exception. In particular, we want to show that
\[\left[ h _ {\mathcal {S} _ {t}, \mathcal {U} _ {t}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {1}} = \left[ g _ {\mathcal {S} _ {t}, \mathcal {U} _ {t}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {2}} = 0\]
for , and ，
\[\left[ h _ {\mathcal {Z} _ {t - 1}, \mathcal {U} _ {t}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {1}} = \left[ g _ {\mathcal {Z} _ {t - 1}, \mathcal {U} _ {t}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {2}} = \left[ h _ {\Sigma_ {t - 1}, \mathcal {U} _ {t}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {1}} = \left[ g _ {\Sigma_ {t - 1}, \mathcal {U} _ {t}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {2}} = [ h _ {\mathcal {U} _ {t}, \mathcal {U} _ {t}} ] _ {j _ {1}, j _ {2}} ^ {i _ {1}} = [ g _ {\mathcal {U} _ {t}, \mathcal {U} _ {t}} ] _ {j _ {1}, j _ {2}} ^ {i _ {2}} = 0\]
for , and , and
\[\left[ h _ {\mathcal {E} _ {t}, \mathcal {U} _ {t}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {1}} = \left[ g _ {\mathcal {E} _ {t}, \mathcal {U} _ {t}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {2}} = 0\]
for , and
Again, we follow the same steps for each part of the result as before and use our previous Öndings regarding which terms are zero.
1. We consider the cross derivative of the element of F with respect to the element of and the element of
\[\begin{array}{r} \left[ F _ {\mathcal {S} _ {t}, \mathcal {U} _ {t}} \right] _ {j _ {1}, j _ {2}} ^ {i} = \left[ f _ {\mathcal {Y} _ {t + 1}} \right] _ {i _ {2}} ^ {i} \left(\left[ g _ {\mathcal {S} _ {t}} \right] _ {i _ {1}} ^ {i _ {2}} \left[ h _ {\mathcal {S} _ {t}, \mathcal {U} _ {t}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {1}} + \left[ g _ {\mathcal {S} _ {t}, \Sigma_ {t - 1}} \right] _ {i _ {1}, j _ {2}} ^ {i _ {2}} \left[ h _ {\mathcal {S} _ {t}} \right] _ {j _ {1}} ^ {i _ {1}} \left(1 - \vartheta_ {j _ {2}} ^ {2}\right) ^ {\frac {1}{2}} \eta_ {j _ {2}}\right) \\ + \left[ f _ {\mathcal {Y} _ {t}} \right] _ {i _ {2}} ^ {i} \left[ g _ {\mathcal {S} _ {t}, \mathcal {U} _ {t}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {2}} + \left[ f _ {\mathcal {S} _ {t + 1}} \right] _ {i _ {1}} ^ {i} \left[ h _ {\mathcal {S} _ {t}, \mathcal {U} _ {t}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {1}} = 0 \end{array}\]
for , and . Since for , and , this is a homogeneous system on and .Therefore
\[\left[ h _ {\mathcal {S} _ {t}, \mathcal {U} _ {t}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {1}} = \left[ g _ {\mathcal {S} _ {t}, \mathcal {U} _ {t}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {2}} = 0\]
for
2. We consider the cross-derivative of the element of F with respect to the element
of and the element of
\[\begin{array}{c} {\left[ F _ {\mathcal {Z} _ {t - 1}, \mathcal {U} _ {t}} \right] _ {j _ {1}, j _ {2}} ^ {i} =} \\ {\left[ f _ {\mathcal {Y} _ {t + 1}} \right] _ {i _ {2}} ^ {i} \left( \begin{array}{c} {[ g _ {\mathcal {S} _ {t}} ] _ {i _ {1}} ^ {i _ {2}} [ h _ {\mathcal {Z} _ {t - 1}, \mathcal {U} _ {t}} ] _ {j _ {1}, j _ {2}} ^ {i _ {1}} + [ g _ {\mathcal {S} _ {t}, \Sigma_ {t - 1}} ] _ {i _ {1}, j _ {2}} ^ {i _ {2}} [ h _ {\mathcal {Z} _ {t}} ] _ {j _ {1}} ^ {i _ {1}} (1 - \vartheta_ {j _ {2}} ^ {2}) ^ {\frac {1}{2}} \eta_ {j _ {2}}} \\ + [ g _ {\mathcal {Z} _ {t - 1}, \Sigma_ {t - 1}} ] _ {j _ {1}, j _ {2}} ^ {i _ {2}} \rho_ {j _ {1}} (1 - \vartheta_ {j _ {2}} ^ {2}) ^ {\frac {1}{2}} \eta_ {j _ {2}} \end{array} \right)} \\ + [ f _ {\mathcal {Y} _ {t}} ] _ {i _ {2}} ^ {i} [ g _ {\mathcal {Z} _ {t - 1}, \mathcal {U} _ {t}} ] _ {j _ {1}, j _ {2}} ^ {i _ {2}} + [ f _ {\mathcal {S} _ {t + 1}} ] _ {i _ {1}} ^ {i} [ h _ {\mathcal {Z} _ {t - 1}, \mathcal {U} _ {t}} ] _ {j _ {1}, j _ {2}} ^ {i _ {1}} = 0 \end{array}\]
for , and . Since for , and , this is a homogeneous system on and for , and Therefore
\[\left[ h _ {\mathcal {Z} _ {t - 1}, \mathcal {U} _ {t}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {1}} = \left[ g _ {\mathcal {Z} _ {t - 1}, \mathcal {U} _ {t}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {2}} = 0\]
for
3. We consider the cross-derivative of the element of with respect to the element of and the element of
\[\begin{array}{c} \left[ F _ {\Sigma_ {t - 1}, \mathcal {U} _ {t}} \right] _ {j _ {1}, j _ {2}} ^ {i} = \\ \left[ f _ {\mathcal {Y} _ {t + 1}} \right] _ {i _ {2}} ^ {i} \left(\left[ g _ {\mathcal {S} _ {t}} \right] _ {i _ {1}} ^ {i _ {2}} \left[ h _ {\Sigma_ {t - 1}, \mathcal {U} _ {t}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {1}} + \left[ g _ {\Sigma_ {t - 1}, \Sigma_ {t - 1}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {2}} \vartheta_ {j _ {1}} \left(1 - \vartheta_ {j _ {2}} ^ {2}\right) ^ {\frac {1}{2}} \eta_ {j _ {2}}\right) \\ + \left[ f _ {\mathcal {Y} _ {t}} \right] _ {i _ {2}} ^ {i} \left[ g _ {\Sigma_ {t - 1}, \mathcal {U} _ {t}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {2}} + \left[ f _ {\mathcal {S} _ {t + 1}} \right] _ {i _ {1}} ^ {i} \left[ h _ {\Sigma_ {t - 1}, \mathcal {U} _ {t}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {1}} = 0 \end{array}\]
