Minimax Instrumental Variable Regression and $L_2$ Convergence Guarantees without Identification or Closedness

TL;DR

This paper introduces a new penalized minimax estimator for nonparametric instrumental variable regression that achieves $L_2$ convergence guarantees without requiring identification, closedness, or smoothness conditions, broadening applicability.

Contribution

It proposes the first method to avoid the limitations of identification, closedness, and metric restrictions in IV regression, enabling convergence analysis under weaker assumptions.

Findings

Achieves $L_2$ error bounds under lax conditions.

Handles multiple solutions in IV estimation.

Does not require the closedness condition.

Abstract

In this paper, we study nonparametric estimation of instrumental variable (IV) regressions. Recently, many flexible machine learning methods have been developed for instrumental variable estimation. However, these methods have at least one of the following limitations: (1) restricting the IV regression to be uniquely identified; (2) only obtaining estimation error rates in terms of pseudometrics (\emph{e.g.,} projected norm) rather than valid metrics (\emph{e.g.,} $L_2$ norm); or (3) imposing the so-called closedness condition that requires a certain conditional expectation operator to be sufficiently smooth. In this paper, we present the first method and analysis that can avoid all three limitations, while still permitting general function approximation. Specifically, we propose a new penalized minimax estimator that can converge to a fixed IV solution even when there are multiple solutions, and we derive a strong $L_2$ error rate for our estimator under lax conditions. Notably, this guarantee only needs a widely-used source condition and realizability assumptions, but not the so-called closedness condition. We argue that the source condition and the closedness condition are inherently conflicting, so relaxing the latter significantly improves upon the existing literature that requires both conditions. Our estimator can achieve this improvement because it builds on a novel formulation of the IV estimation problem as a constrained optimization problem.