Estimating standard errors in regular vine copula models

Abstract

We describe a new algorithm for the computation of the score function and observed information in regular vine (R-vine) copula models. R-vine copulas are constructed hierarchically from bivariate copulas as building blocks only, and the algorithm exploits this hierarchical nature for subsequent computation of log-likelihood derivatives. This allows to routinely estimate standard errors of parameter estimates, and overcomes reliability and accuracy issues associated with numerical differentiation in multidimensional models. Results obtained using the proposed methods are discussed in the context of the asymptotic efficiency of different estimation methods and of an application to exchange rate data.

Appendices

Appendix 1: Algorithm for the calculation of second derivatives

In Sect. 4 we introduced the seven possible cases of dependence which can occur during the calculation of the second log-likelihood derivative. In the following, we illustrate these cases in detail. In case 1 we determine

$$\begin{aligned}&\frac{\partial ^2}{\partial \theta \partial \gamma } \ln \left( c_{U,V\vert \mathbf Z }\left( F_{U \vert \mathbf Z }(u \vert \mathbf z , \theta ), F_{V \vert \mathbf Z }(v \vert \mathbf z , \gamma )\right) \right) \nonumber \\&\quad = \frac{\partial _1 \partial _2 c_{U,V\vert \mathbf Z }\left( F_{U \vert \mathbf Z }(u \vert \mathbf z , \theta ), F_{V \vert \mathbf Z }(v \vert \mathbf z , \gamma )\right) }{ c_{U,V\vert \mathbf Z }\left( F_{U \vert \mathbf Z }(u \vert \mathbf z , \theta ), F_{V \vert \mathbf Z }(v \vert \mathbf z , \gamma )\right) } \times \left( \frac{\partial }{\partial \theta } F_{U \vert \mathbf Z }(u \vert \mathbf z , \theta ) \right) \nonumber \\&\qquad \times \left( \frac{\partial }{\partial \gamma } F_{V \vert \mathbf Z }(v \vert \mathbf z , \gamma ) \right) \nonumber \\&\quad \quad - \left( \frac{\partial }{\partial \theta } \ln \left( c_{U,V\vert \mathbf Z }\left( F_{U \vert \mathbf Z }(u \vert \mathbf z , \theta ), F_{V \vert \mathbf Z }(v \vert \mathbf z , \gamma )\right) \right) \right) \nonumber \\&\quad \quad \times \left( \frac{\partial }{\partial \gamma } \ln \left( c_{U,V\vert \mathbf Z }\left( F_{U \vert \mathbf Z }(u \vert \mathbf z , \theta ), F_{V \vert \mathbf Z }(v \vert \mathbf z , \gamma )\right) \right) \right) , \end{aligned}$$

(24)

for case 2

$$\begin{aligned}&\frac{\partial ^2}{\partial \theta \partial \gamma } \ln \left( c_{U,V\vert \mathbf Z }\left( F_{U \vert \mathbf Z }(u \vert \mathbf z , \theta ), F_{V \vert \mathbf Z }(v \vert \mathbf z , \theta , \gamma )\right) \right) \nonumber \\&\quad = \frac{\partial _1 \partial _2 c_{U,V\vert \mathbf Z }\left( F_{U \vert \mathbf Z }(u \vert \mathbf z , \theta ), F_{V \vert \mathbf Z }(v \vert \mathbf z , \theta , \gamma )\right) }{ c_{U,V\vert \mathbf Z }\left( F_{U \vert \mathbf Z }(u \vert \mathbf z , \theta ), F_{V \vert \mathbf Z }(v \vert \mathbf z , \theta , \gamma )\right) } \times \left( \frac{\partial }{\partial \theta } F_{U \vert \mathbf Z }(u \vert \mathbf z , \theta ) \right) \nonumber \\&\qquad \times \left( \frac{\partial }{\partial \gamma } F_{V \vert \mathbf Z }(v \vert \mathbf z , \theta , \gamma ) \right) \nonumber \\&\qquad - \left( \frac{\partial }{\partial \theta } \ln \left( c_{U,V\vert \mathbf Z }\left( F_{U \vert \mathbf Z }(u \vert \mathbf z , \theta ), F_{V \vert \mathbf Z }(v \vert \mathbf z , \theta , \gamma )\right) \right) \right) \nonumber \\&\qquad \times \left( \frac{\partial }{\partial \gamma } \ln (\left( c_{U,V\vert \mathbf Z }\left( F_{U \vert \mathbf Z }(u \vert \mathbf z , \theta ), F_{V \vert \mathbf Z }(v \vert \mathbf z , \theta , \gamma )\right) \right) \right) \nonumber \\&\qquad + \frac{\partial _2 \partial _2 c_{U,V\vert \mathbf Z }\left( F_{U \vert \mathbf Z }(u \vert \mathbf z , \theta ), F_{V \vert \mathbf Z }(v \vert \mathbf z , \theta , \gamma )\right) }{ c_{U,V\vert \mathbf Z }\left( F_{U \vert \mathbf Z }(u \vert \mathbf z , \theta ), F_{V \vert \mathbf Z }(v \vert \mathbf z , \theta , \gamma )\right) } \times \left( \frac{\partial }{\partial \theta } F_{V \vert \mathbf Z }(v \vert \mathbf z , \theta , \gamma ) \right) \nonumber \\&\qquad \times \left( \frac{\partial }{\partial \gamma } F_{V \vert \mathbf Z }(v\vert \mathbf z , \theta , \gamma ) \right) \nonumber \\&\qquad + \frac{ \partial _2 c_{U,V\vert \mathbf Z }\left( F_{U \vert \mathbf Z }(u \vert \mathbf z , \theta ), F_{V \vert \mathbf Z }(v \vert \mathbf z , \theta , \gamma )\right) }{c_{U,V\vert \mathbf Z }\left( F_{U \vert \mathbf Z }(u \vert \mathbf z , \theta ), F_{V \vert \mathbf Z }(v \vert \mathbf z , \theta , \gamma )\right) } \times \left( \frac{\partial ^2}{\partial \theta \partial \gamma } F_{V \vert \mathbf Z }(v \vert \mathbf z , \theta , \gamma ) \right) ,\nonumber \\ \end{aligned}$$

