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Applied Probability Trust (19 October 2015)
TIMEVARYING COPULA MODELS FOR FINANCIAL
TIME SERIES
R
¨
UDIGER KIESEL,
∗
University DuisburgEssen
MAGDA MROZ,
∗∗
Ulm University
ULRICH STADTM
¨
ULLER,
∗∗∗
Ulm University
Abstract
We perform an analysis of the potential timeinhomogeneity in the dependence
between multiple ﬁnancial time series. In order to do so we use the framework of
copula theory and tackle the question whether dependencies in such a case can
be assumed constant throughout time or rather have to be modelled in a time
inhomogeneous way. This implies that through a changing copula parameter
also the dependency is not constant, but may be stronger or weaker depending
on the point of time. We focus on suitable inference techniques in the context
of a special copulabased multivariate time series model. A recent result due to
[6] is used to derive the joint limiting distribution of local maximum likelihood
estimators on overlapping samples. By restricting the overlap to be ﬁxed, we
succeed in establishing the limiting law of the maximum of the estimator series.
Based on the limiting distributions we develop statistical homogeneity tests and
investigate their local power properties.
A Monte Carlo simulation study demonstrates that bootstrapped variance
estimates are needed in ﬁnite samples. Empirical analyses on realworld
ﬁnancial data ﬁnally conﬁrm that timevarying parameters are rather an
exception than the rule.
Keywords: Multivariate Time Series; Copula; Binomial Test; Extremal Test
2010 Mathematics Subject Classiﬁcation: Primary 62M10
Secondary 62G32;91B84
1. Introduction
Our epigraph is part of a citation in [16] in support of a historical perspective
when applying copulas. Keeping this in mind “copulas form a most useful concept
for a lot of applied modeling [...]” ([16, p. 13]). Recent textbooks such as [9, 27, 28]
contain theoretical background and a multiple of examples for applications supporting
this view. Also, [32] contains a recent overview with an exhaustive list of references.
We are interested in the dependency of multivariate time series and in particular in
possible changes in the dependency structure. In the spirit of ([16, p. 13]) we develop
our methodology based on some wellknown and wellstudied mathematical concepts
∗
Postal address: Chair for Energy Trading and Finance, University DuisburgEssen, Campus Essen,
Universit¨atsstraße 12, 45141 Essen, Germany
∗∗
Postal address: Institut f¨ur Zahlentheorie und Wahrscheinlichkeitstheorie, 89069 Ulm, Germany
∗∗∗
Postal address: Institut f¨ur Zahlentheorie und Wahrscheinlichkeitstheorie, 89069 Ulm, Germany
1
2 Kiesel, Mroz, Stadtm¨uller
while focusing on suitable inference techniques in the context of a special copulabased
multivariate time series model.
Owing to its special structure, the semiparametric copulabased multivariate dy
namic (SCOMDY) model due to [6] directly retains many ﬁndings presented in the
univariate case. Although there are more general approaches for multivariate het
eroscedasticity, we decided in favor of the SCOMDY model due to its parsimonious
parametrization and its statistical tractability. In particular, we are able to derive
the tail behavior of the ﬁnitedimensional distribution of the SCOMDY model by
embedding it into a class of multivariate models for which this result was recently shown
by [17]. A possible timevarying dependence structure is captured by the parameter of
the underlying copula family in the SCOMDY model. Timedependent copula models
have been used in [8] for analysing equity markets. For multivariate time series [4]
develops tests on local nonparametric copula estimates, but strong mixing is assumed.
For an an overview on goodnessofﬁt tests for copulas and further references the reader
should consult [21].
In Chapter 2 we start by deﬁning an estimator series of local estimators on over
lapping subsamples of length b = O(T ) and derive the basic joint distributional results
as the overall observation horizon T tends to ∞ in Section 2.2. Chapter 3 contains
our original contribution in pursuit of the objective to detect inhomogeneities in the
dependence in multivariate time series. In Section 3.1 a Binomial test based on
independent subsamples is constructed. We discuss the notion of statistical size and
local power in detail, and show that the Binomial test has local power of order
√
b. The
Extremal test presented in Section 3.2 is based on subsamples that overlap by a certain
ﬁxed amount of observations. The considerations concerning its local power of order
p
b log (T/b) entail an extreme value limit result for asymptotically normal random
variables. Finally, a third test that uses the whole of the moving window estimators
and has local power of order
√
T is presented in section 4.4 in [30]. A summary of the
advantages and drawbacks of the diﬀerent homogeneity tests concludes.
We apply the Binomial and the Extremal test in Chapter 4 to realworld ﬁnancial
data. We ﬁnd that inhomogeneities in the dependence structure come along with
unusual market events and can be remedied by careful univariate modeling. This is
in accordance with the general perception that changes in the dependency are rather
an exception than the rule, and stem from extreme economic events. In most cases
a copula that is constant throughout time is suﬃcient to capture the dependence
among the components of a multivariate time series model, and there is no gain in
choosing a more complicated and statistically unhandy model with timevarying copula
parameters.
2. The SCOMDY model and its estimators
2.1. Deﬁnition of the SCOMDY model
We now deﬁne a multivariate time series model which is linked to copula theory. As
we cannot provide every technical detail here we have to refer on several occasions to
[30]. The model was ﬁrst introduced by [7], and subsequently used for further statistical
developments such as valueatrisk calculations in ﬁnancial portfolios in [25] or local
changepoint tests of the copula parameter in [22] among others.
Deﬁnition 1. Let {Y
t
}
t≥1
be a vectorvalued process such that Y
t
= Σ
1/2
t
ε
t
with
Copula Models 3
vectorvalued residuals ε
t
= (ε
1t
, . . . , ε
dt
) and conditional covariance matrices Σ
t
=
(σ
2
ijt
)
i,j=1,...,d
.
1. Let Σ
t
= diag(σ
2
1t
, . . . , σ
2
dt
) be such that for every j = 1, . . . , d a univariate
GARCH(p, q) speciﬁcation holds
σ
2
jt
= α
j0
+
p
X
i=1
α
ji
Y
2
j,t−i
+
q
X
k=1
β
jk
σ
2
j,t−k
(1)
with γ
j
= (α
j0
, α
j1
, . . . , α
jp
, β
j1
, . . . , β
jq
) ∈ (0, ∞) × [0, ∞)
p+q
. Furthermore,
let ε
t
, t ≥ 1, be independent and identically distributed according to a dvariate
distribution with distribution function H = C(F
1
, . . . , F
d
; θ) where F
1
, . . . , F
d
are
arbitrary marginal distributions and C( . ; θ) belongs to a parametric copula fam
ily with parameter θ ∈ Θ ⊆ R. Let E (ε
t
) = 0 and cov ε
t
= R
t
= (r
ijt
)
i,j=1,...,d
such that the variables are normalized, i.e., for all i = 1, . . . , d
r
iit
= var ε
it
= 1. (2)
2. Let ˜γ
j
be the quasi maximum likelihood estimator deﬁned using ﬁnite sums (see
[30], §3.1) and let ˜ε
jt
= Y
jt
/˜σ
jt
be the empirical residuals in every component
j = 1, . . . , d. With ν = ν(T ) being an oﬀset, set
˜
U
jt
=
1
T −ν + 1
T
X
s=ν
1
{˜ε
js
≤˜ε
jt
}
(3)
and consider
˜
U
t
= (
˜
U
1t
, . . . ,
˜
U
dt
) for t = ν, . . . , T as a pseudosample from the
copula. As an estimator for the copula parameter θ consider
ˆ
θ = argmax
θ∈Θ
1
T −ν + 1
T
X
t=ν
log c(
˜
U
1t
, . . . ,
˜
U
dt
; θ), (4)
a canonical maximum likelihood estimator oﬀset by ν.
The process {Y
t
}
t≥1
with the time series speciﬁcation in (1) and the estimation
procedure described in (2) is called SCOMDY model (stemming from Semiparametric
COpulabased Multivariate DYnamic model).
Remark 1. In Deﬁnition 1, paragraph 1 the parameter of the copula family is uni
variate so that the testing procedures can be applied immediately  in case of a multi
dimensional parameter space, i.e. θ ∈ Θ ⊆ R
d
where d ≥ 1, joint conﬁdence areas
have to be constructed. As usual, the sizes of the marginal tests have to be adjusted
in such a way that the overall size of the multiple tests meets the theoretical size.
