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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 ofunconditional moment restrictions. Previous works addressed this problem byextending the generalized method of moments (GMM) to continuum momentrestrictions. In contrast, generalized empirical likelihood (GEL) provides amore general framework and has been shown to enjoy favorable small-sampleproperties compared to GMM-based estimators. To benefit from recentdevelopments in machine learning, we provide a functional reformulation of GELin which arbitrary models can be leveraged. Motivated by a dual formulation ofthe resulting infinite dimensional optimization problem, we devise a practicalmethod and explore its asymptotic properties. Finally, we provide kernel- andneural network-based implementations of the estimator, which achievestate-of-the-art empirical performance on two conditional moment restrictionproblems.

Functional Generalized Empirical Likelihood Estimation for Conditional Moment Restrictions