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1、PracticeProblemsThefollowinginformationrelatestoquestions1-5Youareajunioranalystatanassetmanagementirm.Ybursupervisorasksyoutoanalyzethereturndriversforoneoftheirm,sportfolios.Sheasksyoutoconstructaregressionmodeloftheportfoliosmonthlyexcessreturns(RET)againstthreefactors:themarketexcessreturn(MRKT)
2、,avaluefactor(HML),andthemonthlypercentagechangeinavolatilityindex(VIX).Youcollectthedataandruntheregression,andtheresultingmodelisYret=-999+1.817XMRKT+0489XHML+0.037Xy.Youthencreatesomediagnosticchartstohelpdeterminethemodelit.sujmJ SSQUXBoHod%ChangeinvolatilityfactorRETvsMRKTSErUSSB3x20HJodMarket
3、excess returnsRET predicted valuess(snp3J A. Determinethetypeofregressionmodelyoushoulduse.B. 1.ogisticregressionC. SimplelinearregressionD. Multiplelinearregression1. Determinewhichoneofthefollowingstatementsaboutthecoeficientofthevolatilityfactor(VIX)istrue.A. A1.0%increaseinXqxwouldresultina-0.96
4、2%decreaseinYret-B. A0.037%increaseinXvVXWOUIdresultina1.0%increaseinYret-C. A1.0%increaseinXytholdingalltheotherindependentvariablesconstant,wouldresultina0.037%increaseinYRE2. IdentifytheregressionassumptionthatmaybeviolatedbasedonChart1,RETvs.VIX.A. IndependenceoferrorsB. Independenceofindependen
5、tvariablesC. 1.inearitybetweendependentvariableandexplanatoryvariables3. Identifywhichchart,amongCharts2,3,and4,ismostlikelytobeusedtoassesshomoskedasticityA. Chart2B. Chart3C. Chart45. Identifywhichchart,amongCharts2,3,and4,ismostlikelytobeusedtoassessindependenceofindependentvariables.A. Chart2B.
6、Chart3C. Chart41. Ciscorrect.Youshoulduseamultiplelinearregressionmodelsincethedependentvariableiscontinuous(notdiscrete)andthereismorethanoneexplanatoryvariable.Ifthedependentvariablewerediscrete,thenthemodelshouldbeestimatedasalogisticregression.2. Ciscorrect.Thecoeficientofthevolatilityfactor(Xy)
7、is0.037.Itshouldbeinterpretedtomeanthatholdingalltheotherindependentvariablesconstant,a1%increase(decrease)wouldresultina0.037%increase(decrease)inthemonthlyportfolioexcessreturn(V)3. Ciscorrect.Chart1isascatterplotofRETversusVIX.Linearitybetweenthedependentvariableandtheindependentvariablesisanassu
8、mptionunderlyingmultiplelinearregression.AsshowninthefollowingRevisedChart1,therelationshipappearstobemorecurved(i.e.,quadratic)thanlinearSUJruB- O=OjHOd%Changeinvolatilityfactor4. Ciscorrect.Tbassesshomoskedasticitwemustevaluatewhetherthevarianceoftheregressionresidualsisconstantforallobservations.
9、Chart4isascatterplotoftheregressionresidualsversusthepredictedvalues,soitisveryusefulforvisuallyassessingtheconsistencyofthevarianceoftheresidualsacrosstheobservations.Anyclustersofhighand/orlowvaluesoftheresidualsmayindicateaviolationofthehomoskedasticityassumption.5. Biscorrect.Chart3isascatterplo
10、tcomparingthevaluesoftwooftheindependentvariables,MRKTandHML.Thischartwouldmostlikelybeusedtoassesstheindependenceoftheseexplanatoryvariables.EvaluatingRegressionModelFitandInterpretingModelResults1.earningOutcomesThecandidateshouldbeableto: evaluatehowwellamultipleregressionmodelexplainsthedependen
11、tvariablebyanalyzingANOVAtableresultsandmeasuresofgoodnessofit formulatehypothesesonthesigniicanceoftwoormorecoeficientsinamultipleregressionmodelandinterprettheresultsofthejointhypothesistests calculateandinterpretapredictedvalueforthedependentvariable,giventheestimatedregressionmodelandassumedvalu
