Adaptive Estimation and Inference in Conditional Moment Models via the Discrepancy Principle

TL;DR

This paper introduces a discrepancy-principle-based adaptive framework for hyperparameter tuning in ill-posed conditional moment models, enabling optimal inference without prior smoothness knowledge.

Contribution

It proposes a novel adaptive hyperparameter selection method applicable to existing estimators, achieving optimal rates in ill-posed inverse problems without smoothness assumptions.

Findings

Framework applies to RDIV and TRAE estimators

Achieves optimal convergence rates in weak and strong metrics

Constructs a fully adaptive doubly robust estimator

Abstract

We study adaptive estimation and inference in ill-posed linear inverse problems defined by conditional moment restrictions. Existing regularized estimators such as Regularized DeepIV (RDIV) require prior knowledge of the smoothness of the nuisance function, typically encoded by a beta source condition to tune their regularization parameters. In practice, this smoothness is unknown, and misspecified hyperparameters can lead to suboptimal convergence or instability. We introduce a discrepancy-principle-based framework for adaptive hyperparameter selection that automatically balances bias and variance without relying on the unknown smoothness parameter. Our framework applies to both RDIV (Li et al. [2024]) and the Tikhonov Regularized Adversarial Estimator (TRAE) (Bennett et al. [2023a]) and achieves the same rates in both weak and strong metrics. Building on this, we construct a fully adaptive doubly robust estimator for linear functionals that attains the optimal rate of the better-conditioned primal or dual problem, providing a practical, theoretically grounded approach for adaptive inference in ill-posed econometric models.