An Asymtotic Theory of Bayesian Inference for Time Series

This paper develops an asymptotic theory of Bayesian inference for time series. A
limiting representation of the Bayesian data density is obtained and shown to be of the
same general exponential form for a wide class of likelihoods and prior distributions.
Continuous time and discrete time cases are studied. In discrete time, an embedding
theorem is given which shows how to embed the exponential density in a continuous time
process. From the embedding we obtain a large sample approximation to the model of the
data that corresponds to the exponential density. This has the form of discrete observations
drawn from a nonlinear stochastic differential equation driven by Brownian motion.
No assumptions concerning stationarity or rates of convergence are required in the
asymptotics. Some implications for statistical testing are explored and we suggest tests
that are based on likelihood ratios (or Bayes factors) of the exponential densities for
discriminating between models.
KEYWORDSA:utoregression, Bayesian data measure, data density process, Doltans
exponential, exponential data density, likelihood, martingale, posterior process, prior
density, quadratic variation process, stochastic differential equation, unit root.
1. INTRODUCTION
THE BAYESIANAP PROACH TO MODELING and inference in time series econometrics
have become increasingly popular in recent years. Time series applications
raise concerns that deserve special attention, like the nature of prior information
in time series models, the treatment of initial conditions, and nonstationarity.
These concerns are the subject of two recent themed issues of the Journal of
Applied Econometrics (1991) and Econometric Theory (1994). The focus of
attention in the present paper is not the aforementioned concerns per se, but
the development of a general asymptotic theory of Bayesian inference for time
series. As a complement to the literature on formulating "uninformative priors"
for time series, this paper is aimed at obtaining asymptotic results whereby the
prior is dominated by the data.