Title: Minimal Approximately Balancing Weights: Asymptotic Properties and Practical Considerations,
Abstract: In observational studies and sample surveys, and regression settings,
weighting methods are widely used to adjust for or balance observed covariates.
Recently, a few weighting methods have been proposed that focus on directly
balancing the covariates while minimizing the dispersion of the weights. In
this paper, we call this class of weights minimal approximately balancing
weights (MABW); we study their asymptotic properties and address two
practicalities. We show that, under standard technical conditions, MABW are
consistent estimates of the true inverse probability weights; the resulting
weighting estimator is consistent, asymptotically normal, and
semiparametrically efficient. For applications, we present a finite sample
oracle inequality showing that the loss incurred by balancing too many
functions of the covariates is limited in MABW. We also provide an algorithm
for choosing the degree of approximate balancing in MABW. Finally, we conclude
with numerical results that suggest approximate balancing is preferable to
exact balancing, especially when there is limited overlap in covariate
distributions: the root mean squared error of the weighting estimator can be
reduced by nearly a half.