Bayesian Estimation of DSGE Models

by Edward P. Herbst, Frank Schorfheide

Overview

Dynamic stochastic general equilibrium (DSGE) models have become one of the workhorses of modern macroeconomics and are extensively used for academic research as well as forecasting and policy analysis at central banks. This book introduces readers to state-of-the-art computational techniques used in the Bayesian analysis of DSGE models. The book covers Markov chain Monte Carlo techniques for linearized DSGE models, novel sequential Monte Carlo methods that can be used for parameter inference, and the estimation of nonlinear DSGE models based on particle filter approximations of the likelihood function. The theoretical foundations of the algorithms are discussed in depth, and detailed empirical applications and numerical illustrations are provided. The book also gives invaluable advice on how to tailor these algorithms to specific applications and assess the accuracy and reliability of the computations.

Bayesian Estimation of DSGE Models is essential reading for graduate students, academic researchers, and practitioners at policy institutions.

Product Details

ISBN-13:	9781400873739
Publisher:	Princeton University Press
Publication date:	12/29/2015
Series:	The Econometric and Tinbergen Institutes Lectures
Sold by:	Barnes & Noble
Format:	eBook
Pages:	296
File size:	9 MB

About the Author

Edward P. Herbst is an economist in the Division of Research and Statistics at the Federal Reserve Board. Frank Schorfheide is Professor of Economics at the University of Pennsylvania and research associate at the National Bureau of Economic Research. He also is a fellow of the Penn Institute for Economic Research, a visiting scholar at the Federal Reserve Banks of Philadelphia and New York, and a coeditor of Quantitative Economics. For more, see edherbst.net and sites.sas.upenn.edu/schorf.

Read an Excerpt

Bayesian Estimation of DSGE Models

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By Edward P. Herbst, Frank Schorfheide

PRINCETON UNIVERSITY PRESS

Copyright © 2016 Edward P. Herbst and Frank Schorfheide
All rights reserved.
ISBN: 978-1-4008-7373-9

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CHAPTER 1

DSGE Modeling

Estimated dynamic stochastic general equilibrium (DSGE) models are now widely used by academics to conduct empirical research macroeconomics as well as by central banks to interpret the current state of the economy, to analyze the impact of changes in monetary or fiscal policy, and to generate predictions for key macroeconomic aggregates. The term DSGE model encompasses a broad class of macroeconomic models that span the real business cycle models of Kydland and Prescott (1982) and King, Plosser, and Rebelo (1988) as well as the New Keynesian models of Rotemberg and Woodford (1997) or Christiano, Eichenbaum, and Evans (2005), which feature nominal price and wage rigidities and a role for central banks to adjust interest rates in response to inflation and output fluctuations. A common feature of these models is that decision rules of economic agents are derived from assumptions about preferences and technologies by solving intertemporal optimization problems. Moreover, agents potentially face uncertainty with respect to aggregate variables such as total factor productivity or nominal interest rates set by a central bank. This uncertainty is generated by exogenous stochastic processes that may shift technology or generate unanticipated deviations from a central bank's interest-rate feedback rule.

The focus of this book is the Bayesian estimation of DSGE models. Conditional on distributional assumptions for the exogenous shocks, the DSGE model generates a likelihood function, that is, a joint probability distribution for the endogenous model variables such as output, consumption, investment, and inflation that depends on the structural parameters of the model. These structural parameters characterize agents' preferences, production technologies, and the law of motion of the exogenous shocks. In a Bayesian framework, this likelihood function can be used to transform a prior distribution for the structural parameters of the DSGE model into a posterior distribution. This posterior is the basis for substantive inference and decision making. Unfortunately, it is not feasible to characterize moments and quantiles of the posterior distribution analytically. Instead, we have to use computational techniques to generate draws from the posterior and then approximate posterior expectations by Monte Carlo averages.

In Section 1.1 we will present a small-scale New Keynesian DSGE model and describe the decision problems of firms and households and the behavior of the monetary and fiscal authorities. We then characterize the resulting equilibrium conditions. This model is subsequently used in many of the numerical illustrations. Section 1.2 briefly sketches two other DSGE models that will be estimated in subsequent chapters.

