In this paper, we consider estimation of average treatment effect on the treated (ATT), an interpretable and relevant causal estimand to policy makers when treatment assignment is endogenous. By considering shadow variables that are unrelated to the treatment assignment but related to the outcomes of interest, we establish identification of the ATT. Then we focus on efficient estimation of the ATT by characterizing the geometric structure of the likelihood, deriving the semiparametric efficiency bound for ATT estimation and proposing an estimator that can achieve this bound. We rigorously establish the theoretical results of the proposed estimator. The finite sample performance of the proposed estimator is studied through comprehensive simulation studies as well as an application to our motivating study.

翻译：本文考虑处理组平均处理效应（ATT）的估计问题——当处理分配为内生时，该指标对政策制定者而言具有可解释性和相关性。通过引入与处理分配无关但与目标结果相关的影子变量，我们建立了ATT的可识别性。随后通过刻画似然函数的几何结构，推导出ATT估计的半参数效率界，并提出能实现该界的估计量，从而聚焦于ATT的高效估计。我们严格论证了所提估计量的理论性质，并通过全面的模拟研究及对实证研究的应用验证了其有限样本性能。