for , and . Since for , and , this is a homogeneous system on and for , and . Therefore
\[\left[ h _ {\Sigma_ {t - 1}, \mathcal {U} _ {t}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {1}} = \left[ g _ {\Sigma_ {t - 1}, \mathcal {U} _ {t}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {2}} = 0\]
for
4. We consider the cross-derivative of the element of with respect to the element
of and the element of
\[\begin{array}{c} {[ F _ {\mathcal {U} _ {t}, \mathcal {U} _ {t}} ] _ {j _ {1}, j _ {2}} ^ {i} =} \\ {\left[ f _ {\mathcal {Y} _ {t + 1}} \right] _ {i _ {2}} ^ {i} \left(\left[ g _ {\Sigma_ {t - 1}, \Sigma_ {t - 1}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {2}} \left(1 - \vartheta_ {j _ {1}} ^ {2}\right) ^ {\frac {1}{2}} \eta_ {j _ {1}} \left(1 - \vartheta_ {j _ {2}} ^ {2}\right) ^ {\frac {1}{2}} \eta_ {j _ {2}} + [ g _ {\mathcal {S} _ {t}} ] _ {i _ {1}} ^ {i _ {2}} [ h _ {\mathcal {U} _ {t}, \mathcal {U} _ {t}} ] _ {j _ {1}, j _ {2}} ^ {i _ {1}}\right)} \\ + [ f _ {\mathcal {Y} _ {t}} ] _ {i _ {2}} ^ {i} [ g _ {\mathcal {U} _ {t}, \mathcal {U} _ {t}} ] _ {j _ {1}, j _ {2}} ^ {i _ {2}} + \left[ f _ {\mathcal {S} _ {t + 1}} \right] _ {i _ {1}} ^ {i} [ h _ {\mathcal {U} _ {t}, \mathcal {U} _ {t}} ] _ {j _ {1}, j _ {2}} ^ {i _ {1}} = 0 \end{array}\]
for and . Since for and , this is a homogeneous system on and for , and . Therefore
\[[ h _ {\mathcal {U} _ {t}, \mathcal {U} _ {t}} ] _ {j _ {1}, j _ {2}} ^ {i _ {1}} = [ g _ {\mathcal {U} _ {t}, \mathcal {U} _ {t}} ] _ {j _ {1}, j _ {2}} ^ {i _ {2}} = 0\]
for
5. Finally, consider the cross-derivative of the element of with respect to the element of and the element of
\[\begin{array}{c} {[ F _ {\mathcal {E} _ {t}, \mathcal {U} _ {t}} ] _ {j _ {1}, j _ {2}} ^ {i} =} \\ {\left[ f _ {\mathcal {Y} _ {t + 1}} \right] _ {i _ {2}} ^ {i} \left( \begin{array}{c} {[ g _ {\mathcal {S} _ {t}} ] _ {i _ {1}} ^ {i _ {2}} [ h _ {\mathcal {E} _ {t}, \mathcal {U} _ {t}} ] _ {j _ {1}, j _ {2}} ^ {i _ {1}} + \left[ g _ {\mathcal {S} _ {t}, \Sigma_ {t - 1}} \right] _ {i _ {1}, j _ {2}} ^ {i _ {2}} [ h _ {\mathcal {E} _ {t}} ] _ {j _ {1}} ^ {i _ {1}} \left(1 - \vartheta_ {j _ {2}} ^ {2}\right) ^ {\frac {1}{2}} \eta_ {j _ {2}}} \\ + \left[ g _ {\mathcal {Z} _ {t - 1}, \Sigma_ {t - 1}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {2}} \sigma_ {j _ {1}} \exp^ {\vartheta_ {j _ {1}} \log \sigma_ {j _ {1}, t - 1}} \left(1 - \vartheta_ {j _ {2}} ^ {2}\right) ^ {\frac {1}{2}} \eta_ {j _ {2}}} \end{array} \right) \\ + [ f _ {\mathcal {Y} _ {t}} ] _ {i _ {2}} ^ {i} [ g _ {\mathcal {E} _ {t}, \mathcal {U} _ {t}} ] _ {j _ {1}, j _ {2}} ^ {i _ {2}} + \left[ f _ {\mathcal {S} _ {t + 1}} \right] _ {i _ {1}} ^ {i} [ h _ {\mathcal {E} _ {t}, \mathcal {U} _ {t}} ] _ {j _ {1}, j _ {2}} ^ {i _ {1}} = 0 \end{array}\]
for and . Since for , and , this is a homogeneous system on and for , and if . Therefore
\[\left[ h _ {\mathcal {E} _ {t}, \mathcal {U} _ {t}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {2}} = \left[ g _ {\mathcal {E} _ {t}, \mathcal {U} _ {t}} \right] _ {j _ {1}, j _ {2}} ^ {i _ {1}} = 0\]
for , and
Note that if , we have that
\[\begin{array}{c} [ F _ {\mathcal {E} _ {t}, \mathcal {U} _ {t}} ] _ {j _ {1}, j _ {1}} ^ {i} = \\ \left[ f _ {\mathcal {Y} _ {t + 1}} \right] _ {i _ {2}} ^ {i} \left( \begin{array}{c} \left(\left[ g _ {\mathcal {Z} _ {t - 1}, \Sigma_ {t - 1}} \right] _ {j _ {1}, j _ {1}} ^ {i _ {2}} + \left[ g _ {\mathcal {Z} _ {t - 1}} \right] _ {j _ {1}} ^ {i _ {2}}\right) \sigma_ {j _ {1}} \exp^ {\vartheta_ {j _ {1}} \log \sigma_ {j _ {1}, t - 1}} \left(1 - \vartheta_ {j _ {1}} ^ {2}\right) ^ {\frac {1}{2}} \eta_ {j _ {1}} \\ + \left[ g _ {\mathcal {S} _ {t}, \Sigma_ {t - 1}} \right] _ {i _ {1}, j _ {1}} ^ {i _ {2}} [ h _ {\mathcal {E} _ {t}} ] _ {j _ {1}} ^ {i _ {1}} \left(1 - \vartheta_ {j _ {1}} ^ {2}\right) ^ {\frac {1}{2}} \eta_ {j _ {1}} + [ g _ {\mathcal {S} _ {t}} ] _ {i _ {1}} ^ {i _ {2}} [ h _ {\mathcal {E} _ {t}, \mathcal {U} _ {t}} ] _ {j _ {1}, j _ {1}} ^ {i _ {1}} \\ + [ f _ {\mathcal {Y} _ {t}} ] _ {i _ {2}} ^ {i} [ g _ {\mathcal {E} _ {t}, \mathcal {U} _ {t}} ] _ {j _ {1}, j _ {1}} ^ {i _ {2}} + \left[ f _ {\mathcal {S} _ {t + 1}} \right] _ {i _ {1}} ^ {i} [ h _ {\mathcal {E} _ {t}, \mathcal {U} _ {t}} ] _ {j _ {1}, j _ {1}} ^ {i _ {1}} \\ + \left([ f _ {\mathcal {Z} _ {t}} ] _ {j _ {1}} ^ {i} + \rho_ {j _ {1}} \left[ f _ {\mathcal {Z} _ {t + 1}} \right] _ {j _ {1}} ^ {i}\right) \sigma_ {j _ {1}} \exp^ {\vartheta_ {j _ {1}} \log \sigma_ {j _ {1}, t - 1}} \left(1 - \vartheta_ {j _ {1}} ^ {2}\right) ^ {\frac {1}{2}} \eta_ {j _ {1}} = 0 \end{array} \right. \end{array}\]
and since and are di§erent from zero in general for and ; we have that this system is not homogeneous and hence
\[[ h _ {\mathcal {E} _ {t}, \mathcal {U} _ {t}} ] _ {j _ {1}, j _ {1}} ^ {i _ {2}} = [ g _ {\mathcal {E} _ {t}, \mathcal {U} _ {t}} ] _ {j _ {1}, j _ {1}} ^ {i _ {1}} \neq 0\]
for , and ■
3. Equilibrium
In this section we describe the equilibrium conditions of the model. First, we introduce the ones related to the household, then the ones related to the Örm and the monetary authority, and Önally we present the market clearing and aggregation conditions.
