Risks of Using Improper Priors with Gibbs Sampling and Autocorrelated Errors

Abstract

In this article we examine the use of Gibbs sampling to estimate the autocorrelation coefficient in a linear regression model. Researchers had previously experienced difficulty with moderate-to-high positive autocorrelated errors; estimates could be unstable and sometimes failed to converge. We show that the cause of this problem is that the use of an improper prior leads to an improper posterior, although the conditionals are proper, and hence a formal Gibbs sampler can be constructed. The problem is solved by the use of a vague but proper prior. In this simple case many of the calculations can be done analytically and it serves as a warning as to the uncritical use of improper priors with Gibbs sampling.