The fine structure of spectral properties for random correlation matrices: an application to financial markets

Livan, Giacomo and Alfarano, Simone and Scalas, Enrico (2011): The fine structure of spectral properties for random correlation matrices: an application to financial markets.

Abstract

We study some properties of eigenvalue spectra of financial correlation matrices. In particular, we investigate the nature of the large eigenvalue bulks which are observed empirically, and which have often been regarded as a consequence of the supposedly large amount of noise contained in financial data. We challenge this common knowledge by acting on the empirical correlation matrices of two data sets with a filtering procedure which highlights some of the cluster structure they contain, and we analyze the consequences of such filtering on eigenvalue spectra. We show that empirically observed eigenvalue bulks emerge as superpositions of smaller structures, which in turn emerge as a consequence of cross-correlations between stocks. We interpret and corroborate these findings in terms of factor models, and and we compare empirical spectra to those predicted by Random Matrix Theory for such models.

• Item Type: MPRA Paper
• Language: English
• Keywords: random matrix theroy; financial econometrics; correlation matrix
• Subjects: C - Mathematical and Quantitative Methods > C5 - Econometric Modeling > C51 - Model Construction and Estimation; G - Financial Economics > G1 - General Financial Markets > G11 - Portfolio Choice ; Investment Decisions; C - Mathematical and Quantitative Methods > C0 - General > C01 - Econometrics

References

[1] E. P. Wigner, Ann. Math. 62, 548 (1955). E. P. Wigner, Ann. Math. 67, 325 (1958) [2] M. Mehta, Random Matrices (Elsevier, Amsterdam, 2004) [3] J. Wishart, Biometrika 20, 32 (1928) [4] V. A. Marˇcenko and L. A. Pastur, Math. USSR-Sb 1, 457 (1967) [5] L. Laloux, P. Cizeau, J.-P. Bouchaud and M. Potters, Phys. Rev. Lett. 83, 1467 (1999) [6] V. Plerou, P. Gopikrishnan, B. Rosenow, L. A. N. Amaral and H. E. Stanley, Phys. Rev. Lett. 83, 1471 (1999) [7] A. N. Kolmogorov, Foundations of the Theory of Probability (Chelsea Publishing, New York, 1956) [8] P. Billingsley, Probability and Measure (John Wiley & Sons, New York, 1995) [9] J. Y. Campbell, A. W. Lo and A. Craig MacKinlay, The Econometrics of Financial Markets (Princeton University Press, Princeton, 1997) [10] Z. Burda, A. Jarosz, J. Jurkiewicz, M. A. Nowak, G. Papp and I. Zahed, e-print arXiv:physics/0603024 [11] J. W. Silverstein and Z. D. Bai, J. Multivariate Anal. 54, 175 (1995) [12] Z. Burda, A. G¨orlich, A. Jarosz and J. Jurkiewicz, Physica A 343, 295 (2004) [13] Z. Burda and J. Jurkiewicz, Physica A 344, 67 (2004) [14] H. Markowitz, J. Financ. 7, 77 (1952) [15] W. Sharpe, J. Financ. 19, 425 (1964) [16] F. Lillo and R. N. Mantegna, Phys. Rev. E 72, 016219 (2005) [17] G. Raffaelli and M. Marsili, J. Stat. Mech.-Theory E. L08001 (2006) [18] M. Marsili, G. Raffaelli and B. Ponsot, J. Econ. Dyn. Control 33, 1170 (2009) [19] J.-P. Bouchaud and M. Potters, Theory of Financial Risk and Derivative Pricing: From Statistical Physics to Risk Man- agement (Cambridge University Press, Cambridge, 2003) [20] S. Pafka and I. Kondor, Eur. Phys. J. B 27, 277 (2002) [21] J. Shlens, A Tutorial on Principal Component Analysis: Derivation, Discussion and Singular Value Decomposition, avail- able online at http://www.cs.princeton.edu/picasso/mats/PCA-Tutorial-Intuition_jp.pdf [22] J. Kwapien, S. Dro˙zd˙z and P. Oswiecimka, Physica A 359, 589 (2006) [23] G. Akemann, J. Fischmann and P. Vivo, Physica A 389, 2566 (2010) [24] V. Plerou, P. Gopikrishnan, B. Rosenow, L. A. N. Amaral, T. Guhr and H. E. Stanley, Phys. Rev. E 65, 066126 (2002) [25] T. Guhr and B. Kalber, J. Phys. A 36, 3009 (2003) [26] O. E. Barndorff-Nielsen and S. Thorbjørnsen, P. Natl. Acad. Sci. USA 99, 16568 (2002) [27] M. Politi, E. Scalas, D. Fulger and G. Germano, Eur. Phys. J. B 73, 13 (2010) [28] C. Itzykson and J.-M. Drouffe, Statistical Field Theory (Cambridge University Press, Cambridge, 1989) [29] Z. Burda, A. G¨orlich, J. Jurkiewicz and B. Wac law, Eur. Phys. J. B 49, 319 (2006) 21 [30] C. A. Tracy and H. Widom, The Distributions of Random Matrix Theory and Their Applications, available online at http://www.math.ucdavis.edu/~tracy/talks/SITE7.pdf [31] D. Paul, Stat. Sinica 17, 1617 (2007) [32] M. A. Stephens, J. Am. Stat. Assoc. 69, 730 (1974) [33] M. S. Aldenderfer and R. K. Blashfield, Cluster Analysis (SAGE Publications Inc., Newbury Park, 1995)