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Correlation, hierarchies, and networks in financial markets

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

Abstract

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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Bibliographic Info

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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Web page: http://www.elsevier.com/locate/jebo

Related research

Keywords: Multivariate analysis Hierarchical clustering Correlation based networks Bootstrap validation Factor models Kullback-Leibler distance;

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References

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  1. Vincenzo Tola & Fabrizio Lillo & Mauro Gallegati & Rosario N. Mantegna, 2005. "Cluster analysis for portfolio optimization," Papers physics/0507006, arXiv.org.
  2. 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.
  3. R. Mantegna, 1999. "Hierarchical structure in financial markets," The European Physical Journal B - Condensed Matter and Complex Systems, Springer, vol. 11(1), pages 193-197, September.
  4. 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, arXiv.org.
  5. Marc Potters & Jean-Philippe Bouchaud & Laurent Laloux, 2005. "Financial Applications of Random Matrix Theory: Old Laces and New Pieces," Science & Finance (CFM) working paper archive 500058, Science & Finance, Capital Fund Management.
  6. Miccichè, Salvatore & Bonanno, Giovanni & Lillo, Fabrizio & N. Mantegna, Rosario, 2003. "Degree stability of a minimum spanning tree of price return and volatility," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 324(1), pages 66-73.
  7. M. Tumminello & T. Di Matteo & T. Aste & R. N. Mantegna, 2007. "Correlation based networks of equity returns sampled at different time horizons," The European Physical Journal B - Condensed Matter and Complex Systems, Springer, vol. 55(2), pages 209-217, 01.
  8. S. Illeris & G. Akehurst, 2001. "Introduction," The Service Industries Journal, Taylor & Francis Journals, vol. 21(1), pages 1-4, January.
  9. 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.
  10. Giovanni Bonanno & Nicolas Vandewalle & Rosario N. Mantegna, 2000. "Taxonomy of Stock Market Indices," Papers cond-mat/0001268, arXiv.org, revised Aug 2000.
  11. Giovanni Bonanno & Fabrizio Lillo & Rosario N. Mantegna, 2000. "High-frequency Cross-correlation in a Set of Stocks," Papers cond-mat/0009350, arXiv.org, revised Nov 2000.
  12. Giulio Biroli & Jean-Philippe Bouchaud & Marc Potters, 2007. "The Student ensemble of correlation matrices: eigenvalue spectrum and Kullback-Leibler entropy," Papers 0710.0802, arXiv.org.
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Citations

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Cited by:
  1. Dror Y. Kenett & Xuqing Huang & Irena Vodenska & Shlomo Havlin & H. Eugene Stanley, 2014. "Partial correlation analysis: Applications for financial markets," Papers 1402.1405, arXiv.org.
  2. Heiberger, Raphael H., 2014. "Stock network stability in times of crisis," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 393(C), pages 376-381.
  3. Kantar, Ersin & Keskin, Mustafa, 2013. "The relationships between electricity consumption and GDP in Asian countries, using hierarchical structure methods," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(22), pages 5678-5684.
  4. Contreras-Reyes, Javier E., 2014. "Asymptotic form of the Kullback–Leibler divergence for multivariate asymmetric heavy-tailed distributions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 395(C), pages 200-208.
  5. Vizgunov, A. & Goldengorin, B. & Zamaraev, V. & Kalyagin, V. & Koldanov, A. & Koldanov, P. & Pardalos, P., 2012. "Applying Market Graphs for Russian Stock Market Analysis," Journal of the New Economic Association, New Economic Association, vol. 15(3), pages 66-81.
  6. Grigory Bautin & Valery Kalyagin & Alexander Koldanov & Petr Koldanov & Panos Pardalos, 2013. "Simple measure of similarity for the market graph construction," Computational Management Science, Springer, vol. 10(2), pages 105-124, June.
  7. Sandoval, Leonidas, 2012. "Pruning a minimum spanning tree," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(8), pages 2678-2711.
  8. 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.
  9. Leonidas Sandoval Junior, 2011. "A Map of the Brazilian Stock Market," Papers 1107.4146, arXiv.org, revised Mar 2013.
  10. Tiago Trancoso, 2013. "Global macroeconomic interdependence: a minimum spanning tree approach," Review of Applied Socio-Economic Research, Pro Global Science Association, vol. 5(1), pages 179-189, June.
  11. Trancoso, Tiago, 2014. "Emerging markets in the global economic network: Real(ly) decoupling?," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 395(C), pages 499-510.
  12. Hao Meng & Wen-Jie Xie & Zhi-Qiang Jiang & Boris Podobnik & Wei-Xing Zhou & H. Eugene Stanley, 2013. "Systemic risk and spatiotemporal dynamics of the US housing market," Papers 1306.2831, arXiv.org.

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