Nonparametric Instrumental Variable Analysis Without Structural Equations: Debiased Inference on Functionals of Inverse Problems with No Solutions

TL;DR

This paper introduces a debiased inference method for functionals of inverse problems that does not require the existence of exact solutions, enhancing robustness in instrumental variable analysis.

Contribution

It proposes a novel approach for inference on inverse problem functionals that remains valid without assuming the existence of exact solutions, applicable to instrumental variables.

Findings

Allows inference without assuming exact solutions exist

Maintains validity of semiparametric inference under model misspecification

Provides a robust framework for instrumental variable analysis

Abstract

We consider debiased inference on finite-dimensional functionals of infinite-dimensional least-squares solutions to inverse problems as a way to avoid having to assume exact solutions exist. Such assumptions are substantive and not innocuous, and their failure may imperil inference when we impose them on the statistical model. Our approach instead allows us to conduct inference on a quantity that is defined regardless of solutions existing and coincides with the usual estimands when they do. For the case of instrumental variables, this means we can motivate the analysis with structural models but these do not need to hold exactly for the semiparametric inferential procedure to remain valid.