Efficient Propensity Score Regression Estimators of Multivalued Treatment Effects for the Treated

Abstract

Matching is a widely used program evaluation estimation method when treatment is assigned at random conditional on observable characteristics. When a multivalued treatment takes on more than two values, valid causal comparisons for a subpopulation who is treated a particular treatment level are based on two propensity scores - one for the treated level and one for the counterfactual level. The main contribution of this paper is propensity score regression estimators for a class of treatment effects for the treated that achieve the semiparametric efficiency bounds under the cases when the propensity scores are unknown and when they are known. We derive the large sample distribution that accounts for the estimation error of the propensity score as generated regressors. We contribute to the binary treatment literature by a new propensity score regression estimator for the average/quantile treatment effect for the treated: our efficient estimator matches on a normalized propensity score that is a combination of the true propensity score and its nonparametric estimate. There are two key findings: (I) The efficiency bound is reduced by knowledge of the propensity scores for the treated levels, but is not affected by knowledge of the propensity score for the counterfactual level. (II) Matching on the nonparametrically estimated propensity score recovers the information contained in matching on the pretreatment variables.

Keywords: Propensity Score, Multivalued Treatment, Semiparametric Efficiency Bound, Unconfoundedness, Generated Regressor

JEL Classification: C14, C21