Bayes inference in regression models with ARMA (p, q) errors

We develop practical and exact methods of analyzing ARMA (p, q) regression error models in a Bayesian framework by using the Gibbs sampling and Metropolis-Hastings algorithms, and we prove that the kernel of the proposed Markov chain sampler converges to the true density. The procedures can be applied to pure ARMA time series models and to determine features of the likelihood function by choosing appropriate diffuse priors. Our results are unconditional on the initial observations. We also show how the algorithm can be further simplified for the important special cases of stationary AR(p) and invertible MA(q) models. Recursive transformations developed in this paper to diagonalized the covariance matrix of the errors should prove useful in frequentist estimation. Examples with simulated and actual economic data are presented.