An optimization -based matching estimator : large and small sample properties

Abstract

This work proposes a novel matching estimator where weights and the choice of neighbors used are endogenously determined by solving an optimization problem. The estimator is non-parametric and is based on nding, for each unit that needs to be matched, sets of observations such that a convex combination of their covariates has the same value of the covariates as the unit to be matched, or with minimized distance between them. Since there is generally more than one set per each unit, the method chooses the one with the closest covariate values. In this work we contribute to the matching literature by linking the choice of matches and weights to the improvement of post-matching covariate balance in a simple way: an optimization problem that optimizes individual covariate balance. It is worth mentioning that the developed method is not an algorithm that iteratively checks covariate balance until convergence. Instead, it incorporates an individual balance criterion in the objective function that determines the weights used in each match. It can be written as a linear program that allows us to use standard optimization techniques to solve the problem quickly. To aid research, we provide a new R library called blopmatching. Regarding asymptotic properties, we shows that our estimator of the ATE attains stan- dard limit properties (consistency and normality), and it has a conditional bias that is Op(Nð€€€2=k). It worth mentioning that this order improves the order N1=k attained by the NN-matching estimator. In fact, Op(Nð€€€2=k) could be attained by the NN-matching estima- tor in the only case in which the conditional expectation of the outcome variable is a linear expression in covariates, a condition under which the conditional bias of our estimators is as good as we want. Besides, even though the proposed estimator of the ATE is not p N- consistent in general, we show that if the number of control units increases faster than the number of treated units, then our matching estimator of the ATT attains the p N-consistency, as its bias rate is better than the one attained by the NN-matching estimator. Finally, as regards nite sample properties, we implement the proposed estimator to data from the National Supported Work Demonstration nding an outstanding performance even though when using alternative control groups from a non-experimental sample. In addition, by performing Monte Carlo experiments with designs based on the related literature that includes misspeci cation of the selection equation, we study its performance in nite samples. We nd that our estimator provides good post-matching balance and performs well in terms of bias and variance when compared to nearest neighbor matching estimators (for both covariates and propensity score) and the normalized inverse probability weighting estimator. Major improvements are observed when there is underspeci cation of the selection equation for estimating the propensity score. Hence, our method gives researchers a new alternative matching estimator that prevents the selection of an arbitrary number of neighbors or the estimation of the propensity score.