Assessing Omitted Variable Bias when the Controls are Endogenous
This paper addresses sensitivity analysis for omitted-variable bias in causal inference, focusing on the critical yet overlooked scenario where omitted variables are endogenous with respect to included controls—a setting neglected by existing methods. Conventional residualization-based approaches suffer from theoretical deficiencies under endogeneity, leading to erroneous robustness assessments; meanwhile, prevailing sensitivity analyses either rely on strong independence assumptions or lack comparable calibration. We formally prove the failure mechanism of residualization and propose a novel sensitivity analysis framework that explicitly accommodates correlation between omitted and observed covariates. Our approach introduces a standardized sensitivity parameter enabling comparable calibration of observable and unobservable selection strength. Theoretical derivation, implementation via a Stata module (regsensitivity), and empirical validation—using historical frontier settlement to instrument cultural beliefs—demonstrate that the framework rectifies fundamental theoretical shortcomings of mainstream methods and delivers a ready-to-use tool for robust causal inference.