Efficient Difference-in-Differences Estimation with Panel Data
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The difference-in-differences (DiD) is a fundamental econometric technique designed to estimate causal effects by comparing the changes in outcomes over time between the treated and control groups. In the context of staggered DiD with multiple treatment groups and periods, conventional estimation based on the two-way fixed effects model suffers from negative weights to average up heterogeneous group-period treatment effects. In this work, we define the overall average treatment effect on the treated (ATT) nonparametrically as a weighted average of heterogeneous group-period treatment effects. We derive the efficient influence function (EIF) for the ATT and propose two estimators for the ATT. The first is the EIF estimating-equation-based estimator, and the second is the targeted minimum-loss-based estimator (TMLE). These two proposed estimators are asymptotically equivalent, enjoying semiparametric efficiency and double robustness. Even if part of the working models is mis-specified, the proposed estimators still consistently estimate the ATT. We apply the methods to study the effect of the admission mechanism of Gaokao: parallel admission versus ordered admission. We find that parallel admission can significantly reduce the justified envy of students.