3.1. Households
We can deÖne two Lagrangian multipliers, , the multiplier associated with the budget constraint, and (the marginal Tobinís Q), the multiplier associated with the investment adjustment constraint normalized by . Thus, the Örst-order conditions of the household problem with respect to , and can be written as:
\[d _ {t} \left(c _ {j t} - h c _ {j t - 1}\right) ^ {- 1} - b \beta \mathbb {E} _ {t} d _ {t + 1} \left(c _ {j t + 1} - h c _ {j t}\right) ^ {- 1} = \lambda_ {j t},\tag{30}\]
\[\lambda_ {j t} = \beta \mathbb {E} _ {t} \{\lambda_ {j t + 1} \frac {R _ {t}}{\Pi_ {t + 1}} \},\tag{31}\]
\[r _ {t} = \mu_ {t} ^ {- 1} \Phi^ {\prime} [ u _ {j t} ],\tag{32}\]
\[q _ {j t} = \beta \mathbb {E} _ {t} \left\{\frac {\lambda_ {j t + 1}}{\lambda_ {j t}} \left((1 - \delta) q _ {j t + 1} + r _ {t + 1} u _ {j t + 1} - \mu_ {t + 1} ^ {- 1} \Phi [ u _ {j t + 1} ]\right) \right\},\tag{33}\]
and
\[1 = q _ {j t} \mu_ {t} \left(1 - V \left[ \frac {x _ {j t}}{x _ {j t - 1}} \right] - V ^ {\prime} \left[ \frac {x _ {j t}}{x _ {j t - 1}} \right] \frac {x _ {j t}}{x _ {j t - 1}}\right) + \beta \mathbb {E} q _ {j t + 1} \mu_ {t + 1} \frac {\lambda_ {j t + 1}}{\lambda_ {j t}} V ^ {\prime} \left[ \frac {x _ {j t + 1}}{x _ {j t}} \right] \left(\frac {x _ {j t + 1}}{x _ {j t}}\right) ^ {2}.\tag{34}\]
The Örst-order conditions of the ìlabor packerîimply a demand function for labor:
\[l _ {j t} = \left(\frac {w _ {j t}}{w _ {t}}\right) ^ {- \eta} l _ {t} ^ {d} \quad \forall j\]
and, together with a zero proÖt condition , an expression for the wage:
\[w _ {t} = \left(\int_ {0} ^ {1} w _ {j t} ^ {1 - \eta} d j\right) ^ {\frac {1}{1 - \eta}}.\]
Households follow a Calvo scheme when they set their wages. At the start of every period, a randomly selected fraction of households can reoptimize their wages. All other households index their nominal wages given past ináation with an indexation parameter
Since we postulated in the main text both complete Önancial markets for the households and separable utility in consumption, the marginal utilities of consumption are the same for all households. Thus, in equilibrium, , and
The last two equalities tell us that the shadow cost of consumption is equated across households and that all households that can reset their wages optimally will do it at the same level . With these two results, and after several steps of algebra, we Önd that the evolution of wages is described by two recursive equations:
\[f _ {t} = \frac {\eta - 1}{\eta} (w _ {t} ^ {*}) ^ {1 - \eta} \lambda_ {t} w _ {t} ^ {\eta} l _ {t} ^ {d} + \beta \theta_ {w} \mathbb {E} _ {t} \left(\frac {\Pi_ {t} ^ {\chi_ {w}}}{\Pi_ {t + 1}}\right) ^ {1 - \eta} \left(\frac {w _ {t + 1} ^ {*}}{w _ {t} ^ {*}}\right) ^ {\eta - 1} f _ {t + 1}\tag{35}\]
and
\[f _ {t} = \psi d _ {t} \varphi_ {t} \left(\frac {w _ {t}}{w _ {t} ^ {*}}\right) ^ {\eta (1 + \vartheta)} \left(l _ {t} ^ {d}\right) ^ {1 + \vartheta} + \beta \theta_ {w} \mathbb {E} _ {t} \left(\frac {\Pi_ {t} ^ {\chi_ {w}}}{\Pi_ {t + 1}}\right) ^ {- \eta (1 + \vartheta)} \left(\frac {w _ {t + 1} ^ {*}}{w _ {t} ^ {*}}\right) ^ {\eta (1 + \vartheta)} f _ {t + 1}\tag{36}\]
on the auxiliary variable .
Taking advantage of the expression for the wage and that, in every period, a fraction of households set as their wage and the remaining fraction partially index their nominal wage by past ináation, we can write the law of motion of real wage as:
\[w _ {t} ^ {1 - \eta} = \theta_ {w} \left(\frac {\Pi_ {t - 1} ^ {\chi_ {w}}}{\Pi_ {t}}\right) ^ {1 - \eta} w _ {t - 1} ^ {1 - \eta} + (1 - \theta_ {w}) w _ {t} ^ {* 1 - \eta}.\tag{37}\]
3.2. Firms
The Önal good producer is perfectly competitive and minimizes its costs subject to the production function (24) and taking as given all intermediate goods prices and the Önal good price The optimality conditions of this problem result in a demand function for each intermediate good with the classic form:
\[y _ {i t} = \left(\frac {p _ {i t}}{p _ {t}}\right) ^ {- \varepsilon} y _ {t} ^ {d} \quad \forall i\]
where is the aggregate demand and a price for the Önal good:
\[p _ {t} = \left(\int_ {0} ^ {1} p _ {i t} ^ {1 - \varepsilon} d i\right) ^ {\frac {1}{1 - \varepsilon}}.\]
Each of the intermediate goods is produced by a monopolistic competitor. Intermediate good producers produce the quantity demanded of the good by renting and at prices and . Then, by minimization, we have a marginal cost of:
\[m c _ {t} = \left(\frac {1}{1 - \alpha}\right) ^ {1 - \alpha} \left(\frac {1}{\alpha}\right) ^ {\alpha} \frac {w _ {t} ^ {1 - \alpha} r _ {t} ^ {\alpha}}{A _ {t}}\tag{38}\]
The marginal cost is constant for all Örms and all production levels given , and .
