Methods for inference in large multiple-equation Markov-switching models

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.