introductory econometrics for finance Chapter8solutions.docx

introductory econometrics for finance Chapter8solutions.docx

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introductoryeconometricsforfinanceChapter8solutions

SolutionstotheReviewQuestionsattheEndofChapter8

1.（a）.AnumberofstylisedfeaturesoffinancialdatahavebeensuggestedatthestartofChapter8andinotherplacesthroughoutthebook:

-Frequency:

Stockmarketpricesaremeasuredeverytimethereisatradeorsomebodypostsanewquote,sooftenthefrequencyofthedataisveryhigh

-Non-stationarity:

Financialdata（assetprices）arecovariancenon-stationary;butifweassumethatwearetalkingaboutreturnsfromhereon,thenwecanvalidlyconsiderthemtobestationary.

-LinearIndependence:

Theytypicallyhavelittleevidenceoflinear（autoregressive）dependence,especiallyatlowfrequency.

-Non-normality:

Theyarenotnormallydistributed–theyarefat-tailed.

-Volatilitypoolingandasymmetriesinvolatility:

Thereturnsexhibitvolatilityclusteringandleverageeffects.

Ofthese,wecanallowforthenon-stationaritywithinthelinear（ARIMA）framework,andwecanusewhateverfrequencyofdataweliketoformthemodels,butwecannothopetocapturetheotherfeaturesusingalinearmodelwithGaussiandisturbances.

（b）GARCHmodelsaredesignedtocapturethevolatilityclusteringeffectsinthereturns（GARCH（1,1）canmodelthedependenceinthesquaredreturns,orsquaredresiduals）,andtheycanalsocapturesomeoftheunconditionalleptokurtosis,sothateveniftheresidualsofalinearmodeloftheformgivenbythefirstpartoftheequationinpart（e）,the

’s,areleptokurtic,thestandardisedresidualsfromtheGARCHestimationarelikelytobelessleptokurtic.StandardGARCHmodelscannot,however,accountforleverageeffects.

（c）Thisisessentiallya“whichdisadvantagesofARCHareovercomebyGARCH”question.ThedisadvantagesofARCH（q）are:

-Howdowedecideonq?

-Therequiredvalueofqmightbeverylarge

-Non-negativityconstraintsmightbeviolated.

WhenweestimateanARCHmodel,werequirei>0i=1,2,...,q（sincevariancecannotbenegative）

GARCH（1,1）goessomewaytogetaroundthese.TheGARCH（1,1）modelhasonlythreeparametersintheconditionalvarianceequation,comparedtoq+1fortheARCH（q）model,soitismoreparsimonious.SincetherearelessparametersthanatypicalqthorderARCHmodel,itislesslikelythattheestimatedvaluesofoneormoreofthese3parameterswouldbenegativethanallq+1parameters.Also,theGARCH（1,1）modelcanusuallystillcaptureallofthesignificantdependenceinthesquaredreturnssinceitispossibletowritetheGARCH（1,1）modelasanARCH（）,solagsofthesquaredresidualsbackintotheinfinitepasthelptoexplainthecurrentvalueoftheconditionalvariance,ht.

（d）Thereareanumberthatyoucouldchoosefrom,andtherelevantonesthatwerediscussedinChapter8,inlcudingEGARCH,GJRorGARCH-M.

Thefirsttwoofthesearedesignedtocaptureleverageeffects.Theseareasymmetriesintheresponseofvolatilitytopositiveornegativereturns.ThestandardGARCHmodelcannotcapturethese,sincewearesquaringthelaggederrorterm,andwearethereforelosingitssign.

TheconditionalvarianceequationsfortheEGARCHandGJRmodelsarerespectively

And

t2=0+1

+t-12+ut-12It-1

whereIt-1=1ifut-10

=0otherwise

Foraleverageeffect,wewouldsee>0inbothmodels.

TheEGARCHmodelalsohastheaddedbenefitthatthemodelisexpressedintermsofthelogofht,sothateveniftheparametersarenegative,theconditionalvariancewillalwaysbepositive.Wedonotthereforehavetoartificiallyimposenon-negativityconstraints.

OneformoftheGARCH-Mmodelcanbewritten

yt=+otherterms+t-1+ut,utN（0,ht）

t2=0+1

+t-12

sothatthemodelallowsthelaggedvalueoftheconditionalvariancetoaffectthereturn.Inotherwords,ourbestcurrentestimateofthetotalriskoftheassetinfluencesthereturn,sothatweexpectapositivecoefficientfor.Notethatsomeauthorsuset（i.e.acontemporaneousterm）.

（e）.Sinceytarereturns,wewouldexpecttheirmeanvalue（whichwillbegivenby）tobepositiveandsmall.Wearenottoldthefrequencyofthedata,butsupposethatwehadayearofdailyreturnsdata,thenwouldbetheaveragedailypercentagereturnovertheyear,whichmightbe,say0.05（percent）.Wewouldexpectthevalueof0againtobesmall,say0.0001,orsomethingofthatorder.Theunconditionalvarianceofthedisturbanceswouldbegivenby0/（1-（1+2））.Typicalvaluesfor1and2are0.8and0.15respectively.Theimportantthingisthatallthreealphasmustbepositive,andthesumof1and2wouldbeexpectedtobelessthan,butcloseto,unity,with2>1.