Remark 2. Note that the residuals ε
t
are no longer white noise with identity co
variance matrix, but iid random vectors with mean zero and covariance matrix R
which corresponds to the correlation matrix due to (2). Apart from a diﬀerent covari
ance matrix, all white noise properties are preserved, so that we denote the residual
4 Kiesel, Mroz, Stadtm¨uller
process by ε
t
∼ WN(0, R). Condition (2) is required for identiﬁability of the pa
rameters in the univariate GARCH(p, q) models and can be implemented into the
SCOMDY model by rescaling the components by the respective variances. For this
consider ε
∗
t
= (ε
∗
1t
, . . . , ε
∗
dt
) ∼ H
∗
= C
∗
(F
∗
1
, . . . , F
∗
d
) with Var(ε
∗
jt
) = v
j
and set
ε
t
= (ε
∗
1t
/v
1
, . . . , ε
∗
dt
/v
d
). Then the joint distribution for all (x
1
, . . . , x
d
) ∈ R
d
is
H(x
1
, . . . , x
d
) = H
∗
F
∗
1
(v
1
x
1
), . . . , F
∗
d
(v
d
x
d
)
, (5)
and the marginal distributions and their (generalized) inverses for all x
j
, y
j
∈ R, j =
1, . . . , d are
F (x
j
) = F
∗
j
(v
j
x
j
) and F
−
j
(y
j
) = F
∗
j
−
(y
j
)/v
j
. (6)
Due to the invariance principle (scaling invariance), the copula of the distribution of
ε
t
is the same as that of ε
∗
t
, namely C. This implies that restricting the matrix R to
have a unit diagonal is merely a condition on the marginal distributions (which can
easily be accomplished by rescaling), and does not impinge on the freedom of choice
as to the desired copula.
Remark 3. There are various dependencies in the data (3). First of all
˜
U
jt
and
˜
U
j,t+h
are dependent for 0 < h < ν but we take care of it below. That we use pseudo data is
causing dependencies and distortion as described in [15]. So the size of the Rosenblatt
transformationtest does not correspond to the theoretical size. We are estimating
parameters and there, this is reﬂected by are larger asymptotic variance than with
copula data (see [20]), but we still have asymptotic normality and this is what we
need.
Consistency and an asymptotic normality result due to [6] can be shown. We restate
the respective theorems in Theorem 1 below, the precise prerequisites and notation
can be found in the appendix of [30], which we silently assume to hold in the sequel.
Furthermore, for x = (x
1
, . . . , x
d
) and y = (y
1
, . . . , y
d
) let the realvalued Gaussian
process U(x) with E U(x) = 0 and E (U(x)U(y)) =
Q
{x
i
∧ y
i
} −
Q
x
i
y
i
be given.
Theorem 1. Let {Y
t
}
t≥1
and
ˆ
θ be given as in Deﬁnition 1.
The following holds true.
1. Suppose that ν(T )/T → 0, T → ∞, then as T → ∞ we have weak consistency of
ˆ
θ, i.e.
ˆ
θ
P
−→ θ. (7)
2. If as T → ∞ we have ν(T )/ log T → ∞ and ν(T )/T → 0, then
√
T (
ˆ
θ − θ)
converges in distribution to Z distributed as
− Σ
−1
(θ)
n
Z
δ(T
−
d
(x
1
, . . . , x
d
); θ)dU (x
1
, . . . , x
d
)
+
d
X
j=1
Z
δ
j
(x
1
, . . . , x
d
; θ)U ((1, . . . , 1, x
j
, 1, . . . , 1)
>
)c(x
1
, . . . , x
d
; θ)dx
1
. . . dx
d
o
,
(8)
where Σ(θ), δ
j
, and and T
d
are given by
Copula Models 5
Σ(θ) =
E
∂
2
∂θ
i
∂θ
j
log c(F
1
(ε
1
), . . . , F
d
(ε
d
); θ)
i,j=1,...,m
. (9)
δ
j
(u
1
, . . . , u
d
; θ) =
∂
∂u
j
c(u
1
, . . . , u
d
; θ). (10)
T
j
(u
1
, . . . , u
j
) = (C
1
(u
1
), C
2
(u
2
u
1
), . . . , C(u
j
u
1
, . . . , u
j−1
)) . (11)
Despite the necessity to fall back on bootstrapping as an auxiliary variance esti
mation, the SCOMDY model is our model of choice and we assume for all further
considerations that we work on the residual vectors ˜ε
t
. The model is not only numeri
cally tractable in the sense that quasi maximum likelihood for the GARCH parameters
and canonical maximum likelihood for the copula parameters are well established
and available in many statistical software implementations. As the evolution in the
single components only depends on the respective component itself, the stationarity
conditions for univariate GARCH(p, q) models confer directly to the marginal con
ditional variance models in SCOMDY. The same is true for the asymptotic results
with respect to the quasi maximum likelihood estimators for the respective parameter
sets. As to the ﬁnitedimensional distributions, the SCOMDY model embeds itself
in the class of constant conditional correlation GARCH(p, q) models, and therefore
the heavytailedness result by [17] transfers. Here the authors show in Theorem 5
that stationarity and heavytailedness of multivariate constant conditional correlation
GARCH(p, q) models is guaranteed by analogous assumptions as in the univariate case.
A proof can be found in [30, Theorem 3.1.5] which develops the arguments along the
lines of [29, Theorem 2.1].
Assume that Y
1
, . . . , Y
T
, T ≥ 1 is a ﬁnite sample coming from a SCOMDY process
{Y
t
}
t≥1
as given in Deﬁnition 1.1. According to Deﬁnition 1.2 we obtain the empirical
residuals
˜
ε
1
, . . . ,
˜
ε
T
as pseudorealizations of the residual distribution H, and pseudo
realizations
˜
U
1
, . . . ,
˜
U
T
of the copula C
θ
through a probability integral transform in
every component of the vectors. As we aim at testing for homogeneity of the copula
parameter throughout time, consider for the moment
˜
U
t
∼ C
θ
t
(12)
where θ
t
is the present parameter for every point in time t = 1, . . . , T . For an arbitrary
θ
0
∈ Θ, the pair of hypotheses we want to test is then
H
∗
0
: {θ
t
≡ θ
0
∀ t} vs. H
∗
A
: {∃t: θ
t
6= θ
0
}. (13)
We address this task nonparametrically through a purely datadriven approach in
Section 2.2. Based on the resulting estimators and their asymptotic distribution
three diﬀerent tests will be constructed and analyzed with respect to their ability
to detect actual deviations from the homogeneity hypotheses. In Section 3.1 we will
present a Binomial multiple test based on independent subsamples of the given data,
in Section 3.2 we will examine whether the extremes of a parameter series based on
overlapping subsamples exceed certain conﬁdence levels.
6 Kiesel, Mroz, Stadtm¨uller
2.2. Moving window estimators
In order to assess θ
t
for 1 ≤ t ≤ T we deﬁne maximum likelihood estimators on
certain subsamples of a bandwidth b in the following.
Deﬁnition 2. Let
˜
U
1
, . . . ,
˜
U
T
, T ≥ 1 be pseudorealizations of a parametric copula
C
θ
according to Deﬁnition 1.2. With a bandwidth 1 ≤ b = b(T ) ≤ T the local or
moving window maximum likelihood estimator
ˆ
θ
t
of θ for every t = b, . . . , T is deﬁned
by
ˆ
θ
t
= argmax
θ∈Θ
t
X
s=t−b+1
log c(
˜
U
1s
, . . . ,
˜
U
ds
; θ). (14)
In Proposition 1 the consistency and asymptotical normality result for the copula
estimators in the SCOMDY model are transferred to the local estimators
ˆ
θ
t
in terms
of the bandwidth b = b(T ).
Proposition 1. Under the conditions of Theorem 1 and under the assumption that
b = b(T ) is chosen such that b → ∞ and b/T → 0 for T → ∞, in shorthand notation
denoted by b = O(T ) (T → ∞), we have the following as T → ∞.
1.
ˆ
θ
t
P
−→ θ.
2.
√
b(
ˆ
θ
t
−θ)
d
−→ N
0, Σ
, where Σ is the asymptotic variance in canonical maximum
likelihood estimation as in [20].
Proof. 1 and 2 correspond to Theorem 1, 1 and 2 respectively. By posing the
condition b = O(T ) (T → ∞) on the bandwidth b, the respective oﬀset conditions on
ν are fulﬁlled.
Now, we consider the joint limiting distribution of (two) moving window estimators
and show that the covariance structure is nondegenerate if the overlap of the sub
samples, which the respective estimators are based on, is asymptotically constant. For
points in time 1 ≤ s < t ≤ T consider
ˆ
θ
t
and
ˆ
θ
s
based on subsamples of bandwidth
b according to (14). Their lag is deﬁned as l
:
= s − t, whereas the overlap of the
subjacent subsamples is b − l.
