Testing Asymmetry in Dependence with Copula-Coskewness

Abstract

A new measure of asymmetry in dependence is proposed that is based on taking the difference between the margin-free coskewness parameters of the underlying copula. The new measure and a related test are applied to both a hydrological and a financial market data sample, and we show that both samples exhibit systematic asymmetric dependence.

Notes

1 Modeling the tail of a distribution is also relevant for various applications across disciplines such as risk management or actuarial science (see, e.g., Peng 2008; Jong 2012; Hua and Xia 2014), which further explains the growing interest in those topics.

2 This is in contrast to asymmetry in tail dependence, which has been extensively studied by, e.g., Demarta and McNeil (2004), Patton (2006), and Christoffersen et al. (2012).

3 The case of possibly discontinuous margins is briefly treated in Remark 2.4.

4 This is not to be confused with the related concept of radial symmetry, , where denotes the survival copula associated with C.

5 An extensive overview of methods how to construct asymmetric copulas from given symmetric ones can be found in Liebscher (2008).

6 One disadvantage of the copula-coskewness measure is that a value of zero does not necessarily imply symmetry in the copula. This, however, also holds for the common skewness parameter of a real-valued distribution. In the same way, independence of two random variables is not implied by a Spearman’s rho that equals zero.

7 Note that μ_U = μ_V = 1/2 and σ²_U = σ_V² = 1/12.

8 One might alternatively think of defining a similar coefficient based on a copula version of the cokurtosis, which employs fourth moments and is also widely used in the finance literature. However, the difference between two cokurtosis parameters cannot be related to the asymmetry in dependence, because values of these parameters occur whenever two random variables deviate in the same direction. For this reason, we do not pursue the investigation of copula-cokurtosis any further here.

9 Note that by Fubini’s theorem.

10 As a consequence, is the unique copula in case of continuity of the margins.

11 See also Siburg et al. (2016) for a related interpretation of asymmetry in bivariate financial data.

12 Note that a negative asymmetry in dependence of X and Y is the same as a positive asymmetric dependence of Y and X.

13 Similarly, we also find asymmetric dependence of the MSCI World Index without U.S. firms and an oil index.

14 We also compare indexes of non-equity markets such as cotton or oil with LME-Aluminium and Gold Bullion LBM indexes. Oil and cotton exhibit a negative value of a_{X, Y} and thus, cotton may be used to hedge against extreme drops in oil prices. The relation of oil prices and our gold or aluminium indexes is also clear. Higher values of our oil index are associated with both high and low prices of aluminium and gold. However, these findings are not as intuitive as other asymmetry coefficients of indexes and could also be the result of a false rejection.