Unobservable Selection and Coefficient Stability: Theory and Evidence

TLDR

Robustness to omitted variable bias is often assessed by observing coefficient changes when adding controls, but this is meaningful only if observable selection reflects unobservable selection, a link known theoretically yet rarely examined empirically. The study extends theory to link omitted‑variable bias explicitly to coefficient stability. The authors demonstrate that accounting for coefficient and R‑squared changes is required and provide a formal bounding argument. Two validation exercises confirm the approach, and its relevance to economics is discussed. Supplementary materials are available online.

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.