Hypothesis testing for tail dependence parameters on the boundary of the parameter space

RESEARCH OUTPUTS / RÉSULTATS DE RECHERCHE

Author(s) - Auteur(s) :

Publication date - Date de publication :

Permanent link - Permalien :

Rights / License - Licence de droit d’auteur :

Bibliothèque Universitaire Moretus Plantin

Institutional Repository - Research Portal

Dépôt Institutionnel - Portail de la Recherche

researchportal.unamur.be

University of Namur

Hypothesis testing for tail dependence parameters on the boundary of the parameter

space

Kiriliouk, Anna

Published in:

Econometrics and Statistics

DOI:

10.1016/j.ecosta.2019.06.001

Publication date:

2020

Document Version

Publisher's PDF, also known as Version of record

Link to publication

Citation for pulished version (HARVARD):

Kiriliouk, A 2020, 'Hypothesis testing for tail dependence parameters on the boundary of the parameter space',

Econometrics and Statistics, vol. 16, pp. 121-135. https://doi.org/10.1016/j.ecosta.2019.06.001

General rights

Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain

• You may freely distribute the URL identifying the publication in the public portal ? Take down policy

If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.

ContentslistsavailableatScienceDirect

Econometrics

and

Statistics

journalhomepage:www.elsevier.com/locate/ecosta

Hypothesis

testing

for

tail

dependence

parameters

on

the

boundary

of

the

parameter

space

Anna

Kiriliouk

Université de Namur, Faculté des sciences économiques, sociales et de gestion, Rempart de la vierge 8, Namur B-50 0 0, Belgium

a

r

t

i

c

l

e

i

n

f

o

Article history:

Received 25 January 2018 Revised 19 June 2019 Accepted 24 June 2019 Available online 29 July 2019

Keywords:

Brown–Resnick model Hypothesis testing Max-linear model Multivariate extremes Stable tail dependence function Tail dependence

a

b

s

t

r

a

c

t

Modellingmultivariatetaildependenceisoneofthekeychallengesinextreme-value the-ory. Multivariate extremes are usually characterized using parametric models, some of whichhavesimplersubmodelsattheboundaryoftheirparameterspace.Hypothesistests areproposedfortaildependenceparametersthat,underthenullhypothesis,areonthe boundaryofthealternativehypothesis.Theasymptoticdistributionoftheweightedleast squaresestimatorisgivenwhenthetrueparametervectorisontheboundaryofthe pa-rameterspace,andtwoteststatisticsareproposed.Theperformanceoftheseteststatistics isevaluatedfortheBrown–Resnickmodelandthemax-linearmodel.Inparticular, simu-lationsshowthatitispossibletorecovertheoptimalnumberoffactorsforamax-linear model.Finally,themethodsareappliedtocharacterizethedependencestructureoftwo majorstockmarketindices,theDAXandtheCAC40.

© 2019EcoStaEconometricsandStatistics.PublishedbyElsevierB.V.Allrightsreserved.

1. Introduction

Extreme-valuetheory isthebranchofstatistics concerned withthecharacterization ofextremeevents.Theseoccur in alargevarietyoffields,suchashydrology,meteorology,financeandinsurance,butalsoinsettingslikehumanlife spanor athleticrecords.Forsomeexamples,see EinmahlandMagnus(2008),Towleretal.(2010),Chavez-Demoulinetal.(2016),

Thomas etal.(2016) andRootzén andZholud(2017).In theunivariate case, the limitingdistribution ofsuitably normal-izedblock maximaorthresholdexceedancescanbecharacterizedentirelyusingthegeneralizedextreme-value(Fisherand Tippett,1928;Gnedenko,1943)andthegeneralizedParetodistribution(BalkemaandDeHaan,1974;Pickandsetal.,1975) respectively. However, manyextremeevents are inherentlymultivariate, and an importantchallenge is to modelthe tail dependencebetweentwoormorerandomvariablesofinterest.Ifdependencedisappearsasthevariablestakeonmoreand moreextremevalues,wesaythattheyareasymptoticallyindependent.Testingforasymptoticindependenceandmodelling asymptotically independentdatahasbeen doneinLedford andTawn (1996),Draismaetal.(2004),HüslerandLi(2009),

Wadsworthetal.(2016) andGuillouetal.(2018),amongothers.Inthefollowing,weconsidertheframeworkofasymptotic dependence.

Thefamilyoflimitingdistributionsformultivariatemaximaorthresholdexceedancesisinfinite-dimensional,sothat re-alisticandcomputationallyfeasibleparametricmodelsneedtobeproposed.ExamplesincludeGumbel(1960),Smith(1990),

Tawn(1990) andKabluchkoetal.(2009).Manydifferentapproachesofestimatingtaildependenceparametersexist.Fitting amultivariate extreme-valuedistribution tocomponentwise block maximaisdone in Padoanetal.(2010),Davisonetal. (2012),Castruccio etal.(2016) andDombryetal.(2017), alongothers.Multivariate thresholdexceedances, whichare the

E-mail address: anna.kiriliouk@unamur.be https://doi.org/10.1016/j.ecosta.2019.06.001

focus of thiswork, can be tackled using the stabletail dependence function (Drees andHuang, 1998) or the multivari-ategeneralizedParetodistribution(Rootzén andTajvidi,2006).Commonestimationmethodsincludemaximumlikelihood (WadsworthandTawn, 2014;Kiriliouket al., 2018a;de Fondevilleand Davison,2018) andminimum distanceestimation (Einmahletal.,2012;2016;2018).

Somepopularparametricmodelshavesimplersubmodelsattheboundaryoftheparameterspace.Themaingoalofthis paperisto proposehypothesis testsfortail dependenceparametersthat, underthenullhypothesis,are ontheboundary ofthealternative hypothesis.A firstexampleisthe Brown–Resnickmodel(Kabluchkoetal., 2009), popularinspatial ex-tremes,whichreducestothesimplerSmithmodel(Smith,1990)whentheshapeparameteroftheformerattainsitsupper bound. A second example is the max-linear model,where each component ofa vector can be interpreted as the maxi-mumshockamongasetofindependentfactors.Recentworkonmax-linearmodelsisnumerous.InGissiblandKlüppelberg (2018) andGissibletal.(2018),a recursivemax-linearmodelona directedacyclic graphisconsidered.InCui andZhang (2018) andZhaoandZhang(2018),max-linearmodelsareusedtomodelalatentfactorstructureforfinancialreturns. Fi-nally,theMarshall–Olkinmodel(Embrechtsetal.,2003;Segers,2012)isasubmodelthatisfrequentlyusedinpractice,see forinstanceBurtschelletal.,(2009),SuandFurman,(2017) orBrigoetal.,(2018).

Becauseoftheirnon-differentiability,max-linearmodelscannotbeestimatedbystandardlikelihoodmethodsandhence we usetheweightedleastsquaresestimatorofEinmahletal.(2018),whichiscentredaround thestabletaildependence function.Thehypothesistestingframeworkdescribedinthatpaperisvalidforparametersintheinterioroftheparameter space, and henceit cannot be usedto test whether one ormore factorloadings are equalto zero. The hypothesis tests proposed inthispapercanbeappliedtodecidehowmanyfactorsarenecessarytomodelacertain datasetortotest ifa morespecificmodel(eg,Marshall–Olkin)issufficient,bypassingthelimitationthatthenumberoffactorshastobechosen upfront.

