Difference-in-Differences Estimators of Intertemporal Treatment Effects

We consider the estimation of the effect of a treatment, using panel data where groups of units are exposed to different doses of the treatment at different times. We consider two sets of parameters of interest. The first are the average effects of having changed treatment for the first time $\ell$ periods ago. Those parameters generalize the average effect of having started receiving the treatment $\ell$ periods ago that has often been estimated in applications with a binary treatment and staggered adoption. We also consider cost-benefit ratios a planner may use to compare the treatments actually assigned to a counterfactual status quo scenario where groups receive their period-one treatment throughout the panel. We show that under common trends conditions, all these parameters are unbiasedly estimated by weighted sums of differences-in-differences. Our estimators are valid if the treatment effect is heterogeneous, contrary to the commonly-used dynamic two-way fixed effects regressions. We also propose estimators of a dynamic linear model, with group-specific but time-invariant effects of the current and lagged treatments, which may be used to evaluate ex-ante the effect of future policies. In an application, we find that our estimators differ substantially from those obtained with dynamic two-way fixed effects regressions