Given the demand function, the intermediate good producers set prices to maximize proÖts. However, when they do so, they follow the same Calvo pricing scheme as households. In each period, a fraction of intermediate good producers reoptimize their prices. All other Örms partially index their prices by past ináation with an indexation parameter
The solution for the Örmís pricing problem has a recursive structure in two new auxiliary variables and that take the form:
\[g _ {t} ^ {1} = \lambda_ {t} m c _ {t} y _ {t} ^ {d} + \beta \theta_ {p} \mathbb {E} _ {t} \left(\frac {\Pi_ {t} ^ {\chi}}{\Pi_ {t + 1}}\right) ^ {- \varepsilon} g _ {t + 1} ^ {1},\tag{39}\]
\[g _ {t} ^ {2} = \lambda_ {t} \Pi_ {t} ^ {*} y _ {t} ^ {d} + \beta \theta_ {p} \mathbb {E} _ {t} \left(\frac {\Pi_ {t} ^ {\chi}}{\Pi_ {t + 1}}\right) ^ {1 - \varepsilon} \left(\frac {\Pi_ {t} ^ {*}}{\Pi_ {t + 1} ^ {*}}\right) g _ {t + 1} ^ {2},\tag{40}\]
and
\[\varepsilon g _ {t} ^ {1} = (\varepsilon - 1) g _ {t} ^ {2}\tag{41}\]
where
\[\Pi_ {t} ^ {*} = \frac {p _ {t} ^ {*}}{p _ {t}}\tag{42}\]
is the ratio between the optimal new price (common across all Örms that can reset their prices) and the price of the Önal good. With this structure, the ináation follows:
\[1 = \theta_ {p} \left(\frac {\Pi_ {t - 1} ^ {\chi}}{\Pi_ {t}}\right) ^ {1 - \varepsilon} + (1 - \theta_ {p}) \Pi_ {t} ^ {* 1 - \varepsilon}.\tag{43}\]
3.3. The Monetary Authority
The model is closed with a monetary authority that sets the nominal interest rates by a modiÖed Taylor rule described in (25).
3.4. Market Clearing and Aggregation
Aggregate demand is given by:
\[y _ {t} ^ {d} = c _ {t} + x _ {t} + \mu_ {t} ^ {- 1} \Phi [ u _ {t} ] k _ {t - 1}.\tag{44}\]
By relying on the observation that the capital-labor ratio is constant across Örms, we can derive that aggregate supply is:
\[y _ {t} ^ {s} = \frac {A _ {t} (u _ {t} k _ {t - 1}) ^ {\alpha} (l _ {t} ^ {d}) ^ {1 - \alpha} - \phi z _ {t}}{v _ {t} ^ {p}}\tag{45}\]
where:
\[v _ {t} ^ {p} = \int_ {0} ^ {1} \left(\frac {p _ {i t}}{p _ {t}}\right) ^ {- \varepsilon} d i\]
is the aggregate loss of e¢ ciency induced by price dispersion of the intermediate goods.
Market clearing requires that
\[y _ {t} = y _ {t} ^ {d} = y _ {t} ^ {s}.\tag{46}\]
By the properties of Calvoís pricing:
\[v _ {t} ^ {p} = \theta_ {p} \left(\frac {\Pi_ {t - 1} ^ {\chi}}{\Pi_ {t}}\right) ^ {- \varepsilon} v _ {t - 1} ^ {p} + (1 - \theta_ {p}) \Pi_ {t} ^ {* - \varepsilon}.\tag{47}\]
Finally, demanded labor is given by:
\[l _ {t} ^ {d} = \frac {1}{\upsilon_ {t} ^ {w}} \int_ {0} ^ {1} l _ {j t} d j = l _ {t}\tag{48}\]
where:
\[v _ {t} ^ {w} = \int_ {0} ^ {1} \left(\frac {w _ {j t}}{w _ {t}}\right) ^ {- \eta} d j.\]
is the aggregate loss of labor input induced by wage dispersion among di§erentiated types of labor. Again, by Calvoís pricing, this ine¢ ciency evolves as:
\[v _ {t} ^ {w} = \theta_ {w} \left(\frac {w _ {t - 1}}{w _ {t}} \frac {\Pi_ {t - 1} ^ {\chi_ {w}}}{\Pi_ {t}}\right) ^ {- \eta} v _ {t - 1} ^ {w} + (1 - \theta_ {w}) (\Pi_ {t} ^ {w *}) ^ {- \eta}.\tag{49}\]
Thus an equilibrium is characterized by equations (30)-(49), the Taylor rule (25), the law of motion for the structural shocks (augmented with the parameter drifts), and the law of motion for the volatility shocks.
