Confidence Intervals for ARMA-GARCH Value-at-Risk

32 Pages·2013·0.22 MB·English

Confidence Intervals for ARMA-GARCH
Value-at-Risk
Laura Spierdijk
(cid:3)
May 31, 2013
UniversityofGroningen,FacultyofEconomicsandBusiness,DepartmentofEconomics,Econometricsand
(cid:3)
Finance,P.O.Box800,NL-9700AVGroningen,TheNetherlands.Email:[email protected]:+3150363
5929.Fax:[phone].
Confidence Intervals for ARMA-GARCH Value-at-Risk
Abstract
The quasi-maximum likelihood (QML) estimators of a GARCH model may not be
normallydistributedwhenthemodelerrorslackafinitefourthmoment.However,conven-
tional methods to obtain confidence intervals for GARCH-based Value-at-Risk estimates
rely on asymptotic normality. This study proposes a residual subsample bootstrap to esti-
mateconfidenceintervalsforQML-basedARMA-GARCHValue-at-Risk(VaR)estimates.
Simulation experiments show that this approach yields confidence intervals with correct
empirical coverage rates, even when conventional methods fail. An application to daily
stockreturnsillustratestheempiricalrelevanceoftheresidualsubsamplebootstrap.
Keywords:quasi-maximumlikelihood,heavytails,subsamplebootstrap
JELclassification:C22,C53,C58
1 Introduction
During the past decades Value-at-Risk (VaR) has become a standard tool in risk manage-
ment.Forexample,thecapitalrequirementsforinsurancecompaniesoperatingintheEuropean
Union,asprescribedbytheSolvencyIIDirective,arebasedonthe99.5%VaRoftheportfolio
ofassetsandliabilitiesmeasuredoveraone-yeartimehorizon(Sandstro¨m,2011).
The"%-VaRisdefinedasthe"%quantileofagivenprofit-and-loss(P-L)distribution.VaR
estimationrequirescertainassumptionsaboutthelatterdistribution.WhentheP-Ldistribution
of a portfolio of securities is the object of interest, it has become fairly standard to assume a
distribution of the Generalized Autoregressive Conditional Heteroskedasticity (GARCH) type
(Engle, 1982; Bollerslev, 1986) GARCH models capture an important stylized fact of asset
returnsknownasvolatilityclustering:high(absolute)returnstendtobefollowedbymorehigh
returns and low (absolute) returns tend to be followed by more low returns. Models of the
GARCH-typeareabletodealwiththisformofconditionalheteroskedasticity.
An attractive property of GARCH models is that consistent estimation of the model pa-
rameters does not require knowledge of the distribution of the model errors. Quasi-maximum
likelihood (QML) is based on the normal likelihood function and yields consistent parame-
ter estimates even when normality is violated, provided that certain regularity conditions hold
(HallandYao,2003;FrancqandZako¨ıan,2004).Consistent(butgenerallyinefficient)estima-
tion of the covariance matrix is possible using the sandwich estimator proposed by Bollerslev
andWooldridge(1992).
VaRestimatesdependontheestimatedparametersoftheunderlyingP-Ldistribution,which
are subject to estimation uncertainty. To judge the accuracy of a VaR estimate, it is crucial to
quantify this estimation uncertainty in terms of a confidence interval. When the confidence
interval around a VAR estimate is found to be wide, caution is required when decisions are to
bebasedontheinaccuratepointestimate.
Several methods can be used to quantify the parameter uncertainty associated with model-
basedVaRestimates.AmongthesemethodsaretheDeltamethod(Duan,2004)andtheresidual
bootstrap(Dowd,2002;ChristoffersenandGonc¸alves,2005).
Hall and Yao (2003) show that non-normality of the QML estimator arises in a GARCH
1
model when the fourth moment of the model errors is not finite and the distribution of the
squaredmodelerrorsisnotinthedomainofattractionoftheNormaldistribution.Itisobvious
that the Delta method is not appropriate in such a case. Furthermore, it is well known that the
conventional bootstrap is generally not consistent when the limiting distribution of a statistic
is not standard Normal after appropriate normalization (Athreya, 1987a,b; Knight, 1989; Hall,
1990; Mammen, 1992). If the model errors belong to the class of regularly-varying distribu-
tionswithproportionaltails,theasymptoticdistributionoftheextremeconditionalquantilesis
available under certain assumptions and can be used to obtain analytical confidence intervals
(Chanetal.,2007).Asusual,theseassumptionsincurtheriskofmisspecification.However,the
literature has not yet shown how to obtain confidence intervals with correct coverage without
additionalassumptions.