(25)

and case 3 yields

$$\begin{aligned}&\frac{\partial ^2}{\partial \theta \partial \gamma } \ln (c_{U,V\vert \mathbf Z }(F_{U \vert \mathbf Z }(u \vert \mathbf z , \theta , \gamma ), F_{V \vert \mathbf Z }(v \vert \mathbf z , \theta , \gamma ))) \nonumber \\&\quad = \frac{\partial _1 \partial _2 c_{U,V\vert \mathbf Z }\left( F_{U \vert \mathbf Z }(u \vert \mathbf z , \theta , \gamma ), F_{V \vert \mathbf Z }(v \vert \mathbf z , \theta , \gamma )\right) }{c_{U,V\vert \mathbf Z }\left( F_{U \vert \mathbf Z }(u \vert \mathbf z , \theta , \gamma ), F_{V \vert \mathbf Z }(v \vert \mathbf z , \theta , \gamma )\right) }\nonumber \\&\qquad \times \left( \frac{\partial }{\partial \theta } F_{U \vert \mathbf Z }(u \vert \mathbf z , \theta , \gamma ) \right) \left( \frac{\partial }{\partial \gamma } F_{V \vert \mathbf Z }(v \vert \mathbf z , \theta , \gamma ) \right) \nonumber \\&\quad \quad - \left( \frac{\partial }{\partial \theta } \ln \left( c_{U,V\vert \mathbf Z }\left( F_{U \vert \mathbf Z }(u \vert \mathbf z , \theta , \gamma ), F_{V \vert \mathbf Z }(v \vert \mathbf z , \theta , \gamma )\right) \right) \right) \nonumber \\&\quad \quad \times \left( \frac{\partial }{\partial \gamma } \ln (\left( c_{U,V\vert \mathbf Z }\left( F_{U \vert \mathbf Z }(u \vert \mathbf z , \theta , \gamma ), F_{V \vert \mathbf Z }(v \vert \mathbf z , \theta , \gamma )\right) \right) \right) \nonumber \\&\quad \quad + \frac{\partial _1 \partial _1 c_{U,V\vert \mathbf Z }\left( F_{U \vert \mathbf Z }(u \vert \mathbf z , \theta , \gamma ), F_{V \vert \mathbf Z }(v \vert \mathbf z , \theta , \gamma )\right) }{c_{U,V\vert \mathbf Z }\left( F_{U \vert \mathbf Z }(u \vert \mathbf z , \theta , \gamma ), F_{V \vert \mathbf Z }(v \vert \mathbf z , \theta , \gamma )\right) }\nonumber \\&\quad \quad \times \left( \frac{\partial }{\partial \theta } F_{U \vert \mathbf Z }(u \vert \mathbf z , \theta , \gamma ) \right) \left( \frac{\partial }{\partial \gamma } F_{U \vert \mathbf Z }(u\vert \mathbf z , \theta , \gamma ) \right) \nonumber \\&\quad \quad + \frac{\partial _1 \partial _2 c_{U,V\vert \mathbf Z }\left( F_{U \vert \mathbf Z }(u \vert \mathbf z , \theta , \gamma ), F_{V \vert \mathbf Z }(v \vert \mathbf z , \theta , \gamma )\right) }{c_{U,V\vert \mathbf Z }\left( F_{U \vert \mathbf Z }(u \vert \mathbf z , \theta , \gamma ), F_{V \vert \mathbf Z }(v \vert \mathbf z , \theta , \gamma )\right) }\nonumber \\&\quad \quad \times \left( \frac{\partial }{\partial \theta } F_{V \vert \mathbf Z }(v \vert \mathbf z , \theta , \gamma ) \right) \left( \frac{\partial }{\partial \gamma } F_{U \vert \mathbf Z }(u\vert \mathbf z , \theta , \gamma ) \right) \nonumber \\&\quad \quad + \frac{\partial _1 \partial _2 c_{U,V\vert \mathbf Z }\left( F_{U \vert \mathbf Z }(u \vert \mathbf z , \theta , \gamma ), F_{V \vert \mathbf Z }(v \vert \mathbf z , \theta , \gamma )\right) }{c_{U,V\vert \mathbf Z }\left( F_{U \vert \mathbf Z }(u \vert \mathbf z , \theta , \gamma ), F_{V \vert \mathbf Z }(v \vert \mathbf z , \theta , \gamma )\right) }\nonumber \\&\quad \quad \times \left( \frac{\partial }{\partial \theta } F_{U \vert \mathbf Z }(u \vert \mathbf z , \theta , \gamma ) \right) \left( \frac{\partial }{\partial \gamma } F_{V \vert \mathbf Z }(v\vert \mathbf z , \theta , \gamma ) \right) \nonumber \\&\quad \quad + \frac{\partial _2 \partial _2 c_{U,V\vert \mathbf Z }\left( F_{U \vert \mathbf Z }(u \vert \mathbf z , \theta , \gamma ), F_{V \vert \mathbf Z }(v \vert \mathbf z , \theta , \gamma )\right) }{c_{U,V\vert \mathbf Z }\left( F_{U \vert \mathbf Z }(u \vert \mathbf z , \theta , \gamma ), F_{V \vert \mathbf Z }(v \vert \mathbf z , \theta , \gamma )\right) }\nonumber \\&\quad \quad \times \left( \frac{\partial }{\partial \theta } F_{V \vert \mathbf Z }(v \vert \mathbf z , \theta , \gamma ) \right) \left( \frac{\partial }{\partial \gamma } F_{V \vert \mathbf Z }(v\vert \mathbf z , \theta , \gamma ) \right) \nonumber \\&\quad \quad + \frac{ \partial _1 c_{U,V\vert \mathbf Z }\left( F_{U \vert \mathbf Z }(u \vert \mathbf z , \theta , \gamma ), F_{V \vert \mathbf Z }(v \vert \mathbf z , \theta , \gamma )\right) }{c_{U,V\vert \mathbf Z }\left( F_{U \vert \mathbf Z }(u \vert \mathbf z , \theta , \gamma ), F_{V \vert \mathbf Z }(v \vert \mathbf z , \theta , \gamma )\right) } \times \left( \frac{\partial ^2}{\partial \theta \partial \gamma } F_{V \vert \mathbf Z }(v \vert \mathbf z , \theta , \gamma ) \right) \nonumber \\&\qquad + \frac{ \partial _2 c_{U,V\vert \mathbf Z }\left( F_{U \vert \mathbf Z }(u \vert \mathbf z , \theta , \gamma ), F_{V \vert \mathbf Z }(v \vert \mathbf z , \theta , \gamma )\right) }{c_{U,V\vert \mathbf Z }\left( F_{U \vert \mathbf Z }(u \vert \mathbf z , \theta , \gamma ), F_{V \vert \mathbf Z }(v \vert \mathbf z , \theta , \gamma )\right) } \times \left( \frac{\partial ^2}{\partial \theta \partial \gamma } F_{V \vert \mathbf Z }(v \vert \mathbf z , \theta , \gamma ) \right) .\nonumber \\ \end{aligned}$$

