Bayesian moment-based inference in a regression model with misclassification error

We present a Bayesian analysis of a regression model with a binary covariate that may have classification (measurement) error. Prior research demonstrates that the regression coefficient is only partially identified. We take a Bayesian approach which adds assumptions in the form of priors on the unknown misclassification probabilities. The approach is intermediate between the frequentist bounds of previous literature and strong assumptions which achieve point identification, and thus preferable in many settings. We present two simple algorithms to sample from the posterior distribution when the likelihood function is not fully parametric but only satisfies a set of moment restrictions. We focus on how varying amounts of information contained in a prior distribution on the misclassification probabilities change the posterior of the parameters of interest. While the priors add information to the model, they do not necessarily tighten the identified set. However, the information is sufficient to tighten Bayesian inferences. We also consider the case where the mismeasured binary regressor is endogenous. We illustrate the use of our Bayesian approach in a simulated data set and an empirical application investigating the association between narcotic pain reliever use and earnings.