1

Beyond dimension two: A test for higher-order tail risk

Carsten Bormann
Karlsruhe Institute of Technology, Germany
carsten.bormann@kit.edu
Tel. +49 721 608-48113
Schlossbezirk 12, Geb. 20.12
D-76128 Karlsruhe

Melanie Schienle
Karlsruhe Institute of Technology, Germany

Julia Schaumburg
VU University Amsterdam and Tinbergen Institute, The Netherlands

2

Abstract
In practice, multivariate dependencies between extreme risks are often only assessed in a pairwise
way. We propose a test for detecting situations when such pairwise measures are inadequate andgive incomplete results. This occurs when a significant portion of the multivariate dependence
structure in the tails is of higher dimension than two. Our test statistic is based on adecomposition of the stable tail dependence function describing multivariate tail dependence.
The asymptotic properties of the test are provided and a bootstrap based finite sample versionof the test is proposed. A simulation study documents good size and power properties of the test
including settings with time-series components and factor models. In an application to stock
indices for non-crisis times, pairwise tail models seem appropriate for global markets while thetest finds them not admissible for the tightly interconnected European market. From 2007/08
on, however, higher order dependencies generally increase and require a multivariate tail modelin all cases.
Keywords: decomposition of multivariate tail dependence, multivariate extreme values,
stable tail dependence function, extreme dependence modeling
JEL classification: C01, C46, C58
Introduction
Studying extreme co-movements in multidimensional systems is a key concern in finance andinsurance. However, tail dependence structures of multivariate distributions are mostly treatedin bivariate setups, see for instance Poon et al. (2004) and Klugman and Parsa (1999), butalso Straetmans et al. (2008), Li (2013), Rodriguez (2007), among many others. Pairwise
simplification is not only standard when analyzing financial systems but is also widely used forstudying extreme environmental and weather risks (see de Haan and de Ronde (1998) and Ghosh(2010)). This is due to the fact that in practice, bivariate models are more easily tractable andcomputationally more appealing. But also from a theoretical point of view, statistical propertiesof a large group of estimators are only known up to dimension two (Coles et al. (1991), Joe et al.
(1991), de Haan et al. (2008), Guillotte et al. (2011)). Yet, for a variety of empirical settings,
there are periods in time during which a pairwise approach is too restrictive, as joint extremes
occur in cross–sections of dimension three or higher. In particular during the recent financial
crisis, markets became increasingly dependent. The financial contagion literature provides a lotof evidence that the major part of this rising interconnectedness was due to complex higher orderinterdependencies, which could not have been detected by standard pairwise tail dependence
measures (see, e.g., Longstaff (2010), Brunnermeier and Pedersen (2009)). In such situations,the most common bivariate measures for tail dependence, such as the tail dependence coefficient3(see Straetmans et al. (2008), Poon et al. (2004), Hartmann et al. (2004)), bivariate copulas (see,e.g. Li (2013), Rodriguez (2007), and references therein), or simple product moment correlation
coefficients and correlation matrices fail to explicitly account for a large amount of the complex
dependence structure among extreme risks in the system. This leads to severe underestimationsof the effects of extreme comovements. For a discussion of the limitations of common bivariate
measures of dependence, see also Embrechts (2009) and Mikosch (2006).
We propose a test that indicates whether pairwise modeling of multivariate tail dependenceof a d–dimensional random vector X = (X (1) , ..., X (d) )0 with d > 2 is adequate, or whether it
implies significantly different and thus incomplete tail dependence structures. The test is basedon the stable tail dependence function (STDF), which was first introduced in Huang (1992) (seealso de Haan and Ferreira (2006) and Einmahl et al. (2012)). The STDF maps the univariate
tails of a random vector to their joint limit distribution, and therefore completely describes
their extremal dependence structure. It is a general and flexible concept of tail dependence
and allows for straightforward non-parametric estimation, bearing a smaller risk of model
misspecification than alternative parametric approaches. Furthermore, its statistical properties
are well understood for X of dimension beyond two (Einmahl et al. (2012), Bücher et al. (2014)).
Moreover, its rather conservative definition of multivariate extreme events fits the needs of(financial) risk management (Segers (2012)).
The main idea of the test is to decompose the STDF for X into probabilities of univariate
extreme events, the STDFs of all possible bivariate pairs within X, and a remainder term
capturing extreme events in dimensions three to d. We refer to the latter as higher order taildependencies (HOTDs), and denote tail events as multivariate when they comprise three or more
extremes in the cross-section. If an estimate of the remainder term is not significantly different
from zero, we conclude that tail dependence in dimension d can be captured sufficiently wellby analyzing only bivariate tails. However, if we reject the null hypothesis that HOTDs haveno influence, ignoring high-dimensional joint extreme events leads to underestimation of theactual tail risk dependence, which is then driven by a substantial portion of joint extremes indimension three and higher. The asymptotic properties of the test statistic are derived anda bootstrap implementation scheme for finite samples is proposed. Simulation studies withstandard multivariate risk structures for the iid and ARMA–GARCH cases document good sizeand power properties of the test in finite samples. Moreover, our simulations highlight the needto filter the data for conditional heteroskedasticity before applying the test to financial timeseries.
Our empirical application deals with the influence of HOTDs in international stock markets.
Asset allocation and portfolio diversification, as well as systemic risk assessment require a most

4

accurate picture of tail dependencies between financial markets. Univariate tail losses withina portfolio can be diversified by holding tail independent assets. Bivariate tail dependence
eliminates such tail risk diversification opportunities between two assets, as large losses tend
to occur simultaneously. The same reasoning applies to higher–dimensional tail risk: Wheneverextreme losses of three or more assets coincide, multivariate tail risk cannot be diversified
anymore. Ang and Chen (2002), Poon et al. (2004), Chollete et al. (2011) and others estimate
bivariate tail measures for indices of international stock markets. The common conclusion is thatleft bivariate tails, i.e. extreme losses, are dependent, especially intra-continentally. Right tails,
however, tend to be independent. We test for HOTDs within two separate sets of stock market
indices. In a global portfolio including US, Asian-Pacific and European stock indices, we find noevidence for HOTDs in both left and right tails, until the rise of the financial crisis of 2007. Thisfinding suggests that global tail diversification possibilities are limited ever since, a finding thathas also been made by Christoffersen et al. (2012) using a dynamic copula approach. Testing
against HOTDs in a multi-country European portfolio, we find strong evidence for HOTDs
during the last decades, which can only partly be explained by serial correlation, time variation,
and a factor reflecting the development of global markets. Our results therefore contribute to
the empirical international finance literature in three points: First, we find that the extent of
intra-European tail dependence is more severe than discovered in former contributions. Second,higher-order tail effects in European markets are time-varying, and have increased during therecent financial crisis. Third, multivariate effects in extreme losses on the global level become
relevant in the course of the financial crisis, while extreme gains are largely not affected by
HOTDs. We conclude our empirical application by quantifying the share of HOTDs in taildependence. We find time periods in which up to 70% of all bivariate extreme events are in fact
multivariate. Also, in recent years, this share has doubled for losses and even tripled for gains
on the European portfolio.
The rest of the paper is organized as follows. Section 1.1 discusses necessary concepts
from multivariate extreme value theory. Section 1.2 introduces and formalizes test idea, test
asymptotics and finite sample implementation. Finite sample properties are studied in Section
2. Section 3 studies HOTDs between international stock indices. Section 4 concludes. The
Appendix contains supplementary and theoretical results.

5

1

Econometric methodology

1.1 Multivariate dependence in extreme tails
For our analysis of extreme risks, we use techniques from multivariate extreme value theory
which we introduce and motivate in the following. Denote by X := (X (1) , ..., X (d) )0 a d–dimensional random vector with continuous joint cumulative distribution function (CDF)
FX (x), x := (x(1) , ..., x(d) ). Its univariate marginal CDFs are denoted by Fj (x(j) ), j = 1, ..., d.
Suppose we observe a sample of n iid draws from the random vector X, collected in the
(1)

(d)

(j)

(j)

(j)

(n × d) sample matrix X = (Xn , ..., Xn ) with Xn = (X1 , ..., Xn )0 , j = 1, ..., d. We write
(j)

(j)

(j)

max(Xn ) = max(X1 , ..., Xn ) for the sample maximum of margin j. For each marginal, we
(j)

(j)

assume that there exist normalizing constants an ∈ R+ , bn ∈ R, j = 1, ..., d, and a limiting
distribution GX (x), such that
(1)

lim P

n→∞

(1)

max(Xn ) − bn
(1)

an

(d)

(1)

≤ x , ...,

(d)

!

(d)

max(Xn ) − bn

≤x

(d)

an

= GX (x),(1)

for all continuity points of GX (x). Then, GX (x) is a multivariate extreme value distribution,
and FX (x) is said to be in the domain of attraction of GX (x), which is denoted by FX ∈ D(GX ),see de Haan and Ferreira (2006) and Resnick (1987). Necessary and sufficient conditions for
FX ∈ D(GX ) can be found in de Haan and Resnick (1977), Beirlant et al. (2004, p.287), de Haan
and Ferreira (2006), and Resnick (1987). Throughout the paper, we assume that they are
fulfilled. In general, closed-form expressions for GX (x) do not exist. Equation (1) can be writtenas(1)(d) (d)lim FXn (a(1)+ b(1)+ b(d)n x
n , ..., an xn ) = GX (x),

n→∞

(2)

implying that the univariate marginals converge individually to one–dimensional extreme value
distributions Gj (x(j) ), which have the standard Fisher–Tippett form

(j)
(j)
(j) −1/γj
lim Fjn (a(j)+ b(j), j = 1, ..., d,n xn ) = Gj (x ) = exp −(1 + γj x )

n→∞

(3)

where γj denotes the tail index of margin j (Fréchet (1927), Fisher and Tippett (1928),
Gnedenko (1943)). An equivalent formulation of relation (2), with (3) holding true for all
margins, is given by the concept of the stable tail dependence function (STDF) of X, denotedby `X (x) or `(x) (Huang (1992), Einmahl et al. (2012)). Equivalent characterizations of GX (x),and thus `(x), can be obtained via the spectral measure and the exponent measure (de Haan
and Ferreira 2006, Chapter 6) but are less intuitive in interpretation and decomposition. The6

STDF `(x) : Rd 7→ R+ is defined asx(1)`(x) = − log GX

−γ1

γ1

−1

, ...,

x(d)

−γd

γd

−1

!
.

