Large sample properties of posterior densities, Bayesian information criterion and the likelihood principle in nonstationary time series models

Asymptotic normality of the posterior is a well understood result for dynamic as well as nondynamic models based on sets of abstract conditions whose actual applicability is hardly known especially for the case of nonstationarity. In this paper we provide a set of conditions by which we can relatively easily prove the asymptotic posterior normality under quite general situations of possible nonstationarity. This result reinforces and generalizes the point of Sims and Uhlig (1991) that inference based on the likelihood principle, explained by Berger and Wolpert (1988), will be unchanged regardless of whether the data are generated by a stationary process or by a unit root process. On the other hand, our conditions allow us to generalize the Bayesian information criterion known as the Schwarz criterion to the case of possible nonstationarity. In addition, we have shown that consistency of the maximum likelihood estimator, not the asymptotic normality of the estimator, with some minor additional assumptions is sufficient for asymptotic posterior normality.