(26)

Similarly, we have for case 4 that

$$\begin{aligned}&\frac{\partial ^2}{\partial \theta \partial \gamma } \ln \left( c_{U,V\vert \mathbf Z }\left( F_{U \vert \mathbf Z }(u \vert \mathbf z , \theta ), F_{V \vert \mathbf Z }(v \vert \mathbf z )\vert \gamma \right) \right) \nonumber \\&\quad = \left( \frac{\partial }{\partial \theta } \ln \left( c_{U,V\vert \mathbf Z }\left( F_{U \vert \mathbf Z }(u \vert \mathbf z , \theta ), F_{V \vert \mathbf Z }(v \vert \mathbf z )\vert \gamma \right) \right) \right) \nonumber \\&\quad \quad \times \frac{- \partial _\gamma c_{U,V\vert \mathbf Z }\left( F_{U \vert \mathbf Z }(u \vert \mathbf z , \theta ), F_{V \vert \mathbf Z }(v \vert \mathbf z )\vert \gamma \right) }{ c_{U,V\vert \mathbf Z }\left( F_{U \vert \mathbf Z }(u \vert \mathbf z , \theta ), F_{V \vert \mathbf Z }(v \vert \mathbf z )\vert \gamma \right) } \nonumber \\&\quad \quad + \frac{\partial _\gamma \partial _1 c_{U,V\vert \mathbf Z }\left( F_{U \vert \mathbf Z }(u \vert \mathbf z , \theta ), F_{V \vert \mathbf Z }(v \vert \mathbf z )\vert \gamma \right) }{ c_{U,V\vert \mathbf Z }\left( F_{U \vert \mathbf Z }(u \vert \mathbf z , \theta ), F_{V \vert \mathbf Z }(v \vert \mathbf z )\vert \gamma \right) } \times \left( \frac{\partial }{\partial \theta } F_{U \vert \mathbf Z }(u\vert \mathbf z , \theta ) \right) , \end{aligned}$$

