An Econometric Study of Vine Copulas

Dominique Guegan, Pierre-André Maugis

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 Documents de Travail du
Centre d’Economie de la Sorbonne

An Econometric Study of Vine Copulas

Dominique GUEGAN, Pierre-André MAUGIS

2010.40

Maison des Sciences Économiques, 106-112 boulevard de L'Hôpital, 75647 Paris Cedex 13
http://ces.univ-paris1.fr/cesdp/CES-docs.htm
ISSN : 1955-611X
 An Econometric Study Of Vine Copulas

D. Guégan∗and P.A. Maugis†

PSE, Université Paris 1 Panthéon-Sorbonne,

106 boulevard de l’Hopital 75647 Paris Cedex 13, France

Abstract

We present a new recursive algorithm to construct vine copulas based

on an underlying tree structure. This new structure is interesting to
compute multivariate distributions for dependent random variables. We
proove the asymptotic normality of the vine copula parameter estimator
and show that all vine copula parameter estimators have comparable vari-
ance. Both results are crucial to motivate any econometrical work based
on vine copulas. We provide an application of vine copulas to estimate
the V aR of a portfolio, and show they offer significant improvement as
compared to a benchmark estimator based on a GARCH model.

Keywords: Vines Copulas – Conditional Copulas – Risk management.

JEL: D81 - C10 - C40 - C 52

1 Introduction

For almost ten years now, copulas have been used in econometrics and finance.
They became an essential tool for pricing complex products, managing portfolios
and evaluating risks in banks and insurance companies. For instance, they can
be used to compute V aR (Value at Risk) and ES (Expected shortfall), Artzner
et al. (1997). Moreover, copulas appear to be a very flexible tool, allowing
for semi-parametric estimation, fast parameter optimisation and time varying
parameters. These advantages make them a very interesting tool, although one
major shortcoming is their use in high dimension. Indeed, elliptical copulas can
be expended to higher dimension, but they are unable to model for financial tail
dependences (Patton, 2009), and the Archimedean copulas are not satisfactory
∗ email: dominique.guegan@univ-paris1.fr.
† e-mail: pierre-andre.maugis@malix.univ-paris1.fr.

Documents de Travail du Centre d'Economie de la Sorbonne - 2010.40

as models to describe multivariate dependence in dimensions higher than 2 (Joe,
1997).
The objective of this paper is twofold. In a first step we introduce a new re-
cursive algorithm to construct vine copulas. This new type of copulas permits
to estimate the dependence between random variables in any dimensions (Joe
(1997) and Bedford and Cooke (2002, 2001)). Vine copulas have already been
studied by several authors, focussing on information optimisation and algorithm
efficiency. They are introduced as decomposition of a multivariate random vec-
tor density based on a graph structure called "vines", Bedford and Cooke (2002,
2001) and Cooke (1997). We propose an other approach considering an algo-
rithm based on step-by-step factorisation of the density function in a product of
bivariate copulas. This method permits to exhibit the underlying tree structure
of vine copulas that will be central in the proofs of the theorems of convergence.
In a second part, we provide new result on the estimator of the vine copulas
built in the previous step. Indeed, denoting θ the parameter of a vine copula,
we
√ prove the asymptotic normality of the estimator with a convergence rate of
T , where T is the sample size. This new result justifies the use of vine copulas
in economic applications (Aas et al. (2009), Berg and Aas (2010), Czado et al.
(2009), Fischer et al. (2007), Chollete et al. (2008) and Guégan and Maugis
(2010)), and also provides confidence intervals. Finally, we study the variance
behavior of the estimates across vine copulas and show that any two vine copulas
estimates have comparable asymptotic variance. Our result proves that all vine
copula should be used, and that there are no statistical ground for favoring a
subset of them over another one since they are all efficient estimators in terms of
rate and speed of convergence. It confirms the fact that using all possible vine
copulas permits to describe more varied dependence (Guégan and Maugis, 2010).
Our results are prooved under a regular set of hypothesis commonly found when
using copulas (Patton, 2009) and holds for any type of bivariate copulas, this
includes conditional copulas, Markov switching copulas and mixture copulas.
The paper is organised as follow: In section 2 we construct the set of vines
decomposition we work on and present its underlying tree structure. In section
3 we derive the asymptotic properties of the vines as estimators and bound their
relative variance. Section 4 presents an application and section 5 concludes.
We now recall the definition of a copula, Sklar (1959). Let X = (X1 , X2 , ..., Xn )
be a vector of random variables, with joint distribution F and marginal distri-
butions F1 , . . . , Fn , then there exists a function C – called copula – mapping the
individual distribution to the joint distribution:
F (x1 , . . . , xn ) = C(F1 (x1 ), F2 (x2 ), ..., Fn (xn ))
Let Y be another vector of random variables. We call F̃ the distribution function
of (X∣Y ). Patton (2006) defines the conditional copula of (X∣Y ) as the function
mapping the individual distributions to the conditional distribution:
F̃ (x1 , . . . , xn ∣y1 , . . . , yp ) = C(F1 (x1 ), F2 (x2 ), ..., Fn (xn )∣y1 , . . . , yp )

