Chapter4_solutions

SolutionstotheReviewQuestionsattheEndofChapter41.Inthesamewayaswemakeassumptionsaboutthetruevalueofbetaandnottheestimatedvalues,wemakeassumptionsaboutthetrueunobservabledisturbancetermsratherthantheirestimatedcounterparts,theresiduals.Weknowtheexactvalueoftheresiduals,sincetheyaredefinedby.Sowedonotneedtomakeanyassumptionsabouttheresidualssincewealreadyknowtheirvalue.Wemakeassumptionsabouttheunobservableerrortermssinceitisalwaysthetruevalueofthepopulationdisturbancesthatwearereallyinterestedin,althoughweneveractuallyknowwhattheseare.2.Wewouldliketoseenopatternintheresidualplot!Ifthereisapatternintheresidualplot,thisisanindicationthatthereisstillsome“action”orvariabilityleftinytthathasnotbeenexplainedbyourmodel.Thisindicatesthatpotentiallyitmaybepossibletoformabettermodel,perhapsusingadditionalorcompletelydifferentexplanatoryvariables,orbyusinglagsofeitherthedependentorofoneormoreoftheexplanatoryvariables.Recallthatthetwoplotsshownonpages157and159,wheretheresidualsfollowedacyclicalpattern,andwhentheyfollowedanalternatingpatternareusedasindicationsthattheresidualsarepositivelyandnegativelyautocorrelatedrespectively.Anotherproblemifthereisa“pattern”intheresidualsisthat,ifitdoesindicatethepresenceofautocorrelation,thenthismaysuggestthatourstandarderrorestimatesforthecoefficientscouldbewrongandhenceanyinferenceswemakeaboutthecoefficientscouldbemisleading.3.Thet-ratiosforthecoefficientsinthismodelaregiveninthethirdrowafterthestandarderrors.Theyarecalculatedbydividingtheindividualcoefficientsbytheirstandarderrors.=0.638+0.402x2t-0.891x3t(0.436)(0.291)(0.763)t-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.x2t/x3t).Other,moreadhocmethodsfordealingwiththepossibleexistenceofnearmulticollinearitywerediscussedinChapter4:·Ignoreit:ifthemodelisotherwiseadequate,i.e.statisticallyandintermsofeachcoefficientbeingofaplausiblemagnitudeandhavinganappropriatesign.Sometimes,theexistenceofmulticollinearitydoesnotreducethet-ratiosonvariablesthatwouldhavebeensignificantwithoutthemulticollinearitysufficientlytomaketheminsignificant.ItisworthstatingthatthepresenceofnearmulticollinearitydoesnotaffecttheBLUEpropertiesoftheOLSestimator–i.e.itwillstillbeconsistent,unbiasedandefficientsincethepresenceofnearmulticollinearitydoesnotviolateanyoftheCLRMassumptions1-4.However,inthepresenceofnearmulticollinearity,itwillbehardtoobtainsmallstandarderrors.Thiswillnotmatteriftheaimofthemodel-buildingexerciseistoproduceforecastsfromtheestimatedmodel,sincetheforecastswillbeunaffectedbythepresenceofnearmulticollinearitysolongasthisrelationshipbetweentheexplanatoryvariablescontinuestoholdovertheforecastedsample.·Droponeofthecollinearvariables-sothattheproblemdisappears.However,thismaybeunacceptabletotheresearcheriftherewerestrongaprioritheoreticalreasonsforincludingbothvariablesinthemodel.Also,iftheremovedvariablewasrelevantinthedatageneratingprocessfory,anomittedvariablebiaswouldresult.·Transformthehighlycorrelatedvariablesintoaratioandincludeonlytheratioandnottheindividualvariablesintheregression.Again,thismaybeunacceptableiffinancialtheorysuggeststhatchangesinthedependentvariableshouldoccurfollowingchangesintheindividualexplanatoryvariables,andnotaratioofthem.4.(a)Theassumptionofhomoscedasticityisthatthevarianceoftheerrorsisconstantandfiniteovertime.Technically,wewrite.