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.

Introduction

Correlation is an important measure in finance research which is widely applied in many finance applications, such as risk management, banking business, pricing, etc. There are two kinds of correlation relationships involved in financial data: One kind is a linear relationship which mainly exists in regression models measured by covariance and correlation coefficient, and it is based on the multivariate normal distribution to model the dependence structure between financial returns [31]. The potent problem is that the method of linear relationship ignores some fluctuations such as high peak and fat tail relative to kurtosis and skewness, which have been frequently reported in financial data analyses. If the sample is not large enough, the normality assumption is untenable or, at least, questionable. Under this situation, it is impossible to measure nonlinear dependence and it is possible to underestimate risk [10], [22], [31].

The other type of dependence relationship is measured by copula due to Sklar [29], which has been observed in nonlinear dependence researches and has received a lot of attention [1], [10], [21], [22], [31]. A copula is a function that links together univariate distribution functions to form a flexible multivariate distribution instead of a multivariate normal distribution. The theorem shows that any n-dimensional joint distribution function may be decomposed into n marginal distributions and the copula completely describes the dependence between the n variables [19]. Thus, the copula is a special multivariate distribution function which fully captures asymmetric dependence and tail dependence of the data used in many studies. Thus, the implication is that correlation represents an illustration of dependence, but dependence should not be taken to be synonymous with correlation.

Several recent papers have applied the copula theory to study the dependence among financial variables. Examples include, but are not limited to, these studies: Patton [21] who studies the dependence among major foreign exchange rates and develops conditional copulas which allow for a time-varying dependence space; Hu [10] who uses mixed copula functions to examine the dependence among financial markets; and Xu [31] who uses a copula-based model to structure the portfolio model.

In this paper, we use copula and the definition of entropy to structure nonlinear dependence. The entropy value is obtained from the copula function in order to develop our copula entropy models.

The definition of “entropy” is generated from the field of thermodynamics and statistical mechanics to represent a measure of disorder. Shannon [27] uses a similar measure for information in his theory of communication and devises an entropy function to estimate the average information content associated with a random variable. After that, entropy has been considered as a useful statistical tool to measure volatilities from observed market prices in economics and finance studies [8], [9], [25], [30]. Unlike the variance and the regression model, entropy has the advantage of depending on more moments than just the second moment which allows only for measuring the dispersion around the mean. Consequently, entropy depends on much more information about a vector of random variables than its variance–covariance matrix.

Entropy optimization models, which are based on the Jaynes’s principle [11] of maximum entropy, have been widely used in uncertainty and risk measurement recently. The rationale behind the proposed principle of maximum entropy is that the desired probability distribution has maximum uncertainty (minimum information content), subject to representative and known information (some of which can be explicitly stated).

Virtually all the applications of entropy in finance have focused on forecasting and measurement research with the probabilistic description of the future market price of a common stock traded in an open securities market, such as portfolio optimization, derivatives pricing, credit risk measurement, etc. The principle of maximum entropy provides a method of generating a probability distribution from limited information. Philippatos and Wilson [23] refer entropy to portfolio and market risk measurement and propose a model with entropy, instead of using the traditional mean–variance approach to portfolios for inferring risk-neutral probabilities. It was considered as the first attempt to use entropy to measure the risk in finance. Since then, many relative studies [8], [9], [16], [18], [24] have been born.

Entropy was used as a criterion to choose the most fitted copula function in the copula family [1], [4]. As a matter of fact, the entropy principle could be used jointly with copula. Thus, the objective of this research is to present a copula entropy approach based on the entropy theory and copula theory to measure the dependence relationship between the financial variables, with practical applications.

The paper is organized as follows. Section 2 presents the definition of copula entropy, its mathematical properties, and the extensions of Jaynes’s entropy theory. In Section 3, we compare the copula entropy approach with mutual information and the linear correlation coefficient. Section 4 provides empirical evidence of the copula entropy approach. Finally, Section 5 concludes the paper.