Documents de Travail du Centre d'Economie de la Sorbonne - 2010.40

2 Vine Construction

In this section, we introduce a new algorithm to build vine copulas. Our ap-
proach has the advantage of being able to coherently describe a large set of vine
copulas – N = n! n−3
2 ∏i=1
i! in dimension n – while also being a simple recursive
algorithm. Moreover the tree structure and the algorithm are fully recursive so
they can be easily expanded to any dimension.

2.1 Formula

Let us consider a vector X = (X1 , X2 , . . . , Xn ) of random variables characterised

by a joint distribution function FX and we assume it has a density function fX .
We introduce some notations:

• X −α = (X1 , . . . , Xα−1 , Xα+1 , . . . , Xn ) is the set of variables except the α-th.

• We denote fα the density of Xα . In the same fashion fα∣β is the density of
(Xα ∣Xβ ), f−α is the density of X −α and fα∣−β is the density of Xα ∣X −β .
We use the similar notation for the distribution function F : for instance
Fα∣−β is the distribution function of Xα ∣X −β .
• cα,β∣γ = cXα ,Xβ ∣Xγ (FXα ∣Xγ (Xα ∣Xγ ), FXβ ∣Xγ (Xβ ∣Xγ )) is the copula density
of (Xα , Xβ ∣Xγ ) as defined in Sklar (1959). Similarly we denote as cα,β∣−γ
the copula density of (Xα , Xβ ∣X−γ ):

cα,β∣−γ = cXα ,Xβ ∣X −γ (FXα ∣X −γ (Xα ∣X −γ ), FXβ ∣X −γ (Xβ ∣X −γ )).

We also use C with the same notations.

Our objective is to compute c1,...,n , the copula density associated with the vector
X. This will be done by factorizing fX in the following form:

fX = ∏ fi ⋅ c1,...,n .
i=1,...n

By construction for n = 2 we have: fα,β = fα .fβ .cα,β . Using this property we

consider the following factorisation of the joint density fX :
2
∀α, β ∈ {1, . . . , n} , α ≠ β
fα,β∣−(α,β) fα∣−(α,β) .fβ∣−(α,β)
fX = f−α .fα∣−α = f−α . = f−α . .cα,β∣−(α,β)
fβ∣−(α,β) fβ∣−(α,β)
f−α .f−β
= .cα,β∣−(α,β) .
f−(α,β)
(1)

Documents de Travail du Centre d'Economie de la Sorbonne - 2010.40

Formula (1) allows the computation of an n-variate density with a bivariate
copula, two (n − 1)-and one (n − 2)-variate densities. Using this factorisation
recursively, insuring that the denominators cancel at each step, we produce a
factorisation of the n-variate density as a product of univariate and bivariate
copula densities. Using this algorithm we can produce all N possible vine cop-
ulas (Napoles, 2007)1,2 .

At each step of the algorithm we associate a tree construction. The root of this
tree is the new copula density term: cα,β∣−(α,β) in expression (1), and the leaves
are the new (n − 1)-variates densities: f−α and f−β in (1). The tree associated
with expression (1) is:

cα,β∣−(α,β)

f−α f−β

This tree structure is also fully recursive. To each term f−α and f−β we could
apply (1), and produce trees. These trees would then be inserted inside the
previous tree replacing the two leaves f−α and f−β by the two new tree roots.
We explain this mechanism further in an example.

