We study the nonparametric instrumental variable regression (NPIV) with observed covariates, termed NPIV-O. Compared to standard NPIV, the additional covariates aid causal identification and allow for heterogeneous causal effect estimation. However, they introduce two challenges for theoretical analysis. First, they induce a partial identity structure, making previous NPIV analyses based on ill-posedness, stability, or link conditions inapplicable. Second, they impose anisotropic smoothness on the structural function. To address the first challenge, we introduce a novel Fourier measure of partial smoothing; for the second, we extend the existing kernel 2SLS instrumental variable algorithm (KIV-O) to incorporate Gaussian kernel lengthscales adaptive to anisotropic smoothness. We prove upper L2-learning rates for KIV-O and the first L2-minimax lower bounds for NPIV-O. Both rates interpolate between known optimal rates for NPIV and nonparametric regression. We also identify a gap between the upper and lower bounds, arising 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 with the same conditional moment restriction as NPIV-O.
