Weighting methods are a common tool to de-bias estimates of causal effects. And though there are an increasing number of seemingly disparate methods, many of them can be folded into one unifying regime: causal optimal transport. This new method directly targets distributional balance by minimizing optimal transport distances between treatment and control groups or, more generally, between a source and target population. Our approach is model-free but can also incorporate moments or any other important functions of covariates that the researcher desires to balance. We find that the 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 study examining the effect of misoprostol versus oxytocin for treatment of post-partum hemorrhage.
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