BoothSchoolofBusiness,UniversityofChicagoBusiness41202,SpringQuarter2012,Mr.RueyS.Tsay
SolutionstoMidterm
ProblemA:(34pts)Answerbrie?ythefollowingquestions.Eachquestionhastwopoints.
1.DescribetwoimprovementsoftheEGARCHmodelovertheGARCHvolatilitymodel.
Answer:(1)allowsforasymmetricresponsetopastpositiveornegativereturns,i.e.leveragee?ect,(2)useslogvolatilitytorelaxparameterconstraint.2.DescribetwomethodsthatcanbeusedtoinfertheexistenceofARCHe?ectsinareturnseries,i.e.,volatilityisnotconstantovertime.
Answer:(1)ThesampleACF(orPACF)ofthesquaredresidualsofthemeanequation,(2)usetheLjung-Boxstatisticsonthesquaredresiduals.
2
=3.ConsidertheIGARCH(1,1)volatilitymodel:at=σt??twithσt
22
α0+β1σt?1+(1?β1)at?1.Oftenonepre-?xesα0=0.Why?Also,supposethatα0=0andthe1-stepaheadvolatilitypredictionattheforecastoriginhis16.2%(annualized),i.e.,σh(1)=σh+1=16.2forthepercentagelogreturn.Whatisthe10-stepaheadvolatilityprediction?Thatis,whatisσh(10)?
Answer:(1)Fixingα0=0basedonthepriorknowledgethatvolatilityismeanreverting.(2)σh(10)=16.2.
4.(Questions4to8)ConsiderthedailylogreturnsofAmazonstockfromJanuary3,2007toApril27,2012.SomesummarystatisticsofthereturnsaregivenintheattachedRoutput.Istheexpected(mean)returnofthestockzero?Why?
Answer:Thedatadoesnotprovidesu?cientevidencetosuggestthatthemeanreturnisnotzero,becausethe95%con?denceintervalcon-tainszero.5.Letkbetheexcesskurtosis.TestH0:k=0versusHa:k=0.Writedowntheteststatisticanddrawtheconclusion.
1
Answer:t-ratio=√9.875
24/1340
=73.79,whichishighlysigni?cantcom-
paredwithχ21distribution.
6.Arethereserialcorrelationsinthelogreturns?Why?
Answer:No,theLjung-BoxstatisticQ(10)=10.69withp-value0.38.7.ArethereARCHe?ectsinthelogreturnseries?Why?
Answer:Yes,theLjung-BoxstatistofsquaredresidualsgivesQ(10)=39.24withp-valuelessthan0.05.8.Basedonthesummarystatisticsprovided,whatisthe22-stepaheadpointforecastofthelogreturnattheforecastoriginApril27,2012?Why?
Answer:ThepointforecastrT(22)=0becausethemeanisnotsignif-icantlydi?erentfromzero.[Givestudents1pointiftheyusesamplemean.]9.Givetworeasonsthatexplaintheexistenceofserialcorrelationsinob-servedassetreturnsevenifthetruereturnsarenotseriallycorrelated.Answer:Anytwoof(1)bid-askbounce,(2)nonsynchronoustrading,(3)dynamicdependenceofvolaitlityviariskpremuim.10.Givetworeasonsthatmayleadtousingmoving-averagemodelsin
analyzingassetreturns.
Answer:(1)Smoothing(ormanipulation),(2)bid-askbounceinhighfrequencyreturns.11.Describetwomethodsthatcanbeusedtocomparedi?erentmodels
foragiventimeseries.
Answer:(1)InformationcriteriasuchasAICorBIC,(2)backtestingorout-of-sampleforecasting.12.(Questions12to14)LetrtbethedailylogreturnsofStockA.
Assumethatrt=0.004+at,whereat=σt??twith??tbeingiidN(0,1)
2
randomvariatesandσt=0.017+0.15a2t?1.Whatistheunconditionalvarianceofat?
.017
=0.02.Answer:Var(at)=10
?0.1513.Supposethatthelogpriceatt=100is3.912.Also,attheforecast
origint=100,wehavea100=?0.03andσ100=0.025.Computethe
2
1-stepaheadforecastofthelogprice(notlogreturn)anditsvolatilityforStockAattheforecastorigint=100.Answer:r100(1)=0.??004sothat??p100(1)=3.912+0.004=3.916.The
2
(1)=0.017+0.15(?0.03)2=0.131.volatilityforecastisσ100
14.Computethe30-stepaheadforecastofthelogpriceanditsvolatility
ofStockAattheforecastorigint=100.Answer:p100(30)=3.912+0.√004×30=4.032andthevoaltilityistheunconditionalstantarderror0.02=0.141.15.Assetvolatilityhasmanyapplicationsin?nance.Describetwosuch
applications.
Answer:Anytwoof(1)pricingderivative,(2)riskmanagement,(3)assetallocation.16.SupposethelogreturnrtofStockAfollowsthemodelrt=at,at=σt??t,
22
=α0+α1a2andσtt?1+β1σt?1,where??tareiidN(0,1).Underwhat
conditionthatthekurtosisofrtis3?Thatis,statetheconditionunderwhichtheGARCHdynamicsfailtogenerateanyadditionalkurtosisoverthatof??t.Answer:α1=0.17.Whatisthemainconsequenceinusingalinearregressionanalysiswhen
theserialcorrelationsoftheresidualsareoverlooked?