The STDF describes the complete dependence structure of the tails of the univariate marginals.
As one can express `(x) as−1`(x) = lim t P
t→0

d
[


{Fi−1 (1 − tx(i) ) ≤ X (i) } , t ∈ R+ ,

(4)

i=1

the stable tail dependence function (STDF) is an asymptotic measure which can be interpreted
as the scaled asymptotic probability that at least one element of X exceeds an extreme quantile,that is, X (i) exceeds Fi−1 (1 − tx(i) ), as t → 0. From this representation, a direct non-parametricestimate of the STDF can be derived. Also, `(x) can be decomposed into component STDFs ofdimensions lower than d.
There is a rich statistical literature on general properties of the STDF and its estimators (e.g.
Huang (1992), Dietrich et al. (2003), Einmahl et al. (2006), Drees et al. (2006), Einmahl et al.
(2012), Bücher et al. (2014)). Importantly, the STDF is a convex function and homogeneousPof degree one, i.e. `(λx) = λ`(x) for λ ∈ R. Moreover, `(x) ∈ [max(x), x0 1 = di=1 x(i) ] with
1 representing a d-vector of ones. The lower (upper) bound is attained if X is perfectly
tail dependent (independent), that is, extremes of univariate marginals always (never) occur
simultaneously (Beirlant et al. (2004), de Haan and Ferreira (2006)). Tail (in)dependence is often
also denoted as asymptotic (in)dependence. Numerical values of `(x) close to max(x) indicatethat tails of X are strongly interconnected. Values of `(x) close to x0 1 mark the opposite. In
practice, perfect tail dependence is rare.
It is important to note the connection, but also the difference, of the STDF to the so–called
tail copula (TC) which is a closely related metric for tail dependence. The TC of X is definedas−1

R(x) = lim t P
t→0

d
\


{Fi−1 (1 − tx(i) ) ≤ X (i) } .

i=1

O It only considers joint exceedances to characterize tail dependence, see Schmidt andStadtmüller (2006). Sibuya (1960), Joe (1997) and Coles et al. (1998) analyze bivariate taildependence by means of the tail dependence coefficient, which corresponds to the bivariate TC
at the point x = (1, 1). Roughly speaking, it describes the tendency of two random variables
to jointly exceed a high threshold. In two dimensions, there is a one-to-one mapping between
the TC and the STDF. Due to the lack of natural ordering in higher dimensions, however, the7definition of a multivariate extreme event depends on the research objective. There are several
reasons why we prefer the STDF over the TC for our purpose: Firstly, the TC captures only (themost extreme) parts of the multivariate tail dependence in dimensions d > 2, while the STDF
completely describes it (see Subsection 1.2 for the relationship between the two). Secondly, apractical issue for large d is that joint d–dimensional exceedances are rarely observed in finite
samples. Unless a sample contains an observation with all marginals being extreme, the TC
indicates tail independence. That is, the TC only considers the most extreme events when all
marginals are simultaneously extreme, and disregards more likely tail events. On the other hand,
the STDF incorporates events in which a single component of X becomes extreme, and hence
finite samples provide more relevant observations. Segers (2012) interpret `(x) as ”trouble in
the air”, whereas R(x) only considers events as extreme when ”the sky is falling”. The STDF
is therefore an important ingredient for a conservative risk monitoring approach, in the sensethat not only the "most extreme" extremes are considered.
1.2 A new test for higher-order tail dependence
We aim to detect the share which HOTDs contribute to overall tail dependence. Hence, we
decompose the STDF for dimension d into TCs for dimensions two to d. In dimension d = 2, fromequation (4) we have that `(x) is the limiting probability of a union of two events; since P(A ∪B) = P(A) + P(B) − P(A ∩ B) for events A and B. Therefore, we have `(x(1) , x(2) ) = x(1) +x(2) − R(x(1) , x(2) ). For similar decompositions in arbitrary dimension 2 < d < ∞, additional
notation is required. For I ⊂ {1, ..., d} define the subvectors X(I) := (X (i) )i∈I , x(I) := (x(i) )i∈I ,and according STDFs as `I (x(I) ). Then, in R2<d<∞ , using the inclusion–exclusion principle, we
have`(x) =dX
i=1

x(i) −X

X

Rij (x(i,j) ) +i<j≤dRhij (x(h,i,j) ) − ... + (−1)d+1 R(x),(5)

h<i<j≤d

|

{z

=:A

}

where A denotes the portion HOTDs contribute to ”global” tail dependence in X, that is,
the tail dependence of the entire random vector X. Provided that global tail dependence
is only caused by bivariate extreme events, i.e. by the first two terms of equation (5), A
equals zero. In this case, higher dimensional joint extremes are irrelevant. When substituting
Rij (x(i) , x(j) ) = x(i) + x(j) − `ij (x(i) , x(j) ), i < j ≤ d, equation (5) yields`(x) = (2 − d)

d
X
i=1

x(i) +

Xi<j≤d`ij (x(i,j) ) + A,

(6)

8

which decomposes global tail dependence into asymptotic probabilities for univariate extremes
and STDFs for any bivariate combination and HOTDs.
Using equation (6) we can test whether extreme events in dimensions larger than two havea statistically significant impact, that is, if two–dimensional tails explain tail dependence indimension d > 2 sufficiently well. Formally, if A = 0, we have

∆ := `(x) − (2 − d)dXXx(i) −

i=1`ij (x(i,j) ) = 0.

(7)

i<j≤d

In this case, bivariate extreme relations are sufficient for capturing the full global taildependence. Hence, the null hypothesis that the impact of higher–order tail dependencies isnegligible can be formulated as
H0 : ∆ = 0.

(8)

If ∆ substantially deviates from zero, the null is rejected. With x = 1, it is possible to showPthat ∆ ∈ [0, d−2i=1 i], d > 2.
The following proposition clarifies that testing for ∆ = 0 is not equivalent to testing whether
X is tail independent. Thus, multivariate distributions exist which are globally tail dependent
but have ∆ = 0. Hence their global tail dependence is exclusively caused by bivariate tails. A
test for tail independence is proposed in Draisma et al. (2004).
Proposition 1.1 a
If X is tail independent, that is if all bivariate tails of X are tail independent, then ∆ = 0. The
reverse does not hold.
This can, e.g., be easily shown for the family of distributions which we use in the simulation
setting in Section 2.
In order to apply the test, we have to estimate the STDF of X, `X (x), and the STDFs for(i)bivariate pairs. Let Xn:m denote the m–th largest order statistic of margin X (i) , and let 1(C)be the indicator function for event C. In equation (4), replacing the running variable t by k/n(i)and the extreme quantiles Fi−1 (1 − tx(i) ) by Xn:n+0.5−kx(i) we use the following non-parametricestimator for the STDF (see Huang (1992) and Einmahl et al. (2012))n
do
1 X n [ (j)k(j)ˆ`(x) ={Xi ≥ Xn:n+0.5−kx(j) } , n → ∞, k → ∞, → 0,
1
k i=1
n
j=1

x = (x(1) , ..., x(d) ). Under some technical conditions, the empirical process√(9)ˆ − `(x))
k(`(x)

converges to a sum of a centered Gaussian field and univariate centered Gaussian processes with

9

given covariance structure (Einmahl et al. (2012), Bücher et al. (2014)). If X is asymptoticallyˆ
independent, `(x)is still asymptotically normal but with degenerate variance (Hüsler and Liˆ2009). Note, `(x)is invariant against monotone transformations. For simplicity, we fix x = 1,
which is standard in the applied extreme value literature, see e.g. Hartmann et al. (2004). In(i)this case, for each marginal, the threshold equals Xn:n+1/2−k . The asymptotic distribution ofˆ
`(1)
simplifies to√ dˆ − `(1) →k `(1)N (0, σ`2ˆ ),where closed form expressions of σ`2ˆ can be reconstructed from theorem 4.6 in Einmahl et al.
ˆ(2012). Plugging `(1)
into ∆ yields the empirical test statistic
ˆ − 2d + d2 −b := `(1)∆X`ˆij (1) .

(10)

i<j≤d

These considerations lead us to the asymptotic distribution of the test statistic, which is
given next.
Proposition 1.2 Assume FX ∈ D(GX ). Furthermore, let the following assumptions hold:(A1) There exists a constant β > 0 such that for t ↓ 0 it holds that t−1 P(

Sd

−1
(i)
i=1 Fi (1 − tx ) ≤X (i) ) = `(x) + O(tβ ) uniformly on the unit simplex in Rd .
(A2) The threshold parameter k → ∞ for n → ∞ with k = o(n2β/(1+2β) ) with β from (A1).
Then,

√
d
2
b − ∆) →k(∆
N (0, σ∆
b ),

(11)

2
where σ∆b is the sum of all entries of the covariance matrix of




ˆ
ˆ
`(1), (`ij (1, 1))i<j≤d .