1.1 A Small-Scale New Keynesian DSGE Model

We begin with a small-scale New Keynesian DSGE model that has been widely studied in the literature (see Woodford (2003) or Gali (2008) for textbook treatments). The particular specification presented below is based on An and Schorfheide (2007). The model economy consists of final goods producing firms, intermediate goods producing firms, households, a central bank, and a fiscal authority. We will first describe the decision problems of these agents, then describe the law of motion of the exogenous processes, and finally summarize the equilibrium conditions. The likelihood function for a linearized version of this model can be quickly evaluated, which makes the model an excellent showcase for the computational algorithms studied in this book.

1.1.1 Firms

Production takes place in two stages. There are monopolistically competitive intermediate goods producing firms and perfectly competitive final goods producing firms that aggregate the intermediate goods into a single good that is used for household and government consumption. This two-stage production process makes it fairly straightforward to introduce price stickiness, which in turn creates a real effect of monetary policy.

The perfectly competitive final good producing firms combine a continuum of intermediate goods indexed by j [member of] [0, 1] using the technology

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.1)

The final good producers take input

prices Pt([j]) and output prices Pt as given. The revenue from the sale of the final good is PtYt and the input costs incurred to produce Yt are ∫10Pt(j)Yt (j)dj. Maximization of profits

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)

with respect to the inputs Yt]([j]) implies that the demand for intermediate good j is given by

Yt(j) = (Pt(j) Pt-11/ν Yt (1.3)

Thus, the parameter 1/v represents the elasticity of demand for each intermediate good. In the absence of an entry cost, final good producers will enter the market until profits are equal to zero. From the zero-profit condition, it is possible to derive the following relationship between the intermediate goods prices and the price of the final good:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.4)

Intermediate good j is produced by a monopolist who has access to the following linear production technology:

Y(t(t) = AtNt(j), (1.5)

where At is an exogenous productivity process that is common to all firms and Nt](j) is the labor input of firm j. To keep the model simple, we abstract from capital as a factor or production for now. Labor is hired in a perfectly competitive factor market at the real wage Wt.

In order to introduce nominal price stickiness, we assume that firms face quadratic price adjustment costs

ACt(j) = φ/2 (Pt(j)/Pt-1 - π)2 Yt(j), (1.6)

where / governs the price rigidity in the economy and π is the steady state inflation rate associated with the final good. Under this adjustment cost specification it is costless to change prices at the rate π. If the price change deviates from π, the firm incurs a cost in terms of lost output that is a quadratic function of the discrepancy between the price change and π. The larger the adjustment cost parameter /, the more reluctant the intermediate goods producers are to change their prices and the more rigid the prices are at the aggregate level. Firm j chooses its labor input Nt(j) and the price Pt(j) to maximize the present value of future profits

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.7)

Here, Qt+s|t is the time t value of a unit of the consumption good in period t + s to the household, which is treated as exogenous by the firm.

1.1.2 Households

The representative household derives utility from consumption Ct relative to a habit stock (which is approximated by the level of technology At) and real money balances Mt/Pt. The household derives disutility from hours worked Ht and maximizes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.8)

where β is the discount factor, 1/τ is the intertemporal elasticity of substitution, and XM and XH are scale factors that determine steady state money balances and hours worked. We will set XH = 1. The household supplies perfectly elastic labor services to the firms, taking the real wage Wt as given. The household has access to a domestic bond market where nominal government bonds Bt are traded that pay (gross) interest Rt. Furthermore, it receives aggregate residual real profits Dt from the firms and has to pay lump-sum taxes Tt. Thus, the household's budget constraint is of the form

Pt Ct + Bt + Mt + Tt = Pt Wt Ht + Rt-1 Bt-1 + Mt-1 + PtDt + PtSCt (1.9)

where SCt is the net cash inflow from trading a full set of state-contingent securities.