Thepaperisorganizedasfollows.Section2 presentsbackgroundonmultivariateextremesandintroducestheweighted leastsquaresestimator.Section3 establishesasymptoticnormalityfortheweightedleastsquaresestimatorwhenthetrue parameter vector lies onthe boundary of theparameter space.Using results fromAndrews (1999),Andrews(2001) and

Andrews(2002),a deviance-anda Wald-type test statisticare proposed,whose asymptoticdistributions are easily com-putablewhenthedimensionoftheparametervectorontheboundaryismoderate.InSection4.2,weperformsimulations toevaluatetheperformanceoftheteststatisticsfortheBrown–Resnickandthemax-linearmodels.Finally,Section5 illus-tratesourmethodsontwomajorstockmarketindices,theDAXandtheCAC40.Technicaldefinitionsandproofsaredeferred totheappendix.

2. Background

2.1. Multivariateextreme-valuetheory

LetXi=

(

Xi1,...,Xid

)

,i∈

{

1,...,n

}

,be randomvectorsinRd withcumulative distributionfunctionFandmarginal

cu-mulative distribution functionsF1,...,Fd. LetMn:=

(

Mn,1,...,Mnd

)

withMn j:=max

(

X1j,...,Xn j

)

for j=1,...,d.We say

thatFisinthemax-domainofattractionofanextreme-valuedistributionGifthereexistsequencesofnormalizingconstants

an=

(

an1,...,and

)

>0andbn=

(

bn1,...,bnd

)

∈Rd suchthat

P



Mn− bn an ≤ x



=Fn

₍

_a nx+bn

)

d −→G

(

x

)

, asn→∞. (1)

Themargins,G1,...,Gd,ofGareunivariateextreme-valuedistributions,

Gj

(

xj

)

=exp



−



1+

γ

xj−

μ

j

σ

j



−1/γj +



,

σ

j>0,

γ

j,

μ

j∈R,

wherex₊:=max

(

x,0

)

.ThefunctionGisdeterminedby

G

(

x

)

=exp

{−

(

− logG1

(

x1

)

,. . .,− logGd

(

xd

))

}

,

where:[0,∞)d_{→[0,∞)iscalledthe(stable)taildependencefunction,}



(

x

)

:=lim

t↓0t

−1_{P[1− F}

1

(

X11

)

≤ tx1 or ... or1− Fd

(

X1d

)

≤ txd]. (2)

ThecumulativedistributionfunctionFisinthemax-domainofattractionofad-variateextremevaluedistributionGifand onlyifthelimitin(2) existsandthemarginaldistributionsin(1) convergetounivariateextreme-valuedistributions.Inwhat follows,weonlyassumeexistenceof(2),whichconcernsthedependencestructureofF,butnotthemarginaldistributions F1,...,Fd.

Becausetheclassofstabletaildependencefunctionsisinfinite-dimensional,oneusuallyconsidersparametricmodelsfor .Henceforthweassumethatbelongstoaparametricfamily{(· ;

θ

):

θ

∈



}with



⊂ Rp_{.Some examplescanbefound}

below,seealsodeHaanandFerreira(2006),Falketal.(2010),Segers(2012) andreferencestherein. Example 2.1. Thed-dimensionallogisticmodel(Gumbel,1960)hasstabletaildependencefunction

If

θ

₌1_,thevariablesare(asymptotically)independent,while

θ↓

0correspondstothesituationofcompletedependence. Example 2.2. Thed-dimensionalBrown–Resnickmodeldefinedonspatiallocationss1,...,sd∈R2hasstabletaildependence

function 

(

x;

θ

)

= d j=1 xj



d−1

(

η

(j)

(

1/x

)

;

ϒ

(j)

)

, where

η

(j)

₍

_x

₎

₌

₍

_η

(j) 1

(

x1,xj

)

,...,

η

(j−1j)

(

xj−1,xj

)

,

η

(j+1j)

(

xj+1,xj

)

,...,

η

d(j)

(

xd,xj

))

∈Rd−1,

η

(j) l

(

xl,xj

)

=



γ

(

sj− sl

)

2 + log

(

xl/xj

)



2

γ

(

sj− sl

)

∈R,

γ

(

s

)

₌

(

||

s

||

_/

ρ

)

α and

ϒ

(l)_∈R(d−1)×(d−1)_{is the correlation matrix with entries}

ϒ

(j) lk =

γ

(

sj− sl

)

+

γ

(

sj− sk

)

−

γ

(

sl− sk

)

2



γ

(

sj− sl

)

γ

(

sj− sk

)

, l,k=1,...,d; l,k= j,

(Kabluchko etal., 2009; HuserandDavison, 2013). The parametervector is

θ

=

(

ρ

,

α

)

∈

(

0,∞

)

×

(

0,2].The Smith model (Smith,1989)isobtainedwhen

α

₌2.

Example 2.3. Themax-linearmodelwithrfactorshasstabletaildependencefunction



(

x;

θ

)

= r t=1 max j=1,...,dbjtxj, x∈[0,∞

)

d_{, (3)}

wherethefactorloadingsb_jtarenon-negativeconstantssuchthat r_t=1b_jt=1forevery j_∈

{

1_,...,d

}

andallcolumnsums ofthed× rmatrixB:=(bjt)j,t arepositive(Einmahletal.,2012).SincetherowsofBsumuptoone,theparametermatrix

hasonlyd×

(

r− 1

)

free elements. RearrangingthecolumnsofB willnot changethe valueof thestabletaildependence function.Foridentificationpurposes,wedefinetheparametervector

θ

bystackingthecolumnsofBindecreasingorderof theirsums, leavingout thecolumnwiththe lowestsum. Anexampleof arandom vectorZ=

(

Z1,...,Zd

)

thathasstable

taildependencefunction(3) is

Zj= max

t=1,...,rbjtSt, for j∈

{

1,...,d

}

,

whereS1,...,SrareindependentunitFréchetvariables,P[Sj≤ x]=exp

(

−1/x

)

forx>0and j∈

{

1,...,d

}

.

Example 2.4. LetI1,...,Id denoterandom(possiblydependent)Bernoulli randomvariables withpj=P[Ij=1]∈

(

0,1]for

j∈

{

1,...,d

}

. Let, for J⊂

{

1,...,d

}

, p

(

J

)

=P[

{

j=1,...,d:Ij=1

}

=J], so that

(

p

(

J

))

J⊂{1,...,d} is a probability distribution. ThemultivariateMarshall–Olkinmodel(Embrechtsetal.,2003;Segers,2012)hasstabletaildependencefunction



(

x;

θ

)

= ∅=J⊂{1,...,d} p

(

J

)

max j∈J



xj pj



.

Inthismodel,anysubsetofcomponentsofthevectorisassignedashockthatinfluencesallcomponentsofthatsubset.An interpretationintermsofcreditriskmodellingcanbefoundinEmbrechtsetal.(2003).

