Backtesting Value at Risk: From Dynamic Quantile to

Description

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

Sujets

• Dumitrescu
• Var (département)
• Maastricht University
• Violation
• Linear model
• Linear regression

Informations

• Publié par: profil-mupheg-2012
• Nombre de lectures: 42
• Langue: English

Extrait

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:[e-mail]
†
UniversityofOrle´ans,(LEO,UMRCNRS7322).Email:[e-mail].
‡
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