ThispaperproposesanewmethodtoobtainconfidenceintervalsforQML-basedVaResti-
mates,derivedfromARMA-GARCHmodelswithunknownerrordistribution.Nopriorknowl-
edgeaboutthedistributionofthemodelerrorsisassumed,apartfromsomeminimalregularity
conditions.Thefourthmomentoftheerrordistributionisallowedtobeeitherfiniteorinfinite.
TheresidualsubsamplebootstrapthatweproposeboilsdownthesubsamplebootstrapofSher-
manandCarlstein(2004),appliedtotheresidualsoftheARMA-GARCHmodel.Anattractive
feature of our approach is that it is based on an ‘omnibus’ procedure. Regardless of the distri-
bution and the moments of the model errors, the subsample bootstrap will produce confidence
intervalswithcorrectempiricalcoveragerates.
Given the empirical evidence for heavy-tailed asset returns (see e.g. McNeil and Frey
(2000))andassetreturnsthatlackafinitefourthmoment(Pagan,1996;Cont,2001;Embrechts
et al., 1997), there is a substantial risk that conventional methods will fail. Fortunately, the
residual subsample bootstrap provides an omnibus approach to obtain confidence intervals for
VaRestimates.
By means of a simulation study we show that the residual subsample bootstrap approach
yields confidence intervals with accurate coverage in all situations, but particularly when con-
ventionalmethodsfailduetoalackofasymptoticnormalityoftheQMLestimators.Theresid-
ual subsample bootstrap also performs satisfactory when sample sizes are relatively modest.
2
Rule-of-thumb choices for the length of the subsample, as proposed by Sherman and Carlstein
(2004), turn out to work well in an ARMA-GARCH context. We compare the coverage rates
of our method with those of the Normal approximation method of Chan et al. (2007). Like our
method, the latter approach does not rely on asymptotic normality of the QML estimator. Also
thiscomparisonturnsoutfavorablefortheconfidenceintervalsbasedontheresidualsubsample
bootstrap.
Application of the residual subsample bootstrap when the VaR estimator is asymptotically
normally distributed results in confidence intervals with correct empirical coverage rates. But
these confidence intervals are relatively wide: Inefficiency is the price that has to be paid for
the omnibus nature of the residual subsample bootstrap. Because we prefer to avoid such inef-
ficiencies,werecommendsomesimpleteststoassessthefinitenessofthemodelerrors’fourth
moment and the shape of the VaR estimator’s asymptotic distribution. Given the outcomes of
these tests, the researcher can choose between one of the conventional methods relying on
asymptoticnormalityoftheQMLestimatorsandthe‘omnibus’subsamplebootstrap.Weillus-
trate these tests in the empirical part of the paper, where we estimate VaR for heavy-tailed and
skewedstockreturns.
The setup of the remainder of this paper is as follows. Section 2 describes how VaR is
estimated using the QML estimator of an ARMA-GARCH model. The residual subsample
bootstrapisdiscussedinSection3.Bymeansofsimulationthissectionevaluatestheempirical
coverage rates of the residual subsample bootstrap’s confidence intervals. Section 4 discusses
alternative methods to estimate confidence intervals for VaR estimates, followed by more sim-
ulations. Section 5 uses daily stock returns to estimate VaR and subsequently compares the
confidenceintervalsobtainedbytheresidualsubsamplebootstrapandthealternativemethods.
Finally, Section 6 concludes. A separate appendix with supplementary background material is
available.
3
2 Estimation of Value-at-Risk
ARMA-GARCH specifications are location-scale models belonging to the class of time-series
models with GARCH errors (McNeil and Frey, 2000; Li et al., 2002). Alternative VaR models
havebeenstudiedbye.g.EngleandManganelli(2004)andKoenkerandXiao(2006).
ForthesakeofexpositionweconfinetheanalysistotheARMA(1,1)-GARCH(1,1)model,
but we emphasize that our approach can be applied to ARMA-GARCH models of any order.