(27)

and

$$\begin{aligned}&\frac{\partial ^2}{\partial \theta \partial \gamma } \ln \left( c_{U,V\vert \mathbf Z }\left( F_{U \vert \mathbf Z }(u \vert \mathbf z , \theta ), F_{V \vert \mathbf Z }(v \vert \mathbf z , \theta )\vert \gamma \right) \right) \nonumber \\&\quad = \left( \frac{\partial }{\partial \theta } \ln \left( c_{U,V\vert \mathbf Z }\left( F_{U \vert \mathbf Z }(u \vert \mathbf z , \theta ), F_{V \vert \mathbf Z }(v\vert \mathbf z , \theta )\vert \gamma \right) \right) \right) \nonumber \\&\quad \quad \times \frac{- \partial _\gamma c_{U,V\vert \mathbf Z }\left( F_{U \vert \mathbf Z }(u \vert \mathbf z , \theta ), F_{V \vert \mathbf Z }(v\vert \mathbf z , \theta )\vert \gamma \right) }{ c_{U,V\vert \mathbf Z }\left( F_{U \vert \mathbf Z }(u \vert \mathbf z , \theta ), F_{V \vert \mathbf Z }(v\vert \mathbf z , \theta )\vert \gamma \right) } \nonumber \\&\quad \quad + \frac{\partial _\gamma \partial _1 c_{U,V\vert \mathbf Z }\left( F_{U \vert \mathbf Z }(u \vert \mathbf z , \theta ), F_{V \vert \mathbf Z }(v\vert \mathbf z , \theta )\vert \gamma \right) }{ c_{U,V\vert \mathbf Z }\left( F_{U \vert \mathbf Z }(u \vert \mathbf z , \theta ), F_{V \vert \mathbf Z }(v\vert \mathbf z , \theta )\vert \gamma \right) } \times \left( \frac{\partial }{\partial \theta } F_{U \vert \mathbf Z }(u\vert \mathbf z , \theta ) \right) \nonumber \\&\quad \quad + \frac{\partial _\gamma \partial _2 c_{U,V\vert \mathbf Z }\left( F_{U \vert \mathbf Z }(u \vert \mathbf z , \theta ), F_{V \vert \mathbf Z }(v\vert \mathbf z , \theta )\vert \gamma \right) }{ c_{U,V\vert \mathbf Z }\left( F_{U \vert \mathbf Z }(u \vert \mathbf z , \theta ), F_{V \vert \mathbf Z }(v\vert \mathbf z , \theta )\vert \gamma \right) } \times \left( \frac{\partial }{\partial \theta } F_{V \vert \mathbf Z }(v\vert \mathbf z , \theta ) \right) ,\quad \quad \end{aligned}$$

(28)

for the fifth case.Finally,

$$\begin{aligned}&\frac{\partial ^2}{\partial \theta \partial \gamma } \ln \left( c_{U,V\vert \mathbf Z }\left( F_{U \vert \mathbf Z }(u \vert \mathbf z , \theta ,\gamma ), F_{V \vert \mathbf Z }(v \vert \mathbf z ) \right) \right) \nonumber \\&\quad = \frac{\partial _1 \partial _1 c_{U,V\vert \mathbf Z }\left( F_{U \vert \mathbf Z }(u \vert \mathbf z , \theta ,\gamma ), F_{V \vert \mathbf Z }(v \vert \mathbf z ) \right) }{ c_{U,V\vert \mathbf Z }\left( F_{U \vert \mathbf Z }(u \vert \mathbf z , \theta ,\gamma ), F_{V \vert \mathbf Z }(v \vert \mathbf z ) \right) }\nonumber \\&\quad \quad \times \left( \frac{\partial }{\partial \theta } F_{U \vert \mathbf Z }(u\vert \mathbf z , \theta ,\gamma ) \right) \times \left( \frac{\partial }{\partial \gamma } F_{U \vert \mathbf Z }(u\vert \mathbf z , \theta ,\gamma ) \right) \nonumber \\&\quad \quad - \left( \frac{\partial }{\partial \gamma } \ln \left( c_{U,V\vert \mathbf Z }\left( F_{U \vert \mathbf Z }(u \vert \mathbf z , \theta ,\gamma ), F_{V \vert \mathbf Z }(v \vert \mathbf z ) \right) \right) \right) \nonumber \\&\quad \quad \times \left( \frac{\partial }{\partial \theta } \ln \left( c_{U,V\vert \mathbf Z }\left( F_{U \vert \mathbf Z }(u \vert \mathbf z , \theta ,\gamma ), F_{V \vert \mathbf Z }(v \vert \mathbf z ) \right) \right) \right) \nonumber \\&\quad \quad + \frac{\partial _1 c_{U,V\vert \mathbf Z }\left( F_{U \vert \mathbf Z }(u \vert \mathbf z , \theta ,\gamma ), F_{V \vert \mathbf Z }(v \vert \mathbf z ) \right) }{ c_{U,V\vert \mathbf Z }\left( F_{U \vert \mathbf Z }(u \vert \mathbf z , \theta ,\gamma ), F_{V \vert \mathbf Z }(v \vert \mathbf z ) \right) }\nonumber \\&\quad \quad \times \left( \frac{\partial ^2}{\partial \gamma \partial \theta } F_{U \vert \mathbf Z }(u\vert \mathbf z , \theta ,\gamma ) \right) ,\end{aligned}$$

