3. Tail Coefficients
In the bivariate case (i.e.,
), lower and upper tail coefficients are defined, respectively, as
if the limits above exist. Throughout the text, when defining these and other tail coefficients, we will assume the existence of the limits involved. The general idea behind the tail coefficients is to measure how likely a random variable is extreme, given that another variable is extreme. These coefficients can take values between 0 and 1, since they are probabilities.
For extreme-value copulas, tail coefficients can be expressed as functions of Pickands dependence function
corresponding to the copula
as
see [
11]. That is, unless the studied copula is the comonotonicity copula, extreme-value copulas do not possess any lower tail dependence. Recall that, when
, the corresponding copula must be the independence copula
. Therefore, an extreme-value copula possesses upper tail dependence, unless the copula is the independence copula.
In case of Archimedean copulas, the tail coefficients can be expressed via the corresponding generator
as
see [
14]. Note that both tail coefficients only depend on the behavior of the generator
in proximity of the points 0 and
. Recall that, in the case of strict Archimedean generators, the latter is equal to
∞.
Given their meaning and mathematical expression, tail coefficients cannot be generalized in general dimension in a straightforward and unique way. We first propose a set of desirable properties that are expected to hold for any multivariate tail coefficient and for any d-variate copulas and , . The following properties are stated under the working condition that all tail coefficients (, , , and so on) exist.
- (T1)
(Normalization) ,
- (T2)
(Continuity) If , then as ,
- (T3)
(Permutation invariance) for every permutation ,
- (T4)
(Addition of an independent component) For
independent of
Property could be formulated in a slightly stricter way, as
For
, independent of
, there exists a constant
not depending on
such that
Because both lower and upper tail dependence are of interest, usually we consider that has actually two versions and focusing on either upper tail (variables simultaneously large) or lower tail (variables simultaneously small) dependence respectively. Thus we can also consider the following property
- (T5)
(Duality) .
In general, some of the desirable properties above are easy to be enforced. If one starts with a candidate coefficient
, property
can be achieved by defining
Property
can be achieved by taking an average of the candidate coefficient
over all of the permutations
where
denotes all of the permutations of the set
. Note, however, that, especially for high dimensions, this significantly increases computational complexity. In the case of property
, we can simply use it to define an upper tail coefficient from the lower tail one (or the other way around).
In the following, we briefly review multivariate tail coefficients proposed in the literature and elaborate on their behavior with respect to the desirable properties –. For brevity of presentation, we refer to or its variant as the “addition property”. To simplify the notation, the subscript d of , denoting the dimension, will sometimes be omitted in the text, the dimension being clear from an argument of a functional .
3.1. Frahm’s Extremal Dependence Coefficient
Frahm (see [
3]) considered lower and upper extremal dependence coefficients
, respectively, defined as
given the limits exist, where
and
. These coefficients are not equal to
, respectively, in the bivariate case. More specifically, for any copula
(see [
3])
Thus, we can consider it more as a different type of tail dependence coefficient than a generalization of bivariate tail coefficients.
For extreme-value copulas, extremal dependence coefficients can be stated in terms of Pickands dependence function. Let
be an extreme-value copula with Pickands dependence function
and denote the Pickands dependence function of the marginal copula
as
. Subsequently,
where
if
and
otherwise. As opposed to (
8), expression (9) only involves the overall
d-dimensional Pickands dependence function. This might be helpful, for example, during estimation, since not all of the lower-dimensional Pickands dependence functions in (
8) need to be estimated.
Thus, for the lower extremal dependence coefficient, one obtains
because the polynomial (in
t) in the denominator contains lower-degree terms than the polynomial in the numerator. We can see that this behavior resembles
for bivariate extreme-value copulas, since the only extreme-value copula possessing lower tail dependence is the comonotonicity copula.
For the upper extremal dependence coefficient, we can calculate
where, as above,
if
and
otherwise.
We next look into the tail coefficients (
7) for Archimedean copulas. Let
be a sequence of
d-dimensional Archimedean copulas with (the same) generator
. Subsequently,
The corresponding derivatives, if they exist, are
Afterwards, the extremal dependence coefficients can be expressed as
where we used L’Hospital’s rule to get to the equation in (
12), and the second equation in the derivation towards (13). Recall that
and
. One can see that using L’Hospital’s rule does not solve the
limit problem for general
and knowledge of the precise behavior of
is thus crucial for calculating the coefficients
and
.
As will be illustrated in
Section 6, Archimedean copulas can have both extremal dependence coefficients non-zero, depending on the generator. For
, one additional assumption regarding a generator
may become useful. Because (from the definition of the generator)
, if the additional condition
is fulfilled, we get
using that from Lemma 1
cannot be equal to zero. In other words, if
, then the corresponding Archimedean copula is upper tail independent, for every dimension.
