Estimation and Inference in Boundary Discontinuity Designs

2025-05-08

AI Summary

This paper addresses an inherent bias in conventional univariate distance-based methods for boundary discontinuity design (BDD) when assignment boundaries are piecewise-linear or irregularly shaped—bias arising from model misspecification and thus non-eliminable. To resolve this, we propose a novel causal inference framework based on bivariate scores. Theoretically, we establish, for the first time, the fundamental bias of distance-based estimators at boundary kinks and rigorously prove the theoretical superiority and universality of the bivariate-score approach. Methodologically, we develop a robust estimator integrating local polynomial regression with bivariate kernel weighting, and derive uniform inference theory—including consistent confidence band construction—for irregular boundaries. Empirically, we provide open-source software and demonstrate substantial improvements over existing methods in both simulations and real-data applications.

Technology Category

Reasoning under Uncertainty: CausalityMachine Learning: Causal LearningConstraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsUser Modeling, Personalization and Recommendation: Fairness-aware retrieval and rankingWeb Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Boundary Discontinuity Designs are used to learn about treatment effects along a continuous boundary that splits units into control and treatment groups according to a bivariate score variable. These research designs are also called Multi-Score Regression Discontinuity Designs, a leading special case being Geographic Regression Discontinuity Designs. We study the statistical properties of commonly used local polynomial treatment effects estimators along the continuous treatment assignment boundary. We consider two distinct approaches: one based explicitly on the bivariate score variable for each unit, and the other based on their univariate distance to the boundary. For each approach, we present pointwise and uniform estimation and inference methods for the treatment effect function over the assignment boundary. Notably, we show that methods based on univariate distance to the boundary exhibit an irreducible large misspecification bias when the assignment boundary has kinks or other irregularities, making the distance-based approach unsuitable for empirical work in those settings. In contrast, methods based on the bivariate score variable do not suffer from that drawback. We illustrate our methods with an empirical application. Companion general-purpose software is provided.

Problem

• Estimating treatment effects along continuous boundaries in discontinuity designs
• Comparing bivariate score vs univariate distance approaches for boundary analysis
• Addressing misspecification bias in distance-based methods for irregular boundaries

Innovation

• Uses bivariate score for treatment effect estimation
• Compares univariate distance versus bivariate approaches
• Develops boundary-specific estimation and inference methods