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FinancialEconometrics1Part2:
VolatilityModellingandForecasting
Sigit Wibowo
April7, 2015
Contents1 Motivations 2
1.1 StylisedFactsinFinancialData . . . . . . . . . . . . . . . .
. 21.2 TypesofNon-linearModels . . . . . . . . . . . . . . . . . .
. 41.3 Non-linearityTests . . . . . . . . . . . . . . . . . . . . .
. . . 5
2 EWMA 72.1 EWMA Specication . . . . . . . . . . . . . . . . . .
. . . . . 72.2 TheAdvantages . . . . . . . . . . . . . . . . . . .
. . . . . . . 7
3 ARCH 73.1 ARCH Specication . . . . . . . . . . . . . . . . . .
. . . . . . 73.2 ARCH EffectTests . . . . . . . . . . . . . . . . .
. . . . . . . . 93.3 ProblemswithARCH()Models . . . . . . . . . . .
. . . . . . 10
4 GARCH 104.1 GARCH Specication . . . . . . . . . . . . . . . .
. . . . . . . 104.2 TheML Estimation . . . . . . . . . . . . . . .
. . . . . . . . . 12
5 ARCH/GARCH ModelExtensions 185.1 EGARCH Model . . . . . . . .
. . . . . . . . . . . . . . . . . . 195.2 GJR-GARCH Model . . . . .
. . . . . . . . . . . . . . . . . . . 195.3 GARCH-inMeanModel . . .
. . . . . . . . . . . . . . . . . . 21
1
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6 VolatilityForecasting 216.1 Overview . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 216.2 VolatilityForecasting . . . .
. . . . . . . . . . . . . . . . . . . 226.3
TheUseofVolatilityForecasts . . . . . . . . . . . . . . . . . .
236.4 TestingNon-linearRestrictions . . . . . . . . . . . . . . . .
. . 24
7 VolatilityEstimationUsingOxMetrics 25
References[1] ChrisBrooks IntroductoryEconometricsforFinance,
2ndedition. Cam-
bridgeUniversityPress, 2008.[2] Ronald J.Wonnacott
andThomasH.Wonnacott Econometrics, 2nd
edition. JohnWiley&Sons, Inc., 1979.
TimetableWeek Topics References8
Stylisedfactsofnancialtimeseriesvolatility [1], Ch. 89
ExponentialWeightedMovingAverage
ARCH &GARCH Models10 AsymmetricGARCH models [1], Ch. 811
IntegratedGARCH models
OtherunivariateARCH/GARCH models12
Volatilitymodelsusingnancialdata [1], Ch. 6, 713
ForecastingwithARCH/GARCH models [2], Ch. 1814
Review(studentpresentation*)
2
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1 Motivations1.1
StylisedFactsinFinancialDataNon-linearityFeaturesinFinancialData
Thelinearstructural(andtimeseries)modelscannotexplainanumberofimportantfeaturescommontomuchnancialdata
Leptokurtosisorfattails Volatilityclusteringorvolatilitypooling
Leverageeffects
Commontraditionalstructuralmodel = + + ... + + (1)
or = +
whereweassumedthat (0, )
Whatisvolatility? Volatilityissimplydenedasstandarddeviation
Varianceisoftenpreferredbecauseitmeasuresinthesameunitsasoriginaldata
Moredenition: Conditionalvolatility
=
1
=
( )
Realisedvolatility[+]
Impliedvolatility, e.g. Black-Scholesmodel: = (, , , , )
Annualisedvolatility, e.g.252
3
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FinancialAssetReturnsTimeSeries
2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014
201510
5
0
5rjkse
2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014
2015
5
0
5
10 rsnp500
Figure 1: DailyReturnsofJKSE andS&P500, 2002-2014
Non-linearModelsCampbell, LoandMacKinlay(1997)
A non-lineardatageneratingprocesscanbewritten = (, , , ...)
