Normality of Posterior Distribution Under Misspecification and Nonsmoothness, and Bayes Factor for Davies' Problem

Abstract

We examine the large sample properties of Bayes procedures in a general framework, where data may be dependent and models may be misspecified and nonsmooth. The posterior distribution of parameters is shown to be asymptotically normal, centered at the quasi maximum likelihood estimator, under mild conditions. In this framework, the Bayes factor for the test problem of Davies (1997, 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:

• Asymptotics
• Dependent data
• Loss of identification
• Quasi likelihood
• Stochastic differentiability

JEL Classification:

• C10
• C13
• C12
• C11

ACKNOWLEDGMENT

This paper has benefited from conversations with Kevin Fox, Paolo Giordani, Robert Kohn, and Adrian Pagan. I would like to thank two referees and the editors for their comments and suggestions that have led to many improvements. All remaining errors are mine. The support from the Australian Research Council Linkage Grant LP0882468 is gratefully acknowledged.

Notes

Bunke and Milhaud (1998) consider misspecified smooth models with independent data. They study the asymptotics of a pseudo-Bayes estimator (such as posterior mean), whereas we examine the asymptotics of the posterior distribution itself. They find that the asymptotic variance of the pseudo-Bayes estimator from a mis-specified model generally differs from that of the correct model.

In the case of the unit root regression, however, it is well known that the asymptotic distribution of the QML estimator is non-normal (see Dickey and Fuller, 1976, and Phillips, 1987, 1991, among others).

See Lele et al. (2010) for an example of using posterior distributions to approximate maximum likelihood estimators.

For example, Clarke and Barron (1990) use the Taylor expansion to consider the asymptotic relative entropy between the true distribution and the posterior distribution.

See Andrews (1999) for an extensive discussion on the quadratic approximation of estimation objective functions.

See Ebrahimi et al. (2010) for the Bayesian measures of sample information about parameters.

See (Bauwens et al. (1999), pp. 57–61) for the IG ₂, the normal-IG ₂ mix and the corresponding posterior distribution.

A structural break is defined as the change in a parameter after a date in time. A threshold model involves parameters that change when a key variable exceeds a cut-off value (threshold). A Markov switching model is one in which the parameters follow a finite Markov chain.