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Backtesting Value at Risk: From Dynamic Quantile to

De
33 pages
Niveau: Supérieur, Master
Backtesting Value-at-Risk: From Dynamic Quantile to Dynamic Binary Tests Elena-Ivona Dumitrescu?, Christophe Hurlin†, and Vinson Pham‡ February 2012 Abstract In this paper we propose a new tool for backtesting that examines the quality of Value-at- Risk (VaR) forecasts. To date, the most distinguished regression-based backtest, proposed by Engle and Manganelli (2004), relies on a linear model. However, in view of the di- chotomic character of the series of violations, a non-linear model seems more appropriate. In this paper we thus propose a new tool for backtesting (denoted DB) based on a dy- namic binary regression model. Our discrete-choice model, e.g. Probit, Logit, links the sequence of violations to a set of explanatory variables including the lagged VaR and the lagged violations in particular. It allows us to separately test the unconditional coverage, the independence and the conditional coverage hypotheses and it is easy to implement. Monte-Carlo experiments show that the DB test exhibits good small sample properties in realistic sample settings (5% coverage rate with estimation risk). An application on a portfolio composed of three assets included in the CAC40 market index is finally proposed. • Keywords : Value-at-Risk; Risk Management; Dynamic Binary Choice Models • J.

  • violation

  • no violation

  • can thus

  • conditional coverage

  • sample properties

  • linear model

  • linear regression

  • difference hypothesis


Voir plus Voir moins

BacktestingValue-at-Risk:FromDynamicQuantileto

DynamicBinaryTests

Elena-IvonaDumitrescu,

ChristopheHurlin,

andVinsonPham

February2012

Abstract

InthispaperweproposeanewtoolforbacktestingthatexaminesthequalityofValue-at-
Risk(VaR)forecasts.Todate,themostdistinguishedregression-basedbacktest,proposed
byEngleandManganelli(2004),reliesonalinearmodel.However,inviewofthedi-
chotomiccharacteroftheseriesofviolations,anon-linearmodelseemsmoreappropriate.
Inthispaperwethusproposeanewtoolforbacktesting(denoted
DB
)basedonady-
namicbinaryregressionmodel.Ourdiscrete-choicemodel,
e.g.
Probit,Logit,linksthe
sequenceofviolationstoasetofexplanatoryvariablesincludingthelaggedVaRandthe
laggedviolationsinparticular.Itallowsustoseparatelytesttheunconditionalcoverage,
theindependenceandtheconditionalcoveragehypothesesanditiseasytoimplement.
Monte-Carloexperimentsshowthatthe
DB
testexhibitsgoodsmallsampleproperties
inrealisticsamplesettings(5%coverageratewithestimationrisk).Anapplicationona
portfoliocomposedofthreeassetsincludedintheCAC40marketindexisfinallyproposed.


Keywords
:Value-at-Risk;RiskManagement;DynamicBinaryChoiceModels


J.E.LClassification
:C22,C25,C52,G28


Correspondingauthor:MaastrichtUniversityandUniversityofOrle´ans(LEO,UMRCNRS7322),Ruede
Blois,BP6739,45067Orle´ansCedex2,France.Email:elena.dumitrescu@univ-orleans.fr

UniversityofOrle´ans,(LEO,UMRCNRS7322).Email:christophe.hurlin@univ-orleans.fr.

UniversityofCaliforniaatSantaCruz(UCSA).VinsonPhambenefitedfromagrantfromtheEuropean
Program
Atlantis
AIME”ExcellenceinMobility”forhisvisitattheUniversityofOrle´ans.

1Introduction

Thereisanintenseacademicdebateonthevalidityofriskmeasuresingeneralandonthe

validityoftheValue-at-Risk(hereafterVaR)inparticular.Indeed,thisisaparticularproblem,

sincetheVaRisnotobservable,andthereforewehavetorelyupontheanalysisofthebehaviour

oftheviolationssoastotestitsvalidity.Aviolationisactuallydefinedasasituationwhere

thelossobservedex-postgoesbeyondtheex-antevalueoftheVaRinabsolutevalue.Amodel

ishencevalidiftheviolationprocesssatisfiesthemartingaledifferencehypothesis.

TherearethreemainissuesgenerallyemphasizedwhenonecomestoevaluatingVaRse-

quences.First,thepowerofthebacktestingtest,
theprobabilityofrejectingamodelthatisnot

valid
,especiallyinsmallsamples(250to500observations,or,toputitdifferently,1-2yearsof

VaRforecasts)playsakeyrole.Ithasbeenshownthatgenerallythesetestshavelowpower,as

thebacktestingprocedureistoooptimisticinthesensethatitdoesnotrejectthevalidityofa

modelasoftenasitshould(seeHurlinandTokpavi,2008).

