2013/09/17 by Ying Zhu, Zhu, Ying
Computer Science · Engineering · Mathematics · #Distributed Sensor Networks and Detection Algorithms #FOS: Mathematics #Sparse and Compressive Sensing Techniques #Statistical Methods and Inference #Statistics Theory (math.ST) #math.ST #stat.TH
paper · pdf · doi:10.48550/arxiv.1309.4193
arxiv created 2013/09/17 · openalex publication_date 2013/09/17 · arxiv updated 2013/09/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper explores the validity of the two-stage estimation procedure for sparse linear models in high-dimensional settings with possibly many endogenous regressors. In particular, the number of endogenous regressors in the main equation and the instruments in the first-stage equations can grow with and exceed the sample size n. The analysis concerns the exact sparsity case, i.e., the maximum number of non-zero components in the vectors of parameters in the first-stage equations, k1, and the number of non-zero components in the vector of parameters in the second-stage equation, k2, are allowed to grow with n but slowly compared to n. I consider the high-dimensional version of the two-stage least square estimator where one obtains the fitted regressors from the first-stage regression by a least square estimator with l1-regularization (the Lasso or Dantzig selector) when the first-stage regression concerns a large number of instruments relative to n, and then construct a similar estimator using these fitted regressors in the second-stage regression. The main theoretical results of this paper are non-asymptotic bounds from which I establish sufficient scaling conditions on the sample size for estimation consistency in l2-norm and variable-selection consistency. A technical issue regarding the so-called "restricted eigenvalue (RE) condition" for estimation consistency and the "mutual incoherence (MI) condition" for selection consistency arises in the two-stage estimation from allowing the number of regressors in the main equation to exceed n and this paper provides analysis to verify these RE and MI conditions. Depending on the underlying assumptions, the upper bounds on the l2-error and the sample size required to obtain these consistency results differ by factors involving k1 and/or k2. Simulations are conducted to gain insight on the finite sample performance.