(29)

$$\begin{aligned}&\frac{\partial ^2}{\partial \theta \partial \gamma } \ln \left( c_{U,V\vert \mathbf Z }\left( F_{U \vert \mathbf Z }(u \vert \mathbf z ), F_{V \vert \mathbf Z }(v \vert \mathbf z ) \vert \theta ,\gamma \right) \right) \nonumber \\&\quad = \frac{\partial _\theta \partial _\gamma c_{U,V\vert \mathbf Z }\left( F_{U \vert \mathbf Z }(u ), F_{V \vert \mathbf Z }(v \vert \mathbf z ) \vert \theta ,\gamma \right) }{ c_{U,V\vert \mathbf Z }\left( F_{U \vert \mathbf Z }(u \vert \mathbf z ), F_{V \vert \mathbf Z }(v \vert \mathbf z )\vert \theta ,\gamma \right) } \nonumber \\&\quad \quad - \frac{\partial _\theta c_{U,V\vert \mathbf Z }\left( F_{U \vert \mathbf Z }(u ), F_{V \vert \mathbf Z }(v \vert \mathbf z ) \vert \theta ,\gamma \right) \times \partial _\gamma c_{U,V\vert \mathbf Z }\left( F_{U \vert \mathbf Z }(u \vert \mathbf z ), F_{V \vert \mathbf Z }(v) \vert \theta ,\gamma \right) }{ c_{U,V\vert \mathbf Z }\left( F_{U \vert \mathbf Z }(u \vert \mathbf z ), F_{V \vert \mathbf Z }(v \vert \mathbf z )\vert \theta ,\gamma \right) ^2}.\nonumber \\ \end{aligned}$$

(30)

Appendix 2: Calculation of the covariance matrix in the Gaussian case

While analytical results on the Fisher information for the multivariate normal distribution are well known (Mardia and Marshall 1984) we will now illustrate how the matrices $\varvec{\mathcal{K}}_{\theta }$ and $\varvec{\mathcal{J}}_{\theta }$ (Eqs. 7 and 8) can be calculated. We consider a 3-dimensional Gaussian distribution

$$\begin{aligned} \begin{pmatrix} X_1\\ X_2\\ X_3\\ \end{pmatrix} \sim N_3(\varvec{0},\varvec{\varSigma }), \ \ \ \varvec{\varSigma } = \begin{pmatrix} 1 &{}\quad \rho _{12} &{}\quad \rho _{13} \\ \rho _{12} &{}\quad 1 &{}\quad \rho _{23} \\ \rho _{13} &{}\quad \rho _{23} &{}\quad 1 \\ \end{pmatrix}, \end{aligned}$$

with density $f_{123}$ and corresponding copula $c_{123}$. Exampli gratia, we show the computation for the entry $(2,1)$ in $\varvec{\mathcal{K}}_{\theta }$ in detail. The other entries in $\varvec{\mathcal{K}}_{\theta }$ and $\varvec{\mathcal{J}}_{\theta }$ are obtained similarly. The first step is to calculate the following integral:

$$\begin{aligned}&\int \limits _{[0,1]^{3}} \left( \frac{\partial }{\partial \rho _{12}} \ln (c_{12}(u_1,u_2|\rho _{12}))\right) \nonumber \\&\quad \times \left( \frac{\partial }{\partial \rho _{23}} \ln (c_{23}(u_2,u_3|\rho _{23}))\right) c_{123}(u_1,u_2,u_3) du_1du_2 du_3, \end{aligned}$$

(31)

where $c_{12}$ and $c_{23}$ are the corresponding copulas to the bivariate marginal distributions $f_{12}$ and $f_{23}$, respectively. Since the integral is independent of the univariate marginal distributions, we can compute it using standard normal margins (see Smith 2007):

$$\begin{aligned}&\int \limits _\mathbb{R ^3} \left( \frac{\partial }{\partial \rho _{12}} \ln (f_{12}(x_1,x_2|\rho _{12}))\right) \nonumber \\&\quad \times \left( \frac{\partial }{\partial \rho _{23}} \ln (f_{23}(x_2,x_3|\rho _{23}))\right) f_{123}(x_1,x_2,x_3) dx_1dx_2dx_3, \end{aligned}$$