TheMarshall–Olkinmodelisobtainedasaspecialcaseofthemax-linearmodel(3) bysettingbjt=



p

(

Jt

)

/pj



1

(

j∈Jt

)

,

whereJt∈P

(

{

1,...,d

}

)

, Jt=∅,andP

(

A

)

denotesthepower setofA.The Marshall–Olkinmodeliswell-knownind=2,

where B=



_b 11 0 1− b11 0 b22 1− b22



=

⎛

⎜

⎝

P[I1=1,I2=0] P[I1=1] 0 P[I1=1,I2=1] P[I1=1] 0 P[I1=0,I2=1] P[I2=1] P[I1=1,I2=1] P[I2=1]

⎞

⎟

⎠

.

2.2.Estimationofthestabletaildependencefunction

2.2.1. Nonparametricestimationofthestabletaildependencefunction

Letk=kn∈

(

0,n]besuchthatk→∞andk/n→0asn→∞.Astraightforwardnonparametricestimatorofisobtained

byreplacingPandF1,. . .,Fdin(2) bythe(modified)empiricaldistributionfunctionsandreplacingtbyk/n,yielding

_n,k

₍

x

)

:=1 k n i=1 1

{

Ri1>n+1/2− kx1or ... orRid>n+1/2− kxd

}

. Here,Ri j= tn=11



Xt j≤ Xi j



denotestherankofXijamongX1j,...,Xn j.Thisestimator,theempiricaltaildependencefunction,

n₊1_/2 rather than nallows forbetter finite-sample properties, that is,for a lower mean squarederror (Einmahl etal., 2012).

Anothernonparametricestimatoristhebetataildependencefunction,whichwasveryrecentlyproposedinKiriliouketal. (2018b).Contrarytotheempiricaltaildependencefunction,thebetataildependencefunctionisasmoothestimatorwhich leadstosomeimprovementinitsfinite-samplebehavior.Itsnamestemsfromthefactthatitisbasedontheempiricalbeta copula(Segersetal.,2017).Wedefine

 β_n,k

(

x

)

=n k



1− Cβn



1−kx n



, where Cβn

(

u

)

= 1 n n i=1 d  j=1 Fn,Ri j

(

uj

)

, andfor r∈

{

1,...,n

}

, Fn,r

(

u

)

= rs=1



ns

us

₍

_{1− u}

₎

n−s _{isthe cumulative distribution function ofa Beta}

₍

_{r,n+1− r}

₎

_random

variable.

Adrawbackof_n,k or_β_n

,k mightbetheir possiblygrowingbias ask increases:Fougèresetal.(2015) showthat, under

suitableconditions,theestimatorn,ksatisfiestheasymptoticexpansion



n,k

(

x

)

− 

(

x

)

≈ k−1/2Z

(

x

)

+

α

(

n/k

)

M

(

x

)

,

whereZ_isacontinuouscenteredGaussianprocess,

α

isafunctionsuchthatlimx→∞

α

(

x

)

=0andMisacontinuous

func-tion.When√k

α

(

n/k

)

tendstoanon-zeroconstant,anasymptoticbiasappears.InFougèresetal.(2015) andBeirlantetal. (2016),thisbiasisestimatedandsubtractedfromtheestimatorn,k

(

x

)

inordertoproposenewbias-correctedestimators.

Finally, manyother nonparametric estimators existof thePickands dependence function, whichis therestriction of  to the unit simplex. See,for instance,Capéraà et al.(1997),Zhang etal. (2008),Gudendorf andSegers (2012), Berghaus etal.(2013),Vettorietal.(2016) andMarconetal.(2017).Whilethesecan be transformedinto estimatorsofthe stable taildependencefunction,theyrelyonstrongerassumptions,i.e.,theyarebasedondatafromanextreme-valuedistribution, andwewillnotconsiderthemhere.

2.2.2. Semi-parametricestimationofthestabletaildependencefunction

Nonparametricestimationformsasteppingstoneforsemi-parametricestimation(Einmahletal.,2012;2016;2018).Here, we focusonthemethodproposed inEinmahletal.(2018).Letn,kdenoteanyinitialestimatorofbasedon X1,...,Xn.

Letc1,...,cq∈[0,∞

)

d,withcm=

(

cm1,...,cmd

)

form=1,...,q,beqpointsinwhichwewillevaluateandn,k.Consider

for

θ

∈



theq× 1columnvectors Ln,k:=



n,k

(

cm

)

q m=1, L

(

θ

)

:=





(

cm;

θ

)

q m=1, Dn,k

(

θ

)

:=Ln,k− L

(

θ

)

. (4)

The pointsc1,...,cq need to be chosen insuch a waythat themap L:



→Rq isone-to-one, i.e.,

θ

is identifiablefrom



(

c1;

θ

)

,...,

(

cq;

θ

)

.Inparticular,wewillneedtoassumethatq≥ p.

For

θ

∈



,let

(

θ

)beasymmetric,positivedefiniteq× qmatrixanddefine

f_n,k

(

θ

)

:=DT

n,k

(

θ

)

(

θ

)

Dn,k

(

θ

)

. (5)

Thecontinuousupdatingweightedleastsquaresestimatorfor

θ

0isdefinedas



_θ

n,k:=argmin

θ∈ fn,k

(

θ

)

=argθ∈min



Dn,k

(

θ

)

T

(

θ

)

Dn,k

(

θ

)



. (6)

InSection 4,wewillstudytheperformance ofthisestimatorfor

(

θ

)

=Iq,theq× qidentity matrix.Expression(6) then

simplifiesto 

_θ

n,k=argmin θ∈ q m=1



_ n,k

(

cm

)

− 

(

cm;

θ

)

₂ .

Einmahletal.(2018) showtheconsistencyandasymptoticnormalityof

θ

_n,kundertheassumptionthatthetrue param-etervector

θ

0isintheinteriorof



.Inthenextsectionwegivetheasymptoticdistributionof

θ

n,kwithoutthisrestriction.

3. Inference on tail dependence parameters on the boundary of the parameter space

TheasymptoticresultspresentedinSection3.1 buildonAndrews(1999),whoestablishedtheasymptoticdistributionof ageneralextremumestimatorwhenoneormoreparameterslieontheboundaryoftheparameterspace.Themethodology to obtaintheasymptoticdistribution of√k

(



θ

n,k−

θ

0

)

canbe summarizedasfollows:first, oneshowsthat asn→∞,the

criterion functionf_n,k(

θ

) in(5) isequaltoaquadraticfunctionq_n,k

(

√k

(

θ

₋

θ

0

))

plus atermthat doesnot dependon

θ

.If



_θ

_n

,k isconsistent,then its asymptoticdistributionisonly affectedby thepartof



closeto

θ

0;equivalently,we areonly

concerned withthe shifted parameterspace



₋

θ

0 nearthe origin.If



−

θ

0 canbe approximated nearthe originby a

convex cone



, one can show that minimizing fn,k(

θ

) over

θ

∈



is asymptotically equivalentto minimizing qn,k(

λ

) over

InSection3.2,weshowhowaclosed-formexpressioncanbeobtainedfortheasymptoticdistributionof√k

(



θ

n,k−

θ

0

)

.

For

β

⊂

θ

,weshowinSection3.3 howtotestH0:

β

=

β

∗againstH1:

β

=

β

∗when,underthenullhypothesis,

β

∗isonthe

boundaryofthealternativehypothesis.