Second,thebacktestingmethodologyhastobemodel-free.Indeed,theevaluationprocedure

mustbeimplementablewhateverthemodelusedtogeneratethesequenceofVaR,soasto

reachadiagnosticregardingthevalidityoftheVaR.Third,estimationriskmustbetakeninto

account.VaRseriescanbeestimatedusingvariousmodels,somemore,otherslesscomplicated,

withafewornumerousparameters,accordingtothespecificmethodologyofacertainfinancial

institution.TestingprocedurescanthussuccessfullyanswerthequestionofVaRvalidityonly

bytakingintoaccountestimationerror,astheriskofestimationerrorpresentintheestimates

oftheparameterspollutesVaRforecasts.Conditionalonallowingfortheseerrors,weshould

observenoparticularorientationofthediagnosticofthebacktestinthesenseofunder-rejecting

orover-rejectingtoooften.

Variousbacktestshavebeenproposedsoastosatisfythesethreerequirements(highpower,

model-free,introduceestimationrisk).Theycanbeclassifiedintofourcategories.First,in

thepioneerworksofChristoffersen(1998)thevalidityofVaRforecastsistestedthroughpa-

2
rameterrestrictionsonthetransitionprobabilitymatrixassociatedwithatwo-statesMarkov

chainmodel(violation/noviolation).Tobemoreprecise,twoassumptionsarederivedfrom

themartingaledifferencehypothesis,namelytheunconditionalcoverageandtheindependence

hypotheses.Second,testsrelyingonthedurationbetweentwoconsecutiveviolationsareput

forwardbyChristoffersenandPelletier(2004),Haas(2005)andCandelonetal.(2008)ina

likelihood-ratioframework.Atthesametime,themartingaledifferenceassumptionistested

directlybyBerkowitzetal.(2011),HurlinandTokpavi(2007)orPerignonandSmith(2008).

Lastbutnotleast,sometestsarebasedonregressionmodels(seeEngleandManganelli,2004).

ThegeneralideaistoprojectVaRviolationsontoasetofexplanatoryvariablesandsubse-

quentlytestdifferentrestrictionsontheparametersoftheregressionmodel,thatcorrespondto

theconsequencesofthemartingaledifferenceassumption.Insuchacontext,bothlinearand

non-linearregressionmodelscanbeconsidered.Forexample,therecentpaperofGaglianoneet

al.(2011)proposestoevaluatethevalidityoftheVaRbyrelyingonquantileregression,which

allowsthemtoidentifywhyandwhenaVaRmodelismisspecified.

Nevertheless,themostpopulartestofthiscategoryisEngleandManganelli’sDynamic

Quantiletest(2004),hereafter
DQ
.
1
Itconsistsintestingsomelinearrestrictionsinalinear

modelthatlinkstheviolationstoasetofexplanatoryvariables.However,thedependentvariable

isbynatureabinaryone.Itfollowsthatlinearregressionmodelsarenotthemostappropriate

choiceallowingtoinferontheparametersandconsequentlyonthehypothesisofvalidityofthe

VaR.Thelinearmodelhasseveralshortcomingsinthiscontext.Theinnovationsofthelatent

modelareassumedtofollowadiscretedistribution.Theyarealsoheteroscedasticinaway

thatdependsontheestimatedparameters.Atthesametime,constrainingtherightpartof

theregressiontothe0-1intervalimpliesnegativevariancesandnonsenseprobabilities.Still,

itistechnicallypossibletotestthesignificanceoftheslopeparametersinthecaseofabinary

dependentvariablebyrelyingonlinearmodels(seeGourieroux,2000).

InthispaperweproposeanewtoolforbacktestingVaRforecasts.LikeEngleandMan-

ganelli,weconsideraregressionmodelthatlinkstheviolationstoasetofexplanatoryvariables.
1
Notethatthe
DQ
backtestisnotrelatedtothequantileregressionmethodusedintheCAViaRmethodto
forecasttheVaR(EngleandManganelli,2004).

