計量經(jīng)濟學基礎-Further-Inference-in-the-Multiple-Regression-Model-課件

Chapter6FurtherInferenceintheMultipleRegressionModel6.1JointHypothesisTesting6.2TheUseofNonsampleInformation6.3ModelSpecification6.4PoorData,Collinearity,andInsignificance6.5PredictionChapterContentsEconomistsdevelopandevaluatetheoriesabouteconomicbehaviorHypothesistestingproceduresareusedtotestthesetheoriesThetheoriesthateconomistsdevelopsometimesprovidenonsampleinformationthatcanbeusedalongwiththeinformationinasampleofdatatoestimatetheparametersofaregressionmodelAprocedurethatcombinesthesetwotypesofinformationiscalledrestrictedleastsquares6.1JointHypothesisTestingAnullhypothesiswithmultipleconjectures,expressedwithmorethanoneequalsign,iscalledajointhypothesisExample:Shouldagroupofexplanatoryvariablesshouldbeincludedinaparticularmodel?Example:Doesthequantitydemandedofaproductdependonthepricesofsubstitutegoods,oronlyonitsownprice?6.1JointHypothesisTesting6.1TestingJointHypothesesBothexamplesareoftheform:ThejointnullhypothesisinEq.6.1containsthreeconjectures(threeequalsigns):β4=0,β5=0,andβ6=0AtestofH0isajointtestforwhetherallthreeconjecturesholdsimultaneouslyEq.6.16.1JointHypothesisTesting6.1TestingJointHypothesesConsiderthemodel:Testwhetherornotadvertisinghasaneffectonsales–butadvertisingisinthemodelastwovariables6.1.1TestingtheEffectofAdvertising:TheF-TestEq.6.26.1JointHypothesisTestingAdvertisingwillhavenoeffectonsalesifβ3=0andβ4=0Advertisingwillhaveaneffectifβ3≠0orβ4≠0orifbothβ3andβ4arenonzeroThenullhypothesesare:6.1JointHypothesisTesting6.1.1TestingtheEffectofAdvertising:TheF-TestRelativetothenullhypothesisH0:β3=0,β4=0themodelinEq.6.2iscalledtheunrestrictedmodelTherestrictionsinthenullhypothesishavenotbeenimposedonthemodelItcontrastswiththerestrictedmodel,whichisobtainedbyassumingtheparameterrestrictionsinH0aretrue6.1JointHypothesisTesting6.1.1TestingtheEffectofAdvertising:TheF-TestWhenH0istrue,β3=0andβ4=0,andADVERTandADVERT2dropoutofthemodelTheF-testforthehypothesisH0:β3=0,β4=0isbasedonacomparisonofthesumsofsquarederrors(sumsofsquaredleastsquaresresiduals)fromtheunrestrictedmodelinEq.6.2andtherestrictedmodelinEq.6.3ShorthandnotationforthesetwoquantitiesisSSEUandSSER,respectivelyEq.6.36.1JointHypothesisTesting6.1.1TestingtheEffectofAdvertising:TheF-TestTheF-statisticdetermineswhatconstitutesalargereductionorasmallreductioninthesumofsquarederrors whereJisthenumberofrestrictions,NisthenumberofobservationsandKisthenumberofcoefficientsintheunrestrictedmodelEq.6.46.1JointHypothesisTesting6.1.1TestingtheEffectofAdvertising:TheF-TestIfthenullhypothesisistrue,thenthestatisticFhaswhatiscalledanF-distributionwithJnumeratordegreesoffreedomandN-KdenominatordegreesoffreedomIfthenullhypothesisisnottrue,thenthedifferencebetweenSSERandSSEUbecomeslargeTherestrictionsplacedonthemodelbythenullhypothesissignificantlyreducetheabilityofthemodeltofitthedata6.1JointHypothesisTesting6.1.1TestingtheEffectofAdvertising:TheF-TestTheF-testforoursalesproblemis:Specifythenullandalternativehypotheses:ThejointnullhypothesisisH0:β3=0,β4=0.ThealternativehypothesisisH0:β3≠0orβ4≠0botharenonzeroSpecifytheteststatisticanditsdistributionifthenullhypothesisistrue:HavingtworestrictionsinH0

meansJ=2Also,recallthatN=75:6.1JointHypothesisTesting6.1.1TestingtheEffectofAdvertising:TheF-TestTheF-testforoursalesproblemis(Continued):SetthesignificancelevelanddeterminetherejectionregionCalculatethesamplevalueoftheteststatisticand,ifdesired,thep-valueThecorrespondingp-valueisp=P(F(2,71)

