Causal effects based on distributional distances

TL;DR

This paper introduces methods for measuring causal effects using distributional distances, providing estimators, confidence bands, and hypothesis tests for counterfactual distributions.

Contribution

It proposes novel estimators for counterfactual outcome densities and distributional causal effects, with theoretical analysis and practical bootstrap-based inference methods.

Findings

Proposed a doubly robust estimator for counterfactual density.

Developed a bootstrap confidence band for distributional effects.

Analyzed asymptotic properties and error bounds of estimators.

Abstract

Comparing counterfactual distributions can provide more nuanced and valuable measures for causal effects, going beyond typical summary statistics such as averages. In this work, we consider characterizing causal effects via distributional distances, focusing on two kinds of target parameters. The first is the counterfactual outcome density. We propose a doubly robust-style estimator for the counterfactual density and study its rates of convergence and limiting distributions. We analyze asymptotic upper bounds on the $L_q$ and the integrated $L_q$ risks of the proposed estimator, and propose a bootstrap-based confidence band. The second is a novel distributional causal effect defined by the $L_1$ distance between different counterfactual distributions. We study three approaches for estimating the proposed distributional effect: smoothing the counterfactual density, smoothing the $L_1$ distance, and imposing a margin condition. For each approach, we analyze asymptotic properties and error bounds of the proposed estimator, and discuss potential advantages and disadvantages. We go on to present a bootstrap approach for obtaining confidence intervals, and propose a test of no distributional effect. We conclude with a numerical illustration and a real-world example.