Bayesian Estimation and Inference with Generated Regressors

Department of Decision Sciences and Managerial Economics

We consider a two-stage hybrid estimation method that combines a certain frequentist estimation in the first stage and the Bayesian approach in the second stage. The study is motivated from structural discrete choice models with generated regressors, in which the second-stage estimation is conducted using state-of-the-art Bayesian algorithms. We present theoretical properties of the resulting posterior distributions in a general framework that allows for nonparametric components in both stages. Considering the marginal posterior for the finite-dimensional parameter, we prove that the corresponding Bayesian credible set does not have the right coverage. Nonetheless, the Bayesian point estimator, such as the posterior mean or median, is asymptotically equivalent to a frequentist two-stage estimator. We show that proper bootstrap or subsampling methods that account for the first-stage estimation uncertainty are valid for inference.