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

Abstract

Regression discontinuity designs (RDDs) are widely used for causal inference in observational studies with cutoff-based treatment assignment, primarily for Euclidean outcomes. We propose the geodesic regression discontinuity design (GRDD), which extends RDDs to complex non-Euclidean outcomes, including networks, compositional data, functional data, and other random objects in geodesic metric spaces. Since algebraic operations are unavailable in such spaces, we define the causal effect at the cutoff as the geodesic connecting the local Fréchet means of untreated and treated outcomes, recovering the classical local average treatment effect in the scalar case. Estimation is conducted intrinsically via local Fréchet regression to preserve geometric validity and interpretability. For inference, we adopt an extrinsic approach by embedding the metric space into a Hilbert space, enabling tractable asymptotic analysis. We establish asymptotic normality and develop bootstrap-based procedures for hypothesis testing and confidence intervals for the treatment effect magnitude. We also propose a data-adaptive bandwidth selection method tailored to RDDs in metric spaces and study its empirical performance. Applications include compositional voting outcomes in UK elections and daily CO concentration curves after the Taipei metro introduction, and we extend the framework to fuzzy designs with imperfect compliance.

arXiv comment75 pages, 6 figures