(32)

where $f_{12}$ and $f_{23}$ are the according bivariate normal distributions. The 3-dimensional and bivariate normal densities in (31) and (32) can be expressed as

$$\begin{aligned}&f_{123}(x_1,x_2,x_3) = \frac{\sqrt{2}}{{\pi }^{3/2}\sqrt{2\,\rho _{13}\rho _{12}\rho _{23}-{\rho _{13}}^{2}-{\rho _{12}}^{2}+1-{\rho _{23}}^{2}}} \nonumber \\&\quad \times \exp \Bigg \{-\frac{1}{2}{\frac{-{x_{{1}}}^{2}\!-\!{x_{{2}}}^{2}\!-\!{x_{{3}}}^{2} \!+\!{x_{{1}}}^{2}{\rho _{23}}^{2} \!+\!{x_{{2}}}^{2}{\rho _{13}}^{2} \!+\!{x_{{3}}}^{2}{\rho _{12}}^{2} \!+\!2\,x_{{1}}x_{{2}}\rho _{12} \!+\!2\,x_{{1}}x_{{3}}\rho _{13} }{-2\,\rho _{13}\rho _{12}\rho _{23}\!+\!{\rho _{13}}^{2} \!+\!{\rho _{12}}^{2}\!-\!1\!+\!{\rho _{23}}^{2}}}\Bigg \}\nonumber \\&\quad \times \exp \Bigg \{\frac{ +2\,x_{{2}}x_{{3}}\rho _{23} -2\,x_{{1}}x_{{2}}\rho _{13}\rho _{23} -2\,x_{{1}}x_{{3}}\rho _{12}\rho _{23} -2\,x_{{2}}x_{{3}}\rho _{13}\rho _{12}}{-2\,\rho _{13}\rho _{12}\rho _{23} +{\rho _{13}}^{2}+{\rho _{12}}^{2}-1+{\rho _{23}}^{2}}\Bigg \}, \end{aligned}$$

(33)

and

$$\begin{aligned} f_{12}(x_1,x_2) = \frac{1}{2\pi }{\frac{1}{\sqrt{1-{\rho _{12}}^{2}}}} \,{\exp \left\{ -\frac{1}{2}\,{\frac{-{x_{{1}}}^{2}+2\,x_{{1}}x_{{2}}\rho _{12} -{x_{{2}}}^{2}}{ \left( -1+\rho _{12} \right) \left( \rho _{12}+1 \right) }}\right\} }. \end{aligned}$$

(34)

Further, the derivatives needed in Eq. (32) are

$$\begin{aligned} \frac{\partial }{\partial \rho _{12}} \ln (f_{12}(x_1,x_2|\rho _{12})) = -{\frac{ {\rho _{12}}^{3}-x_{{1}}x_{{2}}{\rho _{12}}^{2}+{x_{{2}}}^{2}\rho _{12} -\rho _{12}+{x_{{1}}}^{2}\rho _{12}-x_{{1}}x_{{2}} }{ \left( -1+\rho _{12} \right) ^{2} \left( \rho _{12}+1 \right) ^{2}} },\nonumber \\ \end{aligned}$$

(35)

and

$$\begin{aligned} \frac{\partial }{\partial \rho _{23}} \ln (f_{23}(x_2,x_3|\rho _{23})) \!=\! -{\frac{ {\rho _{23}}^{3}\!-\!x_{{2}}x_{{3}}{\rho _{23}}^{2}\!+\!{x_{{3}}}^{2}\rho _{23} \!-\!\rho _{23}\!+\!{x_{{2}}}^{2}\rho _{23}-x_{{2}}x_{{3}} }{ \left( -1+\rho _{23} \right) ^{2} \left( \rho _{23}\!+\!1 \right) ^{2}} }.\qquad \end{aligned}$$

(36)

Using (33), (35) and (36) in (32) we get

$$\begin{aligned} (32)&= \int \limits _\mathbb{R ^3} {\frac{ {\rho _{12}}^{3}-x_{{1}}x_{{2}}{\rho _{12}}^{2}+{x_{{2}}}^{2}\rho _{12} -\rho _{12}+{x_{{1}}}^{2}\rho _{12}-x_{{1}}x_{{2}} }{ \left( -1+\rho _{12} \right) ^{2} \left( \rho _{12}+1 \right) ^{2}} } \nonumber \\&\times {\frac{ {\rho _{23}}^{3}-x_{{2}}x_{{3}}{\rho _{23}}^{2}+{x_{{3}}}^{2}\rho _{23} -\rho _{23}+{x_{{2}}}^{2}\rho _{23}-x_{{2}}x_{{3}} }{ \left( -1+\rho _{23} \right) ^{2} \left( \rho _{23}+1 \right) ^{2}} }\nonumber \\&\times f_{123}(x_1,x_2,x_3) dx_1dx_2dx_3. \end{aligned}$$

(37)

