3.3 Calculation of the Tail-dependence Coefficient


3.3.1 Archimedean Copulae

Archimedean copulae form an important class of copulae which are easy to construct and have good analytical properties. A bivariate Archimedean copula has the form $ C(u,v)=\psi^{[-1]}\{\psi(u)+\psi(v)\}$ for some continuous, strictly decreasing, and convex generator function $ \psi:[0,1]\rightarrow[0,\infty]$ such that $ \psi (1)=0$ and the pseudo-inverse function $ \psi^{[-1]}$ is defined by:

$\displaystyle \psi^{[-1]}(t)=\left\{\begin{array}{lc} \psi^{-1}(t), & 0\leq t\leq \psi(0),\\
0, & \psi(0)< t\leq \infty. \end{array}\right.
$

We call $ \psi$ strict if $ \psi(0)=\infty.$ In that case $ \psi^{[-1]}=\psi^{-1}.$ Within the framework of tail dependence for Archimedean copulae, the following result can be shown (Schmidt; 2003). Note that the one-sided derivatives of $ \psi$ exist, as $ \psi$ is a convex function. In particular, $ \psi'(1)$ and $ \psi'(0)$ denote the one-sided derivatives at the boundary of the domain of $ \psi.$ Then:

i)
upper tail-dependence implies $ \psi'(1)=0$ and $ \lambda_U=2-(\psi^{-1}\circ 2\psi)'(1),$
ii)
$ \psi'(1)<0$ implies upper tail-independence,
iii)
$ \psi'(0)>-\infty$ or a non-strict $ \psi$ implies lower tail-independence,
iv)
lower tail-dependence implies $ \psi'(0)=-\infty,$ a strict $ \psi,$ and
$ \lambda_L=(\psi^{-1}\circ 2\psi)'(0).$

Tables 3.1 and 3.2 list various Archimedean copulae in the same ordering as provided in Nelsen (1999, Table 4.1, p. 94) and in Härdle, Kleinow, and Stahl (2002, Table 2.1, p. 42) and the corresponding upper and lower tail-dependence coefficients (TDCs).


Table 3.1: Various selected Archimedean copulae. The numbers in the first column correspond to the numbers of Table 4.1 in Nelsen (1999), p. 94.

Number & Type $ C(u, v)$ Parameters
(1) \begin{displaymath}\begin{array}{l}\textrm{Clayton} \end{array}\end{displaymath} $ \max\Big\{(u^{-\theta}+v^{-\theta}-1)^{-1/\theta},0\Big\}$ $ \theta\in[-1,\infty)\backslash\{0\}$
(2) $ \max\Big[1-\Big\{(1-u)^{\theta}+(1-v)^{\theta}\Big\}^{1/\theta},0\Big]$ $ \theta\in[1,\infty)$
(3) \begin{displaymath}\begin{array}{l}\textrm{Ali-} \\ \textrm{Mikhail-Haq}
\end{array}\end{displaymath} $ \displaystyle {\frac{uv}{1-\theta(1-u)(1-v)}}$ $ \theta\in[-1,1)$
(4) \begin{displaymath}\begin{array}{l}\textrm{Gumbel-} \\ \textrm{Hougaard}
\end{array}\end{displaymath} $ \exp\Big[-\big\{(-\log u)^{\theta}+(-\log
v)^{\theta}\big\}^{1/\theta}\Big]$ $ \theta\in[1,\infty)$
(12) $ \Big[1+\Big\{(u^{-1}-1)^{\theta}+(v^{-1}-1)^{\theta}\Big\}^{1/\theta}\Big]^{-1}$ $ \theta\in[1,\infty)$
(14) $ \Big[1+\Big\{(u^{-1/\theta}-1)^{\theta}+(v^{-1/\theta}-1)^{\theta}\Big\}^{1/\theta}\Big]^{-\theta}$ $ \theta\in[1,\infty)$
(19) $ \theta/\log\big(e^{\theta/u}+e^{\theta/v}-e^{\theta}\big)$ $ \theta\in(0,\infty)$