2.2 Example

In this example, we illustrate the unwinding of the previous algorithm for n = 4,

providing the joint density function f1,2,3,4 . Our aim is to compute c1,2,3,4 : the
joint copula density. We describe the three steps of the algorithm using the
previous notations. With this example, we detail the construction of the tree
associated with a specific vine:

• First step:
f1,2,3 .f1,2,4
f1,2,3,4 = .c3,4∣1,2 (2)
f1,2
c3,4∣1,2

f1,2,3 f1,2,4
1 This is the number of "vine" type graph with n nodes, which is also the number of vine

copulas (see Bedford and Cooke (2002, 2001)). The proof of the formula relies heavily on the
graph structure of vines.
2 Our algorithm can produce more varied decompositions, however we do not consider those

additional copulas as they are not efficient estimators, Bedford and Cooke (2002, 2001).

Documents de Travail du Centre d'Economie de la Sorbonne - 2010.40

 • Second step: we apply the relationship (1) to f1,2,3 and f1,2,4 and produce
two sub-trees:
f1,2 .f1,3
f1,2,3 = .c2,3∣1 (3)
f1
c2,3∣1

f1,2 f1,3
and
f1,2 .f2,4
f1,2,4 = .c1,4∣2 (4)
f2
c1,4∣2

f2,4 f1,2
• Third step: We merge formulas (2),(3) and (4), we simplify the f1,2 term
and we expand the bivariate densities using the formula: fα,β = fα .fβ .cα,β .
We also merge the trees and underline the simplified term in the tree.
f1,2,3,4 = f1 .f2 .f3 .f4 .c1,3 .c1,2 .c2,4 .c2,3∣1 .c1,4∣2 .c3,4∣1,2 (5)
c3,4∣1,2

c2,3∣1 c1,4∣2

c1,2 c1,3 c2,4 c1,2

f1 f2 f1 f3 f2 f4 f1 f2

We have now factorised the density f1,2,3,4 into a product of four univariate
densities and six bivariate copula densities. By construction, this means that
the copula density of (X1 , X2 , X3 , X4 ) can be factorised as follows:
c1,2,3,4 = c1,3 .c1,2 .c2,4 .c2,3∣1 .c1,4∣2 .c3,4∣1,2 .

The unwinding of this algorithm can yield other vine copulas if other parameters
are used. For instance the following formula is another possible vine copula.
c1,2,3,4 = c2,3 .c3,4 .c1,4 .c2,4∣3 .c1,3∣4 .c1,2∣3,4
And its associated tree:
c1,2∣3,4

c1,3∣4 c2,4∣3

c1,4 c3,4 c2,3 c3,4

f1 f4 f3 f4 f2 f3 f3 f4

Documents de Travail du Centre d'Economie de la Sorbonne - 2010.40

3 Vine Analysis
This section addresses the question of the estimation of a vine copula. For in-
stance if we estimate the vine copula density c1,2,3,4 constructed above with a
4-dimensional vector (X1 , X2 , X3 , X4 ), we need to estimate the bivariate cop-
ulas: c1,3 , c1,2 , c2,4 and the bivariate conditional copula densities: c2,3∣1 , c1,4∣2
and c3,4∣1,2 . To estimate the formers we use standard methods as described in
Patton (2009). We now turn to the problem of estimating the latters. Recall
that for instance:
c3,4∣1,2 = c3,4∣1,2 (F3∣1,2 (X3 ∣X1 , X2 ), F4∣1,2 (X4 ∣X1 , X2 )).
Thus, to estimate c3,4∣1,2 , we need to estimate F3∣1,2 and F4∣1,2 . These condi-
tional distribution functions can be built as follows: for i, k1 , . . . , kp ∈ ⟦1, n⟧p+1
and j ∈ ⟦1, p⟧:
∂Ci,kj ∣k1 ...,kj−1 ,kj+1 ,...,kp
Fi∣k1 ,...,kp = , (6)
∂Fkj ∣k1 ...,kj−1 ,kj+1 ,...,kp
(Joe, 1997). We use the previous tree algorithm to choose the copula C in
formula (6). In our example to compute F3∣1,2 we will use C2,3∣1 and for F4∣1,2
we will use C1,4∣2 .

3.1 The Estimator

Here we describe the statistical procedure to estimate vine copulas and pro-
vide asymptotic results. For n ∈ N, n > 2, we consider a n-variate vector
X = (Xi )i=1,⋯n and for all i = 1, . . . , n we assume that we have an indepen-
dent identically distributed (i.i.d) T -sample3 . Our purpose is to estimate the
parameter θ ∈ Θ of the vine copula density c1,...,n (.; θ). As soon as the random
variables are i.i.d, we use the canonical maximum of likelihood method (White,
1994). We denote L the likelihood:
L(θ) = ET [c1...,n (F1 (X1 ), . . . , Fn (Xn ); θ)] .
ET denotes the sample expectation operator (ET = T −1 ∑Tt=1 ). We denote θ0 the
pseudo true value of the parameter and θ̂T the maximum-likelihood estimator:
θ̂T = argmax L(θ).
θ∈Θ

We introduce some technical assumptions:

• A1 : θ0 is interior to Θ and Θ is bounded.