Next, we investigate which of the desirable properties – are satisfied for Frahm’s extremal dependence coefficients and .
Proposition 2. Frahm’s extremal dependence coefficients and satisfy normalization property , permutation invariance property , and addition property , with for every , and .
Proof. Normalization property
follows from straightforward calculations
Permutation invariance property
follows immediately from the fact that the coefficients only depend on
and
, which do not depend on the order of components of the random vector.
Look now into the addition of an independent component, i.e., property
. To be able to distinguish between the dimensions, we use the notation
and
. For
independent of
, we have
and
for every
. Further,
and similarly
. Thus,
which means that the property about adding an independent component
holds with constants
for every
.
We next look into duality
. Using relation (
1) between the survival function and the survival copula, coefficients
and
can be rewritten as
and thus
where substitution
was used. This proves the validity of duality property
. □
We suspect that the continuity property
does not hold in its full generality for most multivariate tail coefficients. To obtain insight into this, consider the following example with a sequence of copulas
given by
Note that the distribution that is given by
is uniform on the set
and it corresponds to the upper Fréchet’s bound
otherwise. Note that
is a copula with an ordinal sum representation, see [
8] (Section 3.2.2).
It is easily seen that as uniformly on . Note that for each . On the other hand, . Hence, for this sequence of copulas, the continuity property does not hold.
However, a continuity property may hold, in general, under more specific conditions on the copula sequences. One such condition is that of a sequence of contaminated copulas, defined as follows.
Let
and
, for
be
d-variate copulas, and let
be a sequence of numbers in
. One considers the sequence of contaminated copulas
Note that
is a convex combination of the copulas
and
and, hence, is also a copula, see e.g., [
8]. The interest is to investigate the behavior of a tail coefficient for the sequence
when
, as
.
Proposition 3 establishes a continuity property for Frahm’s extremal dependence coefficient.
Proposition 3. Suppose that, for any d-variate copulas and , , there exist , such thatFurther assume that exists for every . Subsequently, as . In particular, condition (15) is satisfied for a sequence of contaminated copulas, as in (14), for which , as , and provided exists.
Proof. Assumption (15) allows for us to use the Moore–Osgood theorem to interchange the limits and, thus
Suppose now that we have a sequence of contaminated copulas, for which
, as
. Subsequently, one calculates
One next realizes that
and
. Furthermore, with the help of Formula (
2) for the survival function of a copula one gets
. Thus, one can bound
which implies (
15). □
Analogously, a similar result could be stated for .
3.2. Li’s Tail Dependence Parameter
Suppose that
is a subset of indices, such that
and
. Subsequently, Li [
4] (Def. 1.2) defines so-called lower and upper tail dependence parameters, as follows
given the expressions exist. It is evident that these coefficients heavily depend on the choice of the set
. Additionally, this generalization includes the usual bivariate tail dependence coefficients
and
, by letting
,
and
or the other way around. Li [
4] further states that
and, therefore, duality property
is fulfilled.
One can also notice that, for exchangeable copulas (i.e., symmetric in their arguments), the dependence parameters are in fact functions of cardinality
h rather than particular contents of
. Especially in this case, it is worth introducing another notation being
In paper [
15], it is shown that these coefficients can be rewritten while using one-sided derivatives of the diagonal section
of the corresponding copula in the following way:
where
denotes the diagonal section of copula
.
Additionally, the authors in [
15] comment on the connection with Frahm’s extremal dependence coefficients
and
, which can be expressed as
if all of the above quantities exist.
De Luca and Rivieccio [
6] (Def. 2) also use this way to measure tail dependence, although they consider it as a measure for Archimedean copulas only since we can express the measures while using the generator, as
where we applied l’Hospital’s rule for obtaining the equation in (17) and (18). In contrast to the Frahm’s coefficient, here the additional condition that
is not helpful, since it leads to
and numerator and denominator are both equal to zero here.
Proposition 4. Li’s tail dependence parameters and satisfy normalization property , addition property , and duality property .
Proof. Duality property
was shown in [
4]. Normalization property
follows from straightforward calculations while using (17) and (18)
For
, it follows from duality property
.
We now check property
, the addition of an independent random component. Suppose that the added independent component belongs to the set
. Subsequently,
If the added independent component belongs to the set
, then from the definition of the coefficient
Showing the duality property for
is analogous. □
The proof of Proposition 4 shows that, in fact, property is fulfilled if one distinguishes two cases. If the added independent component belongs to the set , then holds with for every . However, if the added independent component belongs to the set , then for every .
Permutation invariance
does not hold in general. However, if one would restrict to only permutations that permute indices within
and within
and not across these two sets,
and
would be invariant with respect to such permutations. Further, we might consider the special case when
, which is if we condition only on one variable. Subsequently, for any permutation
and analogously for
, we have
.
