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.

翻译：我们研究了具有观测协变量的非参数工具变量回归问题，简称NPIV-O。与标准的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支持异质性因果效应估计。然而，观测协变量的存在为其理论分析带来了两个挑战。首先，它引入了部分恒等结构，使得先前基于不适定性度量、稳定性条件或连接条件的NPIV分析方法不再适用。其次，它对结构函数施加了各向异性平滑性。针对第一个挑战，我们提出了一种新的傅里叶部分平滑度量；针对第二个挑战，我们将现有的带观测协变量的核二阶段最小二乘工具变量算法（KIV-O）扩展为采用适应各向异性平滑性的高斯核长度尺度。我们证明了KIV-O的$L^2$上界学习率以及NPIV-O的首个$L^2$极小极大下界学习率。这两个学习率在已知的NPIV与非参数回归（NPR）最优学习率之间插值。有趣的是，我们发现了上界与下界之间存在间隙，这源于为最小化投影风险而调整的核长度尺度选择。我们的理论分析同样适用于近端因果推断——这是一个新兴的因果效应估计框架，与NPIV-O共享相同的条件矩约束。