Table 3.2: Tail-dependence coefficients (TDCs) and generators $ \psi _{\theta }$ for various selected Archimedean copulae. The numbers in the first column correspond to the numbers of Table 4.1 in Nelsen (1999), p. 94.
Number & Type$ $ $ \psi_{\theta}(t)$ Parameter $ \theta$ Upper-TDC Lower-TDC
(1) $ \begin{array}{l}\textrm{Pareto} \end{array}$ $ t^{-\theta}-1$ $ [-1,\infty)\backslash\{0\}$ 0 for $ \theta>0$ $ 2^{-1/\theta}$
for $ \theta>0$
(2) $ (1-t)^{\theta}$ $ [1,\infty)$ $ 2-2^{1/\theta}$ 0
(3) $ \begin{array}{l}\textrm{Ali-} \\ \textrm{Mikhail-Haq}
\end{array}\!\!\!\!\!\!$ $ \displaystyle {\log\frac{1-\theta(1-t)}{t}}$ $ [-1,1)$ 0 0
(4) $ \begin{array}{l}\textrm{Gumbel-} \\ \textrm{Hougaard}
\end{array}\!\!\!\!\!\!$ $ (-\log t)^{\theta}$ $ [1,\infty)$ $ 2-2^{1/\theta}$ 0
(12) $ \Big(\frac{1}{t}-1\Big)^{\theta}$ $ [1,\infty)$ $ 2-2^{1/\theta}$ $ 2^{-1/\theta}$
(14) $ \big(t^{-1/\theta}-1\big)^{\theta}$ $ [1,\infty)$ $ 2-2^{1/\theta}$ $ \displaystyle {\frac{1}{2}}$
(19) $ e^{\theta/t}-e^{\theta}$ $ (0,\infty)$ 0 $ 1$



3.3.2 Elliptically-contoured Distributions

In this section, we calculate the tail-dependence coefficient for elliptically-contoured distributions (briefly: elliptical distributions). Well-known elliptical distributions are the multivariate normal distribution, the multivariate $ t$-distribution, the multivariate logistic distribution, the multivariate symmetric stable distribution, and the multivariate symmetric generalized-hyperbolic distribution.

Elliptical distributions are defined as follows: Let $ X$ be an $ n$-dimensional random vector and $ \Sigma\in \mathbb{R}^{n\times n}$ be a symmetric positive semi-definite matrix. If $ X-\mu$, for some $ \mu\in \mathbb{R}^n,$ possesses a characteristic function of the form $ \phi_{X-\mu}(t)=\Psi(t^\top\Sigma t)$ for some function $ \Psi:\mathbb{R}_0^+\rightarrow\mathbb{R}$, then $ X$ is said to be elliptically distributed with parameters $ \mu$ (location), $ \Sigma$ (dispersion), and $ \Psi.$ Let $ E_n(\mu,\Sigma,\Psi)$ denote the class of elliptically-contoured distributions with the latter parameters. We call $ \Psi$ the characteristic generator. The density function, if it exists, of an elliptically-contoured distribution has the following form:

$\displaystyle f(x)=\vert\Sigma\vert^{-1/2}g\big\{(x-\mu)^\top\Sigma^{-1}(x-\mu)\big\},\quad x\in\mathbb{R}^n,$ (3.5)

for some function $ g:\mathbb{R}_0^+\rightarrow\mathbb{R}_0^+,$ which we call the density generator.

Observe that the name ``elliptically-contoured distribution'' is related to the elliptical contours of the latter density. For a more detailed treatment of elliptical distributions see the monograph of Fang, Kotz, and Ng (1990) or Cambanis, Huang, and Simon (1981).

In connection with financial applications, Bingham and Kiesel (2002) and Bingham, Kiesel, and Schmidt (2002) propose a semi-parametric approach for elliptical distributions by estimating the parametric component ( $ \mu,\Sigma$) separately from the density generator $ g.$ In their setting, the density generator is estimated by means of a nonparametric statistics.

Schmidt (2002b) shows that bivariate elliptically-contoured distributions are upper and lower tail-dependent if the tail of their density generator is regularly varying, i.e. the tail behaves asymptotically like a power function. Further, a necessary condition for tail dependence is given which is more general than regular variation of the latter tail: More precisely, the tail must be O-regularly varying (see Bingham, Goldie, and Teugels (1987) for the definition of O-regular variation). Although the equivalence of tail dependence and regularly-varying density generator has not been shown, all density generators of well-known elliptical distributions possess either a regularly-varying tail or a not O-regularly-varying tail. This justifies a restriction to the class of elliptical distributions with regularly-varying density generator if tail dependence is required. In particular, tail dependence is solely determined by the tail behavior of the density generator (except for completely correlated random variables which are always tail dependent).

