IV regression with distribution-valued outcomes

2026-05-27

AI Summary

This study addresses regression problems where the outcome variable is a probability distribution and covariates are endogenous. The authors propose IV Fréchet regression, the first method to extend the instrumental variable (IV) framework to the 2-Wasserstein space. Their approach constructs weighted quantile curves using instrumental variables and projects them onto the space of valid probability distributions to ensure the fitted distributions remain legitimate. Consistent inference is achieved via a multiplier bootstrap procedure. Theoretical analysis and simulations demonstrate that the proposed method reduces integrated mean squared error by up to 63% compared to existing approaches. In an empirical application, the authors find that import competition significantly affects only the 10th to 35th percentiles of the wage distribution, with confidence bands narrowing by 9–10%.

Abstract

We develop IV Fréchet regression (IVFR), an instrumental-variable (IV) method for settings where the outcome is an entire distribution. Framing the problem as an IV regression in 2-Wasserstein space, IVFR extends global Fréchet regression to the case with endogenous covariates. IVFR projects IV-weighted quantile curves onto the space of valid distributions and then recovers the corresponding regression coefficient functions. The projection provably reduces the estimation error in finite samples and guarantees valid fitted distributions. We show that the IVFR estimator converges weakly to a mean-zero Gaussian process and establish the validity of a multiplier bootstrap procedure for uniform inference. In simulations, the projection reduces the integrated mean squared error (IMSE) by up to 63% relative to existing methods. Revisiting the effects of Chinese import competition on the wage distribution within commuting zones, the proposed method produces 9-10% narrower confidence bands than existing methods. Using our novel uniform confidence bands, we find no evidence that import competition reduced wages at the very bottom of the distribution, but only between the 10th and 35th quantile. We also revisit the effect of county food stamp programs on the county's birth weight distribution and find no significant effects.

Problem

• instrumental variable
• distribution-valued outcomes
• endogeneity
• Fréchet regression
• 2-Wasserstein space

Innovation

• IV Fréchet regression
• distribution-valued outcomes
• 2-Wasserstein space
• instrumental variables
• uniform inference