1.1.3 Monetary and Fiscal Policy

Monetary policy is described by an interest rate feedback rule of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.10)

where εR,t is a monetary policy shock and R*t is the (nominal) target rate:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.11)

Here r is the steady state real interest rate (defined below), πt is the gross inflation rate defined as πt = Pt/Pt-1, and π* target inflation rate. in (1.11) is the level of output that would prevail in the absence of nominal rigidities.

We assume that the fiscal authority consumes a fraction ζt of aggregate output Yv, that is Gt = ζYv, and that ITLζITL [member of] [0, 1] follows an exogenous process specified below. The government levies a lump-sum tax Tt (subsidy) to finance any shortfalls in government revenues (or to rebate any surplus). The government's budget constraint is given by

Pt Gt + Rt-1 Bt-1 + Mt-1 = Tt + Bt + Mt. (1.12)

1.1.4 Exogenous Processes

The model economy is perturbed by three exogenous processes. Aggregate productivity evolves according to

ln At = ln γ + ln At-1 + ln zt ln zt = ρz ln zt-1 + εz,t (1.13)

Thus, on average technology grows at the rate γ and zt captures exogenous fluctuations of the technology growth rate. Define gt = 1/(1 - ζt) where ζt was previously defined as the fraction of aggregate output purchased by the government.

We assume that

ln gt = (1 - ρg) ln g + ρ g ln gt-1 + εg,t. (1.14)

Finally, the monetary policy shock εR,t is assumed to be serially uncorrelated. The three innovations are independent of each other at all leads and lags and are normally distributed with means zero and standard deviations σz, σg, and σR respectively.

1.1.5 Equilibrium Relationships

We consider the symmetric equilibrium in which all intermediate goods producing firms make identical choices so that the j subscript can be omitted. The market clearing conditions are given by

Yt = Ct + Gt + ACt and Ht = Nt. (1.15)

Because the households have access to a full set of state-contingent claims, it turns out that Qt+r|tt in (1.7) is

Qt+s|t = (Ct+s/Ct)τ (At/At+s)1-τ. (1.16)

Thus, in equilibrium households and firms are using the same stochastic discount factor. Moreover, it can be shown that output, consumption, interest rates, and inflation have to satisfy the following optimality conditions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.17)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.18)

Equation (1.17) is the consumption Euler equation which reflects the first-order condition with respect to the government bonds Bt. In equilibrium, the household equates the marginal utility of consuming a dollar today with the discounted marginal utility from investing the dollar, earning interest Rv, and consuming it in the next period. Equation (1.18) characterizes the profit maximizing choice of the intermediate goods producing firms. The first-order condition for the firms' problem depends on the wage Wt. We used the households' labor supply condition to replace Wt by a function of the marginal utility of consumption. In the absence of nominal rigidities (/ = 0) aggregate output is given by

Y*t = (1 - ν)1/τ/ Atgt, (1.19)

which is the target level of output that appears in the monetary policy rule (1.11).

In Section 2.1 of Chapter 2 we will use a solution technique for the DSGE model that is based on a Taylor series approximation of the equilibrium conditions. A natural point around which to construct this approximation is the steady state of the DSGE model. The steady state is attained by setting the innovations εR,v epsilon]g,v and εz,t to zero at all times. Because technology ln At evolves according to a random walk with drift ln γ, consumption and output need to be detrended for a steady state to exist. Let ct = Ct/At and yt = Yt/At, and y*t = Y*t/At. Then the steady state is given by

π = π*, r = γ/β, R = rπ*, c = (1 - v)1/τ, y = gc = y*.

Steady state inflation equals the targeted inflation rate π*; the real rate depends on the growth rate of the economy γ and the reciprocal of the households' discount factor β and finally steady state output can be determined from the aggregate resource constraint. The nominal interest rate is determined by the Fisher equation; the dependence of the steady state consumption on v reflects the distortion generated by the monopolistic competition among intermediate goods producers. We are now in a position to rewrite the equilibrium conditions by expressing each variable in terms of percentage deviations from its steady state value. Let [??]t = ln(xt/x) and write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.21)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.22)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.23)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.24)

[??]t = ρg [??]t-1 + εg,t (1.25)

[??]t = ρz [??]t-1 + εz,t (1.26)

The equilibrium law of motion of consumption, output, interest rates, and inflation has to satisfy the expectational difference equations (1.21) to (1.26).

