Causal effects based on distributional distances

In this paper we develop a framework for characterizing causal effects via distributional distances. In particular we define a causal effect in terms of the $L_1$ distance between different counterfactual outcome distributions, rather than the typical mean difference in outcome values. Comparing entire counterfactual outcome distributions can provide more nuanced and valuable measures for exploring causal effects beyond the average treatment effect. First, we propose a novel way to estimate counterfactual outcome densities, which is of independent interest. Then we develop an efficient estimator of our target causal effect. We go on to provide error bounds and asymptotic properties of the proposed estimator, along with bootstrap-based confidence intervals. Finally, we illustrate the methods via simulations and real data.