On Large Sample Properties of Bayes Procedures: Misspecification, Non-Smoothness and Davies' Problem

Abstract

From a frequentist perspective, we examine the large sample properties of Bayes procedures in a general framework, where data may be dependent and models may be misspecified and non-smooth. The posterior distribution of parameters is shown to be asymptotically normal, centered at the quasi maximum likelihood estimator, under mild conditions. However, its asymptotic variance matrix is generally different from that of the quasi maximum likelihood estimator that possesses asymptotic normality. In this framework, the Bayes factor for the test problem of Davies (1977, 1987), where a parameter is unidentified under the null hypothesis, is analyzed. The probability that the Bayes factor leads to a correct conclusion about the hypotheses in Davies' problem is shown to approach to one.

Keywords: quasi likelihood, stochastic differentiability, asymptotic normality, loss of identification, dependent data

JEL Classification: C10, C13, C12, C11