Iterative Estimation of Solutions to Noisy Nonlinear Operator Equations in Nonparametric Instrumental Regression

TL;DR

This paper introduces a regularized Newton-type iterative method for solving noisy nonlinear integral equations in nonparametric instrumental regression, enabling the estimation of non-identifiable models under stronger independence assumptions.

Contribution

It proposes a novel iterative regularization approach with convergence guarantees for solving noisy nonlinear operator equations in nonparametric instrumental regression.

Findings

The method converges with established rates.

It can estimate non-identifiable models with binary instruments.

Validated through simulated data examples.

Abstract

This paper discusses the solution of nonlinear integral equations with noisy integral kernels as they appear in nonparametric instrumental regression. We propose a regularized Newton-type iteration and establish convergence and convergence rate results. A particular emphasis is on instrumental regression models where the usual conditional mean assumption is replaced by a stronger independence assumption. We demonstrate for the case of a binary instrument that our approach allows the correct estimation of regression functions which are not identifiable with the standard model. This is illustrated in computed examples with simulated data.