3
However,giventhedichotomiccharacteroftheseriesofviolations,weuseanon-linearmodel

and,morespecifically,aDynamicBinary(hereafter
DB
)regressionmodel.Theissueaddressed

inthispaperishencetheimprovementofthefinitesamplepropertiesofthebacktests,particu-

larlythepowerofthesetests,whenusingalinkfunctionthatismoreappropriateforthebinary

dimensionoftheregressand.Besides,thesenewtestsareexpectedtoberobusttoestimation

.ksir

Byproposingdynamicbinarymodels,whichrelyonrecentextensionsadvocatedinthe

EarlyWarningSystem
literature,thepotentialcorrelationbetweentheviolations(clusters)is

takenintoaccountintheestimation.Consequently,thetestsusedtoassesstheindependence

assumptionfortheviolationsandimplicitlytheonestestingtheconditionalcoveragehypothesis

areexpectedtoexhibithigherpowerthantheonespreviouslyproposedintheliterature.To

bemoreprecise,weproposesevendifferentspecifications,denotedby
DB
1
to
DB
7
,inspired
fromtheCAViaRspecificationsputforwardbyEngleandManganelli(2004).Thesubspaceof

explanatoryvariablesincludesseverallagsoftheviolationsseriesandoftheVaR,towhichthe

laggedindexisaddedinviewofthedynamicnatureofthemodels.Totesttheaccuracyofthe

VaRsequence,atwo-stepframeworkisthusimplemented.First,theseven
DB
specifications

areestimatedbyconstraintmaximum-likelihood(KauppiandSaikonnen,2008).Subsequently,

likelihood-ratiostatisticsareusedtoassessthejointsignificanceoftheparametersandthusthe

validityoftheVaR.

Notethatthistesthasseveraladvantages.First,itcanbeeasilyimplemented.Second,it

allowsustoseparatelytesttheunconditionalcoverage,theindependenceandtheconditional

coveragehypotheses.Third,Monte-Carloexperimentsshowthatbytakingintoaccountesti-

mationrisk,ourconditionalcoveragetestexhibitsgoodfinitesamplepropertiesinverysmall

samples(250observations)fora5%coveragerate.

AmainissueinVaRliteratureregardstheconsequencesofthepotentialcorrelationamongst

assetsontheconstructionofriskmeasures.WethusproposetotestthevalidityoftheVaRob-

tainedbyestimatingbothmultivariatemodels,
i.e.
modelsthattakeintoaccountthecorrelation

amongassetsandunivariatemodels,
i.e.
modelsthatdonotcareforthepossiblecorrelation

4
amongassets.Toachievethisaim,weconsideraportfolioconstitutedfromthreeassetsincluded
intheCAC40marketindexfortheperiodJune1,2007-June1,2009.Ourbacktestshows
thatthetwoapproachesleadustoriskmeasuresthatarevalidfromtheconditionalcoverage
hypothesisviewpoint.ThesefindingsgoalongthelinesofBerkowitzandO’Brien’sdiagnostic
(2002).
Therestofthispaperisorganizedasfollows.Section2presentsthetestingframework.
Insection3thebinaryregression-basedbacktestsarepresentedwhileinsection4theirsmall-
samplepropertiesaregauged.Section5revealsthemainresultsofanempiricalapplicationon
athree-assetillustrativeportfolio.

2Environmentandtestablehypotheses

Letusdenoteby
r
t
thereturnofanassetorofaportfolioattime
t
andby
VaR
t
|
t

1
(
α
)the
ex-
ante
VaRforan
α
%coveragerateforecastconditionallyonaninformationset
F
t

1
.Following
theactuarialconventionofapositivesignforaloss(seeGourierouxetal.,2000andScaillet,
2003,
interalii
),theconditionalVaRisactuallydefinedasfollows:

Pr[
r
t
<

VaR
t
|
t

1
(
α
)]=
α
,

t

Z
.

()1

Let
I
t
(
α
)bethebinaryvariableassociatedwiththe
ex-post
observationofan
α
%VaRviolation
attime
t
,
i.e.
:



1if
r
t
<

VaR
t
|
t

1
(
α
)
I
t
(
α
)=.


0otherwise

)2(

AsstressedbyChristoffersen(1998),VaRforecastsarevalidifandonlyiftheviolationprocess
I
t
(
α
)satisfiesthefollowingtwohypotheses:


Theunconditionalcoverage(UCthereafter)hypothesis:theprobabilityofan
ex-post
return
exceedingtheVaRforecastmustbeequaltothe
α
coveragerate,

Pr[
I
t
(
α
)=1]=
E
[
I
t
(
α
)]=
α
.

