Efficient Estimation for Staggered Rollout Designs

This paper studies efficient estimation of causal effects when treatment is (quasi-) randomly rolled out to units at different points in time. We solve for the most efficient estimator in a class of estimators that nests two-way fixed effects models and other popular generalized difference-in-differences methods. A feasible plug-in version of the efficient estimator is asymptotically unbiased with efficiency (weakly) dominating that of existing approaches. We provide both $t$-based and permutation-test based methods for inference. We illustrate the performance of the plug-in efficient estimator in simulations and in an application to \citet{wood_procedural_2020}'s study of the staggered rollout of a procedural justice training program for police officers. We find that confidence intervals based on the plug-in efficient estimator have good coverage and can be as much as eight times shorter than confidence intervals based on existing state-of-the-art methods. As an empirical contribution of independent interest, our application provides the most precise estimates to date on the effectiveness of procedural justice training programs for police officers.