Is completeness necessary? Estimation in nonidentified linear models
This work addresses a fundamental problem in econometrics and machine learning for researchers dealing with nonidentified models, offering a theoretical framework for practical estimation.
The paper tackles the challenge of estimating structural parameters in linear models when identification fails due to incomplete data, showing that regularized estimators can provide consistent estimation even under such conditions, with results illustrated through simulations.
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.