Minimal dispersion approximately balancing weights: asymptotic properties and practical considerations

Weighting methods are widely used to adjust for covariates in observational studies, sample surveys, and regression settings. In this paper, we study a class of recently proposed weighting methods, which find the weights of minimum dispersion that approximately balance the covariates. We call these weights 'minimal weights' and study them under a common optimization framework. Our key observation is that finding weights which achieve approximate covariate balance is equivalent to performing shrinkage estimation of the inverse propensity score. This connection leads to both theoretical and practical developments. From a theoretical standpoint, we characterize the asymptotic properties of minimal weights and show that, under standard smoothness conditions on the propensity score function, minimal weights are consistent estimates of the true inverse probability weights. In addition, we show that the resulting weighting estimator is consistent, asymptotically normal and semiparametrically efficient. From a practical standpoint, we give a finite-sample oracle inequality that bounds the loss incurred by balancing more functions of the covariates than strictly needed. This inequality shows that minimal weights implicitly bound the number of active covariate balance constraints. Finally, we provide a tuning algorithm for choosing the degree of approximate balance in minimal weights. The paper concludes with an empirical study which suggests that approximate balance is preferable to exact balance, especially when there is limited overlap in covariate distributions. Further studies show that the root mean squared error of the weighting estimator can be reduced by as much as a half with approximate balance.