  In the absence of unobserved confounders, matching and weighting methods are
widely used to estimate causal quantities including the Average Treatment
Effect on the Treated (ATT). Unfortunately, these methods do not necessarily
achieve their goal of making the multivariate distribution of covariates for
the control group identical to that of the treated, leaving some (potentially
multivariate) functions of the covariates with different means between the two
groups. When these "imbalanced" functions influence the non-treatment potential
outcome, the conditioning on observed covariates fails, and ATT estimates may
be biased. Kernel balancing, introduced here, targets a weaker requirement for
unbiased ATT estimation, specifically, that the expected non-treatment
potential outcome for the treatment and control groups are equal. The
conditional expectation of the non-treatment potential outcome is assumed to
fall in the space of functions associated with a choice of kernel, implying a
set of basis functions in which this regression surface is linear. Weights are
then chosen on the control units such that the treated and control group have
equal means on these basis functions. As a result, the expectation of the
non-treatment potential outcome must also be equal for the treated and control
groups after weighting, allowing unbiased ATT estimation by subsequent
difference in means or an outcome model using these weights. Moreover, the
weights produced are (1) precisely those that equalize a particular
kernel-based approximation of the multivariate distribution of covariates for
the treated and control, and (2) equivalent to a form of stabilized inverse
propensity score weighting, though it does not require assuming any model of
the treatment assignment mechanism. An R package, KBAL, is provided to
implement this approach.
