Conditional Diffusion for Nonparametric Instrumental Variable Quantile Regression

2026-08-08

AI Summary

This study addresses the estimation challenges of nonparametric instrumental variable quantile regression (IVQR) under heavy-tailed distributions and unbounded support in high-dimensional settings with both covariates and instruments. The authors propose a two-stage approach: in the first stage, a variance-preserving conditional diffusion model is introduced to estimate the joint conditional distribution of the endogenous and outcome variables given the instruments; in the second stage, Monte Carlo sampling combined with a kernel-smoothed indicator function approximates the conditional moment operator, and the structural quantile function is estimated via empirical risk minimization using deep neural networks. This work is the first to integrate conditional diffusion models into the nonparametric IVQR framework, establishing a unified theoretical analysis covering both heavy- and light-tailed scenarios, providing an end-to-end total variation error bound under unbounded support, and demonstrating substantial empirical advantages over existing methods—particularly in high-dimensional regimes.

Abstract

This work proposes deep nonparametric Instrumental variable quantile regression (IVQR), a two-stage estimator that combines conditional diffusion modeling with a kernel-smoothed conditional moment formulation. In the first stage, we estimate the joint conditional distribution of the outcome and endogenous covariates given the instrument using a variance-preserving conditional diffusion model. In the second stage, we approximate the conditional moment operator through Monte Carlo sampling and a kernel-smoothed surrogate for the indicator function, and then estimate the structural quantile function by empirical risk minimization over deep neural networks. We establish an excess-risk bound for the proposed estimator and derive end-to-end total variation guarantees for the conditional diffusion model under unbounded support, explicitly accounting for score estimation, early stopping, and discretization errors. Our theory is developed under a polynomial-tail envelope on the data distribution and degenerates continuously to the exponential setting: as the tail index grows, the obtained excess-risk rate converges to the minimax-optimal rate of nonparametric regression, thus our heavy-tailed theory covers the classical light-tailed nonparametric guarantees as a limiting case. Simulation studies and a real-data application demonstrate that the proposed method outperforms existing nonparametric IVQR approaches, with gains that become increasingly pronounced as the dimensionality of the covariates and instruments increases.

Problem

• Instrumental Variable Quantile Regression
• Nonparametric Estimation
• Endogeneity
• Conditional Diffusion
• Structural Quantile Function

Innovation

• conditional diffusion
• instrumental variable quantile regression
• nonparametric estimation
• deep neural networks
• conditional moment restriction