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A Note on Doubly Robust Estimator in Regression Discontinuity Designs

Masahiro Kato

TL;DR

This work tackles the sensitivity of regression discontinuity (RD) estimates to nonparametric specification by introducing a doubly robust (DR) RD estimator for sharp RD designs. The method combines a first-stage estimate of the conditional outcome $\mu_0(d\mid W,Z)$ with a second-stage estimate of the conditional residual, forming a two-stage, kernel-weighted augmentation that remains consistent if either stage is correctly specified. The paper proves consistency under the double-robustness condition and provides practical implementations that blend local/global and linear/nonlinear modeling (e.g., sieve with neural nets). By not relying on overlap assumptions in the RD setting and by accommodating flexible nuisance estimation, the DR-RD estimator broadens the toolkit for robust causal inference at the RD cutoff.

Abstract

This note introduces a doubly robust (DR) estimator for regression discontinuity (RD) designs. RD designs provide a quasi-experimental framework for estimating treatment effects, where treatment assignment depends on whether a running variable surpasses a predefined cutoff. A common approach in RD estimation is the use of nonparametric regression methods, such as local linear regression. However, the validity of these methods still relies on the consistency of the nonparametric estimators. In this study, we propose the DR-RD estimator, which combines two distinct estimators for the conditional expected outcomes. The primary advantage of the DR-RD estimator lies in its ability to ensure the consistency of the treatment effect estimation as long as at least one of the two estimators is consistent. Consequently, our DR-RD estimator enhances robustness of treatment effect estimators in RD designs.

A Note on Doubly Robust Estimator in Regression Discontinuity Designs

TL;DR

This work tackles the sensitivity of regression discontinuity (RD) estimates to nonparametric specification by introducing a doubly robust (DR) RD estimator for sharp RD designs. The method combines a first-stage estimate of the conditional outcome with a second-stage estimate of the conditional residual, forming a two-stage, kernel-weighted augmentation that remains consistent if either stage is correctly specified. The paper proves consistency under the double-robustness condition and provides practical implementations that blend local/global and linear/nonlinear modeling (e.g., sieve with neural nets). By not relying on overlap assumptions in the RD setting and by accommodating flexible nuisance estimation, the DR-RD estimator broadens the toolkit for robust causal inference at the RD cutoff.

Abstract

This note introduces a doubly robust (DR) estimator for regression discontinuity (RD) designs. RD designs provide a quasi-experimental framework for estimating treatment effects, where treatment assignment depends on whether a running variable surpasses a predefined cutoff. A common approach in RD estimation is the use of nonparametric regression methods, such as local linear regression. However, the validity of these methods still relies on the consistency of the nonparametric estimators. In this study, we propose the DR-RD estimator, which combines two distinct estimators for the conditional expected outcomes. The primary advantage of the DR-RD estimator lies in its ability to ensure the consistency of the treatment effect estimation as long as at least one of the two estimators is consistent. Consequently, our DR-RD estimator enhances robustness of treatment effect estimators in RD designs.

Paper Structure

This paper contains 14 sections, 1 theorem, 20 equations.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Existing RD Methods
  4. Our Contributions
  5. Organization
  6. Notation:
  7. The DR-RD Estimator
  8. First stage.
  9. Second stage.
  10. Consistency and Double Robustness
  11. Examples of the DR-RD Estimators
  12. Integration of Local and Global Approaches
  13. Integration of Linear and Non-Linear Estimators
  14. Conclusion

Key Result

Theorem 3.1

Suppose that as $n\to\infty$, $\widehat{\mu}_n \xrightarrow{\mathrm{p}} \mu^\dagger$, and hold, where $\mu^\dagger$, $\zeta^\dagger$, and $\eta^\dagger$ are constant functions or parameters independent of $n$. Additionally, suppose that either $\mu^\dagger = \mu_0$ or holds for each $d \in \{1, 0\}$. If Assumption asm:continuity holds, then we have

Theorems & Definitions (1)

  • Theorem 3.1: Consistency and Double Robustness