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Testing Independence Among a Large Number of High-Dimensional Random Vectors

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  • Guangming Pan
  • Jiti Gao
  • Yanrong Yang

Abstract

Capturing dependence among a large number of high-dimensional random vectors is a very important and challenging problem. By arranging n random vectors of length p in the form of a matrix, we develop a linear spectral statistic of the constructed matrix to test whether the n random vectors are independent or not. Specifically, the proposed statistic can also be applied to n random vectors, each of whose elements can be written as either a linear stationary process or a linear combination of independent random variables. The asymptotic distribution of the proposed test statistic is established for the case of as n → ∞. To avoid estimating the spectrum of each random vector, a modified test statistic, which is based on splitting the original n vectors into two equal parts and eliminating the term that contains the inner structure of each random vector or time series, is constructed. The facts that the limiting distribution is normal and there is no need to know the inner structure of each investigated random vector result in simple implementation of the constructed test statistic. Simulation results demonstrate that the proposed test is powerful against several commonly used dependence structures. An empirical application to detecting dependence of the closed prices from several stocks in the S&P500 also illustrates the applicability and effectiveness of our provided test. Supplementary materials for this article are available online.

Suggested Citation

  • Guangming Pan & Jiti Gao & Yanrong Yang, 2014. "Testing Independence Among a Large Number of High-Dimensional Random Vectors," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 109(506), pages 600-612, June.
  • Handle: RePEc:taf:jnlasa:v:109:y:2014:i:506:p:600-612
    DOI: 10.1080/01621459.2013.872037
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    References listed on IDEAS

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    1. M. Hashem Pesaran, 2021. "General diagnostic tests for cross-sectional dependence in panels," Empirical Economics, Springer, vol. 60(1), pages 13-50, January.
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    Citations

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    Cited by:

    1. Merlevède, F. & Peligrad, M., 2016. "On the empirical spectral distribution for matrices with long memory and independent rows," Stochastic Processes and their Applications, Elsevier, vol. 126(9), pages 2734-2760.
    2. Loubaton, Philippe & Rosuel, Alexis & Vallet, Pascal, 2023. "On the asymptotic distribution of the maximum sample spectral coherence of Gaussian time series in the high dimensional regime," Journal of Multivariate Analysis, Elsevier, vol. 194(C).
    3. Bo Zhang & Guangming Pan & Jiti Gao, 2016. "CLT for Largest Eigenvalues and Unit Root Tests for High-Dimensional Nonstationary Time Series," Monash Econometrics and Business Statistics Working Papers 11/16, Monash University, Department of Econometrics and Business Statistics.
    4. Bo Zhang & Jiti Gao & Guangming Pan, 2020. "Estimation and Testing for High-Dimensional Near Unit Root Time Series," Monash Econometrics and Business Statistics Working Papers 12/20, Monash University, Department of Econometrics and Business Statistics.
    5. Yayi Yan & Jiti Gao & Bin Peng, 2020. "A Class of Time-Varying Vector Moving Average Models: Nonparametric Kernel Estimation and Application," Papers 2010.01492, arXiv.org.
    6. Bo Zhang & Jiti Gao & Guangming Pan & Yanrong Yang, 2019. "Spiked Eigenvalues of High-Dimensional Separable Sample Covariance Matrices," Monash Econometrics and Business Statistics Working Papers 31/19, Monash University, Department of Econometrics and Business Statistics.
    7. He, Yi & Jaidee, Sombut & Gao, Jiti, 2023. "Most powerful test against a sequence of high dimensional local alternatives," Journal of Econometrics, Elsevier, vol. 234(1), pages 151-177.
    8. Jiti Gao & Xiao Han & Guangming Pan & Yanrong Yang, 2017. "High dimensional correlation matrices: the central limit theorem and its applications," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(3), pages 677-693, June.
    9. Yi He & Sombut Jaidee & Jiti Gao, 2020. "Most Powerful Test against High Dimensional Free Alternatives," Monash Econometrics and Business Statistics Working Papers 13/20, Monash University, Department of Econometrics and Business Statistics.
    10. Yayi Yan & Jiti Gao & Bin peng, 2020. "A Class of Time-Varying Vector Moving Average (infinity) Models," Monash Econometrics and Business Statistics Working Papers 39/20, Monash University, Department of Econometrics and Business Statistics.
    11. Bo Zhang & Jiti Gao & Guangming Pan, 2019. "A Near Unit Root Test for High-Dimensional Nonstationary Time Series," Monash Econometrics and Business Statistics Working Papers 10/19, Monash University, Department of Econometrics and Business Statistics.

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