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

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1、SolutionstotheReviewQuestionsattheEndofChapter41. Inthesamewayaswemakeassumptionsaboutthetruevalueofbetaandnottheestimatedvalues,wemakeassumptionsaboutthetrueunobservabledisturbancetermsratherthantheirestimatedcounterparts,theresiduals.Weknowtheexactvalueoftheresiduals,sincetheyaredefinedbyli=H一.Sow

2、edonotneedtomakeanyassumptionsabouttheresidualssincewealreadyknowtheirvalue.Wemakeassumptionsabouttheunobservableerrortermssinceitisalwaysthetruevalueofthepopulationdisturbancesthatwearereallyinterestedin,althoughweneveractuallyknowwhattheseare.2. Wewouldliketoseenopatternintheresidualplot!Ifthereis

3、apatternintheresidualplot,thisisanindicationthatthereisstillsomeaction”orvariabilityleftiny?thathasnotbeenexplainedbyourmodel.Thisindicatesthatpotentiallyitmaybepossibletoformabettermodel,perhapsusingadditionalorcompletelydifferentexplanatoryvariables,orbyusinglagsofeitherthedependentorofoneormoreof

4、theexplanatoryvariables.Recallthatthetwoplotsshownonpages157and159,wheretheresidualsfollowedacyclicalpattern,andwhentheyfollowedanalternatingpatternareusedasindicationsthattheresidualsarepositivelyandnegativelyautocorrelatedrespectively.Anotherproblemifthereisa,patternz,intheresidualsisthat,ifitdoes

5、indicatethepresenceofautocorrelation,thenthismaysuggestthatourstandarderrorestimatesforthecoefficientscouldbewrongandhenceanyinferenceswemakeaboutthecoefficientscouldbemisleading.3. Theratiosforthecoefficientsinthismodelaregiveninthethirdrowafterthestandarderrors.Theyarecalculatedbydividingtheindivi

6、dualcoefficientsbytheirstandarderrors.=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,sothatthereg

7、ressiontakenasawholeseemstoindicateagoodfit.Thislookslikeaclassicexampleofwhatwetermnearmulticollinearity.Thisiswheretheindividualregressorsareverycloselyrelated,sothatitbecomesdifficulttodisentangletheeffectofeachindividualvariableuponthedependentvariable.Thesolutiontonearmulticollinearitythatisusu

8、allysuggestedisthatsincetheproblemisreallyoneofinsufficientinformationinthesampletodetermineeachofthecoefficients,thenoneshouldgooutandgetmoredata.Inotherwords,weshouldswitchtoahigherfrequencyofdataforanalysis(e.g.weeklyinsteadofmonthly,monthlyinsteadofquarterlyetc.).Analternativeisalsotogetmoredata

9、byusingalongersampleperiod(i.e.onegoingfurtherbackintime),ortocombinethetwoindependentvariablesinaratio(e.g.xztW).Other;moreadhocmethodsfordealingwiththepossibleexistenceofnearmulticollinearitywerediscussedinChapter4:-Ignoreit:ifthemodelisotherwiseadequate,i.e.statisticallyandintermsofeachcoefficien

10、tbeingofaplausiblemagnitudeandhavinganappropriatesign.Sometimes,theexistenceofmulticollinearitydoesnotreducetheratiosonvariablesthatwouldhavebeensignificantwithoutthemulticollinearitysufficientlytomaketheminsignificantItisworthstatingthatthepresenceofnearmulticollinearitydoesnotaffecttheBLUEproperti

11、esoftheOLSestimator-i.e.itwillstillbeconsistent,unbiasedandefficientsincethepresenceofnearmulticollinearitydoesnotviolateanyoftheCLRMassumptions1-4.However,inthepresenceofnearmulticollinearity,itwillbehardtoobtainsmallstandarderrors.Thiswillnotmatteriftheaimofthemodel-buildingexerciseistoproducefore

12、castsfromtheestimatedmodel,sincetheforecastswillbeunaffectedbythepresenceofnearmulticollinearitysolongasthisrelationshipbetweentheexplanatoryvariablescontinuestoholdovertheforecastedsample.-Droponeofthecollinearvariables-sothattheproblemdisappears.However,thismaybeunacceptabletotheresearcheriftherew

13、erestrongaprioritheoreticalreasonsforincludingbothvariablesinthemodel.Also,iftheremovedvariablewasrelevantinthedatageneratingprocessforytanomittedvariablebiaswouldresult.-Transformthehighlycorrelatedvariablesintoaratioandincludeonlytheratioandnottheindividualvariablesintheregression.Again,thismaybeu

14、nacceptableiffinancialtheorysuggeststhatchangesinthedependentvariableshouldoccurfollowingchangesintheindividualexplanatoryvariables,andnotaratioofthem.4. (a)TheassumptionofKomoscedasticityisthatthevarianceoftheerrorsisconstantandfiniteovertime.Technically,wewrite(b) Thecoefficientestimateswouldstill

15、bethe“correctones(assumingthattheotherassumptionsrequiredtodemonstrateOLSoptimalityaresatisfied),buttheproblemwouldbethatthestandarderrorscouldbewrong.Henceifweweretryingtotesthypothesesaboutthetrueparametervalues,wecouldendupdrawingthewrongconclusions.Infact,forallofthevariablesexcepttheconstant,th

16、estandarderrorswouldtypicallybetoosmall,sothatwewouldenduprejectingthenullhypothesistoomanytimes.(c) Thereareanumberofwaystoproceedinpractice,including-UsingKeteroscedasticityrobuststandarderrorswhichcorrectfortheproblembyenlargingthestandarderrorsrelativetowhattheywouldhavebeenforthesituationwheretheerrorvarianceispositivelyrelatedtooneoftheexplanatoryvariables.-Transformingthedataintologs,whichhastheeffectofreducingtheeffectoflargeerrorsrela