CFA+2024++L2数量课后习题及详解.docx

PracticeProblemsThefollowinginformationrelatestoquestions1-5Youareajunioranalystatanassetmanagementirm.Ybursupervisorasksyoutoanalyzethereturndriversforoneoftheirm,sportfolios.Sheasksyoutoconstructaregressionmodeloftheportfolio'smonthlyexcessreturns(RET)againstthreefactors:themarketexcessreturn(MRKT),avaluefactor(HML),andthemonthlypercentagechangeinavolatilityindex(VIX).Youcollectthedataandruntheregression,andtheresultingmodelisYret=-°999+1.817XMRKT+0489XHML+0.037Xy.Youthencreatesomediagnosticchartstohelpdeterminethemodelit.sujmJ SSQUXB°oHod%ChangeinvolatilityfactorRETvsMRKTSErUSSB3x20HJodMarket excess returnsRET predicted valuess(snp3J £A. Determinethetypeofregressionmodelyoushoulduse.B. 1.ogisticregressionC. SimplelinearregressionD. Multiplelinearregression1. Determinewhichoneofthefollowingstatementsaboutthecoeficientofthevolatilityfactor(VIX)istrue.A. A1.0%increaseinXqxwouldresultina-0.962%decreaseinYret-B. A0.037%increaseinXvVXWOUIdresultina1.0%increaseinYret-C. A1.0%increaseinXytholdingalltheotherindependentvariablesconstant,wouldresultina0.037%increaseinYRE2. IdentifytheregressionassumptionthatmaybeviolatedbasedonChart1,RETvs.VIX.A. IndependenceoferrorsB. IndependenceofindependentvariablesC. 1.inearitybetweendependentvariableandexplanatoryvariables3. Identifywhichchart,amongCharts2,3,and4,ismostlikelytobeusedtoassesshomoskedasticityA. Chart2B. Chart3C. Chart45. Identifywhichchart,amongCharts2,3,and4,ismostlikelytobeusedtoassessindependenceofindependentvariables.A. Chart2B. Chart3C. Chart41. Ciscorrect.Youshoulduseamultiplelinearregressionmodelsincethedependentvariableiscontinuous(notdiscrete)andthereismorethanoneexplanatoryvariable.Ifthedependentvariablewerediscrete,thenthemodelshouldbeestimatedasalogisticregression.2. Ciscorrect.Thecoeficientofthevolatilityfactor(Xy)is0.037.Itshouldbeinterpretedtomeanthatholdingalltheotherindependentvariablesconstant,a1%increase(decrease)wouldresultina0.037%increase(decrease)inthemonthlyportfolioexcessreturn(V)3. Ciscorrect.Chart1isascatterplotofRETversusVIX.Linearitybetweenthedependentvariableandtheindependentvariablesisanassumptionunderlyingmultiplelinearregression.AsshowninthefollowingRevisedChart1,therelationshipappearstobemorecurved(i.e.,quadratic)thanlinearSUJruB-× O=OjHOd%Changeinvolatilityfactor4. Ciscorrect.Tbassesshomoskedasticit¾wemustevaluatewhetherthevarianceoftheregressionresidualsisconstantforallobservations.Chart4isascatterplotoftheregressionresidualsversusthepredictedvalues,soitisveryusefulforvisuallyassessingtheconsistencyofthevarianceoftheresidualsacrosstheobservations.Anyclustersofhighand/orlowvaluesoftheresidualsmayindicateaviolationofthehomoskedasticityassumption.5. Biscorrect.Chart3isascatterplotcomparingthevaluesoftwooftheindependentvariables,MRKTandHML.Thischartwouldmostlikelybeusedtoassesstheindependenceoftheseexplanatoryvariables.EvaluatingRegressionModelFitandInterpretingModelResults1.earningOutcomesThecandidateshouldbeableto: evaluatehowwellamultipleregressionmodelexplainsthedependentvariablebyanalyzingANOVAtableresultsandmeasuresofgoodnessofit formulatehypothesesonthesigniicanceoftwoormorecoeficientsinamultipleregressionmodelandinterprettheresultsofthejointhypothesistests