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Causal effects based on distributional distances

Kwangho Kim, Jisu Kim, Edward H. Kennedy

TL;DR

A novel framework for estimating causal effects based on the discrepancy between unobserved counterfactual distributions is developed, and methods to construct confidence intervals for the unknown mean distribution distance are proposed.

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.

Causal effects based on distributional distances

TL;DR

A novel framework for estimating causal effects based on the discrepancy between unobserved counterfactual distributions is developed, and methods to construct confidence intervals for the unknown mean distribution distance are proposed.

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 and the integrated risks of the proposed estimator, and propose a bootstrap-based confidence band. The second is a novel distributional causal effect defined by the distance between different counterfactual distributions. We study three approaches for estimating the proposed distributional effect: smoothing the counterfactual density, smoothing the 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.

Paper Structure

This paper contains 34 sections, 45 theorems, 370 equations, 3 figures, 2 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Setup and Identification
  4. Bootstrap and Stochastic Convergence of Empirical Process
  5. Counterfactual Density Estimation
  6. Proposed Estimator and Asymptotic Properties
  7. Risk Bounds
  8. Connections to non-counterfactual minimax bounds
  9. Density Effect based on $L_1$-distance
  10. Smoothing Counterfactual Density
  11. Smoothing $L_1$ Distance
  12. Under Margin Condition
  13. Inference
  14. Uniform Inference for Counterfactual Density
  15. Inference for Density Effect and Testing No Effect
  16. ...and 19 more sections

Key Result

Theorem 2.1

(Kosorok2008) $\sqrt{n}(\mathbb{P}_{n}-\mathbb{P})\to\mathbb{G}$ weakly in $\ell_{\infty}(\mathcal{F})$ if and only if $\sqrt{n}(\mathbb{P}_{n}^{*}-\mathbb{P}_{n})\to\mathbb{G}$ a.s. weakly in $\ell_{\infty}(\mathcal{F})$ for a limit process $\mathbb{G}$. If either convergence happens, the limit pro

Figures (3)

  • Figure 1: Densities for two counterfactual distributions, which have exactly the same mean and variance, but differ in $L_1$ distance (shaded area) by 0.628. The chance of having an outcome in the three black intervals along the x-axis is only 22.5% under treatment, but 53.9% under control, so the $L_1$ distance between the two distributions is 0.628.
  • Figure 2: RMSE versus nuisance error parameter $r$ across different sample sizes
  • Figure 3: Estimated counterfactual densities with $99\%$ confidence bands for the achievement gap in Math and ELA subjects between White and non-White students.

Theorems & Definitions (116)

  • Remark 1.1
  • Theorem 2.1
  • Remark 3.1
  • Remark 3.2: Sample splitting
  • Theorem 3.1
  • Theorem 3.2
  • Remark 3.3
  • Theorem 3.3
  • Corollary 3.1
  • Theorem 4.1
  • ...and 106 more