On Singular Values of Large Dimensional Lag- τ Sample Auto-Correlation MatricesJournal of Multivariate Analysis

Authors

Zhanting Long
Zeng Li
Ruitao Lin
Jiaxin Qiu

Publication Date

9-1-2023

Journal

Journal of Multivariate Analysis

Abstract

We study the limiting behavior of singular values of a lag-τ" role="presentation" style="box-sizing: inherit; display: inline-block; line-height: 0; font-size: 18.08px; font-size-adjust: none; overflow-wrap: normal; text-wrap-mode: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; margin: 0px; padding: 1px 0px; color: rgb(33, 33, 33); font-family: BlinkMacSystemFont, -apple-system, "Segoe UI", Roboto, Oxygen, Ubuntu, Cantarell, "Fira Sans", "Droid Sans", "Helvetica Neue", sans-serif; position: relative;">ττ sample auto-correlation matrix Rτϵ" role="presentation" style="box-sizing: inherit; display: inline-block; line-height: 0; font-size: 18.08px; font-size-adjust: none; overflow-wrap: normal; text-wrap-mode: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; margin: 0px; padding: 1px 0px; color: rgb(33, 33, 33); font-family: BlinkMacSystemFont, -apple-system, "Segoe UI", Roboto, Oxygen, Ubuntu, Cantarell, "Fira Sans", "Droid Sans", "Helvetica Neue", sans-serif; position: relative;">RϵτRτϵ of large dimensional vector white noise process, the error term ϵ" role="presentation" style="box-sizing: inherit; display: inline-block; line-height: 0; font-size: 18.08px; font-size-adjust: none; overflow-wrap: normal; text-wrap-mode: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; margin: 0px; padding: 1px 0px; color: rgb(33, 33, 33); font-family: BlinkMacSystemFont, -apple-system, "Segoe UI", Roboto, Oxygen, Ubuntu, Cantarell, "Fira Sans", "Droid Sans", "Helvetica Neue", sans-serif; position: relative;">ϵϵ in the high-dimensional factor model. We establish the limiting spectral distribution (LSD) that characterizes the global spectrum of Rτϵ" role="presentation" style="box-sizing: inherit; display: inline-block; line-height: 0; font-size: 18.08px; font-size-adjust: none; overflow-wrap: normal; text-wrap-mode: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; margin: 0px; padding: 1px 0px; color: rgb(33, 33, 33); font-family: BlinkMacSystemFont, -apple-system, "Segoe UI", Roboto, Oxygen, Ubuntu, Cantarell, "Fira Sans", "Droid Sans", "Helvetica Neue", sans-serif; position: relative;">RϵτRτϵ, and derive the limit of its largest singular value. All the asymptotic results are derived under the high-dimensional asymptotic regime where the data dimension and sample size go to infinity proportionally. Under mild assumptions, we show that the LSD of Rτϵ" role="presentation" style="box-sizing: inherit; display: inline-block; line-height: 0; font-size: 18.08px; font-size-adjust: none; overflow-wrap: normal; text-wrap-mode: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; margin: 0px; padding: 1px 0px; color: rgb(33, 33, 33); font-family: BlinkMacSystemFont, -apple-system, "Segoe UI", Roboto, Oxygen, Ubuntu, Cantarell, "Fira Sans", "Droid Sans", "Helvetica Neue", sans-serif; position: relative;">RϵτRτϵ is the same as that of the lag-τ" role="presentation" style="box-sizing: inherit; display: inline-block; line-height: 0; font-size: 18.08px; font-size-adjust: none; overflow-wrap: normal; text-wrap-mode: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; margin: 0px; padding: 1px 0px; color: rgb(33, 33, 33); font-family: BlinkMacSystemFont, -apple-system, "Segoe UI", Roboto, Oxygen, Ubuntu, Cantarell, "Fira Sans", "Droid Sans", "Helvetica Neue", sans-serif; position: relative;">ττ sample auto-covariance matrix. Based on this asymptotic equivalence, we additionally show that the largest singular value of Rτϵ" role="presentation" style="box-sizing: inherit; display: inline-block; line-height: 0; font-size: 18.08px; font-size-adjust: none; overflow-wrap: normal; text-wrap-mode: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; margin: 0px; padding: 1px 0px; color: rgb(33, 33, 33); font-family: BlinkMacSystemFont, -apple-system, "Segoe UI", Roboto, Oxygen, Ubuntu, Cantarell, "Fira Sans", "Droid Sans", "Helvetica Neue", sans-serif; position: relative;">RϵτRτϵ converges almost surely to the right end point of the support of its LSD. Based on these results, we further propose two estimators of total number of factors with lag-τ" role="presentation" style="box-sizing: inherit; display: inline-block; line-height: 0; font-size: 18.08px; font-size-adjust: none; overflow-wrap: normal; text-wrap-mode: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; margin: 0px; padding: 1px 0px; color: rgb(33, 33, 33); font-family: BlinkMacSystemFont, -apple-system, "Segoe UI", Roboto, Oxygen, Ubuntu, Cantarell, "Fira Sans", "Droid Sans", "Helvetica Neue", sans-serif; position: relative;">ττ sample auto-correlation matrices in a factor model. Our theoretical results are fully supported by numerical experiments as well.

Keywords

Auto-correlation matrix, Auto-covariance matrix, Largest eigenvalue, Limiting spectral distribution, Random matrix theory

DOI

10.1016/j.jmva.2023.105205

PMID

37388905

PMCID

PMC10306325

PubMedCentral® Posted Date

9-1-2024

PubMedCentral® Full Text Version

Author MSS

Published Open-Access

yes