The following closed-form expression exists (Schmidt; 2002b) for the upper and lower tail-dependence coefficient of an elliptically-contoured random vector $ (X_1,X_2)^{\top}\in
E_2(\mu,\Sigma,\Psi)$ with positive-definite matrix

$\displaystyle \Sigma=\left(\begin{array}{cc}\sigma_{11} & \sigma_{12}\\
\sigma_{11} & \sigma_{12}\end{array}\right),$

having a regularly-varying density generator $ g$ with regular variation index $ -\alpha/2-1<0:$

$\displaystyle \lambda\stackrel{\mathrm{def}}{=}\lambda_U=\lambda_L=\frac{\displ...
... {\displaystyle{\int_{0}^{1}\displaystyle{\frac{u^{\alpha}}{\sqrt{1-u^2}}}du}},$ (3.6)

where $ \rho=\sigma_{12}/\sqrt{\sigma_{11}\sigma_{22}}$ and $ h(\rho)=\Big\{1+\frac{(1-\rho)^2}{1-\rho^2}\Big\}^{-1/2}.$

Figure 3.3: Tail-dependence coefficient $ \lambda $ versus regular variation index $ \alpha $ for ``correlation'' coefficients $ \rho=0.5,\;0.3,\;0.1$.

Note that $ \rho$ corresponds to the ``correlation'' coefficient when it exists (Fang, Kotz, and Ng; 1990). Moreover, the upper tail-dependence coefficient $ \lambda_U$ coincides with the lower tail-dependence coefficient $ \lambda_L$ and depends only on the ``correlation'' coefficient $ \rho$ and the regular variation index $ \alpha,$ see Figure 3.3.

Table 3.3 lists various elliptical distributions, the corresponding density generators (here $ c_n$ denotes a normalizing constant depending only on the dimension $ n$) and the associated regular variation index $ \alpha $ from which one easily derives the tail-dependence coefficient using formula (3.6).


Table 3.3: Tail index $ \alpha $ for various density generators $ g$ of multivariate elliptical distributions. $ K_{\nu }$ denotes the modified Bessel function of the third kind (or Macdonald function).
  Density generator $ g$ or   $ \alpha $ for
Number & Type characteristic generator $ \Psi$ Parameters $ n=2$
     
(23) Normal $ g(u)=c_n\exp(- u/2)$ -- $ \infty$
       
(24) $ t$ $ g(u)=\displaystyle {c_n\Big(1+\frac{t}{\theta}\Big)^{-(n+\theta)/2}}$ $ \theta>0$ $ \theta$
       
(25) $ \begin{array}{l}\textrm{Symmetric} \\ \textrm{general.}\\
\textrm{hyperbolic} \end{array}$ $ g(u)=\displaystyle {c_n\frac{K_{\lambda-\frac{n}{2}}\{\sqrt{\varsigma(\chi+u)}\}}
{(\sqrt{\chi+u})^{\frac{n}{2}-\lambda}}}$ \begin{displaymath}\begin{array}{c}\varsigma,\chi>0\\ \lambda\in\mathbb{R}\end{array}\end{displaymath} $ \infty$
       
(26) $ \begin{array}{l}\textrm{Symmetric} \\ \textrm{$\theta$-stable
}\end{array}$ $ \Psi(u)=\exp\Big\{-\Big(\frac{1}{2}u\Big)^{\theta/2}\Big\}$ $ \theta\in(0,2]$ $ \theta$
       
(27) logistic $ \displaystyle{g(u)=c_n\frac{\exp(-u)}{\{1+\exp(-u)\}^2}}$ -- $ \infty$
       



3.3.3 Other Copulae

For many other closed form copulae one can explicitly derive the tail-dependence coefficient. Tables 3.4 and 3.5 list some well-known copula functions and the corresponding lower and upper TDCs.




