Efficient semiparametric estimation of multi-valued treatment effects under ignorability

This paper studies the efficient estimation of a large class of multi-valued treatment effects as implicitly defined by a collection of possibly over-identified non-smooth moment conditions when the treatment assignment is assumed to be ignorable. Two estimators are introduced together with a set of sufficient conditions that ensure their n-consistency, asymptotic normality and efficiency. Under mild assumptions, these conditions are satisfied for the Marginal Mean Treatment Effect and the Marginal Quantile Treatment Effect, estimands of particular importance for empirical applications. Previous results for average and quantile treatments effects are encompassed by the methods proposed here when the treatment is dichotomous. The results are illustrated by an empirical application studying the effect of maternal smoking intensity during pregnancy on birth weight, and a Monte Carlo experiment.