Unobservable selection and coefficient stability: Theory and validation

A common heuristic for evaluating robustness of results to omitted variable bias is to observe coefficient movements after inclusion of controls. This heuristic is informative only if selection on observables is informative about selection on unobservables. I formalize this link through a proportional selection assumption. I show that it is necessary to take into account coefficient movements and movements in R-squared values in identifying omitted variable bias. I further demonstrate that in the empirically common case with multiple observed controls it is also necessary to account for the share of the variation in treatment accounted for by control variables. I describe a formal bounding argument for omitted variable bias under the proportional selection assumption. I show two validation exercises suggesting that this performs well empirically. I discuss application of this procedure to a large set of publications in economics, and use evidence from randomized studies to draw guidelines for bounding values