Selection and Parallel Trends

One of the perceived advantages of difference-in-differences (DiD) methods is that they do not explicitly restrict how units select into treatment. However, when justifying DiD, researchers often argue that the treatment is "quasi-randomly" assigned. We investigate what selection mechanisms are compatible with the parallel trends assumptions underlying DiD. We derive necessary and sufficient conditions for parallel trends that clarify whether and how selection can depend on time-invariant and time-varying unobservables. We also suggest a menu of interpretable primitive sufficient conditions for parallel trends, thereby providing the formal underpinnings for justifying DiD based on contextual information about selection into treatment. We provide results for both separable and nonseparable outcome models and show that this distinction has implications for the use of covariates in DiD analyses. Building on our analysis of nonseparable models, we connect DiD to the literature on nonparametric identification in panel models.