Estimation and inference for high-dimensional nonparametric additive instrumental-variables regression

The method of instrumental variables provides a fundamental and practical tool for causal inference in many empirical studies where unmeasured confounding between the covariates of interest and the outcome is present. Modern data such as the genetical genomics data from these studies are often high-dimensional. The high-dimensional linear instrumental-variables regression has been considered in the literature due to its simplicity albeit a true nonlinear relationship may exist. We propose a more data-driven approach by considering the nonparametric additive models between the instruments and the endogenous covariates while keeping a linear model between the covariates and the outcome so that the coefficients therein can directly bear causal interpretation. We provide a two-stage framework for estimation and inference under this more general setup. The group lasso regularization is firstly employed to select optimal instruments from the high-dimensional additive models, and the outcome variable is then regressed on the fitted values from the additive models to identify and estimate important covariate effects. We provide non-asymptotic analysis of the estimation error of the proposed estimator. A debiasing procedure is further employed to yield valid inference. Extensive numerical experiments show that our method can rival or outperform existing approaches in the literature. We finally analyze the mouse obesity data and discuss new findings from our method.