Proposition 2. Let 1 ≤ s, t ≤ T and let
ˆ
θ
t
and
ˆ
θ
s
be the respective moving window
estimators. Let 1 ≤ l
:
= s − t < b(T ) be such that l/b(T ) → µ
l
∈ (0,1) as T → ∞.
Under the conditions of Theorem 1 the following joint limiting law holds
√
b
(
ˆ
θ
s
,
ˆ
θ
t
)
>
− (θ, θ)
>
d
−→ N
0, Σ
¯
M
(15)
with matrix
¯
M =
1 1 − µ
l
1 − µ
l
1
and Σ the scalar variance as in [20].
Proof. The proof can be done by ﬁrst order Taylor expansion of the loglikelihood
and the classical central limit theorem (for details see Proposition 4.1.3 in [30]).
The moving window estimators
ˆ
θ
t
are deﬁned for all t = b, . . . , T and a given
realization Y
1
, . . . , Y
T
of a SCOMDY model. For the subsequent sections we will
regard the estimators as an estimator time series, and proceed in testing the null
hypothesis (13) depending on the index set for the times t. To this end, consider the
following deﬁnitions.
Copula Models 7
Deﬁnition 3. Let a SCOMDY sample of size T ≥ 1 and the respective moving window
estimators
ˆ
θ
t
with bandwidth b = b(T ) and for all t = b, . . . , T be given. Consider
the thinning constant c ∈ [1/b, 1] and the series length n
(c)
= b
T −b
cb
c, where bxc
:
=
max{k ∈ Z : k ≤ x} for all x ∈ R. Set the index set as I
(c)
:
= {t
k
= bT − (n
(c)
−
k)cbc: k = 1, . . . , n
(c)
} and consider the estimator series {
ˆ
θ
t
}
t∈I
(c)
.
1. For c = 1, {
ˆ
θ
t
}
t∈I
(c)
is called independent estimator series. In particular, we set
n
:
= n
(1)
= b
T −b
b
c and
ˆ
θ
k
:
=
ˆ
θ
t
k
for all t
k
= T −(n − k)b, k = 1, . . . , n.
2. For ﬁxed c with 1/b < c < 1, the estimator series {
ˆ
θ
t
}
t∈I
(c)
is called thinned.
3. For c = c(b) = 1/b, we refer to {
ˆ
θ
t
}
t∈I
(c)
as full estimator series. In this case, we
stress the dependence on the bandwidth by writing n(b)
:
= n
(1/b)
= T − b and
t
k
(b) = t
(1/b)
k
= k + b.
Along with this threestep deﬁnition we will proceed by presenting a suitable approach
to test for parameter homogeneity in the independent estimator series in Section 3.1.
3. Homogeneity tests
3.1. Binomial test
Let
ˆ
θ
1
, . . . ,
ˆ
θ
n
be an independent estimator series as in Deﬁnition 3, 1. The general
null hypothesis is replaced by a ﬁrst approximation in order to take advantage of the
independence of the individual estimators
ˆ
θ
k
. We aim at testing
H
0
: {θ
t
1
= . . . = θ
t
n
= θ
0
: t
k
= T −(n − k)b} vs. H
A
: {∃t
k
: θ
t
k
6= θ
0
}. (16)
We will clarify the notation and the statistical concepts for one individual estimator
in § 3.1.1. In § 3.1.2 we will perform a multiple simultaneous test of the partial null
hypotheses to arrive at the hypothesis given in (16).
3.1.1. Partial tests Consider an arbitrary k = 1, . . . , n and set
ˆ
θ =
ˆ
θ
k
as well as θ = θ
t
k
in the following. Furthermore, consider Σ as the asymptotic variance in canonical
maximum likelihood estimation given in [20]. Note that the copula parameter estimator
ˆ
θ is determined locally on subsamples preceding the point in time t
k
. The following
asymptotically normal test statistic for b(T ) → ∞ (T → ∞)
T
b
:
=
√
b(
ˆ
θ − θ
0
)
√
Σ
d
−→ N(0,1) (17)
can be constructed to test the partial null hypothesis H
0
: {θ = θ
0
}. In applications θ
0
is replaced by the global estimate
ˆ
θ (see (4)) without disturbing the asymptotic results
below since
√
b(
ˆ
θ − θ
0
)
p
→ 0 as T → ∞ under the null hypotheses.
As noted in [26, p. 433], local power considerations for a given test have to be
conducted in every special case of test statistic, asymptotic distribution and alternative
hypothesis. Subsequently, we elaborate the deﬁnitions in terms of the test statistic T
b
given in (17) and the partial null hypothesis H
0
: {θ = θ
0
}. For the power calculations
we consider a ﬁxed and a local alternative hypothesis. On the one hand, we have
H
A
(η): {θ = θ(η) = θ
0
+ η} as the ﬁxed alternative, on the other hand,
¯
H
A
(η): {θ =
8 Kiesel, Mroz, Stadtm¨uller
θ
b
(η) = θ
0
+η/
√
b} as the local alternative. Both can be regarded as onesided if η > 0
or η < 0 respectively, and as twosided if η 6= 0.
For the onesided alternatives, H
0
is rejected to the signiﬁcance level α ∈ (0,1) if
T
b
> q
α
or T
b
< q
−α
, whereas for the twosided alternative, rejection occurs if T
b
 >
q
α/2
. Here we denote by q
α
the (1 −α)quantile of the standard normal distribution,
i.e. P (Z ≤ q
α
) = 1 − α for Z ∼ N(0,1). For the critical region K this implies
K(η) =
(q
α
, ∞), for η > 0
(−∞, −q
α
), for η < 0
(−∞, −q
α/2
) ∪ (q
α/2
, ∞), for η 6= 0.
Depending on whether the alternative is local or ﬁxed, we will derive the power of
both the one and the twosided tests in the following lemma. Without loss of generality
we assume that θ
0
= 0 and Σ = 1 for notational ease. The results remain unchanged
under general θ
0
∈ Θ and Σ > 0.
Lemma 1. By setting the null hypothesis as H
0
: {θ = 0} we obtain the ﬁxed alterna
tive H
A
(η): {θ = η} and the local alternative
¯
H
A
(η): {θ = η/
√
b}. Then we have the
following.
1. Let the sample size b ≥ 1 be ﬁxed and η ∈ R. Then for γ
b
(η) = P
H
A
(η)
(T
b
∈ K(η))
we have
P
H
A
(η)
(T
b
∈ K(η)) = (1 + O(1))·
1 − Φ(q
α
−
√
bη), η > 0
Φ(−q
α
−
√
bη), η < 0
1 − Φ(q
α/2
−
√
bη) + Φ(−q
α/2
−
√
bη), η 6= 0.
Note that the asymptotic equality of the lefthand and the righthand side is with
respect to b → ∞.
2. For η ∈ R we have asymptotically
¯γ(η) = lim
b→∞
P
¯
H
A
(η)
(T
b
∈ K(η)) =
1 − Φ(q
α
− η), η > 0
Φ(−q
α
− η), η < 0
1 − Φ(q
α/2
− η) + Φ(−q
α/2
− η), η 6= 0.
Φ is the distribution function of the standard normal law and P
H
A
denotes the proba
bility under the (alternative) hypothesis H
A
.
The proof is obvious by plugging in the (local) alternatives into γ
b
(η) and ¯γ(η)
respectively (see Lemma 4.2.2 in [30]). Note that for the ﬁxed alternative we consider
γ
b
, i.e. the power of the test depending on the sample size b. For b → ∞ we have
γ
b
(η) → 1 for any η in any case of either onesided or twosided testing. The advantage
of local alternatives becomes apparent here as the power ¯γ is independent of the sample
size and the limit for b → ∞ can be established right away.
Furthermore, we see that γ(0) = α and γ(η) > α if η 6= 0. Consequently the
following theorem formally shows that the test is unbiased, which in case of the local
alternative can be used for a result concerning the local power and its order.
Copula Models 9
Theorem 2. Let γ
b
(η) and ¯γ(η) be given as in Lemma 1 1. and 2., i.e. the ﬁxed and
local power of the test to the level α ∈ (0,1) of H
0
: {θ = 0} against H
A
(η): {θ = η}
and
¯
H
A
(η): {θ = η/
√
b}, respectively. Then we have for η 6= 0
γ
b
(η) > α and ¯γ(η) > α, (18)
i.e., the test is asymptotically unbiased against both alternatives H
A
and
¯
H
A
. Further
more, the test against
¯
H
A
has local power of order
√
b.