Wesetupsome notationfirst.LetB_ε(

θ

)denoteanopen ballcenteredat

θ

withradius

ε

andletC_ε(

θ

)denotean open cubecenteredat

θ

withsidesoflength2

ε

.Letcl(



)denotetheclosureof



.Aset



⊂ Rp_{issaidtobelocallyequaltoa}

set



_{⊂ R}p_if

_

_∩B

ε

(

0

)

₌



_∩B_ε

(

0

)

forsome

ε

_>0.Finally,aset



_{⊂ R}p_isaconeif

λ

_∈

_

_impliesa

λ

_∈

_

_foralla>0.

3.1. Estimation,consistencyandasymptoticnormality

When

θ

0isontheboundaryof



,themapLin(4) isnotdefinedandthusnotdifferentiableonaneighbourhoodof

θ

0.

Wewillneedthefollowingassumption.

(A1)



includesaset



+ suchthat



+−

θ

0 equalstheintersectionofaunionoforthantsandanopencube Cε(0)for

some

ε

_>0.Moreover,



_{∩ Bε1}

(

θ

0

)

⊂



+forsome

ε

1>0.

If



−

θ

0 happenstobelocallyequaltoaunionoforthants,wecansimplyset



+=



∩Cε

(

θ

0

)

.Thisisthecaseforthe

modelsconsideredinSection4.Wewillassume existenceoftheso-calledleft/right(l/r)partialderivativeson



+;aformal definitionisgivenintheappendix.Theshapeof



+issuchthatthesecanalwaysbedefined.

WriteL˙:=

(

∂

/

∂

θ

)

L

(

θ

)

∈Rq×p _for

_θ

_∈

_

+_,whereL˙ denotesthematrixofl/rpartialderivatives.Let

λ

1(

θ

)>0denotethe

smallesteigenvalueof

(

θ

).

Theorem 3.1 (Existence, uniqueness and consistency). Let c1,...,cq∈[0,∞

)

d be q≥ p points such that the map L:

θ

→

(



(

cm;

θ

))

q_m=1 is a homeomorphism. Let

θ

0∈cl(



) and assume that(A1) holds, that each element of L(

θ

) has continuous l/r

partialderivativesofordertwoon



+,thatL˙

(

θ

0

)

isoffullrank,that

:



→Rq×qhascontinuousl/rpartialderivativeson



+

andthatinf_θ∈

λ

1(

θ

)>0.Finallyassume,form=1,...,q,

_n,k

₍

cm

)

p

−→

(

cm;

θ

0

)

, asn→∞. (7)

Thenwithprobabilitytendingtoone,theminimizer

θ

n,kin(6)existsandisunique.Moreover,



_θ

_n

,k

p

−→

θ

0, asn→∞.

WeomittheproofofthistheoremsinceitisdirectlyobtainedbyreplacingB_ε(

θ

0)by



+ intheproofofEinmahletal. (2018,Theorem1).

Whenf_n,kisnotdefinedonaneighbourhoodof

θ

0,butthereexistsaset



+whichsatisfies(A1),aTaylorexpansionof

fn,k(

θ

)aroundfn,k(

θ

0)holds(Andrews,1999,Theorem6).Foreach

θ

∈



+,wehave f_n,k

(

θ

)

=f_n,k

(

θ

0

)

+D fn,k

(

θ

0

)

T

(

θ

−

θ

0

)

+

(

θ

−

θ

0

)

TD2fn,k

(

θ

0

)(

θ

−

θ

0

)

/2+Rn,k

(

θ

)

,

whereDf_n,kandD2_f

n,karebasedonl/rpartialderivativesandRn,k(

θ

)istheremainderterm.Wesuppressdependenceof

,

˙

LandDn,kon

θ

0foreaseofnotation.FromEinmahletal.(2018,ProofofTheorems3.2.1and3.2.2)weknowthat D fn,k

(

θ

0

)

=−2DnT,k

L˙+op

(

1

)

, D2fn,k

(

θ

0

)

=2L˙T

L˙+op

(

1

)

,

sothequadraticexpansionaboveisequalto

fn,k

(

θ

)

=fn,k

(

θ

0

)

− 2DTn,k

L˙

(

θ

−

θ

0

)

+

(

θ

−

θ

0

)

TL˙T

L˙

(

θ

−

θ

0

)

+Rn,k

(

θ

)

. Define J:=L˙T

_L˙∈Rp×p_{, Y} n,k:=



kJ−1L˙T

_D n,k∈Rp, and qn,k

(

λ

)

=

(

λ

− Yn,k

)

TJ

(

λ

− Yn,k

)

.

Thenfn,k(

θ

)canbewrittenas

fn,k

(

θ

)

=fn,k

(

θ

0

)

− k−1YTn,kJYn,k+k−1qn,k



k

(

θ

−

θ

0

)



+Rn,k

(

θ

)

.

Weseethatfn,k(

θ

)isequaltoaquadraticfunction plusatermthatdoesnot dependon

θ

plusRn,k(

θ

).InAndrews(2002,

Lemma3),itisshownthatundertheassumptionsofTheorem3.1 and(A3)below,theremaindertermRn,k(

θ

)issufficiently

smallsothatminimizing thequadraticapproximationtofn,k(

θ

)isequivalenttominimizingqn,k



√k

(

θ

−

θ

0

)

.Now,assume that

Then inf θ∈qn,k



k

(

θ

−

θ

0

)



= inf λ∈qn,k

(

λ

)

+op

(

1

)

,

forsomecone



.Finally,assumethat (A3)√kDn,k

(

θ

0

)

d

−→D∼ Nq

(

0,

(

θ

0

))

asn→∞,forsomecovariancematrix



(

θ

0).

Assumptions(A3)and(7) holdforthenonparametricestimatorspresentedinSection2.2.1 undersomegeneralregularity conditions.Thematrix



canbeexpressedintermsof_anditsfirst-orderpartialderivatives.Wehave

Yn,k d −→Y:=J−1L˙T

_{D, q} n,k

(

λ

)

d −→q

(

λ

)

:=

(

λ

− Y

)

T_J

₍

_λ

_{− Y}

₎

_, forall

λ

_∈Rp_.

Theorem 3.2 (Asymptoticnormality). Ifalloftheaboveassumptionshold,



k

(



θ

_n,k−

θ

0

)

d

−→

λ

=argmin

λ∈ q

(

λ

)

, asn→∞.

Here

λ

can beinterpretedastheprojectionofYon



withrespecttothe norm



y



J:=yTJy.Thistheoremisa special

caseofAndrews(1999,Theorem3).Intheappendix,weverifythatourassumptionsaresufficient.Notethatif



includes aneighbourhoodof

θ

0,then



=RpandwefindthesamedistributionasinEinmahletal.(2018,Theorem2)since

λ

=Y.