>8.44)=0.00056.1JointHypothesisTesting6.1.1TestingtheEffectofAdvertising:TheF-TestTheF-testforoursalesproblemis(Continued):StateyourconclusionSinceF=8.44>Fc=3.126,werejectthenullhypothesisthatbothβ3=0andβ4=0,andconcludethatatleastoneofthemisnotzeroAdvertisingdoeshaveasignificanteffectuponsalesrevenue6.1JointHypothesisTesting6.1.1TestingtheEffectofAdvertising:TheF-TestConsideragainthegeneralmultipleregressionmodelwith(K-1)explanatoryvariablesandKunknowncoefficientsToexaminewhetherwehaveaviableexplanatorymodel,wesetupthefollowingnullandalternativehypotheses:6.1.2TestingtheSignificanceoftheModelEq.6.5Eq.6.66.1JointHypothesisTestingSincewearetestingwhetherornotwehaveaviableexplanatorymodel,thetestforEq.6.6issometimesreferredtoasatestoftheoverallsignificanceoftheregressionmodel.Giventhatthet-distributioncanonlybeusedtotestasinglenullhypothesis,weusetheF-testfortestingthejointnullhypothesisinEq.6.66.1JointHypothesisTesting6.1.2TestingtheSignificanceoftheModelTheunrestrictedmodelisthatgiveninEq.6.5Therestrictedmodel,assumingthenullhypothesisistrue,becomes:Eq.6.76.1JointHypothesisTesting6.1.2TestingtheSignificanceoftheModelTheleastsquaresestimatorofβ1inthisrestrictedmodelis:TherestrictedsumofsquarederrorsfromthehypothesisEq.6.6is:6.1JointHypothesisTesting6.1.2TestingtheSignificanceoftheModelThus,totesttheoverallsignificanceofamodel,butnotingeneral,theF-teststatisticcanbemodifiedandwrittenas:Eq.6.86.1JointHypothesisTesting6.1.2TestingtheSignificanceoftheModelForourproblem,note:Wearetesting:IfH0istrue:6.1JointHypothesisTesting6.1.2TestingtheSignificanceoftheModelForourproblem,note(Continued):Usinga5%significancelevel,wefindthecriticalvaluefortheF-statisticwith(3,71)degreesoffreedomisFc=2.734.Thus,werejectH0

ifF≥2.734.TherequiredsumsofsquaresareSST=3115.482andSSE=1532.084whichgiveanF-valueof:p-value=P(F≥24.459)=0.00006.1JointHypothesisTesting6.1.2TestingtheSignificanceoftheModelForourproblem,note(Continued):Since24.459>2.734,werejectH0

andconcludethattheestimatedrelationshipisasignificantoneNotethatthisconclusionisconsistentwithconclusionsthatwouldbereachedusingseparatet-testsforthesignificanceofeachofthecoefficients6.1JointHypothesisTesting6.1.2TestingtheSignificanceoftheModelWeusedtheF-testtotestwhetherβ3=0andβ4=0in:SupposewewanttotestifPRICEaffectsSALES6.1.3RelationshipBetweenthet-andF-TestsEq.6.9Eq.6.106.1JointHypothesisTestingTheF-valuefortherestrictedmodelis:The5%criticalvalueisFc=F(0.95,1,71)=3.976WerejectH0:β2=06.1JointHypothesisTesting6.1.3RelationshipBetweenthet-andF-TestsUsingthet-test:Thet-valuefortestingH0:β2=0againstH1:β2≠0ist=7.640/1.045939=7.30444Itssquareist=(7.30444)2=53.355,identicaltotheF-value6.1JointHypothesisTesting6.1.3RelationshipBetweenthet-andF-TestsTheelementsofanF-testThenullhypothesisH0

