The Variational Method of Moments

TL;DR

This paper introduces the variational method of moments (VMM), a flexible framework for estimating conditional moments that can handle infinitely-many moments, with theoretical guarantees and practical algorithms for inference.

Contribution

It proposes a general class of VMM estimators, including kernel and neural network variants, with theoretical analysis and algorithms for consistent, efficient inference.

Findings

VMM estimators are consistent and asymptotically normal.

Neural network and kernel-based VMM methods perform well in synthetic experiments.

VMM enables controlling infinitely-many moments effectively.

Abstract

The conditional moment problem is a powerful formulation for describing structural causal parameters in terms of observables, a prominent example being instrumental variable regression. A standard approach reduces the problem to a finite set of marginal moment conditions and applies the optimally weighted generalized method of moments (OWGMM), but this requires we know a finite set of identifying moments, can still be inefficient even if identifying, or can be theoretically efficient but practically unwieldy if we use a growing sieve of moment conditions. Motivated by a variational minimax reformulation of OWGMM, we define a very general class of estimators for the conditional moment problem, which we term the variational method of moments (VMM) and which naturally enables controlling infinitely-many moments. We provide a detailed theoretical analysis of multiple VMM estimators, including ones based on kernel methods and neural nets, and provide conditions under which these are consistent, asymptotically normal, and semiparametrically efficient in the full conditional moment model. We additionally provide algorithms for valid statistical inference based on the same kind of variational reformulations, both for kernel- and neural-net-based varieties. Finally, we demonstrate the strong performance of our proposed estimation and inference algorithms in a detailed series of synthetic experiments.