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Global Testing in Multivariate Regression Discontinuity Designs

Artem Samiahulin

TL;DR

This paper develops a robust global testing framework for multivariate regression discontinuity designs by integrating multivariate machine learning estimators with a distance-based aggregation to detect any discontinuity along a multidimensional boundary, especially in modest samples. The method operates in two stages: first, it uses a multivariate estimator (Local Linear Forest for treatment effects; a random-forest density estimator for density) to recover boundary information and the signs of local effects, and then it aggregates these into a univariate test statistic with cross-fitting and multiplier bootstrap for valid inference. Simulations show near-nominal size and strong power across settings where standard multivariate RD methods struggle, and an empirical application to the Bolsa Família boundary demonstrates how the approach complements existing multivariate RD estimators. The contribution extends global testing and manipulation checks to multivariate RD, providing a practical tool for researchers facing high-dimensional running variables and limited effective sample sizes.

Abstract

Regression discontinuity (RD) designs with multiple running variables arise in a growing number of empirical applications, including geographic boundaries and multi-score assignment rules. Although recent methodological work has extended estimation and inference tools to multivariate settings, far less attention has been devoted to developing global testing methods that formally assess whether a discontinuity exists anywhere along a multivariate treatment boundary. Existing approaches perform well in large samples, but can exhibit severe size distortions in moderate or small samples due to the sparsity of observations near any particular boundary point. This paper introduces a complementary global testing procedure that mitigates the small-sample weaknesses of existing multivariate RD methods by integrating multivariate machine learning estimators with a distance-based aggregation strategy, yielding a test statistic that remains reliable with limited data. Simulations demonstrate that the proposed method maintains near-nominal size and strong power, including in settings where standard multivariate estimators break down. The procedure is applied to an empirical setting to demonstrate its implementation and to illustrate how it can complement existing multivariate RD estimators.

Global Testing in Multivariate Regression Discontinuity Designs

TL;DR

This paper develops a robust global testing framework for multivariate regression discontinuity designs by integrating multivariate machine learning estimators with a distance-based aggregation to detect any discontinuity along a multidimensional boundary, especially in modest samples. The method operates in two stages: first, it uses a multivariate estimator (Local Linear Forest for treatment effects; a random-forest density estimator for density) to recover boundary information and the signs of local effects, and then it aggregates these into a univariate test statistic with cross-fitting and multiplier bootstrap for valid inference. Simulations show near-nominal size and strong power across settings where standard multivariate RD methods struggle, and an empirical application to the Bolsa Família boundary demonstrates how the approach complements existing multivariate RD estimators. The contribution extends global testing and manipulation checks to multivariate RD, providing a practical tool for researchers facing high-dimensional running variables and limited effective sample sizes.

Abstract

Regression discontinuity (RD) designs with multiple running variables arise in a growing number of empirical applications, including geographic boundaries and multi-score assignment rules. Although recent methodological work has extended estimation and inference tools to multivariate settings, far less attention has been devoted to developing global testing methods that formally assess whether a discontinuity exists anywhere along a multivariate treatment boundary. Existing approaches perform well in large samples, but can exhibit severe size distortions in moderate or small samples due to the sparsity of observations near any particular boundary point. This paper introduces a complementary global testing procedure that mitigates the small-sample weaknesses of existing multivariate RD methods by integrating multivariate machine learning estimators with a distance-based aggregation strategy, yielding a test statistic that remains reliable with limited data. Simulations demonstrate that the proposed method maintains near-nominal size and strong power, including in settings where standard multivariate estimators break down. The procedure is applied to an empirical setting to demonstrate its implementation and to illustrate how it can complement existing multivariate RD estimators.
Paper Structure (25 sections, 7 theorems, 86 equations, 3 figures, 6 tables)

This paper contains 25 sections, 7 theorems, 86 equations, 3 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Simple Simulation Example
  3. Setup and Notation
  4. Identification
  5. Treatment Effect Heterogeneity
  6. Density Manipulation
  7. Estimation and Inference
  8. Treatment Effect Heterogeneity
  9. Density Test
  10. Accounting for Random Split Selection
  11. Simulations
  12. Treatment Effect Heterogeneity Simulations
  13. Density Manipulation Simulations
  14. Application
  15. Conclusion
  16. ...and 10 more sections

Key Result

Theorem 1

Under Assumptions 1-3, 5, and 7, we have: where $dS_\Omega$ denotes a Hausdorff measure over $\partial \Omega$.

Figures (3)

  • Figure 1: Treatment and control regions for simulated treatment rule.
  • Figure 2: Graph of the DGP 1 function. Darker colors refer to lower values and lighter colors refer to higher values.
  • Figure 3: Treatment effects for pro-poor spending outcome using a local polynomial regression. The darker region represents pointwise bands and the lighter region represents uniform bands.

Theorems & Definitions (14)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 1
  • Theorem 4
  • Lemma 2
  • Theorem 5
  • proof
  • proof
  • proof
  • ...and 4 more