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,whichhastheeffectofreducingtheeffectoflargeerrorsrela