Is completeness necessary? Estimation in nonidentified linear models

Understand

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.