Uniform inference for kernel instrumental variable regression
Marvin Lob, Rahul Singh, Suhas Vijaykumar
TL;DR
The paper develops a practical, nonparametric inference framework for kernel instrumental variable regression (KIV) by deriving valid and sharp uniform confidence bands. It introduces an anti-symmetric Gaussian multiplier bootstrap that requires only a single run of the KIV estimator, preserving the closed-form kernel structure. Central to the theory are low effective dimension, source and link smoothness conditions, and a strong instrument assumption, which together enable nonasymptotic Gaussian and bootstrap couplings in an ill-posed inverse problem. The results yield $ ext{H}_x$-norm confidence sets that translate into uniform sup-norm bands, providing researchers with reliable inference for nonlinear causal relationships estimated by KIV across nonstandard data types. This work thus substantially enhances the practical uptake of kernel methods for causal analysis in the social and biomedical sciences.
Abstract
Instrumental variable regression is a foundational tool for causal analysis across the social and biomedical sciences. Recent advances use kernel methods to estimate nonparametric causal relationships, with general data types, while retaining a simple closed-form expression. Empirical researchers ultimately need reliable inference on causal estimates; however, uniform confidence sets for the method remain unavailable. To fill this gap, we develop valid and sharp confidence sets for kernel instrumental variable regression, allowing general nonlinearities and data types. Computationally, our bootstrap procedure requires only a single run of the kernel instrumental variable regression estimator. Theoretically, it relies on the same key assumptions. Overall, we provide a practical procedure for inference that substantially increases the value of kernel methods for causal analysis.