3.2. Simplifyingtheasymptoticdistribution

The goalofthissectionis tosimplifytheasymptoticdistribution of√k

(



θ

n,k−

θ

0

)

andtogive conditionsunderwhich

(part of) 

λ

has a closed-form expression. LetG:=_L˙T

_{D, i.e.,Y=J}−1_{G. We start by partitioningthe vector}

_θ

_0∈Rp _into

twosubvectors,

β

_0∈Rc _and

δ

0∈Rp−c forc∈

{

1,...,p

}

,where

δ

0 consistsofall parametersthatare intheinteriorofthe

parameterspace.Wepartition

θ

n,k,Y,J,Gand

λ

accordingly:

θ

n,k=



β

n,k

δ

n,k



, Y=



Y_β Y_δ



, J=



J_β J_βδ J_δβ J_δ



, G=



G_β G_δ



,

λ

=



λ

β

λ

δ



.

LetIcdenotethec× cidentitymatrix.ForH:=



Ic:0

∈Rc×p_,define q_β

(

λ

_β

)

:=

(

λ

β− Yβ

)

T



HJ−1HT

−1

(

λ

β− Yβ

)

. Supposethat

(A4)Thecone



ofassumption(A2)isequaltotheproductset



_β× Rp−c_,where

_

β⊂ Rcisacone. Corollary 3.3. Ifalloftheaboveassumptionshold,then



k

(



β

n,k−

β

0

)

d −→

λ

β:=argmin λβ∈β q_β

(

λ

_β

)

,



k

(



δ

n,k−

δ

0

)

d −→J_δ−1G_δ− J−1 δ Jδβ

λ

β.

When



_β isdefined byequality and/or inequality constraints,a closed-formexpressionfor

λ

_β canbe computed; we givesomeexamplesthatarerelevantforSection4.Foramoreformalsolution,seeAndrews(1999,Theorem5).

Example 3.1 ( c=1). Suppose that



=[0,1]p_{. If}

_β

0=0, then



β=[0,∞

)

and

λ

β=max

(

Yβ,0

)

. If

β

0=1, then



β=

(

−∞,0]and

λ

_β=min

(

Y_β,0

)

.

Example 3.2 ( c₌2). Supposethat



₌[0_,1]p_.If

β

0=

(

1,1

)

,then



β=

(

−∞,0]2.Let

ρ

:=HJ−1HT.Then



λ

β:=1



Y_β1<0,Yβ2<0



Y_β+1



Y_β1−

ρ

21Yβ2<0,Yβ2≥ 0



(

Y_β1−

ρ

21Yβ2,0

)

T +1



Y_β1≥ 0,Yβ2−

ρ

12Y_β1<0



(

0,Y_β2−

ρ

12Y_β1

)

T.

If

β

₀=

(

0,0

)

,

β

0=

(

0,1

)

or

β

0=

(

1,0

)

,similar expressionscan be obtainedbyreversing thefirst and/or thesecond

3.3.Hypothesistestingfortaildependenceparametersontheboundaryoftheparameterspace

Weareinterestedinconstructinghypothesistestsforparametervaluesthat,underthenullhypothesis,areonthe bound-aryofthealternativehypothesis.Weproposetwotest statistics,whoseasymptoticdistributionfollowsfromtheresultsin

Andrews(2001).

Assumethattherearenonuisanceparametersontheboundary,i.e.,all componentsof

θ

0 thatlieontheboundaryare

partofthenullhypothesis.Weareinterestedintesting

H0:

β

=

β

_∗ vs H1:

β

=

β

_∗,

β

_∗∈Rc.

Let



0:=

{

θ

∈



:

θ

=

(

β

∗,

δ

)

forsome

δ

∈Rp−c

}

denotetherestrictedparameterspace.Assumethat

(A5)Forall

θ

∈



0,



isaproductsetwithrespectto(

β

,

δ

)localto

θ

.Thatis,forall

θ

∈



0,



∩B_ε

(

θ

)

=

(

B×

)

∩B_ε

(

θ

)

forsomeB⊂ Rc_,

_

_{⊂ R}p−c_and

_ε

_>0.

Define

θ

(_n0_,k):₌argmin_θ∈

0 fn,k

(

θ

)

.Adevianceteststatisticcanbedefinedas

T_n(_,k1):=k



f_n,k

(



θ

n(0,k)

)

− fn,k

(



θ

n,k

)



.

Corollary 3.4. Suppose

θ

0∈



0 andallpreviousassumptionshold.Then T_n(_,k1)−→d 

λ

Tβ



HJ−1HT

−1

λ

β.

Recall that G=_L˙T

_{D andlet J denote its covariance matrix,i.e., G∼ N}

p

(

0 ,J

)

with J=L˙T



L˙∈Rp×p. Note that

J=J if and only if

₌



−1. If



_β=Rc _{(no parameters on the boundary) andJ =J, then}

T_n,k(1)−→d YT

β



HJ−1HT

−1Yβ∼

χ

c2,

where

χ

2

c denotesachi-squaredrandomvariablewithcdegreesoffreedom.

AWald-typeteststatisticcanbebasedonthequadraticformin

β

n,k−

β

∗.LetVn−1:=

(

HJn−1HT

)

−1∈Rc×cdenoteaweight

matrixwhereJn:=J

(



θ

n,k

)

.Define

T_n(_,k2):=k





β

_n,k−

β

∗



T_V−1 n





β

n,k−

β

∗



.

Corollary 3.5. Suppose

θ

0∈



0 andalloftheaboveassumptionshold.Then T_n(_,k2)−→d 

λ

T

β



HJ−1HT

−1

λ

β.

Corollary 3.4 is a special caseof Andrews(2001,Theorem 4c) andCorollary 3.5 is a special case ofAndrews (2001, Theorem6d);theirproofsaredirectandthusomitted.

4. Simulation studies

Weconduct simulation experimentswhere wesimulate 1000 samplesof sizen=5000froma parametric modeland weassessthequality ofthe weightedleastsquaresestimatorintermsofitsrootmeansquarederror(RMSE) forsettings whereoneormoreofthe parametersare ontheboundaryof theparameterspace.Wetake k∈

{

25,50,...,300

}

andwe usetheempiricalandthebetataildependencefunction asinitial estimators.Next,we studythe performanceofthetwo teststatisticsintroduced inSection3.3 intermsofempiricallevelandpowerfork∈{25,50,75,100}, aswe observethat higherkleadstoasteadilygrowingbiasthatquicklydeterioratestheperformanceofthehypothesistest.Inallexperiments, wetake

=Iq,theq× q identitymatrix,becauseusingan optimalweightmatrixhasa veryminoreffectonthe quality

ofestimationwhileseverelyslowingdowntheestimationprocedure.Moreover,inverting



maybehinderedbynumerical problemsinthecaseofamax-linearmodel(Einmahletal.,2018).

4.1. Brown–Resnickmodel

Wesimulatedatafroma Brown–Resnickmodelwith

α

=2and

ρ

=1onaregular 3× 2grid (d=6)andona regular 4_{× 4}grid(d₌16).AsinEinmahletal.(2018),weletcm∈{0,1}dsuchthatexactlytwocomponentsofcmareequaltoone

(meaningthat weconsider apairwise estimator)andwefocusonpairs ofneighbouringlocationsonly,i.e., locationsthat areatmostadistanceof√2apart.Thisleadstoq=11andq=42respectively.

