# Is completeness necessary? Estimation in nonidentified linear models

arXiv: https://arxiv.org/abs/1709.03473 · HTML: https://mlsanity.com/paper/1709.03473/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Sep 2017.

## Built on by (most cited first, 6 of 6)

- [Minimax Instrumental Variable Regression and L2 Convergence Guarantees without Identification or Closedness](https://mlsanity.com/paper/2302.05404/) (2023)
- [Inference on Strongly Identified Functionals of Weakly Identified Functions](https://mlsanity.com/paper/2208.08291/) (2022)
- [Nonparametric Instrumental Regression via Kernel Methods is Minimax Optimal](https://mlsanity.com/paper/2411.19653/) (2024)
- [Stochastic Optimization Algorithms for Instrumental Variable Regression with Streaming Data](https://mlsanity.com/paper/2405.19463/) (2024)
- [Econometrics of Machine Learning Methods in Economic Forecasting](https://mlsanity.com/paper/2308.10993/) (2023)
- [Functional Partial Least-Squares: Adaptive Estimation and Inference](https://mlsanity.com/paper/2402.11134/) (2024)

## Abstract (arXiv)

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.