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Covariance matrix selection and estimation via penalised normal likelihood

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  • Jianhua Z. Huang
  • Naiping Liu
  • Mohsen Pourahmadi
  • Linxu Liu
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    Abstract

    We propose a nonparametric method for identifying parsimony and for producing a statistically efficient estimator of a large covariance matrix. We reparameterise a covariance matrix through the modified Cholesky decomposition of its inverse or the one-step-ahead predictive representation of the vector of responses and reduce the nonintuitive task of modelling covariance matrices to the familiar task of model selection and estimation for a sequence of regression models. The Cholesky factor containing these regression coefficients is likely to have many off-diagonal elements that are zero or close to zero. Penalised normal likelihoods in this situation with L-sub-1 and L-sub-2 penalities are shown to be closely related to Tibshirani's (1996) LASSO approach and to ridge regression. Adding either penalty to the likelihood helps to produce more stable estimators by introducing shrinkage to the elements in the Cholesky factor, while, because of its singularity, the L-sub-1 penalty will set some elements to zero and produce interpretable models. An algorithm is developed for computing the estimator and selecting the tuning parameter. The proposed maximum penalised likelihood estimator is illustrated using simulation and a real dataset involving estimation of a 102 � 102 covariance matrix. Copyright 2006, Oxford University Press.

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    Bibliographic Info

    Article provided by Biometrika Trust in its journal Biometrika.

    Volume (Year): 93 (2006)
    Issue (Month): 1 (March)
    Pages: 85-98

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    Handle: RePEc:oup:biomet:v:93:y:2006:i:1:p:85-98

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    Cited by:
    1. Natalia Bailey & M. Hashem Pesaran & L. Vanessa Smith, 2014. "A Multiple Testing Approach to the Regularisation of Large Sample Correlation Matrices," CESifo Working Paper Series 4834, CESifo Group Munich.
    2. Pesaran, M. H. & Yamagata, T., 2012. "Testing CAPM with a Large Number of Assets (Updated 28th March 2012)," Cambridge Working Papers in Economics 1210, Faculty of Economics, University of Cambridge.
    3. Verzelen, N. & Villers, F., 2009. "Tests for Gaussian graphical models," Computational Statistics & Data Analysis, Elsevier, vol. 53(5), pages 1894-1905, March.
    4. Fan, Jianqing & Fan, Yingying & Lv, Jinchi, 2008. "High dimensional covariance matrix estimation using a factor model," Journal of Econometrics, Elsevier, vol. 147(1), pages 186-197, November.
    5. Clifford Lam & Jianqing Fan, 2009. "Sparsistency and rates of convergence in large covariance matrix estimation," LSE Research Online Documents on Economics 31540, London School of Economics and Political Science, LSE Library.
    6. Chen, Songxi, 2012. "Two Sample Tests for High Dimensional Covariance Matrices," MPRA Paper 46026, University Library of Munich, Germany.
    7. M Hashem Pesaran & Takashi Yamagata, 2012. "Testing CAPM with a Large Number of Assets," Discussion Papers 12/05, Department of Economics, University of York.
    8. Natalia Bailey & Vanessa Smith & Hashem Pesaran, 2014. "A multiple testing approach to the regularisation of large sample correlation matrices," Cambridge Working Papers in Economics 1413, Faculty of Economics, University of Cambridge.
    9. Fisher, Thomas J. & Sun, Xiaoqian, 2011. "Improved Stein-type shrinkage estimators for the high-dimensional multivariate normal covariance matrix," Computational Statistics & Data Analysis, Elsevier, vol. 55(5), pages 1909-1918, May.
    10. Xi Luo, 2011. "Recovering Model Structures from Large Low Rank and Sparse Covariance Matrix Estimation," Papers 1111.1133, arXiv.org, revised Mar 2013.
    11. Song Liu & Yuhong Yang, 2012. "Combining models in longitudinal data analysis," Annals of the Institute of Statistical Mathematics, Springer, vol. 64(2), pages 233-254, April.
    12. Xue, Lingzhou & Zou, Hui, 2013. "Minimax optimal estimation of general bandable covariance matrices," Journal of Multivariate Analysis, Elsevier, vol. 116(C), pages 45-51.
    13. Daye, Z. John & Jeng, X. Jessie, 2009. "Shrinkage and model selection with correlated variables via weighted fusion," Computational Statistics & Data Analysis, Elsevier, vol. 53(4), pages 1284-1298, February.
    14. Clifford Lam, 2008. "Estimation of large precision matrices through block penalization," LSE Research Online Documents on Economics 31543, London School of Economics and Political Science, LSE Library.

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