🤖 AI Summary
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.
📝 Abstract
Omitted variables are one of the most important threats to the identification of causal effects. Several widely used approaches, including Oster (2019), assess the impact of omitted variables on empirical conclusions by comparing measures of selection on observables with measures of selection on unobservables. These approaches either (1) assume the omitted variables are uncorrelated with the included controls, an assumption that is often considered strong and implausible, or (2) use a method called residualization to avoid this assumption. In our first contribution, we develop a framework for objectively comparing sensitivity parameters. We use this framework to formally prove that the residualization method generally leads to incorrect conclusions about robustness. In our second contribution, we then provide a new approach to sensitivity analysis that avoids this critique, allows the omitted variables to be correlated with the included controls, and lets researchers 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.