4. Non-linearities in Parameter Drifting
We argued in the main text that, since we consider the e§ects of stochastic volatility on our model, it was of the essence to deal with higher-order approximations. In this section we argue that higher-order approximations are also key to dealing with parameter drifting in the Taylor rule. The reason is that parameter drifting disappears from a linear solution. To see this, take the Taylor rule deÖned in (25) (assuming only in this paragraph and to simplify notation that and log and let us rewrite it:
\[f \left(\widehat {R} _ {t}, \widehat {R} _ {t - 1}, \widehat {\Pi} _ {t}, \widehat {\gamma} _ {\Pi , t}, \varepsilon_ {\xi t}\right) = \exp^ {\widehat {R} _ {t}} - \exp^ {\gamma_ {R} \widehat {R} _ {t - 1} + (1 - \gamma_ {R}) \gamma_ {\Pi} \exp^ {\widehat {\gamma} _ {\Pi , t}} \widehat {\Pi} _ {t} + \sigma_ {\xi} \varepsilon_ {\xi t}} = 0.\]
where we have expressed each variable in terms of log deviation with respect to the steady state, . The log-linear approximation of f around the steady state is
\[f \left(\widehat {R} _ {t}, \widehat {R} _ {t - 1}, \widehat {\Pi} _ {t}, \widehat {\gamma} _ {\Pi , t}, \varepsilon_ {\xi t}\right) \simeq f _ {1} (\mathbf {0}) \widehat {R} _ {t} + f _ {2} (\mathbf {0}) \widehat {R} _ {t - 1} + f _ {3} (\mathbf {0}) \widehat {\Pi} _ {t} + f _ {4} (\mathbf {0}) \widehat {\gamma} _ {\Pi , t} + f _ {5} (\mathbf {0}) \varepsilon_ {\xi t}.\]
where is the Örst derivative of the function f evaluated at (0; 0; 0; 0; 0) with respect to the variable i. Note that:
\[f _ {4} \left(\widehat {R} _ {t}, \widehat {R} _ {t - 1}, \widehat {\Pi} _ {t}, \widehat {\gamma} _ {\Pi , t}, \varepsilon_ {\xi t}\right) = - \left(1 - \gamma_ {R}\right) \gamma_ {\Pi} \widehat {\Pi} _ {t} \exp^ {\widehat {\gamma} _ {\Pi , t}} \exp^ {\gamma_ {R} \widehat {R} _ {t - 1} + (1 - \gamma_ {R}) \gamma_ {\Pi} \exp^ {\widehat {\gamma} _ {\Pi , t}} \widehat {\Pi} _ {t} + \sigma_ {\xi} \varepsilon_ {\xi t}}.\]
Clearly, and does not play any role in this Örst-order approximation. This is the consequence of one variable (t) being raised to another variable . Thus, the log-linear approximation of the Taylor rule is:
\[f \left(\widehat {R} _ {t}, \widehat {R} _ {t - 1}, \widehat {\Pi} _ {t}, \widehat {\gamma} _ {\Pi , t}, \varepsilon_ {m t}\right) \simeq \widehat {R} _ {t} - \gamma_ {R} \widehat {R} _ {t - 1} - (1 - \gamma_ {R}) \gamma_ {\Pi} \widehat {\Pi} _ {t} + \sigma_ {\xi} \varepsilon_ {\xi t}\tag{50}\]
which does not depend on ; but only on the steady state and it is exactly the same expression as the one without parameter drifting. Hence, in order to capture parameter drifting in the Taylor rule, we need, at least, to perform a second-order approximation.
5. Construction of Data
When we estimate the model, we make the series provided by the National Income and Product Accounts (NIPA) consistent with the deÖnition of variables in the theory. The main adjustment we undertake is to express both real output and real gross investment in consumption units. Our model implies that there is a numeraire in terms of which all the other prices need to be quoted. We pick consumption as the numeraire. The NIPA, in comparison, uses an index of all prices to transform nominal GDP and investment into real values. In the presence of changing relative prices, such as the ones we have seen in the U.S. over the last several decades with the fall in the relative price of capital, NIPAís procedure biases the valuation of di§erent series in real terms.
We map theory into the data by computing our own series of real output and real investment. To do so, we use the relative price of investment, deÖned as the ratio of an investment deáator and a deáator for consumption. The denominator is easily derived from the deáators of nondurable goods and services reported in the NIPA. It is more complicated to obtain the numerator because, historically, NIPA investment deáators were poorly constructed. Instead, we rely on the investment deáator computed by Fisher (2006). Since the series ends early in 2000.Q4, we have extended it to 2007.Q1 by following Fisherís methodology.
For the real output per capita series, we Örst deÖne nominal output as nominal consumption plus nominal gross investment. We deÖne nominal consumption as the sum of personal consumption expenditures on non-durable goods and services. We deÖne nominal gross investment as the sum of personal consumption expenditures on durable goods, private residential investment, and non-residential Öxed investment. Per capita nominal output is equal to the ratio between our nominal output series and the civilian non-institutional population between 16 and 65. To obtain per capita values, we divide the previous series by the civilian non-institutional population between 16 and 65. Finally, real wages are deÖned as compensation per hour in the non-farm business sector divided by the CPI deáator.
6. Additional Empirical Results
6.1. Determinacy of Equilibrium
We mentioned in the main text that the estimated value of (1.045 in levels) guarantees local determinacy of the equilibrium. To see this, note that, for local determinacy, the relevant part of the solution of the model is only the linear Örst-order component. This component depends on the mean policy response, and not on the current value of . The economic intuition is that local unicity survives even if temporarily violates the Taylor principle as long as there is reversion to the mean in the policy response and, thus, the agents have the expectation that will satisfy the Taylor principle on average. For a related result in models with Markov-switching regime changes, see Davig and Leeper (2006). While we cannot Önd an analytical expression for the determinacy region, numerical experiments show that, conditional on the other point estimates, values of above 0.98 ensure uniqueness. Since the likelihood assigns zero probability to values of lower than 1.01, well inside the determinacy region, multiplicity of local equilibria is not an issue in our application.
6.2. Impulse Response Functions
As a check of our estimates, we plot the IRFs generated by the model to a monetary policy shock. This exercise is an important test. If the IRFs match the shapes and sizes of those gathered by time series methods such as SVARs, it will strengthen our belief in the rest of our results. Otherwise, we should at least understand where the di§erences come from.
Auspiciously, the answer is positive: our model generates dynamic responses that are close to the ones from SVARs (see, for instance, Sims and Zha, 2006). The left panel of Ögure 6.1 plots the IRFs to a monetary shock of three variables commonly discussed in monetary models: the federal funds rate, output growth, and ináation. Since we have a non-linear model, in all the Ögures in this section, we report the generalized IRFs starting from the mean of the ergodic distribution (Koop, Pesaran, and Potter, 1996). After a one-standard-deviation shock to the federal funds rate, ináation goes down in a hump-shaped pattern for many quarters and output growth drops.
The right panel of Ögure 6.1 plots the IRFs after a one-standard-deviation innovation to the monetary policy shock computed conditional on Öxing to the estimated mean during the tenure of each of three di§erent chairmen of the Board of Governors: the combination Burns-Miller, Volcker, and Greenspan. This exercise tells us how the variation in the systematic component of monetary policy has a§ected the dynamics of aggregate variables. Furthermore, it allows a comparison with numerous similar exercises done in the literature with SVARs where the IRFs are estimated on di§erent subsamples.
The most interesting di§erence is that the response of output growth under Volcker was the mildest: the estimated average stance of monetary policy under his tenure reduces the volatility of output. Ináation responds moderately as well since the agents have the expectation that future shocks will be smoothed out by the monetary authority. This Önding also explains why the IRFs of the interest rate are nearly on top of each other for all three periods: while we estimate that monetary policy responded more during Volckerís years for any given level of ináation than under Burns-Miller or Greenspan, this policy lowers ináation deviations and hence moderates the actual movement along the equilibrium path of the economy. Moreover, this second set of IRFs already points out one important result of this paper: we estimate that monetary policy under Burns-Miller and Greenspan was similar, while it was di§erent under Volcker. This Önding will be reinforced by the results we present in the main body of the text.