SFor the proof, please see the Appendix. Assumption (A1) imposes that t−1 P( di=1 Fi−1 (1 −tx(i) ) ≤ X (i) ) exists for t small and converges to the STDF at a certain speed. This second-ordercondition refines the base assumption of max-domain attraction of FX . The second assumption
restricts the speed with which k grows to infinity, and in combination with (A1) guarantees
that an asymptotic bias term for the left hand side of equation (11) vanishes (see Resnickand de Haan (1996), Einmahl et al. (2008) for details). According to Bücher et al. (2014) asmoothness assumption for the STDF is not required. In particular, we do not need to imposethat partial derivatives of ` exist for the asymptotic result to hold. Such an assumption might
be too rigid, as it would, e.g., exclude factor models, which are practically important in financial

10

applications. For obtaining Proposition 1.2, we therefore rely on asymptotic results by Bücher
et al. (2014), which do not require the existence of partial derivatives of the STDF, but whichˆ
are also no longer uniform and yield convergence of `(x)in a wider sense.1
In both the simulation study and the empirical application in Sections 2 and 3, we restrict
the test to dimension 7. However, if X exhibits tail dependence in dimension d larger than 7,
it necessarily exhibits tail dependence in dimensions 3 ≤ g < d. Thus, the asymptotic power of
the test also increases with larger dimensions. Subsection 2.2 further discusses these details, inthe context of the results on the empirical power in the simulation settings. Also, the test canbe readily adapted to detect whether joint extremes of dimension 3 < g ≤ d are significant.
1.3 Finite sample version of the test
Although it is possible to derive the explicit form and calculate empirical versions of theasymptotic variance of the test statistic, a bootstrap version is practically superior. The reason2is that bootstrapping σ∆
b works under milder conditions, in particular if X exhibits asymptotic
2
dependence (Bücher and Dette 2013). In contrast, direct estimation of σ∆b may require theestimation of partial derivatives of the STDF and of covariances between the different STDFs.
In principle, a weighted least squares based estimator for such partial derivatives of the STDF
exists, but its statistical properties have only been established for dimension d = 2 so far (seePeng and Qi 2006). Furthermore, smoothness assumptions for the STDF might not be met. In
such cases, estimating the partial derivatives is not admissible (Bücher and Dette 2013).
As our goal is to bootstrap extremal observations, we do not resample from the full sample,
but only from a subsample (Politis and Romano (1994)). Otherwise, an asymptotically vanishingb inherited from `ˆX (see Huang (1992)), might distort the bootstrap distribution.
bias term of ∆,Peng (2010) propose a similar approach and successfully employ a subsample size of n0.95 . Qi
(2008), El-Nouty and Guillou (2000), Danielsson et al. (2001), Geluk and de Haan (2002)
generally document the benefits of subsampling for pointwise extreme value statistics. Web with theconstruct rejection regions for the test from the asymptotic normal distribution of ∆resampled form of the variance. We explicitly mark if an estimator θ̂ depends on the thresholdparameter k by writing θ̂(k). In summary, we proceed along the following six steps for obtaining
a test decision:b from the sample X.
(1) Choose the threshold parameter, denoted by k ∗ , for ∆1√ In particular, Einmahl et al. (2012) show weak convergence of the empirical processˆ − `(x)) for bounded functions in the sup-norm, while Bücher et al. (2014) showk(`(x)
convergence for locally bounded functions in the so-called hypi-semimetric.

11ˆ ∗ ), and any `ˆi (k ∗ ), i ∈ I (d) , to determine the full sample test statistic ∆(kb ∗)(2) Calculate `(k(2)
from X.
(3) Draw at least B = 500 bootstrap samples with replacement from X with sample sizen∗ = n0.95 and denote the resulting bootstrap samples by X∗1 , ..., X∗B .
b ∗ ) from the bootstrap samples X∗1 , ..., X∗ , yielding B(4) For j = 1, ..., B, estimate ∆(k
B
∗
∗
b
b
bootstrapped estimates ∆(k )1 , ..., ∆(k )B .
2
(5) Estimate σ∆b from the bootstrapped estimates in the previous step by its empirical analogue.

b ∗ ) + z α σ̂ b ∗ , where z α denotes the(6) On a 1 − α confidence level reject H0 : ∆ = 0 if 0 < ∆(k∆(k )α quantile of the standard normal distribution.2
A theoretically optimal, data driven choice of the threshold parameter k should balance thebias–variance trade–off that is inherent in the estimation of `(x). Finding such a solution andderiving its optimality properties is non-standard even in the univariate case and is thus beyond
the scope of this paper. In our simulations we choose k randomly from an interval in order tominimize possible distortions from a poorly chosen k. In the application, we estimate ∆ overa grid of different values for k and calculate the median over this set of estimates.3 Further
details can be found in the respective sections. For alternative, purely data-driven proceduresfor determining k in a univariate setup, we refer to Frahm et al. (2005) and Schmidt andStadtmüller (2006).
For time series data, issues of short-range serial dependence can be addressed by
implementing a blocked version of the bootstrap providing appropriate up to second momentadjustments, see, e.g., Straetmans et al. (2008) with an asymptotically optimal choice of block
length of order n1/3 according to Hall et al. (1995). Instead, however, we use appropriate
GARCH-type filtered observations before applying the test. With this we also control for andamend higher order moment effects and volatility clustering of heteroskedastic financial data
(McNeil and Frey (2000), Poon et al. (2004)). See Section 3.3. for details.
2

b is theoretically not justified under tail independence,
Note, a normal approximation for ∆
Pd
(i)
2
ˆ
b
i.e. `(x) = i=1 x ; then, it holds σ`2ˆ = σ∆b = 0 and the distributions of both ` and ∆ aredegenerate, while the theoretical ∆ is zero, i.e. the null is true. However, in such situations, thetest typically indicates the correct decision not to reject the null.
3

Specifically, for a sample size of 750, k ∈ {8, 9, ..., 48} in dimension three and k ∈ {8, 9, ..., 30}
in dimension 7.

12

2

Simulation study

2.1 Size and power
In this subsection, we evaluate the empirical size and power of the test in finite samples in an
iid setting. Results for time series data are presented in Subsection 3.3. We simulate from two
types of distribution families with various subspecifications, for which we know whether the nullof no significant HOTDs is true. In particular, we focus on the class of meta t–distributions andmax factor models, which are both commonly used in financial risk management (McNeil et al.
(2006), Fama and French (1992)). The meta t–distribution is a generalization of the multivariatet–distribution and the t–copula, and max factor models have the same tail dependence structure
as factor models (Einmahl et al. (2012)). We employ the finite sample version of the test
introduced in Subsection 1.3. All simulations are repeated S = 500 times.
Model dimensions are d ∈ {3, 5, 7}. For a power analysis, considering larger dimensions
is often not necessary, as higher order tail dependencies of moderate order are sufficient for
concluding that HOTDs are significant.
t(x) denote the t–copula with ν degrees of freedom, and dispersion matrix P .
Let Cν,P

Following Demarta and McNeil (2005),

t
Cν,P
(u) =

Z t−1
(d) )
ν (u

Z t−1
(1) )
ν (u
···
−∞

−∞

Γ((ν + d)/2)p(1 + ν −1 (x0 P −1 x))−(ν+d)/2 dx,dΓ(ν/2) (πν) |P |

(12)

(i)
where t−1
ν (u ) denotes the quantile transform of a t-distribution with ν degrees of freedom

for margin i, and Γ(·) is the gamma function. According to Hua and Joe (2011), the t-copula
is of second-order regular variation and thus fulfills the assumptions of Proposition 1.2. In
contrast to a classical t–copula, meta t–distributions allow the degrees of freedom of marginals(i)νm to differ from the degrees of freedom of the copula, denoted by νC . For the simulation,(i)

we choose νC ∈ {5, 10, 15, 20}, νm := νm = 5, i = 1, ..., d, and P = (0.5)i6=j , Pii = 1. Thus, weconsider equicorrelated t–distributions with common degrees of freedom νm that are linked by
the t–copula with νC degrees of freedom. Exploiting results from Demarta and McNeil (2005)
and Nikoloulopoulos et al. (2009, theorem 2.3), it is possible to show, that for the classicalmultivariate t–distribution the theoretical values of our test statistic ∆ are larger than zeroas the t-copula is capable of producing joint extremes in dimension d > 2; ∆ increases if thedegrees of freedom of the copula decreases, and/or if pairwise correlation increases. It equals
zero if the correlation parameter equals −1. A meta t–distribution comprises the widely used
multivariate t–distribution whenever νC = νm .4
4Theoretically, the dependence structure is only governed by the parametrization of the copulaand not by distributional properties of the univariate tails, i.e. νm . In additional simulations

13

In finance, often factor models are applied, in which asset returns X (j) depend on common
factors Z (i) in a linear fashion. Max factor models assume X (j) can be modeled as the maximumof the factors times a parameter amj , the so called factor loadings. Both models have thesame tail dependence structure (Einmahl et al. (2012)). Let Z := (Z (1) , ..., Z (r) )0 be a random
vector of independent Fréchet random variables (ν =1). A d-dimensional max factor model for
X = (X (1) , ..., X (d) )0 is then defined by
X (j) := max(a1j Z (1) , ..., arj Z (r) ), j = 1, ..., d,
with

Pd

A
j=1 amj = 1, amj ≥ 0. The loading matrix Bd := (amj ) governs the dependence between

the tails of X. Employed calibrations of BdA can be found in the Appendix. In the notation of theloading matrix the subscript denotes the dimension d of X and the superscript denotes whether
the model fulfills the null (B 0 ) or the specific kind of alternative (B A ). The null is fulfilled if
at most two entries within a row of the loading matrix are non-zero as then tail dependence isonly caused by pairs. For example, given the parametrization
B30 =



0.5 0.5 0
0.5 0 0.5
0 0.5 0.5,

we have that the STDFs for bivariate pairs are `12 (1, 1) = `13 (1, 1) = `23 (1, 1) = 1.5, whileP
P
R123 (1, 1, 1) = 0. Einmahl et al. (2012) show that `(x) = ri=1 maxj=1,...,d (aij /( ri=1 aij ))x(j) ,and thus `123 (1, 1, 1) = 1.5 and ∆ = 0. If more than two elements within a row are non-zero,
there exist common factors that induce three or more components of X to become simultaneously
extreme. Thus, tail dependence is also caused by higher–dimensional joint extremes, and thenull would be false. This is the case for B3A1 , B5A2 , B5A2 , B7A1 , B7A2 , B7A3 . Specifically, the number of
non-zero entries per row describes the dimension in which joint extremes occur. Model notationis chosen such that with increasing index of A the order of tail events increases, i.e. B5A1 allows
for joint extremes of X (1) , X (2) and X (3) (first row) while in case of B5A2 also four-dimensional
joint extremes of X (1) , X (2) , X (3) and X (4) can occur (first row).
In extreme value statistics, simulation results are usually sensitive to the choice of theb whereas a small k
threshold parameter k. Large values of k cause a systematic bias of ∆,induces a large variance. We use a data-driven approach to the threshold choice in our simulation
study. Within a reasonable interval, k is chosen randomly within each simulation replication.
This interval is defined as [0.01n, cn1/2 ], c ∈ [1, 2]. By several simulation runs, we found thethat are not reported here, we found finite sample properties of the test are robust againstchanging the marginal degrees of freedom.