• A2 : All bivariate copula densities are bounded, C 1 in X and a.s C 2 in their
parameter4 . In a neighborhood of θ0 , all bivariate copula densities are C 2
in their parameter.
3 In practical applications, univariate models would be fit to the marginal process to filter

the data into an i.i.d sample.

4 A map f is C k if it is k times differentiable and its k-th derivative is continuous.

Documents de Travail du Centre d'Economie de la Sorbonne - 2010.40

Theorem 1 Under assumptions A1 and A2 there exist a bounded matrix V
such that: √ D
T V (θ̂T − θ0 ) Ð→ N (0, Ik ).
D 1/2
"Ð→" means convergence in distribution and V = C0 ⋅H0 where H0 = (H0i,j )i,j
2
∂ L
2
with H0i,j = ∣
∂θi ∂θj θ
and C0 = (C0i )i with (C0 )i = ( ∂θ
∂L
i
) ∣ .
0 θ0

Proof The proof can be found in the Annex.

√
This theorem provides the asymptotic normality of θ̂T with a T convergence
rate. This result is central to any econometric applications. We denote now
̂ θ̂t ) the estimator of the covariance matrix of θ̂T , and we have the following
Cov(
result:

Corollary 1 Under assumptions A1 and A2 :

̂ θ̂T ) = H −1 ⋅ C ⋅ H −1 ′ .
Cov(
2
∂2L
H = (Hi,j )i,j with Hi,j = ∣
∂θi ∂θj θ̂
and C = (Ci )i with Ci = ( ∂θ
∂L
i
) ∣ .
T θ̂T

Proof The proof can be found in the Annex.

Now we focus on the heterogeneity within the set of vine copulas in terms of
asymptotic variance.

3.2 Variance comparison

In estimation theory when faced with the choice between two estimators we
privilege the estimator with the smallest variance. Indeed such an estimator will
require a smaller sample size T to obtain significant results. Thus we compare
the variance of two different vine copulas’ estimators.

Theorem 2 Under assumptions A1 , A2 and A3 . Let θ̂T and θ̂T′ be the esti-
mator associated with two different vine copulas, then:

∣∣Cov(θ̂T ) − Cov(θ̂T′ )∣∣ < , (7)

where  is a small real number.

Documents de Travail du Centre d'Economie de la Sorbonne - 2010.40

Proof The proof can be found in the Annex where we specify assumption A3
and provide details on the choice of .

The consequence of this theorem is that there is no evidence indicating that one
should favor one type of vine copula over another. This result has important
practical consequences that are discussed in Guégan and Maugis (2010).

The previous results confirm that we have a good estimators for vine copulas in
terms of rate of convergence and that all vine copulas have comparable variance,
thus their use is justified for applications. We now provide such an example.

4 Application to the CAC40 index

Given the five main assets composing the CAC40, the French leading index,
using the methodology described above, we estimate their joint density in order
to compute the V aR5 of a portfolio composed of the five assets.

The dataset is taken from Datastream, daily quotes from Total, BNP - Paribas,
Sanofi - Synthelabo, GDF-Suez and France Telecom from 25/4/08 to 21/11/08.
This period is marked by the 2008 crisis, and our purpose is to test the resilience
of our model to this shock and change of regimes. We estimate the parameters
and select the vine copula based on the data ranging from 25/4/08 to 9/12/08,
while the V aR is computed from the remaining dates. The portfolio we consider
is as follows: Total (33%), BNP - Paribas (20%), Sanofi - Synthelabo (20%),
GDF-Suez (14%) and France Telecom (13%).

For each dataset a GARCH(p, q) process is selected using the AIC criterion
(Akaike, 1974) and estimated using pseudo likelihood. On the residuals we esti-
mated the vine copula parameters using maximum likelihood. The parametric
copula families used in this exercise are chosen among a panel of copulas which
take into account most features commonly found in financial time series (Patton,
2009): they are the Clayton, Gumbel, Student and Gaussian copulas.