Answer:Thet-ratiosofcoe?cientestimatesarenotreliable.ProblemB.(30pts)ConsiderthedailylogreturnsofAmazonstockfromJanuary3,2007toApril27,2012.Severalvolatilitymodelsare?ttedtothedataandtherelevantRoutputisattached.Answerthefollowingquestions.1.(2points)Avolatilitymodel,calledm1inR,isentertained.Writedownthe?ttedmodel,includingthemeanequation.Isthemodeladequate?Why?
Answer:ARCH(1)model.rt=0.0018+at,at=σt??twith??tbeingiid
2
N(0,1)andσt=7.577×10?4+0.188a2t?1.Themodelisinadequatebecausethenormalityassumptionisclearlyrejected.2.(3points)Anothervolatilitymodel,calledm2inR,is?ttedtothereturns.Writedownthemodel,includingallestimatedparameters.
3
Answer:ARCH(1)model.rt=4.907×10?4+at,at=σt??t,where??t～
?
t?3.56withtvdenotingstandardizedStudent-tdistributionwithvdegrees
2
offreedom.Thevolatilityequationisσt=7.463×10?4+0.203a2t?1.3.(2points)Basedonthe?ttedmodelm2,testH0:ν=5versusHa:ν=5,whereνdenotesthedegreesoffreedomofStudent-tdistribution.Performthetestanddrawaconclusion.
562?5
=?3.93,whichcomparedwith1.96ishighlyAnswer:t-ratio=3.0.366signi?cant.Ifyoucomputethep-value,itis8.53×10?5.Therefore,v=5isrejected.4.(3points)Athirdmodel,calledm3inR,isalsoentertained.Writedownthemodel,includingthedistributionalparameters.Isthemodeladequate?Why?
Answer:AnotherARCH(1)model.rt=0.0012+at,at=σt??t,where??tareiidandfollowaskewstandardizedStudent-tdistributionwithskewparameter1.065anddegreesoffreedom3.591.Thevolatilityequation
2isσt=7.418×10?4+0.208a2t?1.Eceptfortheinsigicantmeanvalue,the?ttedARCH(1)modelappearstobeadequatebasedonthemodelcheckingstatisticsshown.5.(2points)Letξbetheskewparameterinmodelm3.Doestheestimateofξcon?rmthatthedistributionofthelogreturnsisskewed?Why?Performthetesttosupportyouranswer.
065?1
=1.67,whichissmallerthan1.96.Thus,Answer:Thet-ratiois1.0.039thenullhypothesisofsymmetricinnovationscannotberejectedatthe5%level.6.(3points)Afourthmodel,calledm4inR,isalso?tted.Writedownthe?ttedmodel,includingthedistributionoftheinnovations.
Answer:aGARCH(1,1)model.rt=0.0017+at,at=σt??t,where??tareiidandfollowaskewstandardizedStudent-tdistributionwithskewparameter1.101anddegreesoffreedom3.71.Thevolatilityequation
22isσt=1.066×10?5+0.0414a2t?1+0.950σt?1.7.(2points)Basedonmodelm4,isthedistributionofthelogreturnsskewed?Why?Performatesttosupportyouranswer.
101?1
Answer:Thet-ratiois1.0=2.349,whichisgreaterthan1.96..043Thus,thedistributionisskewatthe5%level.
4
8.(2points)Amongmodelsm1,m2,m3,m4,whichmodelispreferred?Statethecriterionusedinyourchoice.
Answer:Model4ispreferredasithasasmallerAICvalue.?1isverycloseto1,weconsider9.(2points)Sincetheestimatesα?1+β
anIGARCH(1,1)model.Writedownthe?ttedIGARCH(1,1)model,calledm5.
22
Answer:rt=at,at=σt??t,whereσt=3.859×10?5+0.85σt?1+0.15a2t?1.10.(2points)UsetheIGARCH(1,1)modelandtheinformationprovidedtoobtain1-stepand2-stepaheadpredictionsforthevolatilityofthelogreturnsattheforecastorigint=1340.
22
Answer:Fromtheoutputσ1340(1)=σ1341=3.859×10?5+0.85×
2
(0.02108)2+0.15(.146)2=0.00361.Therefore,σ1340(2)=3.859×10?5+2
(1)=0.00365.Thevolatilityforecastsarethen0.0601and0.0604,σ1340
respectively.11.(2points)AGARCH-Mmodelisentertainedforthepercentagelog
returns,calledm6intheRoutput.Basedonthe?ttedmodel,istheriskpremiumstatisticalsigni?cant?Why?
Answer:Theriskpremiumparameteris?0.112witht-ratio?0.560,whichislessthan1.96inmodulus.Thus,theriskpremiumisnotstatisticalsigni?cantatthe5%level.12.(3points)Finally,aGJR-typemodelisentertained,calledm7.Write
downthe?ttedmodel,includingallparameters.
Answer:ThisisanAPARCHmodel.Themodelisrt=0.0014+at,at=σt??t,where??tareiidandfollowaskewstandardizedStudent-tdistributionwithskewparameter1.098anddegreesoffreedom3.846.Thevolatilityequationis
22σt=7.583×10?6+0.0362(|at?1|?0.478at?1)2+0.953σt?1.
13.(2points)Basedonthe?ttedGJR-typeofmodel,istheleveragee?ectsigni?cant?Why?
Answer:Yes,theleverageparameterγ1issign?icantlydi?erentfromzerosothatthereisleveragee?ectinthelogreturns.
5
exam12s经典教材《金融时间序列分析》Ruey S. Tsay 英文第三版2012年试题及答案高清版