A continuity property can be shown under a specific condition on the copula sequence as is established in Proposition 5.
Proposition 5. Suppose that, for any d-variate copulas and , , there exist , such thatFurther assume that exists for every , as well as . Subsequently, as . In particular, condition (20)
holds for a sequence of contaminated copulas, see (14)
, for which , as , andand exists. Proof. The first part of Proposition 5 is proven along the same lines as the proof of Proposition 3 and hence omitted here.
Consider now a sequence of contaminated copulas satisfying in addition (21). We need to show that (20) holds. To see this, note that, similarly as in (16), one gets
Further note that, for all sufficiently large
m for all
Combining (21), (22) and (23) now yields that (for all sufficiently large
m)
where the
-term does not depend on
u. Thus, one can conclude that condition (20) of Proposition 5 is satisfied. □
An analogous result as the one stated in Proposition 5 can be stated for .
3.3. Schmid’s and Schmidt’s Tail Dependence Measure
Schmid and Schmidt (see [
5] (Sec. 3.3)) considered a generalization of tail coefficients based on a multivariate conditional version of Spearman’s rho, which is defined as
for some non-negative measurable function
g provided that the integrals exist. The choice
leads to
and the multivariate tail dependence measure is defined as
provided the existence of the limit. Similarly, they define
Additionally, these coefficients are not equal to
, respectively, in the bivariate case, so we can consider it more as a different type of tail dependence coefficient rather than a generalization.
Proposition 6. Schmid’s and Schmidt’s tail dependence measure satisfies normalization property , permutation invariance property , and addition property , with for every .
Proof. Normalization property
and permutation invariance
follow from the normalization property and permutation invariance of Spearman’s rho, see, for example [
16]. When adding an independent component, one gets
This finishes the proof. □
In order for duality property
to hold, the upper version should rather be defined as
This seems to be more logical, since
can only be expressed, after substituting
into (25), as
which cannot be further simplified. It is easy to show that in the bivariate case (i.e.,
) the coefficients
and
coincide. For a general dimension
however they can differ.
The continuity property cannot be shown in full generality, but a continuity property is fulfilled in the special case of a sequence of contaminated copulas, as in (14).
Proposition 7. Consider a sequence of contaminated copulas, , such that , as , and exists. Afterwards, as , Proof. Direct calculation gives
since
is bounded. □
3.4. Tail Dependence of Extreme-Value Copulas
As stated in (
6), bivariate tail coefficients for extreme-value copulas can be simply expressed using the corresponding Pickands dependence function. Thus tail dependence is fully determined by the Pickands dependence function
in the point
The range of values for
is limited by
, which also gives us
where the bivariate tail coefficient
gets larger when
is closer to
. On the other hand,
means tail independence. Following this idea and given that also for the
d-dimensional Pickands dependence function
associated to a copula
we have
, a measure of tail dependence for
d-dimensional extreme-value copulas could be based on the difference
. After proper standardization, this leads to
Note that such a coefficient is equal to a translation of the extremal coefficient given in [
17] or [
7] and defined as
. This coefficient
was termed extremal coefficient in [
17]. Schlather and Town (see [
18]) give a simple interpretation of
, related to the amount of independent variables that are involved in the corresponding
d-variate random vector.
Proposition 8. The multivariate tail dependence coefficient in (28) satisfies normalization property , continuity property , permutation invariance property , and addition property , with for every .
Proof. Normalization
and permutation invariance
follow immediately from the definition of
. If
, and then also
, which proves the validity of
. For
independent of
, we can use Example 1 and obtain
□
Remark 1. The duality property is not applicable, since the survival copula of an extreme-value copula does not have to be an extreme-value copula.
3.5. Tail Dependence Using Subvectors
A common element of the multivariate tail dependence measures discussed in
Section 3.1,
Section 3.2 and
Section 3.3 is that they focus on extremal behavior of all
d components of a random vector
. However, one could also be interested in knowing whether there is any kind of tail dependence present in the vector, which is even for subvectors of
. An interesting observation to be made is for tail dependence measures that satisfy property
with
for every
. Assume that
X and
Y are independent random variables. Then any tail measure
would be zero for the random couple
and no matter which random component we add the tail measure for the extended random vector would stay 0. In other words, for any such tail dependence measure, this leads to tail independence of the
d-dimensional random vector
, no matter what
d is. Considering tail dependence of subvectors would be of particular interest in this case.
Suppose that we have a multivariate tail coefficient
that can be calculated for general dimension
. Suppose further that this coefficient only depends on the strength of tail dependence when all of the components of a random vector are simultaneously large or small. This is the case for all multivariate tail coefficients mentioned in
Section 3.1,
Section 3.2 and
Section 3.3. Subsequently, we can introduce a tail coefficient given by
where
can be interpreted as an average tail dependence measure per dimension, and where
. This measure deals with a disadvantage of current multivariate tail coefficients that assign a value of 0 to the copulas, where
components are highly dependent in their tail, and the
d-th component is independent. When dealing with possible stock losses, for example, this situation should be also captured by a tail coefficient.