14best choices for c concerning test size are 1.75 in d = 3, 1.4 in d = 5, and 1.1 in d = 7.5
For comparability of results across increasing dimensions d, we let c decrease with d. With
increasing dimensions, the range of `X and the number of possible univariate extremes increase.
(i)

To achieve comparability across dimensions, higher cut-off values X[n:n+0.5−k] are chosen forhigher dimensions. Generally, in our simulation experiments, we find that the power of the test
is fairly robust against changes in c.
For the simulation in each specification, we employ five sample sizes which are standard
for analyzing daily financial data (n1 = 200, n2 = 500, n3 = 1000, n4 = 1500, n5 = 2000). Table(1) contains the empirical rejection rates of the test in each of the model classes at a nominal
significance level of 5%. For max factor models, we find that the empirical power of the test is
generally high in all considered dimensions. For models with only a slight impact of HOTDs,
however, the test requires sample sizes larger than 1000 in d = 5 in order to yield satisfactory
power, which appears adequate given the difficulty of the problem in small samples. But
empirical power quickly converges to one for larger sample sizes. And empirical sizes appear close
to the nominal level and plateaus around 5% for n sufficiently large. Depending on the exact
model specification, this can occur already for the smallest sample size of 200. While empirical
power is robust against the choice of k, we found that empirical sizes vary substantially whenaltering the domain of k. Generally, the test rejects too often if k tends to be small, thus
empirical sizes are systematically smaller than nominal levels. In financial risk management,
however, one would prefer a test with a larger false positive rate over a test that tends to falsely
overlook prevalent HOTDs. Still, as we model k as a uniform random variable defined over aninterval of reasonable possible values, reported sizes are more robust with respect to k than if
k was a fixed value.
For the meta t-distribution, increasing dimensions and decreasing degrees of freedom of the
copula imply high empirical rejection rates. This is to be expected given the above discussionof the properties of the meta t–distribution.
For all specifications, empirical power monotonously converges to one as n increases. For
perfectly tail dependent DGPs (B3A1 ), and meta t–distributions with small νC , empirical poweris always very high, irrespective of the dimension. Conditional on the choice of k, empiricalsizes are also close to α for the DGPs characterized by B30 , B50 , and B70 , again irrespective of thedimension. Hence, up to dimension 7, the usual curse of dimensionality often encountered whenemploying non-parametric methods appears not to play a role for our test. For small sample
sizes, empirical size is slightly larger than the nominal size α. Furthermore, if ∆ is close to zero(e.g. for a meta t–distribution with νC = 20), larger sample sizes such as n3 = 1000 are required
5

In dimensions d = 4, and d = 6 we found c = 1.5, and c = 1.2, respectively, to perform best.

15

Table 1: Empirical rejection rates: Max factor models and iid t–copula (df = νC , ρ = 0.5) with
t– distributed marginals (df = 5).
200

500

d=3
1000 1500 2000

200

500

d=5
1000 1500 2000

200

500

d=7
1000 1500 2000

35.2
29.8
25.6
24.0

51.2
39.0
32.8
30.0

66.2
45.4
42.0
41.6

74.2
55.4
45.8
40.0

78.6
63.2
57.8
51.6

60.2
54.8
54.2
60.4

79.4
68.0
57.2
62.2

91.6
78.2
73.6
73.2

96.2
88.2
83.0
81.2

99.6
93.4
91.8
87.0

70.4
59.2
58.4
56.6

82.2
77.0
72.2
69.0

97.0
91.0
85.0
86.0

100
97.6
94.0
92.0

100
99.8
98.8
96.6

B30
B3A1

5.2
100

4.2
100

4.2
100

5.0
100

4.8
100

–
–

–
–

–
–

–
–

–
–

–
–

–
–

–
–

–
–

–
–

B50
B5A1
B5A2
B5A3

–
–
–
–

–
–
–
–

–
–
–
–

–
–
–
–

–
–
–
–

7.2
20.4
59.2
100

6.8
34.4
76.8
100

7.6
48.2
94.4
100

5.6
60.0
97.4
100

5.2
70.2
99.4
100

–
–
–
–

–
–
–
–

–
–
–
–

–
–
–
–

–
–
–
–

B70
B7A1
B7A2
B7A3

–
–
–
–

–
–
–
–

–
–
–
–

–
–
–
–

–
–
–
–

–
–
–
–

–
–
–
–

–
–
–
–

–
–
–
–

–
–
–
–

4.0
49.2
59.6
68.8

4.4
73.8
80.6
89.4

2.4
95.4
98.6
99.0

5.2
100
100
99.8

5.8
100
100
100

t-distr.
νC
5
10
15
20
max factor

for the test to accurately identify the presence of HOTDs.
2.2 Local power analysis
In this subsection, we study the performance of the test under a series of local deviations fromthe null hypothesis. In contrast to the fixed alternatives of the subsection before, alternatives
here are very close to the null and their distance to the null can shrink with increasing sample
size, revealing the power optimality properties of the test. Thus, we evaluate the ability of thetest to detect a violation of the null if the nature of the underlying distribution of X is suchthat fewer and fewer joint extremes in dimension ≥ 3 occur in finite samples. Following Bergand Quessy (2009) and Kojadinovic and Yan (2010), such distributions are generated by mixing
distributions that violate the null, denoted by FX,H1 , with distributions that comply with thenull, denoted by FX,H0 . We define the mixture distribution by
FX,λ(n) (x) := (1 − λ(n))FX,H0 (x) + λ(n)FX,H1 (x),(13)

where λ(n) decreases to zero for increasing sample size n and FX,H0 (x) satisfying ∆ = 0,
FX,H1 (x) satisfying ∆ > 0, and FX,H0 (x) ≤ FX,H1 (x), ∀x, ensuring realizations from FX,H1 enter
the extreme part of the sample. Denote the test statistic resulting from the mixture distribution

16

p
FX,λ(n) (x) by ∆λ(n) . For λ(n) = O(( k(n))−1 ), we can show that, asymptotically,√db λ(n) − ∆λ(n) ) → N (0, σ 2b
k(∆
∆

),
λ(n)

where the asymptotic variance can again be obtained analytically from theorem 4.3 in Einmahl
et al. (2012). Thus, the test has power against any local alternatives if and only if thesepalternatives are at least of order ( k(n))−1 apart from the null.
In the following simulations, we illustrate this result. Hence, we are interested in rejection
rates of ∆ = 0 from mixture distributions defined in equation (13) for λ(n) := λk(n)−1/2 , with
0 < λ ≤ k(n)−1/2 . We determine k as in the simulations before. In order to calculate local powers
pn , we generate S = 1000 samples from a DGP of mixture distribution type, with fixed samplePb λ(n) > z α σ̂ bsize and increasing λ. Local power is estimated by p̂n = 1/S S 1{∆
k −1/2 } fori=1∆,λ(n)
2
every λ(n). The asymptotic variance σ̂∆,λ(n)is estimated by the bootstrap procedure presented inbSection (1.3). Berg and Quessy (2009), Kojadinovic and Yan (2010) carry out similar analyses
for goodness–of–fit tests of parametric (extreme value) copulas. For the sake of brevity, weconcentrate on dimensions d ∈ {3, 5}, sample size n = 2000, and we let λ increase. For d = 3,
FX,λ(n) (x) = (1 − λ(n))FY (y) + λ(n)FW (w),(14)

where FY (y) and FW (w) are the distribution functions of the max factor model B30 and B3A1 ,respectively. For d = 5, we mix the distribution function of B50 and B5A3 := ( 1/5 1/5 1/5 1/5 1/5 ).
If, for example, λ(n) = 0.1, then 10% of the extreme part of the sample should be generated by
the FW which violates the null. Figure 1 shows estimated local powers with α = 0.05. The test
successfully detects minor violations from the null. Even for small λ, when the impact of theperturbating DGP is small, rejection rates quickly converge to one. Increasing the dimension daccelerates the convergence speed of empirical power.
2.3 Size and power for serially dependent and conditionally heteroskedastic components
While Proposition 1.2 assumes iid data, financial time series, in particular asset returns, featuresmall autocorrelations and time–varying conditional volatility, and thus cannot be considerediid. In order to apply our test to detect the cross-sectional tail dependence structure of financial
time series, the data have to be pre-filtered. We use autoregressive moving average (ARMA)
models for the mean, and the class of generalized autoregressive conditional heteroskedasticity(GARCH) models for the variance, (Bollerslev (1986)). After filtering, we expect that theresulting standardized residuals are largely free of serial dependence in first and second moments,
and are thus nearly iid. We therefore apply the test to these pre-filter residuals instead of the raw

17

Figure 1: Empirical test power for the mixture distributions defined in equation (14) with sample
size n = 2000 at level 5%.
(b) d = 5
1.00
0.75
0.50
0.25
0.00

0.00

0.25

0.50

0.75

1.00

(a) d = 3

0.01

0.25
λ(n)

0.5

0.01

0.25
λ(n)

0.5

observations. In the applied extreme value literature, this approach is common when dealing
with time-series effects, see e.g. McNeil and Frey (2000) in a univariate setting for extreme
quantile estimation, and Poon et al. (2004) for estimating bivariate tail dependencies between
financial time series.
While it is intuitively clear that√n-consistent parametric pre-filtering should not impact

the consistency of the slower converging non–parametric estimator of the STDF, there are
no formal theoretical results on the asymptotic properties of such dependence estimates forpre-estimated residuals available yet. In comparison to semi-parametric and non-parametric
distribution copula estimation (see e.g. Chen and Fan (2006), Rémillard (2010), Oh and Patton(2013)) such results for non-parametric tail dependence estimation would require completely
different empirical process techniques for respective rank statistics which do not exist andare extremely challenging to develop. In what follows, we therefore focus on the finite sample
performance of the test in such settings. In particular, we explore if and how empirical size andpower of the DGPs from Section 2 change when introducing autocorrelation and time–varying
conditional volatility.