WethusconsiderthefollowingARMA(1,1)-GARCH(1,1)modelfor t 1;:::;n:
D
r (cid:22) (cid:27) z (1)
t t t t
D C I
(cid:22) (cid:22) (cid:13)r (cid:18).r (cid:22) / (2)
t t 1 t 1 t 1
D C (cid:0) C (cid:0) (cid:0) (cid:0) I
(cid:27)2 ! (cid:11).r (cid:22) /2 (cid:12)(cid:27)2 (3)
t D C t(cid:0)1 (cid:0) t(cid:0)1 C t(cid:0)1I
z D.0;1/ iid; (4)
t
(cid:24)
where D.0;1/ denotes a distribution with mean 0 and variance 1. Covariance-stationarity is
imposed by assuming that (cid:13) < 1, ! > 0, (cid:11) > 0, (cid:12) > 0, and (cid:11) (cid:12) < 1. We also make the
j j C
standardassumptionthatIE.z2/ < (Chanetal.,2007).
t 1
Given a times series sample r ;:::;r , we estimate the ARMA-GARCH model by means
1 n
ofQML,yieldingthevectorofestimatedmodelparameters(cid:21) .(cid:22);(cid:13);(cid:18);!;(cid:11);(cid:12)/.Wereferto
O O O
D O O O O
z .r (cid:22) /=(cid:27) asthe‘modelerrors’orthe‘standardizedreturns’.
t t t t
D (cid:0)
Because our main interest goes to losses, we focus on the left tail of the P-L distribution.
Weconsiderthe "%one-stepaheadconditionalVaR,denoted(cid:24)" andsatisfying
nn 1
j C
IP.r < (cid:24)" I / "=100; (5)
nC1 njnC1 j n D
where I denotes all information available up to time n. Conditional on I , the "% quantile of
n n
thereturndistributionequals
(cid:24)" (cid:22) (cid:27) Q"; (6)
njnC1 D nC1 C nC1 z
where Qy denotes the y% quantile of the standardized returns z . A point estimate of (cid:24)" is
z t nn 1
j C
obtainedbyreplacing (cid:22) ,(cid:27) andQ" bytheirforecastedcounterparts:
n 1 n 1 z
C C
(cid:24)" (cid:22) (cid:27) Q": (7)
bnjnC1 D OnC1 C OnC1 Oz
4
We consider static forecasts of (cid:22) and (cid:27) , which are obtained by iterating Equations (2)
n 1 n 1
C C
and (3) using (cid:21) and the observed values of r ;:::;r . We estimate the quantiles of the stan-
O 1 n
dardized return distribution from the empirical distribution of z .r (cid:22) /=(cid:27) , yielding the
t t t t
O D (cid:0) O O
sample quantiles Q" (Christoffersen and Gonc¸alves, 2005; Goa and Song, 2008). Because the
Oz
resultingVaRestimateissubjecttoparameteruncertaintyvia(cid:22) ,(cid:27) ,andQ",itisimportantto
Ot Ot Oz
provideaconfidenceintervalinadditiontothepointestimate.
Awell-knownlimitationofVaRasariskmeasureitthatitdoesnotsatisfythesub-additivity
property. Although other risk measures, such as expected shortfall, may be preferable from a
theoreticalpointofview,VaRremainsamongthemostwidelyusedriskmeasures.Wetherefore
limitouranalysistoVaR,butweemphasizethatitisstraightforwardtoextendtheapproachto
alternativeriskmeasures.
3 Residual subsample bootstrap
The goal of this section is to show that the residual subsample bootstrap yields confidence
intervals with correct coverage rates for a wide range of ARMA-GARCH specifications, but
particularly when the ARMA-GARCH errors lack a finite fourth moment. We start with a
descriptionoftheresidualsubsamplebootstrap,followedbysomesimulationresults.
3.1 Methodology
Given a possibly dependent sample r ;:::;r , Sherman and Carlstein (2004) consider a gen-
1 n
eral statistic s s .r ;:::;r / as a consistent point estimator of a target parameter s. Their
n n 1 n
D
subsample bootstrap assumes that a .s s/ converges in distribution for n and some
n n
(cid:0) ! 1
sequencea ,butdoesneitherrequireknowledgeof.a /norofthelimitingdistribution.
n n
! 1
The ‘omnibus’ nature of the subsample bootstrap makes this method attractive for confidence
intervalestimation.
