Two Approaches to Direct Estimation of Riesz Representers

TL;DR

This paper compares two methods for directly estimating Riesz representers in semiparametric statistics, revealing their equivalence under certain models and proposing a novel constrained optimization approach for complex machine learning models.

Contribution

It demonstrates the equivalence of two estimation approaches under linear and kernel models and introduces a new constrained optimization method for neural networks and Lasso.

Findings

Estimators are equivalent for linear, sieve, and RKHS models.

Different regularization schemes lead to non-equivalent estimators.

A new constrained optimization approach for complex models is proposed.

Abstract

The Riesz representer is a central object in semiparametric statistics and debiased/doubly-robust estimation. Two literatures in econometrics have highlighted the role for directly estimating Riesz representers: the automatic debiased machine learning literature (as in Chernozhukov et al., 2022b), and an independent literature on sieve methods for conditional moment models (as in Chen et al., 2014). These two literatures solve distinct optimization problems that in the population both have the Riesz representer as their solution. We show that with unregularized or ridge-regularized linear, sieve, or RKHS models, the two resulting estimators are numerically equivalent. However, for other regularization schemes such as the Lasso, or more general machine learning function classes including neural networks, the estimators are not necessarily equivalent. In the latter case, the Chen et al. (2014) formulation yields a novel constrained optimization problem for directly estimating Riesz representers with machine learning. Drawing on results from Birrell et al. (2022), we conjecture that this approach may offer statistical advantages at the cost of greater computational complexity.