非参数工具变量回归与观测协变量
Nonparametric Instrumental Variable Regression with Observed Covariates
AI总结：
本文提出了一种结合观测协变量的非参数工具变量回归方法，通过引入傅里叶测度和自适应核长度尺度，实现了对异质因果效应的高效估计，并建立了理论学习速率的上下界。
摘要：
我们研究了带有观测协变量的非参数工具变量回归问题，称为NPIV-O。与标准非参数工具变量回归（NPIV）相比，额外的观测协变量有助于因果识别，并使异质因果效应估计成为可能。然而，观测协变量的存在给其理论分析带来了两个挑战。首先，它诱导出部分恒等结构，这使得基于不规范性、稳定性条件或链接条件的先前NPIV分析不再适用。其次，它对结构函数施加各向异性光滑性。为解决第一个挑战，我们引入了一种新的傅里叶测度部分平滑度；为解决第二个挑战，我们将现有的带有观测协变量的核2SLS工具变量算法（称为KIV-O）扩展以适应各向异性光滑性。我们证明了KIV-O的上$ L^2 $-学习速率以及NPIV-O的第一个$ L^2 $-最小最大下界学习速率。两者在已知的NPIV和非参数回归（NPR）最优速率之间插值。有趣的是，我们发现上界和下界之间存在差距，这源于核长度尺度的选择，以最小化投影风险。我们的理论分析也适用于近端因果推断，这是一种新兴的因果效应估计框架，与NPIV-O共享相同的条件矩限制。
英文摘要：
We study the problem of nonparametric instrumental variable regression with observed covariates, which we refer to as NPIV-O. Compared with standard nonparametric instrumental variable regression (NPIV), the additional observed covariates facilitate causal identification and enables heterogeneous causal effect estimation. However, the presence of observed covariates introduces two challenges for its theoretical analysis. First, it induces a partial identity structure, which renders previous NPIV analyses - based on measures of ill-posedness, stability conditions, or link conditions - inapplicable. Second, it imposes anisotropic smoothness on the structural function. To address the first challenge, we introduce a novel Fourier measure of partial smoothing; for the second challenge, we extend the existing kernel 2SLS instrumental variable algorithm with observed covariates, termed KIV-O, to incorporate Gaussian kernel lengthscales adaptive to the anisotropic smoothness. We prove upper $L^2$-learning rates for KIV-O and the first $L^2$-minimax lower learning rates for NPIV-O. Both rates interpolate between known optimal rates of NPIV and nonparametric regression (NPR). Interestingly, we identify a gap between our upper and lower bounds, which arises from the choice of kernel lengthscales tuned to minimize a projected risk. Our theoretical analysis also applies to proximal causal inference, an emerging framework for causal effect estimation that shares the same conditional moment restriction as NPIV-O.
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