calculateandinterpretapredictedvalueforthedependentvariable,giventheestimatedregressionmodelandassumedvaluesfortheindependentvariablePracticeProblemsThefollowinginformationrelatestoquestions1-5Youareajunioranalystatanassetmanagementirm.Ybursupervisorasksyoutoanalyzethereturndriversforoneoftheirm,sportfolios.Sheasksyoutoconstructaregressionmodeloftheportfolio'smonthlyexcessreturns(RET)againstthreefactors:themarketexcessreturn(MRKT),avaluefactor(HML),andthemonthlypercentagechangeinavolatilityindex(VIX).Youcollectthedataandruntheregression.Aftercompletingtheirstregression(Model1),youreviewtheANOVAresultswithyoursupervisorThen,sheasksyoutocreatetwomoremodelsbyaddingtwomoreexplanatoryvariables:asizefactor(SMB)andamomentumfactor(MOM).Yburthreemodelsareasfollows:Model1:RETj-bq+ffMRKTz+IjhmlHMLj+byV!X/+/.Model2:RET/=bq+bMRcMRKTj+bMLHML+byVlX÷bMBSMB/+z.Model3:RETj=bo+ffMRKT/+1>hmlHML,+byVlX/+bsMBSMB/+OmomMOM/+/.TheregressionstatisticsandANOVAresultsforthethreemodelsareshowninExhibit1,Exhibit2,andExhibit3.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.0.817HML0.4890.1184.1330.000StandardError3.438VIX0.0370.0182.1220.037Observations96.000ANOVADfSSMSFSigniicanceFRegression35058.4301686.143142.6280.000Residual921087.61811.822Total956146.048Exhibit2:ANOVATableforModel2RET/=o+MRKTMRKTi+bfMLHML+byjVIXi+bsMBSMBi+£iRegressionStatisticsCoeficientStd.Errort-Stat.P-ValueMultipleR0.923Intercept-0.8200.383-2.1390.035R-SqUared0.852MRKT1.6490.12113.6830.000RETi=bq+bMRMRKT/+1)hmlHML+byVIXf+bsMBSMB/+；RegressionStatisticsCoeficientStd.Errort-Stat.P-VaIueAdjustedR-Sq.0.846HML0.4340.1093.9700.000StandardError3.161VIX0.0250.0161.5160.133Observations96.000SMB0.5630.1334.2230.000ANOVADfSSMSFSigniicanceFRegression4Residual91Total955236.6351309.159131.0000.000909.4139.9946146.048Exhibit3: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.0.844HML0.4120.1382.9890.004StandardError3.177VIX0.0260.0171.5320.129Observations96.000SMB0.5530.1393.9870.000MOM-0.0670.242-0.2760.783ANOVADfSSMSFSigniicanceFRegression55237.4021047.480103.7510.000Residual90908.64710.096Total956146.048Ybursupervisorasksforyourassessmentofthemodelthatprovidesthebestitaswellasthemodelthatisbestforpredictingvaluesofthemonthlyportfolioreturn.So,youcalculateAkaike,sinformationcriterion(AIC)andSchwarz,sBayesianinformationcriterion(BIC)forallthreemodels,asshowninExhibit4.Exhibit4:Goodness-of-FitMeasuresAICBlCModel1241.03251.29Model2225.85238.67Model3227.77243.161. DeterminewhichoneofthefollowingreasonsforthechangeinadjustedR2fromModel2toModel3ismostlikelytobecorrect.A. AdjustedR2decreasessinceaddingMOMdoesnotimprovetheoverallexplanatorypowerofModel3.B. AdjustedR2increasessinceaddingSMBimprovestheoverallexplanatorypowerofModel2.C. AdjustedR2decreasessinceaddingMOMimprovestheoverallexplanatorypowerofModel3.2. 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-statisticanddeterminewhetherSMBandMOMtogethercontributetoexplainingRETinModel3ata1%signiicancelevel(useacriticalvalueof4.849).A. 2.216,soSMBandMOMtogetherdonotcontributetoexplainingRETB. 8.863,soSMBandMOMtogetherdocontributetoexplainingRETC. 