Figure 6.1: IRFs to a Monetary Policy Shock, Unconditional and Conditional
For completeness, we also plot, in Ögure 6.2, the IRFs to each of the other four shocks in our model: the two preference shocks (intertemporal and intratemporal) and the two technology shocks (investment-speciÖc and neutral). The behavior of the model is standard. A one-standarddeviation intertemporal preference shock raises output growth and ináation because there is an increase in the desire for consumption in the current period. The intratemporal shock lowers output because labor becomes less attractive, driving up the marginal costs and, with it, prices. The two supply shocks raise output growth and lower ináation by increasing productivity. All of those IRFs show that the behavior of the model is standard.
Figure 6.2: IRFs of ináation, output growth, and the federal funds rate to an intertemporal demand shock, an intratemporal demand shock, an investment-speciÖc shock, and a neutral technology shock. The responses are measured as log di§erences with respect to the mean of the ergodic distribution.
6.3. Model Comparison
Another use of our procedure to evaluate the likelihood function is to compare our model against alternative models -or alternative versions of the same model. For instance, a natural question is to compare our benchmark model with stochastic volatility and parameter drifting with a version without parameter drifting but with stochastic volatility. That is: once we have included stochastic volatility, is it still important to allow for changes in monetary policy to account for the time-varying volatility of aggregate data in the U.S.?
Given our Bayesian framework, a natural approach for model comparison is the computation of log marginal data densities (log MDD) and log Bayes factors. Let us focus on the proposed example of comparing the full model with stochastic volatility and parameter drifting with a version without parameter drifting (no drift) but with stochastic volatility. In this second case, we have two fewer parameters, and (but we still have . To ease notation, we partition the parameter vector as ; where is the vector of all the other parameters,
common to the two versions of the model.
Given that our priors are 1) uniform, 2) independent of each other, and 3) cover all the areas where the likelihood is (numerically) positive, and that 4) the priors on are common across the two speciÖcations of the model, we can write
\[\log p \left(\mathbb {Y} ^ {T} = \mathbb {Y} ^ {\text {data}, T}; d r i f t\right) = \log \int p \left(\mathbb {Y} ^ {T} = \mathbb {Y} ^ {\text {data}, T}; \gamma , d r i f t\right) d \gamma + \log p (\widetilde {\gamma}) + \log p \left(\rho_ {\gamma_ {\Pi}}\right) + \log p \left(\sigma_ {\pi}\right),\]
where log , log , and log are constants and
\[\log p \left(\mathbb {Y} ^ {T} = \mathbb {Y} ^ {d a t a, T}; n o d r i f t\right) = \log \int p \left(\mathbb {Y} ^ {T} = \mathbb {Y} ^ {d a t a, T}; \widetilde {\gamma}, n o d r i f t\right) d \widetilde {\gamma} + \log p \left(\widetilde {\gamma}\right).\]
Thus
\[\begin{array}{c} \log B _ {d r i f t, n o d r i f t} = \log p \left(\mathbb {Y} ^ {T} = \mathbb {Y} ^ {d a t a, T}; d r i f t\right) - \log p \left(\mathbb {Y} ^ {T} = \mathbb {Y} ^ {d a t a, T}; n o d r i f t\right) \\ = \log \int p \left(\mathbb {Y} ^ {T} = \mathbb {Y} ^ {d a t a, T}; \gamma , d r i f t\right) d \widetilde {\gamma} - \log \int p \left(\mathbb {Y} ^ {T} = \mathbb {Y} ^ {d a t a, T}; \widetilde {\gamma}, n o d r i f t\right) d \widetilde {\gamma} \\ - \log p \left(\rho_ {\gamma_ {\Pi}}\right) - \log p \left(\sigma_ {\pi}\right). \end{array}\]
The di§erence
\[\log \int p \left(\mathbb {Y} ^ {T} = \mathbb {Y} ^ {d a t a, T}; \gamma , d r i f t\right) d \gamma - \log \int p \left(\mathbb {Y} ^ {T} = \mathbb {Y} ^ {d a t a, T}; \widetilde {\gamma}, n o d r i f t\right) d \widetilde {\gamma}\]
tells us how much better the version with parameter drift Öts the data in comparison with the version with no drift. The last two terms, log ; penalize for the presence of two extra parameters.
We estimate the log MDDs following Gewekeís (1998) harmonic mean method. This requires us to generate a new draw of the posterior of the model for the speciÖcation with no parameter drift to compute log . After doing so, we Önd that
\[\log B _ {d r i f t, n o d r i f t} = 1 2 6. 1 3 3 1 + \log p \left(\rho_ {\gamma_ {\Pi}}\right) + \log p \left(\sigma_ {\pi}\right)\]
This expression shows a potential problem of Bayes factors: by picking uniform priors for and spread out over a su¢ ciently large interval, we could overcome any di§erence in
Öt. But the prior for is pinned down by our desire to keep that process stationary, which imposes natural bounds in [ 1; 1] and makes log . Thus, there is only one degree of freedom left: our choice of log . Any sensible prior for will only put mass in a relatively small interval: the point estimate is 0:1479, the standard deviation is 0:002, and the likelihood is numerically zero for values bigger than 0.2. Hence, we can safely impose that log would imply a uniform prior between 0 and 2:7183, a considerably wider support than any evidence in the data), and conclude that log . This is conventionally considered overwhelming evidence in favor of the model with parameter drift (Je§reys, 1961, for instance, suggests that di§erences bigger than 5 are decisive). Thus, even after controlling for stochastic volatility, the data strongly prefer a speciÖcation where monetary policy has changed over time. This Önding, however, does not imply that volatility shocks did not play an important role in the time-varying volatilities of U.S. aggregate time series. In fact, as we will see in the next section of this appendix, they were a key mechanism in accounting for it.
A formal comparison with the case without stochastic volatility is more di¢ cult, since we are taking advantage of its presence to evaluate the likelihood. Fortunately, Justiniano and Primiceri (2008) and Fern·ndez-Villaverde and Rubio-RamÌrez (2007) estimate models similar to ours with and without stochastic volatility (in the Örst case, using only a Örst-order approximation to the decision rules of the agents and in the second with measurement errors). Both papers Önd that the Öt of the model improves substantially when we include stochastic volatility. Finally, Fern·ndez-Villaverde and Rubio-RamÌrez (2008) compare a model with parameter drifting and no stochastic volatility with a model without parameter drifting and no stochastic volatility and report that parameter drifting is also strongly preferred by the likelihood.
