Fully Bayesian endogenous variable estimation with many instrumental variables

Abstract

Endogeneity remains a thorny problem in applications of regression analysis, where the correlation with random error may introduce large bias into the parameter estimates. Instrumental variables are an elegant solution to the problem, but only when the instrumental variables affect the response solely through the endogenous variable. If any subset of the instrumental variables are included directly in the response model, bias is reintroduced through these invalid instruments. In practice, it is impossible to know for certain which instruments may be invalid, but much work has been done to estimate the validity of candidate instruments through penalized regression methods. We introduce a Bayesian alternative to these methods with oracle properties which also accounts for model uncertainty through Bayesian model averaging. Our model contains a sparsity assumption on the invalid instruments, specifically that the invalid instruments are less than half of the total set of candidate instruments, and we introduce an alternative construction of the Gibbs sampler to account for the sparsity constraint. Our estimator demonstrates MSE comparable to oracle estimators in several simulation studies. We also apply the estimator to a real-world dataset on global trade to identify invalid instruments.