Functional Generalized Empirical Likelihood Estimation for Conditional Moment Restrictions

TL;DR

This paper introduces a functional reformulation of generalized empirical likelihood (GEL) for conditional moment restrictions, enabling the use of machine learning models like neural networks and kernels, with strong empirical results.

Contribution

It develops a novel functional GEL framework for continuum moment restrictions, incorporating machine learning models and analyzing its asymptotic properties.

Findings

Achieves state-of-the-art empirical performance on conditional moment problems.

Provides kernel- and neural network-based implementations of the estimator.

Offers a practical method with theoretical asymptotic analysis.

Abstract

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.