Recall that the weight
corresponds to an importance given to the average tail dependence within all the
j-dimensional subvectors of
. Because tail dependence in a higher dimension is more severe, as more extremes occur simultaneously, it is natural to assume
. However, such an assumption excludes other approaches to measure tail dependence. For example, setting
and
for
would lead to the construction of a tail dependence measure as the average of all pairwise measures. If the underlying bivariate measure satisfies
,
,
, and
with
only, these properties are carried over to the pairwise measure. Additionally,
can be shown similarly as in Proposition 1 in [
16]. Despite possibly fulfilling the desirable properties, all of the higher dimensional dependencies are ignored, being a clear drawback of such a pairwise approach. In the sequel, we focus on the setting that
.
Proposition 9. Suppose that the tail dependence measures satisfy normalization property , continuity property , permutation invariance property , and duality property , for . Further assume that . Subsequently, the coefficient in (29) also satisfies properties , , , and .
Proof. Clearly and . The continuity, permutation invariance, and duality properties follow from the continuity, permutation invariance, and duality properties of . □
What happens in case of the addition of an independent component (property
) is not so straightforward, since the weights differ depending on the overall dimension
d. The addition of an independent component increases dimension and, thus, possibly changes all of the weights. However, one could try to come up with a weighting scheme that guarantees fulfilment of property
. Consider
independent of
. Suppose that the input tail dependence measures
satisfy property
, with
for
. First, we express
for the random vector
, as
Now using property
in (30) together with the fact that for index
, the corresponding summand is
and, thus, this index can be omitted, one obtains
which is equal to
with weights given as
for every
. A sufficient criterion for fulfillment of property
would thus be to have
for every
. Knowing the values
,
,
, for
, and
, one can check (31).
One rather general method of weight selection can then be as follows. Suppose that one wants to achieve that proportions of weights
and
corresponding to two subdimensions
and
do not depend on the overall dimension
d. This can be achieved by setting recursively
for
and
. The initial condition is obviously given as
. To obtain
, one needs that
for every
. Values of
closer to 0 give more weight to the
d-th dimension, values close to
limit its influence. If we further assume that
, which is
does not depend on
d, this further simplifies to
for
and
. We next check the condition in (31) for this particular weight selection. Condition (31) can be rewritten as
If
for every
j as in one case of Li’s tail dependence parameter, condition (32) allows for only one selection of
r, which is
. On the other hand, if
for every
j,
r can take any values in
. Looking from the other perspective, if
, then condition (32) is satisfied if
Let us recall that these conditions can only be seen as sufficient, not necessary. A precise study of what happens when an independent component is added requires knowledge of the weighting scheme and knowledge of the underlying input tail dependence measure.
In summary, the above discussion reveals that a measure that is able to detect tail dependence not only in a random vector as a whole, but also in lower-dimensional subvectors, can be constructed. A simple and interpretable weighting scheme proposed above can be used, such that several desirable properties of the tail dependence measure are guaranteed.
3.6. Overview of Multivariate Tail Coefficients and Properties
For convenience of the reader, we list in
Table 1 all of the discussed tail dependence measures, with reference to their section number, and indicate which properties they satisfy.
4. Multivariate Tail Coefficients: Further Properties
In
Section 3, the focus was on properties
–
. In this section, we aim at exploring some further properties that might be of special interest. We, in particular, investigate the following type of properties. Here,
denotes a multivariate tail coefficient for
. When needed, we specify whether it concerns a lower or upper tail coefficient, referring to them as
and
, respectively.
Expansion property ().
Given is a random vector with copula . One adds one random component to . Denote the copula of the expanded random vector by . How does compare to ? Does it hold that ?
Monotonicity property ().
Consider two copulas . Does the following hold?
- (i)
If for in some neighborhood of , then .
- (ii)
If for in some neighborhood of , then .
Convex combination property ().
Suppose that the copula can be written as for , and . What can we say about the comparison between and ?
For extreme-value copulas, we look into geometric combinations instead.
The logic behind property comes from the perception of a tail coefficient as a probability of extreme events of components of the random vector to happen simultaneously. Thus, when another component is added, the probability of having extreme events cannot increase. However, there is no such a limitation from below and adding a component can immediately lead to a decrease of the coefficient to zero.
In the next subsections, we briefly discuss these properties for the multivariate tail coefficients discussed in
Section 3.
4.1. Expansion Property (P1)
For Frahm’s coefficient, it holds that
and analogously for the upper coefficient. This result can be found in Proposition 2 of [
3].