The integral (37) can be solved using well known results on product moments of multivariate normal distributions (see Isserlis 1918).

$$\begin{aligned} (37)&= \frac{ \rho _{23}{\rho _{12}}^{3}+{\rho _{12}}^{3}{\rho _{23}}^{3} -3\,{\rho _{23}}^{2}{\rho _{12}}^{2}\rho _{13}-{\rho _{12}}^{2}\rho _{13} +2\,\rho _{23}{\rho _{13}}^{2}\rho _{12} }{ \left( \rho _{23}+1 \right) ^{2} \left( -1+\rho _{23} \right) ^{2} \left( -1+{\rho _{12}}^{2} \right) ^{2} } \nonumber \\&\quad +\frac{-\rho _{12}\rho _{23}+{\rho _{23}}^{3} \rho _{12}+\rho _{13}-{\rho _{23}}^{2}\rho _{13}}{ \left( \rho _{23}+1 \right) ^{2} \left( -1+\rho _{23} \right) ^{2} \left( -1+{\rho _{12}}^{2} \right) ^{2} } \end{aligned}$$

(38)

Since $\left( \rho _{23}+1 \right) ^{2} \left( -1+\rho _{23} \right) ^{2} = (1-\rho _{23}^2)^2$ we can simplify Eq. (38) to

$$\begin{aligned} (31) = (38)&= \frac{(\rho _{13}-\rho _{12}\rho _{23})(1-\rho _{12}^2)(1-\rho _{23}^2) + 2\rho _{12}\rho _{23}(\rho _{13}-\rho _{12}\rho _{23})^2 }{ (1-\rho _{12}^2)^2(1-\rho _{23}^2)^2 } \\&= \frac{\rho _{13}-\rho _{12} \rho _{23}}{(1-\rho _{12}^2)(1-\rho _{23}^2)} +2\rho _{12}\rho _{23}\frac{(\rho _{13}-\rho _{12}\rho _{23})^2}{(1-\rho _{12}^2)^2(1-\rho _{23}^2)^2} \\&= \frac{k_{12}}{(1-\rho _{12}^2)(1-\rho _{23}^2)}, \end{aligned}$$

with

$$\begin{aligned} k_{12} = (\rho _{13}-\rho _{12}\rho _{23})\left( 1+2\rho _{12}\rho _{23} \frac{\rho _{13}-\rho _{12}\rho _{23}}{(1-\rho _{12}^2)(1-\rho _{23}^2)} \right) . \end{aligned}$$

For the computation of terms corresponding to parameter $\rho _{13|2}$, note that

$$\begin{aligned} \rho _{13|2} = \frac{\rho _{13} - \rho _{12}\rho _{23}}{\sqrt{(1-\rho _{12}^2)(1-\rho _{23}^2)}}, \end{aligned}$$

and

$$\begin{aligned} \rho _{13} = \rho _{13|2}\sqrt{(1-\rho _{12}^2)(1-\rho _{23}^2)} + \rho _{12}\rho _{23}, \end{aligned}$$

which means that (33), (35) and (36) can easily be re-parametrized.

The final matrices are

$$\begin{aligned} \varvec{\mathcal{K}}_{\theta }&= \begin{pmatrix} \frac{1+\rho _{12}^2}{(1-\rho _{12}^2)^2} &{} \frac{k_{12}}{(1-\rho _{12}^2)(1-\rho _{23}^2)} &{} 0 \\ \frac{k_{12}}{(1-\rho _{12}^2)(1-\rho _{23}^2)} &{} \frac{1+\rho _{23}^2}{(1-\rho _{23}^2)^2} &{} 0 \\ 0 &{} 0 &{} \frac{1+\rho _{13|2}}{(\rho _{13|2}^2-1)^2} \\ \end{pmatrix},\\ \varvec{\mathcal{J}}_{\theta }&= \begin{pmatrix} \frac{1+\rho _{12}^2}{(1-\rho _{12}^2)^2} &{} 0 &{} 0 \\ 0 &{} \frac{1+\rho _{23}^2}{(1-\rho _{23}^2)^2} &{} 0 \\ \frac{\rho _{13|2}\rho _{12}}{(\rho _{12}^2-1)(\rho _{13|2}^2-1)} &{} \frac{\rho _{13|2}\rho _{23}}{(\rho _{23}^2-1)(\rho _{13|2}^2-1)} &{} \frac{1+\rho _{13|2}}{(\rho _{13|2}^2-1)^2} \end{pmatrix}. \end{aligned}$$

Appendix 3: Selected model for the exchange rate data

To obtain marginally uniformly distributed copula data on $[0,1]^8$, we conduct a pre-analysis as described in (Schepsmeier (2010), Chapter 5). AR(1)-GARCH(1,1) models (Table 3) are selected for the marginal time series, and the resulting standardized residuals are transformed using the non-parametric rank transformation (see Genest et al. 1995). We could also employ the probability integral transformation based on the parametric error distributions (IFM, Joe and Xu 1996) but since we are only interested in dependence properties here, we choose the non-parametric alternative which is more robust with respect to misspecification of marginal error distributions.

