C-vines, D-vines, and Regular Vines

Summary

A regular vine (R-vine) is a nested sequence of $d-1$ trees in which the edges of one tree become the nodes of the next, subject to a proximity condition; each edge carries a pair-copula. Two graphically simple special cases dominate financial applications: the C-vine (canonical vine) — each tree has one pivotal node connected to all others (star shape), good when one variable governs the system — and the D-vine (drawable vine) — each node touches at most two edges (a path/line), good when variables have a natural ordering.

Overview

Bedford and Cooke introduced regular vines as a graphical model for choosing which pair-copulae appear in a PCC. The C-vine and D-vine are the two structures originally used in finance (Kurowicka & Cooke 2006); a general R-vine is anything in between.

Main Content

Regular vine definition (Bedford-Cooke; Def. 4.4 of Kurowicka-Cooke 2006)

An R-vine $\mathcal{V}$ on $d$ variables is a set of trees $T_1,\dots ,T_{d-1}$ with node sets $N_i$ and edge sets $E_i$ satisfying:

1. Tree $T_1$ has nodes $N_1=\{1,\dots ,d\}$ and edges $E_1$.
2. For $i=2,\dots ,d-1$, the nodes of $T_i$ are the edges of $T_{i-1}$: $N_i=E_{i-1}$.
3. Proximity condition: if two edges of $T_i$ are joined as nodes by an edge in $T_{i+1}$, they must share a common node in $T_i$.

Each edge $e\in E_i$ is associated with a bivariate copula $C_{j(e),k(e)\mid D(e)}$; $\{j(e),k(e)\}$ are the conditioned nodes, $D(e)$ the conditioning set, and $\{j(e),k(e),D(e)\}$ the constraint set. There are $d(d-1)/2$ edges (pair-copulae) in total.

C-vine (canonical vine)

In a C-vine, each tree $T_j$ has a unique pivotal node connected to $n-j$ edges (a star). One variable is the “root” of tree 1 and conditions everything; the next variable roots tree 2, and so on. The $n$-dimensional C-vine density is

$$\prod _{k=1}^{n}f(x_k)\prod _{j=1}^{n-1}\prod _{i=1}^{n-j}{c}_{j,j+i\mid 1,\dots ,j-1}\{F(x_j\mid x_1,\dots ,x_{j-1}),F(x_{j+i}\mid x_1,\dots ,x_{j-1})\}.$$

Use it when a particular variable is known to govern interactions in the dataset; place that key variable at the root (e.g. a market index driving individual stocks).

D-vine (drawable vine)

In a D-vine, no node in any tree is connected to more than two edges — each tree is a path/line. Variables sit in a sequence and only “neighbouring” conditioned pairs are linked. The $n$-dimensional D-vine density is

$$\prod _{k=1}^{n}f(x_k)\prod _{j=1}^{n-1}\prod _{i=1}^{n-j}{c}_{i,i+j\mid i+1,\dots ,i+j-1}\{F(x_i\mid x_{i+1},\dots ,x_{i+j-1}),F(x_{i+j}\mid x_{i+1},\dots ,x_{i+j-1})\},$$

where index $j$ identifies the tree and $i$ runs over the edges in each tree. Use it when variables have a natural linear ordering (e.g. a time/serial order, or a maturity ladder) and no single dominant variable exists.

When each structure is appropriate

• C-vine — one pivotal/key variable; star trees; conditioning sets grow $\{1\},\{1,2\},\dots$
• D-vine — natural ordering, no dominant variable; path trees; conditioning sets are the variables “between” the conditioned pair.
• General R-vine — neither star nor path; the most flexible structure, selected by Dißmann’s algorithm. For $d=3$ all three coincide.

Examples

5-dimensional D-vine (Fig. 2)

Trees: $T_1$ edges $52,23,34,41$; $T_2$ edges $53\mid 2,24\mid 3,31\mid 4$; $T_3$ edges $54\mid 23,21\mid 34$; $T_4$ edge $51\mid 234$. Density:

$$\begin{array}{rl}f(x_1,\dots ,x_5)=& f_1{f}_2{f}_3{f}_4{f}_5\\ & \cdot c_{52}\{F_5,F_2\}{c}_{23}\{F_2,F_3\}{c}_{34}\{F_3,F_4\}{c}_{41}\{F_4,F_1\}\\ & \cdot c_{53\mid 2}\{F(x_5\mid x_2),F(x_3\mid x_2)\}{c}_{24\mid 3}\{F(x_2\mid x_3),F(x_4\mid x_3)\}{c}_{31\mid 4}\{F(x_3\mid x_4),F(x_1\mid x_4)\}\\ & \cdot c_{54\mid 23}\{F(x_5\mid x_2,x_3),F(x_4\mid x_2,x_3)\}{c}_{21\mid 34}\{F(x_2\mid x_3,x_4),F(x_1\mid x_3,x_4)\}\\ & \cdot c_{51\mid 234}\{F(x_5\mid x_2,x_3,x_4),F(x_1\mid x_2,x_3,x_4)\}.\end{array}$$

5-dimensional C-vine with Variable 1 at the root (Fig. 3)

Trees: $T_1$ edges $12,13,14,15$ (variable 1 pivotal); $T_2$ edges $23\mid 1,24\mid 1,25\mid 1$; $T_3$ edges $34\mid 12,35\mid 12$; $T_4$ edge $45\mid 123$. Density:

$$\begin{array}{rl}f(x_1,\dots ,x_5)=& f_1{f}_2{f}_3{f}_4{f}_5\\ & \cdot c_{12}\{F_1,F_2\}{c}_{13}\{F_1,F_3\}{c}_{14}\{F_1,F_4\}{c}_{15}\{F_1,F_5\}\\ & \cdot c_{23\mid 1}\{F(x_2\mid x_1),F(x_3\mid x_1)\}{c}_{24\mid 1}\{F(x_2\mid x_1),F(x_4\mid x_1)\}{c}_{25\mid 1}\{F(x_2\mid x_1),F(x_5\mid x_1)\}\\ & \cdot c_{34\mid 12}\{F(x_3\mid x_1,x_2),F(x_4\mid x_1,x_2)\}{c}_{35\mid 12}\{F(x_3\mid x_1,x_2),F(x_5\mid x_1,x_2)\}\\ & \cdot c_{45\mid 123}\{F(x_4\mid x_1,x_2,x_3),F(x_5\mid x_1,x_2,x_3)\}.\end{array}$$

Here Variable 1 (e.g. a market factor) is conditioned on in every higher tree.

Connections

• Pair-Copula Constructions — the general R-vine density these structures specialize.
• Vine Copulas - Overview — where C/D/R-vines sit among high-dimensional copulae.
• Estimation and Structure Selection for Vines — choosing among the $2^{\left(\genfrac{}{}{0ex}{}{d-2}{2}\right)-1}d!$ possible R-vines.
• The Simplifying Assumption — applies to the conditional pair-copulae in every higher tree.

See Also

• Factor Copulas - Overview — a C-vine rooted at a market variable parallels a one-factor model in spirit.
• Dependence Modeling