Modeling Dependence with Copulas Clause Samples

Modeling Dependence with Copulas. ‌ Since we are testing the joint distribution of two random variables, the simulation pro- cedures involve in generating random samples from some joint distribution with certain dependence structure. Copulas are probably the most commonly used method to model the complete dependence structure between random variables (see ▇▇▇▇▇▇, 2004; ▇▇▇▇▇▇▇▇▇, 2007; ▇▇▇▇▇▇▇, 2008, for some applications of copulas in finance). ▇▇▇▇▇ (1959) proves that all bivariate distribution functions F (x1, x2) can be completely described by the univariate marginal distributions F1(x1) and F2(x2) and a copula function C : [0, 1]2 '→ [0, 1]. Copula, a word chosen by ▇▇▇▇▇, is a multivariate probability distribution function that describes such dependence structure between the two (or more) marginal distributions (see ▇▇▇▇▇▇, 1999, for a more detailed introduction to copulas). Many copulas with different dependence structures have been developed and commonly applied in the literature. Some of those parametric copulas, such as Gaussian, Student’s t and ▇▇▇▇▇ ▇▇▇▇▇▇▇, are known to have symmetric tail dependence structure. Some copulas are constructed to have asymmetric tail dependence. For example, ▇▇▇▇▇▇▇ copula is known for strong left tail dependence, whereas ▇▇▇▇▇▇ copula shows strong right tail dependence. As stock returns usually show stronger left tail dependence than right tail dependence with the market return (see ▇▇▇ and ▇▇▇▇, 2002), ▇▇▇▇▇▇▇ ▇▇▇▇▇▇ seems to be a natural choice. However, it is not wise to completely rule out those copulas with symmetric de- pendence. Figure 1.2 gives the scatter plots of random samples generated by Gaussian, ▇▇▇▇▇▇▇ and mixed Gaussian-▇▇▇▇▇▇▇ ▇▇▇▇▇▇▇, as well as the actual data plots of the small- est decile size portfolio returns. It is clear that ▇▇▇▇▇▇▇ copula generated data with strong left tail dependence, as the plots are highly concentrated at the left tail, but the dependence seems to be much stronger than that is actually reflected in the scatter plot of the smallest decile size portfolio. Comparing to the smallest size portfolio, which has shown to have the strongest asymmetric dependence in the following section and in Hong, Tu, and ▇▇▇▇ (2007), the generated data plots do not look much like the actual data plots. As shown in subfigure (C), the scatter plots generated by equal-weighted mixed Gaussian-▇▇▇▇▇▇▇ cop- ula look more similar to the actual data plots in subfigure (D). Therefore, I choose to use those mixed copulas as the dat...