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

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  • 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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    References listed on IDEAS

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    1. Jorge Cadima & Francisco Lage Calheiros & Isabel Preto, 2010. "The eigenstructure of block-structured correlation matrices and its implications for principal component analysis," Journal of Applied Statistics, Taylor & Francis Journals, vol. 37(4), pages 577-589.
    2. Ilya Archakov & Peter Reinhard Hansen, 2021. "A New Parametrization of Correlation Matrices," Econometrica, Econometric Society, vol. 89(4), pages 1699-1715, July.
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    Cited by:

    1. Ilya Archakov & Peter Reinhard Hansen & Asger Lunde, 2020. "A Multivariate Realized GARCH Model," Papers 2012.02708, arXiv.org, revised May 2024.

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