IDEAS home Printed from https://ideas.repec.org/a/eee/jmvana/v98y2007i3p568-587.html
   My bibliography  Save this article

Simultaneous modelling of the Cholesky decomposition of several covariance matrices

Author

Listed:
  • Pourahmadi, Mohsen
  • Daniels, Michael J.
  • Park, Trevor

Abstract

A method for simultaneous modelling of the Cholesky decomposition of several covariance matrices is presented. We highlight the conceptual and computational advantages of the unconstrained parameterization of the Cholesky decomposition and compare the results with those obtained using the classical spectral (eigenvalue) and variance-correlation decompositions. All these methods amount to decomposing complicated covariance matrices into "dependence" and "variance" components, and then modelling them virtually separately using regression techniques. The entries of the "dependence" component of the Cholesky decomposition have the unique advantage of being unconstrained so that further reduction of the dimension of its parameter space is fairly simple. Normal theory maximum likelihood estimates for complete and incomplete data are presented using iterative methods such as the EM (Expectation-Maximization) algorithm and their improvements. These procedures are illustrated using a dataset from a growth hormone longitudinal clinical trial.

Suggested Citation

  • Pourahmadi, Mohsen & Daniels, Michael J. & Park, Trevor, 2007. "Simultaneous modelling of the Cholesky decomposition of several covariance matrices," Journal of Multivariate Analysis, Elsevier, vol. 98(3), pages 568-587, March.
    • Handle: RePEc:eee:jmvana:v:98:y:2007:i:3:p:568-587
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0047-259X(05)00197-1
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. W. J. Krzanowski, 1984. "Principal Component Analysis in the Presence of Group Structure," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 33(2), pages 164-168, June.
    2. P. Diggle & M. G. Kenward, 1994. "Informative Drop‐Out in Longitudinal Data Analysis," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 43(1), pages 49-73, March.
    3. Michael J. Daniels & Joseph W. Hogan, 2000. "Reparameterizing the Pattern Mixture Model for Sensitivity Analyses Under Informative Dropout," Biometrics, The International Biometric Society, vol. 56(4), pages 1241-1248, December.
    4. Bollerslev, Tim, 1990. "Modelling the Coherence in Short-run Nominal Exchange Rates: A Multivariate Generalized ARCH Model," The Review of Economics and Statistics, MIT Press, vol. 72(3), pages 498-505, August.
    5. Raymond J. Carroll, 2003. "Variances Are Not Always Nuisance Parameters," Biometrics, The International Biometric Society, vol. 59(2), pages 211-220, June.
    6. Michael J. Daniels, 2002. "Bayesian analysis of covariance matrices and dynamic models for longitudinal data," Biometrika, Biometrika Trust, vol. 89(3), pages 553-566, August.
    7. Robert J. Boik, 2003. "Principal component models for correlation matrices," Biometrika, Biometrika Trust, vol. 90(3), pages 679-701, September.
    8. Robert J. Boik, 2002. "Spectral models for covariance matrices," Biometrika, Biometrika Trust, vol. 89(1), pages 159-182, March.
    9. Fraley C. & Raftery A.E., 2002. "Model-Based Clustering, Discriminant Analysis, and Density Estimation," Journal of the American Statistical Association, American Statistical Association, vol. 97, pages 611-631, June.
    Full references (including those not matched with items on IDEAS)

    Citations

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


    Cited by:

