Difference-in-Differences with Sample Selection

Abstract

We consider the identification of average treatment effects on the treated (ATT) in difference-in-differences (DiD) settings in the presence of endogenous sample selection. We first establish that the conventional DiD estimand generally fails to recover causally meaningful treatment effects, even if selection and treatment assignment are independent. We then partially identify the ATT for individuals whose outcomes would be observed post-treatment under either counterfactual treatment state, and derive sharp bounds on this parameter under different sets of assumptions on the relationship between sample selection and treatment assignment. These identification results are extended to allow for covariates, repeated cross-section data, and two-by-two comparisons in staggered adoption designs. Furthermore, we present identification results for the ATT of three additional empirically relevant latent groups by imposing outcome mean dominance assumptions that have intuitive appeal in applications. Finally, two empirical illustrations demonstrate the approach's usefulness by revisiting (i) the effect of a job training program on earnings and (ii) the effect of a working-from-home policy on employee performance.