Fig.1 showstheRMSEoftheparameterestimatesbasedontheempiricaltaildependencefunction(solidlines)andthe

betataildependencefunction(dashedlines).Weseethat theempiricaltaildependencefunctionoutperformsthebetatail dependencefunctionfor

α

,whiletheoppositeistruefor

ρ

,althoughdifferencesareveryminor.Forsmallvaluesofk,the RMSEislowerford=16thanford=6.

Table 1

Empirical level of T _n,k(2) _{based on the empirical and the beta tail dependence} function for a significance level of 0.05.

k = 25 k = 50 k = 75 k = 100

emp beta emp beta emp beta emp beta

d = 6 6.0 4.3 5.1 5.0 7.0 5.5 8.8 8.0 d = 16 6.7 10.3 8.3 10.2 9.8 12.4 13.2 16.0 50 100 150 200 250 300 0.00 0.10 0.20 d = 6 k RMSE of α 50 100 150 200 250 300 0.00 0.04 0.08 0.12 d = 6 k RMSE of ρ 50 100 150 200 250 300 0.00 0.10 0.20 d = 16 k RMSE of α 50 100 150 200 250 300 0.00 0.04 0.08 0.12 d = 16 k RMSE of ρ

Fig. 1. RMSE for parameter estimators of the Brown–Resnick model based on the empirical tail dependence function (solid lines) and the beta tail depen- dence function (dashed lines) for (α, ρ) = (2 , 1) ; d = 6 (left) and d = 16 (right).

1.5 1.6 1.7 1.8 1.9 2.0 02 0 4 0 60 80 100 d = 6, k = 25 α P o w e r in % 1.5 1.6 1.7 1.8 1.9 2.0 02 0 40 60 80 100 d = 6, k = 50 α P o w e r in % 1.5 1.6 1.7 1.8 1.9 2.0 02 0 4 0 60 80 100 d = 6, k = 75 α P o w e r in % 1.5 1.6 1.7 1.8 1.9 2.0 0 20 40 60 80 100 d = 6, k = 100 α P o w e r in % 1.5 1.6 1.7 1.8 1.9 2.0 02 0 4 0 60 80 100 d = 16, k = 25 α P o w e r in % 1.5 1.6 1.7 1.8 1.9 2.0 02 0 40 60 80 100 d = 16, k = 50 α P o w e r in % 1.5 1.6 1.7 1.8 1.9 2.0 02 0 4 0 60 80 100 d = 16, k = 75 α P o w e r in % 1.5 1.6 1.7 1.8 1.9 2.0 0 20 40 60 80 100 d = 16, k = 100 α P o w e r in %

Fig. 2. Empirical power of T _n,k(2) _{based on the empirical tail dependence function (solid lines) and the beta tail dependence function (dashed lines) for}

k ∈ {25, 50, 75, 100}; d = 6 (top) and d = 16 (bottom).

Next, we study the empirical level and power of the test H0:

α

=2. Assumtion (A4) holds with



=

(

−∞,0]× R. Table 1 showsthe empirical level of the test statistic T_n,k(2) based on a significance level of

α

=0.05. We do not display T_n,k(1)asitsbehaviourisidenticaltothat ofT_n,k(2).Surprisingly,theempiricaltaildependencefunctionispreferredford=16 whilethebetataildependencefunctionperformsbestford₌6.Ingeneral,wecanconcludethatlowvaluesofkaretobe preferredandthatthetestperforms betterforlower dimension.Fig.2 showstheempiricalpowerofT_n,k(2) asafunction of

α

∈[1.5,2).Thepowerincreasesbothwiththedimensionandwithk.

Table 2

Empirical level of T _n,k(1) _{and T n,k}(2) _{for models 1–3 (M1–M3) based on the empirical and the beta tail dependence function for a} significance level of 0.05.

T_n,k(1) k = 25 k = 50 k = 75 k = 100

emp beta emp beta emp beta emp beta

M1 2.9 2.1 6.1 4.2 5.9 5.3 8.0 6.4

M2 0.3 2.3 8.7 5.0 12.4 12.6 40.3 27.7

M3 3.3 7.9 7.9 9.7 6.5 9.8 9.8 10.0

T_n,k(2) k = 25 k = 50 k = 75 k = 100

emp beta emp beta emp beta emp beta

M1 3.0 2.9 6.2 5.0 6.1 6.1 8.5 7.1

M2 0.3 2.4 9.2 5.3 12.5 13.0 41.3 28.3

M3 5.4 9.9 10.3 10.2 8.4 10.7 10.7 11.1

4.2.Max-linearmodel

Theconstantsc1,. . .,cqneedtobechosensuchthatthemax-linearmodelisidentifiable.InEinmahletal.(2018) itwas

observedthattakingextremalcoefficientsforcm,i.e., cm∈{0,1}dwhichatleasttwonon-zeroelements,isnotenoughfor

identifiability.Instead,oneneedstochoosecm suchthatitsnon-zeroelementsareunequal.Forsome theoretical

consider-ations,see alsoEinmahletal.(2018,AppendixB). Simulationexperiments showedthatin practice,takingalarge gridof values(meaningthat q p) on [0,1]d _{willlead to goodestimators interms ofRMSE. Hence, inall following simulation}

experiments,wechoosec1,...,cqsuchthat

cm∈

{

0,0.01,0.1,0.2,...,0.8,0.9,0.99,1

}

2, m∈

{

1,...,q

}

.

4.2.1. Testingthestructureofamax-linearmodel Weconsiderthreedifferentscenarios:

Model 1 Letr=2andd=3,sothat



=[0,1]3_.Let

_θ

0=

(

b11,b21,b31

)

=

(

1,0.7,0.2

)

.WetestH0:b11=1.Assumption

(A4)holdswith



=

(

−∞,0]× R2_.

Model 2 Letr₌2andd₌3_,sothat



₌[0_,1]3_.Let

_θ

0=

(

b11,b21,b31

)

=

(

1,0.7,0

)

.WetestH0:

(

b11,b31

)

=

(

1,0

)

.

As-sumption(A4)holdswith



=

(

−∞,0]× [0,∞

)

× R. Model 3 Let r=3and d=2,so that



=[0,1]4_{. Let}

_θ

0=

(

b11,b21,b12,b22

)

=

(

0,0.8,0.6,0

)

. We test H0:

(

b11,b22

)

=

(

0_,0

)

.Assumption(A4)holdswith



₌[0_,∞

)

_{× R}2_{× [0,∞}

₎

_{.Ifthenullhypothesiscannotberejected,itmeansthat}

aMarshall–Olkinmodelsuffices.

Fig.3 showstheRMSEoftheparameterestimatesofthethreemax-linearmodelsexplainedabove,basedonthe

empiri-caltaildependencefunction(solidlines)andthebetataildependencefunction(dashedlines).Weseeagainthatthesetwo initialestimatorsleadtosimilarresults.Parameterswhosetruevaluesareontheboundaryarebetterestimatedforlowk, whilehighervaluesofkarepreferredforparameterswhosetruevaluesareintheinterioroftheparameterspace.

Table2 showstheempirical levelofthetest statisticsT_n,k(1)andT_n,k(2) fork∈{25,50,75,100},usinga significancelevel of0.05.The testsperformwell formodel 1ingeneral,while lowvaluesofk arenecessaryformodels 2and3.Thebeta taildependencefunctionoutperformstheempiricaltaildependencefunctionformodel2,whiletheoppositeisthecasefor model3.Overall,teststatisticT_n(_,k1) performssomewhatbetterthanT_n(_,k2).

