Is completeness necessary? Estimation in nonidentified linear models

TL;DR

This paper explores the necessity of the completeness condition in linear models, demonstrating that regularized estimators can still provide consistent estimates even when identification fails, with nonstandard asymptotic distributions.

Contribution

It develops a comprehensive theory for regularized estimators in nonidentified linear models, including high-dimensional ridge, gradient descent, and PCA methods.

Findings

Regularized estimators can achieve consistent estimation despite identification failure.

Asymptotic distributions in these cases can be nonstandard.

Simulation experiments support the theoretical results.

Abstract

Modern data analysis depends increasingly on estimating models via flexible high-dimensional or nonparametric machine learning methods, where the identification of structural parameters is often challenging and untestable. In linear settings, this identification hinges on the completeness condition, which requires the nonsingularity of a high-dimensional matrix or operator and may fail for finite samples or even at the population level. Regularized estimators provide a solution by enabling consistent estimation of structural or average structural functions, sometimes even under identification failure. We show that the asymptotic distribution in these cases can be nonstandard. We develop a comprehensive theory of regularized estimators, which include methods such as high-dimensional ridge regularization, gradient descent, and principal component analysis (PCA). The results are illustrated for high-dimensional and nonparametric instrumental variable regressions and are supported through simulation experiments.