Testing for Endogeneity: A Moment-Based Bayesian Approach

A standard assumption in the Bayesian estimation of linear regression models is that the regressors are exogenous in the sense that they are uncorrelated with the model error term. In practice, however, this assumption can be invalid. In this paper, using the exponentially tilted empirical likelihood framework, we develop a Bayes factor test for endogeneity that compares a base model that is correctly specified under exogeneity but misspecified under endogeneity against an extended model that is correctly specified in either case. We provide a comprehensive study of the log-marginal exponentially tilted empirical likelihood. We demonstrate that our testing procedure is consistent from a frequentist point of view: as the sample grows, it almost surely selects the base model if and only if the regressors are exogenous, and the extended model if and only if the regressors are endogenous. The methods are illustrated with simulated data, and problems concerning the causal effect of automobile prices on automobile demand and the causal effect of potentially endogenous airplane ticket prices on passenger volume.