Bayes Inference via Gibbs Sampling of Autoregressive Time Series Subject to Markov Mean and Variance Shifts

Abstract

We examine autoregressive time series models that are subject to regime switching. These shifts are determined by the outcome of an unobserved two-state indicator variable that follows a Markov process with unknown transition probabilities. A Bayesian framework is developed in which the unobserved states, one for each time point, are treated as missing data and then analyzed via the simulation tool of Gibbs sampling. This method is expedient because the conditional posterior distribution of the parameters, given the states, and the conditional posterior distribution of the states, given the parameters, all have a form amenable to Monte Carlo sampling. The approach is straightforward and generates marginal posterior distributions for all parameters of interest. Posterior distributions of the states, future observations, and the residuals, averaged over the parameter space are also obtained. Several examples with real and artificial data sets and weak prior information illustrate the usefulness of the methodology.

KEY WORDS:

• Data augmentation
• Hidden Markov models
• Missing data
• Mixture distribution
• Monte Carlo simulation
• Regime shifts