  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.
