introductory-econometrics-for-finance--Chapter4-solutions.docx

SolutionstotheReviewQuestionsattheEndofChapter41. Inthesamewayaswemakeassumptionsaboutthetruevalueofbetaandnottheestimatedvalues,wemakeassumptionsaboutthetrueunobservabledisturbancetermsratherthantheirestimatedcounterparts,theresiduals.Weknowtheexactvalueoftheresiduals,sincetheyaredefinedbyli=H一.Sowedonotneedtomakeanyassumptionsabouttheresidualssincewealreadyknowtheirvalue.Wemakeassumptionsabouttheunobservableerrortermssinceitisalwaysthetruevalueofthepopulationdisturbancesthatwearereallyinterestedin,althoughweneveractuallyknowwhattheseare.2. Wewouldliketoseenopatternintheresidualplot!Ifthereisapatternintheresidualplot,thisisanindicationthatthereisstillsome"action”orvariabilityleftiny?thathasnotbeenexplainedbyourmodel.Thisindicatesthatpotentiallyitmaybepossibletoformabettermodel,perhapsusingadditionalorcompletelydifferentexplanatoryvariables,orbyusinglagsofeitherthedependentorofoneormoreoftheexplanatoryvariables.Recallthatthetwoplotsshownonpages157and159,wheretheresidualsfollowedacyclicalpattern,andwhentheyfollowedanalternatingpatternareusedasindicationsthattheresidualsarepositivelyandnegativelyautocorrelatedrespectively.Anotherproblemifthereisa,patternz,intheresidualsisthat,ifitdoesindicatethepresenceofautocorrelation,thenthismaysuggestthatourstandarderrorestimatesforthecoefficientscouldbewrongandhenceanyinferenceswemakeaboutthecoefficientscouldbemisleading.3. Theratiosforthecoefficientsinthismodelaregiveninthethirdrowafterthestandarderrors.Theyarecalculatedbydividingtheindividualcoefficientsbytheirstandarderrors.=0.638+0.402及L0.891胃=o.96灰？=o.89(0.436)(0.291)(0.763)f-ratios1.461.38-1.17Theproblemappearstobethattheregressionparametersareallindividuallyinsignificant(i.e.notsignificantlydifferentfromzero),althoughthevalueofR2anditsadjustedversionarebothveryhigh,sothattheregressiontakenasawholeseemstoindicateagoodfit.Thislookslikeaclassicexampleofwhatwetermnearmulticollinearity.Thisiswheretheindividualregressorsareverycloselyrelated,sothatitbecomesdifficulttodisentangletheeffectofeachindividualvariableuponthedependentvariable.Thesolutiontonearmulticollinearitythatisusuallysuggestedisthatsincetheproblemisreallyoneofinsufficientinformationinthesampletodetermineeachofthecoefficients,thenoneshouldgooutandgetmoredata.Inotherwords,weshouldswitchtoahigherfrequencyofdataforanalysis(e.g.weeklyinsteadofmonthly,monthlyinsteadofquarterlyetc.).Analternativeisalsotogetmoredatabyusingalongersampleperiod(i.e.onegoingfurtherbackintime),ortocombinethetwoindependentvariablesinaratio(e.g.xztW).Other;moreadhocmethodsfordealingwiththepossibleexistenceofnearmulticollinearitywerediscussedinChapter4:-Ignoreit:ifthemodelisotherwiseadequate,i.e.statisticallyandintermsofeachcoefficientbeingofaplausiblemagnitudeandhavinganappropriatesign.Sometimes,theexistenceofmulticollinearitydoesnotreducetheratiosonvariablesthatwouldhavebeensignificantwithoutthemulticollinearitysufficientlytomaketheminsignificantItisworthstatingthatthepresenceofnearmulticollinearitydoesnotaffecttheBLUEpropertiesoftheOLSestimator-i.e.itwillstillbeconsistent,unbiasedandefficientsincethepresenceofnearmulticollinearitydoesnotviolateanyoftheCLRMassumptions1-4.However,inthepresenceofnearmulticollinearity,itwillbehardtoobtainsmallstandarderrors.Thiswillnotmatteriftheaimofthemodel-buildingexerciseistoproduceforecastsfromtheestimatedmodel,sincetheforecastswillbeunaffectedbythepresenceofnearmulticollinearitysolongasthisrelationshipbetweentheexplanatoryvariablescontinuestoholdovertheforecastedsample.