Threeminimalconditionsforconsistencyarerequired,whichboildownto(1)theassump-
tionthatthelimitdistributionofa .s s/exists,(2)acoupleofstandardassumptionsregard-
n n
(cid:0)
ingthesubsamplesize‘asafunctionofthesamplesizen,and(3)amixingcondition(Sherman
5
and Carlstein, 2004, Conditions R.1 – R.3, p. 127). In general, non-replacement subsampling
methodsworkbecausetheresultingsubsamplesarerandomsamplesofthetruepopulationdis-
tribution instead of estimators of the latter distribution. The formal proof that the subsample
bootstrapofShermanandCarlstein(2004)yieldsasymptoticallycorrectcoverageprobabilities
reliesheavilyontherelatedworkofPolitisandRomano(1994);Bertailetal.(1999)andisnot
repeatedhere.
We propose a residual subsample bootstrap for estimating VaR confidence intervals. This
bootstrap is an application of the subsample bootstrap of Sherman and Carlstein (2004) to
the ARMA-GARCH model residuals z , which are assumed to be iid. Obviously, the mixing
t
O
condition is satisfied when the subsample bootstrap is applied to iid residuals. Sherman and
Carlstein (2004, p. 128) describe how to adjust their subsample bootstrap to non-dependent
data. It is evidently not possible to use all (cid:0)n(cid:1) subsamples of size ‘ from the n residuals. We
‘
therefore randomly draw B 1000 of such subsamples (Politis and Romano, 1994). This
D
choice of B will turn out adequate in Section 3.3, in the sense that a further increase in B has
onlyaminorimpactontheconfidenceintervals.
AfterfittinganARMA-GARCHmodeltotheobservedsampler ;:::;r bymeansofQML
1 n
(resulting in the vector of estimated ARMA-GARCH parameters (cid:21)), the residual subsample
O
bootstrap consists of the following steps. A random subsample of length ‘ is drawn with-
out replacement from the estimated ARMA-GARCH model’s standardized returns z ;:::;z ,
1 n
O O
yielding z ;:::;z . Next, (cid:21) and Equations (1) – (3) are used to recursively generate values
1 ‘ O
Q Q
r ;:::;r . The bootstrap sample r ;:::;r is used to estimate the ARMA-GARCH model by
1 ‘ 1 ‘
Q Q Q Q
means of QML, yielding (cid:21). Subsequently, (cid:21) and the observed r ;:::;r are used to statically
Q Q 1 n
calculate (cid:22) and (cid:27) according to Equations (2) – (3). Q" is defined as the "% sample quantile
Qn Qn Qz
ofz .r (cid:22) /=(cid:27) andusedasabootstrapestimatorofQ".Finally,(cid:24)" iscalculatedfrom
Qt D t (cid:0) Qt Qt z enn 1
j C
Equation(7)using(cid:22) ,(cid:27) ,andQ".
Qn Qn Qz
TheabovestepsarerepeatedB times,resultingin(cid:24)" ;:::;(cid:24)" .Theproposedx%
enn 1;1 enn 1;B
j C j C
confidenceintervalhastheform
h(cid:24)" Q.1 x/=2;(cid:24)" Q.1 x/=2i: (8)
enn 1;1 (cid:0) 1 C enn 1;1 (cid:0) 1 (cid:0)
j C j C
6
HereQy isdefinedastheempiricaly%quantileof
1
(cid:24)" (cid:24)" ;:::;(cid:24)" (cid:24)" ; (9)
enn 1;1 (cid:0)bnn 1 enn 1;B (cid:0)bnn 1
j C j C j C j C
with (cid:24)" as defined in Equation (7). As noted by Sherman and Carlstein (2004), the choice
bnn 1
j C
of(cid:24)" in(8)isarbitraryandcanbereplacedbyany(cid:24)" ,i 2;:::;B.
enn 1;1 enn 1;i D
j C j C
ShermanandCarlstein(2004)provethatthecoveragerateoftheaboveconfidenceinterval
is asymptotically correct. They also show that the width of the confidence interval converges
in probability to zero for n (under their standard assumptions ‘ , ‘=n 0
! 1 ! 1 !
and a =a 0). Intuitively, a larger subsample size results in bootstrap estimators that are
‘ n
!
based on more data. Estimators based on more data are subject to less parameter uncertainty,
resulting in a smaller expected confidence interval width. On the other hand, with a larger
subsample size the subsamples tend to overlap more. At the same time, (cid:24)" and (cid:24)"
enn 1;1 bnn 1
j C j C
becomemoredependent.Thesetwoeffectsresultinahighercoverageprobabilityerror.Hence,
there is a trade-off between expected confidence interval width and coverage probability error.
