Difference-in-Differences with Variation in Treatment Timing

The canonical difference-in-differences (DD) model contains two time periods, “pre” and “post”, and two groups, “treatment” and “control”. Most DD applications, however, exploit at least some variation across units that receive treatment at different times. This paper derives an expression for this general DD estimator, and shows that it is a weighted average of all possible “two-group” DD estimators in the data. This refines the definition of and test for common trends, shows that DD with timing does not estimate a sample-weighted average treatment effect on the treated, and establishes that DD estimates based on timing variation are inconsistent when treatment effects vary over time. I also examine inverse propensity score reweighting, triple-difference models, and unit-specific linear time trends.