Maximum Moment Restriction for Instrumental Variable Regression

We propose a simple framework for nonlinear instrumental variable (IV) regression based on a kernelized conditional moment restriction (CMR) known as a maximum moment restriction (MMR). The MMR is formulated by maximizing the interaction between the residual and the instruments belonging to a unit ball in a reproducing kernel Hilbert space (RKHS). The MMR allows us to reformulate the IV regression as a single-step empirical risk minimization problem, where the risk depends on the reproducing kernel on the instrument and can be estimated by a U-statistic or V-statistic. This simplification not only eases the proofs of consistency and asymptotic normality in both parametric and non-parametric settings, but also results in easy-to-use algorithms with an efficient hyper-parameter selection procedure. We demonstrate the advantages of our framework over existing ones using experiments on both synthetic and real-world data.