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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.

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  • 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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    Cited by:

    1. Sebastiano Michele Zema & Giorgio Fagiolo & Tiziano Squartini & Diego Garlaschelli, 2021. "Mesoscopic Structure of the Stock Market and Portfolio Optimization," Papers 2112.06544, arXiv.org.
    2. Hongli Zeng & R'emi Lemoy & Mikko Alava, 2013. "Financial interaction networks inferred from traded volumes," Papers 1311.3871, arXiv.org.
    3. Cai, Yumei & Cui, Xiaomei & Huang, Qianyun & Sun, Jianqiang, 2017. "Hierarchy, cluster, and time-stable information structure of correlations between international financial markets," International Review of Economics & Finance, Elsevier, vol. 51(C), pages 562-573.
    4. Zdzisław Burda & Andrzej Jarosz & Maciej Nowak & Jerzy Jurkiewicz & Gabor Papp & Ismail Zahed, 2011. "Applying free random variables to random matrix analysis of financial data. Part I: The Gaussian case," Quantitative Finance, Taylor & Francis Journals, vol. 11(7), pages 1103-1124.
    5. Ochiai, Tomoshiro & Nacher, Jose C., 2022. "Unveiling the directional network behind financial statements data using volatility constraint correlation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 600(C).
    6. 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.
    7. Leonidas Sandoval Junior & Italo De Paula Franca, 2011. "Correlation of financial markets in times of crisis," Papers 1102.1339, arXiv.org, revised Mar 2011.
    8. Stefanos Bennett & Mihai Cucuringu & Gesine Reinert, 2022. "Lead-lag detection and network clustering for multivariate time series with an application to the US equity market," Papers 2201.08283, arXiv.org.

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