Important problems in causal inference, economics, and, more generally, robust machine learning can be expressed as conditional moment restrictions, but estimation becomes challenging as it requires solving a continuum of unconditional moment restrictions. Previous works addressed this problem by extending the generalized method of moments (GMM) to continuum moment restrictions. In contrast, generalized empirical likelihood (GEL) provides a more general framework and has been shown to enjoy favorable small-sample properties compared to GMM-based estimators. To benefit from recent developments in machine learning, we provide a functional reformulation of GEL in which arbitrary models can be leveraged. Motivated by a dual formulation of the resulting infinite dimensional optimization problem, we devise a practical method and explore its asymptotic properties. Finally, we provide kernel- and neural network-based implementations of the estimator, which achieve state-of-the-art empirical performance on two conditional moment restriction problems.

翻译：因果推断、经济学以及更广泛意义上的鲁棒机器学习中的关键问题可表述为条件矩约束，但由于需要求解连续统的无条件矩约束，此类估计问题具有挑战性。现有研究通过将广义矩估计法（GMM）扩展至连续统矩约束来处理该问题。相比之下，广义经验似然（GEL）提供了更通用的框架，且相较于基于GMM的估计量，被证明在小样本性质上具有优势。为受益于机器学习的最新进展，我们提出GEL的函数化重新表述，允许在其中利用任意模型。受由此产生的无限维优化问题对偶形式的启发，我们设计了一种实用方法并探讨其渐近性质。最后，我们给出了基于核函数和神经网络的估计量实现，在两类条件矩约束问题上取得了当前最优的实证表现。