Assessing Omitted Variable Bias when the Controls are Endogenous

Omitted variables are one of the most important threats to the identification of causal effects. Several widely used methods, including Oster (2019), have been developed to assess the impact of omitted variables on empirical conclusions. These methods all require an exogenous controls assumption: the omitted variables must be uncorrelated with the included controls. This is often considered a strong and implausible assumption. We provide a new approach to sensitivity analysis that allows for endogenous controls, while still letting researchers calibrate sensitivity parameters by comparing the magnitude of selection on observables with the magnitude of selection on unobservables. 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.