  Imbalance in covariate distributions leads to biased estimates of causal
effects. Weighting methods attempt to correct this imbalance but rely on
specifying models for the treatment assignment mechanism, which is unknown in
observational studies. This leaves researchers to choose the proper weighting
method and the appropriate covariate functions for these models without knowing
the correct combination to achieve distributional balance. In response to these
difficulties, we propose a nonparametric generalization of several other
weighting schemes found in the literature: Causal Optimal Transport. This new
method directly targets distributional balance by minimizing optimal transport
distances between treatment and control groups or, more generally, between any
source and target population. Our approach is semiparametrically efficient and
model-free but can also incorporate moments or any other important functions of
covariates that a researcher desires to balance. Moreover, our method can
provide nonparametric estimate the conditional mean outcome function and we
give rates for the convergence of this estimator. Moreover, we show how this
method can provide nonparametric imputations of the missing potential outcomes
and give rates of convergence for this estimator. We find that Causal Optimal
Transport outperforms competitor methods when both the propensity score and
outcome models are misspecified, indicating it is a robust alternative to
common weighting methods. Finally, we demonstrate the utility of our method in
an external control trial examining the effect of misoprostol versus oxytocin
for the treatment of post-partum hemorrhage.
