Assessing Omitted Variable Bias when the Controls are Endogenous

Abstract: Omitted variables are one of the most important threats to the identification of causal effects. Several widely used methods assess the impact of omitted variables on empirical conclusions by comparing measures of selection on observables with measures of selection on unobservables. The recent literature has discussed various limitations of these existing methods, however. This includes a companion paper of ours which explains issues that arise when the omitted variables are endogenous, meaning that they are correlated with the included controls. In the present paper, we develop a new approach to sensitivity analysis that avoids those limitations, while still allowing researchers to calibrate sensitivity parameters by comparing the magnitude of selection on observables with the magnitude of selection on unobservables as in previous methods. We illustrate our results in an empirical study of the effect of historical American frontier life on modern cultural beliefs. Finally, we implement these methods in the companion Stata module regsensitivity for easy use in practice.