Quasi-Bayesian Estimation and Inference with Control Functions

We consider a quasi-Bayesian method that combines a frequentist estimation
in the first stage and a Bayesian estimation/inference approach in the second stage.
The study is motivated by structural discrete choice models that use the control function
methodology to correct for endogeneity bias. In this scenario, the first stage estimates
the control function using some frequentist parametric or nonparametric approach. The
structural equation in the second stage, associated with certain complicated likelihood
functions, can be more conveniently dealt with using a Bayesian approach. This paper
studies the asymptotic properties of the quasi-posterior distributions obtained from the
second stage. We prove that the corresponding quasi-Bayesian credible set does not have
the desired coverage in large samples. Nonetheless, the quasi-Bayesian point estimator
remains consistent and is asymptotically equivalent to a frequentist two-stage estimator.
We show that one can obtain valid inference by bootstrapping the quasi-posterior that
takes into account the first-stage estimation uncertainty.