ekonometri 1

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

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

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

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

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

Volatility Forecast

Models

Historical Standard

Deviation

ARCH Class

Conditional Volatility

Time Series Volatility

Forecasting

Stochastic Volatility

Option-based

Volatility Forecating

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Figure 2: VolatilityClass, Source: Wibowo(2006)

6

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

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


Thefullmodelcanbewrittenas = + + ... + + , (0, ) (6)

where = +

8

TheARCH modelcanbeextendedintothegeneralcasewheretheerrorvariancedependson lagsofsquarederrors:

= + + + ... + (7)

Theequation (7) isknownasanARCH()model


Inmanyliterature, isusedtoaddressthevarianceoftheerrorsinsteadof

Therefore, = + + ... + + , (0, ) = + + + ... +

(8)


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

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

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


WecanextendtheGARCH(1,1)modeltoaGARCH(, ): = + + + ... +

+ + + ... + +

= +=

+=


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

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

MaximumLikelihoodParameterEstimation(contd)

Theimplicationof (13) Successivevaluesof would traceout the familiarbell-shapedcurve

Sincetheassumptionof isiid, then willalsobeiid Thejointpdfforallthe scanbeexpressedasaproductoftheindividual

densityfunctions:

, , ..., | + , = | + ,

| + , ...

| + ,

==

| + , (14)


Substitutingintoequation (14) forevery fromequation (13)

, , ..., | + , =

1

2 exp

12

=

(15)


Thetypicalsituationwehaveisthatthe and aregivenandwewanttoestimate , ,

14

Ifthisisthecase, then () isknownasthelikelihoodfunction, denoted(, , ), therefore

(, , ) =1

2 exp

12

=

(16)

Maximumlikelihoodestimationcomprisesofchoosingparameterval-ues(, , )thatmaximisethisfunction

Wewanttodifferentiate (17) w.r.t. , , , but (17) isaproductcon-taining terms


Since max () = max log(()), wecantakelogsof (17) Usingthevariouslawsfortransformingfunctionscontaininglogarithms,

wegetthelog-likelihoodfunction,

= log 2 log(2) 12

=

( )

whichisequivalentto

= 2 log 2 log(2)

12

=

( ) (17)


Differentiating (17) w.r.t. , , , weget

= 12

=

( )2 1 (18)

= 12

=

( )2 (19)

=

21 +

12

=

( ) (20)

15


Setting (18)-(20) to zero tominimise the functions, and putting hatsabovetheparameterstodenotethemaximumlikelihoodestimators

From (18),( ) = 0

= 0 = 0

1

1 = 0

= (21)


From (19),( ) = 0

= 0 = 0

= = ( ) = +

=

=

(22)


16

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

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

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

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

..

-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

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


Let , betheonestepaheadforecastfor madeattime

, canbeobtainedbytakingconditionalexpectationof (29):, = + + (32)

Given , , the2-stepaheadforecastfor madeattime canbeob-tainedbytakingtheconditionalexpectationsof (30)

, = + (+| ) + , (33)

where (+| ) istheexpectation, madeatdate , of +, whichisthesquareddisturbanceterm


Weget(+| ) = + (34)

Since + isnotknownatdate , itisreplacedbytheforecastforit,,

Therefore, the2-stepaheadforecastisgivenby, = +

, +

,

= + + , (35)


Usingsimilararguments, the3-stepaheadforecastwillbegivenby, = + + +

= + + ,

= + + + + ,

= + + + + , (36)

23

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

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

..

.

()

.

.

B

.()

.

.

A

.

( )

Figure 4: Comparisonoftestingproceduresundermaximumlikelihood

7 VolatilityEstimationUsingOxMetrics

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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

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

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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.

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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

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