18

We follow Oh and Patton (2013) and generate random draws from the following AR(1)–GARCH(1,1) processes, which are linked by the error term copula Cη :
(i)

(i)

(i) (i)(i)(i) (i)(i) (i)yt = µt + σt ηt = ϕ0 + ϕ1 yt−1 + σt ηt ,

2
2,(i)
(i)
(i)
2,(i)
(i)
(i)
σt = ω + α
yt−1 − µt−1 + β (i) σt−1
η := (η (1) , ..., η (d) ) ∼ iid Fη (x(1) , ..., x(d) ) = Cη (Fη,(1) (η (1) ), ..., Fη,(d) (η (d) )),
(i)

(15)

(i)

t = 1, ..., T . θ(i) = (ϕ0 = 0.01, ϕ1 = 0.05, ω (i) = 0.05, α(i) = 0.1, β (i) = 0.85)0 denotes the vectorof AR–GARCH parameters for marginal i, Fη is the continuous joint distribution function of
the vector of error terms η = (η (1) , ..., η (d) ), and Fη,(i) (η (i) ) are the marginal distributions of theerror terms linked by error term copula Cη . Hence the dependence structure of η is the “true” butunobserved dependence we are interested in, and from which the observed dependence structure(i)

between the realizations yt might differ due to autocorrelation and GARCH effects. See Oh
and Patton (2013) for details on such DGPs.
(1)

(d)

We test for HOTDs in the observed, unfiltered realizations (yt , ..., yt )Tt=1 , and in correctly∗,(1)standardized residuals (η̂t

∗,(i)
(i)
(i)
(i)
∗,(d) T
)t=1 , η̂t := (yt − µ̂t )/σ̂t . To evaluate the size of the, ..., η̂ttest, we choose the max factor model of type B30 , B50 , and B70 as models for the error term copulaCη . Thus, the test is again applied in dimensions 3, 5, and 7. In contrast to the iid setting, we do
not employ a Fréchet(1) distribution, which would produce very extreme observations such thatnumerical GARCH–estimation may fail to converge. As marginal error distributions Fη,(i) (η (i) )we choose t–distributions with degrees of freedom νm ∈ {5, 10, 15, 20} for size analysis. For
power analysis, Cη is the t–copula with degree of freedom νC ∈ {5, 10, 15, 20}, and fixed marginal
degrees of freedom νm = 5, i.e. η follows a meta t–distribution.6 Thereby we can observe how
quickly the test reacts to a steadily diminishing degree of HOTDs.
Simulations are repeated S = 500 times with sample sizes n2 = 500, n3 = 1000, n4 =
1500, n5 = 2000. We do not include n1 = 200 in this section as GARCH estimates for a sample
size of 200 may be unreliable. The parameter vector θ = (θ(1) , ..., θ(d) )0 is estimated by maximum
likelihood, assuming marginal t–distributions with estimated degrees of freedom. Table (2)reports empirical rejection probabilities for the factor copula with ∆ = 0; Table (3) reports
empirical rejection probabilities in case of the t-copula as error term copula for filtered andunfiltered data, respectively. We find that disregarding serial correlation and time–varying
volatility worsens size and power properties, and a correct filter leads to similar results as
6

As in the iid case, empirical power is robust against varying the marginal degrees of freedom.
Yet, we report empirical sizes for different νm in order to have more data points for a moreaccurate comparison between test performances for unfiltered and filtered time series.

19

Table 2: Empirical rejection rates under H0 : Max factor copula as error term copula, t(df = νm )–distributed errors, and GARCH(1,1) volatility model.
500

1000

d=3
1500

2000

500

1000

d=5
1500

2000

500

1000

d=7
1500

2000

4.6
3.4
5.2
3.8

3.0
2.0
5.0
2.6

3.8
2.8
5.4
5.4

5.2
3.8
4.4
4.4

4.4
6.0
4.4
5.6

6.0
4.0
3.6
5.2

6.4
5.6
4.8
5.4

6.2
5.2
4.2
4.4

2.8
2.8
3.0
3.6

2.4
3.6
2.0
1.8

4.2
3.2
3.4
3.6

3.4
2.4
5.0
4.8

2.6
2.2
4.6
2.6

4.4
4.2
4.6
5.6

4.6
4.2
5.8
4.4

4.2
4.8
5.4
6.4

5.4
4.4
4.2
6.0

4.6
5.2
4.8
4.2

2.8
4.6
5.8
7.0

1.4
1.6
3.4
3.0

3.4
3.8
2.0
2.0

1.8
2.8
2.8
2.2

1.6
2.0
1.8
2.8

filtered
νm
5
10
15
20

unfiltered
νm
5
10
15
20

3.4
4.6
4.6
5.2

in the iid case. Empirical rejection rates for the max factor copula indicate that the test isslightly undersized. Yet empirical sizes are still satisfactorily close to 5%.
The effect of serial correlation and GARCH effects becomes clear when comparing thenumber of test rejections of the binomial test H0 : pi = 0.05, where pi denotes a rejection
probability (test level 5%). That is, for each setting we compare all 48 empirical rejection
rates of filtered and unfiltered data with the nominal size of 5%. The correctly specified AR(1)GARCH(1,1) filter leads to 18.2% of all cases in which the empirical rejection probability
significantly differs from the nominal size. Not filtering the data amounts to 31.3% significant
deviations. With a binomial test one can also compare empirical powers of the iid and thenon-iid settings. In 95.8% of comparisons, applying the test to the residuals of the correctly
specified GARCH process produces significantly higher power than testing in the unfilteredreturns. Hence, disregarding the time series properties of the data worsens size and power
results.
Finally, we compare the power results obtained when simulating from the iid meta t–
distribution (Table (1)) with those corresponding to correctly filtered, and unfiltered AR(1)GARCH(1,1) processes connected via the meta t–distribution (Table (3)). In case of the correctfilter, empirical power never differs significantly from the iid case (test level 5%). In the unfilteredseries, however, empirical power is significantly lower in 91.6% of all cases (test level 5%). Hence,
disregarding time–varying volatility amounts to lower power and lower test size, and testing incorrectly filtered series produces nearly identical results as for iid data.

20

Table 3: Empirical rejection rates under H1 : Correctly filtered and unfiltered GARCH processes
with t–copula (df = νC , ρ = 0.5) as error copula, t(νm = 5)–distributed errors.
500

1000

d=3
1500

2000

500

1000

d=5
1500

2000

500

1000

d=7
1500

2000

45.0
49.4
32.2
26.0

63.2
62.2
42.0
39.6

71.8
74.2
45.4
42.2

79.4
83.0
52.8
49.6

77.8
67.0
60.4
56.4

90.8
81.8
74.6
70.6

96.8
86.4
83.0
79.4

99.2
92.4
90.6
86.8

83.6
72.4
70.8
66.6

97.2
91.2
86.0
82.2

99.4
97.4
93.8
93.0

99.8
99.8
100
97.0

27.2
19.2
15.0
12.4

30.2
21.0
14.2
18.0

31.6
23.4
15.4
13.2

51.2
41.6
37.8
36.8

53.8
42.0
39.0
35.2

60.0
42.4
38.4
35.2

61.6
43.4
38.8
38.0

61.8
54.0
45.6
43.0

68.2
55.8
50.8
48.2

76.0
61.4
55.2
51.2

79.4
60.4
56.6
53.2

filtered
νC
5
10
15
20

unfiltered
νC
5
10
15
20

3

24.6
17.0
17.4
16.4

Higher order tail dependencies in global and European stock markets

3.1 Data description
In our empirical application, we study extreme gains and losses of two different sets of stock
indices. First, we test for HOTDs in left and right tails of the weekly stock return distributions
on a global level, while in a second study we focus exclusively on daily European stock returns.
The "global portfolio" consists of three stock indices of the USA, Europe and the Asian Pacificregion, namely the MSCI USA, MSCI Pacific, and MSCI Europe.7 The "European portfolio"
consists of seven individual European MSCI indices, including the largest European economies(United Kingdom, Germany, France, Italy, Spain), as well as smaller economies that played arole during the recent European sovereign debt crisis (Greece, Portugal).8 The two portfolios
are analyzed separately.
The sample period of the global portfolio ranges from 01/30/1970 to 10/29/2014. To
overcome problems arising from different time zones, we use weekly returns. As observationsof MSCI Pacific are only available on a monthly frequency until 12/30/1983, a weekly proxy
for MSCI Pacific during that time period is created by averaging over weekly observations of
the MSCI Japan and MSCI Australia, with weights equal to 2/3 and 1/3, respectively. Thisweighting scheme resembles the current composition of the MSCI Pacific.9 After deleting weeks
with zero returns, the sample features 2335 observations for each index. The sample period of
the European portfolio ranges from 01/04/1988 to 10/29/2014. In this second portfolio time
7

Data are available on Datastream with mnemonics MSUSAML, MSPACF$ and MSEROP$.

