Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Causal-DRF: Conditional Kernel Treatment Effect Estimation using Distributional Random Forest

Jeffrey Näf, Junhyung Park, Herbert Susmann

Abstract

The conditional average treatment effect (CATE) is a commonly targeted statistical parameter for measuring the effect of a treatment conditional on covariates. However, the CATE will fail to capture effects of treatments beyond differences in conditional expectations. Inspired by causal forests for CATE estimation, we develop a forest-based method to estimate the conditional kernel treatment effect (CKTE), based on the recently introduced Distributional Random Forest (DRF) algorithm. Adapting the splitting criterion of DRF, we show how one forest fit can be used to obtain a consistent and asymptotically normal estimator of the CKTE, as well as an approximation of its sampling distribution. This allows to study the difference in distribution between control and treatment group and thus yields a more comprehensive understanding of the treatment effect.

Causal-DRF: Conditional Kernel Treatment Effect Estimation using Distributional Random Forest

Abstract

The conditional average treatment effect (CATE) is a commonly targeted statistical parameter for measuring the effect of a treatment conditional on covariates. However, the CATE will fail to capture effects of treatments beyond differences in conditional expectations. Inspired by causal forests for CATE estimation, we develop a forest-based method to estimate the conditional kernel treatment effect (CKTE), based on the recently introduced Distributional Random Forest (DRF) algorithm. Adapting the splitting criterion of DRF, we show how one forest fit can be used to obtain a consistent and asymptotically normal estimator of the CKTE, as well as an approximation of its sampling distribution. This allows to study the difference in distribution between control and treatment group and thus yields a more comprehensive understanding of the treatment effect.

Paper Structure

This paper contains 15 sections, 11 theorems, 251 equations, 6 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Notation and RKHS Background
  4. Motivational Examples
  5. Causal-Distributional Random Forests
  6. Theoretical Development
  7. Empirical Study: Impact of 401(k) Eligibility
  8. Conclusion
  9. Variables used in the 401(k) eligibility study
  10. Simulation Study
  11. Algorithm and Details on the Splitting Criterion
  12. Preliminaries
  13. Forest Assumptions and a Refinment
  14. Improving upon the proof of Theorem 5 in [Näf et al., 2023]
  15. Main Proofs

Key Result

Theorem 4.1

Assume conditions forestass1--forestass6CausalDRF, forestass1starCausalDRF, dataass1--dataass7, kernelass1--kernelass4 and causalityass1--causalityass2 hold. Then, there exists $\sigma_n \to 0$, such that, for some self-adjoint HS operator $\boldsymbol{\Sigma}_{\mathbf{x}}$, we have

Figures (6)

  • Figure 1: Density and Causal-DRF estimates for Example 1 (left, no effect) and Example 2 (right, mean shift). Blue: Density for the control group $W=0$, Red: Density for the treatment group $W=1$.
  • Figure 2: Density and Causal-DRF estimates for Example 3 (left, variance effect) and Example 4 (right, mean shift + variance effect). Blue: Density for the control group $W=0$, Red: Density for the treatment group $W=1$.
  • Figure 3: Witness functions with confidence bands for Net Financial Assets (Left), Net Non-401(k) Financial Assets (Middle), and Total Wealth (Right) for a person of age 31, 28'146 income, family size of 5, 12 years of education, married, single earner, homeowner, no defined benefit pension status, and no IRA participation.
  • Figure 4: Witness functions with confidence bands for Net Financial Assets (Left), Net Non-401(k) Financial Assets (Middle), and Total Wealth (Right) for a person of age 50, 5'766 income, family size of 1, 14 years of education, unmarried, single earner, no defined benefit pension status, and IRA participation.
  • Figure 5: Simulation setting of both confounding and effect for $n=1000$. The black solid lines are the estimated witness functions, while the dashed lines are the estimated 95$\%$ confidence bands. The true witness function is given in gray. Left: Causal-DRF, Right: DRF.
  • ...and 1 more figures

Theorems & Definitions (21)

  • Theorem 4.1
  • Remark 4.2
  • Theorem 4.3
  • Theorem 4.4
  • Theorem 4.5
  • Lemma D.1: Lemma 9 in DRF-paper
  • Remark D.2
  • Remark D.3
  • Theorem D.4
  • proof
  • ...and 11 more