A Test for Jumps in Metric-Space Conditional Means

TL;DR

This paper introduces a nonparametric test for detecting jumps in the conditional mean of outcomes in complex metric spaces, extending existing methods to non-Euclidean data like distributions and networks.

Contribution

It develops a novel test based on local Fréchet regression for outcomes in metric spaces, with proven asymptotic properties and demonstrated effectiveness through simulations and empirical examples.

Findings

Detects discontinuities in non-Euclidean outcomes

Identifies sharp changes in income-based work-from-home data

Reveals network shifts after trade policy changes

Abstract

Standard methods for detecting discontinuities in conditional means are not applicable to outcomes that are complex, non-Euclidean objects like distributions, networks, or covariance matrices. This article develops a nonparametric test for jumps in conditional means when outcomes lie in a non-Euclidean metric space. Using local Fr\'echet regression, the method estimates a mean path on either side of a candidate cutoff. This extends existing $k$-sample tests to a non-parametric regression setting with metric-space valued outcomes. I establish the asymptotic distribution of the test and its consistency against contiguous alternatives. For this, I derive a central limit theorem for the local estimator of the conditional Fr\'echet variance and a consistent estimator of its asymptotic variance. Simulations confirm nominal size control and robust power in finite samples. Two empirical illustrations demonstrate the method's ability to reveal discontinuities missed by scalar-based tests. I find sharp changes in (i) work-from-home compositions at an income threshold for non-compete enforceability and (ii) national input-output networks following the loss of preferential U.S. trade access. These findings show the value of analyzing regression outcomes in their native metric spaces.