We have shown that the asymptotically normal test statistic T
b
based on an estima
tor
ˆ
θ for the copula parameter θ on a subsample of size b is suitable to test H
0
: {θ = θ
0
}.
The test against the local alternative
¯
H
A
(η): {θ = θ
b
(η) = θ
0
+ η/
√
b} has been shown
to have local power of order
√
b. We will now take advantage of the fact that the
estimators
ˆ
θ
1
, . . . ,
ˆ
θ
n
as in Deﬁnition 3,1. are established on subsamples that do not
overlap, i.e. that are independent, to construct a multiple test.
3.1.2. Multiple test Our aim is to simultaneously test the following pairs of hypotheses
H
0k
: {θ
t
k
= θ
0
} vs.
¯
H
Ak
(η): {θ
t
k
= θ
b
(η)} (19)
for all t
k
= T − (n − k)b, n = b
T −b
b
c. This notation also implies that we limit the
following considerations to the local alternatives θ
b
(t) = θ
0
+ η/
√
b, η 6= 0. For every
k = 1, . . . , n construct the partial test statistics
T
bk
:
=
√
b(
ˆ
θ
k
− θ
0
)
√
Σ
. (20)
Consider the random variables Y
k
= 1
{T
bk
∈K(η)}
that indicate whether the kth partial
test rejects the null hypothesis H
0k
to the level α ∈ (0,1). Summing them up, we
obtain
S
n
=
n
X
k=1
Y
k
∼ Bin(n, α) (21)
under the null hypothesis of parameter constancy. Note that since the partial tests are
asymptotical tests, the size of the above Binomial distribution is only approximately
α. S
n
is now suitable to test the simultaneous null hypothesis H
0
=
T
n
k=1
H
0k
of
all the partial null hypotheses being true. The alternatives might range from one
false hypothesis up to all hypotheses being false. Formally, we specify the alternative
hypothesis for 1 ≤ m ≤ n by H
A,m
(η) =
T
m
k=1
¯
H
Ak
(η) ∩
T
n
k=m+1
H
0k
, that is
H
A,m
(η): {θ
t
k
= θ
b
(η) for k ≤ m, θ
t
k
= θ
0
for k > m}. (22)
Note that in this setup all the partial alternatives have to be of the same absolute
value with respect to η.
The null hypothesis H
0
is rejected to the signiﬁcance level α
∗
∈ (0,1) if S
n
>
c
α
∗
where c
α
∗
is the (1 − α
∗
)quantile of the Binomial distribution with size n and
probability α. The critical region has the form K = {c
α
∗
+ 1, c
α
∗
+ 2, . . . , n}. Note
that with ﬁxed α, α
∗
∈ (0,1) the rejection probability under the null hypothesis H
0
is
in general
P
H
0
(S
n
> c
α
∗
) =
n
X
j=c
α
∗
+1
n
j
α
j
(1 − α)
n−j
≤ α
∗
, (23)
10 Kiesel, Mroz, Stadtm¨uller
i.e. the size of the test is less than the desired signiﬁcance level α
∗
for most of the time.
This is usually the case when the distribution of the test statistic is discrete. In order
to actually attain the proper size, Binomial tests of such kind have to be randomized.
The existence of p in the following modiﬁcation of the sample function ϕ is due to the
fundamental lemma of Neyman and Pearson (see Theorem 3.2.1 in [26]). The lemma
basically states that for every given signiﬁcance level α ∈ (0,1), a test that exactly
attains this level can be constructed.
Deﬁnition 4. Let a pair of hypotheses H
0
vs. H
A
and the signiﬁcance level α ∈ (0,1)
be given. Denote by ∂K the boundary of the respective critical region K and by K its
closure, and set
˜ϕ(x
1
, . . . , x
b
) =
1, (x
1
, . . . , x
b
) ∈ K,
R, (x
1
, . . . , x
b
) ∈ ∂K,
0, (x
1
, . . . , x
b
) /∈ K,
(24)
where R ∼ Bin(1, p) and the value p ∈ (0,1) can be chosen such that P
H
0
( ˜ϕ = 1) = α
and P
H
A
( ˜ϕ = 0) ≥ α. This test is called a randomized test to the level α.
In case of the Binomial test statistic S
n
in (21) we set
˜ϕ(s) =
1, c
α
∗
≤ s − 1,
R, s − 1 < c
α
∗
≤ s,
0, s < c
α
∗
,
(25)
for a realization 0 ≤ s ≤ n of S ∼ Bin(n, α), and choose p ∈ (0,1) such that
1 · P (S ≥ c
α
∗
+ 1) + p · P (S = c
α
∗
) = α
∗
, (26)
i.e. p =
α
∗
− P (S ≥ c
α
∗
+ 1)
/P (S = c
α
∗
). Practically, we sample r from R ∼
Bin(1, p) whenever the speciﬁc realization s of S
n
is equal to c
α
∗
, and reject H
0
if
s + r = c
α
∗
+ 1.
It is highly recommended to avoid randomized tests as the outcome for a speciﬁc
sample with a ﬁxed signiﬁcance level is not deterministic, and hence not reproducible
(see [26, p. 75]). Since we sum over integervalued j in (23), it is possible to ﬁx κ
:
=
c
α
∗
+ 1 ∈ N and then solve
n
X
j=κ
n
j
α
j
(1 − α)
n−j
= α
∗
(27)
for the partial level α ∈ (0,1). With this α, it holds that κ − 1 = c
α
∗
and that
P
H
0
(S
n
> c
α
∗
) = α
∗
. (28)
There cannot be given a clear advice how to choose κ. Its choice depends very much
on the number n of partial hypotheses and on the alternative. We will discuss this
question at the end of the current section. Formula (27) is also given in [35] (note
that there is an error in the lower bound of the second sum due to a misprint of 1 for
l = n − m). They compare the Binomial multiple test to other testing procedures for
multiple comparisons such as the [33] or the [18] test.
Copula Models 11
Now, we will turn to the local power P
H
A,m
(η)
(S
n
> c
α
∗
) of the multiple test and
show that the local power of order
√
b confers to the multiple test regardless of the
number m of false partial alternatives.
Theorem 3. The Binomial test S
n
> c
α
∗
against the alternative H
A,m
(η) is asymp
totically unbiased for η 6= 0, and thus has local power of order
√
b for all m = 1, . . . , n.
For the proof see [30].
Summarizing, we have constructed a multiple Binomial test with proper size and
local power of order
√
b. The multiple null hypothesis is composed of n independent
partial null hypotheses and is rejected when there are at least κ partial rejections. We
are free to choose 1 ≤ κ ≤ bn/2c, and arrive at a test that attains the signiﬁcance level
α
∗
under the null hypothesis. In order to maximize the local power, we should take
into account the number of false partial hypotheses in our choice of κ.
In the subsequent section we turn to another approximation of the global null
hypothesis given in (13) that allows to test against a more general alternative. The
test we will construct makes use of a greater portion of the estimator series {
ˆ
θ
t
} by
allowing the underlying subsamples to overlap.
3.2. Extremal test
The null hypothesis H
∗
0
: {θ
t
≡ θ
0
∀t} means that the copula parameter is constant
for all points in time t. A consequence of this is that in particular max
t
θ
t
= θ
0
. Thus,
we will subsequently test the following hypothesis and its onesided alternative
H
0
0
: {max
t
θ
t
= θ
0
} vs. H
0
A
: {max
t
θ
t
> θ
0
} (29)
for an arbitrary θ
0
∈ Θ. As basis for our test statistic, consider a ﬁxed constant c
with 1/b < c < 1 and the thinned estimator series {
ˆ
θ
t
k
} for t
k
= T − (n
(c)
− k)cb,
k = 1, . . . , n
(c)
= b(T −b)/cbc (cf. Deﬁnition 3, 2.).
For notational convenience we use a centered and scaled version of
ˆ
θ
t
k
for all k =
1, . . . , n
(c)
, such that under H
∗
0
we have
ξ
k
:
=
√
b(
ˆ
θ
t
k
− θ
0
)
d
−→ N
0, Σ
(30)
as b → ∞ due to Proposition 1,2..
In order to test the pair of hypotheses in (29), a test statistic based on the vector
ξ = (ξ
1
, . . . , ξ
n
(c)
) is suitable. For assessing the statistic’s distribution, we ﬁrst need the
joint distribution of the vector ξ, and secondly, extreme value theory for asymptotically
normal variables.
