Measuring and Testing Tail Dependence and Contagion Risk Between Major Stock Markets

Abstract

This paper studies the tail dependence for two smaller stock markets that are Taiwanese Taiex and South Korean Kospi against four larger stock markets that are S& P500, Nikkei, MSCI China, and MSCI Europe. The vector autoregression result indicates that both S&P500 and MSCI China indeed have the greatest impact and significance on the other four stock markets. However, the tail dependence of Taiex and Kospi versus either S&P500 or MSCI China are lower due to unilateral impacts from US or China. The Clayton copula yields the jumps of tail dependence and the elliptical copulas generate the trends of tail dependence. The threshold tests of Clayton Kendall’s taus between most stock markets are significant in both subprime and Greek debt crises while the tests of Student-t Kendall’s taus are only significant for the subprime crisis. It appears that the subprime has changeable trend and jump states of contagion risk while Greek debt has one steady trend state and changeable jump states of contagion risk.

Appendix: The Log Likelihood of Gaussian, Student-t, and Clayton Copula

• 1. The log likelihood of Gaussian copula is

$$\begin{aligned} L({\varvec{\varepsilon }}_t ;R)=1/2\sum _{t=1}^T (\log |\mathbf{R}|+ {\varvec{\varepsilon }}_t^{\prime } (\mathbf{R}^{-1}-I){\varvec{\varepsilon }}_t ), \end{aligned}$$

(20)

where ${\varvec{\varepsilon }}_t=({\phi }^{-1} (u_{1,t}),\ldots ,{\phi }^{-1} (u_{p,t}))$ is the vector of the transformed standardized residuals and $\mathbf{R}$ is the correlation matrix of ${\varvec{\varepsilon }}_t$ and p is the number of residual series.

• 2. The log likelihood of Student-t copula

$$\begin{aligned} L({\varvec{\varepsilon }}_t ;\mathbf{R},v_{c} )= & {} -T\log \frac{\Gamma (\frac{v_c +p}{2})}{\Gamma (\frac{v_c }{2})}-pT\log \frac{\Gamma (\frac{v_c +p}{2})}{\Gamma (\frac{v_c }{2})}-\frac{v_c +p}{2}\sum _{t=1}^T {\log \left( 1+\frac{\mathbf{\varepsilon }_t^{\prime } \mathbf{R}^{-1}{\varvec{\varepsilon }}_t }{v_c }\right) } \nonumber \\&-\sum _{t=1}^T {\log |\mathbf{R}|+\frac{v_c +1}{2}\sum _{t=1}^T {\sum _{i=1}^p {\log \left( 1+\frac{\varepsilon _{i,t}^2 }{v_c }\right) , } } } \end{aligned}$$

(21)

where $v_c$ is the degree of freedom.

• 3. The log likelihood of Clayton copula

$$\begin{aligned} L(\mathbf{u}_t ;\theta )=\sum _{t=1}^T {(\log (1+\theta )(u_{1,t} \cdot u_{2,t})^{-1-\theta }(u_{1t}^{-\theta } +u_{2,t}^{-\theta } -1)^{-2-\frac{1}{\theta }} } ), \end{aligned}$$

(22)

where $\theta$ is solved optimally as $\frac{2\rho _{\tau }}{1-\rho _{\tau }}$ and $\rho _{\tau }$ is Kendall’s tau.