(b)Thecoefficientestimateswouldstillbethe“correct”ones(assumingthattheotherassumptionsrequiredtodemonstrateOLSoptimalityaresatisfied),buttheproblemwouldbethatthestandarderrorscouldbewrong.Henceifweweretryingtotesthypothesesaboutthetrueparametervalues,wecouldendupdrawingthewrongconclusions.Infact,forallofthevariablesexcepttheconstant,thestandarderrorswouldtypicallybetoosmall,sothatwewouldenduprejectingthenullhypothesistoomanytimes.(c)Thereareanumberofwaystoproceedinpractice,including-Usingheteroscedasticityrobuststandarderrorswhichcorrectfortheproblembyenlargingthestandarderrorsrelativetowhattheywouldhavebeenforthesituationwheretheerrorvarianceispositivelyrelatedtooneoftheexplanatoryvariables.-Transformingthedataintologs,whichhastheeffectofreducingtheeffectoflargeerrorsrelativetosmallones.5.(a)Thisiswherethereisarelationshipbetweentheithandjthresiduals.RecallthatoneoftheassumptionsoftheCLRMwasthatsucharelationshipdidnotexist.Wewantourresidualstoberandom,andifthereisevidenceofautocorrelationintheresiduals,thenitimpliesthatwecouldpredictthesignofthenextresidualandgettherightanswermorethanhalfthetimeonaverage!(b)TheDurbinWatsontestisatestforfirstorderautocorrelation.Thetestiscalculatedasfollows.Youwouldrunwhateverregressionyouwereinterestedin,andobtaintheresiduals.ThencalculatethestatisticYouwouldthenneedtolookupthetwocriticalvaluesfromtheDurbinWatsontables,andthesewoulddependonhowmanyvariablesandhowmanyobservationsandhowmanyregressors(excludingtheconstantthistime)youhadinthemodel.Therejection/non-rejectionrulewouldbegivenbyselectingtheappropriateregionfromthefollowingdiagram:(c)Wehave60observations,andthenumberofregressorsexcludingtheconstanttermis3.Theappropriatelowerandupperlimitsare1.48and1.69respectively,sotheDurbinWatsonislowerthanthelowerlimit.Itisthusclearthatwerejectthenullhypothesisofnoautocorrelation.Soitlooksliketheresidualsarepositivelyautocorrelated.(d)Theproblemwithamodelentirelyinfirstdifferences,isthatoncewecalculatethelongrunsolution,allthefirstdifferencetermsdropout(asinthelongrunweassumethatthevaluesofallvariableshaveconvergedontheirownlongrunvaluessothatyt=yt-1etc.)Thuswhenwetrytocalculatethelongrunsolutiontothismodel,wecannotdoitbecausethereisn’talongrunsolutiontothismodel!(e)Theanswerisyes,thereisnoreasonwhywecannotuseDurbinWatsoninthiscase.Youmayhavesaidnoherebecausetherearelaggedvaluesoftheregressors(thexvariables)variablesintheregression.InfactthiswouldbewrongsincetherearenolagsoftheDEPENDENT(y)variableandhenceDWcanstillbeused.6.Themajorstepsinvolvedincalculatingthelongrunsolutionareto-setthedisturbancetermequaltoitsexpectedvalueofzero-dropthetimesubscripts-removealldifferencetermsaltogethersincethesewillallbezerobythedefinitionofthelongruninthiscontext.Followingthesesteps,weobtainWenowwanttorearrangethistohaveallthetermsinx2togetherandsothatyisthesubjectoftheformula:Thelastequationaboveisthelongrunsolution.7.Ramsey’sRESETtestisatestofwhetherthefunctionalformoftheregressionisappropriate.Inotherwords,wetestwhethertherelationshipbetweenthedependentvariableandtheindependentvariablesreallyshouldbelinearorwhetheranon-linearformwouldbemoreappropriate.Thetestworksbyaddingpowersofthefittedvaluesfromtheregressionintoasecondregression.Iftheappropriatemodelwasalinearone,thenthepowersofthefittedvalueswouldnotbesignificantinthissecondregression.IfwefailRamsey’sRESETtest,thentheeasiest“solution”isprobablytotransformallofthevariablesintologarithms.Thishastheeffectofturningamultiplicativemodelintoanadditiveone.Ifthisstillfails,thenwereallyhavetoadmitthattherelationshipbetweenthedependentvariableandtheindependentvariableswasprobablynotlinearafterallsothatwehavetoeitherestimateanon-linearmodelforthedata(whichisbeyondthescopeofthiscourse)orwehavetogobacktothedrawingboardandrunadifferentregressioncontainingdifferentvariables.8.