Priors and Posterior Computation in Linear Endogenous Variable Models with Imperfect Instruments

Estimation in models with endogeneity concerns typically begins by searching for instruments. This search is inherently subjective and identification is generally achieved upon imposing the researcher's strong prior belief that such variables have no conditional impacts on the outcome. Results obtained from such analyses are necessarily conditioned upon the untestable opinions of the researcher, and such beliefs may not be widely shared. In this paper we, like several studies in the recent literature, employ a Bayesian approach to estimation and inference in models with endogeneity concerns by imposing weaker prior assumptions than complete excludability. When allowing for instrument imperfection of this type, the model is only partially identified, and as a consequence, standard estimates obtained from the Gibbs simulations can be unacceptably imprecise. We thus describe a substantially improved \semi-analytic" method for calculating parameter marginal posteriors of interest that only requires use of the well-mixing simulations associated with the identifiable model parameters and the form of the conditional prior. Our methods are also applied in an illustrative application involving the impact of Body Mass Index (BMI) on earnings.