Efficient Estimation of Average Treatment Effects under Treatment-Based Sampling

Nonrandom sampling schemes are often used in program evaluation settings to improve the quality of inference. This paper considers what we call treatment-based sampling, a type of standard stratified sampling where part of the strata are based on treatments. This paper first establishes semiparametric efficiency bounds for estimators of weighted average treatment effects and average treatment effects on the treated. In doing so, this paper illuminates the role of information about the aggregate shares from the original data set. This paper also develops an optimal design of treatment-based sampling that yields the best semiparametric efficiency bound. Lastly, this paper finds that adapting the efficient estimators of Hirano, Imbens, and Ridder (2003) to treatment-based sampling does not always lead to an efficient estimator. This paper proposes different estimators that are efficient in such a situation.