In this paper we examine the problem of testing for heterogeneity and heterosckedasticity in a Bayesian framework. We first show that a model with random coefficients is identical to a model with heteroskedastic residuals. We then consider two approaches for testing. The first one is concerned with the point of view of misspecification. The original model is homoskedastic. One is willing to detect any departure from homoskedasticity. We propose diagnostics based on the examination of the Bayesian residuals, after stressing the differences between classical and Bayesian residuals. In the second approach, the starting point is a precise form of the alternative hypothesis, and the model for inference is heteroskedastic. A test for homoskedasticity is then a test for a parameter restriction. This can be done by looking at highest posterior probability regions or by the use of the posterior odds ratio. As a joint product, we develop the posterior analysis of a heteroskedastic regression model for several classes of prior distributions.