We select the best vine copula using the methodology described in Guégan
and Maugis (2010) with the second test described in Chen et al. (2004). We
recall that this test associates to each estimated density a χ2 sample. In this
application, the retained vine copula is6 :

c1,2,3,4,5 = c2,5 .c4,5 .c1,3 .c1,5 .c1,5∣4 .c1,2∣5 .

5 Given a random variable X, the 10% V aR of X is the value V aR(X) such that:
P (X < V aR(X)) = 0.1.

6 The computation took one hour on a 1.5Ghz processor computer.

Documents de Travail du Centre d'Economie de la Sorbonne - 2010.40

Our final objective is to use this estimated vine copula density to compute
the 10% V aR. We computed it from 9/12/08 to 21/11/08 using Monte-Carlo
based integration and optimisation, see Figure 1. We compared it to a univari-
ate GARCH(p, q) model-based estimate of the V aR computed directly on the
portfolio value time series in the same fashion as Samia et al. (2009). To dis-
criminate between the two approaches, we use the Kupiec test (Kupiec, 1995).
The Kupiec statistic is the number Q of times the out-sample time-series is be-
low the predicted 10% V aR. Under the null of the prediction being a true 10%
V aR the sampling distribution of the statistic follows a binomial distribution of
parameter 0.1.

In our example the vine copula-V aR has a p-value of 0.96 for the Q statistic, so
it is accepted as a true V aR, while the GARCH-V aR has a p-value of 0.00 and
is rejected according to this test7 . Nevertheless, the vine copula approach fails
to predict the major drop during the crisis, but the prediction remains solid
before and after the crisis. These results make the vine copula methodology we
described an interesting approach for risk management in order to estimate the
multivariate (n > 2) density of a portfolio and to compute its associated V aR.

In sample: 04/25/08 to 09/12/08. VaR from 09/12/08 to 11/21/08.

46

44

42

40
Price

38

36

34

32
0 5 10 15 20 25 30 35 40 45 50
Time

Figure 1: In Blue: The CAC40, In Red: Vine VaR Estimation, In Green:

GARCH VaR Estimation.
7Q for the vine copula-V aR is equal to 5 and for the GARCH-V aR is equal to 21.

Documents de Travail du Centre d'Economie de la Sorbonne - 2010.40

5 Conclusion

This paper focuses on the building of vines copulas using a tree-based algorithm.
We provide the asymptotic normality of the vine copula parameter estimate
under regular conditions. We show that, in the case of two competing vine
copulas, no vine copula is better than another one in terms of variance criteria.
This work provides solid statistical ground for the ideas developed in Aas et al.
(2009) and Berg and Aas (2010) and justifies the methodologies used in Czado
et al. (2009), Fischer et al. (2007), Chollete et al. (2008) and Guégan and Maugis
(2010). Moreover, proving that no vine copula or sub-family of vine copula
yields better estimator than others, justifies the use of all N vine copulas which
– as shown in Guégan and Maugis (2010) – enhances vine copulas capacity to
represent more varied distributions. Finally, an application shows vine copulas
usefulness to estimate V aR and provides new and interesting risk management
strategies for managers working with high dimensional portfolios.

Our results opens the possibility for varied uses of vines copulas. Most inter-
estingly, the use of conditional copulas as described in Patton (2006), permits
to relax the conditional independence hypothesis common throughout the vine
copulas literature. This allows for very varied and interesting applications in all
fields of economics and risk management.

6 Acknowledgment

This work has already been presented at the MIT Econometric Seminar (Cam-
bridge, USA, 2009) and at the International Symposium in Computational Eco-
nomics and Finance (Sousse, Tunisia, 2010). We thank the participants of the
seminars and our reviewers for their helpful comments. All remaining errors are
still ours. P.A. Maugis thank Arun Chandrasekhar, Victor Chernozhukov and
Miriam Sofronia for their helpful comments.

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7 Annex

7.1 Proof of Theorem 1

Before providing the proof of Theorem 1, we introduce some definitions and

notations.

7.1.1 Definitions and Notations

Let n ∈ N, n > 2 and X = (Xi )i=1,⋯n . For all i we consider an independent

identically distributed T -sample.

• We note E the distribution expectation operator and ET the sample ex-

pectation operator (ET = T −1 ∑Tt=1 ).
• We use the del operator ∇: for a function f (x1 , . . . , xn ), ∇f is equal to
(∂x1 f, . . . , ∂xn f ).
• For i, j, k1 , . . . , kp ∈ ⟦1, n⟧p+2 , we call s = {i, j∣k1 , . . . , kp } the index of the
following copula density:

cs = ci,j∣k1 ,...,kp (F (Xi ∣(Xl )l=k1 ,...,kp ), F (Xj ∣(Xl )l=k1 ,...,kp )).

