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Random matrix ensembles of time-lagged correlation matrices: derivation of eigenvalue spectra and analysis of financial time-series

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  • Christoly Biely
  • Stefan Thurner

Abstract

We derive the exact form of the eigenvalue spectra of correlation matrices derived from a set of time-shifted, finite Brownian random walks (time-series). These matrices can be seen as real, asymmetric random matrices where the time-shift superimposes some structure. We demonstrate that, for large matrices, the associated eigenvalue spectrum is circular symmetric in the complex plane. This fact allows us to exactly compute the eigenvalue density via an inverse Abel-transform of the density of the symmetrized problem. We demonstrate the validity of this approach numerically. Theoretical findings are then compared with eigenvalue densities obtained from actual high-frequency (5 min) data of the S&P 500 and the observed deviations are discussed. We identify various non-trivial, non-random patterns and find asymmetric dependencies associated with eigenvalues departing strongly from the Gaussian prediction in the imaginary part. For the same time-series, with the market contribution removed, we observe strong clustering of stocks into causal sectors. We finally comment on the stability of the observed patterns.

Suggested Citation

  • Christoly Biely & Stefan Thurner, 2008. "Random matrix ensembles of time-lagged correlation matrices: derivation of eigenvalue spectra and analysis of financial time-series," Quantitative Finance, Taylor & Francis Journals, vol. 8(7), pages 705-722.
  • Handle: RePEc:taf:quantf:v:8:y:2008:i:7:p:705-722
    DOI: 10.1080/14697680701691477
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    References listed on IDEAS

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    Cited by:

    1. Sandoval, Leonidas & Franca, Italo De Paula, 2012. "Correlation of financial markets in times of crisis," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(1), pages 187-208.
    2. Hongli Zeng & R'emi Lemoy & Mikko Alava, 2013. "Financial interaction networks inferred from traded volumes," Papers 1311.3871, arXiv.org.
    3. repec:eee:reveco:v:51:y:2017:i:c:p:562-573 is not listed on IDEAS

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