5(3)


Theindependence(INDthereafter)hypothesis:VaRviolationsobservedattwodifferent

datesforthesamecoverageratemustbedistributedindependently.Formally,thevari-

able
I
t
(
α
)associatedwithaVaRviolationattime
t
foran
α
%coveragerateshouldbe

independentofthevariable
I
t

k
(
α
),

k
6
=0.Inotherwords,pastVaRviolationsshould

notbeinformativeaboutcurrentandfutureviolations.

TheUChypothesisisastraightforwardone.Indeed,ifthefrequencyofviolationsobserved

over
T
periodsissignificantlylower(respectivelyhigher)thanthecoveragerate
α
,thenthe

VaRmodeloverestimates(respectivelyunderestimates)thetruelevelofrisk.However,the

UChypothesisshadesnolightonthepossibledependenceofVaRviolations.Therefore,the

independencepropertyofviolationsisanessentialone,becauseitisrelatedtotheabilityofa

VaRmodeltoaccuratelymodelthehigher-orderdynamicsofreturns.Infact,amodelwhich

doesnotsatisfytheindependencepropertycanleadtoclusteringsofviolations(foragiven

period)evenifithasthecorrectaveragenumberofviolations.Consequently,theremustbeno

dependenceintheviolationsvariable,whateverthecoveragerateconsidered.

WhentheUCandINDhypothesesaresimultaneouslyvalid,VaRforecastsaresaidtohave

acorrectconditionalcoverage(CCthereafter),andtheVaRviolationprocessisamartingale

difference:

E
[
I
t
(
α
)
|F
t

1
]=
α.

(4)

ThislastpropertyisatthecoreofmostofthebacktestsforVaRmodelsavailableintheliterature

(Christoffersen,1998;EngleandManganelli,2004;Berkowitzetal.,2011;etc.).

Actually,thisbacktestingapproach,basedontheUC,INDandCCassumptions,canbecon-

sideredasaspecialcaseoftheeventprobabilityforecastevaluationandcanbeimplemented

throughfourmaintypesoftests.Tobemoreexact,thefirstbacktestingtests(Kupiec,1995;

Christoffersen,1998)werebasedonaMarkovchaintypemodelwithtwostates(violation/no

violation).Inthisframework,theUC,INDandCCassumptionsaresimplytestedthrough

parameterrestrictionsonthetransitionprobabilitymatrixassociatedwiththeMarkovrepre-

sentation.Othertests(ChristoffersenandPelletier,2004;Haas,2005;Candelonetal.,2008)

relyonthedurationbetweentwoconsecutiveviolations.UndertheCChypothesis,thedura-

6
tionvariablefollowsageometricdistributionwithparameter
α
.Exploitingthisproperty,itis

straightforwardtodevelopalikelihoodratio(
LR
)testforthenullhypothesisofconditional

coverage.Thegeneralideaconsistsinspecifyingalifetimedistributionthatneststhegeometric,

sothatthememorylesspropertycanbetestedbymeansof
LR
tests.Third,someothertests

checkthemartingaledifferenceassumptiondirectly,throughitsmaincorrelationbasedimpli-

cations.Itis,forinstance,thecaseofBerkowitzetal.(2011),HurlinandTokpavi(2007)or

PerignonandSmith(2008).Finally,sometestsarebasedonregressionmodels.Inthiscontext,

themostpopularbacktestingtest,oftencalled
DQ
(DynamicQuantile)test,wasproposedby

EngleandManganelli(2004).Itisbasedonalinearregressionmodelofthehitsvariableona

setofexplanatoryvariablesincludingaconstant,thelaggedvaluesofthehitvariable,andany

functionofthepastinformationsetsuspectedofbeinginformative.
Moreformally,letusdenoteby
Hit
t
(
α
)=
I
t
(
α
)

α
thedemeanedprocessofviolation,
thattakesthevalue1

α
everytimes
r
t
islessthanthe
ex-ante
VaRand

α
otherwise.From
thedefinitionoftheVaR,theconditionalexpectationof
Hit
t
(
α,
)giventheinformationknown
at
t

1mustbezero.Inparticular,undertheCCassumption,thevariable
Hit
t
(
α
)must
beuncorrelatedwithitsownlaggedvaluesandwithanyotherlaggedvariable(includingpast

returns,pastVaR,etc.),anditsexpectedvaluemustbeequaltozero.Toputitanotherway,

theCCassumptioncanbetestedinthefollowinglinearregressionmodel:

(5)