For instance {2, 3∣1} is the index of c2,3∣1 (F (x2 ∣x1 ), F (x3 ∣x1 )).

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 • To each vine copula we associate a set M . Let M be the set of the
indexes of the bivariate copulas and marginal densities used to estimated
the associated vine copula. It is also the set of the labels of the nods of
the tree associated with a vine copula. We call Mn the set of all possible
models in dimension n.
For instance for n = 3:

f1,2,3 = f1 ⋅ f2 ⋅ f3 ⋅ c1,2 ⋅ c1,3 ⋅ c2,3∣1 .

Then the model M associated with this vine copula is:

M = {{1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3∣1}}.

• We introduce respectively the following parametric families: for each uni-

variate processes, bivariate copulas and conditional copula parameter,

{γ(.; θ)∣θ ∈ Θ′ } ; {ϕ(., .; θ)∣θ ∈ Θ′′ } ; {ξ(.; θ)∣θ ∈ Θ′′′ } .

We now construct the likelihood functions used to estimate the parameters. For
M ∈ Mn we define the likelihood functions {ψsM T }s∈M as:

∀i, j, k, ∈ ⟦1, n⟧3 i ≠ j ≠ k, ∀θ = (θi , θj , θi,k , θj,k , θi,j∣k ) ∈ Θ′ × Θ′′ × Θ′′′ :

2 2

ψiM T (θ) = ET [log γ M (Xi ; θi )]

M
ψi,j T
(θ) = ET [log ϕM (FiM (Xi ; θi ), FjM (Xj ; θj ); θi,j )]
M
ψi,j∣k (θ) =
T
ET [log ϕM (Fi∣k
M M
(Xi ∣Xk ; θi , θk , θi,k ), Fj∣k (Xj ∣Xk ; θj , θk , θj,k ); ξ(Xk ; θi,j∣k ))]
... = ...

We use M as superscript to denote that the estimation is done according to the

model M . The conditional distribution functions are computed using formula
(6). Using the previous notations the log-likelihood is equal to:

LM (θ) = ∑ ψsM T (θ).

s∈M

Then:

θ̂TM = argmax LM (θ)

θ∈Θ

θ0M = 0
[θM (i) ]i∈⟦1,∣M ∣⟧

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7.1.2 Assumptions

To prove the convergence we make some assumptions that are verified for a vast
majority of parametric copula families:

• θ0 is interior to Θ, the set of possible parameters.

• ∀s ∈ M ψsM , ∇θ,θ ψsM and ∇2θ,θ ψsM are bounded and a.s uniformly contin-
uous in θ and uniformly continuous in a neighborhood of θ0 .
• All ϕ and ξ used for estimation are a.s C 1 in (X)i≤n and a.s C 2 in θ and
are C 1 in (X)i≤n and C 2 in θ in a neighborhood of θ0 .

These assumptions are weaker than assumptions A1 and A2 and are implied by
A1 and A2 .

We introduce the exponent 0 to specify the function are evaluated at θ0M , and
denote:

ϕ0i,j∣k = ϕ(Fi∣k
M
(Xi ∣Xk ; {θ0M }i,k,{i,k} ), Fj∣k
M
(Xj ∣Xk ; {θ0M }j,k,{j,k} ); ξ(Xk ; θ0M i,j∣k )).

Also:

ϕi M,0 = ϕM (Xi ; θi0 ),M ϕi,j 0t = ϕM (F M (Xi ), F M (Xj ); θi0 , θj0 , θi,j
0
),

when we work at θ0M .

7.1.3 Operators

To compute the derivates of LM , we use the following property:

∀M ∈ Mn , ∀i ∈ ⟦1, ∣M ∣⟧, ∀s ∈ M ∖ m(M (i)), ∇θM (i) ψsM T ≡ 0.

Indeed if s ∈ M ∖ m(M (i)) then ψsM T is not dependent in θM (i) so that the
gradient is null.