(a)Itisimportanttonotethatwedidnotneedtoassumenormalityinordertoderivethesampleestimatesof(and(orincalculatingtheirstandarderrors.Weneededthenormalityassumptionatthelaterstagewhenwecometotesthypothesesabouttheregressioncoefficients,eithersinglyorjointly,sothattheteststatisticswecalculatewouldindeedhavethedistribution(torF)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,itimpliesthatthecoefficientestimateswouldbedifferentforsomesub-samplesofthedatacomparedtoothers.Thisisclearlynotwhatwewanttofindsincewhenweestimatearegression,weareimplicitlyassumingthattheregressionparametersareconstantovertheentiresampleperiodunderconsideration.(b)1981M1-1995M12rt=0.0215+1.491rmtRSS=0.189T=1801981M1-1987M10rt=0.0163+1.308rmtRSS=0.079T=821987M11-1995M12rt=0.0360+1.613rmtRSS=0.082T=98(c)Ifwedefinethecoefficientestimatesforthefirstandsecondhalvesofthesampleas(1and(1,and(2and(2respectively,thenthenullandalternativehypothesesareH0:(1=(2and(1=(2andH1:(1((2or(1((2(d)TheteststatisticiscalculatedasTeststat.=ThisfollowsanFdistributionwith(k,T-2k)degreesoffreedom.F(2,176)=3.05atthe5%level.Clearlywerejectthenullhypothesisthatthecoefficientsareequalinthetwosub-periods.10.Thedatawehaveare1981M1-1995M12rt=0.0215+1.491RmtRSS=0.189T=1801981M1-1994M12rt=0.0212+1.478RmtRSS=0.148T=1681982M1-1995M12rt=0.0217+1.523RmtRSS=0.182T=168First,theforwardpredictivefailuretest-i.e.wearetryingtoseeifthemodelfor1981M1-1994M12canpredict1995M1-1995M12.TheteststatisticisgivenbyWhereT1isthenumberofobservationsinthefirstperiod(i.e.theperiodthatweactuallyestimatethemodelover),andT2isthenumberofobservationswearetryingto“predict”.TheteststatisticfollowsanF-distributionwith(T2,T1-k)degreesoffreedom.F(12,166)=1.81atthe5%level.Sowerejectthenullhypothesisthatthemodelcanpredicttheobservationsfor1995.Wewouldconcludethatourmodelisnouseforpredictingthisperiod,andfromapracticalpointofview,wewouldhavetoconsiderwhetherthisfailureisaresultofa-typicalbehaviouroftheseriesout-of-sample(i.e.during1995),orwhetheritresultsfromagenuinedeficiencyinthemodel.Thebackwardpredictivefailuretestisalittlemoredifficulttounderstand,althoughnomoredifficulttoimplement.TheteststatisticisgivenbyNowweneedtobealittlecarefulinourinterpretationofwhatexactlyarethe“first”and“second”sampleperiods.ItwouldbepossibletodefineT1asalwaysbeingthefirstsampleperiod.ButIthinkiteasiertosaythatT1isalwaysthesampleoverwhichweestimatethemodel(eventhoughitnowcomesafterthehold-out-sample).ThusT2isstillthesamplethatwearetryingtopredict,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.8/8“IntroductoryEconometricsforFinance”©ChrisBrooks2008_1077939193.unknown_1077941873.unknown_1077942039.unknown_1077942555.unknown_1077942762.unknown_1077941937.unknown_1077941655.unknown_1077941820.unknown_1077941301.unknown_1077938447.unknown_1077938459.unknown_1067766412.unknown_1077938443.unknown