3.2.1. Multivariate central limit theorem We will trace back the structure of ξ =
(ξ
1
, . . . , ξ
n
(c)
) to the summands in the likelihood estimation, and then establish its
asymptotic distribution. To this end we will work on a random quantity S
b
exhibiting
a special structure that we need to obtain our results. Although the subsample size b
grows with the overall observation horizon T , the overlap proportion remains ﬁxed. A
reordering of the original summation enables us to apply a recent result on convergence
rates in the multivariate central limit theorem that depends on the involved dimension
(due to [3]). Together with bounds on the elements in the squareroot inverse of the
occurring covariance matrices (a result by [12]), this ﬁnally leads to conditions on b
12 Kiesel, Mroz, Stadtm¨uller
and T that guarantee distributional convergence of S
b
to a Gaussian distribution (the
relevant covariance structure is discussed in [30]).
Now we can proceed to the distributional convergence of S
b
as the sample size
increases to ∞. In a recent publication of [3], Lyapunovtype BerryEsseen bounds in
the multivariate central limit theorem are given inheriting a dimensional dependence.
A version of this result suitable for our purposes is stated below.
Theorem 4. Let Y
(1)
, . . . , Y
(n)
∈ R
d
be independent random vectors with E Y
(k)
= 0
for all k. Let S = Y
(1)
+ . . . + Y
(n)
and let C
2
= cov S. Denote by Φ
d,C
2
the
distribution function of the dvariate centered Gaussian distribution with covariance
matrix C
2
.
Then there exists a positive constant C
2
such that
∆
:
= sup
x∈R
d
F
S
(x) − Φ
d,C
2
(x) ≤ C
2
· d
1/4
· β, (31)
where F
S
is the dvariate distribution function of S and β = β
1
+ . . . + β
n
with β
k
=
E C
−1
Y
(k)

3
.
Proof. See Theorem 1.1 in [3].
Besides this convergence result, the special structure of the covariance matrix will
now be exploited to derive the limiting distribution of the target variable S
b
as well as
a convergence rate depending on the sample sizes T and b.
S
b
can now be seen as the sum of the iid vectors Y
(k)
(with a special structure as
deﬁned in §4.3.1 in [30])
S
b
=
1
√
b
Y
(1)
+ . . . + Y
(cb)
(32)
with covariance matrix
C
2
= cov S
b
= c ΣM (33)
where c is the thinning constant. We now have (see Theorem 4.3.5 in [30])
Theorem 5. For b = T
δ
, δ ∈ (0, 1) and n
(c)
= O (T/b) , let ∆
b
be deﬁned in analogy
to equation (31), namely as
∆
b
= sup
x∈R
n
(c)
F
S
b
(x) − Φ
n
(c)
,C
2
(x).
If δ > 7/9, then ∆
b
−→ 0 as T → ∞.
To see that ∆
b
actually converges to 0 as T → ∞ we apply Theorem 4 to ∆
b
and show
that β is bounded from above by const · n
(c)3/2
·b
−1/2
by applying H¨older’s inequality
two times and the result by [12] to the Cholesky squareroot of M .
Note that this speciﬁc ratio of the subsample size b to the overall sample size T
ensures that the vector of normalized subsample sums is asymptotically jointly normal,
even if the samples overlap by a certain ﬁxed amount of b.
Copula Models 13
3.2.2. Asymptotic extreme value theory We show that the maximal component of the
asymptotically Gaussian target vector S
b
converges in law. To this end, recall that the
kth component of S
b
is
S
bk
=
1
√
b
S
k
=
1
√
b
t
k
X
τ=t
k
−b+1
X
τ
.
Furthermore, the covariance matrix is denoted by C
2
= cov(S
b1
, . . . , S
bN
) = cov S
b
=
c ΣM as in equation (33). Then the following theorem holds.
Theorem 6. If T
7/9
/b = o(1) and n
(c)
=
T −b
cb
then
P
max
k=1,...,n
(c)
S
bk
≤ z
n
(c)
→ e
− e
−z
(T → ∞)
with z
n
(c)
= a
n
(c)
z + d
n
(c)
for suitable z ∈ R. where
a
n
(c)
=
q
Σ/(2 log n
(c)
) and d
n
(c)
=
√
Σ
p
2 log n
(c)
−(log log n
(c)
+log 4π)/
q
2 log n
(c)
)
.
(34)
Proof. Let n
(c)
= n
(c)
(T ) =
T −b
cb
= O(T/b) and recall that c =
1
K
, K ∈ N.
For C
2
= (c
2
ij
) it holds that
c
2
ij
= γ
i−j
=
(
Σ
1 −
i−j
K
, 0 ≤ i − j ≤ K,
0, i − j > K.
This means that {S
bk
}
k≥1
is a Kdependent sequence. In particular, we have
lim
h→∞
γ
h
log h = 0.
Moreover it is stationary, i.e. S
bk
∼ F
b
for all k, since the summation extends over iid
variables X
i
. Deﬁne z
n
(c)
= (z
n
(c)
, . . . , z
n
(c)
) as the n
(c)
dimensional vector with all
components being z
n
(c)
as above. According to Theorem 4 it holds that
P
max
k=1,...,n
(c)
S
bk
≤ z
n
(c)
= P (S
b1
≤ z
n
(c)
, . . . , S
bn
(c)
≤ z
n
(c)
)
= F
S
b
(z
n
(c)
)
≤ Φ
n
(c)
,C
2
(z
n
(c)
) + ∆
b
= P (Z
1
≤ z
n
(c)
, . . . , Z
n
(c)
≤ z
n
(c)
) + ∆
b
,
where the vector (Z
1
, . . . , Z
n
(c)
) is n
(c)
variate Gaussian with mean vector 0 and
covariance matrix C
2
.
Theorem 5 in § 3.2.1 and the assumption that T
7/9
/b = o(1) ensure that the second
summand ∆
b
converges to zero as T → ∞. For the ﬁrst summand, a standard argument
on max of random variables can be applied. Since n
(c)
tends to inﬁnity as T → ∞ and
due to the appropriate covariance structure (see equation 33) it holds
P (Z
1
≤ z
n
(c)
, . . . , Z
n
(c)
≤ z
n
(c)
) = P (M
n
(c)
≤ a
n
(c)
z + d
n
(c)
) −−−−−→
(T →∞)
e
− e
−z
This concludes the proof.
14 Kiesel, Mroz, Stadtm¨uller
3.2.3. Local power Now, we are ready to state an asymptotic distributional result for
the extremes of ξ
k
as an application of Theorem 6.
Theorem 7. Let a
n
(c)
and d
n
(c)
be as in (34). If T
7/9
/b = o(1), we have
P
max
k=1,...,n
(c)
ξ
k
≤ a
n
(c)
z + d
n
(c)
−→ e
− e
−z
(35)
for all z ∈ R as T → ∞.
Proof. Proposition 2 in § 2.2 can be extended beyond the bivariate case to show
that the n
(c)
variate vector (ξ
1
, . . . , ξ
n
(c)
) is asymptotically normally distributed with
mean 0 and covariance matrix c ΣM. To be precise, we know that
(ξ
1
, . . . , ξ
n
(c)
)
d
−→ (Z
1
, . . . , Z
n
(c)
) ∼ N
0, c ΣM
.
By assumption, it holds that δ > 7/9, so by setting S
bk
= ξ
k
, Theorem 6 is directly
applicable to establish (35).
This result can now be exploited to construct a global test of parameter constancy
over the observation horizon T . Deﬁne M
n
(c)
:
= max
k=1,...,n
(c)
ξ
k
and consider the test
statistic
T
0
n
(c)
= (M
n
(c)
− d
n
(c)
)/a
n
(c)
d
−→ Λ (36)
as n
(c)
→ ∞ where Λ denotes the Gumbel extremal distribution. The Extremal test
rejects the null hypothesis H
0
0
against the alternative that the maximal value of the
thinned parameter series exceeds a ﬁxed value θ
0
∈ Θ to the conﬁdence level α ∈ (0,1) if
T
n
(c)
> λ
α
where λ
α
denotes the (1 −α)quantile of the Gumbel extremal distribution.
The critical region of the test is then K
0
= (λ
α
, ∞). We are again interested in the
local power of the test, and thus consider the following local hypothesis for η > 0
H
0
A
(η):
n
θ
t
k
= θ
0
(η) = θ
0
+ η
q
Σ /(b log n
(c)
)
o
. (37)
The Extremal test’s properties with respect to its statistical power are condensed
in the following theorem.
