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A Canonical Representation of Block Matrices with Applications to Covariance and Correlation Matrices

Author

Listed:
  • Ilya Archakov

    (York University and University of Vienna)

  • Peter Reinhard Hansen

    (University of North Carolina and Copenhagen Business School)

Abstract

We obtain a canonical representation for block matrices. The representation facilitates simple computation of the determinant, the matrix inverse, and other powers of a block matrix, as well as the matrix logarithm and the matrix exponential. These results are particularly useful for block covariance and block correlation matrices, where evaluation of the Gaussian log-likelihood and estimation are greatly simplified. We illustrate this with an empirical application using a large panel of daily asset returns. Moreover, the representation paves new ways to model and regularize large covariance/correlation matrices, test block structures in matrices, and estimate regressions with many variables.

Suggested Citation

  • Ilya Archakov & Peter Reinhard Hansen, 2024. "A Canonical Representation of Block Matrices with Applications to Covariance and Correlation Matrices," The Review of Economics and Statistics, MIT Press, vol. 106(4), pages 1099-1113, July.
    • Handle: RePEc:tpr:restat:v:106:y:2024:i:4:p:1099-1113
    DOI: 10.1162/rest_a_01258
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    Cited by:

    1. Chen Tong & Peter Reinhard Hansen & Ilya Archakov, 2026. "Cluster GARCH," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 44(1), pages 148-161, January.
      • Chen Tong & Peter Reinhard Hansen & Ilya Archakov, 2024. "Cluster GARCH," Papers 2406.06860, arXiv.org.
    2. Chen, Han & Fei, Yijie & Yu, Jun, 2025. "Multivariate stochastic volatility models based on generalized Fisher transformation," Journal of Econometrics, Elsevier, vol. 251(C).
    3. Wu, Yongqiang & Zhang, Jun & Lan, Wei, 2025. "Structured covariance matrix estimation under volatility constraint," Finance Research Letters, Elsevier, vol. 85(PD).
    4. Neville Francis & Peter Reinhard Hansen & Chen Tong, 2025. "Principled Identification of Structural Dynamic Models," Papers 2512.17005, arXiv.org, revised Apr 2026.
    5. Archakov, Ilya & Hansen, Peter Reinhard & Lunde, Asger, 2026. "A multivariate realized GARCH model," Journal of Econometrics, Elsevier, vol. 254(PA).
    6. Han Chen & Yijie Fei & Jun Yu, 2026. "Multivariate Stochastic Volatility Model with Block Correlations," Working Papers 202638, University of Macau, Faculty of Business Administration.
    7. Ilya Archakov, 2026. "A Robust Similarity Estimator," Papers 2601.12198, arXiv.org.
    8. Chen Tong & Peter Reinhard Hansen, 2025. "Dynamic Factor Correlation Model," Papers 2503.01080, arXiv.org.

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