KXHit
t
(
α
)=
δ
+
β
k
Hit
t

k
(
α
)
1=kKX+
γ
k
g
[
Hit
t

k
(
α
)
,Hit
t

k

1
(
α
)
,...,z
t

k
,z
t

k

1
,...
]+
ε
t
,
1=k

where
ε
t
isadiscrete
i.i.d.
processandwhere
g
(
.
)isafunctionofpastviolationsandofthe
variables
z
t

k
belongingtotheentireinformationalsetavailable
F
t

1
.Tobemoreexact,we
canconsidervariableslikepastreturns
r
t

k
,thesquareofpastreturns
r
t
2

k
,thepredicted
Value-at-Risk
VaR
t

k
|
t

k

1
(
α
)orevenimplicitvolatilitydata.Nevertheless,testingforthenull
hypothesisofconditionalefficiencyisequivalenttotestingthejointnullityofthecoefficients
β
k
,
γ
k
,

k
=1
,..,K,
andthatoftheintercept
δ
,independentoftheVaRmodelspecification

7
considered:
H
0
:
δ
=
β
1
=
...
=
β
k
=
γ
1
=
...
=
γ
k
=0
,

k
=1
,..,K

(6)

Atthesametime,presentviolationsoftheVaRarenotcorrelatedwithpastviolationsif
β
1
=
...
=
β
k
=
γ
1
=
...
=
γ
k
=0(asanimplicationoftheindependencehypothesis),whilethe
unconditionalcoveragehypothesisisfulfillediftheconstant
δ
isnull.Indeed,underthenull
hypothesis,
E
[
Hit
t
(
α
)]=
E
(
ε
t
)=0
,
whichmeansthatbydefinitionPr[
I
t
(
α
)=1]=
E
[
I
t
(
α
)]=
α.
Allinall,itresultsthatthetestofjointnullityofallcoefficientscorrespondstoaconditional
efficiencytest.
0Therefore,ifwedenotebyΨ=(
δ,β
1
,..,β
K

1
,..,γ
K
)thevectorofthe2
K
+1parameters
ofthemodelandby
Z
thematrixofexplanatoryvariablesofmodel(5),theteststatistic
designatedby
DQ
CC
andassociatedwiththeconditionalcoveragetest
2
satisfiesthefollowing
relation:

Ψ
b
0
Z
0
Z
Ψ
b
L
2
DQ
CC
=
α
(1

α
)
T




χ
(2
K
+1)

)7(

SimilartoChristoffersen’stest,the
DQ
testcanbedecomposedsothatwecanverify,for
example,onlythehypothesisofindependenceofviolations.Theteststatistic
DQ
IND
associated
withtheindependencehypothesis
H
0
:
β
k
=
γ
k
=0satisfies:

Ψ
b
0
R
0

R
(
Z
0
Z
)

1
R
0


1
R
Ψ
b
L
2DQ
IND
=
α
(1

α
)
T




χ
(2
K
)

)8(

0where
R
=[0:
I
2
K
]isa(2
K
+1
,
2
K
)matrixsothat
R
Ψ=
β,
where
β
=(
β
1
,..,β
K

1
,..,γ
K
)
.

3ADynamicBinaryResponsemodel

Asaforementioned,EngleandManganelli(2004)implementatestbasedonasimplelinearre-
gressionwhichlinksthecurrenthit(violation)tothelaggedviolations.Nevertheless,itiswell
knownthatthistypeofmodelisnotsuitablefordichotomousdependentvariables(Gourieroux,
2000)sincetheresidualsfollowadiscretedistributionandtheyareheteroscedasticbyconstruc-
2
Underthenullhypothesis,theresiduals
ε
t
correspondtotheviolationprocess
Hit
t
(
α
),whichfollowsa
Bernouillidistributionofparameter
α
andvariance
α
(1

α
)
.

