First draft: September 2007, This version: August 2008.
previously circulated under the title, Semiparametric Bayesian Regression: Large Sample Theory.
We consider a semiparametric Bayesian regression model with iid errors. We do not impose a parametric form for the likelihood function; rather, we treat the true density function of error terms as an infinite dimensional nuisance parameter and estimate it via quantile regression. Once the likelihood of parameters of interest is constructed based on the estimated error density, one can conduct a conventional parametric Bayesian inference using MCMC methods. We derive the asymptotic properties of the resulting estimator. In particular, we identify conditions under which our two-step Bayes estimator has the same asymptotic normal distribution that is enjoyed by the Bayes estimator that could be obtained if we knew the true density. Hence, we establish that in certain Bayesian models it is possible to obtain parallel results to the adaptive estimation literature, that is, asymptotically there is no efficiency loss due to not knowing the functional form of underlying distribution.
The paper is available upon request.
: presented at the 2007 Midwest Econometrics Group Meetings (Saint Louis, IL).