Christopher A. Sims, Daniel F. Waggoner, and Tao Zha
Working Paper 2006-22
November 2006
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The inference for hidden Markov chain models in which the structure is a multiple-equation macroeconomic model raises a number of difficulties that are not as likely to appear in smaller models. One is likely to want to allow for many states in the Markov chain without allowing the number of free parameters in the transition matrix to grow as the square of the number of states but also without losing a convenient form for the posterior distribution of the transition matrix. Calculation of marginal data densities for assessing model fit is often difficult in high-dimensional models and seems particularly difficult in these models. This paper gives a detailed explanation of methods we have found to work to overcome these difficulties. It also makes suggestions for maximizing posterior density and initiating Markov chain Monte Carlo simulations that provide some robustness against the complex shape of the likelihood in these models. These difficulties and remedies are likely to be useful generally for Bayesian inference in large time-series models. The paper includes some discussion of model specification issues that apply particularly to structural vector autoregressions with a Markov-switching structure.
JEL classification: C32, C52, E52
Key words: volatility, coefficient changes, discontinuous shifts, Lucas critique, independent Markov processes
The authors thank Tim Cogley, John Geweke, Michel Juillard, Ulrich Mueller, and Frank Schorfheide for helpful discussions and comments. Eric Wang provided excellent research assistance in computation on the Linux operating system. The authors also acknowledge the technical support on parallel and grid computation from the Computing College of the Georgia Institute of Technology. The views expressed here are the authors’ and not necessarily those of the Federal Reserve Bank of Atlanta or the Federal Reserve System. Any remaining errors are the authors’ responsibility.
Please address questions regarding content to Christopher A. Sims, Professor of Economics and Banking, Department of Economics, 104 Fisher Hall, Princeton University, Princeton, NJ 08544-1021, 609-258-4033, sims@princeton.edu; Daniel Waggoner, Research Economist and Associate Policy Adviser, Research Department, Federal Reserve Bank of Atlanta, 1000 Peachtree Street, N.E., Atlanta, GA 30309-4470, 404-498-8278, daniel.f.waggoner@atl.frb.org; or Tao Zha, Research Economist and Senior Policy Adviser, Research Department, Federal Reserve Bank of Atlanta, 1000 Peachtree Street, N.E., Atlanta, GA 30309-4470, 404-498-8353, tzha@earthlink.net.
For further information, contact the Public Affairs Department, Federal Reserve Bank of Atlanta, 1000 Peachtree Street, N.E., Atlanta, Georgia 30309-4470, 404-498-8020.