# Convergence Analysis of a Collapsed Gibbs Sampler for Bayesian Vector Autoregressions

## TL;DR

This paper analyzes the convergence of a collapsed Gibbs sampler for Bayesian vector autoregressions, demonstrating geometric ergodicity regardless of sample size and providing conditions for stability as data grows.

## Contribution

It establishes the geometric ergodicity of the sampler under broad conditions and analyzes its stability with increasing data, a novel result for practical MCMC algorithms.

## Key findings

- Geometric ergodicity holds regardless of sample size.
- Convergence rate remains bounded away from one asymptotically.
- Results hold under near arbitrary model misspecification.

## Abstract

We study the convergence properties of a collapsed Gibbs sampler for Bayesian vector autoregressions with predictors, or exogenous variables. The Markov chain generated by our algorithm is shown to be geometrically ergodic regardless of whether the number of observations in the underlying vector autoregression is small or large in comparison to the order and dimension of it. In a convergence complexity analysis, we also give conditions for when the geometric ergodicity is asymptotically stable as the number of observations tends to infinity. Specifically, the geometric convergence rate is shown to be bounded away from unity asymptotically, either almost surely or with probability tending to one, depending on what is assumed about the data generating process. This result is one of the first of its kind for practically relevant Markov chain Monte Carlo algorithms. Our convergence results hold under close to arbitrary model misspecification.