A copula entropy approach to correlation measurement at the country level

Abstract The entropy optimization approach has widely been applied in finance for a long time, notably in the areas of market simulation, risk measurement, and financial asset pricing. In this paper, we propose copula entropy models with two and three variables to measure dependence in stock markets, which extend the copula theory and are based on Jaynes’s information criterion. Both of them are usually applied under the non-Gaussian distribution assumption. Comparing with the linear correlation coefficient and the mutual information, the strengths and advantages of the copula entropy approach are revealed and confirmed. We also propose an algorithm for the copula entropy approach to obtain the numerical results. With the experimental data analysis at the country level and the economic circle theory in international economy, the validity of the proposed approach is approved; evidently, it captures the non-linear correlation, multi-dimensional correlation, and correlation comparisons without common variables. We would like to make it clear that correlation illustrates dependence, but dependence is not synonymous with correlation. Copulas can capture some special types of dependence, such as tail dependence and asymmetric dependence, which other conventional probability distributions, such as the normal p.d.f. and the Student’s t p.d.f., cannot.

[1]  Michael A. H. Dempster,et al.  EMPIRICAL COPULAS FOR CDO TRANCHE PRICING USING RELATIVE ENTROPY , 2007 .

[2]  R. Reesor Relative entropy, distortion, the bootstrap and risk , 2001 .

[3]  George C. Philippatos,et al.  Entropy, market risk, and the selection of efficient portfolios , 1972 .

[4]  Paul M. B. Vitányi,et al.  Clustering by compression , 2003, IEEE Transactions on Information Theory.

[5]  Douglas J. Miller,et al.  On the recovery of joint distributions from limited information , 2002 .

[6]  E. Parzen On Estimation of a Probability Density Function and Mode , 1962 .

[7]  L. Gulko THE ENTROPY THEORY OF BOND OPTION PRICING , 1999 .

[8]  E. Miller Risk, Uncertainty, and Divergence of Opinion , 1977 .

[9]  Ling Hu Dependence patterns across financial markets: a mixed copula approach , 2006 .

[10]  M. Sklar Fonctions de repartition a n dimensions et leurs marges , 1959 .

[11]  E. Jaynes Information Theory and Statistical Mechanics , 1957 .

[12]  Andrew J. Patton Applications of copula theory in financial econometrics , 2002 .

[13]  S. Rachev Handbook of heavy tailed distributions in finance , 2003 .

[14]  Kun Liu,et al.  A note on a minimax rule for portfolio selection and equilibrium price system , 2009, Appl. Math. Comput..

[15]  Donald P. Morgan,et al.  Rating Banks: Risk and Uncertainty in an Opaque Industry , 2000 .

[16]  Rick L. Jenison,et al.  The Shape of Neural Dependence , 2004, Neural Computation.

[17]  T. Ané,et al.  Dependence Structure and Risk Measure , 2003 .

[18]  R. Nelsen An Introduction to Copulas , 1998 .

[19]  C. E. SHANNON,et al.  A mathematical theory of communication , 1948, MOCO.

[20]  Radko Mesiar,et al.  Discrete Copulas , 2006, IEEE Transactions on Fuzzy Systems.

[21]  Mohamed Saleh Estimating market shares in each market segment using the information entropy concept , 2007, Appl. Math. Comput..

[22]  Ba Chu,et al.  Recovering copulas from limited information and an application to asset allocation , 2011 .

[23]  Silviu Guiaşu,et al.  Information theory with applications , 1977 .

[24]  Adrianus M. H. Meeuwissen,et al.  Minimally informative distributions with given rank correlation for use in uncertainty analysis , 1997 .

[25]  Andrew J. Patton Copula-Based Models for Financial Time Series , 2009 .

[26]  Richard A. Davis,et al.  Handbook of Financial Time Series , 2009 .