Time Series Modeling with a Bayesian Frame of Reference: Concepts, Illustrations and Asymptotics

This paper offers an approach to time series modeling that attempts to reconcile classical and Bayesian methods. The central idea put forward to achieve this reconciliation is that the Bayesian approach relies implicitly on a frame of reference for the data generating mechanism that is quite different from the one that is employed in the classical approach. Differences in inferences from the two approaches are therefore to be expected unless the altered frame of reference is taken into account. We show that the new frame of reference in Bayesian inference is a consequence of a change of measure that arises naturally in the application of Bayes theorem. Our paper explores this change of measure and its consequences using martingale methods. Examples are given to illustrate its practical implications. No assumptions concerning stationarity or rates of convergence are required in the development of our asymptotic theory. Some implications for statistical testing are explored and we suggest new tests, which we call Bayes model tests, for discriminating between models. A posterior odds version of these tests is developed and shown to have good finite sample properties. This is the test that we recommend for practical use. Autoregressive models with multiple lags and deterministic trends are considered and explicit forms are given for the posterior odds tests for the presence of a unit root and for joint tests for the presence of a unit root, drift and trend. This paper emphasizes the new conceptual framework for thinking about Bayesian methods in time series and provides illustrations of its use in some common models for possibly nonstationary time series. A sequel to the present paper develops a general and more abstract theory that will have a wider range of applications.