Table 2 Selected copula models and their features. B is the beta function
From: Relationships among return and liquidity of cryptocurrencies
Symbol | Elliptical | Parameter range | Kendall’s \({\uptau }\) | Tail dependence | |
|---|---|---|---|---|---|
E1 | Gaussian | \(\theta \in \left( { - 1, 1} \right)\) | \(\frac{2}{\pi }arc\sin \theta\) | 0 | |
E2 | Student’s T | \(\theta \in \left( { - 1, 1} \right), \nu > 2\) | \(\frac{2}{\pi }arc\sin \theta\) | \(2t_{\nu + 1} \left( { - \sqrt {\nu + 1} \sqrt {\frac{1 - \theta }{{1 + \theta }}} } \right)\) | |
Symbol | Archimedean | Generator function \(\Psi\) | Parameter range | Kendall’s \({\uptau }\) | Tail dependence (lower, upper) |
|---|---|---|---|---|---|
A1 | Independence | \(- \log t\) | \({\text{NA}}\) | \({\text{NA}}\) | \({\text{NA}}\) |
A2 | Clayton | \(\frac{1}{\theta }\left( {t^{ - \theta } - 1} \right)\) | \({\uptheta } > 0\) | \(\frac{\theta }{\theta + 2}\) | \(\left( {2^{{ - \frac{1}{\theta }}} ,{ }0} \right)\) |
A3 | Gumbel | \(( - \log t)^{\theta }\) | \({\uptheta } \ge 1\) | \(1 - \frac{1}{\theta }\) | \(\left( {0, 2 - 2^{{\frac{1}{\theta }}} } \right)\) |
A4 | Frank | \(- {\text{log}}\left( {\frac{{\exp \left( { - \theta t} \right) - 1}}{{\exp \left( { - \theta } \right) - 1}}} \right)\) | \({\uptheta } \in {\text{R}}\) | \(1 - \frac{{4\left[ {1 - D_{1} \left( \theta \right)} \right]}}{\theta },D_{1} \left( \theta \right) = \frac{1}{\theta }\int_{0}^{\theta } {\frac{t}{{e^{t} - 1}}dt}\) | \(\left( {0,0} \right)\) |
A5 | Joe | \(- {\text{log}}\left( {1 - \left( {1 - t} \right)^{\theta } } \right)\) | \({\uptheta } \ge 1\) | \(1 + \frac{4}{{\theta^{2} }}\int_{0}^{1} {t\log \left( t \right)\left( {1 - t} \right)\frac{{2\left( {t - \theta } \right)}}{\theta }dt}\) | \(\left( {0,{ }2 - 2^{{\frac{1}{\theta }}} } \right)\) |
A6 | BB1 | \((t^{ - \theta } - 1)^{\delta }\) | \({\uptheta } > 0,{\updelta } \ge 1\) | \(1 - \frac{2}{{\delta \left( {\theta + 2} \right)}}\) | \(\left( {2^{{ - 1/\left( {\theta \delta } \right)}} ,2 - 2^{1/\delta } } \right)\) |
A7 | BB6 | \(\left( { - \log \left[ {1 - \left( {1 - t} \right)^{\theta } } \right]} \right)^{\delta }\) | \({\uptheta } \ge 1,{\updelta } \ge 1\) | \(1 + 4\int_{0}^{1} {\left[ { - \log \left( {1 - t^{\theta } } \right)\frac{{t - t^{1 - \theta } }}{\theta \delta }} \right]dt}\) | \((0,2 - 2^{{1/\left( {\theta \delta } \right)}}\) |
A8 | BB7 | \([1 - \left( {1 - t} \right)^{\theta } ]^{ - \delta } - 1\) | \({\uptheta } \ge 1,{\updelta } > 0\) | \(1 - \frac{2}{{\delta \left( {2 - \theta } \right)}} + \frac{4}{{\theta^{2} \delta }}B\left( {\frac{2 - \theta }{\theta },\delta + 2} \right)\) | \(2^{ - 1/\delta } ,2 - 2^{1/\theta }\) |
A9 | BB8 | \(- {\text{log}}\left[ {\frac{{1 - \left( {1 - \delta t} \right)^{\theta } }}{{1 - \left( {1 - \delta } \right)^{\theta } }}} \right]\) | \({\uptheta } \ge 1,0 < {\updelta } \le 1\) | \(1 + 4\int_{1 - \delta }^{1} {\left[ {\log \left( {\frac{{\left( {1 - \delta } \right)^{\theta } - 1}}{{t^{\theta } - 1}}} \right)\frac{{t - t^{1 - \theta } }}{{\theta \delta^{2} }}} \right]dt}\) | (0,0) |