(2)
where isanon-linearfunctionand isaniiderrorterm
Morespecicdenition
= (, , ...) + (, , ...) (3)where
isafunctionofpasterrortermsonlyand isavarianceterm
Modelswithnonlinear () arenon-linearinmean,
whilethosewithnonlinear () arenon-linearinvariance
4
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1.2 TypesofNon-linearModelsTypesofNon-linearModels
Manynon-linearrelationshipscanbechangedintolinearusingaspe-cictransformation
However, manyrelationshipsinnanceareintrinsicallynon-linear
Manytypesofnon-linearmodels,
ARCH/GARCH switchingmodels bilinearmodels
1.3 Non-linearityTestsTestingforNon-linearity
Thetraditionaltoolsoftimeseriesanalysismayndnoevidencethatwecouldusealinearmodel,
butthedatastillmaynotbeindependent
Portmanteautestfornon-lineardependence RamseysRESET
testcanbeusedtotestnon-lineardependence:
= + + + ... + +
Othertests: theBDS testandthebispectrumtest
HeteroscedasticityRevisited
A structuralmodelcanbewrittenasfollows: = + + + +
where (0, ) Thevarianceoferrorsisassumedtobeconstantandisknownas
homoscedasticity
or () =
5
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Volatility Forecast
Models
Historical Standard
Deviation
ARCH Class
Conditional Volatility
Time Series Volatility
Forecasting
Stochastic Volatility
Option-based
Volatility Forecating
Ra
nd
om
Wa
lk
His
torica
l Ave
rag
e
Mo
vin
g A
ve
rag
e
Exp
on
en
tia
l S
mo
oth
ing
Sm
oo
th T
ran
sitio
n
Sim
ple
Re
gre
ssio
n
Au
to-R
eg
ressiv
e M
ovin
g
Ave
rag
e (
AR
MA
)
Th
resh
old
Au
to-R
eg
ressiv
eIn
teg
rate
d
Au
to-R
eg
ressiv
e M
ovin
g
Ave
rag
e (
AR
IMA
)
Fra
tio
na
lly I
nte
gra
ted
Au
to-R
eg
ressiv
e M
ovin
g
Ave
rag
e (
AR
FIM
A)
Exp
on
en
tia
lly W
eig
hte
d
Mo
vin
g A
ve
rag
e
AR
CH
Ge
ne
raliz
ed
AR
CH
Th
resh
old
or
GJR
AR
CH
Exp
on
en
tia
l AR
CH
Asym
me
tric
Po
we
r
GA
RC
H
Dia
go
na
l-V
ec M
GA
RC
H
Co
nsta
nt
Co
rre
latio
n
Ba
ba
, E
ng
le,
Kra
ft,
Kro
ne
r (B
EK
K)
Asym
me
try
Tim
e-v
ary
ing
Ris
k
Pre
miu
m
Mu
ltiv
aria
te G
AR
CH
GA
RC
H-in
-Me
an
Dyn
am
ic C
on
ditio
na
l
Co
rre
latio
n (
DC
C)
Qu
ad
ratic G
AR
CH
Figure 2: VolatilityClass, Source: Wibowo(2006)
6
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If () , thenheteroscedasticityexistsand
thestandarderrorestimatescouldbeincorrect
Fornancialdata,
thevarianceoftheerrorsisnotlikelytobeconstantovertime
2 EWMA2.1 EWMA
SpecicationEWMAExponentiallyWeightedMovingAverage
= (1 )=
( ) (4)
where denotestheestimateofthevarianceforperiod
andalsobecomes
theforecastoffuturevolatilityforallperiods
denotestheaveragereturnestimatedovertheobservations
denotesthedecayfactorwhichdetermineshowmuchweightisgiven
torecentversusolderobservation
2.2 TheAdvantagesEWMAExponentiallyWeightedMovingAverage
Twoadvantages: Volatilityislikelyinuencedbyrecentevents,
whichcarrymoreweights
Theeffectonvolatilityofasingleobservationdeclinesatanexpo-nentialrateasweightsattractedtorecenteventsfall
7
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3 ARCH3.1 ARCH
SpecicationARCHAutoregressiveConditionallyHeteroscedastic
Ifheteroscedasticityexists,
useamodelwhichdoesnotassumethevari-anceisconstant
Recallthedenitionofthevariance : = (|, , , ...)
= [( ())|, , ...]
Thevarianceisusuallyassumedtobe () = 0 = (|, , ...)
= [ |, , ...]
ARCHAutoregressiveConditionallyHeteroscedastic(contd)
Whatcouldthecurrentvalueofthevarianceoftheerrorspossiblyde-pendupon?