12、esfortheindependentvariablePracticeProblemsThefollowinginformationrelatestoquestions1-5Youareajunioranalystatanassetmanagementirm.Ybursupervisorasksyoutoanalyzethereturndriversforoneoftheirm,sportfolios.Sheasksyoutoconstructaregressionmodeloftheportfoliosmonthlyexcessreturns(RET)againstthreefactors:
13、themarketexcessreturn(MRKT),avaluefactor(HML),andthemonthlypercentagechangeinavolatilityindex(VIX).Youcollectthedataandruntheregression.Aftercompletingtheirstregression(Model1),youreviewtheANOVAresultswithyoursupervisorThen,sheasksyoutocreatetwomoremodelsbyaddingtwomoreexplanatoryvariables:asizefact
14、or(SMB)andamomentumfactor(MOM).Yburthreemodelsareasfollows:Model1:RETj-bq+ffMRKTz+IjhmlHMLj+byV!X/+/.Model2:RET/=bq+bMRcMRKTj+bMLHML+byVlXbMBSMB/+z.Model3:RETj=bo+ffMRKT/+1hmlHML,+byVlX/+bsMBSMB/+OmomMOM/+/.TheregressionstatisticsandANOVAresultsforthethreemodelsareshowninExhibit1,Exhibit2,andExhibit
15、3.Exhibit1:ANOVATableforModel1RET尸bo+bRMRKT；+1)hmlHML/+byVIXj+RegressionStatisticsCoeficientStd.Errort-Stat.P-ValueMultipleR0.907Intercept-0.9990.414-2.4110.018R-SqUared0.823MRKT1.8170.12414.6830.000AdjustedR-Sq.0.817HML0.4890.1184.1330.000StandardError3.438VIX0.0370.0182.1220.037Observations96.000A
16、NOVADfSSMSFSigniicanceFRegression35058.4301686.143142.6280.000Residual921087.61811.822Total956146.048Exhibit2:ANOVATableforModel2RET/=o+MRKTMRKTi+bfMLHML+byjVIXi+bsMBSMBi+iRegressionStatisticsCoeficientStd.Errort-Stat.P-ValueMultipleR0.923Intercept-0.8200.383-2.1390.035R-SqUared0.852MRKT1.6490.12113
17、.6830.000RETi=bq+bMRMRKT/+1)hmlHML+byVIXf+bsMBSMB/+；RegressionStatisticsCoeficientStd.Errort-Stat.P-VaIueAdjustedR-Sq.0.846HML0.4340.1093.9700.000StandardError3.161VIX0.0250.0161.5160.133Observations96.000SMB0.5630.1334.2230.000ANOVADfSSMSFSigniicanceFRegression4Residual91Total955236.6351309.159131.
18、0000.000909.4139.9946146.048Exhibit3:ANOVATableforModel3RETi=bq+bMRMRKT/+b11LHML；+bvVIXf+bsMBSMB/+b0MMOMz+,RegressionStatisticsCoeficientStd.Errort-Stat.P-ValueMultipleR0.923Intercept-0.8230.385-2.1360.035R-SqUared0.852MRKT1.7190.2806.1300.000AdjustedR-Sq.0.844HML0.4120.1382.9890.004StandardError3.1
19、77VIX0.0260.0171.5320.129Observations96.000SMB0.5530.1393.9870.000MOM-0.0670.242-0.2760.783ANOVADfSSMSFSigniicanceFRegression55237.4021047.480103.7510.000Residual90908.64710.096Total956146.048Ybursupervisorasksforyourassessmentofthemodelthatprovidesthebestitaswellasthemodelthatisbestforpredictingval
20、uesofthemonthlyportfolioreturn.So,youcalculateAkaike,sinformationcriterion(AIC)andSchwarz,sBayesianinformationcriterion(BIC)forallthreemodels,asshowninExhibit4.Exhibit4:Goodness-of-FitMeasuresAICBlCModel1241.03251.29Model2225.85238.67Model3227.77243.161. Determinewhichoneofthefollowingreasonsforthec
21、hangeinadjustedR2fromModel2toModel3ismostlikelytobecorrect.A. AdjustedR2decreasessinceaddingMOMdoesnotimprovetheoverallexplanatorypowerofModel3.B. AdjustedR2increasessinceaddingSMBimprovestheoverallexplanatorypowerofModel2.C. AdjustedR2decreasessinceaddingMOMimprovestheoverallexplanatorypowerofModel
22、3.2. Identifythemodelthatprovidesthebestit.A. Model1B. Model2C. Model33. Identifythemodelthatshouldbeusedforpredictionpurposes.A. Model1B. Model2C. Model34. CalculatethepredictedRETforModel3giventheassumedfactorvalues:MRKT=3,HML=-2,VIX=-5,SMB=IfMOM=3.A. 3.732B. 3.992C. 4.5555. CalculatethejointF-sta
23、tisticanddeterminewhetherSMBandMOMtogethercontributetoexplainingRETinModel3ata1%signiicancelevel(useacriticalvalueof4.849).A. 2.216,soSMBandMOMtogetherdonotcontributetoexplainingRETB. 8.863,soSMBandMOMtogetherdocontributetoexplainingRETC. 9.454,soSMBandMOMtogetherdocontributetoexplainingRET1. Aiscor
