Asymptotic Normality in Linear Regression with Approximately Sparse Structure

Saulius Jokubaitis and Remigijus Leipus
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Saulius Jokubaitis: Faculty of Mathematics and Informatics, Institute of Applied Mathematics, Vilnius University, Naugarduko 24, LT-03225 Vilnius, Lithuania
Remigijus Leipus: Faculty of Mathematics and Informatics, Institute of Applied Mathematics, Vilnius University, Naugarduko 24, LT-03225 Vilnius, Lithuania

Mathematics, 2022, vol. 10, issue 10, 1-28

Abstract: In this paper, we study the asymptotic normality in high-dimensional linear regression. We focus on the case where the covariance matrix of the regression variables has a KMS structure, in asymptotic settings where the number of predictors, p , is proportional to the number of observations, n . The main result of the paper is the derivation of the exact asymptotic distribution for the suitably centered and normalized squared norm of the product between predictor matrix, X , and outcome variable, Y , i.e., the statistic ∥ X ′ Y ∥ 2 2 , under rather unrestrictive assumptions for the model parameters β j . We employ variance-gamma distribution in order to derive the results, which, along with the asymptotic results, allows us to easily define the exact distribution of the statistic. Additionally, we consider a specific case of approximate sparsity of the model parameter vector β and perform a Monte Carlo simulation study. The simulation results suggest that the statistic approaches the limiting distribution fairly quickly even under high variable multi-correlation and relatively small number of observations, suggesting possible applications to the construction of statistical testing procedures for the real-world data and related problems.

Keywords: linear regression; sparsity; asymptotic normality; variance-gamma distribution (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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