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Estimation and Inference for the Average Treatment Effect in a Score-Explained Heterogeneous Treatment Effect Model

Kevin Christian Wibisono, Debarghya Mukherjee, Moulinath Banerjee, Ya'acov Ritov

TL;DR

The paper addresses non-random treatment allocation with a score threshold and heterogeneous individual treatment effects. It develops a residual-differencing and nearest-neighbor residual-matching estimator that uses all observations to estimate the ATT at the population level and extends naturally to estimate the ITE and CATE nonparametrically. The authors prove $\sqrt{n}$-consistency and asymptotic normality for the ATT estimator, provide a tractable form for the asymptotic variance, and validate the approach through simulations and real-data applications, with bootstrap shown to effectively approximate inference. The method robustly handles endogeneity via latent confounders and delivers practical tools for inference and heterogeneous effect analysis in settings where treatment is instructionally threshold-based.

Abstract

In many practical situations, randomly assigning treatments to subjects is uncommon due to feasibility constraints. For example, economic aid programs and merit-based scholarships are often restricted to those meeting specific income or exam score thresholds. In these scenarios, traditional approaches to estimating treatment effects typically focus solely on observations near the cutoff point, thereby excluding a significant portion of the sample and potentially leading to information loss. Moreover, these methods generally achieve a non-parametric convergence rate. While some approaches, e.g., Mukherjee et al. (2021), attempt to tackle these issues, they commonly assume that treatment effects are constant across individuals, an assumption that is often unrealistic in practice. In this study, we propose a differencing and matching-based estimator of the average treatment effect on the treated (ATT) in the presence of heterogeneous treatment effects, utilizing all available observations. We establish the asymptotic normality of our estimator and illustrate its effectiveness through various synthetic and real data analyses. Additionally, we demonstrate that our method yields non-parametric estimates of the conditional average treatment effect (CATE) and individual treatment effect (ITE) as a byproduct.

Estimation and Inference for the Average Treatment Effect in a Score-Explained Heterogeneous Treatment Effect Model

TL;DR

The paper addresses non-random treatment allocation with a score threshold and heterogeneous individual treatment effects. It develops a residual-differencing and nearest-neighbor residual-matching estimator that uses all observations to estimate the ATT at the population level and extends naturally to estimate the ITE and CATE nonparametrically. The authors prove -consistency and asymptotic normality for the ATT estimator, provide a tractable form for the asymptotic variance, and validate the approach through simulations and real-data applications, with bootstrap shown to effectively approximate inference. The method robustly handles endogeneity via latent confounders and delivers practical tools for inference and heterogeneous effect analysis in settings where treatment is instructionally threshold-based.

Abstract

In many practical situations, randomly assigning treatments to subjects is uncommon due to feasibility constraints. For example, economic aid programs and merit-based scholarships are often restricted to those meeting specific income or exam score thresholds. In these scenarios, traditional approaches to estimating treatment effects typically focus solely on observations near the cutoff point, thereby excluding a significant portion of the sample and potentially leading to information loss. Moreover, these methods generally achieve a non-parametric convergence rate. While some approaches, e.g., Mukherjee et al. (2021), attempt to tackle these issues, they commonly assume that treatment effects are constant across individuals, an assumption that is often unrealistic in practice. In this study, we propose a differencing and matching-based estimator of the average treatment effect on the treated (ATT) in the presence of heterogeneous treatment effects, utilizing all available observations. We establish the asymptotic normality of our estimator and illustrate its effectiveness through various synthetic and real data analyses. Additionally, we demonstrate that our method yields non-parametric estimates of the conditional average treatment effect (CATE) and individual treatment effect (ITE) as a byproduct.

Paper Structure

This paper contains 22 sections, 7 theorems, 136 equations, 4 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Our contributions
  3. Methodology
  4. Individual treatment effect estimation
  5. Theoretical results
  6. Estimation of asymptotic variance of ATT
  7. Simulation studies
  8. Data generating process
  9. Simulation 1: Estimation of the ATT
  10. Simulation 2: Estimation of the asymptotic variance via bootstrapping
  11. Simulation 3: Estimation of the ITE
  12. Case I: $\alpha_0$ only depends on $X$
  13. Case II: $\alpha_0$ depends on $(X, \eta)$
  14. Real data analysis
  15. Effect of Islamic party on women's education
  16. ...and 7 more sections

Key Result

Proposition 1

Consider the estimator $\hat{\beta}$ of $\beta_0$ following the first four steps of Algorithm alg:est. Under Assumptions asm:momentofxz to asm:fbounded, the estimator is $\sqrt{{n}}$-CAN: where the explicit value of $\Sigma_\beta$ can be found in Equation eq:betahat_var of Appendix app:proof-prop.

Figures (4)

  • Figure 1: A graphical representation of the variables of Equations \ref{['eq:model_1']} and \ref{['eq:model_2']}, with $X=Z$.
  • Figure 2: The histogram of the $\zeta_k^{\textrm{cf}}$'s looks fairly normal and centered around 0.
  • Figure 3: Actual vs. predicted ITE for $(X_2, X_3) = (\pm 0.7, \pm 0.2)$.
  • Figure 4: Actual vs. predicted ITE for $(X_1, X_2, X_3, X_4) = (0.1, \pm 0.2, \pm 0.8, 1.5)$.

Theorems & Definitions (16)

  • Remark 1: Cross-fitting
  • Remark 2: Applications to CATE and fixed treatment effects
  • Remark 3: Comparison between our method and mukherjee2021estimation
  • Proposition 1: $\hat{\beta}$ is $\sqrt{{n}}$-CAN
  • Remark 4: $\sqrt{\tilde{n}}$ versus $\sqrt{n}$
  • Theorem 3.1: $\hat{\theta}$ is $\sqrt{{n}}$-CAN
  • Remark 5: Cross-fitting
  • Lemma 1
  • proof
  • Lemma 2
  • ...and 6 more