用于非参数工具变量分位数回归的条件扩散模型

中文摘要 AI 辅助

本研究提出了深度非参数工具变量分位数回归（IVQR），这是一种两阶段估计方法，将条件扩散建模与核平滑条件矩公式相结合。第一阶段，我们使用保方差条件扩散模型，估计给定工具变量下因变量与内生协变量的联合条件分布；第二阶段，通过蒙特卡洛采样和指示函数的核平滑代理近似条件矩算子，再通过深度神经网络的经验风险最小化估计结构分位数函数。我们为所提估计量建立了超额风险界，并在无界支撑下推导了条件扩散模型的端到端总变差保证，明确考虑了得分估计、早停和离散化误差。该理论基于数据分布的多项式尾包络开发，可连续退化为指数设置：随着尾指数增大，所得超额风险率收敛至非参数回归的极小极大最优率，因此我们的重尾理论涵盖了经典轻尾非参数保证作为极限情况。模拟研究和实际数据应用表明，所提方法优于现有非参数IVQR方法，且随着协变量和工具变量维度的增加，性能提升愈发显著。

英文摘要

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.

发表机构

• Institute of Data Science and Statistics, Shanghai University of Finance and Economics（上海财经大学数据科学与统计研究院）
• School of Mathematics and Statistics, Wuhan University（武汉大学数学与统计学院）
• School of Artificial Intelligence, Wuhan University（武汉大学人工智能学院）

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