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.

翻译：本文研究一种两阶段准贝叶斯方法：第一阶段采用频率学派估计，第二阶段采用贝叶斯估计/推断方法。该研究源于使用控制函数方法纠正内生性偏差的结构性离散选择模型。在此框架下，第一阶段通过频率学派参数或非参数方法估计控制函数；第二阶段的结构方程涉及某些复杂似然函数，更适合采用贝叶斯方法处理。本文研究第二阶段获得的准后验分布的渐近性质。我们证明，相应的准贝叶斯可信集在大样本下不具备理想覆盖概率。尽管如此，准贝叶斯点估计量仍具相合性，且渐近等价于频率学派两阶段估计量。研究表明，通过考虑第一阶段估计不确定性的准后验自助法，可以获得有效的统计推断。