Previoussquarederrorterms Thisleadstothe AutoRegressive
Conditionally Heteroscedasticmodel
forthevarianceoftheerrors: = + (5)
Theequation (5) isknownasanARCH(1)model
ARCHAutoregressiveConditionallyHeteroscedastic(contd)
Thefullmodelcanbewrittenas = + + ... + + , (0, ) (6)
where = +
8
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TheARCH
modelcanbeextendedintothegeneralcasewheretheerrorvariancedependson
lagsofsquarederrors:
= + + + ... + (7)
Theequation (7) isknownasanARCH()model
ARCHAutoregressiveConditionallyHeteroscedastic(contd)
Inmanyliterature,
isusedtoaddressthevarianceoftheerrorsinsteadof
Therefore, = + + ... + + , (0, ) = + + + ... +
(8)
ARCHAutoregressiveConditionallyHeteroscedastic(contd)
Insteadof (8), wecanwrite = + + ... + + = (0, 1) = +
(9)
(8) and (9) aredifferentwaytoexpressexactlythesamemodel (8)
iseasiertounderstand (9) isrequiredtosimulateanARCH model
3.2 ARCH EffectTestsTestingARCH Effect
1. Runanypostulatedlinearregressione.g. = ++ ...
++,andthensavetheresiduals,
9
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2. Squaretheresiduals, andregressthemon
ownlagstotestforARCHoforder , forexample
= + + + ... + +
where isiid. Alsoobtain fromthisregression3.
Theteststatisticisdenedas orthenumberofobservationsmulti-
pliedbythecoefcientofmultiplecorrelationfromthelastregressionandisdistributedasa
()
TestingARCH Effect(contd)
4. Thenullandhypothesesare = 0 and = 0 and = 0 and ... and = 0 0
or 0 or 0 or ... or 0
Ifthevalueoftheteststatisticisgreaterthanthecriticalvaluefromthe
distribution, thenthenullhypothesisisrejected
Notethat theARCH
testisalsosometimesapplieddirectlytoreturnsinsteadoftheresidualsfromStage1
3.3 ProblemswithARCH()ModelsProblemswithARCH()Models
Howdowedecideon ? Therequiredvalueof mightbeverylarge
Non-negativityconstraintsmightbeviolated
Since thevariancecannotbenegative, we require > 0 =1, 2, ...,
toestimateanARCH model
A
naturalexpansionofanARCH()modelwhichgetsaroundsomeoftheseproblemsisaGARCH
model
10
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4 GARCH4.1 GARCH SpecicationGeneralisedARCH (GARCH) Models
Bollerslev(1986)proposedanewmodelwhichallowsvariancetobedependentuponpreviousownlag
= + + (10)whichisnowasaGARCH(1,1)model
Wecanalsowrite = + + = + +
Substitutinginto (10) for = + + + +
= + + + + (11)
GeneralisedARCH (GARCH) Models(contd)
Substitutinginto (11) for
= + + + + + + = + + + + + + = 1 + + + 1 + + +
Aninnitenumberofsuccessivesubstitutionswouldyield
= 1 + + + 1 + + +
Therefore, theGARCH(1,1)modelcanbewrittenasaninniteorderARCH
model
11
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GeneralisedARCH (GARCH) Models(contd)
WecanextendtheGARCH(1,1)modeltoaGARCH(, ): = + + + ... +
+ + + ... + +
= +=
+=
GeneralisedARCH (GARCH) Models(contd)
Ingeneral,
aGARCH(1,1)modelwillbesufcienttocapturethevolatil-ityclusteringinthe(nancial)data
WhyisGARCH betterthanARCH? moreparsimonious-avoidsovertting
lesslikelytobreachnon-negativityconstraints
GARCH SpecicationTheUnconditionalVariance
Theunconditionalvarianceof isgivenby
() =
1 ( + )(12)
when + < 1 Non-stationarityinvarianceisgivenby + 1
Theconditionalvarianceforecastswillnotconvergeontheirun-conditionalvalueasthehorizonincreases
IntegratedGARCH isgivenby + = 1
12
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4.2 TheML EstimationEstimationofARCH/GARCH Models
Becausethemodelisnon-linearform, OLS cannotbeused Therefore,
maximumlikelihoodtechniqueisutilised
Themethodworksbyndingthemostlikelyvaluesoftheparam-etersgiventheactualdata
Wespecicallyformalog-likelihoodfunctionandmaximiseit
EstimationofARCH/GARCH ModelsTheProcedure
1. Specify theappropriateequations for themeanand thevariance,
forexampleAR(1)GARCH(1,1)model:
= + + , (0, ) = + +
2. Specifythelog-likelihoodfunctiontomaximise:
= 2 log(2) 12
=
log( ) 12
=
( )
3.
