Model Selection and Adaptive Markov chain Monte Carlo for Bayesian Cointegrated VAR model

In this paper, we develop novel Markov chain Monte Carlo sampling methodology for Bayesian Cointegrated Vector Auto Regression (CVAR) models. Here we focus on two novel extensions to the sampling methodology for the CVAR posterior distribution. The first extension we develop replaces the popular sampling methodology of the griddy Gibbs sampler with an automated alternative which is based on an Adaptive Metropolis-Hastings algorithm. This is particularly relevant to automate the proposal mechanism in the MCMC algorithm in settings where griddy Gibbs is impractical such as when the dimension of the CVAR series is large, e.g. d > 5. We also treat the rank of the CVAR model as a random variable and perform joint inference on the rank and model parameters. This is achieved with a Bayesian posterior distribution defined over both the rank and the CVAR model parameters, and inference is made via a Savage-Dickey density estimator for the Bayes Factor analysis of rank.