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.

翻译：我们研究了一种基于核化条件矩约束（称为最大矩约束）的非线性工具变量回归的简单目标函数。该MMR目标通过最大化残差与属于再生核希尔伯特空间中单位球的工具变量之间的交互作用来构建。首先，它允许我们将工具变量回归简化为经验风险最小化问题，其中风险泛函依赖于工具变量的再生核，并可通过U统计量或V统计量进行估计。其次，基于这一简化，我们能够在参数和非参数设定下提供一致性和渐近正态性结果。最后，我们给出了易于使用的工具变量回归算法，并附带高效的超参数选择流程。通过合成数据和真实数据实验，我们验证了所提算法的有效性。