Instrumental variable regression via kernel maximum moment loss

Abstract

We investigate a simple objective for nonlinear instrumental variable (IV) regression based on a kernelized conditional moment restriction known as a maximum moment restriction (MMR). The MMR objective is formulated by maximizing the interaction between the residual and the instruments belonging to a unit ball in a reproducing kernel Hilbert space. First, it allows us to simplify the IV regression as an empirical risk minimization problem, where the risk function depends on the reproducing kernel on the instrument and can be estimated by a U-statistic or V-statistic. Second, on the basis this simplification, we are able to provide consistency and asymptotic normality results in both parametric and nonparametric settings. Finally, we provide easy-to-use IV regression algorithms with an efficient hyperparameter selection procedure. We demonstrate the effectiveness of our algorithms using experiments on both synthetic and real-world data.