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On singular values of large dimensional lag- sample auto-correlation matrices.

Zhanting Long1, Zeng Li1, Ruitao Lin2

  • 1Southern University of Science and Technology.

Journal of Multivariate Analysis
|June 30, 2023
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Summary
This summary is machine-generated.

This study analyzes singular values of auto-correlation matrices in high-dimensional factor models. We establish their limiting spectral distribution and largest singular value, aiding in factor number estimation.

Keywords:
Auto-correlation matrixAuto-covariance matrixLargest eigenvalueLimiting spectral distributionPrimary 60B20Random matrix theorySecondary 62H25

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Area of Science:

  • Statistics
  • Econometrics
  • Time Series Analysis

Background:

  • High-dimensional factor models are crucial for analyzing large datasets.
  • Understanding the behavior of auto-correlation matrices is key in these models.
  • Previous research has focused on auto-covariance matrices, with less attention to auto-correlation.

Purpose of the Study:

  • To investigate the limiting spectral distribution (LSD) of lag-k sample auto-correlation matrices.
  • To determine the asymptotic behavior of the largest singular value of these matrices.
  • To develop estimators for the total number of factors using auto-correlation matrices.

Main Methods:

  • Derivation of asymptotic results under a high-dimensional regime (dimension and sample size grow proportionally).
  • Establishing the limiting spectral distribution (LSD) of the auto-correlation matrix.
  • Analyzing the convergence of the largest singular value.

Main Results:

  • The LSD of the lag-k sample auto-correlation matrix is shown to be identical to that of the lag-k sample auto-covariance matrix.
  • The largest singular value converges almost surely to the endpoint of the LSD's support.
  • Two novel estimators for the total number of factors are proposed.

Conclusions:

  • The asymptotic equivalence between auto-correlation and auto-covariance matrices simplifies analysis.
  • The derived results provide a theoretical foundation for factor number estimation.
  • Numerical experiments validate the theoretical findings and the proposed estimators.