Covariate adjustment in experiments with matched pairs

Abstract

This paper studies inference for the average treatment effect (ATE) in experiments in which treatment status is determined according to “matched pairs” and it is additionally desired to adjust for observed, baseline covariates to gain further precision. By a “matched pairs” design, we mean that units are sampled i.i.d. from the population of interest, paired according to observed, baseline covariates, and finally, within each pair, one unit is selected at random for treatment. Importantly, we presume that not all observed, baseline covariates are used in determining treatment assignment. We study a broad class of estimators based on a “doubly robust” moment condition that permits us to study estimators with both finite-dimensional and high-dimensional forms of covariate adjustment. We find that estimators with finite-dimensional, linear adjustments need not lead to improvements in precision relative to the unadjusted difference-in-means estimator. This phenomenon persists even if the adjustments interact with treatment; in fact, doing so leads to no changes in precision. However, gains in precision can be ensured by including fixed effects for each of the pairs. Indeed, we show that this adjustment leads to the minimum asymptotic variance of the corresponding ATE estimator among all finite-dimensional, linear adjustments. We additionally study an estimator with a regularized adjustment, which can accommodate high-dimensional covariates. We show that this estimator leads to improvements in precision relative to the unadjusted difference-in-means estimator and also provides conditions under which it leads to the “optimal” nonparametric, covariate adjustment. A simulation study confirms the practical relevance of our theoretical analysis, and the methods are employed to reanalyze data from an experiment using a “matched pairs” design to study the effect of macroinsurance on microenterprise.