Bayesian Comparison of ARIMA and Stationary ARMA Models

Time series analysts have long been concerned with distinguishing stationary "generating processes" from processes for which differencing is required to induce stationarity. In practical applications, this issue is addressed almost invariably through formal hypothesis testing. In this paper, we explore some aspects of the Bayesian approach to the problem, leading to the calculation of posterior odds ratios. Interesting features arise in the simplest possible variant of the problem, where a choice has to be made between a random walk and a stationary first order autoregressive model. We discuss in detail the analysis of this case, and also indicate how our approach extends to the more general comparison of an ARIMA model with a stationary competitor.