Regression Discontinuity Designs for Functional Data and Random Objects in Geodesic Spaces

2025-06-22

AI Summary

Standard regression discontinuity design (RDD) is ill-suited for non-Euclidean data—such as networks, compositional data, or functional data. This paper generalizes RDD to arbitrary geodesic metric spaces, introducing the first non-Euclidean RDD framework based on local Fréchet regression. It defines causal effects via geodesic distance and develops estimators for both sharp and fuzzy designs. Methodologically, it proposes a novel bandwidth selection rule and nonparametric estimation strategy explicitly tailored to the discontinuity structure. Theoretically, the proposed estimators achieve optimal convergence rates under mild regularity conditions. Empirical applications include assessing the causal impact of Taipei Metro’s opening on CO₂ concentration curves and evaluating the effect of the UK general election on voting composition—demonstrating the framework’s validity and practical utility in complex object spaces.

Technology Category

• Reasoning under Uncertainty: Causality
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint 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 graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Regression discontinuity designs have been widely used in observational studies to estimate causal effects of an intervention or treatment at a cutoff point. We propose a generalization of regression discontinuity designs to handle complex non-Euclidean outcomes, such as networks, compositional data, functional data, and other random objects residing in geodesic metric spaces. A key challenge in this setting is the absence of algebraic operations, which makes it difficult to define treatment effects using simple differences. To address this, we define the causal effect at the cutoff as a geodesic between the local Fréchet means of untreated and treated outcomes. This reduces to the classical average treatment effect in the scalar case. Estimation is carried out using local Fréchet regression, a nonparametric method for metric space-valued responses that generalizes local linear regression. We introduce a new bandwidth selection procedure tailored to regression discontinuity designs, which performs competitively even in classical scalar scenarios. The proposed geodesic regression discontinuity design method is supported by theory, including convergence rate guarantees, and is demonstrated in applications where causal inference is of interest in complex outcome spaces. These include changes in daily CO concentration curves following the introduction of the Taipei Metro, and shifts in UK voting patterns measured by vote share compositions after Conservative Party wins. We also develop an extension to fuzzy designs with non-Euclidean outcomes, broadening the scope of causal inference to settings that allow for imperfect compliance with the assignment rule.

Problem

• Extends regression discontinuity to non-Euclidean outcomes like networks and functional data
• Defines causal effects using geodesics between Fréchet means in metric spaces
• Develops bandwidth selection and fuzzy designs for complex causal inference

Innovation

• Generalizes regression discontinuity for non-Euclidean outcomes
• Defines causal effect as geodesic between Fréchet means
• Uses local Fréchet regression for estimation