24、rect.AdjustedR2inModel3decreasesto0.844from0.846inModel2.Model3includesallindependentvariablesfromModel2,whileaddingMOM.AddingvariablestoaregressionmodelalwayseitherincreasesR2orcausesittostaythesame.ButadjustedR2onlyincreasesifthenewvariablemeetsathresholdofsigniicance,t-statistic1.MOMdoesnotmeetth
25、isthreshold,indicatingitdoesnotimprovetheoverallexplanatorypowerofModel3.2. Biscorrect.BICisthepreferredmeasurefordeterminingwhichmodelprovidesthebestit,andalowerBICisbetterSinceModel2hasthelowestBICvalue,itprovidesthebestitamongthethreemodels.3. Biscorrect.AICisthepreferredmeasurefordeterminingthem
26、odelthatisbestusedforpredictionpurposes.AswithBIC,alowerAICisbetterModel2alsohasthelowestAICvalueamongthethreemodels;thus,itshouldbeusedforpredictionpurposes.4. Aiscorrect.TheregressionequationforModel3isRET=-0.823+1.719MRKT+0.412HML+0.026VIX+0.553SMB-0.067MOM.Usingtheassumedvaluesfortheindependentv
27、ariables,wehaveRET=-0.823+(1.719)(3)+(0.412)(-2)+(0.026)(-5)+(0.553)(1)-(0.067)(3)=3.732.5. Biscorrect.TbdeterminewhetherSMBandMOMtogethercontributetotheexplanationofRETatleastoneofthecoeficientsmustbenon-zero.So,Ho：mb=mom=andHa:bsMB0and/orbM0MWeusetheF-statistic,whereF_(SSEOfrestrictedmodel-SSEOfUn
28、reStriCtedmodelPo-SSEofunrestrictedmodel(n-k-)withq=2andn-k-1=90degreesoffreedom.Thetestisone-tailed,rightside,with=1%,sothecriticalF-valueis4.849.Model1doesnotincludeSMBandMOM,soitistherestrictedmodel.Model3includesallofthevariablesofModel1aswellasSMBandMOM,soitistheunrestrictedmodel.UsingdatainExh
29、ibits1and3,thejointF-statisticiscalculatedasF_(1087618-908647y2_89.485_一908,647/9010,096-oodo,Since8.8634.849,werejectH.Thus,SMBandMOMtogetherdocontributetotheexplanationofRETinModel3ata1%signiicancelevel.ModelMisspecification1.earningOutcomesThecandidateshouldbeableto: describehowmodelmisspeciicati
30、onaffectstheresultsofaregressionanalysisandhowtoavoidcommonformsofmisspeciication explainthetypesofheteroskedasticityandhowitaffectsstatisticalinference explainserialcorrelationandhowitaffectsstatisticalinference explainmulticollinearityandhowitaffectsregressionanalysisPracticeProblemsThefollowingin
31、formationrelatestoquestions1-4Youareajunioranalystatanassetmanagementirm.Ybursupervisorasksyoutoanalyzethereturndriversforoneoftheirm,sportfolios.Sheasksyoutoconstructthreeregressionmodelsoftheportfoliosmonthlyexcessreturns(RET),startingwiththefollowingfactors:themarketexcessreturn(MRKT),avaluefacto
32、r(HML),andthemonthlypercentagechangeinavolatilityindex(VIX).Nextyouaddasizefactor(SMB)1andinallyyouaddamomentumfactor(MOM).Yburthreemodelsareasfollows:Model1:RETz=o+ktMRKT/+1)hmlHMLf+bylX/+/.Model2:RET,=o+ffMRKT/+1hmlHML/+byjVlX+bsMBSMB/+/.Model3:RET=o+ffMRKT/+1)hmlHML,+byjVlX+bsM8SMB/+OmomMOM/+/.Yb
33、ursupervisorisconcernedaboutconditionalKeteroskedasticityinModel3andasksyoutoperformtheBreusch-Pagan(BP)test.Ata5%conidencelevel,theBPcriticalvalueis11.07.YouruntheregressionfortheBPtest;theresultsareshowninExhibit1.RegreSSionStatiStiCSMultipleR0.25517R-Squared0.06511Adjusted/？-Squared0.01317Standar
34、dError18.22568Observations96Nowthechiefinvestmentoficer(CIO)joinsthemeetingandasksyoutoanalyzetworegressionmodels(AandB)fortheportfoliohemanages.Hegivesyouthetestresultsforeachofthemodels,showninExhibit2.TestTypeTestStatisticCriticalValueIndependentVariableIsLaggedValueofDependentVariableModelBreusc
35、h-12.1243.927YesAGodfreyModelDurbin-5.0884.387NoBWatsonTheCIOalsoasksyoutotestafactormodelformulticollinearityamongitsfourexplanatoryvariables.YoucalculatethevarianceinIationfactor(VIF)foreachofthefourfactors;theresultsareshowninExhibit3.Variable_R2_VIFT10.7483.968X20.4511.82030.94217.257JU0.92613.4
