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Correlation, hierarchies, and networks in financial markets

  • Tumminello, Michele
  • Lillo, Fabrizio
  • Mantegna, Rosario N.

We discuss some methods to quantitatively investigate the properties of correlation matrices. Correlation matrices play an important role in portfolio optimization and in several other quantitative descriptions of asset price dynamics in financial markets. Here, we discuss how to define and obtain hierarchical trees, correlation based trees and networks from a correlation matrix. The hierarchical clustering and other procedures performed on the correlation matrix to detect statistically reliable aspects of it are seen as filtering procedures of the correlation matrix. We also discuss a method to associate a hierarchically nested factor model to a hierarchical tree obtained from a correlation matrix. The information retained in filtering procedures and its stability with respect to statistical fluctuations is quantified by using the Kullback-Leibler distance.

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Article provided by Elsevier in its journal Journal of Economic Behavior & Organization.

Volume (Year): 75 (2010)
Issue (Month): 1 (July)
Pages: 40-58

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Handle: RePEc:eee:jeborg:v:75:y:2010:i:1:p:40-58
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  1. G. Bonanno & F. Lillo & R. N. Mantegna, 2001. "High-frequency cross-correlation in a set of stocks," Quantitative Finance, Taylor & Francis Journals, vol. 1(1), pages 96-104.
  2. M. Tumminello & T. Di Matteo & T. Aste & R. N. Mantegna, 2006. "Correlation based networks of equity returns sampled at different time horizons," Papers physics/0605251,, revised Apr 2007.
  3. M. Tumminello & F. Lillo & R. N. Mantegna, 2007. "Shrinkage and spectral filtering of correlation matrices: a comparison via the Kullback-Leibler distance," Papers 0710.0576,
  4. Salvatore Miccich\`e & Giovanni Bonanno & Fabrizio Lillo & Rosario N. Mantegna, 2002. "Degree stability of a minimum spanning tree of price return and volatility," Papers cond-mat/0212338,
  5. Michele Tumminello & Fabrizio Lillo & Rosario Nunzio Mantegna, 2007. "Kullback-Leibler distance as a measure of the information filtered from multivariate data," Papers 0706.0168,
  6. S. Illeris & G. Akehurst, 2001. "Introduction," The Service Industries Journal, Taylor & Francis Journals, vol. 21(1), pages 1-4, January.
  7. Giovanni Bonanno & Nicolas Vandewalle & Rosario N. Mantegna, 2000. "Taxonomy of Stock Market Indices," Papers cond-mat/0001268,, revised Aug 2000.
  8. Tola, Vincenzo & Lillo, Fabrizio & Gallegati, Mauro & Mantegna, Rosario N., 2008. "Cluster analysis for portfolio optimization," Journal of Economic Dynamics and Control, Elsevier, vol. 32(1), pages 235-258, January.
  9. C. Coronnello & M. Tumminello & F. Lillo & S. Miccich\`e & R. N. Mantegna, 2005. "Sector identification in a set of stock return time series traded at the London Stock Exchange," Papers cond-mat/0508122,
  10. Rosario N. Mantegna, 1998. "Hierarchical Structure in Financial Markets," Papers cond-mat/9802256,
  11. Ledoit, Olivier & Wolf, Michael, 2003. "Improved estimation of the covariance matrix of stock returns with an application to portfolio selection," Journal of Empirical Finance, Elsevier, vol. 10(5), pages 603-621, December.
  12. M. Potters & J. P. Bouchaud & L. Laloux, 2005. "Financial Applications of Random Matrix Theory: Old Laces and New Pieces," Papers physics/0507111,
  13. Schäfer Juliane & Strimmer Korbinian, 2005. "A Shrinkage Approach to Large-Scale Covariance Matrix Estimation and Implications for Functional Genomics," Statistical Applications in Genetics and Molecular Biology, De Gruyter, vol. 4(1), pages 1-32, November.
  14. Giulio Biroli & Jean-Philippe Bouchaud & Marc Potters, 2007. "The Student ensemble of correlation matrices: eigenvalue spectrum and Kullback-Leibler entropy," Papers 0710.0802,
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