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.

• 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 st-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

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