-Droponeofthecollinearvariables-sothattheproblemdisappears.However,thismaybeunacceptabletotheresearcheriftherewerestrongaprioritheoreticalreasonsforincludingbothvariablesinthemodel.Also,iftheremovedvariablewasrelevantinthedatageneratingprocessforytanomittedvariablebiaswouldresult.-Transformthehighlycorrelatedvariablesintoaratioandincludeonlytheratioandnottheindividualvariablesintheregression.Again,thismaybeunacceptableiffinancialtheorysuggeststhatchangesinthedependentvariableshouldoccurfollowingchangesintheindividualexplanatoryvariables,andnotaratioofthem.4. (a)TheassumptionofKomoscedasticityisthatthevarianceoftheerrorsisconstantandfiniteovertime.Technically,wewrite(b) Thecoefficientestimateswouldstillbethe“correct"ones(assumingthattheotherassumptionsrequiredtodemonstrateOLSoptimalityaresatisfied),buttheproblemwouldbethatthestandarderrorscouldbewrong.Henceifweweretryingtotesthypothesesaboutthetrueparametervalues,wecouldendupdrawingthewrongconclusions.Infact,forallofthevariablesexcepttheconstant,thestandarderrorswouldtypicallybetoosmall,sothatwewouldenduprejectingthenullhypothesistoomanytimes.(c) Thereareanumberofwaystoproceedinpractice,including-UsingKeteroscedasticityrobuststandarderrorswhichcorrectfortheproblembyenlargingthestandarderrorsrelativetowhattheywouldhavebeenforthesituationwheretheerrorvarianceispositivelyrelatedtooneoftheexplanatoryvariables.-Transformingthedataintologs,whichhastheeffectofreducingtheeffectoflargeerrorsrelativetosmallones.5.(a)ThisiswherethereisarelationshipbetweenthehandTthresiduals.RecallthatoneoftheassumptionsoftheCLRMwasthatsucharelationshipdidnotexist.Wewantourresidualstoberandom,andifthereisevidenceofautocorrelationintheresiduals,thenitimpliesthatwecouldpredictthesignofthenextresidualandgettherightanswermorethanhalfthetimeonaverage!(b) TheDurbinWatsontestisatestforfirstorderautocorrelation.Thetestiscalculatedasfollows.Youwouldrunwhateverregressionyouwereinterestedin,andobtaintheresiduals.Thencalculatethestatistic(-J2DW=2=22r-2YouwouldthenneedtolookupthetwocriticalvaluesfromtheDurbinWatsontables,andthesewoulddependonhowmanyvariablesandhowmanyobservationsandhowmanyregressors(excludingtheconstantthistime)youhadinthemodel.Therejection/non-rejectionrulewouldbegivenbyselectingtheappropriateregionfromthefollowingdiagram:Reject：positiveInconclusiveautocorrelationIIIDonotrejectRejectH:NOevidenceInconclusivenegativeofautocorrelationautocorrelationIIIIodLdu24-du4-dL4(c) Wehave60observations,andthenumberofregressorsexcludingtheconstanttermis3.Theappropriatelowerandupperlimitsare1.48and1.69respectively,sotheDurbinWatsonislowerthanthelowerlimit.Itisthusclearthatwerejectthenullhypothesisofnoautocorrelation.Soitlooksliketheresidualsarepositivelyautocorrelated.(d) a±=四+Bax+故3,+BaZ+w,Theproblemwithamodelentirelyinfirstdifferences,isthatoncewecalculatethelongrunsolution,allthefirstdifferencetermsdropout(asinthelongrunweassumethatthevaluesofallvariableshaveconvergedontheirownlongrunvaluessothatyt=yt-etc.)Thuswhenwetrytocalculatethelongrunsolutiontothismodel,wecannotdoitbecausethereisn,talongrunsolutiontothismodel!