Selection and parallel trends

Abstract

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 conditions for parallel trends that clarify whether and how selection can depend on time-invariant and time-varying unobservables. Motivated by these necessary conditions, we suggest a menu of interpretable 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.

Not All Differences-in-Differences Are Equally Compatible With Outcome-Based Selection Models

“…Note that when T = 2, the selection equation in Point 2 of Assumption 6 is similar to a selection equation previously considered by Ashenfelter and Card (1985) and Ghanem, Sant'Anna and Wüthrich (2022). Under that selection equation and a model similar to that in Point 1 of Assumption 6, Ghanem, Sant'Anna and Wüthrich (2022) show that the standard parallel trends assumption with two periods holds.…”

Identifying and Estimating Effects of Sustained Interventions under Parallel Trends Assumptions

“…Second, there is a notable similarity between the proposed parallel trends assumption and the discrete-time independent censoring assumption from linear increments methods for missing data (Diggle et al, 2007); this connection may be informative for suggesting new estimators, particularly in the case of nonmonotonic treatments. Finally, though parallel trends may be considered more plausible than sequential exchangeability in some settings, strategies for evaluating the plausibility of the assumption (including any implications regarding effect heterogeneity) using domain knowledge are needed (e.g., Ghanem et al, 2022). While the parallel trends assumption in this paper avoid restricting effect heterogeneity by considering only one regime, structural models that allow parallel trends for one regime but not others may be difficult to justify in practice (Shahn et al, 2022).…”

Double-Robust Identification for Causal Panel Data Models

“…See [RSBP23, RR23, Jak21] for a discussion and bounds based on limits on the deviations from parallel trends. Bridging some of the gap between design and sampling based approaches [RS23,GSW22] show how parallel trends can be implied by random assignment of treatment. They also discuss the sensitivity to transformations of the parallel trend assumption.…”