consistsofoneormoreequalityrestrictionsonthemodelparametersβk

ThealternativehypothesisstatesthatoneormoreoftheequalitiesinthenullhypothesisisnottrueTheteststatisticistheF-statisticin(6.4)Ifthenullhypothesisistrue,FhastheF-distributionwithJnumeratordegreesoffreedomandN-KdenominatordegreesoffreedomWhentestingasingleequalitynullhypothesis,itisperfectlycorrecttouseeitherthet-orF-testprocedure:theyareequivalent6.1JointHypothesisTesting6.1.3RelationshipBetweenthet-andF-TestsTheconjecturesmadeinthenullhypothesiswerethatparticularcoefficientsareequaltozeroTheF-testcanalsobeusedformuchmoregeneralhypothesesAnynumberofconjectures(≤K)involvinglinearhypotheseswithequalsignscanbetested6.1.4MoreGeneralF-Tests6.1JointHypothesisTestingConsidertheissueoftesting:IfADVERT0=$1,900permonth,then: orEq.6.116.1JointHypothesisTesting6.1.4MoreGeneralF-TestsNotethatwhenH0istrue,β3=1–3.8β4sothat: orEq.6.126.1JointHypothesisTesting6.1.4MoreGeneralF-TestsThecalculatedvalueoftheF-statisticis:Forα=0.05,thecriticalvalueisFc=3.976SinceF=0.9362<Fc=3.976,wedonotrejectH0

Weconcludethatanadvertisingexpenditureof$1,900permonthisoptimaliscompatiblewiththedata6.1JointHypothesisTesting6.1.4MoreGeneralF-TestsThet-valueist=0.9676F=0.9362isequaltot2=(0.9676)2Thep-valuesareidentical:6.1JointHypothesisTesting6.1.4MoreGeneralF-TestsSupposewehave:Inthiscase,wecannolongerusetheF-testBecauseF=t2,theF-testcannotdistinguishbetweentheleftandrighttailsasisneededforaone-tailtestWerestrictourselvestothet-distributionwhenconsideringalternativehypothesesthathaveinequalitysignssuchas<or>6.1.4aOne-tailTestEq.6.136.1JointHypothesisTestingMostsoftwarepackageshavecommandsthatwillautomaticallycomputet-andF-valuesandtheircorrespondingp-valueswhenprovidedwithanullhypothesisThesetestsbelongtoaclassoftestscalledWaldtests6.1.5UsingComputerSoftware6.1JointHypothesisTestingSupposeweconjecturethat:Weformulatethejointnullhypothesis:BecausethereareJ=2restrictionstotestjointly,weuseanF-testAt-testisnotsuitable6.1JointHypothesisTesting6.1.5UsingComputerSoftware6.2TheUseofNonsampleInformationInmanyestimationproblemswehaveinformationoverandabovetheinformationcontainedinthesampleobservationsThisnonsampleinformationmaycomefrommanyplaces,suchaseconomicprinciplesorexperienceWhenitisavailable,itseemsintuitivethatweshouldfindawaytouseit6.2TheUseofNonsampleInformationConsiderthelog-logfunctionalformforademandmodelforbeer:Thismodelisaconvenientonebecauseitprecludesinfeasiblenegativeprices,quantities,andincome,andbecausethecoefficientsβ2