(e) Ayr=AI+0axli+Psx2t-+3-+匕Theanswerisyes,thereisnoreasonwhywecannotuseDurbinWatsoninthiscase.Youmayhavesaidnoherebecausetherearelaggedvaluesoftheregressors(thexvariables)variablesintheregression.InfactthiswouldbewrongsincetherearenolagsoftheDEPENDENT引variableandhenceDWcanstillbeused.6. Ayr=+Qx*+B4n+A-1÷Ar2r-1+Bs%+A-4÷%Themajorstepsinvolvedincalculatingthelongrunsolutionareto-setthedisturbancetermequaltoitsexpectedvalueofzero-dropthetimesubscripts-removealldifferencetermsaltogethersincethesewillallbezerobythedefinitionofthelongruninthiscontext.Followingthesesteps,weobtain=A+Ay+52+Py+P3Wenowwanttorearrangethistohaveallthetermsinxztogetherandsothatyisthesubjectoftheformula:iys2ft3P3Ay=-5×2-(A÷Pi)£v_BBSX(A+A)yx9x1BaBJAThelastequationaboveisthelongrunsolution.7. Ramsey1sRESETtestisatestofwhetherthefunctionalformoftheregressionisappropriate.Inotherwords,wetestwhethertherelationshipbetweenthedependentvariableandtheindependentvariablesreallyshouldbelinearorwhetheranon-linearformwouldbemoreappropriate.Thetestworksbyaddingpowersofthefittedvaluesfromtheregressionintoasecondregression.Iftheappropriatemodelwasalinearone,thenthepowersofthefittedvalueswouldnotbesignificantinthissecondregression.IfwefailRamsey,sRESETtest,thentheeasiest“solution“isprobablytotransformallofthevariablesintologarithms.Thishastheeffectofturningamultiplicativemodelintoanadditiveone.Ifthisstillfails,thenwereallyhavetoadmitthattherelationshipbetweenthedependentvariableandtheindependentvariableswasprobablynotlinearafterallsothatwehavetoeitherestimateanon-linearmodelforthedata(whichisbeyondthescopeofthiscourse)orwehavetogobacktothedrawingboardandrunadifferentregressioncontainingdifferentvariables.8. (a)Itisimportanttonotethatwedidnotneedtoassumenormalityinordertoderivethesampleestimatesofaandorincalculatingtheirstandarderrors.Weneededthenormalityassumptionatthelaterstagewhenwecometotesthypothesesabouttheregressioncoefficients,eithersinglyorjointly,sothattheteststatisticswecalculatewouldindeedhavethedistribution(forF)thatwesaidtheywould.(b)Onesolutionwouldbetouseatechniqueforestimationandinferencewhichdidnotrequirenormality.Butthesetechniquesareoftenhighlycomplexandalsotheirpropertiesarenotsowellunderstood,sowedonotknowwithsuchcertaintyhowwellthemethodswillperformindifferentcircumstances.Onepragmaticapproachtofailingthenormalitytestistoplottheestimatedresidualsofthemodel,andlookforoneormoreveryextremeoutliers.Thesewouldberesidualsthataremuch“bigger”(eitherverybigandpositive,orverybigandnegative)thantherest.Itis,fortunatelyforus,oftenthecasethatoneortwoveryextremeoutlierswillcauseaviolationofthenormalityassumption.Thereasonthatoneortwoextremeoutlierscancauseaviolationofthenormalityassumptionisthattheywouldleadthe(absolutevalueofthe)skewnessand/orkurtosisestimatestobeverylarge.Oncewespotafewextremeresiduals,weshouldlookatthedateswhentheseoutliersoccurred.Ifwehaveagoodtheoreticalreasonfordoingso,wecanaddinseparatedummyvariablesforbigoutlierscausedby,forexample,wars,changesofgovernment,stockmarketcrashes,changesinmarketmicrostructure(e.g.the"bigbang”of1986).Theeffectofthedummyvariableisexactlythesameasifwehadremovedtheobservationfromthesamplealtogetherandestimatedtheregressionontheremainder.Ifweonlyremoveobservationsinthisway,thenwemakesurethatwedonotloseanyusefulpiecesofinformationrepresentedbysamplepoints.9. (a)Parameterstructuralstabilityreferstowhetherthecoefficientestimatesforaregressionequationarestableovertime.Iftheregressionisnotstructurallystable,itimpliesthatthecoefficientestimateswouldbedifferentforsomesubsamplesofthedatacomparedtoothers.Thisisclearlynotwhatwewanttofindsincewhenweestimatearegression,weareimplicitlyassumingthattheregressionparametersareconstantovertheentiresampleperiodunderconsideration.