It has been noted that the estimation of log MDDs is dangerous because of numerical instabilities in the evaluation of the integral log marginal data density (log MDD). This concern is particularly relevant in our case, since we have a large model saddled with burdensome computation. Thus, as a robustness analysis, we also computed the Bayesian Information Criterion (BIC) (Schwarz, 1978). The BIC, which avoids the need to handle the integral in the log MDD, can be understood as an asymptotic approximation of the Bayes factor that also automatically penalizes for extra parameters. The BIC of model i is deÖned:
\[B I C _ {i} = - 2 \ln p \left(\mathbb {Y} ^ {T} = \mathbb {Y} ^ {\text { data }, T}; \widehat {\gamma}, i\right) + k _ {i} \ln n\]
where is the maximum likelihood estimator (or, given our áat priors, the mode of the posterior), is the number of parameters, and n is the number of observations. Then, the BIC of the model with stochastic volatility and parameter drifting is If we eliminate parameter drifting and the parameters and associated with it (and, of course, with a new point estimate of the other parameters) 7; 484:7. The di§erence is, therefore, over 138 log points, which is again overwhelming evidence in favor of the model with parameter drifting.
6.4. Historical Counterfactuals
One important additional exercise is to quantify how much of the observed changes in the volatility of aggregate U.S. variables can be accounted for by changes in the standard deviations of shocks and how much by changes in policy. To accomplish this, we build a number of historical counterfactuals. In these exercises, we remove one source of variation at a time and we measure how aggregate variables would have behaved when hit only by the remaining shocks. Since our model is structural in the sense of Hurwicz (1962) (it is invariant to interventions, including shocks by nature such as the ones we are postulating), we will obtain an answer that is robust to the Lucas critique.
In the next two subsections, we will always plot the same three basic variables that we used in section 5 of the main text: ináation, output growth, and the federal funds rate. Counterfactual histories of other variables could be built analogously. Also, we will have vertical bars for the tenure of each chairman, following the same color scheme as in section 5.
6.4.1. Counterfactual I: Switching Chairmen
In our Örst counterfactual, we move one chairman from his mandate to an alternative time period. For example, we appoint Greenspan as chairman during the Burns-Miller years. By that, we mean that the Fed would have followed the policy rule dictated by the average estimated during Greenspanís time while starting from the same states as Burns-Miller and su§ering the same shocks (both structural and of volatility). We repeat this exercise with all the other possible combinations: Volcker in the Burns-Miller decade, Burns-Miller in Volckerís mandate, Greenspan in Volckerís time, Burns-Miller in the Greenspan years, and, Önally, Volcker in Greenspanís time.
It is important to be careful in interpreting this exercise. By appointing Greenspan at Volckerís time, we do not literally mean Greenspan as a person, but Greenspan as a convenient label for a particular monetary policy response to shocks that according to our model were observed during his tenure. The real Greenspan could have behaved in a di§erent way, for example, as a result of some non-linearities in monetary policy that are not properly captured by a simple rule such as the one postulated in section 3. The argument could be pushed one step further and we could think about the appointment of Volcker as an endogenous response of the political-economic equilibrium to high ináation. In our model agents have a probability distribution regarding possible changes in monetary policy in the next periods, but those changes are uncorrelated with current conditions. Therefore, our model cannot capture the endogeneity of policy selection.
Another issue that we sidelined is the evolution of expectations. In our model, agents have rational expectations and observe the changes in monetary policy parameters. This hypothesis may be a poor approximation of the agentsíbehavior in real life. It could be the case that was high in 1984, even though ináation was already low by that time, because of the high ináationary expectations that economic agents held during most of the 1980s (this point is also linked to issues of commitment and credibility that our model does not address). While we see all these arguments as interesting lines of research, we Önd it important to focus Örst on our basic counterfactual.
Moments In table 6.1, we report the mean and the standard deviation of ináation, output growth, and the federal funds rate in the observed data and in the sets of counterfactual data. Ináation was high with Burns-Miller, fell with Volcker, and stayed low with Greenspan. Output growth went down during the Volcker years to recover with Greenspan. The federal funds rate reached its peak with Volcker. The standard deviation of output growth fell from 4.7 in Burns-Millerís time to 2.45 with Greenspan, a cut in half. Similarly, ináation volatility fell nearly 54 percent and the federal funds rate volatility 5 percent.
But table 6.1 also tells us one important result: time-varying monetary policy signiÖcantly a§ected average ináation. In particular, Volckerís response to ináation was strong and switching him to either Burns-Millerís or Greenspanís time would have reduced average ináation dramatically. But it also tells us other things: contrary to the conventional wisdom, our estimates suggest that the stance of monetary policy against ináation under Greenspan was not strong. In Burns-Millerís time, the monetary policy under Greenspan would have delivered slightly higher average ináation, 6.83 versus the observed 6.23, accompanied by a lower federal funds rate and lower output growth, 1.89 versus the observed 2.03. The di§erence is even bigger in Volckerís time, during which average ináation would have been nearly 1.4 percent higher, while output growth would have been virtually identical (1.34 versus 1.38). The key for this Önding is in the behavior of the federal funds rate, which would have increased only by 9 basis points, on average, if Greenspan had been in charge of the Fed instead of Volcker. Given the higher ináation in the counterfactual, the higher nominal interest rates would have meant much lower real rates. The counterfactual of Burns-Miller in Greenspanís and Volckerís time casts doubt on the malignant reputations of these two short-lived chairmen, at least when compared with Greenspan. Burns-Miller would have brought even slightly lower ináation than Greenspan, thanks to a higher real federal funds rate and a bit higher output growth. However, Burns-Miller would have delivered higher ináation than Volcker.
Table 6.1: Switching Chairmen, Data versus Counterfactual Histories
| Means | Standard Deviations | |||||
| Inflation | Output Gr. | FFR | Inflation | Output Gr. | FFR | |
| BM (data) | 6.2333 | 2.0322 | 6.5764 | 2.7347 | 4.7010 | 2.2720 |
| Greenspan to BM | 6.8269 | 1.8881 | 6.5046 | 3.3732 | 4.6781 | 2.0103 |
| Volcker to BM | 4.3604 | 1.5010 | 7.6479 | 2.4620 | 4.6219 | 2.3470 |
| Volcker (data) | 5.3584 | 1.3846 | 10.3338 | 3.1811 | 4.4811 | 3.4995 |
| BM to Volcker | 6.4132 | 1.3560 | 10.4126 | 2.9728 | 4.4220 | 3.0648 |
| Greenspan to Volcker | 6.7284 | 1.3423 | 10.4235 | 2.9824 | 4.3730 | 2.8734 |
| Greenspan (data) | 2.9583 | 1.5177 | 4.7352 | 1.2675 | 2.4567 | 2.1887 |
| BM to Greenspan | 2.3355 | 1.5277 | 4.4529 | 1.5625 | 2.4684 | 2.4652 |
| Volcker to Greenspan | -0.4947 | 1.3751 | 3.6560 | 1.7700 | 2.4705 | 2.7619 |
This is an important empirical Önding: according to our model, time-varying monetary policy signiÖcantly a§ected average ináation. While Volckerís response to ináation was strong, Greenspanís response was milder and he seems to have behaved quite similarly to how Burns-Miller would have behaved.