For Li’s tail dependence parameters, we need to distinguish two cases. If we add the new component to the set
, then we have
However, if the component is added to the set , no relationship can be shown, in general. A special situation occurs when the component added to the set is just a duplicate of a component, which is already included in . Subsequently, obviously .
For Schmid’s and Schmidt’s tail dependence measures, one cannot say, in general, how the coefficient behaves when compared to . As can be seen from (24), the integral expression decreases with increasing dimension d, but, at the same time, the normalizing constant increases with d.
For the tail coefficient for extreme-value copulas,
it follows from Example 7 in
Section 6 that the addition of another component can lead to an increase in this coefficient. See, in particular, also Figure 5.
4.2. Monotonicity Property (P2)
Concerning the monotonicity property it is easily seen that (i) holds for Frahm’s lower dependence coefficient if we additionally assume that for in some neighborhood of . Similarly, we need to assume that for in some neighborhood of in order to show that (ii) holds.
For Li’s tail dependence parameters, property
does not hold in general. This is illustrated via the following example in case
. Consider a random vector
with uniform marginals and with distribution function a Clayton copula with parameter
(see Example 6), given by
(see (39)). We denote this first copula by
. Note that the random vector
has as joint distribution a three-dimensional Clayton copula with parameter
, which we denote by
. The vector
has the same joint distribution
. Next, we consider the copula of the random vector
that we denote by
. One has that, for all
,
In Example 6 we calculate Li’s lower tail dependence parameter for a
d-variate Clayton copula, which equals
(see (41)). Applying this in the setting of the current example leads to
which thus contradicts monotonicity property
(i).
From the definition of Schmid’s and Schmidt’s tail dependence measure, it is immediate that the monotonicity property holds.
For the tail coefficient for extreme-value copulas,
defined in (28) the monotonicity property
holds. To see this, recall from (
3), that, for an extreme-value copula
, we can express its stable tail dependence function as
and, hence, using that
, it follows that
. The same inequality holds for Pickands dependence function
, which is a restriction of the stable tail dependence function
on the unit simplex. Hence,
also implies that
. From the definition of the tail coefficient in (28) it thus follows
.
4.3. Investigation of a Tail Coefficient for a Convex/Geometric Combination (Property (P3))
Consider a copula that is a convex combination of two copulas and , i.e., for . For the survival function, we then also have .
Before stating the results for the various multivariate tail coefficients, we first make the following observation. For
, it is straightforward to show that
Frahm’s lower extremal dependence coefficient for the copula
is given by
Using (34), it then follows that, if
, then
The same conclusion can be found for Frahm’s upper extremal dependence coefficient
.
Li’s lower tail dependence parameter for
, a convex mixture of copulas, equals
and an application of (34) gives that, if
, then
. The same conclusion can be found for Li’s upper tail dependence parameter
.
Schmid’s and Schmidt’s lower tail dependence measure for a convex mixture of copulas is
For an extreme-value copula, it does not make sense to look at convex combinations of two extreme-value copulas, since it cannot be shown, in general, that such a convex combination would again be an extreme-value copula. A more natural way to combine two extreme-value copulas
and
is by means of a geometric combination, i.e., by considering
, with
. In, for example, Falk et al. [
19] (p. 123) it was shown that a convex combination of two Pickands dependence functions is also a Pickands dependence function. Denoting by
and
, the Pickands dependence functions of
and
, respectively, it then follows from (33) that the Pickands dependence function
for
, is given by
. From this it is seen that
is again an extreme-value copula. For the tail dependence coefficient for extreme-value copulas, it thus holds that
i.e., the coefficient
of a geometric mean of two extreme-value copulas is equal to the corresponding convex combination of the coefficients of the concerned two copulas.
5. Tail Coefficients for Archimedean Copulas in Increasing Dimension
A natural question to examine is an influence of increasing dimension on possible multivariate tail dependence. If one restricts to the class of Archimedean copulas, several results can be achieved, despite that similar problems with interchanging limits occur while studying the continuity property . First, let us formulate a useful lemma that describes the behavior of the main diagonal of Archimedean copulas when the dimension increases.
Lemma 2. Let be a sequence of d-dimensional Archimedean copulas with (the same) generator ψ. Then for and Proof. The proof is along the same lines as the proof of Proposition 9 in [
16]. □
This lemma can be used in the following statements that focus on individual multivariate tail coefficients. The first one to be examined is the Frahm’s extremal dependence coefficient .
Proposition 10. Let be a sequence of d-dimensional Archimedean copulas with (the same) generator ψ. Further assume thatThen Proof. The statement follows by the direct application of Lemma 2, since then
□
An analogous result could be stated for .