1.2 Other DSGE Models Considered in This Book

In addition to the small-scale New Keynesian DSGE model, we consider two other models: the widely used Smets-Wouters (SW) model, which is a more elaborate version of the small-scale DSGE model that includes capital accumulation as well as wage rigidities, and a real business cycle model with a detailed characterization of fiscal policy. We will present a brief overview of these models below and provide further details as needed in Chapter 6.

1.2.1 The Smets-Wouters Model

The Smets and Wouters (2007) model is a more elaborate version of the small-scale DSGE model presented in the previous section. In the SW model capital is a factor of intermediate goods production, and in addition to price stickiness the model features nominal wage stickiness. In order to generate a richer autocorrelation structure, the model also includes investment adjustment costs, habit formation in consumption, and partial dynamic indexation of prices and wages to lagged values. The model is based on work by Christiano, Eichenbaum, and Evans (2005), who added various forms of frictions to a basic New Keynesian DSGE model in order to capture the dynamic response to a monetary policy shock as measured by a structural vector autoregression (VAR). In turn (the publication dates are misleading), Smets and Wouters (2003) augmented the Christiano-Eichenbaum-Evans model by additional exogenous structural shocks (among them price markup shocks, wage markup shocks, preference shocks, and others) to be able to capture the joint dynamics of Euro Area output, consumption, investment, hours, wages, inflation, and interest rates.

The Smets and Wouters (2003) paper has been highly influential, not just in academic circles but also in central banks because it demonstrated that a modern DSGE model that is usable for monetary policy analysis can achieve a time series fit that is comparable to a less restrictive vector autoregression (VAR). The 2007 version of the SW model contains a number of minor modifications of the 2003 model in order to optimize its fit on U.S. data. We will use the 2007 model exactly as it is presented in Smets and Wouters (2007) and refer the reader to that article for details. The log-linearized equilibrium conditions are reproduced in Appendix A.1. By now, the SW model has become one of the workhorse models in the DSGE model literature and in central banks around the world. It forms the core of most large-scale DSGE models that augment the SW model with additional features such as a multi-sector production structure or financial frictions. Because of its widespread use, we will consider its estimation in this book.

1.2.2 A DSGE Model for the Analysis of Fiscal Policy

In the small-scale New Keynesian DSGE model and in the SW model, fiscal policy is passive and non-distortionary. The level of government spending as a fraction of GDP is assumed to evolve exogenously and an implicit money demand equation determines the amount of seignorage generated by the interest rate feedback rule. Fiscal policy is passive in the sense that the government raises lump-sum taxes (or distributes lump-sum transfers) to ensure that its budget constraint is satisfied in every period. These lump-sum taxes are non-distortionary, because they do not affect the decisions of households and firms. The exact magnitude of the lump-sum taxes and the level of government debt are not uniquely determined, but they also do not matter for macroeconomic outcomes. Both the small-scale DSGE model and the SW model were explicitly designed for the analysis of monetary policy and abstract from a realistic representation of fiscal policy.

(Continues...)

