  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.
