  We investigate a simple objective for nonlinear instrumental variable (IV)
regression based on a kernelized conditional moment restriction (CMR) 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 (RKHS). First, it allows
us to simplify the IV regression as an empirical risk minimization problem,
where the risk functional depends on the reproducing kernel on the instrument
and can be estimated by a U-statistic or V-statistic. Second, based on this
simplification, we are able to provide the consistency and asymptotic normality
results in both parametric and nonparametric settings. Lastly, we provide
easy-to-use IV regression algorithms with an efficient hyper-parameter
selection procedure. We demonstrate the effectiveness of our algorithms using
experiments on both synthetic and real-world data.
