  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.