36、341. CalculatetheBPteststatisticusingthedatainExhibit1anddeterminewhetherthereisevidenceofHeteroskedasticityA. 1.264,sothereisnoevidenceofheteroskedasticityB. 6,251,sothereisnoevidenceofheteroskedasticityC. 81.792,sothereisevidenceofheteroskedasticity2. Identifythetypeoferroranditsimpactsonregressio
37、nModelAindicatedbythedatainExhibit2.A. Serialcorrelation,invalidcoeficientestimates,anddeIatedstandarderrors.B. Heteroskedasticityvalidcoeficientestimates,anddeIatedstandarderrors.C. Serialcorrelation,validcoeficientestimates,andinIatedstandarderrors.3. DetermineusingExhibit3whichoneofthefollowingst
38、atementsismostlikelytobecorrect.Multicollinearityissuesexistforvariables:AXlandX2.B.X2andX3.C.X3andX4.4. Identifythecorrectanswerrelatedtothefollowingstatement.PossiblesolutionsforaddressingthemulticollinearityissuesidentiiedinExhibit3include:1. excludingoneormoreoftheregressionvariables.2. usingadi
39、fferentproxyforoneofthevariables.3. increasingthesamplesize.A. OnlySolution1iscorrect.B. OnlySolution2iscorrect.C. Solutions1,2,and3areeachcorrect.1. Biscorrect.TheBPteststatisticiscalculatedasnR2,wherenisthenumberofobservationsandR2isfromtheregressionfortheBPtest.So,theBPteststatistic=960.06511=6.2
40、51.Thisislessthanthecriticalvalueof11.07,sowecannotrejectthenullhypothesisofnoHeteroskedasticity.Thus,thereisnoevidenceofKeteroskedasticity.2. Aiscorrect.TheBreusch-Godfrey(BG)testisforserialcorrelation,andforModelA,theBGteststatisticexceedsthecriticalvalue.Inthepresenceofserialcorrelation,iftheinde
41、pendentvariableisalaggedvalueofthedependentvariable,thenregressioncoeficientestimatesareinvalidandcoeficients,standarderrorsaredelated,sot-statisticsareinlated.3. Ciscorrect.AVIFabove10indicatesseriousmulticollinearityissuesrequiringcorrection,whileaVIFabove5warrantsfurtherinvestigationofthegivenvar
42、iable.SinceX3andX4eachhaveVIFsabove10,seriousmulticollinearityexistsforthesetwovariables.VIFsforXlandX2arebothwellbelow5,somulticollinearitydoesnotappeartobeanissuewiththesevariables.4. Ciscorrect.Possiblesolutionsforaddressingmulticollinearityissuesincludeallofthesolutionsmentioned:excludingoneormo
43、reoftheregressionvariables,usingadifferentproxyforoneofthevariables,andincreasingthesamplesize.PracticeProblemsThefollowinginformationrelatestoquestions1-5Thechiefinvestmentoficerasksyoutoanalyzeoneoftheirm,sportfoliostoidentifyinIuentialoutliersthatmightbeskewingregressionresultsofitsreturndrivers.
44、Foreachobservation,youcalculateleverage,thestudentizedresidual,andCook,sD.Thereare96observationsandtwoindependentvariables(k=2),andthecriticalt-statisticis2.63ata1%signiicancelevel.PartialresultsofyourcalculationsareshowninExhibit1.hiiStudentizedResidualCook,sDObservation10.0432.7840.161Observation2
45、0.022-0.1030.000Observation30.036-0.7310.010Observation40.059-0.1220.000Observation50.011-0.6600.002Observation60.101-2.9060.347Observation450.0422.1170.094Observation460.0130.1720.000Observation470.015-0.6720.003Observation480.012-0.7340.003Observation490.0640.4750.008Observation500.141-2.7880.594O
46、bservation510.0111.6790.016Observation520.023-1.2180.017Observation910.035-1.2600.029Observation920.0253.0010.106Observation930.0171.4830.019Observation940.097-0.1720.001Observation950.0170.0460.000Observation960.0111.8190.019Finally;youaretaskedwithinvestigatingwhetherthereisanymonthlyseasonalityintheexcessportfolioreturns.Youconstructaregressionm