Thecomputerwillmaximisethefunctionandgenerateparametervaluesandtheirstandarderrors
MaximumLikelihoodParameterEstimation
Forsimplicity,
letsconsiderthebivariateregressioncasewithhomoscedas-ticerrors:
= + + Assumethat (0, ) then ( + , ) Therefore, theprobability
density function for a normally distributed
randomvariablewiththemeanandvarianceisgivenby
| + , =1
2exp
12
(13)
13
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MaximumLikelihoodParameterEstimation(contd)
Theimplicationof (13) Successivevaluesof would traceout the
familiarbell-shapedcurve
Sincetheassumptionof isiid, then willalsobeiid
Thejointpdfforallthe scanbeexpressedasaproductoftheindividual
densityfunctions:
, , ..., | + , = | + ,
| + , ...
| + ,
==
| + , (14)
MaximumLikelihoodParameterEstimation(contd)
Substitutingintoequation (14) forevery fromequation (13)
, , ..., | + , =
1
2 exp
12
=
(15)
MaximumLikelihoodParameterEstimation(contd)
Thetypicalsituationwehaveisthatthe and
aregivenandwewanttoestimate , ,
14
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Ifthisisthecase, then () isknownasthelikelihoodfunction,
denoted(, , ), therefore
(, , ) =1
2 exp
12
=
(16)
Maximumlikelihoodestimationcomprisesofchoosingparameterval-ues(,
, )thatmaximisethisfunction
Wewanttodifferentiate (17) w.r.t. , , , but (17)
isaproductcon-taining terms
MaximumLikelihoodParameterEstimation(contd)
Since max () = max log(()), wecantakelogsof (17)
Usingthevariouslawsfortransformingfunctionscontaininglogarithms,
wegetthelog-likelihoodfunction,
= log 2 log(2) 12
=
( )
whichisequivalentto
= 2 log 2 log(2)
12
=
( ) (17)
MaximumLikelihoodParameterEstimation(contd)
Differentiating (17) w.r.t. , , , weget
= 12
=
( )2 1 (18)
= 12
=
( )2 (19)
=
21 +
12
=
( ) (20)
15
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MaximumLikelihoodParameterEstimation(contd)
Setting (18)-(20) to zero tominimise the functions, and putting
hatsabovetheparameterstodenotethemaximumlikelihoodestimators
From (18),( ) = 0
= 0 = 0
1
1 = 0
= (21)
MaximumLikelihoodParameterEstimation(contd)
From (19),( ) = 0
= 0 = 0
= = ( ) = +
=
=
(22)
MaximumLikelihoodParameterEstimation(contd)
16
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From (20), =
=
( )
= 1
=
( )
= 1 (23)
TheML andOLS Estimators HowdotheseformulaecomparewiththeOLS
estimators
(21) and (22) areidenticaltoOLS (24) isdifferentwheretheOLS
estimatorwas
= 1
TheML estimatorofthevarianceofthedisturbancesisconsistent,
butitisbiased
Howdoesthishelpusinestimatingheteroscedasticmodels?
EstimationofGARCH ModelsUsingML Nowwehave
= + + , (0, ) = + +
= 2 log(2) 12
=
log( ) 12
=
( )
However, theLLF
foramodelwithtime-varyingvariancescannotbemaximisedanalytically,
exceptinthesimplestofcases
Therefore,
anumericalprocedureisutilisedtomaximisethelog-likelihoodfunction
A potentialproblem:
localoptimaormultimodalitiesinthelikelihoodsurface
17
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LocalOptimainML Estimation
..
.
()
.A
. B.C
Figure 3: TheproblemoflocaloptimainML estimation
EstimationofGARCH ModelsUsingMLOptimisationProcedure
1. SetupLLF2.
Useregressiontogetinitialguessesforthemeanparameters3.
Choosesomeinitialguessesfortheconditionalvarianceparameters4.
