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Joint Inference for the Regression Discontinuity Effect and Its External Validity

Yuta Okamoto

TL;DR

This paper tackles the external validity of regression discontinuity designs by proposing a joint inference framework for the RD effect at the cutoff, $\tau(0)$, and its local external validity via the treatment effect derivative, $\tau'(0)$, within a robust bias-corrected approach. It then augments this with a locally linear treatment effects (LLTE) assumption to enable direct extrapolation of RD effects and to construct a uniform confidence band for the extrapolated RD effects, all without requiring additional covariates or design changes. The authors derive a joint asymptotic distribution for $(\tau_{SRD},\tau'_{SRD})$, provide an RBC-based confidence region (an ellipse), and develop LLTE-based uniform bands $\mathcal{U}_{1-\alpha}(x)$ that adapt to the extrapolation window $[-\delta_1,\delta_2]$, while ensuring validity for any subinterval where local linearity holds. They demonstrate the method on two empirical RD applications (Colombia's SPP and California textbook funding) and conduct extensive simulations showing good finite-sample coverage and the practical reliability of bandwidth choices. Overall, the work offers a practical, transparent tool to assess external validity and to conduct informative extrapolation in RD analyses, enhancing policy relevance and comparability across studies.

Abstract

The external validity of regression discontinuity designs is crucial for informing policy but is rarely examined in applied work. To advance empirical practice, we propose a joint inference procedure for the treatment effect and its local external validity, captured by the treatment effect derivative (TED), within a robust bias correction framework. We further introduce a locally linear treatment effects assumption, which extends the scope of the TED and enables identification and the construction of a uniform confidence band for extrapolated effects. These methods apply to most empirical studies. Empirical illustrations demonstrate their practical usefulness.

Joint Inference for the Regression Discontinuity Effect and Its External Validity

TL;DR

This paper tackles the external validity of regression discontinuity designs by proposing a joint inference framework for the RD effect at the cutoff, , and its local external validity via the treatment effect derivative, , within a robust bias-corrected approach. It then augments this with a locally linear treatment effects (LLTE) assumption to enable direct extrapolation of RD effects and to construct a uniform confidence band for the extrapolated RD effects, all without requiring additional covariates or design changes. The authors derive a joint asymptotic distribution for , provide an RBC-based confidence region (an ellipse), and develop LLTE-based uniform bands that adapt to the extrapolation window , while ensuring validity for any subinterval where local linearity holds. They demonstrate the method on two empirical RD applications (Colombia's SPP and California textbook funding) and conduct extensive simulations showing good finite-sample coverage and the practical reliability of bandwidth choices. Overall, the work offers a practical, transparent tool to assess external validity and to conduct informative extrapolation in RD analyses, enhancing policy relevance and comparability across studies.

Abstract

The external validity of regression discontinuity designs is crucial for informing policy but is rarely examined in applied work. To advance empirical practice, we propose a joint inference procedure for the treatment effect and its local external validity, captured by the treatment effect derivative (TED), within a robust bias correction framework. We further introduce a locally linear treatment effects assumption, which extends the scope of the TED and enables identification and the construction of a uniform confidence band for extrapolated effects. These methods apply to most empirical studies. Empirical illustrations demonstrate their practical usefulness.

Paper Structure

This paper contains 31 sections, 5 theorems, 39 equations, 5 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Empirical Practices in Recent RD Studies
  3. Sample and Procedure
  4. Results
  5. External Validity and Extrapolation under Sharp RD
  6. Setup
  7. Robust Local Polynomial Confidence Region
  8. Inference under a Locally Linear Treatment Effects Assumption
  9. Remarks
  10. Extrapolation Region.
  11. Testing the Local Linearity.
  12. MISE-Optimal Bandwidth.
  13. Fuzzy RD Design.
  14. Empirical Illustrations
  15. Londoño-Vélez, Rodríguez, and Sánchez (2020)
  16. ...and 16 more sections

Key Result

Lemma 1

Suppose Assumptions assumption: regularity-assumption: kernel hold. If $S\geq3$, $\max\{h,b\}<\kappa_0$, $n \min\{h^5,b^5\}\times\max\{h^2,b^2\}\to0$, $n\min\{h,b\}\to\infty$, then provided that $\Omega$ is invertible, where and $\mathrm{V}_{\mathtt{SRD}}$, $\mathrm{V}_{\mathtt{SRD}}^\prime$, and $\mathrm{C}_{\mathtt{SRD}}$ are provided in the Online Appendix Section S2.1. Furthermore, $\Omega$

Figures (5)

  • Figure 1: Same RD Effect, Different Policy Implication
  • Figure 2: External Validity and Extrapolation of Treatment Effects
  • Figure 3: External Validity and Extrapolation of Treatment Effects
  • Figure 4: Joint Confidence Region & Confidence Band \ref{['eq: dgp2']}
  • Figure S2: Data Generating Processes

Theorems & Definitions (5)

  • Lemma 1
  • Proposition 1
  • Lemma 2
  • Proposition 2
  • Lemma 3