8
tion.Additionally,thescatterplotoftheobservationscannotbecorrectlyadjustedwithasingle
lineasinthecaseofcontinuousvariables.
Asolutionconsistsinusingnon-linearregressionmodelssuitedforlimited-dependentvari-
ables.Indeed,weproposeadichotomicmodel(logit,probit,cloglog,...)toestablishalink
betweenthecurrentviolationsofVaRandthesetofexplanatoryvariables.Weactuallyexpect
thattheuseofthisappropriatenon-linearmethodologyforbinarydependentvariablesinthe
backtestingregressionwillimprovethefinitesamplepropertiesofthebacktestingtests.
ItshouldbenotedthatthebacktestingVaRtopicresemblesatdifferentlevelsthe
Early
WarningSystems
methodologywhichaimsatforecastingfinancialcrisesorrecessions.First,
theobjectofthestudyhasthesameform,sinceinbothcasesthedependentvariableisbinary,
e.g.
thelossisgreaterthentheVaRattime
t
ornot;thereisacrisisinacertainperiodor
not.Itfollowsthataspecialeconometricmethodologymustbeimplementedsoastoobtain
consistentestimators.Furthermore,bothevents(lossorcrisis)canbepersistentasitisnot
alwayspossibletoshiftfromlosstoprofitorfromcrisistocalminjustoneperiod.Thisdynamics
shouldbetakenintoaccountinaVaRbacktestregressionmodelsasithasbeenrecentlydonein
EarlyWarningSystems
byKauppiandSaikonnen(2008).Therefore,weexplicitlyimplementa
dynamicmodelwhosemainadvantageisthatitallowsustotestbothhypothesesofindependence
andconditionalcoverage.
Inthispartofthepaperournewmethodologybasedonadynamicbinaryresponsemodel
isintroduced.First,thedifferentmodelspecificationsconsideredarepresented.Then,the
estimationmethodisdevelopedandfinallythebacktestingteststatisticsareintroduced.

3.1ModelSpecification

Letusconsiderthefollowingdynamicbinaryresponsemodel,inwhichtheconditionalprobability
ofviolationattime
t
isgivenby:

Pr[
I
t
(
α
)=1
|F
t

1
]=
E
[
I
t
(
α
)
|F
t

1
]=
F
(
π
t
)
.

9(9)

where
F
(
.
)denotesac.d.f.(whichdoesnotnecessarilycorrespondtothedistributionofthe
returns)and
F
t

1
isasetofinformationavailableat
t

1.Weassumethattheindex
π
t
satisfies
thefollowingautoregressiverepresentation:

q
X
1
q
X
2
q
X
3
q
X
4
π
t
=
c
+
β
j
π
t

j
+
δ
j
I
t

j
(
α
)+
ψ
j
l
(
x
t

j

)+
γ
j
l
(
x
t

j

)
I
t

j
,
(10)
j
=1
j
=1
j
=1
j
=1

where
l
(
.
)isafunctionofafinitenumberoflaggedvaluesofobservables,and
x
t
isavectorof
explicativevariables.Theroleof
l
(
.
),istolinktheindex
π
t
totheobservablevariablesthat
belongtotheinformationset.Anaturalchoicefor
x
t

j
isgivenbythelaggedreturnsorthe
laggedVaR.
Consequently,weproposesevenspecifications,denotedby
DB
1
to
DB
7
,(where
DB
stands
for
DynamicBinaryModel
):

DB
1:
π
t
=
c
+
β
1
π
t

1
,
DB
2:
π
t
=
c
+
β
1
π
t

1
+
δ
1
I
t

1
(
α
)
,
DB
3:
π
t
=
c
+
β
1
π
t

1
+
δ
1
I
t

1
+
δ
2
I
t

2
(
α
)
,
DB
4:
π
t
=
c
+
β
1
π
t

1
+
δ
1
I
t

1
+
δ
2
I
t

2
+
δ
3
I
t

3
(
α
)
,
DB
5:
π
t
=
c
+
β
1
π
t

1
++
ψ
1
VaR
t

1
,
DB
6:
π
t
=
c
+
β
1
π
t

1
+
δ
1
I
t

1
(
α
)+
ψ
1
VaR
t

1
,
DB
7:
π
t
=
c
+
β
1
π
t

1
+
δ
1
I
t

1
(
α
)+
ψ
1
VaR
t

1
+
γ
1
VaR
t

1
I
t

1
.

)11()21()31(()41)51()61()71(

Thefirstfourspecificationscorrespondtoadynamicbinaryresponsemodelincludingthelagged
indexasanexplanatoryvariableandsomeadditionalinformationthroughtheobservedpast
valuesoftheviolationprocess.Thefifthandsixthmodelarederivedfromtheautoregressive
quantilespecificationsCAViaRusedbyEngleandMangenelli(2004),andmorespecificallyfrom
theirsymmetricabsolutevaluespecification.Tobemoreprecise,theindexisassumedtorespond
symmetricallytolaggedVaRvalues.Thelastspecification
DB
7
introducesanasymmetryin
theresponseoftheindextopastVaRaccordingtotheappearanceofaVaRviolation.

01

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