Pair-copula constructions of multiple dependence
Kjersti Aas†
The Norwegian Computing Center, Oslo, Norway
Claudia Czado
Technische Universit
¨
at, M
¨
unchen, Germany
Arnoldo Frigessi
University of Oslo and The Norwegian Computing Center, Norway
Henrik Bakken
The Norwegian University of Science and Technology, Trondheim,Norway‡
Abstract. Building on the work of Bedford, Cooke and Joe, we show how multivariate data,
which exhibit complex patterns of dependence in the tails, can be modelled using a cascade
of pair-copulae, acting on two variables at a time. We use the pair-copula decomposition of
a general multivariate distribution and propose a method to perform inference. The model
construction is hierarchical in nature, the various levels corresponding to the incorporation
of more variables in the conditioning sets, using pair-copulae as simple building blocs. Pair-
copula decomposed models also represent a very ¤exible way to construct higher-dimensional
coplulae. We apply the methodology to a £nancial data set. Our approach represents the
£rst step towards developing of an unsupervised algorithm that explores the space of possible
pair-copula models, that also can be applied to huge data sets automatically.
1. Introduction
The pioneering work of Bedford and Cooke (2001b, 2002), also based on Joe (1996), which
introduces a probabilistic construction of multivariate distributions based on the simple
building blocs called pair-copulae, has remained completely overseen. It represents a rad-
ically new way of constructing complex multivariate highly dependent models, which par-
allels classical hierarchical modelling (Green et al., 2003). There, the principle is to model
dependency using simple local building blocs based on conditional independence, e.g. cliques
in random fields. Here, the building blocs are pair-copulae. The modelling scheme is based
on a decomposition of a multivariate density into a cascade of pair copulae, applied on
original variables and on their conditional and unconditional distribution functions. In this
paper, we show that this decomposition can be a central tool in model building, not requir-
ing conditional independence assumptions when these are not natural, but maintaining the
logic of building complexity by simple elementary bricks. We present some of the theory of
†Address for correspondence: The Norwegian Computing Center, P.O. Box 114 Blindern, N-0314
Oslo, Norway, Kjersti.Aas@nr.no
‡Henrik Bakkens part of the work described in this paper was conducted at the Norwegian
Computing Center while he worked on his diploma thesis.
2 K. Aas, C. Czado, A. Frigessi, H. Bakken
Bedford and Cooke (2001b, 2002) from a more practical point of view, as a general mod-
elling approach, concentrating on inference based on n variables repeatedly observed, say
over time.
Building higher-dimensional copulae is generally recognised as a difficult problem. There
is a huge number of parametric bivariate copulas, but the set of higher-dimensional copu-
lae is rather limited. There have been some attempts to construct multivariate extensions
of Archimedean bivariate copulae, see e.g. Embrechts et al. (2003) and Savu and Trede
(2006). However, it is our opinion that the pair-copula decomposition treated in this pa-
per represents a more flexible and intuitive way of extending bivariate copulae to higher
dimensions.
The paper is organised as follows. In Section 2 we introduce the pair-copula decompo-
sition of a general multivariate distribution and illustrate this with some simple examples.
In Section 3 we see the effect of conditional independence, if assumed, on the pair-copula
construction. Section 4 describes how to simulate from pair-copula decomposed models.
In Section 5 we describe our estimation procedure, while Section 6 reviews several basic
pair-copulae useful in model constructions. In Section 7 we discuss aspects of the model
selection process. In Section 8 we apply the methodology, and discuss its limitations and
difficulties in the context of a financial data set. Finally, Section 9 contains some concluding
remarks.
2. A pair-copula decomposition of a general multivariate distribution
Consider n random variables X = (X
1
, . . . , X
n
) with a joint density function f(x
1
, . . . , x
n
).
This density can be factorised as
f(x
1
, . . . , x
n
) = f(x
n
) · f(x
n−1
|x
n
) · f(x
n−2
|x
n−1
, x
n
) . . . · f(x
1
|x
2
, . . . , x
n
), (1)
and this decomposition is unique up to a relabelling of the variables.
In a sense every joint distribution function implicitly contains both a description of the
marginal behaviour of individual variables and a description of their dependency structure.
Copulae provide a way of isolating the description of their dependency structure. A copula is
multivariate distribution, C, with uniformly distributed marginals U(0, 1) on [0,1]. Sklar’s
theorem (Sklar, 1959) states that every multivariate distribution F with marginals F
1
,
F
2
,. . . ,F
n
can be written as
F (x
1
, . . . , x
n
) = C(F
1
(x
1
), F
2
(x
2
), ...., F
n
(x
n
)), (2)
for some apropriate n-dimensional copula C. In fact, the copula from (2) has the expression
C(u
1
, . . . , u
n
) = F (F
−1
1
(u
1
), F
−1
2
(u
2
), . . . , F
−1
n
(u
n
)),
where the F
−1
i
’s are the inverse distribution functions of the marginals.