We introduce M
GradT as the gradient of LM :
M
GradT = ∇LM = (∇θM (i) LM )i∈⟦1,∣M ∣⟧

⎛ ⎞
= ∇θM (i) ∑ ψsM T
⎝ s∈m(M (i)) ⎠
i∈⟦1,∣M ∣⟧

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We define M
HessT , the Hessian matrix of LM :
M
HessT = ∇2 LM = (∇θM (i) ,θM (j) LM )i,j∈⟦1,∣M ∣⟧2

⎛ ⎞
= ∇θM (i) ,θM (j) ∑ ψsM T
⎝ s∈m(M (i))∩m(M (j)) ⎠
i,j∈⟦1,∣M ∣⟧2

√ M
We denote M
CT0 the covariance matrix of T ⋅ GradT (θ0 ):

M M M
CT0 = (ET (( ∑ ∇θM (i) logϕ0s ) ( ∑ ∇θM (j) logϕ0s )))
s∈M s∈M i,j∈⟦1,∣M ∣⟧2

⎛ ⎛ M M ⎞⎞
= ET ∑ ∇θM (i) logϕ0s ∇θM (j) logϕ0s
⎝ ⎝s∈m(M (i))∩m(M (j)) ⎠⎠
i,j∈⟦1,∣M ∣⟧2

Finally we define H0M and C0M as follows:

H0M = E [M Hess0T ] ; C0M = E [M CT0 ]

7.1.4 Rate of Convergence

We now √ prove that the rate of convergence for the maximum likelihood estimate
θ̂TM is T for all models M :

0 = M Grad0T (θ̂TM ) = M Grad0T (θ0M ) + M Hess0T (θT )(θ̂TM − θ0M )

Where θT belongs to [θ̂TM , θ0M ]

M
Hess0T (θT )(θ̂TM − θ0M ) = −M Grad0T (θ0M )
√ −1/2 M −1/2 √ M
T C0M ⋅ Hess0T (θT )(θ̂TM − θ0M ) = −C0M T Grad0T (θ0M )
√ −1/2 −1/2 √
T C0M ⋅ H0M (θ̂TM − θ0M ) = −C0M T M Grad0T (θ0M ) + op (1)
√ −1/2 D
T C0M ⋅ H0M (θ̂TM − θ0M ) → N (0, Ik )

Using the central limit theorem and the uniform bounds on ψ’s derivatives we
obtain the convergence in distribution (we can use these bounds because θ0 is
interior to Θ). The proof of Theorem 1 is complete.

7.2 Proof of Theorem 2

Before providing the proof of Theorem 2, we introduce some definitions and

notations.

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7.2.1 Defininition

• For each vine decomposition we define a function m that associates to

each element s in M the set of indexes of bivariate copulas and marginal
densities necessary to estimates cs

m ∶ M → P(M )

Where P(M ) is the set of subsets of M .

• For all M in Mn we order their elements according to the underlying tree
structure. M (1) is the index of the copula which is at the root of the tree,
M (2) is the index of the copula of left leaf from the root of the tree, M (3)
is the label of the copula of the right leaf of the tree, and so on.

Example Consider the following trivariate vine copula:

f1,2,3 = f1 ⋅ f2 ⋅ f3 ⋅ c1,2 ⋅ c1,3 ⋅ c2,3∣1 .

c2,3∣1

c1,2 c1,3

f1 f2 f1 f3

Then the map m and the indexes in M (.) defined above are:

m ({1}) = {1}; m ({2}) = {2}; m ({3}) = {3}

m ({1, 2}) = {{1}, {2}, {1, 2}}; m ({1, 3}) = {{1}, {3}, {1, 3}}
m ({2, 3∣1}) = {{1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3∣1}}
M (1) = {2, 3∣1}; M (2) = {1, 2}; M (3) = {1, 3}
M (4) = {1}; M (5) = {2}; M (6) = {3}

7.2.2 Proof

The key point in this analysis is the following: as soon as the ordering of the
model M is done according to the tree structure, and this structure is the same
for all models, then:

∀M, M ′ ∈ M2n , ∀i, j ∈ ⟦1, ∣M ∣⟧2

−1
M −1 (m(M (i)) ∩ m(M (j))) = M ′ (m′ (M ′ (i)) ∩ m′ (M ′ (j))),

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where M −1 denotes the function that maps a copula to its index number in M .
This equality links the two copulas numbered i and j through the tree structure.
We now define Ki,j and ki,j as follows:
∀M ∈ Mn , ∀i, j ∈ ⟦1, ∣M ∣⟧2 Ki,j = m(M (i)) ∩ m(M (j)),
∀M ∈ Mn , ∀i, j ∈ ⟦1, ∣M ∣⟧2 ki,j = ∣m(M (i)) ∩ m(M (j))∣.
This definition allows us to control the covariance matrix of two different models
by controlling the difference of each term of the sums over Ki,j for all i and j.
We now introduce a distance that permits to compare two models M and M ′ :

Definition

• ≃ is defined by:
∀A, B ∈ E 2 (R, ∣∣.∣∣) A ≃ B ⇔ ∣∣A − B∣∣ ≤  ⇔ ∃ν ∈ E, A = B + ν, ∣∣ν∣∣ ≤ .

