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Cleaning large correlation matrices: tools from random matrix theory

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  • Joel Bun
  • Jean-Philippe Bouchaud
  • Marc Potters

Abstract

This review covers recent results concerning the estimation of large covariance matrices using tools from Random Matrix Theory (RMT). We introduce several RMT methods and analytical techniques, such as the Replica formalism and Free Probability, with an emphasis on the Marchenko-Pastur equation that provides information on the resolvent of multiplicatively corrupted noisy matrices. Special care is devoted to the statistics of the eigenvectors of the empirical correlation matrix, which turn out to be crucial for many applications. We show in particular how these results can be used to build consistent "Rotationally Invariant" estimators (RIE) for large correlation matrices when there is no prior on the structure of the underlying process. The last part of this review is dedicated to some real-world applications within financial markets as a case in point. We establish empirically the efficacy of the RIE framework, which is found to be superior in this case to all previously proposed methods. The case of additively (rather than multiplicatively) corrupted noisy matrices is also dealt with in a special Appendix. Several open problems and interesting technical developments are discussed throughout the paper.

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  • Joel Bun & Jean-Philippe Bouchaud & Marc Potters, 2016. "Cleaning large correlation matrices: tools from random matrix theory," Papers 1610.08104, arXiv.org.
    • Handle: RePEc:arx:papers:1610.08104
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    2. Longfeng Zhao & Wei Li & Andrea Fenu & Boris Podobnik & Yougui Wang & H. Eugene Stanley, 2017. "The q-dependent detrended cross-correlation analysis of stock market," Papers 1705.01406, arXiv.org, revised Jun 2017.
    3. Sebastien Valeyre, 2022. "Optimal trend following portfolios," Papers 2201.06635, arXiv.org.
    4. Soufiane Hayou, 2017. "On the overestimation of the largest eigenvalue of a covariance matrix," Papers 1708.03551, arXiv.org.
    5. Jean-Philippe Bouchaud, 2021. "Radical Complexity," Papers 2103.09692, arXiv.org.

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