Efficient Estimation of Average Treatment Effects Under Treatment-Based Sampling, Second Version

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 treatment status. This paper establishes semiparametric efficiency bounds for estimators of weighted average treatment effects and average treatment effects on the treated. 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 efficient estimators that involve a different form of propensity score-weighting. Finally, this paper establishes an optimal design of treatment-based sampling that minimizes the semiparametric efficiency bound over the sampling designs.