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Shrinkage estimation in the frequency domain of multivariate time series

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  • Bhm, Hilmar
  • von Sachs, Rainer
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    Abstract

    In this paper on developing shrinkage for spectral analysis of multivariate time series of high dimensionality, we propose a new nonparametric estimator of the spectral matrix with two appealing properties. First, compared to the traditional smoothed periodogram our shrinkage estimator has a smaller L2 risk. Second, the proposed shrinkage estimator is numerically more stable due to a smaller condition number. We use the concept of "Kolmogorov" asymptotics where simultaneously the sample size and the dimensionality tend to infinity, to show that the smoothed periodogram is not consistent and to derive the asymptotic properties of our regularized estimator. This estimator is shown to have asymptotically minimal risk among all linear combinations of the identity and the averaged periodogram matrix. Compared to existing work on shrinkage in the time domain, our results show that in the frequency domain it is necessary to take the size of the smoothing span as "effective sample size" into account. Furthermore, we perform extensive Monte Carlo studies showing the overwhelming gain in terms of lower L2 risk of our shrinkage estimator, even in situations of oversmoothing the periodogram by using a large smoothing span.

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    Bibliographic Info

    Article provided by Elsevier in its journal Journal of Multivariate Analysis.

    Volume (Year): 100 (2009)
    Issue (Month): 5 (May)
    Pages: 913-935

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    Handle: RePEc:eee:jmvana:v:100:y:2009:i:5:p:913-935

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    Related research

    Keywords: 62H12 62M10 62M15 Multivariate time series Shrinkage Spectral analysis Regularization Condition number;

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    References

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    1. Jushan Bai & Serena Ng, 2000. "Determining the Number of Factors in Approximate Factor Models," Boston College Working Papers in Economics 440, Boston College Department of Economics.
    2. Mario Forni & Marc Hallin & Marco Lippi & Lucrezia Reichlin, 2000. "The Generalized Dynamic-Factor Model: Identification And Estimation," The Review of Economics and Statistics, MIT Press, vol. 82(4), pages 540-554, November.
    3. Ledoit, Olivier & Wolf, Michael, 2004. "A well-conditioned estimator for large-dimensional covariance matrices," Journal of Multivariate Analysis, Elsevier, vol. 88(2), pages 365-411, February.
    4. Li, Baibing & Martin, Elaine B. & Morris, A. Julian, 2002. "On principal component analysis in L1," Computational Statistics & Data Analysis, Elsevier, vol. 40(3), pages 471-474, September.
    5. Ombao, Hernando & von Sachs, Rainer & Guo, Wensheng, 2005. "SLEX Analysis of Multivariate Nonstationary Time Series," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 519-531, June.
    6. Yin, Y. Q., 1986. "Limiting spectral distribution for a class of random matrices," Journal of Multivariate Analysis, Elsevier, vol. 20(1), pages 50-68, October.
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