Table 3.4: Various copulae. Copulae BBx are provided in Joe (1997).
Number & Type $ $ $ C(u, v)$ Parameters
(28) Raftery $ \begin{array}{c}
g\left\{\min(u,v),\max(u,v);\theta\right\}\;\textrm{with}\\
...
...a}
x^{1/(1-\theta)}\Big(y^{-\theta/(1-\theta)}-y^{1/(1-\theta)}\Big)\end{array}$ $ \theta\in[0,1]$
(29) BB1 $ \displaystyle {\Big[1+\Big\{(u^{-\theta}-1)^{\delta}+(v^{-\theta}-1)^{\delta}\Big\}^{1/\delta}\Big]^{-1/\theta}}$ \begin{displaymath}\begin{array}{c} \theta\in(0,\infty)\\ \delta\in[1,\infty)\end{array}\end{displaymath}
(30) BB4 $ \displaystyle {\begin{array}{c} \Big[u^{-\theta}+v^{-\theta}-1-\\
-\Big\{(u^{...
...elta}+(v^{-\theta}-1)^{-\delta}\Big \}^{-1/\delta}\Big]^{-1/\theta}\end{array}}$ \begin{displaymath}\begin{array}{c} \theta\in[0,\infty)\\ \delta\in(0,\infty)\end{array}\end{displaymath}
(31) BB7 $ \displaystyle {\begin{array}{c} 1-\Big(1-\Big[\big\{1-(1-u)^{\theta}\big\}^{-\...
...1-(1-v)^{\theta}\big\}^{-\delta}-1\Big]^{-1/\delta}\Big)^{1/\theta}\end{array}}$ \begin{displaymath}\begin{array}{c} \theta\in[1,\infty)\\ \delta\in(0,\infty)\end{array}\end{displaymath}
(32) BB8 \begin{displaymath}\begin{array}{c} \displaystyle {\frac{1}{\delta}\Big(1-\Big[1...
...{1-(1-\delta
v)^{\theta}\big\}\Big]^{1/\theta}\Big) \end{array}\end{displaymath} \begin{displaymath}\begin{array}{c} \theta\in[1,\infty)\\ \delta\in[0,1]\end{array}\end{displaymath}
(33) BB11 $ \theta\min(u,v)+(1-\theta)uv$ $ \theta\in[0,1]$
(34) \begin{displaymath}\begin{array}{l} C_{\Omega} \textrm{ in } \\
\textrm{Junker and } \\
\textrm{May (2002) }\end{array}\end{displaymath} $ \begin{array}{c} \beta
C^s_{(\bar{\theta},\bar{\delta})}(u,v)-(1-\beta)C_{(\th...
...is the survival copula with param.}\;\; (\bar{\theta},\bar{\delta})
\end{array}$ \begin{displaymath}\begin{array}{c} \theta,\bar{\theta}\in\mathbb{R}\backslash\{0\} \\
\delta,\bar{\delta}\geq 1 \\
\beta\in[0,1]
\end{array} \end{displaymath}





Table 3.5: Tail-dependence coefficients (TDCs) for various copulae. Copulae BBx are provided in Joe (1997).
Number & Type $ $ Parameters upper-TDC lower-TDC
       
(28) Raftery $ \theta\in[0,1]$ 0 $ \displaystyle {\frac{2\theta}{1+\theta}}$
       
(29) BB1 \begin{displaymath}\begin{array}{c} \theta\in(0,\infty)\\ \delta\in[1,\infty)\end{array}\end{displaymath} $ 2-2^{1/\delta}$ $ 2^{-1/(\theta\delta)}$
       
(30) BB4 \begin{displaymath}\begin{array}{c} \theta\in[0,\infty)\\ \delta\in(0,\infty)\end{array}\end{displaymath} $ 2^{-1/\delta}$ \begin{displaymath}\begin{array}{c} (2-\\ 2^{-1/\delta})^{-1/\theta} \end{array}\end{displaymath}
       
(31) BB7 \begin{displaymath}\begin{array}{c} \theta\in[1,\infty)\\ \delta\in(0,\infty)\end{array}\end{displaymath} $ 2-2^{1/\theta}$ $ 2^{-1/\delta}$
       
(32) BB8 \begin{displaymath}\begin{array}{c} \theta\in[1,\infty)\\ \delta\in[0,1]\end{array}\end{displaymath} \begin{displaymath}\begin{array}{c} 2-\\ -2(1-\delta)^{\theta-1} \end{array}\end{displaymath} 0
       
       
(33) BB11 $ \theta\in[0,1]$ $ \theta$ $ \theta$
       
(34) \begin{displaymath}\begin{array}{l} C_{\Omega} \textrm{ in } \\
\textrm{Junker and } \\
\textrm{May (2002) }\end{array}\end{displaymath} \begin{displaymath}\begin{array}{c} \theta,\bar{\theta}\in\mathbb{R}\backslash\{0\} \\
\delta,\bar{\delta}\geq 1 \\
\beta\in[0,1]
\end{array} \end{displaymath} \begin{displaymath}\begin{array}{c} (1-\beta)\cdot\\ \cdot(2-2^{1/\delta}) \end{array}\end{displaymath} $ \beta (2-2^{1/\bar{\delta}}) $