Passing to the joint density function f, for an absolutely continous F with strictly
increasing, continuous marginal densities F
1
, . . . F
n
(McNeil et al., 2006), we have
f(x
1
, . . . , x
n
) = c
12···n
(F
1
(x
1
), . . . F
n
(x
n
)) · f
1
(x
1
) ···f
n
(x
n
) (3)
for some (uniquely identified) n-variate copula density c
12···n
(·). In the bivariate case (3)
simplifies to
f(x
1
, x
2
) = c
12
(F
1
(x
1
), F
2
(x
2
)) · f
1
(x
1
) · f
2
(x
2
),
Pair-copula constructions 3
where c
12
(·, ·) is the appropriate pair-copula density for the pair of transformed variables
F
1
(x
1
) and F
2
(x
2
). For a conditional density it easily follows that
f(x
1
|x
2
) = c
12
(F
1
(x
1
), F
2
(x
2
)) · f
1
(x
1
),
for the same pair-copula. For example, the second factor, f (x
n−1
|x
n
), in the right hand
side of (1) can be decomposed into the pair-copula c
(n−1)n
(F (x
n−1
), F (x
n
)) and a marginal
density f
n
(x
n
). For three random variables X
1
, X
2
and X
3
we have that
f(x
1
|x
2
, x
3
) = c
12|3
(F
1|3
(x
1
|x
3
), F
2|3
(x
2
|x
3
)) · f(x
1
|x
3
), (4)
for the appropriate pair-copula c
12|3
, applied to the transformed variables F (x
1
|x
3
) and
F (x
2
|x
3
). An alternative decomposition is
f(x
1
|x
2
, x
3
) = c
13|2
(F
1|2
(x
1
|x
2
), F
3|2
(x
3
|x
2
)) · f(x
1
|x
2
), (5)
where c
13|2
is different from the pair-copula in (4). Decomposing f (x
1
|x
2
) in (5) further,
leads to
f(x
1
|x
2
, x
3
) = c
13|2
(F
1|2
(x
1
|x
2
), F
3|2
(x
3
|x
2
)) · c
12
(F
1
(x
1
), F
2
(x
2
)) · f
1
(x
1
),
where two pair-copulae are present.
It is now clear that each term in (1) can be decomposed into the appropriate pair-copula
times a conditional marginal density, using the general formula
f(x|v) = c
xv
j
|v
−j
(F (x|v
−j
), F (v
j
|v
−j
)) · f(x|v
−j
),
for a d-dimensional vector v. Here v
j
is one arbitrarily chosen component of v and v
−j
denotes the v-vector, excluding this component. In conclusion, under appropriate regularity
conditions, a multivariate density can be expressed as a product of pair-copulae, acting on
several different conditional probability distributions. It is also clear that the construction
is iterative in its nature, and that given a specific factorisation, there are still many different
reparameterisations.
The pair-copula construction involves marginal conditional distributions of the form
F (x|v). For every j, Joe (1996) showed that
F (x|v) =
∂ C
x,v
j
|v
−j
(F (x|v
−j
), F (v
j
|v
−j
))
∂F (v
j
|v
−j
)
, (6)
where C
ij|k
is a bivariate copula distribution function. For the special case where v is
univariate we have
F (x|v) =
∂ C
xv
(F
x
(x), F
v
(v))
∂F
v
(v)
.
In Sections 4-7 we will use the function h(x, v, Θ) to represent this conditional distribution
function when x and v are uniform, i.e. f(x) = f(v) = 1, F (x) = x and F (v) = v. That is,
h(x, v, Θ) = F (x|v) =
∂ C
x,v
(x, v, Θ)
∂v
, (7)
where the second parameter of h(·) always corresponds to the conditioning variable and Θ
denotes the set of parameters for the copula of the joint distribution function of x and v.
Further, let h
−1
(u, v, Θ) be the inverse of the h-function with respect to the first variable
u, or equivalently the inverse of the conditional distribution function.
4 K. Aas, C. Czado, A. Frigessi, H. Bakken
1 2 3 4 5
12 23 34 45
12 23 34 45
13|2 24|3 35|4
13|2 24|3 35|4
14|23 25|34
14|23 25|34
15|234
T
1
T
2
T
3
T
4
Figure 1. A D-vine with 5 variables, 4 trees and 10 edges. Each edge may be may be associated
with a pair-copula.
2.1. Vines
For high-dimensional distributions, there are a significant number of possible pair-copulae
constructions. For example, as will be shown in Section 2.4, there are 240 different construc-
tions for a five-dimensional density. To help organising them, Bedford and Cooke (2001b,
2002) have introduced a graphical model denoted the regular vine. The class of regular vines
is still very general and embraces a large number of possible pair-copula decompositions.
Here, we concentrate on two special cases of regular vines; the canonical vine and the D-vine
(Kurowicka and Cooke, 2004). Each model gives a specific way of decomposing the density.
The specification may be given in form of a nested set of trees. Figure 1 shows the specifi-
cation corresponding to a five-dimensional D-vine. It consists of four trees T
j
, j = 1, . . . 4.
Tree T
j
has 6 − j nodes and 5 − j edges. Each edge corresponds to a pair-copula density
and the edge label corresponds to the subscript of the pair-copula density, e.g. edge 14|23
corresponds to the copula density c
14|23
(·). The whole decomposition is defined by the
n(n −1)/2 edges and the marginal densities of each variable. The nodes in tree T
j
are only
necessary for determining the labels of the edges in tree T
j+1
. As can be seen from Figure
1, two edges in T
j
, which become nodes in T
j+1
, are joined by an edge in T
j+1
only if these
edges in T
j
share a common node. Note that the tree structure is not strictly necessary
for applying the pair-copula methodology, but it helps identifying the different pair-copula
decompositions.
Bedford and Cooke (2001b) give the density of an n-dimensional distribution in terms
of a regular vine, which we specialise to a D-vine and a canonical vine. The density