Counterfactual Paths An alternative way to analyze our results is to plot the whole counterfactual histories summarized in table 6.1. We Önd it interesting to plot the whole history because changes in the economyís behavior in one period will propagate over time and we want to understand, for example, how Greenspanís legacy would have molded Volckerís tenure. Also, plotting the whole history allows us to track the counterfactual response of monetary policy to large economic events such as the oil shocks.
In Ögure 6.3, we move to Burns-Miller being reappointed in Greenspanís time. This plot suggests that the di§erences in monetary policy under Greenspan and Burns-Miller may have been overstated by the literature.
Figure 6.3: Burns-Miller during the Greenspan years
In Ögure 6.4, we plot the counterfactual of Burns-Miller extending their tenure to 1987. The results are very similar to the case in which we move Greenspan to the same period: slower disináation and no improvement in output growth.
Figure 6.4: Burns-Miller during the Volcker years
A particularly interesting exercise is to check what would have happened if Reagan had decided to reappoint Volcker and not appoint Greenspan. We plot these results in Ögure 6.5. The quick answer is: lower ináation and interest rates. Our estimates also suggest that Volcker would have reduced price increases with little cost to output.
Figure 6.5: Volcker during the Greenspan years
Our Önal exercise is to plot, in Ögure 6.6, the counterfactual in which we move Volcker to the time of Burns-Miller. The main Önding is that ináation would have been rather lower, especially because the e§ects of the second oil shock would have been much more muted. This counterfactual is plausible: other countries, such as Germany, Switzerland, and Japan, that undertook a more aggressive monetary policy during the late 1970s were able to keep ináation under control at levels below 5 percent at an annual rate, while the U.S. had peaks of price increases over 10 percent.
Figure 6.6: Volcker during the Burns-Miller years
6.4.2. Counterfactual II: No Volatility Changes
In our second historical counterfactual, we compute how the economy would have performed in the absence of changes in the volatility of the shocks, that is, if the volatility of the innovation of the structural shocks had been Öxed at its historical mean. To do so, we back out the smoothed structural shocks as we did in section 5.8 and we feed them to the model, given our parameter point estimates and the historical mean of volatility, to generate series for ináation, output, and the federal funds rate.
Moments Table 6.2 reports the moments of the data (in annualized terms) and the moments from the counterfactual history (no s.v. in the table stands for ìno stochastic volatilityî). In both cases, we include the moments for the whole sample and for the sample divided before and after
1984.Q1, a conventional date for the start of the great moderation (McConnell and PÈrez-QuirÛs, 2000). In the last two rows of the table, we compute the ratio of the moments after 1984.Q1 over the moments before 1984.Q1. The benchmark model with stochastic volatility plus parameter drifting replicates the data exactly.
Some of the numbers in table 6.2 are well known. For instance, after 1984, the standard deviation of ináation falls by nearly 60 percent, the standard deviation of output growth falls by 44 percent, and the standard deviation of the federal funds rate falls by 39 percent. In terms of means, after 1984, there is less ináation and the federal funds rate is lower, but output growth is also 15 percent lower.
Table 6.2: No Volatility Changes, Data versus Counterfactual History
| Means | Standard Deviations | |||||
| Inflation | Output Growth | FFR | Inflation | Output Growth | FFR | |
| Data | 3.8170 | 1.8475 | 6.0021 | 2.6181 | 3.5879 | 3.3004 |
| Data, pre 1984.1 | 4.6180 | 1.9943 | 6.7179 | 3.2260 | 4.3995 | 3.8665 |
| Data, after 1984.1 | 2.9644 | 1.6911 | 5.2401 | 1.3113 | 2.4616 | 2.3560 |
| No s.v. | 2.5995 | 0.7169 | 6.9388 | 3.5534 | 3.1735 | 2.4128 |
| No s.v., pre-1984.1 | 2.0515 | 0.9539 | 6.3076 | 3.7365 | 3.4120 | 2.7538 |
| No s.v., after-1984.1 | 3.1828 | 0.4647 | 7.6106 | 3.2672 | 2.8954 | 1.7673 |
| Data, post-1984.1/pre-1984.1 | 0.6419 | 0.8480 | 0.7800 | 0.4065 | 0.5595 | 0.6093 |
| No s.v., post-1984.1/pre-1984.1 | 1.5515 | 0.4871 | 1.2066 | 0.8744 | 0.8486 | 0.6418 |
The table also reáects the fact that without volatility shocks, the reduction in volatility observed after 1984 would have been noticeably smaller. The standard deviation of ináation would have fallen by only 13 percent, the standard deviation of output growth would have fallen by 16 percent, and the standard deviation of the federal funds rate would have fallen by 35 percent, that is, only 33, 20, and 87 percent, respectively, of how much they would have fallen otherwise. We must resist here the temptation to undertake a standard variance decomposition exercise. Since we have a second-order approximation to the policy function and its associated cross-product terms, we cannot neatly divide total variance among the di§erent shocks as we could do in the linear case.
Table 6.2 documents that, while time-varying policy is reáected in changes of average ináation, volatility shocks a§ect the standard deviation of the observed series. Without time-varying volatility the decrease in observed volatility would not have been nearly as big as we observed in the data. Hence, while changes in the systematic component of monetary policy account for changes in average ináation, volatility shocks account for changes in the standard deviation of ináation, output growth and interest rates observed after 1984. Also, without stochastic volatility, output growth would have been quite lower on average.
Counterfactual Paths Figure 6.7 compares the whole path of the counterfactual history (blue line) with the observed one (red line). Figure 6.7 tells us that volatility shocks mattered throughout the sample. The run-up in ináation would have been much slower in the late 1960s (ináation would have actually been negative during the last years of Martinís tenure) with small e§ects on output growth or the federal funds rate (except at the very end of the sample). Ináation would not have picked up nearly as much during the Örst oil shock, but output growth would have su§ered. During Volckerís time, ináation would also have fallen faster with little cost to output growth. These are indications that both Burns-Miller and Volcker su§ered from large and volatile shocks to the economy. In comparison, during the 1990s, ináation would have been more volatile, with a big increase in the middle of the decade. Similarly, during those years, output growth would have been much lower, with a long recession between 1994 and 1998, and the federal funds rate would have been prominently higher. ConÖrming the results presented in the paper, this is another manifestation of how placid the 1990s were for policy makers.
Figure 6.7: Counterfactual ìNo Changes in Volatilityî.
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