Dependence Of Random Variables And Its Applications

By using properties of Copulas, this thesis attempts to discuss tail dependence coefficient of two class of skew distributions from theoretical perspective, and to apply dependent random variables to model real data sets from empirical point of view. The paper is divided into three chapters.The first chapter is to introduce the basic theory of Copula and the advantages of Copulas in study of dependence.For the second chapter, we study the tail dependence of bivariate skew Laplace distribution and bivariate skew Cauchy distribution, respectively. Through basic properties of Copulas, the upper tail dependence coefficient can be expressed as the limits of two conditional probabilities. The results shows that the bivariate skew Laplace distribution is asymptotically independent, while the bivariate skew Cauchy distribution are asymptotic dependent.In the third chapter, we apply GARCH-Copula model to investigate the impact of the 2008 world financial crisis on the dependence structure among industry sectors of China’s equity market. We make an empirical analysis by using the industry sectors’stock-indices of the Shanghai stock market and dividing the time series into pre-crisis period and post-crisis period. The approach chosen to estimate the model is two-stage maximum likelihood (IFM):model each marginal return series by AR(p)-GARCH(1.1) model and estimate related parameters; then, four copu-las and time-varying dependence are considered, the parameters are obtained given the estimated marginal parameters. The empirical analysis shows that leverage ef-fect does not exist in return series. Moreover. t-Copula and mixed Gumbel Copula are more suitable to fit the dependence structure of the mentioned time series ac-cording to the AIC criteria even though empirical copulas prefer asymmetry tail dependence structure. The GARCH-Copula model shows that each markets tend to co-movement and there exists stronger dependence structure after financial crisis.