Efficient Estimation for Staggered Rollout Designs

Researchers are frequently interested in the causal effect of a treatment that is (quasi-)randomly rolled out to different units at different points in time. This paper studies how to efficiently estimate a variety of causal parameters in a Neymanian-randomization based framework of random treatment timing. We solve for the most efficient estimator in a class of estimators that nests two-way fixed effects models as well as several popular generalized difference-in-differences methods. The efficient estimator is not feasible in practice because it requires knowledge of the optimal weights to be placed on pre-treatment outcomes. However, the optimal weights can be estimated from the data, and in large datasets the plug-in estimator that uses the estimated weights has similar properties to the "oracle" efficient estimator. We illustrate the performance of the plug-in efficient estimator in simulations and in an application to Wood et al. (2020a,b)'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 five times shorter than confidence intervals based on existing 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.