Advanced Search
MyIDEAS: Login to save this article or follow this journal

Covariance matrix selection and estimation via penalised normal likelihood


Author Info

  • Jianhua Z. Huang
  • Naiping Liu
  • Mohsen Pourahmadi
  • Linxu Liu
Registered author(s):


    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.

    Download Info

    If you experience problems downloading a file, check if you have the proper application to view it first. In case of further problems read the IDEAS help page. Note that these files are not on the IDEAS site. Please be patient as the files may be large.
    File URL:
    Download Restriction: Access to full text is restricted to subscribers.

    As the access to this document is restricted, you may want to look for a different version under "Related research" (further below) or search for a different version of it.

    Bibliographic Info

    Article provided by Biometrika Trust in its journal Biometrika.

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

    as in new window
    Handle: RePEc:oup:biomet:v:93:y:2006:i:1:p:85-98

    Contact details of provider:
    Postal: Oxford University Press, Great Clarendon Street, Oxford OX2 6DP, UK
    Fax: 01865 267 985
    Web page:

    Order Information:

    Related research



    No references listed on IDEAS
    You can help add them by filling out this form.


    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as in new window

    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,, 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.


    This item is not listed on Wikipedia, on a reading list or among the top items on IDEAS.


    Access and download statistics


    When requesting a correction, please mention this item's handle: RePEc:oup:biomet:v:93:y:2006:i:1:p:85-98. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Oxford University Press) or (Christopher F. Baum).

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If references are entirely missing, you can add them using this form.

    If the full references list an item that is present in RePEc, but the system did not link to it, you can help with this form.

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your profile, as there may be some citations waiting for confirmation.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.