8

Data are available on Datastream with mnemonics MSUTDKL, MSGERML, MSFRNCL,MSITALL, MSGREEL, MSSPANL and MSPORDL.
9

Data are available on Datastream with mnemonics MSJPANL and MSAUSTL.

21

zone effects do not matter, so we can use daily returns. After discarding days with zero returns,
the sample has 6889 observations for each index.
Both samples are tested against HOTDs with rolling windows containing n = 750 observations, corresponding to roughly 15 years in the global portfolio, and roughly three and a halfyears in the European portfolio. Simulation studies document appropriate test performance forsuch a sample size, and we aim to keep the window length as short as reasonably possible.
The test is applied bi–weekly for the global portfolio and every fifth day for the European
portfolio. We test against HOTDs in raw observations, and in standardized ARMA–GARCH
residuals, along the lines of model (15) in order to eliminate the effects of serial correlation andtime–varying volatility. Returns are thus modeled by
(i)

(i)

(i) (i)(i)(i)(i) iidyt = µt + σt ηt , i = 1, ..., 7, ηt ∼ t(νm ).
∗,(i)Standardized residuals η̂t(i)= (yt − µ̂t )/σ̂t

are re–estimated in each window to address

potential parameter changes. In every window, each time series is fitted to an ARMA(p ≤2, q ≤ 2) model with automatic order choice according to the Schwarz information criterion.
Subsequently, the data are fitted to a threshold- (T)GARCH(1,1) (see Glosten et al. (1993))
model.10 A TGARCH(1,1) model is given by
2
2
2
2
2
σt2 = ω + ασt−1
ηt−1+ δ1{ηt−1 < 0}σt−1
ηt−1+ βσt−1,

t = 1, ..., T.

Notably, a TGARCH model is able to capture asymmetric impacts of positive and negative
shocks. Hence, in each window and for both return losses and return gains, we test against
HOTDs in raw returns and in standardized ARMA(p,q)–TGARCH(1,1) residuals.
b we have to choose the number of upper order statistics k. The rolling
In order to calculate ∆,√window scheme complicates a manual choice for each window. Thus, for each k ∈ [0.01n, c n]b
we compute ∆(k)and take the median thereof as the final estimator for ∆ (see Sections 1.3 andb 90% confidence intervals for ∆ and test
2 for details). Figures 2 and 3 show the evolution of ∆,decisions for the global and the European portfolio at each time point; a confidence interval iscolored gray whenever H0 : ∆ = 0 has to be rejected, i.e. whenever multivariate tail risk is not
only bivariate. 11
10

Results obtained by using a standard GARCH model were qualitatively very similar and are
therefore not reported.
11

Note, as ∆ ≥ 0, theoretically, confidence interval lower bounds should not become negative
yet this bound decides whether ∆ is significantly larger than zero. Furthermore, as we areconducting one-sided tests, the shaded areas within the 90% confidence intervals refer to test

22

3.2 Results and economic implications
Regarding global portfolio gains, the TGARCH(1,1) filtered series never allows rejecting thenull hypothesis of no HOTDs, while for the unfiltered series the null has to be rejected after
2010 with P–values close to 5%. Still, the absolute amount of HOTDs is also small after 2010.
For losses, we detect an accentuated increase in HOTDs after 2006/2007, whereas no significant
HOTDs up to 2006 can be found. The gradually increasing HOTDs appear to be still on the riseat the end of the sample. Although the sample covers major historical events such as the 1970s
oil crises, the Black Monday 1987, the dissolution of the Soviet Union, the Gulf War 1990–91,
the Asian financial crisis in 1997, the introduction of the Euro, the burst of the dot–com bubble,
and 09/11/01, it is the global financial crisis of 2007–08 that marks the start of global HOTDs
becoming significant. Thus the latter is the only event within the sample, that is capable ofherding global high–dimensional extreme losses. Before 2007–08, investors holding a globally
diversified portfolio did not have to pay attention to HOTDs while this has apparently become
an additional challenge in asset allocation on top of bivariate tail dependence nowadays. On
the other hand, throughout the considered time span, investors cannot expect to benefit from
HOTDs between gains: The financial turmoils during 2007–08 caused univariate extreme losses
to trigger joint global extreme losses, whereas univariate extreme gains still do not spread out(unfiltered), or at least not as strongly as extreme losses (filtered).
Losses and gains within the European portfolio, on the other hand, are more prone
to HOTDs. This may be explained by closer economic connections, but also by the factthat now seven indices are considered, implying that extreme connections between three ormore components are more likely than within a three–dimensional portfolio.12 Intra–European
HOTDs appear to be time–varying and are most of the time significant. The TGARCH filterb suggesting that the unstable behavior of ∆b for the unfiltered seriessmoothes the evolution of ∆,can be partly explained by serial correlation and time–varying conditional volatility. However,
the results do not differ qualitatively with respect to whether the filter is employed or not, as
the test decisions on a significance level of 5% are mostly alike for both specifications. Overall,b appears to be constant for both the filtered and the raw data. For
the empirical variance of ∆losses, one observes a decrease of HOTDs from the sample beginning until the mid to mid/end–
1990s; also, HOTDs are not significant between 1994 and 1998. Afterwards, the importance of
HOTDs increases until the beginning of the 2007–08 crisis, remaining on a stable, high levelever since. Interestingly, this movement is continuous and the major political events that fall inrejections on 5% significance level.
12

The latter makes a comparison of test results across both data sets difficult.

23

this period (dot–com crisis, 9/11/01, introduction of the Euro) do not cause discontinuities ofbthe trajectory of ∆.
We conclude that HOTDs in the European portfolio are not driven by one–time events
but rather mirror established, mid- to long-term processes due to the European financial andeconomic integration. This also gives an explanation for why gains HOTDs of the European
portfolio prevail throughout the sample, which stands in contrast to nearly non–existent HOTDs
in gains within the global portfolio. Diversification opportunities of cross–sectional extreme
losses are limited within Europe, as it was also found in Christoffersen et al. (2012). Our
test results for right tails indicate, however, that there is potential to benefit from cross–sectional extreme gains. This generalizes the results in Poon et al. (2004) as the presence of
HOTDs implies their results based on pairwise bivariate analysis. Moreover, we observe thattail dependence, at least within European stock markets, is more severe than assumed so far.
3.3 Factor model for the European stock market
The presence of HOTDs within the European portfolio might be caused by tail events of acommon external factor. To distill truly intra–European HOTDs, we now control for effects of(i)global financial markets. Returns yt are thus modeled by a factor market model
(i)

(i)

yt = ζ (i) Mt + t , i = 1, ..., 7,
(i)

(i)

where Mt denotes a common factor for all marginal returns yt . The disturbance t is often(i)interpreted as the idiosyncratic part of yt . An apparent choice for Mt is the return series of the
MSCI World Ex Europe13 as it is an index of all relevant stock markets except for European
ones.
We repeat the rolling window analysis of the previous section for the European portfolio and
(1)

(7)

test for HOTDs between (unfiltered) factor model residuals (ˆt , ..., ˆt )nt=1 . Furthermore, we
obtain standardized residuals from a ARMA(p,q)–TGARCH(1,1) model for the factor model
residuals. Test decisions for the latter two models thus account for serial correlation, time–varying volatility and the effect of the common risk driver. The remaining dependence structure
can be considered as idiosyncratic to the European stock market system. For all seven indices,
the return model is re–estimated in each window14 and the orders of the ARMA models are
13
14

This index runs under mnemonic MSWXEU$ in Datastream.

Whenever numerical optimization of the likelihood function failed for the given setting, we
first changed the conditional distribution from a t to a normal distribution. In seven out of 8596
estimated models we then only came across convergence problems for 8 TGARCH models. In
these cases we used residuals from the GARCH(1,1) model as substitute. There appears to beb for TGARCH residuals at 4/7/96 where the optimization of the likelihood forone outlier of ∆24b (see equation (10)), together with 90% confidence
Figure 2: Dynamics of the test statistic ∆intervals, the global portfolio, using a rolling window of roughly 15 years. The left panel shows
test decisions for portfolio losses, whereas the right panel shows test decisions for portfolio gains.
Confidence intervals are colored gray whenever H0 : ∆ = 0 has to be rejected.
(b) Gains, unfiltered(a) Losses, unfiltered✞✞✠✡☛✡☞✌ ✍✎ ✏ ✑✒✓✔✕✌ ✠✡☛✡☞✌✖
✟ ✑ ✗ ✘✓✙ ✚✛

✞

✞

✞

✞
✁✂✄

✁✂✁

✁✁✄

✁✁✁

☎✆✆✝

☎✆✆✂

☎✆ ✝

✁✂✄

✞

✠✡☛✡☞✌ ✍✎ ✏ ✑✒✓✔✕✌ ✠✡☛✡☞✌✖
✟ ✑ ✗ ✘✓✙ ✚✛

✞

✞

✞
✁✂✁

✁✁✄

✁✁✁

☎✆✆✝

☎✆✆✂

☎✆ ✝

✁✁✄

✁✁✁

☎✆✆✝

☎✆✆✂

☎✆ ✝

✠✡☛✡☞✌ ✍✎ ✏ ✑✒✓✔✕✌ ✠✡☛✡☞✌✖
✟ ✑ ✗ ✘✓✙ ✚✛

✞

✁✂✄

✁✂✁

(d) Gains, ARMA(p,q)–TGARCH(1,1) filter(c) Losses, ARMA(p,q)–TGARCH(1,1) filter✞✠✡☛✡☞✌ ✍✎ ✏ ✑✒✓✔✕✌ ✠✡☛✡☞✌✖
✟ ✑ ✗ ✘✓✙ ✚✛

✁✂✄

✁✂✁

✁✁✄

✁✁✁

☎✆✆✝

☎✆✆✂

☎✆ ✝

again found with the Schwarz criterion. Test results for both gains and losses of unfiltered data
and ARMA-TGARCH(1,1) filtered data are shown in Figure 4.
Controlling for changes in global stock markets slightly attenuates European HOTDs, yet
results closely resemble the results from the previous subsection (Figure 3). The only major
exception where controlling for the world index alters the test decision, in the sense that it
causes HOTDs to be significant, is for gains during 1990–94, Figure 3 (d) and Figure 4 (d).
the TGARCH model struggles.

