Keywords: Causal Inference, Observational Study, Program Evaluation, Unit Nonresponse, Weighting
In observational studies, weight adjustments are often performed by fitting a model for the propensity score and then inverting the predicted propensities, but recently several approaches have been proposed that instead focus on directly balancing the covariates. In this paper, we study the general class of approximate balancing weights (ABW) by drawing a connection to shrinkage estimation. This allows us to establish the large sample properties of ABW by leveraging asymptotic results from propensity score estimation. In particular, we show that under mild technical conditions ABW are consistent estimates of the true inverse probability weights. To the best of our knowledge, this functional consistency of balancing weights has not been established in the literature even for exact balancing. We also show that the resulting weighting estimator is consistent, asymptotically normal, and semiparametrically efficient. In three empirical studies we show that the root mean squared error of the weighting estimator can be reduced by a half by using approximate balancing instead of exact balancing when the data exhibits poor overlap in covariate distributions.