Pair-Copula Constructions

Summary

A pair-copula construction (PCC) decomposes a $d$-dimensional density into the product of the $d$ marginal densities and $d(d-1)/2$ bivariate (pair) copula densities, each evaluated at conditional distribution functions of pairs of variables given a conditioning set. The conditional margins that serve as the pair-copula arguments are computed by a recursive partial-derivative formula, and — for a regular vine — the needed conditional copulae always appear in earlier trees, so they are available without extra work.

Overview

By Sklar’s theorem a joint density factors into margins times a copula density. The PCC goes further and factors the copula density itself into bivariate pieces. Joe (1996) observed that any multivariate density can be written using only bivariate copulae and conditional distributions; the regular vine organizes which pairs and which conditioning sets appear. The arguments of the pair-copulae are conditional distributions in every tree except the first, where they are the univariate margins.

Main Content

The R-vine density factorization (Eq. 1)

Let $\mathbf{X}=(X_1,\dots ,X_d)$ follow an R-vine distribution with trees $T_1,\dots ,T_{d-1}$, edge sets $E_i$, and for each edge $e$ a conditioned pair $\{j(e),k(e)\}$ and conditioning set $D(e)$ (with ${\mathbf{x}}_{D(e)}$ the corresponding subvector). The joint density is

$$f(x_1,\dots ,x_d)=\left[\prod _{k=1}^d{f}_k(x_k)\right]\times \left[\prod _{i=1}^{d-1}\prod _{e\in E_i}{c}_{j(e),k(e)\mid D(e)}\left(F\right(x_{j(e)}\mid {\mathbf{x}}_{D(e)}),F(x_{k(e)}\mid {\mathbf{x}}_{D(e)}\left)\right)\right].$$

The right factor is a product of exactly $d(d-1)/2$ bivariate copula densities and is called the R-vine copula. Conditioning sets are empty in $T_1$, contain one variable in $T_2$, two in $T_3$, and so on.

Free choice of pair-copulae

The key property: all copulae in the decomposition are bivariate and may belong to different families (Gaussian, $t$, Clayton, Gumbel, …) with their own parameters. There are no restrictions on which copula types can be combined — the resulting multivariate structure is guaranteed to be a valid copula regardless. This is what makes PCCs able to characterize highly heterogeneous, asymmetric, tail-dependent multivariate behaviour.

Recursive conditional-distribution formula (Eq. 2)

The conditional CDFs that serve as pair-copula arguments are obtained recursively (Joe 1996). For an arbitrary component $v_j$ of the conditioning vector $\mathbf{v}$, with ${\mathbf{v}}_{-j}$ denoting $\mathbf{v}$ excluding $v_j$:

$$F(x\mid \mathbf{v})=\frac{\partial C_{x{v}_j\mid {\mathbf{v}}_{-j}}(F(x\mid {\mathbf{v}}_{-j}),F(v_j\mid {\mathbf{v}}_{-j}))}{\partial F(v_j\mid {\mathbf{v}}_{-j})},$$

where $C_{x{v}_j\mid {\mathbf{v}}_{-j}}$ is a bivariate copula. The single-conditioning special case $F(x\mid v)=\partial C_{xv}(F(x),F(v))/\partial F(v)$ is the familiar h-function. By construction of an R-vine, the copula $C_{x{v}_j\mid {\mathbf{v}}_{-j}}$ appears in a preceding tree, so the conditional margins are available without extra computation.

R-vine matrix and density via $M$ (Eqs. 3-4)

To store the index structure efficiently, Morales-Napoles (2011) uses a lower-triangular $d\times d$ R-vine matrix $M=(m_{i,j})$ whose diagonal entries are the first-tree nodes; each row from the bottom up encodes a tree. The conditioned set of a node is read from a diagonal entry and the column entry of the current row; the conditioning set from the column entries below. The density can then be written compactly as

$$f(x_1,\dots ,x_d)=\left[\prod _{k=1}^d{f}_k(x_k)\right]\times \left[\prod _{j=d-1}^1\prod _{i=d}^{j+1}{c}_{m_{j,j},m_{i,j}\mid m_{i+1,j},\dots ,m_{d,j}}\right],$$

with pair-copula arguments $F(x_{m_{j,j}}\mid x_{m_{i+1,j}},\dots ,x_{m_{d,j}})$ and $F(x_{m_{i,j}}\mid x_{m_{i+1,j}},\dots ,x_{m_{d,j}})$. Copula types and parameters are stored in companion matrices shaped like $M$.

Examples

A 4-dimensional D-vine decomposition

Take variables $1-2-3-4$ in a D-vine (path) order. The three trees are

• $T_1$: edges $12,23,34$ (raw margins),
• $T_2$: edges $13\mid 2,24\mid 3$,
• $T_3$: edge $14\mid 23$.

The density factorizes ($d(d-1)/2=6$ pair-copulae) as

$$f(x_1,x_2,x_3,x_4)=f_1{f}_2{f}_3{f}_4\times c_{12}{c}_{23}{c}_{34}\times c_{13\mid 2}\{F(x_1\mid x_2),F(x_3\mid x_2)\}{c}_{24\mid 3}\{F(x_2\mid x_3),F(x_4\mid x_3)\}\times c_{14\mid 23}\{F(x_1\mid x_2,x_3),F(x_4\mid x_2,x_3)\}.$$

The conditional margins are built by the recursion (Eq. 2), e.g. $F(x_1\mid x_2)=\partial C_{12}(F_1,F_2)/\partial F_2$, and $F(x_1\mid x_2,x_3)=\partial C_{13\mid 2}(F(x_1\mid x_2),F(x_3\mid x_2))/\partial F(x_3\mid x_2)$ — each using a copula from a previous tree.

A 3-dimensional construction (both vine types coincide)

For $d=3$ every R-vine is simultaneously a C-vine and a D-vine. With order $1-2-3$:

$$f(x_1,x_2,x_3)=f_1{f}_2{f}_3\times c_{12}{c}_{23}\times c_{13\mid 2}\{F(x_1\mid x_2),F(x_3\mid x_2)\}.$$

Three pair-copulae ($3\cdot 2/2=3$) reconstruct the full trivariate dependence.

Connections

• Vine Copulas - Overview — the big-picture motivation and place among high-dimensional copulae.
• C-vines, D-vines, and Regular Vines — how the choice of trees specializes this general factorization.
• The Simplifying Assumption — the assumption that lets each $c_{\cdot \cdot \mid D}$ ignore the conditioning value.
• Estimation and Structure Selection for Vines — fitting the families and parameters in this product.
• Copula Estimation — estimation of the bivariate building blocks.

See Also

• Factor Copulas - Overview — contrast: a latent-factor rather than pairwise copula factorization.
• Dependence Modeling