Specifyaconvergencecriterion-eitherbycriterionorbyvalue
Non-NormalityandMaximumLikelihood
Recallthattheconditionalnormalityassumptionfor isessential
Thenormalitytestcanbeconductedasfollows:
= (0, 1)
= + + =
Thesamplecounterpartis =
18
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Typically arestillleptokurtic, althoughlesssothanthe
Thisisnotreallyaproblem, aswecanusetheML
witharobustvariance/covarianceestimator
ML
withrobuststandarderrorsiscalledQuasi-MaximumLikeli-hoodorQML
5 ARCH/GARCH ModelExtensionsExtensionstoARCH/GARCH Models
ThreeofthemostimportantARCH/GARCH modelextensions/variants:
EGARCH model GJR orTGARCH model GARCH-M model
ProblemswithGARCH(, )models
Non-negativityconstraintsmaystillbeviolated GARCH
modelscannotaccountforleverageeffects
5.1 EGARCH ModelTheEGARCH Model
Nelson(1991)proposedthevarianceequationcanbeexpressedasfol-lows:
log ( ) = + log () +
+
||
2
(24)
Theadvantagesofthemodel: Becausethemodelhas log( ),
willbepositiveevenifthepa-rametersarenegative
Themodelaccountsfortheleverageeffects:
iftherelationshipbe-tweenvolatilityandreturnsisnegative,
willbenegative
19
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5.2 GJR-GARCH ModelTheGJR-GARCH Model
Glosten, Jaganathan, andRunkle(1993)proposedGJR-GARCH (, ,
)model
A GJR-GARCH (1, 1, 1)canbewrittenasfollows: = + + + (25)
where =
1, if < 00, otherwise
A
GJR-GARCHmodelisconvariancestationaryifandonlyiftheparam-eterrestrictionsaresatisedand
+ + + < 1
Foraleverageeffect, > 0 + 0 and 0
isrequiredfornon-negativity
TheT-GARCH Model
Zakoian(1994)proposedTARCH(, , )model
TARCH(1,1,1)canbewrittenasfollows:
= + || + || + + 0
(26)
where =
1, if < 00, otherwise
TheGJR ModelNewImpactCurve
20
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..
-2
.
-1
.
0
.
1
.
2
.
0.2
.
0.3
.
0.4
.
0.5
.
0.6
.
0.7
.
0.8
..
.
GARCH
.
GJR-GARCH
5.3 GARCH-inMeanModelGARCH-inMean
Theideaistomodelthereturnofasecuritywhichispartlydeterminebyitsrisk
Engle, LilienandRobins(1987)proposedtheARCH-M specication = + +
(0, ) = + +
(27)
canbeinterpretedasasortofriskpremium It is possible to combine
all or part of thesemodels together to get
morecomplexhybridmodels, forexampleanARMA-EGARCH(1,1)-M
model
6 VolatilityForecasting6.1
OverviewWhatUseAreGARCH-typeModels
21
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GARCH
isabletomodelthevolatilityclusteringeffectbecausethecon-ditionalvarianceisautoregressive
Suchmodelscanbealsousedtoforecastvolatility Wecouldshowthat
|, , ... = (|, , ...)
Thereforemodelling willgiveusmodelsandforecastsfor aswell
Varianceforecastsareadditiveovertime
6.2 VolatilityForecastingForecastingVariancesUsingGARCH
Models
ConsiderthefollowingGARCH(1,1)model: = + (0, ) = + +
Weneedtogenerateforecastsof+| , +| , ..., +| (28)
where
denotesallinformationavailableuptoandincludingobser-vation
ForecastingVariancesUsingGARCH Models(contd)
Addingonetoeachofthetimesubscriptsoftheconditionalvarianceequationin
(28), andthentwo, andthenthreewouldproduce
+ = + + (29)+ = + + + + (30)+ = + + + + (31)
22
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ForecastingVariancesUsingGARCH Models(contd)
Let , betheonestepaheadforecastfor madeattime
, canbeobtainedbytakingconditionalexpectationof (29):, = + +
(32)
Given , , the2-stepaheadforecastfor madeattime
canbeob-tainedbytakingtheconditionalexpectationsof (30)
, = + (+| ) + , (33)
where (+| ) istheexpectation, madeatdate , of +,
whichisthesquareddisturbanceterm
ForecastingVariancesUsingGARCH Models(contd)
Weget(+| ) = + (34)
Since + isnotknownatdate , itisreplacedbytheforecastforit,,
Therefore, the2-stepaheadforecastisgivenby, = +
, +
,
= + + , (35)
ForecastingVariancesUsingGARCH Models(contd)
Usingsimilararguments, the3-stepaheadforecastwillbegivenby, = +
+ +
= + + ,
= + + + + ,
= + + + + , (36)
23
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Any -stepaheadforecast( 2)canwrittenas
, = =
+
+ +
, (37)
6.3 TheUseofVolatilityForecastsWhatUseAreVolatilityForecasts
Optionpricing = (, , , , )
Conditionalbetas, =
,,
Dynamichedgeratios
Thesizeofthefuturespositiontothesizeoftheunderlyingexpo-sure, i.e.