Theorem 8. Let T
0
n
(c)
be given as in (36), i.e. the centered and normalized maximum
of the thinned and rescaled estimator series {ξ
k
}
k=1,...,n
(c)
. Under the conditions of
Theorem 7, especially under the assumption that T
7/9
/b = o(1), and under the local
alternative H
0
A
(η) for η ≥ 0, consider the extremal test to the level α ∈ (0,1) as
T
0
n
(c)
> λ
α
.
1. As n
(c)
→ ∞ with b → ∞, the asymptotic local power of the extremal test is
P
H
0
A
(η)
(T
0
n
(c)
> λ
α
) → Γ
0
(η) = 1 − Λ(λ
α
− η
√
2) (38)
where Λ( . ) is the distribution function of the Gumbel extremal distribution.
2. For η = 0, we have Γ
0
(η) = α, and for η > 0, we have Γ
0
(η) > α. Thus, the
extremal test has local power of order
p
b log n
(c)
.
Copula Models 15
Proof. For notational convenience we set P = P
H
0
A
, n = n
(c)
and θ
0
= 0 and consider
the rejection probability under the alternative:
P
T
0
n
> λ
α
= P
ξ
1
> a
n
λ
α
+ d
n
, . . . , ξ
n
> a
n
λ
α
+ d
n
= P
√
b(
ˆ
θ
t
1
− θ
0
(η)) > a
n
λ
α
+ d
n
−
√
b θ
0
(η), . . . ,
√
b(
ˆ
θ
t
n
− θ
0
(η)) > a
n
λ
α
+ d
n
−
√
b θ
0
(η)
= P
˜
ξ
1
> a
n
(λ
α
− η
√
2) + d
n
, . . . ,
˜
ξ
n
> a
n
(λ
α
− η
√
2) + d
n
= P
˜
M
n
> a
n
(λ
α
− η
√
2) + d
n
) → 1 − Λ(λ
α
− η
√
2
,
where a
n
=
p
Σ/2 log n in case of centered Gaussian variables with variance Σ and
Thm.7 was applied.
Note that for any sequence {z
n
}
n∈N
we have that min{z
n
} = −max{−z
n
}, and
furthermore we know that for X ∼ N(0,1) it holds that X
d
= −X. Therefore a similar
result for the minimal limiting law of the estimator series {ξ
k
} holds true, namely for
all z ∈ R we have
P
min
k=1,...,n
(c)
ξ
k
> −a
n
(c)
z − d
n
(c)
−→ e
− e
−z
(39)
with a
n
(c)
and d
n
(c)
as in Theorem 7. This can be used to test H
0
0
against the
alternative H
00
A
: {min
t
θ
t
< θ
0
} with the test statistic T
00
n
(c)
= −(m
n
(c)
+ d
n
(c)
)/a
n
(c)
where m
n
(c)
= min
k=1,...,n
(c)
ξ
k
. Rejecting the null hypothesis upon T
00
n
(c)
< λ
1−α
yields
an asymptotically unbiased test with local power of the same order as in the maximum
case.
Furthermore, as the sequence {ξ
k
}
k=1,...,n
(c)
is Kdependent we have asymptotic
independence of M
n
(c)
and m
n
(c)
, so that we can also test against the alternative
H
000
A
: {min
t
θ
t
< θ
0
or max
t
θ
t
> θ
0
}, i.e. that the extrema of the parameter series
exceed some conﬁdence band. The null hypothesis is then rejected if either T
0
n
(c)
> λ
α/2
or T
00
n
(c)
< λ
1−α/2
. This testing procedure results in a symmetric simultaneous test with
proper asymptotic size and local power of the same order as in the preceding cases.
3.3. Discussion
Conclusively, we discuss the two types of tests from the previous sections with
respect to their ability to detect whether and of which type there are deviations from
the null hypothesis of parameter homogeneity.
In § 3.1 we presented a Binomial test with local power of order
√
b where b = O(T )
as T → ∞. The facts that the test does not achieve the usual local power of order
√
T ,
as might be expected when a sample of size T is available, and that the local estimates
are restricted to be independent, may seem to be a disadvantage at ﬁrst sight. At
the same time, take into account that using moving window estimation we implicitly
presume a locally homogeneous structure in the copula parameter anyway. Under this
perspective, a rejection of one (or more) of the partial hypotheses hints at one (or more)
regime changes in the parameter series. In order to avoid multiple testing the multiple
approach can be used. We again refer to Chapter 4 for an empirical application of the
Binomial test on realworld data.
The Extremal test presented in § 3.2 has local power of order
p
b log n
(c)
where
n
(c)
= b
T −b
cb
c and T
7/9
/b = o(1) with respect to the overall observation horizon T. This
local power is not least among the considered tests but the speed of the approximation
16 Kiesel, Mroz, Stadtm¨uller
in the limit law is low. Furthermore, the applicability of the procedure is limited
in ﬁnite samples due to the restriction T
7/9
/b = o(1). This bandwidth condition is
practically only fulﬁlled in samples over a long observation horizon, say for T in the
order of decades if we consider daily price data. Nevertheless the test is suitable to
test against the most general alternative in our setup, namely whether there is an
exceedance from a long term constant parameter value at all, and if the deviation is in
a sense extreme, i.e. the maximal (or minimal) parameter value peaks high above (or
below) an assumed average level. Motivated by the empirical observations in realworld
data where the estimated parameter series sometimes exhibit high peaks (mostly above
the average parameter level), this test was developed as an omnibus test that allows
to judge if peaks in the parameter value lie within a predeﬁned conﬁdence region. In
the course of the data description and empirical analyses in Chapter 4, we will also
present what the rejection of the extremal null hypothesis H
0
A
could be used for after
that.
Summarizing, each of the presented tests has certain pros and cons, and on its own is
only able to detect the special alternatives that were outlined in the respective sections.
We can though combine the testing of the respective null hypotheses in the same order
as they were presented in this discussion: First, the Extremal test lends itself to decide
whether there is an extremal deviation of an average copula parameter, second, the
Binomial test provides evidence for moderate deviation and/or the location of possible
change points. Finally, a suspected ﬁxed regime change can be veriﬁed through a
socalled forecast regression based test, see e.g. [36, 30].
4. Empirical Results
4.1. Empirical size of the Binomial test
Under the null hypothesis of parameter constancy we simulate copula data of dif
ferent dimensions and with diﬀerent parameter values. Then the Binomial test is run
with various types of base data in order to investigate the test’s empirical size. Within
this scope, the proper determination of the involved estimation variance proves vital
for a correct testing procedure. The main outcome is that bootstrapping the variance
cannot be avoided in our ﬁnitesample setup notwithstanding the theoretical results.
We have to refer to [30] for details.
4.2. Commodity contracts
In recent years the socalled ﬁnancialisation of commodities, which implies a growing
interaction of commodity markets with classical equity and ﬁxed income markets,
has been investigated in a number of studies. We contribute to this discussion be
investigating the dependency structure of typical commodity contracts.
4.2.1. Data description We have chosen to work on socalled second frontmonth for
ward prices for diﬀerent types of commodities such as coal, natural gas, crude oil
and electricity. We have decided to consider the second front month as these are the
contracts that are traded most liquidly. Further information on these contracts can be
found in [5].
The historical time series data for our analysis is retrieved from Bloomberg’s business
database. We consider the following contract identiﬁed by their Bloomberg ticker:
API22MON Index coal, price per metric ton (based on All Published Indices), delivered
Copula Models 17
to the Amsterdam, Rotterdam and Antwerp region (ARA) in Northwest Europe.
API42MON Index coal, same as above, but the delivery takes place to Richards Bay,
South Africa. The pricing source was not Bloomberg, but Credit Suisse.
TTFG2MON Index natural gas, price per megawatt hour, based on the virtual gas hub
covering all entry and exit points in the Netherlands.
NBPG2MON Index natural gas, price per megawatt hour, based on the market move
ments at the virtual gas hub covering mainland Britain.
CO2 Comdty Brent crude oil, price per barrel, supplied at Sullom Voe on Shetland,
Scotland.
BRSWMO2 Index Brent crude oil, price per barrel, calculated by Bloomberg based on
the Intercontinental Exchange (ICE) futures prices.
EL{U,G,B}B2MON Index price per megawatt hour of baseload electricity in the United
Kingdom, Germany and Belgium respectively.
EL{U,G}P2MON Index price per megawatt hour of peakload electricity in the United
Kingdom and Germany respectively.
After cleaning the data (described in §5.2.1 in [30]) the transformed series of logre
turns
R
(j)
t
= log
S
(j)
t
S
(j)
t−1
!
for all j = 1, . . . , 11, t ≥ 1 forms the basis of all further considerations.
