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There are various approaches to the construction of multivariate copulas in the literature. Some of the more popular approaches are considered in this section.

5.2.1 Copula Product Operation

The first approach to the construction of multivariate copulas make use of the product

operation for copulas that was proposed by Darsow et al. (1992). The idea behind

this approach is to deal directly with the compatibility problem usually associated with multivariate copulas.

Theorem 5.1 Let A and B be two copulas, then the product of A and B is the function

A ? B : I2 → I defined as (A ? B) (u, v) = 1 Z 0 ∂A (u, t) ∂t ∂B (t, v) ∂t dt. (5.1)

Please refer to Nelsen (2006) or Laforge (2007a) for a proof that A ? B is a valid copula. Theorem 5.1 can be used to derive copula functions that are compatible with A and B. Please also refer to the work of Durante et al. (2007) and Kolesarova & Sempi (2006), which studies the compatibility of the copulas using the product construction.

The multivariate extension is discussed by Laforge (2007a):

C (u1, . . . , un) = 1 Z 0 n Y i=1 ∂Ci(ui, t) ∂t ! dt. (5.2)

Laforge proposes a new approach to extend bivariate copulas based on the product op- eration of copulas. He uses the example of a three-dimensional copula and suggests that the product operation of copulas can be used to determine the constraints on one of the margins C (u, v, 1), given that C (1, v, w) and C (u, 1, w) are known, to ascertain that a

three-dimensional copula C (u, v, w) exists and has those three bivariate copulas as mar- gins. The solution, however, is not unique and it is not clear how the properties of the bivariate copulas are transferred to the multivariate copula.

5.2.2 Laplace Transform

The second approach to construct Archimedean n-copulas is to note the relationship between the Archimedean copula generator function and the Laplace transform (Joe, 1997; McNeil et al., 2005).

Definition 5.1 Let L = ϕ(−1)and G be a distribution function of a non-negative random

viariable, then the Laplace transform is defined as

L (s) =

Z ∞

0

e−stdG (t)

for s ≥ 0.

The Laplace transform L has these properties : • L (−t) is the moment generating function of G.

• L is continuous and strictly decreasing with L (0) = 1 and we assume L (∞) = 0.

• The functional inverse L−1_{is strictly decreasing with L}−1_{(0) = ∞ and L}−1_{(1) = 0.}

• L is a completely monotonic function.

To derive the n-copula function, let V be a random variable with distribution function

G and let U1, ...Un be a sequence of random variables conditionally independent given V

with conditional distribution function given by F_Ui|V (u|v) = exp −vL−1(u)
. Then

C (u1, · · · , un) = L L−1(u1) + · · · + L−(un)

where C denotes an Archimedean copula with generator ϕ = L−1. Copulas derived

this way are also referred to as LT-Archimedean copulas in the literature (McNeil et al., 2005).

5 Constructing Multivariate Archimedean Copulas: Part I 5.2.3 Copula Vines

The third construction approach is based on the ideas of Kimberling and this is also the focus of this chapter. In Chapter 2 it is shown that the bivariate Archimedean copula is defined in terms of a continuous and strictly decreasing generator function ϕ by

C (u1, u2) = ϕ[−1](ϕ (u1) + ϕ (u2)) (5.3)

where ϕ[−1] is the pseudo-inverse of ϕ. C is a copula if and only if ϕ is convex, that is

ϕ00 > 0 . If ϕ (0) = ∞ then ϕ is called a strict generator and it follows that ϕ[−1](t) =

ϕ(−1)_{(t). The copula derived from a strict generator function is also called a strict}

Archimedean copula (Nelsen, 2006).

Kimberling (1974) shows that for a strict generator function, it follows from equation (5.3) that

C (u1, u2, u3) = C (C (u1, u2) , u3) = ϕ[−1](ϕ (u1) + ϕ (u2) + ϕ (u3))

C (u1, u2, u3, u4) = C (C (u1, u2, u3) , u4) = ϕ[−1](ϕ (u1) + ϕ (u2) + ϕ (u3) + ϕ (u4))

so that in general a multivariate copula can be constructed using

C (u1, · · · , un) = ϕ[−1](ϕ (u1) + · · · + ϕ (un)) . (5.4)

The issue with the Kimberling construction is that the same generator function is used repeatedly to construct a multivariate copula, which leads to a multivariate copula that can only capture very limited types of dependence. Another problem is that only a single parameter α is used to model the dependence over all n dimensions of the copula. This approach cannot be used to accurately capture the specific dependence between different combinations of variables.