thenumberoffuturescontractstobuyorsellperunitofthespotgood
WhatUseAreVolatilityForecasts(contd)
Assumethattheobjectiveofhedgingistominimisethevarianceofthehedgedportfolio,
theoptimalvalueofthehedgeratio
= where
= hedgeratio =
correlationcoefcientbetweenchangeinspotprice()andchangeinfuturesprice()
= standarddeviationof = standarddeviationof
Fortime-varyingcovarianceandcorrelation, thehedgeratiois
= ,,
24
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6.4
TestingNon-linearRestrictionsTestingNon-linearRestrictionsTestingHypothesisaboutNon-linearModels
The and testsarestillvalidinnon-linearmodels,
butthesearenotexibleenough
Threehypothesistestingproceduresbasedonmaximumlikelihoodprin-ciples:
Wald LikelihoodRatio LagrangeMultiplier
Letsconsiderasingleparameter tobeestimated, betheMLE,and
bearestrictedestimate
LikelihoodRatioTests
Estimateunderthenullhypothesisandunderthealternative
ComparethemaximisedvaluesoftheLLF
Estimatetheunconstrainedmodelandachieveagivenmaximisedvalue
oftheLLF,denoted
Estimatethemodelimposingtheconstraint(s)andgetanewvalueof
theLLF,denoted Whichwillbebigger?
comparableto TheLR teststatisticisgivenby
= 2( ) ()
DiagrammaticRepresentation
25
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..
.
()
.
.
B
.()
.
.
A
.
( )
Figure 4:
Comparisonoftestingproceduresundermaximumlikelihood
7 VolatilityEstimationUsingOxMetrics
26
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27
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28
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2003 2004 2005 2006 2007
105
110
115
120
125 JPY
2003 2004 2005 2006 2007
2
1
0
1
2 RJPY
29
-
100.0 102.5 105.0 107.5 110.0 112.5 115.0 117.5 120.0 122.5
125.0 127.5
0.025
0.050
0.075
0.100
DensityJPY N(s=5.69)
2.5 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 2.5
0.5
1.0
1.5
2.0Density
RJPY N(s=0.44)
30
-
31
-
32
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2003 2004 2005 2006 20072
0
2 RJPY Fitted
2003 2004 2005 2006 20075
0
5r:RJPY (scaled)
2003 2004 2005 2006 2007
0.4
0.6CondSD
2007624 71 78 715
0.50
0.25
0.00
0.25
0.50 Forecasts RJPY
2007624 71 78 715
0.0750
0.0775
0.0800
0.0825
0.0850 CondVar Forecasts
33
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34
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35
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36
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References[1] DanielB.Nelson
GeneralizedAutoregressiveConditionalHeteroskedasticity.
JournalofEconomet-
rics, Vol.31, No.2(1991), 307?327.
[2] Robert F. Engle
AutoregressiveConditionalHeteroscedasticitywith Estimates of
theVariance ofUnitedKingdomInation. Econometrica, Vol.50,
No.4(1982), 987-1008
[3] RobertF.Engle, DavidM.LilienandRussellP.Robins
EstimatingTimeVaryingRiskPremiaintheTermStructure: TheArch-M Model.
Econometrica, Vol.55, No.2(1987), 391-407.
[4] LawrenceR.Glosten, RaviJagannathan, DavidE.Runkle
OntheRelationbetweentheExpectedValueandtheVolatilityoftheNominalExcessReturnonStocks.
TheJournalofFinance, Vol.48,No.5(1993), 1779-1801.
[5] DanielB.Nelson ConditionalHeteroskedasticityinAssetReturns:
A NewApproach. Econometrica,Vol.59, No.2(1991), 347-370.
37
1 Motivations1.1 Stylised Facts in Financial Data1.2 Types of
Non-linear Models1.3 Non-linearity Tests
2 EWMA2.1 EWMA Specification2.2 The Advantages
3 ARCH3.1 ARCH Specification3.2 ARCH Effect Tests3.3 Problems
with ARCH(q) Models
4 GARCH4.1 GARCH Specification4.2 The ML Estimation
5 ARCH/GARCH Model Extensions5.1 EGARCH Model5.2 GJR-GARCH
Model5.3 GARCH-in Mean Model
6 Volatility Forecasting6.1 Overview6.2 Volatility
Forecasting6.3 The Use of Volatility Forecasts6.4 Testing
Non-linear Restrictions
7 Volatility Estimation Using OxMetrics