25b (see equation (10)), together with 90% confidence
Figure 3: Dynamics of the test statistic ∆intervals, for the European portfolio, using a rolling window of 3 to 4 years. The left panel shows
test decisions for portfolio losses, whereas the right panel shows test decisions for portfolio gains.
Confidence intervals are colored gray whenever H0 : ∆ = 0 has to be rejected.
(b) Gains, unfiltered(a) Losses, unfiltered✟✟✡☛☞☛✌✍ ✎✏ ✑ ✒✓✔
✕✖✍ ✡☛☞☛✌✍✗✠ ✒ ✘ ✙✔✚ ✛✜

✟

✟

✟

✟
✁✁✂

✁✁✄

✁✁☎ ✆✂✂✂ ✆✂✂✝ ✆✂✂✞ ✆✂ ✂ ✆✂ ✝

✁✁✂✟✡☛☞☛✌✍ ✎✏ ✑ ✒✓✔
✕✖✍ ✡☛☞☛✌✍✗✠ ✒ ✘ ✙✔✚ ✛✜

✟

✟

✟
✁✁✄

✁✁☎ ✆✂✂✂ ✆✂✂✝ ✆✂✂✞ ✆✂ ✂ ✆✂ ✝

✁✁☎ ✆✂✂✂ ✆✂✂✝ ✆✂✂✞ ✆✂ ✂ ✆✂ ✝

✡☛☞☛✌✍ ✎✏ ✑ ✒✓✔
✕✖✍ ✡☛☞☛✌✍✗✠ ✒ ✘ ✙✔✚ ✛✜

✟

✁✁✂

✁✁✄

(d) Gains, ARMA(p,q)–TGARCH(1,1) filter(c) Losses, ARMA(p,q)–TGARCH(1,1) filter✟✡☛☞☛✌✍ ✎✏ ✑ ✒✓✔
✕✖✍ ✡☛☞☛✌✍✗✠ ✒ ✘ ✙✔✚ ✛✜

✁✁✂

✁✁✄

✁✁☎ ✆✂✂✂ ✆✂✂✝ ✆✂✂✞ ✆✂ ✂ ✆✂ ✝

However, the effect of the market factor to HOTDs between European gains has increased
since 2006 which can be seen by comparing Figures 3 (b) and 4 (b). Both do not account for
ARMA–(T)GARCH effects and the only possible source for a difference is the accounting for thecommon factor. Overall, HOTDs between the idiosyncratic risks of European stock markets haveincreased since 2000. Thus, we can reveal that joint extremes are truly due to intra–European
HOTDs. For a practitioner, this provides econometric evidence that losses on portfolios withdifferent European based Exchange-traded funds, or with different single European stocks, arelikely to add up in times of crisis, and diversification effects may fade away in case of tail events26b (see equation (10)), together with 90% confidence
Figure 4: Dynamics of the test statistic ∆intervals, for the European portfolio, using a rolling window of 3 to 4 years, after controllingfor a market factor. The left panel shows test decisions for portfolio losses, whereas the rightpanel shows test decisions for portfolio gains. Confidence intervals are colored gray whenever
H0 : ∆ = 0 has to be rejected.
(b) Gains, unfiltered(a) Losses, unfiltered✟✟✡☛☞☛✌✍ ✎✏ ✑ ✒✓✔
✕✖✍ ✡☛☞☛✌✍✗✠ ✒ ✘ ✙✔✚ ✛✜

✟

✟

✟

✟
✁✁✂

✁✁✄

✁✁☎

✆✂✂✆

✆✂✂✝

✆✂✂✁

✆✂ ✞

✁✁✂

(c) Losses, ARMA(p,q)–TGARCH(1,1) filter✟✡☛☞☛✌✍ ✎✏ ✑ ✒✓✔
✕✖✍ ✡☛☞☛✌✍✗✠ ✒ ✘ ✙✔✚ ✛✜

✟

✟

✟

✟
✁✁☎

✆✂✂✆

✆✂✂✝

✆✂✂✁

✆✂ ✞

✆✂✂✆

✆✂✂✝

✆✂✂✁

✆✂ ✞

✡☛☞☛✌✍ ✎✏ ✑ ✒✓✔
✕✖✍ ✡☛☞☛✌✍✗✠ ✒ ✘ ✙✔✚ ✛✜

✟

✁✁✄

✁✁☎

(d) Gains, ARMA(p,q)–TGARCH(1,1) filter✡☛☞☛✌✍ ✎✏ ✑ ✒✓✔
✕✖✍ ✡☛☞☛✌✍✗✠ ✒ ✘ ✙✔✚ ✛✜

✁✁✂

✁✁✄

✁✁✂

✁✁✄

✁✁☎

✆✂✂✆

✆✂✂✝

✆✂✂✁

✆✂ ✞

for solely stock-based portfolios. As multivariate extreme losses of European stock markets areapparently only slightly affected by events of the market factor, there exist tail diversification
opportunities between both. These opportunities slightly diminish for extreme gains. Besides
the importance for asset allocation, significance of HOTDs also seems to mark periods of distressin the markets, i.e. when stock indices tend to jointly experience large losses.

27

3.4 Importance and share of higher order tail dependencies in practice
To show the importance of testing for HOTDs, we provide some simple descriptive screening
tools in this subsection. In particular, we assess the share of bivariate tail events that cannot
be captured by tail correlations. For this, we use the asymptotic probabilities of two or three
joint extremes, κ2 and κ3 , defined as
κ2 = lim t−1 P
t→0

κ3 = lim t
t→0

[

{X (i) ≥ Fi−1 (1 − tx(i) )} ∩ {X (j) ≥ Fj−1 (1 − tx(j) )} ,
i6=j

−1

P

 [ 

{X (h) ≥ Fh−1 (1 − tx(h) )} ∩ {X (i) ≥ Fi−1 (1 − tx(i) )} ∩ {X (j) ≥ Fj−1 (1 − tx(j) )}.

h6=i6=j

They describe the likelihood of at least two or respectively three assets becoming extreme atonce. Their ratio κ3 /κ2 quantifies the share of bivariate extremes that also amount to a trivariate
extreme event. Similar to the estimation of the STDF, this magnitude can be estimated by its
empirical counterpart. We compare the days featuring a bivariate (κ2 ) or trivariate (κ3 ) extremewith the number of days with at least one univariate extreme,
 (g)
(g)
1
Xt > Xn:n+0.5−k }t=1i6=jg∈{i,j}
κ̂2 =
,
Sd
Pn
(i)
(i)
X
>
X
}
1{
t
n:n+0.5−k
i=1
t=1
 (g)
Pn P
Q
(g)
> Xn:n+0.5−k }t=1
h6=i6=jg∈{h,i,j} 1 Xt
κ̂3 =
.
Sd
Pn
(i)
(i)
X
>
X
}
1{
t
n:n+0.5−k
i=1
t=1
Pn P

Q

(16)

(17)

Section 5.3 in the Appendix provides a small simulation study that shows κ3 /κ2 is indeed areasonable measure for determining the severeness of HOTDs.
Figure 5 shows estimates κ̂2 , κ̂3 and κ̂3 /κ̂2 for the TGARCH filtered European portfolio
without controlling for a common factor. As before, final estimates in each window were found√by taking the estimates’ medians for k ∈ [0.01n, c n].
b (Figures 3 c-d). The probability of
Not surprisingly, trajectories resemble the dynamics of ∆observing trivariate extremes (κ̂3 ) has steadily increased from 10–20% for losses, and 5–10% for
gains, respectively, during the 1990s up to 20–30% for losses, and 30-40% for gains, respectively,
at the peak of the recent financial crisis 2007–2009. However, the share of trivariate extremesin bivariate extremes κ̂3 /κ̂2 steadily declined both for losses and gains during the 1990s (from
60% to 35% for losses, and from 50% to 20% for gains) and has consequently ascended for bothtails until the end of the 2010s (up to 70-80% for both losses and gains). Thus, for losses, theprobability that multivariate extremes occur in larger cross-sections has doubled during the
2000s, while it has even tripled for gains in that time span. This highlights that extremes morethan ever occur not only in bivariate pairs, but also in larger cross-sections.