    1. Zongliang Hu & Zhishui Hu & Kai Dong & Tiejun Tong & Yuedong Wang, 2021. "A shrinkage approach to joint estimation of multiple covariance matrices," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 84(3), pages 339-374, April.
    2. Pousinho, H.M.I. & Silva, H. & Mendes, V.M.F. & Collares-Pereira, M. & Pereira Cabrita, C., 2014. "Self-scheduling for energy and spinning reserve of wind/CSP plants by a MILP approach," Energy, Elsevier, vol. 78(C), pages 524-534.
    3. Paul D. McNicholas, 2016. "Model-Based Clustering," Journal of Classification, Springer;The Classification Society, vol. 33(3), pages 331-373, October.
    4. Chi, Eric C. & Lange, Kenneth, 2014. "Stable estimation of a covariance matrix guided by nuclear norm penalties," Computational Statistics & Data Analysis, Elsevier, vol. 80(C), pages 117-128.
    5. Luca Bagnato & Antonio Punzo, 2021. "Unconstrained representation of orthogonal matrices with application to common principal components," Computational Statistics, Springer, vol. 36(2), pages 1177-1195, June.
    6. A. R. Soltani & A. R. Nematollahi & M. R. Mahmoudi, 2019. "On the asymptotic distribution of the periodograms for the discrete time harmonizable simple processes," Statistical Inference for Stochastic Processes, Springer, vol. 22(2), pages 307-322, July.
    7. 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.
    8. Wagner Hugo Bonat & Bent Jørgensen, 2016. "Multivariate covariance generalized linear models," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 65(5), pages 649-675, November.
    9. Paolo Giordani & Xiuyan Mun & Robert Kohn, 2012. "Efficient Estimation of Covariance Matrices using Posterior Mode Multiple Shrinkage," Journal of Financial Econometrics, Oxford University Press, vol. 11(1), pages 154-192, December.
    10. Joong-Ho Won & Johan Lim & Seung-Jean Kim & Bala Rajaratnam, 2013. "Condition-number-regularized covariance estimation," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 75(3), pages 427-450, June.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Daniels, Michael J., 2006. "Bayesian modeling of several covariance matrices and some results on propriety of the posterior for linear regression with correlated and/or heterogeneous errors," Journal of Multivariate Analysis, Elsevier, vol. 97(5), pages 1185-1207, May.
    2. Joseph W. Hogan & Xihong Lin & Benjamin Herman, 2004. "Mixtures of Varying Coefficient Models for Longitudinal Data with Discrete or Continuous Nonignorable Dropout," Biometrics, The International Biometric Society, vol. 60(4), pages 854-864, December.
    3. Li, Degui, 2024. "Estimation of Large Dynamic Covariance Matrices: A Selective Review," Econometrics and Statistics, Elsevier, vol. 29(C), pages 16-30.
    4. Zongliang Hu & Zhishui Hu & Kai Dong & Tiejun Tong & Yuedong Wang, 2021. "A shrinkage approach to joint estimation of multiple covariance matrices," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 84(3), pages 339-374, April.
    5. Luca Bagnato & Antonio Punzo, 2021. "Unconstrained representation of orthogonal matrices with application to common principal components," Computational Statistics, Springer, vol. 36(2), pages 1177-1195, June.
    6. Jolene Birmingham & Garrett M. Fitzmaurice, 2002. "A Pattern-Mixture Model for Longitudinal Binary Responses with Nonignorable Nonresponse," Biometrics, The International Biometric Society, vol. 58(4), pages 989-996, December.
    7. Boik, Robert J., 2013. "Model-based principal components of correlation matrices," Journal of Multivariate Analysis, Elsevier, vol. 116(C), pages 310-331.
    8. Yu Cao & Nitai D. Mukhopadhyay, 2021. "Statistical Modeling of Longitudinal Data with Non-Ignorable Non-Monotone Missingness with Semiparametric Bayesian and Machine Learning Components," Sankhya B: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 83(1), pages 152-169, May.
    9. Prajamitra Bhuyan, 2019. "Estimation of random-effects model for longitudinal data with nonignorable missingness using Gibbs sampling," Computational Statistics, Springer, vol. 34(4), pages 1693-1710, December.
    10. Bingkai Wang & Xi Luo & Yi Zhao & Brian Caffo, 2021. "Semiparametric partial common principal component analysis for covariance matrices," Biometrics, The International Biometric Society, vol. 77(4), pages 1175-1186, December.
    11. Boik, Robert J., 2005. "Second-order accurate inference on eigenvalues of covariance and correlation matrices," Journal of Multivariate Analysis, Elsevier, vol. 96(1), pages 136-171, September.
    12. Antonio Rubia & Trino-Manuel Ñíguez, 2006. "Forecasting the conditional covariance matrix of a portfolio under long-run temporal dependence," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 25(6), pages 439-458.
    13. David M. Murray & Jonathan L. Blitstein, 2003. "Methods To Reduce The Impact Of Intraclass Correlation In Group-Randomized Trials," Evaluation Review, , vol. 27(1), pages 79-103, February.
    14. Bodart, Vincent & Reding, Paul, 1999. "Exchange rate regime, volatility and international correlations on bond and stock markets," Journal of International Money and Finance, Elsevier, vol. 18(1), pages 133-151, January.
    15. Markus Haas, 2018. "A note on the absolute moments of the bivariate normal distribution," Economics Bulletin, AccessEcon, vol. 38(1), pages 650-656.
    16. Jumah, Adusei & Kunst, Robert M., 2001. "The Effects of Exchange-Rate Exposures on Equity Asset Markets," Economics Series 94, Institute for Advanced Studies.
    17. Claudio Morana, 2014. "Factor Vector Autoregressive Estimation of Heteroskedastic Persistent and Non Persistent Processes Subject to Structural Breaks," Working Papers 273, University of Milano-Bicocca, Department of Economics, revised May 2014.
    18. Pedersen, Rasmus Søndergaard, 2016. "Targeting Estimation Of Ccc-Garch Models With Infinite Fourth Moments," Econometric Theory, Cambridge University Press, vol. 32(2), pages 498-531, April.
    19. Cavit Pakel & Neil Shephard & Kevin Sheppard & Robert F. Engle, 2021. "Fitting Vast Dimensional Time-Varying Covariance Models," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 39(3), pages 652-668, July.
    20. Kamel Helali & Thouraya Boujelbene Dammak, 2019. "Examining the Role of Structural Change in a Phillips Curve: Bivariate GARCH DCC Analysis," Economic Alternatives, University of National and World Economy, Sofia, Bulgaria, issue 3, pages 385-393, September.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:jmvana:v:98:y:2007:i:3:p:568-587. See general information about how to correct material in RePEc.

    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 CitEc recognized a bibliographic reference but did not link an item in RePEc 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 RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/622892/description#description .

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

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.