Toassessthepowerofthetwoteststatistics,weconsiderthemodelsdescribedabove,butwelet

b11∈

{

0.8,0.85,...,1

}

formodel1,

b11∈

{

0.8,0.85,...1

}

,b31∈

{

0,0.05,...0.2

}

formodel2, and b11,b22∈

{

0,0.05,...0.2

}

formodel3.

Fig.4 showstheempiricalpowerofT_n(_,k1)formodels1–3forthebetataildependencefunction(resultsfortheempiricaltail

dependencefunction areverysimilar). Themoststriking resultofthelower-leftpanel isthatthegraphisnotsymmetric, i.e.,thepowerishigherwhenb22 isnearzero(andb11isnot)thanwhenb11 isnearzero(andb22 isnot).

4.2.2. Testingthenumberoffactorsinamax-linearmodel

Westudiedthe performanceof theteststatistics incasewewanted toidentifya specificsubmodel ofthemax-linear model.Wewouldalsoliketoinvestigatethecapabilityoftheteststatisticstocorrectlyretrievethetruenumberoffactors, i.e.,fors∈

{

1,...,r

}

,wewishtotestthehypothesisH0:

(

b1s,...,bds

)

=0 .However,asmentionedinSection2.1,the

max-linearmodelisonly definedforparameter valueswith d_j=1bjs>0foralls∈

{

1,...,d

}

;when thisconditionisnot met,

50 100 150 200 250 300 0.000 0.010 0.020 Model 1, b11 = 1 k RMSE of b 11 50 100 150 200 250 300 0.00 0.04 0.08 Model 1, b21 = 0.7 k RMSE of b 21 50 100 150 200 250 300 0.00 0.04 0.08 Model 1, b31 = 0.2 k RMSE of b31 50 100 150 200 250 300 0.000 0.010 0.020 Model 2, b11 = 1 k RMSE of b 11 50 100 150 200 250 300 0.00 0.04 0.08 Model 2, b21 = 0.7 k RMSE of b 21 50 100 150 200 250 300 0.000 0.010 0.020 Model 2, b31 = 0 k RMSE of b31 50 100 150 200 250 300 0.000 0.010 0.020 0.030 Model 3, b11 = 0 k RMSE of b 11 50 100 150 200 250 300 0.00 0.04 0.08 Model 3, b21 = 0.8 k RMSE of b 21 50 100 150 200 250 300 0.00 0.05 0.10 0.15 Model 3, b12 = 0.6 k RMSE of b12 50 100 150 200 250 300 0.00 0.01 0.02 0.03 0.04 Model 3, b22 = 0 k RMSE of b 22

Fig. 3. RMSE for models 1–3 based on the empirical tail dependence function (solid lines), and the beta tail dependence function (dashed lines).

Table 3

Empirical level of T (2)

n,k based on the empirical and the beta tail dependence function for a significance level of 0.05.

k = 25 k = 50 k = 75 k = 100

emp beta emp beta emp beta emp beta

T_n,k(2) 4.6 23.9 2.7 24.9 2.4 23.8 2.5 20.0

theasymptoticdistributionoftheteststatisticsusing

θ

n,kratherthan

θ

0.Weconsideramodelwithd=2andr=3,with

parametervector

θ

0=

(

b11,b21,b12,b22

)

=

(

0.8,0.6,0.2,0.4

)

,i.e.,thethirdcolumnofBcontainsonlyzeroesandthemodel

haseffectivelytwofactors.Weestimateathree-factormodelandwetestH0:

(

b13,b23

)

=

(

0,0

)

.

Fig. 5 showsthe RMSEof the parameterestimates based on the empiricaltail dependence function (solidlines) and

thebetataildependencefunction(dashedlines).Weseethat, contrarytoprevious experiments,thebetataildependence hasmuchhigherRMSEthantheempiricaltaildependencefunction:oversmoothingleadstoalargebiasfor(b13,b23)(and

hencefortheotherparametersaswell).

Table 3 showsthe empirical level ofthe test statistic T_n(_,k2) usinga significance level of 0.05. We included theresults basedonthebetataildependencefunctionforcompleteness:becauseittendstooverestimate(b13,b23),itrejectsthenull

hypothesisfartoooften.TeststatisticT_n(1)

,k isnotpresentedbecauseforanyk andanyinitial estimator,ithasanempirical

levelof0.Evenwhen

|



θ

n,k− 

θ

(0)

n,k

|

islarge,

|

fn,k

(



θ

n,k

)

− fn,k

(



θ

(0)

n,k

)

|

issmallandhencethedeviance-typetestisnotadapted

0.80

0.85

0.90

0.95

1.00

0

2

04

06

08

0

1

0

0

T

_n,k(1)

for k = 50 and k= 75

b

11

P

o

w

e

r in %

Power of Tn,k (1) based on beta stdf 0.80 0.85 0.90 0.95 1.000.00 0.05 0.10 0.15 0.20 0 20 40 60 80 100 b11 b31 power Power of Tn,k (1) based on beta stdf 0.00 0.05 0.10 0.15 0.20 0.00 0.05 0.10 0.15 0.20 0 20 40 60 80 100 b11 b22 power

Fig. 4. Top: empirical power in % for model 1 and k = 50 (dashed line), k = 75 (solid line), based on the empirical tail dependence function. Bottom: empirical power in % for model 2 (left) and model 3 (right) and k = 50 based on the beta tail dependence function.

50 100 150 200 250 300 0.00 0.10 0.20 0.30 b11 = 0.8 k rm se of b11 50 100 150 200 250 300 0.00 0.10 0.20 b21 = 0.6 k rm se of b21 50 100 150 200 250 300 0.00 0.05 0.10 0.15 0.20 b12 = 0.2 k rm se of b 12 50 100 150 200 250 300 0.00 0.05 0.10 0.15 b22 = 0.4 k rm se of b 22

Fig. 5. RMSE based on the empirical tail dependence function (solid lines) and the beta tail dependence function (dashed lines).

Weremarkthatthepowerissimilarwhenoneparameterisfarfromitsboundaryandwhenbothparametersarefarfrom theirboundary.

5. Application to stock market indices

ConsidertwomajorEuropeanstockmarketindices,theGermanDAXandtheFrenchCAC40.Wetakethedailynegative log-returnsofthepricesofthesetwoindicesfromhttps://finance.yahoo.com fortheperiodofJanuary1st,1997toDecember 31st,2017.Weremove all datesforwhich atleastone ofthetwo serieshasmissingvalues,endingup withasampleof

sizen=5313.Fig.7 showsthetimeseriesplotsoflog-returnsandthedependencestructure forthereturns standardized

Power of T

_n,k(2)

based on empirical stdf, k = 50

0.00 0.05 0.10 0.15 0.20 0.00 0.05 0.10 0.15 0.20 0 20 40 60 80 100 b13 b23 power

Power of T

_n,k(2)

based on empirical stdf, k = 75

0.00 0.05 0.10 0.15 0.20 0.00 0.05 0.10 0.15 0.20 0 20 40 60 80 100 b13 b23 power

Fig. 6. Empirical power in % of T _n,k(2) _{, based on the empirical tail dependence function for k = 50 (left) and k = 75 (right).}

2000 2005 2010 2015 −10 −5 05 1 0

Daily negative log−returns of the DAX index

Year retur ns in % 2000 2005 2010 2015 −10 −5 0 5 10

Daily negative log−returns of the CAC40 index

Year

retur

ns in %

0 2 4 6 8

02468

DAX versus CAC40

Fig. 7. Left and middle: time series plots of daily negative log-returns of the DAX and the CAC40; right: scatterplot of daily negative log-returns of the DAX versus the CAC40 on the unit Pareto scale, plotted on the exponential scale.

