Bayesian regression with heteroscedastic error density and parametric mean function

In this paper we consider Bayesian estimation of restricted conditional moment models with the linear regression as a particular example. A common practice in the Bayesian literature for linear regression and other semi-parametric models is to use flexible families of distributions for the errors and to assume that the errors are independent from covariates. However, a model with flexible covariate dependent error distributions should be preferred for the following reason. Assuming that the error distribution is independent of predictors might lead to inconsistent estimation of the parameters of interest when errors and covariates are dependent. To address this issue, we develop a Bayesian regression model with a parametric mean function defined by a conditional moment condition and flexible predictor dependent error densities. Sufficient conditions to achieve posterior consistency of the regression parameters and conditional error densities are provided. In experiments, the proposed method compares favorably with classical and alternative Bayesian estimation methods for the estimation of the regression coefficients and conditional densities.