Remark 2. The condition on interchanging limits is, in general, difficult to check. However, we discuss some examples in which the condition can be checked. A first example is that of the independence copula for which and . Henceforth, for all . Furthermore, , for all . Consequently, in this example, the condition of interchanging limits holds. A second example is the Gumbel–Hougaard copula also considered in Example 7 in Section 6. For this copula it can be seen that, as in the previous example, the two concerned limits (when and when ) are zero and, hence, interchanging the limits is also valid in this example. Proposition 10 further shows that if we construct estimators (based on values of u close to 0 or close to 1) of the limits above for Archimedean copulas in high dimensions, these will be very close to 0.
For Li’s tail dependence parameters and , the situation is further complicated by the necessary selection of and and, in particular, of the cardinality h. However, if the cardinality of the set is kept constant when the dimension d increases, the following result can be achieved.
Proposition 11. Let be a sequence of d-dimensional Archimedean copulas with (the same) generator ψ and let h in definition of be given as for a constant . Further assume thatSubsequently Proof. Using Lemma 2, we obtain
from which the statement of this proposition follows. □
An analogous statement could be formulated for .
What can one learn from the results in this section? Archimedean copulas may be not very appropriate in high dimensions, because of their symmetry, but they are a convenient class of copulas to use. It is good to be aware though that, when the dimension increases, the tail dependence of Archimedean copulas vanishes, at least from the perspective of , and their upper tail counterparts.
Obtaining similar results for different classes of copulas would also be of interest, for example, for extreme-value copulas with restrictions on Pickands dependence function. However, this is complicated by the fact that, unlike Archimedean copulas, extreme-value copulas do not share a structure that could be carried through different dimensions. Some insights into this behavior are studied using the examples given in
Section 6. This section includes examples on both Archimedean and extreme-value copulas, as well as examples outside these classes.
6. Illustrative Examples
Example 4. Farlie–Gumbel–Morgenstern copula.
Let
be a
d-dimensional Farlie–Gumbel–Morgenstern copula defined as
where the parameters have to satisfy the following
conditions
This copula is neither an Archimedean nor extreme-value copula.
We first consider Frahm’s extremal dependence coefficients
and
. From (35), up to a constant
when
. Further, plugging (35) into (
2) gives that
behaves like a polynomial
when
. Thus,
because the polynomial in the numerator converges to zero faster than the polynomial in the denominator. Similarly, one obtains
While examining
and
, the very same arguments are of use. No matter how one chooses index sets
and
,
since, again, the corresponding limits contain ratios of polynomials, such that the polynomials in the numerators converge to zero faster than the polynomials in the denominators.
To obtain
, the integral
needs to be calculated. Consider now a special case when the only non-zero parameter is
. Then
Going back to general
, we can notice that the resulting integral would always be a polynomial in
p, with the lowest power being
and thus
A similar calculation leads to
. Some further calculations (not presented here) also show that
.
From the perspective of all the above tail dependence coefficients, the Farlie–Gumbel–Morgenstern copula does not possess any tail dependence.
Example 5. Cuadras-Augé copula.
Let
be a
d-variate Cuadras-Augé copula, that is of the form
for
. The Cuadras-Augé copula combines the comonotonicity copula
with the independence copula
. If
, then
becomes
. If
, then
becomes
.
We again start with calculating
and
. From (
2), we find
and Frahm’s lower extremal dependence coefficient
is thus given as
since if
, the polynomial in
u in the numerator converges to zero faster than the polynomial in the denominator. For
, using L’Hospital’s rule leads to
These values coincide with those calculated in [
20] for a more general group of copulas. One can also notice that
In other words, if the parameter
is smaller than 1, any sign of tail dependence disappears when the dimension increases. If
, then
for every
which is no surprise, since, in that case,
is the comonotonicity copula
. This behavior is illustrated in
Figure 3 that details the influence of the parameter
on the speed of decrease of
when
d increases.
A Cuadras–Augé copula is an exchangeable copula, which is invariant with respect to the order of its arguments. Therefore, when calculating Li’s tail dependence parameters, only the cardinality of the index sets
and
plays a role. Subsequently,
and by using L’Hospital’s rule
If
, then
, as expected, and it does not depend on the conditioning sets
and
.
For Schmid’s and Schmidt’s lower tail dependence measure
, defined in (24), we first need to calculate the integral
. A straightforward calculation gives that
where
is the Beta function. We then get
which equals 1 when
and 0 when
. Schmid’s and Schmidt’s lower tail dependence measure thus equals Frahm’s lower extremal dependence coefficient
as well as Li’s lower tail dependence parameter
.
