On rate optimality for ill-posed inverse problems in econometrics

TL;DR

This paper investigates the convergence rates of nonparametric inverse regression models in econometrics, establishing minimax bounds and demonstrating the optimality of certain estimators across various ill-posedness scenarios.

Contribution

It clarifies regularity conditions for convergence rates and proves the minimax optimality of specific estimators in both mildly and severely ill-posed inverse problems.

Findings

Minimax risk lower bounds are established for NPIR and NPIV models.

Projection and sieve estimators achieve these bounds, proving their rate-optimality.

Results hold uniformly over a broad class of functions, covering different ill-posedness levels.

Abstract

In this paper, we clarify the relations between the existing sets of regularity conditions for convergence rates of nonparametric indirect regression (NPIR) and nonparametric instrumental variables (NPIV) regression models. We establish minimax risk lower bounds in mean integrated squared error loss for the NPIR and the NPIV models under two basic regularity conditions that allow for both mildly ill-posed and severely ill-posed cases. We show that both a simple projection estimator for the NPIR model, and a sieve minimum distance estimator for the NPIV model, can achieve the minimax risk lower bounds, and are rate-optimal uniformly over a large class of structure functions, allowing for mildly ill-posed and severely ill-posed cases.