Table 3 Parameters of the AR(1)-GARCH(1,1) models

The R-vine describing the exchange rate data set is specified by the structure matrix $M$, the copula family matrix $\mathcal B$ and the estimated copula parameter matrix $\hat{\varvec{\theta }}^{MLE}$. For simplicity, we use the following abbreviations: 1 = AUD (Australian dollar), 2 = JPY (Japanese yen), 3 = BRL (Brazilian real), 4 = CAD (Canadian dollar), 5 = EUR (Euro), 6 = CHF (Swiss frank), 7 = INR (Indian rupee) and 8 = GBP (British pound).

The pair-copula families in the application were chosen from the elliptical copulas Gauss and Student’s $t$ copula, the Archimedean Clayton, Gumbel, Frank and Joe copula, and their rotated versions. The selection is done using to AIC/BIC.

$$\begin{aligned} M&= \begin{pmatrix} 8\\ 7 &{}\quad 7 \\ 2 &{}\quad 2 &{}\quad 6 \\ 3 &{}\quad 3 &{}\quad 2 &{}\quad 5 \\ 6 &{}\quad 4 &{}\quad 3 &{}\quad 2 &{}\quad 4 \\ 4 &{}\quad 1 &{}\quad 4 &{}\quad 3 &{}\quad 2 &{}\quad 3 \\ 1 &{}\quad 5 &{}\quad 1 &{}\quad 4 &{}\quad 3 &{}\quad 2 &{}\quad 2 \\ 5 &{}\quad 6 &{}\quad 5 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1 &{}\quad 1\\ \end{pmatrix}\end{aligned}$$

(39)

$$\begin{aligned} \hat{\varvec{\theta }}^{MLE}&= \begin{pmatrix} \, &{}\quad &{}\quad &{}\quad &{}\quad &{} \quad &{}\quad 11.85 &{}\quad 8.96 \\ &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad 7.74 \\ &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad 3.76 \\ &{}\quad -0.70 &{}\quad -0.09 &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad 9.97 \\ &{}\quad -0.88 &{}\quad -1.44 &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad 8.41 \\ 0.07 &{}\quad -1.10 &{}\quad -0.73 &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad 7.46 \\ 0.26 &{}\quad -1.23 &{}\quad -1.17 &{}\quad 1.13 &{}\quad 1.08 &{}\quad 0.63 &{}\quad &{}\quad \\ 0.72 &{}\quad 0.55 &{}\quad 0.88 &{}\quad 0.63 &{}\quad 0.54 &{}\quad 0.48 &{}\quad 0.3 &{}\quad \\ \end{pmatrix}\end{aligned}$$

(40)

$$\begin{aligned} \mathcal B&= \begin{pmatrix} \, &{} \\ \text{ Indep. }&{} \\ \text{ Indep. } &{} \text{ Indep. }\\ \text{ Indep. } &{} \text{ Frank } &{} \text{ Gauss } \\ \text{ Indep. } &{} \text{ Frank } &{} \text{ Frank } &{} \text{ Indep. } \\ \text{ Gauss } &{} \text{ r. } \text{ Joe } &{} \text{ Frank } &{} \text{ Indep. } &{} \text{ Indep. } \\ \text{ Student's } \text{-t } &{} \text{ r. } \text{ Gumbel } &{} \text{ r. } \text{ Gumbel } &{} \text{ Gumbel } &{} \text{ Frank } &{} \text{ Frank } \\ \text{ Student's } \text{ t } &{} \text{ Student's } \text{ t } &{} \text{ Student's } \text{ t } &{} \text{ Student's } \text{ t } &{} \text{ Student's } \text{ t } &{} \text{ Student's } \text{ t } &{} \text{ Gauss }\\ \end{pmatrix}\nonumber \\ \end{aligned}$$

(41)

The standard errors corresponding to the parameters in (40) are

$$\begin{aligned} {\varvec{SE}}_n^{MLE} = \begin{pmatrix} \, &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad 5.00 &{}\quad 2.54 \\ &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad 1.95 \\ &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad 0.59 \\ &{}\quad 0.19 &{}\quad 0.03 &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad 3.04 \\ &{}\quad 0.19 &{}\quad 0.20 &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad 2.40 \\ 0.03 &{}\quad 0.03 &{}\quad 0.19 &{}\quad &{}\quad &{}\quad &{}\quad &{}\quad 2.01 \\ 0.03 &{}\quad 0.03 &{}\quad 0.03 &{}\quad 0.02 &{}\quad 0.19 &{}\quad 0.19 &{}\quad &{}\quad \\ 0.01 &{}\quad 0.02 &{}\quad 0.01 &{}\quad 0.02 &{}\quad 0.02 &{}\quad 0.02 &{}\quad 0.03 &{}\quad \\ \end{pmatrix}. \end{aligned}$$

(42)