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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
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
%coveragerateforecastconditionallyonaninformati