Determining Schmid’s and Schmidt’s upper tail dependence measure
in (25) is less straightforward. This dependence measure involves three integrals. Because its expression concerns the limit when
, it suffices to investigate the behavior of the numerator and the denominator of (25) for
p close to 0. From (27) it is easy to see that, for
p close to 0,
and, hence, the denominator of (25) behaves, for
p close to 0, as
For the integral
, note that, since
is an exchangeable copula, we can divide the integration domain
into
d parts depending on which argument from
is minimal. The integrals over each of the
d parts are equal. We get
where the approximation, valid for
p close to 0, is based on a careful evaluation of the integral. For brevity, we do not include the details here. Consequently the numerator of (25) behaves, for
p close to 0, as
Combining (37) and (38) reveals that
, for all
. Other calculations (omitted here for brevity) lead to
.
A Cuadras–Augé copula is also an extreme-value copula. This can be seen through the following calculation, where the notation
is used. One gets
and, thus,
is an extreme-value copula with Pickands dependence function
This allows for calculating the tail coefficient for extreme-value copulas,
, as
In case of the Cuadras–Augé copula, tail dependence measured by
does not depend on the dimension
d. For illustration, the values of
are included in
Figure 3. One can see that
and
behave very differently, both in terms of shapes and values.
Example 6. Clayton copula.
Let
be a
d-variate Clayton family copula defined as
for
. The Clayton copula is an Archimedean copula and the behavior of its generator is studied in Example 2.
For Frahm’s lower extremal dependence coefficient, either using (
12) or by factoring out as below, one obtains
whereas, for Frahm’s upper extremal dependence coefficient, using (13) with the derivative of the Clayton generator
, one finds
Analytical calculation of
is not possible; however, insight can be gained by plotting
as a function of the dimension
d. This is done in
Figure 4. From the plot it is evident that
decreases when the dimension increases. However, for larger parameter values, the decrease seems to be slow.
A Clayton copula is also an exchangeable copula and, thus, when calculating Li’s tail dependence parameters, only the cardinality of the index sets
and
comes into play. Then
If, as in Proposition 11, the cardinality of
is kept constant (equal to
) when the dimension increases, then
In fact, in this example, even a milder condition is sufficient for achieving (42). If
is linked to the dimension such that
, then (42) holds. However, for large values of the parameter
, the convergence in (42) might be very slow. By applying L’Hospital’s rule
times, one can also calculate
Spearman’s rho for the Clayton copula cannot be explicitly calculated and, thus, the values of and are unknown.
Example 7. Gumbel-Hougaard copula.
Let
be a
d-variate Gumbel–Hougaard copula, defined as
where
. The Gumbel-Hougaard copula is the only copula (family) that is both an extreme-value and an Archimedean copula as proved in [
21] (Sec. 2). The behavior of its Archimedean generator is studied in Example 3. Note that
corresponds to the independence copula
and the limiting case
corresponds to the comonotonicity copula
.
As expected (see (
10)), for an extreme-value copula which is not the comonotonicity copula, the Frahm’s lower extremal dependence coefficient is
since the polynomial in
u in the numerator converges to zero faster than the polynomial in the denominator. For the Frahm’s upper extremal dependence coefficient, by using (13) with the derivative of the Gumbel–Hougaard generator
, one obtains
Analytical calculation of
is not possible; however, insights can be gained by plotting
as a function of dimension
d. This is done in
Figure 5. It is evident that
decreases when the dimension increases; but, the decrease seems to be slow for larger parameter values. When comparing
Figure 4 and
Figure 5, one might come to a conclusion that
for the Clayton copula with parameter
is equal to
for the Gumbel–Hougaard copula with the same parameter
. Despite their similarity, that is not true, as can be easily checked by calculating both of the quantities for any pair
.
When calculating Li’s tail dependence parameters, one uses that the Gumbel–Hougaard copula is also an exchangeable copula and, thus, only the cardinality of the index sets
and
plays a role. Then
If
, then
, otherwise by using L’Hospital’s rule
This function of parameter
, dimension
d and cardinality
h is rather involved and it is depicted in
Figure 6 for different parameter choices and also two different selections of
h. In one of the cases,
and thus corresponds to
in Proposition 11. In the other case, the number of components on which we condition
is chosen to increase with
d, specifically
. For
(and thus the setting of Proposition 11), the tail coefficient slowly decreases with dimension, as expected. An interesting behavior is seen for
, where the tail coefficient seems to be, except for instability in low dimensions, constant, independently of the parameter
choice.
Spearman’s rho for a Gumbel–Hougaard copula cannot be calculated explicitly and thus the values of and are unknown.
Pickands dependence function
of a Gumbel–Hougaard copula is
and thus
Note that
. From our perspective, such a behavior is rather counter-intuitive and should be taken into account when using this tail coefficient.
An overview of the results obtained in the illustrative examples is given in
Table 2.
