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High Dimensional Correlation Matrices: CLT and Its Applications

Author

Listed:
  • Jiti Gao
  • Xiao Han
  • Guangming Pan
  • Yanrong Yang

Abstract

Statistical inferences for sample correlation matrices are important in high dimensional data analysis. Motivated by this, this paper establishes a new central limit theorem (CLT) for a linear spectral statistic (LSS) of high dimensional sample correlation matrices for the case where the dimension p and the sample size n are comparable. This result is of independent interest in large dimensional random matrix theory. Meanwhile, we apply the linear spectral statistic to an independence test for p random variables, and then an equivalence test for p factor loadings and n factors in a factor model. The finite sample performance of the proposed test shows its applicability and effectiveness in practice. An empirical application to test the independence of household incomes from different cities in China is also conducted.

Suggested Citation

  • Jiti Gao & Xiao Han & Guangming Pan & Yanrong Yang, 2014. "High Dimensional Correlation Matrices: CLT and Its Applications," Monash Econometrics and Business Statistics Working Papers 26/14, Monash University, Department of Econometrics and Business Statistics.
    • Handle: RePEc:msh:ebswps:2014-26
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    File URL: http://business.monash.edu/econometrics-and-business-statistics/research/publications/ebs/wp26-14.pdf
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    References listed on IDEAS

    as
    1. James R. Schott, 2005. "Testing for complete independence in high dimensions," Biometrika, Biometrika Trust, vol. 92(4), pages 951-956, December.
    2. Silverstein, J. W., 1995. "Strong Convergence of the Empirical Distribution of Eigenvalues of Large Dimensional Random Matrices," Journal of Multivariate Analysis, Elsevier, vol. 55(2), pages 331-339, November.
    3. Bai, Z. D. & Silverstein, Jack W. & Yin, Y. Q., 1988. "A note on the largest eigenvalue of a large dimensional sample covariance matrix," Journal of Multivariate Analysis, Elsevier, vol. 26(2), pages 166-168, August.
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    Cited by:

    1. Guangming Pan & Jiti Gao & Yanrong Yang & Meihui Guo, 2015. "Cross-sectional Independence Test for a Class of Parametric Panel Data Models," Monash Econometrics and Business Statistics Working Papers 17/15, Monash University, Department of Econometrics and Business Statistics.

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    More about this item

    Keywords

    Central limit theorem; equivalence test; high dimensional correlation matrix; independence test; linear spectral statistics.;
    All these keywords.

    JEL classification:

    • C21 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Cross-Sectional Models; Spatial Models; Treatment Effect Models
    • C32 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes; State Space Models

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