28

Figure 5: Dynamics of κ̂2 , κ̂3 , and κ̂3 /κ̂2 (see equations (16) and (17)) for the TGARCH filtered
European portfolio.
(a) Losses(b) Gains✍✎ ✏☛✑✍✎ ✏✌✑✍✎ ✏✌✑ ✍✎ ✏☛✑✁✆✍✎ ✏☛✑✍✎ ✏✌✑✍✎ ✏✌✑ ✍✎ ✏☛✑✁✆✍✎ ✏✌✑ ✍✎ ✏☛✑✁☎✍✎ ✏✌✑ ✍✎ ✏☛✑✁☎✍✎ ✏☛✑✄✁✍✎ ✏☛✑✄✁✂✁✁✍✎ ✏✌✑

✁
✝✞✞✟

4

✂✁

✍✎ ✏✌✑

✝✞✞✠

✝✞✞✡

☛✟✟☛

☛✟✟☞

☛✟✟✞

☛✟✝✌

✝✞✞✟

✝✞✞✠

✝✞✞✡

☛✟✟☛

☛✟✟☞

☛✟✟✞

☛✟✝✌

SummaryThis paper proposes a test that reveals situations in which common bivariate measures
for tail dependence underdiagnose the potential for higher-dimensional extreme events. Test
asymptotics are derived and simulations show the bootstrap implementation routine features
attractive finite sample properties, despite the challenging threshold choice, inherent to extreme
value statistics, which occasionally affects test size. In the case of data that exhibit serial
correlation and GARCH effects, we recommend studying estimated residuals instead observed
realizations, to maintain the good size and power properties.
On global stock markets, we find that cross–sectional extremes become relevant in thecourse of the financial crisis of 2007–08. Multivariate extremes on European stock markets
are historically more intertwined, as the impact of high–dimensional extremes is significant
throughout the considered sample. There appears to be diversification potential of multivariate
extreme losses between European and non–European stock markets, while extreme gains do
not share this feature. Within the European system, left tail events feature no potential fordiversification. We find time periods when up to 80% of extremes are truly multivariate.

29

5

Appendix

5.1 Model specifications
Table 4: Specifications of the max factor models.

B30 =



1/2 1/2 0
1/2 0 1/2
0 1/2 1/2



 1/2 1/2 0

B3A1 = ( 1/3 1/3 1/3 )0 0
1/2 0 1/2 0 0
 1/2 0 0 1/2 0  1/2 0 0 0 1/2 
1/2 0 0 
B50 =  00 1/2
0 1/2 0 
 0 1/2
1/2 0 0 1/2  0 0 1/2 0 1/2 
0 0 1/2 0 1/2
0 0 0 1/2 1/2

 1/2 1/2 0 1/3 1/3 1/3 00
1/2 0 0 1/2 0 1/2 0 0 0 1/2 
B5A1 =  0 1/2 0 1/2 0 
 0 1/2 0 0 1/2 
0 0 1/2 1/2 0
0 0 1/2 0 1/2

 1/4 1/4 1/4 1/4 0 
1/2 0

0

0 1/2
0 1/2 0 1/2 0 1/2
0 0 1/2 1/2

B5A2 =  0 1/2 000



0 0 0 0
1/2 0 1/2 0 0 0 0
 1/2 0 0 1/2 0 0 0  1/2 0 0 0 1/2 0 0  1/2 0 0 0 0 1/2 0  1/2 0 0 0 0 0 1/2  0 1/2 1/2 0 0 0 0 
 0 1/2 0 1/2 0 0 0 
 0 1/2 0 0 1/2 0 0 






1/3 1/3 1/3 0 0 0 0
1/3 1/3 1/3 0 0 0 0
 0 1/2 0 0 0 1/2 0  A1
0A2

0 0 1/2 1/2 0 0 0
B7 =  0 1/2 0 0 0 0 1/2  B7 = 1/3 0 0 1/3 1/3 0 0 B7 =0 1/3 0 0 0 1/3 1/3
0 0 0 1/4 1/4 1/4 1/4
 0 0 1/2 1/2 0 0 0 
 0 0 1/2 0 1/2 0 0 
 0 0 1/2 0 0 1/2 0 
 0 0 1/2 0 0 0 1/2  0 0 0 1/2 1/2 0 0 
 0 0 0 1/2 0 1/2 0 
 0 0 0 1/2 0 0 1/2  0 0 0 0 1/2 1/2 0 
0 0 0 0 1/2 0 1/2
0 0 0 0 0 1/2 1/2

B7A3 =

1/6 1/6 1/6 1/6 1/6 1/6 0
1/2 0 0 0 0 0 1/2
0 1/2 0 0 0 0 1/2
0 0 1/2 0 0 0 1/2!

5.2 Proofs
Proof of proposition 1.1
If X is tail independent, `(x) = x1 ⇔ `i (x(i) ) = x(i) 1, for all possible bivariate combinations
P
i. Plugging this into the general form of ∆, and realizing that in this case i<j≤2 `i (x(i) ) =30(d − 1)

Pd

i=1 x

(i)

, it follows that∆ = `(x) − 2

d
X

x

(i)

+d

i=1

=

=

d
X

x

(i)

−2

d
X

x(i) −

i=1

d
X

x

(i)

+d

d
X

i=1

i=1

d
X

d
X

d
X

x

−2

i=1

x

(i)

i=1

+d

`i (x(i) )

i<j≤2

i=1
(i)

X

x(i) −X`i (x(i) )

i<j≤2

x

(i)

− (d − 1)

d
X

i=1

x(i)

i=1

= 0.
The reverse does not hold true. E.g. let X := (X (1) , X (2) , X (3) ), with X(3) being indepena.s.

dent of X (1) , and let X (1) = X (2) , i.e. X (1) and X (2) are perfectly tail dependent. Thus,`12 (x(1) , x(2) ) ≡ `11 (x(1) , x(1) ) = x(1) , `13 (x(1) , x(3) ) = x(1) + x(3) , and`123 (x(1) , x(1) , x(3) ) = lim tP
t↓0



[


{X (i) ≥ Fi−1 (1 − tx(i) )}i∈{1,2,3}= lim tP {X (1) ≥ F1−1 (1 − tx(1) )} ∪ {X (3) ≥ F3−1 (1 − tx(3) )}t↓0= x(1) + x(3) .
Rewriting ∆ yields∆ = `123 (x(1) , x(1) , x(3) ) − 2(2x(1) + x(3) ) + 3(2x(1) + x(3) )− 2`11 (x(1) , x(1) ) − `13 (x(1) , x(3) )= x(1) + x(3) − 2(2x(1) + x(3) ) + 3(2x(1) + x(3) ) − x(1) − 2(x(1) + x(3) )= 0.
Hence, we have tail dependence in X and ∆ is zero as extreme events in dimension three do notmatter.
Proof of proposition 1.2 The result directly follows from Einmahl et al. (2012), theorem 4.6,√
ˆ − `X (x), x ∈ [0, 1]d , is asymptotic
and Bücher and Dette (2013), Bücher et al. (2014) k(`(x)
normal with zero mean and covariance matrix equal to a sum of a centered Gaussian field andGaussian processes. It is assumed that `X (x) < x0 1 to ensure the asymptotic variance of `ˆX (x)is non–zero. This holds if at least one bivariate pair (X (i) , X (j) ) is asymptotic dependent. In R2 ,

31

where x = (x(i) , x(j) ), it holds that√dk `ˆij (x(i) , x(j) ) → N (`(x(i) , x(j) ), σ`2 ), x(i) , x(j) > 0,

whereσ`2 = `(x(i) , x(j) ) − 2x(i) `∂i (x(i) , x(j) ) − 2x(j) `∂j (x(i) , x(j) ) + x(i) `2∂i (x(i) , x(j) )+ x(j) `2∂j (x(i) , x(j) ) + 2`∂i (x(i) , x(j) )`∂j (x(i) , x(j) )(x(i) + x(j) − `(x(i) , x(j) ),with `∂j (x) := (∂`/∂x(j) )(x) denoting the partial derivative of the STDF with respect tobargument x(j) . According to equations (5) and (6), and setting x = 1, R(x)is also asymptotic
b directly follows from equation (7). Thus,
normal. Asymptotic normality of ∆
d
2
b →
∆
N (∆, σ∆
b ),

with
−1 2
−1
2
σ∆
b = k σ`ˆ + k

X
i<j≤2

σ`2ˆi + 2

 X
i<j≤2

ˆ +
Cov(`ˆi , `)X

Cov(`ˆij , `ˆgh ) ∈ (0, ∞).

i<j≤2;g<h≤2;i6=gWhenever partial derivatives of the STDF do not exist, the same reasoning for the limit law of√b applies using asymptotic results in Bücher et al. (2014).
k∆
5.3 Auxiliary simulations
The ratio κ3 /κ2 gives the share of bivariate extremes that are also extremes in dimension threeor larger; as this ratio conditions on the occurrence of bivariate extremes, the magnitude isdriven by multivariate (d > 2) tails and is not driven by the number of bivariate extremes, as is
the case for κ3 . Table 5 reports averages from 1000 simulation repetitions of all three measures
for the distributions considered in the simulations of Section (2). Sample size, dimension andchoice of k are as in the empirical application of Chapter (3).
Note, the only distribution in dimension 7 that fulfills the null of no HOTDs is the max factor
model with loading matrix B70 . In this case, both κ3 and κ3 /κ2 are close to zero. Theoretically,
they should be exactly zero, however, for a sample size of n = 750 this distortion can be beinterpreted as finite sample bias. Yet in this case, a simple t-test would not indicate a statisticalsignificance (α = 0.05). For the meta t-distribution, κ3 /κ2 grows with decreasing degree of
freedom of the copula, which governs the strength of bivariate and multivariate extremes. Thus,κ̂3 /κ̂2 is indeed capable of reflecting the severeness of HOTDs.

32

Table 5: Means and standard deviations of simulated κ̂3 /κ̂2 , κ̂3 , κ̂2 for max factor models
and meta t-distributions from Section 2 with 1000 repetitions, d = 7, n = 750 and k as in theempirical application.
κ̂3 /κ̂2

κ̂3

κ̂2

0.507
(0.070)0.453(0.071)0.431(0.072)0.425(0.072)0.212(0.040)0.168(0.036)0.151(0.033)0.145(0.032)0.419(0.046)0.371(0.042)0.351(0.041)
0.344(0.040)0.050(0.027)0.590(0.063)0.558(0.062)0.517(0.019)0.040(0.022)0.500(0.035)0.534(0.053)0.328(0.028)0.795(0.048)0.852(0.033)0.958(0.028)0.633(0.048)t-distr.
νC
5
10
15
20
max factor
B70
B7A1
B7A2
B7A3

33

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