• Let  be the the smallest real number such that:

∀M, M ′ ∈ M2n , ∀i, j, r ∈ ⟦1, ∣M ∣⟧3
M M
⎧
⎪
⎪
⎪ ET (∇θM (i) logϕ0M (r) ∇θM (j) logϕ0M (r) ) ≃
⎪
⎪
⎪ M′ M′
⎨ ET (∇θM ′ (i) logϕ0M ′ (r) ∇θM ′ (j) logϕ0M ′ (r) ) ,
⎪
M ′ ,0
⎪
⎪
⎪ M,0
⎩ ∇θM (i) ,θM (j) ψM (r) ≃ ∇θM (i) ,θM (j) ψM (r) .
⎪
⎪
′
In practice  is small as both pairs of continuous functions ϕM
M (r) , ϕM ′ (r) and
M
′

(r) , ψM ′ (r) are equal if the same parametric pair copula is used throughout
M M
ψM
the estimation, which is the case in most applications (Aas et al. (2009), Berg
and Aas (2010) and Czado et al. (2009)).

Assumption A3
∃M ∈ Mn , s.t  < ∣∣M Hess0T ∣∣/∣∣(ki,j )i,j ∣∣

Using the previous assumptions and the definition made above we can write
M
Hess0T as follows:

M ⎛ 0⎞
Hess0T = ∇θM (i) ,θM (j) ∑ ψsM T
⎝ s∈m(M (i))∩m(M (j)) ⎠
i,j∈⟦1,∣M ∣⟧2

⎛ 0⎞
= ∇θM (i) ,θM (j) ∑ ψsM T
⎝ s∈Ki,j ⎠
i,j∈⟦1,∣M ∣⟧2

⎛ ′ ⎞
= ∇θM ′ (i) ,θM ′ (j) ∑ M ψs 0T + (νi,j ki,j )i,j∈⟦1,∣M ∣⟧2
⎝ s∈Ki,j ⎠
i,j∈⟦1,∣M ∣⟧2
′
≃⋅∣∣(ki,j )i,j ∣∣ M Hess0T .

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And M
CT0 can also be rewritten in the following way:

⎛ ⎛ M M ⎞⎞
M
CT0 = ET ∑ ∇θM (i) logϕ0s ∇θM (j) logϕ0s
⎝ ⎝s∈m(M (i))∩m(M (j)) ⎠⎠
i,j∈⟦1,∣M ∣⟧2

⎛ ⎛ M M ⎞⎞
= ET ∑ ∇θM (i) logϕ0s ∇θM (j) logϕ0s
⎝ ⎝s∈Ki,j ⎠⎠
i,j∈⟦1,∣M ∣⟧2

⎛ ⎛ ⎞⎞
≃⋅∣∣(ki,j )i,j ∣∣ ET ∑ ∇θM ′ (i) logϕ0s M ′ ∇θM ′ (j) logϕ0s M ′
⎝ ⎝s∈Ki,j ⎠⎠
i,j∈⟦1,∣M ∣⟧2
M′
≃⋅∣∣(ki,j )i,j ∣∣ CT0

In order to establish formula (7) we will use shorthand notations8 .

′ ′
K = (ki,j )i,j ; H = M Hess0t ; C = M CT0 ; V = H −1 CH −1
; V0 = H −1 KH −1
; I = Id∣M ∣

′
′
̂M
Cov = (H + ν1 K)−1 (G + ν2 K)(H + ν1 K) −1
T
′
= (I + ν1 H −1 K)−1 H −1 (G + ν2 K)H −1 (I + ν1 KH −1 ) −1
∞ ∞
= (I + ∑(−1)i (ν1 H −1 K)i )(V + ν2 V0 )(I + ∑(−1)i (ν1 H −1 K)i )′
i=1 i=1
= (I + Σν1 )(V + ν2 V0 )(I + Σν1 )′
= V + ν2 V0 + 2Σν1 (V + ν2 V0 ) + Σν1 2 (V + ν2 V0 )
< α⋅

The proof of theorem 2 is complete.

8 In Theorem 2 we noted  the value α ⋅ 

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