9.454,soSMBandMOMtogetherdocontributetoexplainingRET1. Aiscorrect.AdjustedR2inModel3decreasesto0.844from0.846inModel2.Model3includesallindependentvariablesfromModel2,whileaddingMOM.AddingvariablestoaregressionmodelalwayseitherincreasesR2orcausesittostaythesame.ButadjustedR2onlyincreasesifthenewvariablemeetsathresholdofsigniicance,t-statistic>1.MOMdoesnotmeetthisthreshold,indicatingitdoesnotimprovetheoverallexplanatorypowerofModel3.2. Biscorrect.BICisthepreferredmeasurefordeterminingwhichmodelprovidesthebestit,andalowerBICisbetterSinceModel2hasthelowestBICvalue,itprovidesthebestitamongthethreemodels.3. Biscorrect.AICisthepreferredmeasurefordeterminingthemodelthatisbestusedforpredictionpurposes.AswithBIC,alowerAICisbetterModel2alsohasthelowestAICvalueamongthethreemodels;thus,itshouldbeusedforpredictionpurposes.4. Aiscorrect.TheregressionequationforModel3isRET=-0.823+1.719MRKT+0.412HML+0.026VIX+0.553SMB-0.067MOM.Usingtheassumedvaluesfortheindependentvariables,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/orbM0M°WeusetheF-statistic,whereF_(SSEOfrestrictedmodel-SSEOfUnreStriCtedmodelPo-SSEofunrestrictedmodel(n-k-)'withq=2andn-k-1=90degreesoffreedom.Thetestisone-tailed,rightside,with=1%,sothecriticalF-valueis4.849.Model1doesnotincludeSMBandMOM,soitistherestrictedmodel.Model3includesallofthevariablesofModel1aswellasSMBandMOM,soitistheunrestrictedmodel.UsingdatainExhibits1and3,thejointF-statisticiscalculatedasF_(1087618-908647y2_89.485_一908,647/9010,096-oodo,Since8.863>4.849,werejectH°.Thus,SMBandMOMtogetherdocontributetotheexplanationofRETinModel3ata1%signiicancelevel.ModelMisspecification1.earningOutcomesThecandidateshouldbeableto: describehowmodelmisspeciicationaffectstheresultsofaregressionanalysisandhowtoavoidcommonformsofmisspeciication explainthetypesofheteroskedasticityandhowitaffectsstatisticalinference explainserialcorrelationandhowitaffectsstatisticalinference explainmulticollinearityandhowitaffectsregressionanalysisPracticeProblemsThefollowinginformationrelatestoquestions1-4Youareajunioranalystatanassetmanagementirm.Ybursupervisorasksyoutoanalyzethereturndriversforoneoftheirm,sportfolios.Sheasksyoutoconstructthreeregressionmodelsoftheportfolio'smonthlyexcessreturns(RET),startingwiththefollowingfactors:themarketexcessreturn(MRKT),avaluefactor(HML),andthemonthlypercentagechangeinavolatilityindex(VIX).Nextyouaddasizefactor(SMB)1andinallyyouaddamomentumfactor(MOM).Yburthreemodelsareasfollows:Model1:RETz=o+ktMRKT/+1)hmlHMLf+bylX/+/.Model2:RET,=o+ffMRKT/+1>hmlHML/+byjVlX+bsMBSMB/+/.Model3:RET=o+ffMRKT/+1)hmlHML,+byjVlX+bsM8SMB/+OmomMOM/+/.YbursupervisorisconcernedaboutconditionalKeteroskedasticityinModel3andasksyoutoperformtheBreusch-Pagan(BP)test.Ata5%conidencelevel,theBPcriticalvalueis11.07.YouruntheregressionfortheBPtest;theresultsareshowninExhibit1.RegreSSionStatiStiCSMultipleR0.25517R-Squared0.06511Adjusted/？-Squared0.01317StandardError18.22568Observations96Nowthechiefinvestmentoficer(CIO)joinsthemeetingandasksyoutoanalyzetworegressionmodels(AandB)fortheportfoliohemanages.Hegivesyouthetestresultsforeachofthemodels,showninExhibit2.TestTypeTestStatisticCriticalValueIndependentVariableIsLaggedValueofDependentVariableModelBreusch-12.1243.927YesAGodfreyModelDurbin-5.0884.387NoBWatsonTheCIOalsoasksyoutotestafactormodelformulticollinearityamongitsfourexplanatoryvariables.YoucalculatethevarianceinIationfactor(VIF)foreachofthefourfactors;theresultsareshowninExhibit3.Variable_R2_VIFT10.7483.968X20.4511.820×30.94217.257JU0.92613.4341. CalculatetheBPteststatisticusingthedatainExhibit1anddeterminewhetherthereisevidenceofHeteroskedasticityA. 