Table 4

Parameter matrix for a bivariate max-linear model with r = 4 factors for k = 40 ; standard errors are in parentheses.

DAX 0.41 (0.11) 0.14 (0.05) 0.43 (0.11) 0.03 (0.05)

CAC40 0.43 (0.12) 0.44 (0.10) 0.13 (0.05) 0.00 (0.01)

Dickey–Fullertest,whichgivesp-valuesbelow0.01forbothseries,andaKPSStest,givingp-valuesabove0.1forbothseries. Hence,theassumptionofstationarityisreasonable.Letujdenotethe0.95%quantileofX1j,...,Xn j for j∈

{

1,...,d

}

.Wefit

ageneralizedParetodistribution(GPD)toXi j− uj

|

Xi j>uj andobtaintheparameterestimates

σ

1=1.12(0.11),

γ

1=0.03

(0.08),

σ

2=1.10(0.10)and

γ

2=0.02(0.07).

Table4 showstheestimatedparametermatrixB=

(

bjt

)

j,t for j=1,2andt=1,2,3,4obtainedbyfittingafour-factor

max-linearmodeltoourdatabasedontheempiricaltaildependencefunction.Valuesofk∈

{

25,30,...,70,75

}

were con-sidered:wechose k=40becauseparameterestimatesarestablearoundthisvalue. Standarderrorswerecalculatedusing theasymptoticvariance matrixoftheestimator. TheresultsinTable4 suggestto testH0:

(

b14,b24

)

=

(

0,0

)

;thevalue of

theteststatisticT_n,k(2) is0.04.Comparingwithacriticalvalueof17.4basedonasignificancelevelof0.05,wefindthatwe cannot rejectthe nullhypothesis.Critical valuesarecalculated bysimulation fromtheasymptoticdistributionof thetest statistics,whichisgiveninCorollaries3.4 and3.5.WedonotconsiderT_n,k(1)orthebetataildependencefunctionbecauseof theirbadperformance(seeSection4.2.2).

Table 5

Parameter matrix for a bivariate max-linear model with r = 3 factors for k = 40 ; stan- dard errors are in parentheses.

DAX 0.41 (0.11) 0.14 (0.06) 0.46 (0.09)

CAC40 0.43 (0.12) 0.44 (0.10) 0.13 (0.05)

DAX versus CAC40

x y 0.2 0.4 0.6 0.8 1 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 200 150 100 50 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Tail dependence coefficient χ^n,k

k 4 6 8 10 12

1e−05

1e−03

1e−01

probability of a joint exceedance

return threshold

Fig. 8. Left: level sets {(x1 , x 2) :  (x1 , x 2) = c} for c ∈ {0.2, 0.4, 0.8, 1} based on the empirical tail dependence function (solid lines) and on the fitted max-linear model (dashed lines); middle: nonparametric (black dots) and model-based (red dots) estimates of the tail dependence coefficient  χn,k for

k ∈ { 200 , 195 , . . . , 30 } ; right: probability of a joint exceedance based on the fitted max-linear model (solid line) and on the empirical excesses (dashed line). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Table5 showstheestimatedparameter matrixB=

(

bjt

)

j,t for j=1,2andt=1,2,3forathree-factormodel.We test

ifa Marshall–Olkinmodel suffices,i.e., H0:

(

b12,b23

)

=

(

0,0

)

. Thevalue ofthe teststatistic T_n(_,k2) is75.9. Comparingto a

criticalvalueof8.03,werejectthenullhypothesisofaMarshall–Olkinmodel.

Fig.8 showstwogoodness-of-fitmeasuresandtheestimatedprobabilityofajointexceedancesofthetwostocks.Onthe

left,weplottedthelevelsets

{

(

x1,x2

)

:

(

x1,x2

)

=c

}

forc∈{0.2,0.4,0.8,1}basedontheempiricaltaildependencefunction

(solidlines)andonthefittedmax-linearmodel(dashedlines).

Acommonsummarymeasureofdependenceisthetaildependencecoefficient,

χ

(

q

)

:=lim

q↑1P[F1

(

X11

)

<q

|

F2

(

X12

)

<q]=2− 

(

1,1

)

,

see(2).Fig.8 (middle)showsnonparametric estimatesofthetaildependencecoefficient (black dots)

χ

n,k=2− n,k

(

1,1

)

anditsmodel-basedcounterpart(reddots)fordecreasingk.Thehorizontalgreylinecorrespondstothevaluek₌40that wasused forthe parameterestimatorsandthetests, correspondingto

χ

_n,k≈ 0.68,whichis equaltothemodel-based

χ

. Thedottedlinescorrespondto95%pointwisebootstrapconfidenceintervals.

TheprobabilityofajointexcessF

(

x1,x2

)

=P[X11>x1,X12>x2]canbeapproximatedforlarge(x1,x2)by F

(

x1,x2

)

=x∗1+x∗2− 1+exp

{

−

(

x∗1,x∗2

)

}

forx∗j=1− Fj

(

xj

)

, j=1,2.

LetH

(

· ;γj,

σ

j

)

denotethesurvivalfunctionofaGPDwithshape

γ

jandscale

σ

j.Forx1>u1 andx2>u2,wecanestimate

F

(

x1,x2

)

by _F

₍

_x 1,x2

)

=x1∗+x∗2− 1+exp



−

(

x∗₁,x∗₂;

θ

n,k

)



,

where

(

· ;

θ

n,k

)

denotes themax-linearstabletaildependencefunctionbasedonthefittedparameterestimatesandx∗j=

0.05× H

(

xj− uj; 

γ

j,

σ

j

)

.Theright-handplotofFig.7 showstheprobabilitiesF

(

x,x

)

onthelog-scaleforx∈

{

2.5,3,...,12

}

(solidline).Thesecanbecomparedtotheempiricalexcessprobabilitiesn−1 n_i=11

{

X1i>x,X2i>x

}

(dashedline),whichare

zeroforx>8.5. 6. Discussion

Weproposedtwoteststatisticsforparametersthat,underthenullhypothesis,areontheboundaryofthealternative hy-pothesis.Theseteststatisticsareconvenientbecausetheirasymptoticdistributionhasanexplicitexpression,andhavebeen showntoperformwellinsimulationstudies.Inpractice,their asymptoticdistributionbecomescumbersomewhentesting fora higherdimensional (c>4) parametervector. Apossible solution mightbe to consider a multipletesting procedure withaBonferroni-typeofcorrection.