Abstract: There is a complex dependent structure among the return rate series of financial assets. Multivariate NBS Copula is not only suitable for two- and higher-dimensional dependence structure modeling, but also contains multivariate normal copulas and multivariate symmetrical NBS Copula subclasses. The correlation parameter matrix, tail parameter and skew parameter vector can also be used to flexibly describe the positive and negative dependence, tail dependence and asymmetric dependence between pairs. It is also necessary to consider the time-varying dependence in the modeling dynamic structures. However, there are few researches on the dynamics of the multivariate skew ellipse Copula,mainly including DCC-GHST Copula model, GAS-GHST Copula model and DCC-NCCN Copula model. DCC-GHST Copula model, GAS-GHST Copula model and DCC-NCCN Copula model. In this study, a new time-varying NBS Copula model, namely DCC-NBS Copula model, is proposed for a detailed visual analysis of the dependent structure of the return of the three major stock index futures in the United States.
Kendall's rank correlation coefficient, QD (Quantitative Dependence) coefficient and the normal score's Maria skewness and kurtosis are used to measure the dependence characteristics of the joint distribution. On this basis, a multivariate NBS model is constructed. The parameter set of its distribution includes a correlation parameter matrix, a tail parameter and a skew parameter vector. In order to make the relevant parameter matrix dynamic, the DCC model is introduced and the multivariate time-varying DCC-NBS Copula model is constructed. For the given dependent structure data, the ML (Maximum Likelihood) method can be used to obtain the parameter estimates.
Taking Dow 30 index futures of the Chicago Board of Trade (CBOT), S&P 500 index futures of the Chicago Mercantile Exchange (CME) and NASDAQ 100 index futures as research objects, daily closing price data from January 1, 2005 to December 31, 2020 are selected to calculate daily logarithmic returns. Each closing price series has 4118 data, so each return series has 4,117 data. Data come from the website https://cn.investing.com and are calculated by Matlab software programming.
In order to verify the relative effect of the DCC-NBS Copula model, the CCC-N Copula, CCC-t Copula, CCC-NBS Copula, DCC-N Copula and DCC-t Copula models are selected for comparison. By fixing all skew parameters of NBS Copula, based on CCC-N Copula model, in the dualistic case, three common Archimedes copulas are also considered, namely the Clayton Copula, Gumbel Copula, and Frank Copula. In order to reflect the characteristics of the dependent structure and the model effect, Kendall rank correlation coefficient and Quantile Dependence coefficient are used for visualization analysis, including global and local dependence, symmetric and asymmetric dependence, static and dynamic dependence of the return series.
According to the descriptive statistics of edge distribution, the yield series of the three major stock index futures in the United States have obvious fat tail, asymmetry and time-varying volatility. According to the fitting results of the marginal distribution, NAGARCH model can describe the dynamic characteristics of the rate of return series well. According to the descriptive statistics of the dependence structure, the three major stock index futures in the United States have positive dependence, fat-tailed dependence, asymmetric dependence and time-varying dependence. Among them, the asymmetric dependence shows that the lower tail dependence is stronger than the upper tail dependence. In the dualistic case, the three kinds of Archimedean Copula perform the worst, and the fitting effects of normal Copula, t Copula and NBS Copula improve in turn. Compared with the CCC Copula model, the time-varying Kendall rank correlation coefficient sequence of the DCC-NBS Copula model is basically consistent with the Kendall rank correlation coefficient sequence of the moving sample. It shows that the binary DCC-NBS Copula model can better describe the time-varying dependence of the sample, therefore, and the fitting effect of CCC Copula and DCC Copula also improves in turn. According to the fitting results of dualistic dependence structures, elliptic Copula is superior to Archimedean Copula, asymmetric ellipse Copula is superior to symmetric ellipse Copula, thick-tailed ellipse Copula is superior to normal ellipse Copula, and time-varying ellipse Copula is superior to static ellipse Copula. Overall, the new DCC-NBS Copula model has the best performance. Further research shows that the fitting effects of multivariate dependent structures are basically consistent with those of dualistic dependence structures.

Key words: DCC model, multivariate NBS Copula, tail dependence, asymmetric dependence, time-varying dependence

摘要： 基于多元NBS(Normal Birnbaum-Saunders)分布构造了一种新的多元偏斜厚尾Copula,即多元NBS Copula,并进一步采用DCC(Dynamic Conditional Correlation)模型构造了时变NBS Copula模型。以美国道琼斯30指数期货、标准普尔500指数期货和纳斯达克100指数期货为例,可视化分析了收益率序列之间的各种相依特征,比较了DCC-NBS Copula模型与其他一些Copula模型在相依结构拟合上的效果差异。实证结果表明:美国三大股指期货之间的相依结构具有正相依性、厚尾相依性、非对称相依性和时变相依性,其中,NAGARCH模型可以较好地描述收益率序列的动态特征,椭圆Copula优于阿基米德Copula,非对称椭圆Copula优于对称椭圆Copula,厚尾椭圆Copula优于正态Copula,时变椭圆Copula优于静态椭圆Copula。综合来看,DCC-NBSCopula模型是所有模型中对相依结构的拟合效果最优的。

关键词: DCC模型, 多元NBS Copula, 尾部相依性, 非对称相依性, 时变相依性