Pair-copula Constuctions Of Multiple Dependence And Empirical Study

The influcences and dependences among finacial markets are gradually increased.It is very meaningful to find the dependence structure and coefficient.But which tools should be used to describe the structure ?It is a problem,which we should pay more attention.Many scholars choose copula as the tool. Copula is a function which combine the distribution fuctions of random variables with their margin distributions.Considering the following two advantages ,we choose copula as the tool,one is that copula is a new and effective tool,with good properties,such as the dependence structure of random variables is unchangeable if we transfer these variables into their strictly increasing functions;the other is copula can be used to construct the multivariate distribution functions of variables,and this way is simple and easy to do.But there are many kinds of copulas,to choose which one as the best copula of given variables,such as stock markets, is a problem.The traditional methods is based on goodness of fit,but these methods pose a lots of problems in pratical application.So it is hard to make us entirely satisfying.We introduce a new method called Bayes molde selection method in this paper.Bayes method is difference with traditional methods,and it is independent with the parameters of copula,that is we treat the copula parameters as nuisance variables.It selects the best copula family by caculating the probabilities of copula families.In order to study the reliability of the method ,we choose the composite index of ShangHai stock market and the Standard Poors Index as example.The results show that comparing with the other copula families,Clayton copula family fits the datas best.And at last we get the lower tail dependence coefficient.Following the discussion above,we introduce a new method,pair copula method , to build the multivariate denpence structure because we usually need to consider the relationships among multi stochatice variables,such as the relationships among the composite index of ShangHai stock market ,the Standard Poors Index and the index of HongKong stock market or the relationships among monetary markets,stock markets,and the real estate markets.Pair copula method transforms the n-variate probability density function into a cascade of pair-copulae.In fact,pair-copula is bivariate copula,so multiple dependence is simplied into two variates.We apply the methodology to a financial data set,four broads of Shanghai Stock Exchange ,and at last got a good result.