Abstract:

We consider Gibbs samplers for Bayesian vector autoregressive processes of order q (VAR(q)) with covariates. We consider nonconjugate priors which lead to an analytically intractable posterior distribution and hence Markov chain Monte Carlo (MCMC) methods are required to use the model for inferential purposes. The particular MCMC sampler appropriate here is a threecomponent collapsed Gibbs sampler. We establish that the Markov chain converges at a geometric rate, which is a crucial step in ensuring the existence of a central limit theorem and hence for rigorous output analysis. However, this analysis breaks down when the number of observations, n, in the VAR(q) process increases without bound. Thus we consider and refine a technique recently developed by Qin and Hobert (2019+, Annals) to show that the collapsed Gibbs sampler enjoys asymptotically stable geometric ergodicity as the number of observations increases without bound.