1.264,sothereisnoevidenceofheteroskedasticityB. 6,251,sothereisnoevidenceofheteroskedasticityC. 81.792,sothereisevidenceofheteroskedasticity2. IdentifythetypeoferroranditsimpactsonregressionModelAindicatedbythedatainExhibit2.A. Serialcorrelation,invalidcoeficientestimates,anddeIatedstandarderrors.B. Heteroskedasticityvalidcoeficientestimates,anddeIatedstandarderrors.C. Serialcorrelation,validcoeficientestimates,andinIatedstandarderrors.3. DetermineusingExhibit3whichoneofthefollowingstatementsismostlikelytobecorrect.Multicollinearityissuesexistforvariables:AXlandX2.B.X2andX3.C.X3andX4.4. Identifythecorrectanswerrelatedtothefollowingstatement.PossiblesolutionsforaddressingthemulticollinearityissuesidentiiedinExhibit3include:1. excludingoneormoreoftheregressionvariables.2. usingadifferentproxyforoneofthevariables.3. increasingthesamplesize.A. OnlySolution1iscorrect.B. OnlySolution2iscorrect.C. Solutions1,2,and3areeachcorrect.1. Biscorrect.TheBPteststatisticiscalculatedasnR2,wherenisthenumberofobservationsandR2isfromtheregressionfortheBPtest.So,theBPteststatistic=96×0.06511=6.251.Thisislessthanthecriticalvalueof11.07,sowecannotrejectthenullhypothesisofnoHeteroskedasticity.Thus,thereisnoevidenceofKeteroskedasticity.2. Aiscorrect.TheBreusch-Godfrey(BG)testisforserialcorrelation,andforModelA,theBGteststatisticexceedsthecriticalvalue.Inthepresenceofserialcorrelation,iftheindependentvariableisalaggedvalueofthedependentvariable,thenregressioncoeficientestimatesareinvalidandcoeficients,standarderrorsaredelated,sot-statisticsareinlated.3. Ciscorrect.AVIFabove10indicatesseriousmulticollinearityissuesrequiringcorrection,whileaVIFabove5warrantsfurtherinvestigationofthegivenvariable.SinceX3andX4eachhaveVIFsabove10,seriousmulticollinearityexistsforthesetwovariables.VIFsforXlandX2arebothwellbelow5,somulticollinearitydoesnotappeartobeanissuewiththesevariables.4. Ciscorrect.Possiblesolutionsforaddressingmulticollinearityissuesincludeallofthesolutionsmentioned:excludingoneormoreoftheregressionvariables,usingadifferentproxyforoneofthevariables,andincreasingthesamplesize.PracticeProblemsThefollowinginformationrelatestoquestions1-5Thechiefinvestmentoficerasksyoutoanalyzeoneoftheirm,sportfoliostoidentifyinIuentialoutliersthatmightbeskewingregressionresultsofitsreturndrivers.Foreachobservation,youcalculateleverage,thestudentizedresidual,andCook,sD.Thereare96observationsandtwoindependentvariables(k=2),andthecriticalt-statisticis2.63ata1%signiicancelevel.PartialresultsofyourcalculationsareshowninExhibit1.hiiStudentizedResidualCook,sDObservation10.0432.7840.161Observation20.022-0.1030.000Observation30.036-0.7310.010Observation40.059-0.1220.000Observation50.011-0.6600.002Observation60.101-2.9060.347Observation450.0422.1170.094Observation460.0130.1720.000Observation470.015-0.6720.003Observation480.012-0.7340.003Observation490.0640.4750.008Observation500.141-2.7880.594Observation510.0111.6790.016Observation520.023-1.2180.017Observation910.035-1.2600.029Observation920.0253.0010.106Observation930.0171.4830.019Observation940.097-0.1720.001Observation950.0170.0460.000Observation960.0111.8190.019Finally;youaretaskedwithinvestigatingwhetherthereisanymonthlyseasonalityintheexcessportfolioreturns.Youconstructaregressionm