8. Real Data Application
In this section, we illustrate the practical use of the multivariate tail coefficients via a real data example. The data concern stock prices of companies that are constituents of the EURO STOXX 50 market index. EURO STOXX 50 index is based on the largest and the most liquid stocks in the eurozone. Daily adjusted prices of these stocks are publicly available on
https://finance.yahoo.com/ (downloaded 19 March 2020). The selected time period is 15 years, starting on 18 March 2005 and ending on 18 March 2020. Note that this period covers both the global financial crisis 2007–2008, as well as the sharp decline of the markets that was caused by COVID-19 coronavirus pandemic in early 2020. All the calculations are done in the statistical software R [
28]. The R codes for the data application, written by the authors, are available at
https://www.karlin.mff.cuni.cz/~omelka/codes.php.
The preprocessing of the data was done, as follows. The stocks are traded on different stock exchanges and thus might differ in trading days. The union of all trading days is used and missing data introduced by this method are filled in by linear interpolation. No data were missing on the first or the last day of the studied time range. Negative log-returns are calculated from the adjusted stock prices and ARMA(1,1)–GARCH(1,1) is fitted to each of the variables (stocks), similarly as for example in [
29]. We also refer therein for detailed model specification. Fitting ARMA(1,1)–GARCH(1,1) model to every stock does not necessarily provide the best achievable model, but residual checks show that the models are adequate. The standardized residuals obtained from these univariate models are used as the final dataset for calculating various tail coefficients. The total number of observations is
.
Table 3 summarizes the stocks used for the analysis.
It is of interest here to discuss tendency of extremely low returns happening simultaneously, which translates into calculating upper tail coefficients while working with negative log-returns. This allows us to use also the methods assuming that the data are coming from an extreme-value copula.
Six different settings are considered: stocks from Group 1 (G1), from Group 2 (G2), from Group 3 (G3), from G1 and G2, from G1 and G3, and finally stocks from G2 and G3. The dimension d is equal to 3 for the first three settings and equal to 6 for the last three settings.
Six different estimators are considered: , , , , and with two different selections of the conditioning sets and . In one case, and we condition on only one variable. The specific choice of that one variable does not impact the result, as follows from (19). The analysis with the conditioning on only one variable shows how the rest of the group is affected by the behavior of one stock. In the other case, we condition on all of the stocks, except for the one with largest market capitalization within the group. This analysis indicates how the largest player is affected by the behavior of the rest of the group.
The estimators that are functions of the amount of data points
k (recall from
Section 7.2 that a common choice is
, with
here) do not provide one specific estimate but rather a function of
k. A selection of in some sense the best possible
k requires further study. Intuitively, one should look at lowest
k for which the estimator is not too volatile. This idea was used in [
30] for estimating bivariate tail coefficients by finding a plateau in the considered estimator as a function of
k. The results of the analysis are summarized in
Figure 7 and
Figure 8 and
Table 4. Examining
Figure 7, it seems that
k around 100 would be a possible reasonable choice for the tail coefficients of Frahm, and Schmid and Schmidt, for these data. For Li’s tail dependence parameters, it appears from
Figure 8 that, when conditioning on more than one variable, a larger value for
k is needed, for example
.
For the tail dependence measurements for extreme-value copulas, we include the coefficients
and the original extremal coefficient
(see [
17]), where the latter can be estimated from the former, since
. Recall that the various tail coefficient estimators estimate different quantities and, therefore, their values should not be compared to each other. However, a few general conclusions can be made based on
Figure 7 and
Figure 8. Clearly, all the studied groups possess a certain amount of tail dependence. The combinations of groups also seem to be tail dependent, although the strength of dependence is smaller. Groups G2 and G3 seem to be slightly more tail dependent than G1, which suggests that sharing industry influences tail dependence more than sharing geographical location.
The estimator of Frahm’s extremal dependence coefficient in
Figure 7a,b is clearly the smallest of all the estimators, which follows its “strict” definition in (
7). The dots, representing the estimates under the assumption of underlying copula being an extreme-value copula, are greater than the fully non-parametric estimators. This indicates that assuming underlying extreme-value copula might not be appropriate.
The estimator of Schmid’s and Schmidt’s tail dependence measure in
Figure 7c,d is much smoother as a function of
k than the other estimators. However, it tends to move towards 0 or 1 for very low
k.
The estimator
in
Figure 8a suggests that, for all three groups, the probability of two stocks having an extremely low return given that the third stock has an extremely low return is approximately
. The estimator
in
Figure 8d on the other hand suggests that, in all three group combinations, the largest company is heavily affected if the remaining five stocks have extremely low returns. For group combinations G1 + G3 and G2 + G3, the estimated tail coefficient is, in fact, equal to 1.
The values of
and
are presented in
Table 4. One can notice that these measures also suggest that groups G2 and G3 are slightly more tail dependent than G1, or, in other words, they likely contain less independent components (see [
18]).