In the subsequent section we will describe the univariate GARCH models for the
logreturn series, how we proceeded to decide on goodnessofﬁt of the diﬀerent copula
classes and the test results of the Binomial and the Extremal test.
4.2.2. SCOMDY modeling and testing After the data retrieval and preparation, the
focus is on ﬁtting appropriate time series models and ﬁnding a proper copula family
to capture the dependence structure.
In a ﬁrst analysis we use one year of historical data in the period from 1803
2008 to 18032009 and consider all of the possible pairwise combinations resulting
in 55 bivariate series vectors. Subsequently we ﬁt several of the copula families
presented in Appendix A in [30] to the ranktransformed logreturns, i.e. we do not
deGARCH the series ﬁrst, but directly investigate the present pairwise dependence
structure. According to the maximum likelihood logic, we choose the minimal value of
the negative loglikelihood function of the whole series for t = 1, . . . , T as the goodness
ofﬁt criterion, i.e. we select a copula family as “best ﬁt” for a logreturn pair when the
respective negative loglikelihood function is minimal compared to the other families.
The result is that the Clayton copula family with a positive parameter (see § A.5 in
[30]) emerges most frequently as “best ﬁt”.
The second step is taken with regard to suitable univariate GARCH models. As
argued in § 3.1 in [30] with reference to [24], it suﬃces in most cases to set up the
18 Kiesel, Mroz, Stadtm¨uller
parsimonious GARCH(1,1) model. Therefore, we assume that for the logreturns we
have
R
(j)
t
= µ
j
+ Y
jt
Y
jt
= σ
jt
ε
jt
(40)
σ
2
jt
= ω
j
+ α
j
Y
2
j,t−1
+ β
j
σ
2
j,t−1
with parameters µ
j
, ω
j
, α
j
, β
j
and univariate white noise residuals ε
jt
∼ WN(0,1) with
respect to t for all j = 1, . . . , 11. This time, we take as basis for the estimation
two years’ time series data from 19112007 to 19112009 and obtain the following
parameter estimates in the univariate models. In Table 1 the respective estimates are
given together with the sum ˆα
j
+
ˆ
β
j
. A value greater than or equal to 1 indicates
that the respective model is not covariancestationary; note however, that this does
not aﬀect the consistency of the quasi maximum likelihood estimates.
j ˆµ
j
ˆω
j
ˆα
j
ˆ
β
j
ˆα
j
+
ˆ
β
j
1 API2 8.7409e04 6.4357e06 0.146726 0.86039960 1.00713
2 API4 1.8548e03 1.3157e05 0.350214 0.70303320 1.05325
3 TTFG 5.7206e04 1.8513e05 0.135967 0.87672740 1.01269
4 NBPG 7.1524e04 2.9631e05 0.077660 0.91306180 0.99072
5 CO2 9.5303e04 8.7841e06 0.071987 0.91476680 0.98675
6 BRSW 9.2096e04 8.2616e06 0.063581 0.92285210 0.98643
7 ELUB 2.1332e04 8.9599e05 0.321522 0.62147820 0.94300
8 ELGB 3.5195e05 8.0567e04 0.211854 0.00000001 0.21185
9 ELBB 2.4977e04 1.0554e03 0.160923 0.03653276 0.19746
10 ELUP 8.6258e05 1.3132e04 0.231266 0.63050610 0.86177
11 ELGP 2.1356e04 8.9710e04 0.358092 0.00000001 0.35809
Table 1: Estimated parameters for univariate commodity GARCH(1,1) models
Note that we abbreviated the tickers. After deGARCHing the original logreturn
series, we obtain 11 univariate series of empirical residuals which are ranktransformed
before the further investigation. All bivariate combinations result in
11
2
= 55 possible
pairs. The pairwise dependence is captured by a bivariate Clayton copula. In Table 2
the global parameter estimates
ˆ
θ are given in the lower left triangle, and the respective
variance estimates
ˆ
Σ obtained by the bootstrap procedure described in § 5.1.2 in [30]
are given in the upper right triangle.
The two considered trading years sum up to the observation horizon T = 539, the
bandwidth is chosen as one tenth of T , namely b = 54 (which accounts for merely
n = 9 independent estimators), the thinning constant is c = 1/4 (which accounts
for N = 37 weakly dependent estimators in the thinned series). With a signiﬁcance
level of α = α
∗
= 5% and the number of local exceedances less than q
α
∗
= 2 the
null hypothesis of parameter constancy cannot be rejected. On the other hand if the
number of exceedances is greater or equal 3, then the null hypothesis is rejected. For
exactly two local exceedances a randomization has to be conducted, i.e. by generating
r as a realization of a Bernoulli random variable with size p as in (26), the null is
Copula Models 19
rejected if r = 1 and cannot be rejected if r = 0. Furthermore, the Extremal test
is conducted in its minmax version, i.e. we test for both deviations of the maximum
above the upper global conﬁdence level, as well as deviations of the minimum below the
lower global conﬁdence level. The test results can be visualized in a very intuitive way
(see Figure 1); by giving two examples in case of the twoyear data we demonstrate
on the one hand socalled global exceedances which are deviations of the extremal
values of the estimator series, and that they possibly may come along with extreme
realworld events. On the other hand we will explain the diﬀerent occurring elements
of the graphics.
0.0 0.5 1.0 1.5 2.0 2.5
24.10.2007 01.05.2008 06.11.2008 18.05.2009 20.11.2009
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
θ
^
= 0.829
●
●
local CI (α = 0.0051)
local exc
global CB
global exc
(a) NBPG2MON vs. ELUB2MON
−0.5 0.0 0.5 1.0
24.10.2007 01.05.2008 06.11.2008 18.05.2009 20.11.2009
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
θ
^
= 0.308
●
●
local CI (α = 0.0051)
local exc
global CB
global exc
(b) API42MON vs. BRSWMO2
Figure 1: Global exceedance and absence of exceedances
The horizontal black lines depict the global parameter estimate
ˆ
θ, the whole time
varying parameter series {
ˆ
θ
t
} is displayed in grey. The solid circles mark the inde
pendent estimators
ˆ
θ
k
, whereas the nonsolid circles mark the thinned series
ˆ
θ
t
k
. The
marked estimators are grey whenever they lie within the local (blue) respectively the
global (green) bands, otherwise they are colored in red. The observation horizon is
given on the time axis, the parameter values on the yaxis. In particular, we see the
3.4427 1.5737 1.5563 2.2915 2.3264 1.4272 1.6389 1.4685 1.3076 1.5001 API2
1.2703 1.5303 1.4775 2.0483 2.1947 1.6920 1.7098 1.7401 1.5123 1.5528 API4
0.1407 0.1635 8.2404 1.6686 1.6423 2.2104 2.1753 1.6393 2.0779 2.4668 TTFG
0.1881 0.1702 1.6905 1.8589 1.7635 2.4607 2.8032 1.8740 1.9904 2.5159 NBPG
0.3345 0.3159 0.1743 0.1895 78.7038 1.8754 1.8699 1.5593 1.6921 1.5389 CO2
0.3095 0.3081 0.1151 0.1514 6.0296 1.7605 1.8124 1.5718 1.5494 1.5377 BRSW
0.2659 0.1791 0.6536 0.8289 0.2813 0.2536 3.0064 2.0581 2.4142 2.4250 ELUB
0.3125 0.2288 0.4261 0.4527 0.1774 0.1587 0.6175 4.4794 3.9574 7.2225 ELGB
0.2395 0.1776 0.4166 0.3651 0.2070 0.1873 0.5471 1.2602 2.4444 2.5953 ELBB
0.1943 0.1685 0.6918 0.7331 0.2387 0.2068 1.6195 0.7405 0.7813 4.6296 ELUP
0.1739 0.1172 0.4409 0.4247 0.1221 0.1088 0.6480 1.6319 1.1894 0.9103 ELGP
API2 API4 TTFG NBPG CO2 BRSW ELUB ELGB ELBB ELUP ELGP
Table 2: Estimated global parameter (lower left) and bootstrapped variance estimate (upper
right) for bivariate SCOMDY models
20 Kiesel, Mroz, Stadtm¨uller
following:
NBPG2MON vs. ELUB2MON The dependence structure between the natural gas and the
UK baseload electricity contract is rather unstable  nevertheless, the binomial
test on independent samples doesn’t detect the timeinhomogeneity, whereas the
extremal test leads to the detection of three exceedances beyond the upper global
conﬁdence band. Let us summarize the special features of this pair which might
help to understand this behavior. The global dependence which stems from a
parameter value of
ˆ
θ = 0.83 is rather high and peaks beyond a value of θ = 2
in its extremes. Last, the so called subprime mortgage crisis in September 2008
falls into the observation horizon, and precisely in this period the dependence
exhibits a peak which exceeds the global bands.
