Unobservable Selection and Coefficient Stability: Theory and Evidence

Abstract

A common approach to evaluating robustness to omitted variable bias is to observe coefficient movements after inclusion of controls. This is informative only if selection on observables is informative about selection on unobservables. Although this link is known in theory in existing literature, very few empirical articles approach this formally. I develop an extension of the theory that connects bias explicitly to coefficient stability. I show that it is necessary to take into account coefficient and R-squared movements. I develop a formal bounding argument. I show two validation exercises and discuss application to the economics literature. Supplementary materials for this article are available online.

KEY WORDS:

• Coefficient stability
• Selection
• Omitted variable bias

ACKNOWLEDGMENTS

Ling Zhong, Unika Shrestha, Damian Kozbur, Guillaume Pouliot, David Birke and Angela Li provided excellent research assistance. The author thanks David Cesarini, Raj Chetty, Todd Elder, Amy Finkelstein, Guido Imbens, Larry Katz, Jonah Gelbach, Matt Gentzkow, Matt Notowidigdo, Chad Syverson, Manisha Shah, Azeem Shaikh, Jesse Shapiro, Bryce Steinberg, Matt Taddy, Heidi Williams, and participants in seminars at Brown University, University of Chicago Booth School of Business, Wharton and Yale for helpful comments. The author is grateful to a number of authors for providing replication files or rerunning analysis by request. The author gratefully acknowledges financial support from the Neubauer Family. Stata code to perform the calculations described in this article is available from the author's website or through ssc under the name psacalc.