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Table of Contents

Figures xi
Tables xiii
Series Editors’ Introduction xv
Preface xvii
I Introduction to DSGE Modeling and Bayesian Inference 1
1 DSGE Modeling 3
1.1 A Small-Scale New Keynesian DSGE Model 4
1.2 Other DSGE Models Considered in This Book 11
2 Turning a DSGE Model into a Bayesian Model 14
2.1 Solving a (Linearized) DSGE Model 16
2.2 The Likelihood Function 19
2.3 Priors 22
3 A Crash Course in Bayesian Inference 29
3.1 The Posterior of a Linear Gaussian Model 31
3.2 Bayesian Inference and Decision Making 35
3.3 A NonGaussian Posterior 43
3.4 Importance Sampling 46
3.5 Metropolis-Hastings Algorithms 52
II Estimation of Linearized DSGE Models 63
4 Metropolis-Hastings Algorithms for DSGE Models 65
4.1 A Benchmark Algorithm 67
4.2 The RWMH-V Algorithm at Work 69
4.3 Challenges Due to Irregular Posteriors 77
4.4 Alternative MH Samplers 81
4.5 Comparing the Accuracy of MH Algorithms 87
4.6 Evaluation of the Marginal Data Density 93
5 Sequential Monte Carlo Methods 100
5.1 A Generic SMC Algorithm 101
5.2 Further Details of the SMC Algorithm 109
5.3 SMC for the Small Scale DSGE Model 125
6 Three Applications 130
6.1 A Model with Correlated Shocks 131
6.2 The Smets-Wouters Model with a Diffuse Prior 141
6.3 The Leeper-Plante-Traum Fiscal Policy Model 150
III Estimation of Nonlinear DSGE Models 161
7 From Linear to Nonlinear DSGE Models 163
7.1 Nonlinear DSGE Model Solutions 164
7.2 Adding Nonlinear Features to DSGE Models 167
8 Particle Filters 171
8.1 The Bootstrap Particle Filter 173
8.2 A Generic Particle Filter 182
8.3 Adapting the Generic Filter 185
8.4 Additional Implementation Issues 191
8.5 Adapting s_t-1 Draws 198
8.6 Application to the Small-Scale DSGE Model 204
8.7 Application to the SW Model 212
8.8 Computational Considerations 216
9 Combining Particle Filters with MH Samplers 218
9.1 The PFMH Algorithm 218
9.2 Application to the Small-Scale DSGE Model 222
9.3 Application to the SW Model 224
9.4 Computational Considerations 229
10 Combining Particle Filters with SMC Samplers 231
10.1 An SM C² Algorithm 231
10.2 Application to the Small-Scale DSGE Model 237
10.3 Computational Considerations 239
Appendix 241
A Model Descriptions 241
A.1 Smets-Wouters Model 241
A.2 Leeper-Plante-Traum-Fiscal Policy Model 247
B Data Sources 249
B.1 Small-Scale-New Keynesian DSGE Model 249
B.2 Smets-Wouters Model 249
B.3 Leeper-Plante-Traum Fiscal Policy Model 251
Bibliography 257
Index 271

What People are Saying About This

From the Publisher

"This book depicts valuable and revealing methods for solving, estimating, and analyzing a class of dynamic equilibrium models of the macroeconomy. It describes formally tractable techniques for the study of macroeconomic models that feature transition mechanisms for a large number of underlying shocks. Both authors have played important roles in developing and applying these techniques. This is a terrific resource for how to use these methods in practice."—Lars Peter Hansen, David Rockefeller Distinguished Service Professor of Economics, University of Chicago, and recipient of the Nobel Prize in economics

"This timely book collects in one place many of the key Markov chain Monte Carlo methods for numerical Bayesian inference along with many of their recent refinements. Written for applied users, it offers clear descriptions of each algorithm and illustrates how it can be used to estimate dynamic stochastic general equilibrium models in macroeconomics."—James D. Hamilton, Professor of Economics, University of California, San Diego

"This is perhaps the most thorough book available on how to estimate DSGE models using sophisticated Bayesian computation tools. It is an excellent resource for professionals and advanced students of the topic."—Serena Ng, Professor of Economics, Columbia University

Editorial Reviews

"Well written and well organized, and the topic analyzed is very interesting and current."—-Manuel Salvador, MathSciNet

"The book is well written and clearly arranged. It can be recommended to economists and researchers who are interested in applying state-of-the-art computational methods to estimate DSGE models as well as to econometricians who want to learn about the application of DSGE models in empirical macroeconomics."—-Peter Hackl, Statistical Papers

From the Publisher