API42MON vs. BRSWMO2 The coal contract versus the oil contract depicted here is an
example for a parameter series which ﬂuctuates around the global parameter in
a very quiet way. Adding the local and global bands conﬁrms the hypothesis of
parameter constancy since there are no exceedances at all.
The numerical results that have been described so far represent the ﬁrst experiments
and a preliminary data survey that we conducted to get a sense for the speciﬁc features
of the data. Now we report on a comprehensive numerical analysis that was conducted
conclusively. We use ﬁve years of historical data from 19122006 to 19122011. After
a preliminary data preparation, we have chosen to apply more sophisticated univariate
GARCH models for the logreturn series. We proceed along the lines of [11] and use an
exponential GARCH(1,1) model due to [31]. We add an autoregressive part and arrive
at
R
(j)
t
= µ
j
+ ρ
j
R
(j)
t−1
+ Y
jt
Y
jt
= σ
jt
ε
jt
(41)
log(σ
2
jt
) = ω
j
+ α
j
ε
j,t−1
+ γ
j
(ε
j,t−1
 − E (ε
j,t−1
)) + β
j
log(σ
2
j,t−1
)
with real parameters µ
j
, ρ
j
, ω
j
, α
j
, γ
j
, β
j
and univariate white noise residuals ε
jt
∼
WN(0,1) with respect to t for all j = 1, . . . , 11. Furthermore, in accordance with
the ideas presented in [10] univariate structural break analyses are performed using
the ﬂuctuation tests due to [2]. Furthermore, the asymptotic properties of estimators
of the break dates are derived. The empirical residuals of the AR(1)eGARCH(1,1)
model given in (41) are subsequently the basis of a pairwise dependence analysis, i.e.
the bivariate vectors of empirical residuals are assumed to be distributed according to
a copulabased multivariate distribution. We condense the results we obtained for the
parameter estimate of bivariate Clayton copulas in Table 3.
Here, we only give the average (∅), the minimal (min) and the maximal (max)
value of the global parameter estimate
ˆ
θ based on the whole sample of about T = 1300
observations, as well as of the variance estimate
ˆ
Σ stemming from the canonical
maximum likelihood estimation and the bootstrapped variance estimate
ˆ
Σ
(BS)
. The
bootstrapped version of the variance estimator was obtained nonparametrically as
based on B = 200 estimations on a bootstrap sample of the same length as the
bandwidth of the moving window estimates, i.e. b = T /10. Furthermore, we give
the respective range of the estimator of Kendall’s tau and the lower tail dependence.
Copula Models 21
ˆ
θ
ˆ
Σ
ˆ
Σ
(BS)
ˆτ =
ˆ
θ
(
ˆ
θ+2)
ˆ
λ
L
= 2
−
1
ˆ
θ
∅ 0.41403 0.00336 0.07040 0.17151 0.18747
min 0.00998 0.00067 0.00405 0.00497 0.00000
max 7.30908 0.10148 2.71164 0.78516 0.90952
Table 3: Summary statistics for Clayton copula estimation with 5year commodity data
The estimation of
ˆ
θ does actually work for 47 of the 55 possible pairs, so that we can
conduct the Binomial and the Extremal test for these commodity pairs. This results in
25 local rejections (53%), i.e. rejections of the null hypothesis H
0
as in (16) based on
the Binomial test, and 27 global rejections (60%), i.e. rejections of H
0
0
as in (29) based
on the Extremal test for the maximum of the parameter series. After reestimation for
the potential breakpoints indicated by the local exceedances, the ration of rejections
shrinks to 22 (42%) in the binomial and 21 (44%) in the extremal test respectively.
In Figure 2, the graphical display of an exemplary test result on the 5year historical
data similar to Figure 1 is given. In subﬁgure 2(a) the estimator series
ˆ
θ
t
together with
the local conﬁdence regions due to the Binomial test and the global conﬁdence bands
due to the Extremal test are depicted. The empirical residuals that form the basis for
the copula estimation stem from a SCOMDY model over the whole observation horizon
t = 1, . . . , T = 1296.
The Binomial test leads to the rejection of the null hypothesis of parameter con
stancy as there is one exceedance beyond the local conﬁdence band at the point in
time τ = 1037, which corresponds to the date 22122010. With this knowledge, we
reestimate the univariate GARCH models with two diﬀerent regimes: the ﬁrst is
for t = 1, . . . , τ and the second for t = τ + 1, . . . , T . Then the empirical residuals
are concatenated and again used as pseudorealizations of the residual copulabased
distribution. Canonical maximum likelihood estimation of the global copula parameter
ˆ
θ and the estimator series
ˆ
θ
t
, as well as bootstrapping a variance estimate
ˆ
Σ
(BS)
based
on this concatenated series then results in a Binomial test that does not reject the null
hypothesis, as subﬁgure 2(b) shows.
This approach is advantageous compared to another attempt to remedy timevariation
in the copula parameter: the respective regimes are detected separately in the univari
ate models in order to reestimate the GARCH parameters on diﬀerent regimes.
We conclude this section with an additional discussion of the possible economic
interpretation of our results.
4.2.3. Discussion Concerning the empirical data, let us note the following: In shorter
time series we observed fewer exceedances in sum, but also some extreme exceedances
that ﬁnally gave the motivation for the Extremal test. This happened especially with
pairs which have the same underlying commodity. In the 5year time series data, the
Binomial test gave more overall rejections, but we had fewer rejections when applying
the Extremal test.
In our subsequent analyses, we found that there are cases when exceedances hint
at some regime changes in the univariate models. Taking times of local (or global)
exceedances as change points in the GARCH parameters, reestimating the univariate
models, and using the concatenated empirical residuals as basis for a repeated canonical
22 Kiesel, Mroz, Stadtm¨uller
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
20.12.2006 24.03.2008 24.06.2009 22.09.2010 19.12.2011
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●
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●
●
●
●
●
●
●
●
●
●
●
●
●
θ
^
= 0.158
●
●
local CI (α = 0.0051)
local exc
global CB
global exc
(a) before change point handling
0.0 0.1 0.2 0.3 0.4 0.5
20.12.2006 24.03.2008 24.06.2009 22.09.2010 19.12.2011
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●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
θ
^
= 0.143
●
●
local CI (α = 0.0051)
local exc
global CB
global exc
(b) after reestimation of univariate models
Figure 2: Binomial test with proper size for API22MON vs. ELGP2MON
maximum likelihood estimation of the global and local copula parameters, we found
that the dependency represented by the copula parameter series exhibited a time
homogeneous structure at least in some of the investigated cases.
However, there were also cases where regimechanged GARCH models did not have
any eﬀect on the heterogeneity in the copula parameter. With a merely graphical
valuation of the results, the estimator series seemed to have either diﬀerent regimes
themselves, or exhibited an autoregressive structure.
From the economic point of view, our observations and test results are in line
with similar econometric ﬁndings. For example, [19] measure a portfolio’s dependence
structure through Spearman’s ρ and ﬁnd that signiﬁcant deviations from the hypothesis
of constancy come along with unusual economic events (their procedure actually detects
the volatility peaks that the ﬁnancial markets exhibited in 2002 and the beginning of
2004).
Also [13] model the conditional dependence structure using copulae and are so able
to detect changes in the dependence structure of bivariate time series such as changes
in the tail of the joint distribution. A timevarying copula approach was already used
in [14], who detect changes in the dependence structure of exchange rates.
A very recent example regarding the assessment of timedynamic dependencies
with the help of copulas is [34]. There a pair copula construction, also called vine
copula, is used to model the empirical residuals of highdimensional ﬁnancial data. By
applying Bayesian estimation techniques to determine Markovswitching parameters,
the authors ﬁnd that diﬀerent regimes in the dependency parameters are often implied
by exceptional economic events (such as the 2008 subprime crisis).
Moreover, [1] investigate the dependence between the DAX 30 constituents with
vine copulas and incorporate timedynamics by the stochastic autoregression due to
[23]. They found that according to the BIC model selection criterion nondynamic
models outperform the timevarying ones in most of the levels of the vine construction.
Copula Models 23
Acknowledgements:
• The authors thank an anonymous referee for many thoughtful suggestions which
improved the paper considerably.
• Access to data was provided by the LBBW Trading Room operated by the
Institute of Financial Mathematics at Ulm University.
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