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Shrinkage estimation in the frequency domain of multivariate time series

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  • Bhm, Hilmar
  • von Sachs, Rainer

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.

Suggested Citation

  • Bhm, Hilmar & von Sachs, Rainer, 2009. "Shrinkage estimation in the frequency domain of multivariate time series," Journal of Multivariate Analysis, Elsevier, vol. 100(5), pages 913-935, May.
    • Handle: RePEc:eee:jmvana:v:100:y:2009:i:5:p:913-935
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    References listed on IDEAS

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    1. 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.
    2. Jushan Bai & Serena Ng, 2002. "Determining the Number of Factors in Approximate Factor Models," Econometrica, Econometric Society, vol. 70(1), pages 191-221, January.
    3. 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.
    4. 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.
    5. 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.
    6. 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.
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    Cited by:

    1. Mark Fiecas & Jürgen Franke & Rainer von Sachs & Joseph Tadjuidje Kamgaing, 2017. "Shrinkage Estimation for Multivariate Hidden Markov Models," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 112(517), pages 424-435, January.
    2. Fiecas, Mark & von Sachs, Rainer, 2012. "Spectral density shrinkage for high-dimensional time series," LIDAM Discussion Papers ISBA 2012037, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    3. von Sachs, Rainer, 2019. "Spectral Analysis of Multivariate Time Series," LIDAM Discussion Papers ISBA 2019008, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    4. Fiecas, Mark & von Sachs, Rainer, 2013. "Data-driven Shrinkage of the Spectral Density Matrix of a High-dimensional Time Series," LIDAM Discussion Papers ISBA 2013044, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).

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