Large sample properties of an optimization-based matching

Abstract

This paper mainly concerns the the asymptotic properties of the BLOP matching estimator introduced by D´ıaz, Rau & Rivera (Forthcoming), showing that this estimator of the ATE attains the standard limit properties, and that its conditional bias is Op(N !2/k), with k the dimension of continuous covariates. Even though this estimator is not p N-consistent in general, when the order of magnitude of the numbers of control units is bigger than the one of treated units, we show that the BLOP matching estimator of ATT is p N-consistent. Finally, for a general nonparametric setting, the conditional bias of matching estimators that use a constant number of matches to perform the potential outcomes cannot attain the aforementioned stochastic orders, regardless of the weighting schemes used to perform the potential outcomes. The proof of these results uses novel contributions in the field of geometric probability theory we provide in this work. Our results improve the obtained by Abadie & Imbens (2006) when studying the limit properties of the well known NN-matching estimator.