(b)1981M1-1995M12rt = 0.0215 + 1.491 rmt?SS=O.189T=1801981M1-1987M1Ort = 0.0163 + 1.308 rmt/?SS= 0.079T=821987M11-1995M12rt = 0.0360 + 1.613 rmt/?SS=0.082 T=98(c) Ifwedefinethecoefficientestimatesforthefirstandsecondhalvesofthesampleasaand,anda2and2respectively,thenthenullandalternativehypothesesareH0:=sandandHi:a2or2(d) TheteststatisticiscalculatedasTeststat.=RSS-(RSS.+RSS,)±(T-2&)0.189-(0.079+0.()82)180-4”，RSSl+RSS2k0.079+0.0822ThisfollowsanFdistributionwith(k,T-2kdegreesoffreedom.尺2,176)=3.05atthe5%level.Clearlywerejectthenullhypothesisthatthecoefficientsareequalinthetwosub-periods.10. Thedatawehaveare1981M1-1995M12 : 0.0215 + 1.491 Rmt/?SS=0.189=1801981M1-1994M12rt = 0.0212 + 1.478 Rmt?SS=O.148上1681982M1-1995M12rt = 0.0217 + 1.523 Rmt?SS=O.182=168First,theforwardpredictivefailuretest-i.e.wearetryingtoseeifthemodelfor1981M1-1994M12canpredict1995M1-1995M12.Theteststatisticisgivenby型g*X ="好3 W = 3.832RSSl0.14812Where71isthenumberofobservationsinthefirstperiod(i.e.theperiodthatweactuallyestimatethemodelover),andTiisthenumberofobservationswearetryingto"predict".Theteststatisticfollowsanfdistributionwith(Ti,Ti-Qdegreesoffreedom.尺12,166)=1.81atthe5%level.Sowerejectthenullhypothesisthatthemodelcanpredicttheobservationsfor1995.Wewouldconcludethatourmodelisnouseforpredictingthisperiod,andfromapracticalpointofview,wewouldhavetoconsiderwhetherthisfailureisaresultofa-typicalbehaviouroftheseriesout-of-sample(i.e.during1995),orwhetheritresultsfromagenuinedeficiencyinthemodel.Thebackwardpredictivefailuretestisalittlemoredifficulttounderstand,althoughnomoredifficulttoimplementTheteststatisticisgivenbyRSS-RSSi.Ti-k0.189-0.182.168-2n_*=*=().)32RSS、T20.18212Nowweneedtobealittlecarefulinourinterpretationofwhatexactlyarethe“first"and"second"sampleperiods.Itwouldbepossibletodefine71asalwaysbeingthefirstsampleperiod.ButIthinkiteasiertosaythatTiisalwaysthesampleoverwhichweestimatethemodel(eventhoughitnowcomesafterthehold-out-sample).ThusTiisstillthesamplethatwearetryingtopredict,eventhoughitcomesfirst.Youcanuseeithernotation,butyouneedtobeclearandconsistent.IfyouwantedtochoosetheotherwaytotheoneIsuggest,thenyouwouldneedtochangethesubscript1everywhereintheformulaabovesothatitwas2,andchangeevery2sothatitwasa1.Eitherway,weconcludethatthereislittleevidenceagainstthenullhypothesis.Thusourmodelisabletoadequatelyback-castthefirst12observationsofthesample.11. Bydefinition,variableshavingassociatedparametersthatarenotsignificantlydifferentfromzeroarenot,fromastatisticalperspective,helpingtoexplainvariationsinthedependentvariableaboutitsmeanvalue.Onecouldthereforearguethatempirically,theyservenopurposeinthefittedregressionmodel.Butleavingsuchvariablesinthemodelwilluseupvaluabledegreesoffreedom,implyingthatthestandarderrorsonalloftheotherparametersintheregressionmodel,willbeunnecessarilyhigherasaresult.Ifthenumberofdegreesoffreedomisrelativelysmall,thensavingacouplebydeletingtwovariableswithinsignificantparameterscouldbeuseful.Ontheotherhand,ifthenumberofdegreesoffreedomisalreadyverylarge,theimpactoftheseadditionalirrelevantvariablesontheothersislikelytobeinconsequential.12. Anoutlierdummyvariablewilltakethevalueoneforoneobservationinthesampleandzeroforallothers.TheChowtestinvolvessplittingthesampleintotwoparts.Ifwethentrytoruntheregressiononboththesub-partsbutthemodelcontainssuchanoutlierdummy,thentheobservationsonthatdummywillbezeroeverywhereforoneoftheregressions.Forthatsub-sample,theoutlierdummywouldshowperfectmulticollinearitywiththeinterceptandthereforethemodelcouldnotbeestimated.