Adaptive Reduction of Curse of Dimensionality in Nonparametric Instrumental Variable Estimation

Nonparametric estimation of instrumental variable treatment effects typically builds on various nonparametric identification results. However, these estimators often face challenges from the curse of dimensionality in practice, as multi-dimensional covariates are common. To address this issue, we investigate the nonparametric identification of a range of treatment effects within different sufficient dimension reduction models. We also examine the efficiency of estimation and find that, unlike fully nonparametric approaches, nonparametric estimators derived from maximal dimension reduction based on identification results may not be efficient. We study the conditions for achieving maximal dimension reduction to ensure efficiency for a binary instrumental variable and extend these results to multivariate and general instrumental variables. The proposed nonparametric sufficient dimension reduction framework imposes no constraints on the distribution of the observed data while mitigating the curse of dimensionality in a data-adaptive manner.