（f）Sincethemodelwasestimatedusingmaximumlikelihood,itdoesnotseemnaturaltotestthisrestrictionusingtheF-testviacomparisonsofresidualsumsofsquares（andat-testcannotbeusedsinceitisatestinvolvingmorethanonecoefficient）.Thusweshoulduseoneoftheapproachestohypothesistestingbasedontheprinciplesofmaximumlikelihood（Wald,LagrangeMultiplier,LikelihoodRatio）.Theeasiestonetousewouldbethelikelihoodratiotest,whichwouldbecomputedasfollows:

1.Estimatetheunrestrictedmodelandobtainthemaximisedvalueofthelog-likelihoodfunction.

2.Imposetherestrictionbyrearrangingthemodel,andestimatetherestrictedmodel,againobtainingthevalueofthelikelihoodatthenewoptimum.NotethatthisvalueoftheLLFwillbelikelytobelowerthantheunconstrainedmaximum.

3.Thenformthelikelihoodratioteststatisticgivenby

LR=-2（Lr-Lu）2（m）

whereLrandLuarethevaluesoftheLLFfortherestrictedandunrestrictedmodelsrespectively,andmdenotesthenumberofrestrictions,whichinthiscaseisone.

4.Ifthevalueoftheteststatisticisgreaterthanthecriticalvalue,rejectthenullhypothesisthattherestrictionsarevalid.

（g）Infact,itispossibletoproducevolatility（conditionalvariance）forecastsinexactlythesamewayasforecastsaregeneratedfromanARMAmodelbyiteratingthroughtheequationswiththeconditionalexpectationsoperator.

WeknowallinformationincludingthatavailableuptotimeT.TheanswertothisquestionwillusetheconventionfromtheGARCHmodellingliteraturetodenotetheconditionalvariancebyhtratherthant2.WhatwewanttogenerateareforecastsofhT+1T,hT+2T,...,hT+sTwhereTdenotesallinformationavailableuptoandincludingobservationT.Adding1then2then3toeachofthetimesubscripts,wehavetheconditionalvarianceequationsfortimesT+1,T+2,andT+3:

hT+1=0+1

+hT

（1）

hT+2=0+1

+hT+1

（2）

hT+3=0+1

+hT+2（3）

Let

betheonestepaheadforecastforhmadeattimeT.Thisiseasytocalculatesince,attimeT,weknowthevaluesofallthetermsontheRHS.Given

howdowecalculate

thatisthe2-stepaheadforecastforhmadeattimeT?

From

（2）,wecanwrite

=0+1ET（

）+

（4）

whereET（

）istheexpectation,madeattimeT,of

whichisthesquareddisturbanceterm.Themodelassumesthattheseriesthaszeromean,sowecannowwrite

Var（ut）=E[（ut-E（ut））2]=E[（ut）2].

Theconditionalvarianceofutisht,so

htt=E[（ut）2]

Turningthisargumentaround,andapplyingittotheproblemthatwehave,

ET[（uT+1）2]=hT+1

butwedonotknowhT+1,sowereplaceitwith

sothat（4）becomes

=0+1

+

=0+（1+）

Whataboutthe3-stepaheadforecast?

Bysimilararguments,

=ET（0+1

+hT+2）

=0+（1+）

=0+（1+）[0+（1+）

]

Andsoon.Thisisthemethodwecouldusetoforecasttheconditionalvarianceofyt.Ifytwere,say,dailyreturnsontheFTSE,wecouldusethesevolatilityforecastsasaninputintheBlackScholesequationtohelpdeterminetheappropriatepriceofFTSEindexoptions.

（h）Ans-stepaheadforecastfortheconditionalvariancecouldbewritten

（x）

Forthenewvalueof,thepersistenceofshockstotheconditionalvariance,givenby（1+）is0.1251+0.98=1.1051,whichisbiggerthan1.Itisobviousfromequation（x）,thatanyvaluefor（1+）biggerthanonewillleadtheforecaststoexplode.Theforecastswillkeeponincreasingandwilltendtoinfinityastheforecasthorizonincreases（i.e.assincreases）.Thisisobviouslyanundesirablepropertyofaforecastingmodel!

Thisiscalled“non-stationarityinvariance”.

For（1+）<1,theforecastswillconvergeontheunconditionalvarianceastheforecasthorizonincreases.For（1+）=1,knownas“integratedGARCH”orIGARCH,thereisaunitrootintheconditionalvariance,andtheforecastswillstayconstantastheforecasthorizonincreases.

2.（a）Maximumlikelihoodworksbyfindingthemostlikelyvaluesoftheparametersgiventheactualdata.Morespecifically,alog-likelihoodfunctionisformed,usuallybaseduponanormalityassumptionforthedisturbanceterms,andthevaluesoftheparametersthatmaximiseitaresought.Maximumlikelihoodestimationcanbeemployedtofindparametervaluesforbothlinearandnon-linearmodels.

（b）Thethreehypothesistestingproceduresavailablewithinthemaximumlikelihoodapproacharelagrangemultiplier（LM）,likelihoodratio（LR）andWaldtests.ThedifferencesbetweenthemaredescribedinFigure8.4,andarenotdefinedagainhere.TheLagrangemultipliertestinvolvesestimationonlyunderthenullhypothesis,thelikelihoodratiotestinvolvesestimationunderboththenullandthealternativehypothesis,whiletheWaldtestinvolvesestimationonlyunderthealternative.Giventhis,itshouldbeevidentthattheLMtestwillinmanycasesbethesimplesttocomputesincetherestrictionsimpliedbythenullhypothesiswillusuallyleadtosometermscancellingouttogiveasimplifiedmode