基于条件概率积分变换的多元Copula函数选择

本文构建了基于条件概率积分变换的Copula函数选择方法,通过对条件概率积分变换下Anderson-Darling(AD)、Kolmogorov-Smirnov(KS)、Cramér-von Mises(CM)这三种统计量的比较,讨论在不同样本容量和变量维数下其对多种Copula函数的拟合效果。利用GSPTSE、INMEX.MX和NDX三大股指样本,将基于条件概率积分变换的Copula函数选择方法与核密度估计和极大似然估计选择法的效果进行系统比较。结果表明,基于条件概率积分变换的检验法可以有效解决多元Copula函数的选择问题,其拟合优度检验更精确、更稳定;核密度估计检验在大样本下比较稳定,而小样本下稳定性较差;相比之下,极大似然值检验法则不稳定。

Selection for Multivariate Copula Based on Conditional Probability Integral Transformation

The dependence structure among multivariate financial assets is a critical factor for achieving accuracy in the integrated risk measurement.Copula function is a very useful tool to describe the dependence structure between risk factors and plays an important role in the field of financial risk management.When using the Copula model,the most important thing is to judge which Copula is more suitable to describe the data dependence structure.Therefore,it is very important to do research on the selection criteria of multivariate Copula models and the goodness-of-fit test methods.However,in the field of financial risk management,most of the research has focused on bivariate cases.There is very little research on the goodness-of-fit and empirical analysis on the multivariate cases.There is still no effective solutions for the selection and goodness-of-fit test of multivariate Copula functions. Therefore,this paper proposes a selection criterion for Copula’s goodness-of-fit based on the method of conditional probability integral transformation.We analyze and compare the Anderson-Darling(AD),Kolmogorov-Smirnov(KS) and Cramér-von Mises(CM) test statistics under the CPIT method with various sample sizes and different variable dimensions.In addition,we use daily data of three stock indices: S&P/TSX Composite index(GSPTSE) in Canada’s stock market,INMEX.MX in Mexico’s stock market and NASDAQ-100(NDX) in America’s stock market.Our samples consist of 1606 adjusted-closing price in each of the three indices.We compare the CPIT test statistics with two other methods based on kernel the density estimate and the maximum likelihood estimate. The empirical studies results show that in terms of the power of goodness-of-fit test,the approach we proposed has a better performance.This method is able to solve the puzzle in selecting multivariate Copula models and its goodness-of-fit test is accurate and stable.Specifically,CM test statistic is more powerful in small samples;however in large samples the test is weaker than AD and KS tests.We also show that the AD test has a strong testing ability in large samples.On the other hand,the statistics based on the kernel density estimate method is more appropriate for selecting the best Copula functions under a large sample.The reason is that in a large sample,the kernel function selection has little effect on the estimated distribution.However,in a small sample,the choice of bandwidth in kernel estimation has a big effect on the estimation of marginal distribution,which may result in an unstable result.Although the test based on maximum likelihood estimate is able to choose Gauss Copula function as the best Copula to describe the correlation pattern between datasets,the method is quite unstable.As a result,its capacity of choosing the optimal Copula function is relatively weak.