The following definition establishes the conditions under which the multivariate copula derived from the Kimberling construction, will be valid (Nelsen, 2006).

Definition 5.2 A decreasing function f (t) is completely monotonic on an interval [a, b]when the function is continuous in that interval and has derivatives of all orders that alternate in sign, in other words

(−1)k d

k

dtkf (t) ≥ 0

for k ∈ N and t ∈ [a, b]. The derivatives of an m-monotonic function are defined and alternate in sign, but only up to order m.

Definition 5.3 The n-copula C : [0, 1] → [0, 1] defined by

C (u1, · · · , un) = ϕ[−1](ϕ (u1) + · · · + ϕ (un))

is a n-copula for all n ≥ 2 if and only if ϕ[−1] is completely monotonic on [0, ∞). If ϕ[−1]

is only m-monotonic on [0, ∞) for a m ≥ 2, then C is an n-copula for 2 ≤ n ≤ m.

McNeil & Nešlehová (2009) explore less restrictive conditions that allow for m-dimensional Archimedean copulas without densities in Definition 5.4.

Definition 5.4 Let ϕ(−1) denote the inverse of an Archimedean copula generator func-

tion, then ϕ(−1) is m-monotone on [0, ∞) if and only if (−1)m−2ϕ−1(m−2)exist on (0, ∞)

and is non-negative, non-increasing and convex there.

A popular way to construct multivariate Archimedean copulas is to use copula vines.

Figure 5.1: Fully nested copula vine

Copula vines provide alternatives to the Kimberling construction, where the n-copula function is derived by adding one dimension at a time using a single generator function. When relaxing the assumption that the same generator is used in each step, it is possible to derive the fully nested copula (Savu & Trede, 2006). The structure of the fully nested copula is illustrated in Figure 5.1 for four variables. The idea is that the first two variables

5 Constructing Multivariate Archimedean Copulas: Part I

C11(u1, u2) = ϕ−111 (ϕ11(u1) + ϕ11(u2))

the third variable is then added to the structure and modelled with copula C21

C (u1, u2, u3) = C21(C11(u1, u2) , u3)

= ϕ−1₂₁ ϕ21◦ ϕ−111 (ϕ11(u1) + ϕ11(u2)) + ϕ21(u3)
. The process leads to an n-copula with a relatively involved equation:

C (u1, · · · , un) = ϕn−1,1−1 (ϕn−1,1◦ ϕ−1n−2,1(ϕn−2,1 · · · ϕ21◦ ϕ−111 (ϕ11(u1) + ϕ11(u2)) + ϕ21(u3)

+ · · · + ϕn−2,1(un−1)) + ϕn−1,1(un)) .

The dependence structure of the fully-nested n-copula is more general than (5.4) in that there are n − 1 parameters to be estimated and thus captures the dependence between the different variable pairs more accurately.

An alternative way to link the bivariate copulas is to use the partially nested copula construction, also sometimes referred to as the hierarchical copula. The 4-copula derived from the partially-nested copula structure is illustrated in Figure 5.2 and is denoted mathematically as:

C (u1, u2, u3, u4) = C21(C11(u1, u2) , C12(u3, u4))

= ϕ−1₂₁ ϕ21◦ ϕ−111 (ϕ11(u1) + ϕ11(u2)) + ϕ21◦ ϕ−112 (ϕ12(u3) + ϕ12(u4))

The density functions of the multivariate copulas constructed with the vines are discussed in Bedford & Cooke (2002) and Czado (2010). Algorithms that can be used to estimate the parameters and simulate values from copula vines can be found in Aas et al. (2009) and Kurowicka & Joe (2011). Specifically the d-vine construction will be used in Chapter 8.

Figure 5.2: Partially nested copula vine

5.3 Parameter Constraints when Modelling Positive