Onthebasisofasimulationstudy,ShermanandCarlstein(2004)recommendasubsamplesize
between (a constant times) n1=2 and (a constant times) n2=3. Throughout, we will consider the
subsample sizes ‘ n2=3, ‘ .n1=2 n2=3/=2, and ‘ n1=2 (all three rounded to the
1 2 3
D D C D
nearestinteger).
In the introduction we already mentioned the possible inefficiency of the subsample boot-
strap. We can make this statement more formal here. The price that is paid for the omnibus
nature of the subsample bootstrap is that the resulting confidence intervals typically have a
length of order a 1 instead of a 1 (where ‘ denotes the subsample size and n the full sample
(cid:0)‘ (cid:0)n
size).WewillcomebacktothisissueinSection4.1.
3.2 Simulation setup
WestartwithanARMA(1,1)-GARCH(1,1)modelwithparameters(cid:21) .0;0:5;0:3;0:01;0:1;0:8/,
D
reflectingrelativelyhighpersistenceinboththeARMAandtheGARCHpartofthemodel.The
distributionofthestandardizedreturnsischosentobeNormal,Student’st,orskewedStudent’s
t (Fernandez and Steel, 1998). In all cases the distribution of the standardized returns is scaled
7
in such a way that it has mean 0 and variance 1. The skewed Student’s t distributionhas skew-
ness parameter (cid:24) 0:5, reflecting negative skewness. The shape parameter (reflecting the
D
numberofdegreesoffreedom)equals(cid:23) 3forbotht distributions.Thet distributionswith3
D
degreesoffreedomhaveafinitesecondmoment(asrequired),butinfinitefourthmoment.
Foragivendistributionofz ,werandomlydrawvaluesz ;:::;z andrecursivelysimulate
t 1 n
N 1000 time series r ;:::;r on the basis of Equations (1) – (3). We estimate (cid:24)" as
D 1 n nn 1
j C
outlined in Section 2 and obtain the associated 90% confidence intervals using the residual
subsample bootstrap of Section 3.1. We calculate the true value (cid:24)" using the true values of
nn 1
j C
(cid:22) and(cid:27) .ThetruevalueofQ" isdefinedasthe"%samplequantileofz ;:::;z .Wedonotuse
t t z 1 n
the true "% quantile of z ’s distribution (i.e., the Normal, t or skewed t distribution), because
t
we want to assess the empirical coverage rate of (cid:24)" exclusive of sampling uncertainty. The
bnn 1
j C
empirical coverage probability is calculated as the fraction of simulations for which the true
value of (cid:24)" falls within the estimated confidence interval. We consider the sample sizes
nn 1
j C
n 300;500;1000 and 5000. The separate appendix with supplementary material provides
D
furthercomputationaldetails.
3.3 Simulation results
Theempiricalcoverageprobabilitiesfor" 5%aredisplayedinthefirstpanelofTable1(cap-
D
tioned ‘Case 1’). This table shows that the subsample bootstrap yields coverage probabilities
closeto90%foralldistributions.
The coverage rates depend on the selected subsample size, but each of the three chosen
subsample sizes results in reasonably accurate coverage rates. For n 1000, the subsample
D
sizes ‘ 100 and ‘ 66 yield the best coverage rates. For n 5000, the most accurate
1 2
D D D
coverage rates are obtained for the subsample sizes ‘ 182 and ‘ 71. For n 300 500,
2 3
D D D I
we only consider the subsample sizes ‘ 45 65 and ‘ 31 43. Even for these modest
1 2
D I D I
samplesizes,thecoverageratesareatleast0.86withthemostfavorablesubsamplesize.
We run a wide range of robustness checks based on different ARMA-GARCH specifica-
tions, involving lower persistence, positive skewness, and t distributions with 4 or 5 instead of
3degreesoffreedom.Wealsoestimate1%and10%VaRinadditiontothe5%VaR.The
8