,β3,β4

,andβ5

areelasticitiesEq.6.146.2TheUseofNonsampleInformationArelevantpieceofnonsampleinformationcanbederivedbynotingthatifallpricesandincomegoupbythesameproportion,wewouldexpecttheretobenochangeinquantitydemandedForexample,adoublingofallpricesandincomeshouldnotchangethequantityofbeerconsumedThisassumptionisthateconomicagentsdonotsufferfrom‘‘moneyillusion’’6.2TheUseofNonsampleInformationHavingallpricesandincomechangebythesameproportionisequivalenttomultiplyingeachpriceandincomebyaconstant,sayλ:Eq.6.156.2TheUseofNonsampleInformationTohavenochangeinln(Q)whenallpricesandincomegoupbythesameproportion,itmustbetruethat:WecallsucharestrictionnonsampleinformationEq.6.166.2TheUseofNonsampleInformationToestimateamodel,startwith:Solvetherestrictionforoneoftheparameters,sayβ4:Eq.6.176.2TheUseofNonsampleInformationSubstitutinggives:Eq.6.186.2TheUseofNonsampleInformationTogetleastsquaresestimatesthatsatisfytheparameterrestriction,calledrestrictedleastsquaresestimates,weapplytheleastsquaresestimationproceduredirectlytotherestrictedmodel:Eq.6.196.2TheUseofNonsampleInformationLettherestrictedleastsquaresestimatesinEq.6.19bedenotedbyb*1,b*2,b*3,andb*5Toobtainanestimateforβ4,weusetherestriction:Byusingtherestrictionwithinthemodel,wehaveensuredthattheestimatesobeytheconstraint:6.2TheUseofNonsampleInformationPropertiesofthisrestrictedleastsquaresestimationprocedure:Therestrictedleastsquaresestimatorisbiased,unlesstheconstraintsweimposeareexactlytrueTherestrictedleastsquaresestimatoristhatitsvarianceissmallerthanthevarianceoftheleastsquaresestimator,whethertheconstraintsimposedaretrueornot6.2TheUseofNonsampleInformation6.3ModelSpecificationInanyeconometricinvestigation,choiceofthemodelisoneofthefirststepsWhataretheimportantconsiderationswhenchoosingamodel?Whataretheconsequencesofchoosingthewrongmodel?Aretherewaysofassessingwhetheramodelisadequate?6.3ModelSpecificationItispossiblethatachosenmodelmayhaveimportantvariablesomittedOureconomicprinciplesmayhaveoverlookedavariable,orlackofdatamayleadustodropavariableevenwhenitisprescribedbyeconomictheory6.3.1OmittedVariables6.3ModelSpecificationConsiderthemodel:Eq.6.206.3ModelSpecification6.3.1OmittedVariablesIfweincorrectlyomitwife’seducation:Eq.6.216.3ModelSpecification6.3.1OmittedVariablesRelativetoEq.6.20,omittingWEDUleadsustooverstatetheeffectofanextrayearofeducationforthehusbandbyabout$2,000Omissionofarelevantvariable(definedasonewhosecoefficientisnonzero)leadstoanestimatorthatisbiasedThisbiasisknownasomitted-variablebias6.3ModelSpecification6.3.1OmittedVariablesWriteageneralmodelas:Omittingx3isequivalenttoimposingtherestrictionβ3=0ItcanbeviewedasanexampleofimposinganincorrectconstraintontheparametersEq.6.226.3ModelSpecification6.3.1OmittedVariablesThebiasis:Eq.6.236.3ModelSpecification6.3.1OmittedVariablesTable6.1CorrelationMatrixforVariablesUsedinFamilyIncomeExample6.3ModelSpecification6.3.1OmittedVariablesNotethat:β3>0becausehusband’seducationhasapositiveeffectonfamilyincome.becausehusband’sandwife’slevelsofeducationarepositivelycorrelated.Thus,thebiasispositive6.3ModelSpecification6.3.1OmittedVariablesNowconsiderthemodel:NoticethatthecoefficientestimatesforHEDUandWEDUhavenotchangedagreatdealThisoutcomeoccursbecauseKL6isnothighlycorrelatedwiththeeducationvariablesEq.6.246.3ModelSpecification6.3.1OmittedVariablesYoutothinkthatagoodstrategyistoincludeasmanyvariablesaspossibleinyourmodel.Doingsowillnotonlycomplicateyourmodelunnecessarily,butmayalsoinflatethevariancesofyourestimatesbecauseofthepresenceofirrelevantvariables.6.3.2IrrelevantVariables6.3ModelSpecificationYoutothinkthatagoodstrategyistoincludeasmanyvariablesaspossibleinyourmodel.Doingsowillnotonlycomplicateyourmodelunnecessarily,butmayalsoinflatethevariancesofyourestimatesbecauseofthepresenceofirrelevantvariables.6.3ModelSpecification6.3.2IrrelevantVariablesConsiderthemodel:Theinclusionofirrelevantvariableshasreducedtheprecisionoftheestimatedcoefficientsforothervariablesintheequation6.3ModelSpecification6.3.2IrrelevantVariablesSomepointsforchoosingamodel:ChoosevariablesandafunctionalformonthebasisofyourtheoreticalandgeneralunderstandingoftherelationshipIfanestimatedequationhascoefficientswithunexpectedsigns,orunrealisticmagnitudes,theycouldbecausedbyamisspecificationsuchastheomissionofanimportantvariableOnemethodforassessingwhetheravariableoragroupofvariablesshouldbeincludedinanequationistoperformsignificancetests6.3.3ChoosingtheModel6.3ModelSpecificationSomepointsforchoosingamodel(Continued):ConsidervariousmodelselectioncriteriaTheadequacyofamodelcanbetestedusingageneralspecificationtestknownasRESET6.3ModelSpecification6.3.3ChoosingtheModelTherearethreemainmodelselectioncriteria:R2AICSC(BIC)6.3.4ModelSelectionCriteria6.3ModelSpecificationAcommonfeatureofthecriteriawedescribeisthattheyaresuitableonlyforcomparingmodelswiththesamedependentvariable,notmodelswithdifferentdependentvariableslikeyandln(y)6.3ModelSpecification6.3.4ModelSelectionCriteriaTheproblemisthatR2canbemadelargebyaddingmoreandmorevariables,evenifthevariablesaddedhavenojustificationAlgebraically,itisafactthatasvariablesareaddedthesumofsquarederrorsSSEgoesdown,andthusR2goesupIfthemodelcontainsN-1variables,thenR2=16.3.4aTheAdjustedCoefficientofDetermination6.3ModelSpecificationAnalternativemeasureofgoodnessoffitcalledtheadjusted-R2,denotedas:Eq.6.256.3ModelSpecification6.3.4aTheAdjustedCoefficientofDeterminationTheAkaikeinformationcriterion(AIC)isgivenby:6.3.4bInformationCriteriaEq.6.266.3ModelSpecificationSchwarzcriterion(SC),alsoknownastheBayesianinformationcriterion(BIC)isgivenby:Eq.6.276.3ModelSpecification6.3.4bInformationCriteriaTable6.2Goodness-of-FitandInformationCriteriaforFamilyIncomeExample6.3ModelSpecification6.3.4bInformationCriteriaAmodelcouldbemisspecifiedif:wehaveomittedimportantvariablesincludedirrelevantoneschosenawrongfunctionalformhaveamodelthatviolatestheassumptionsofthemultipleregressionmodel6.3.5RESET6.3ModelSpecificationRESET(REgressionSpecificationErrorTest)isdesignedtodetectomittedvariablesandincorrectfunctionalform6.3ModelSpecification6.3.5RESETSupposewehavethemodel:Letthepredictedvaluesofybe:Eq.6.286.3ModelSpecification6.3.5RESETNowconsiderthefollowingtwoartificialmodels:Eq.6.29Eq.6.306.3ModelSpecification6.3.5RESETInEq.6.29atestformisspecificationisatestofH0:γ1=0againstthealternativeH1:γ1≠0InEq.6.30,testingH0:γ1=γ2=0againstH1:γ1≠0and/orγ2≠0isatestformisspecification6.3ModelSpecification6.3.5RESETApplyingRESETtoourproblem(Eq.6.24),weget:Inbothcasesthenullhypothesisofnomisspecificationisrejectedata5%significancelevel6.3ModelSpecification6.3.5RESET6.4PoorDataQuality,Collinearity,andInsignificanceWhendataaretheresultofanuncontrolledexperiment,manyoftheeconomicvariablesmaymovetogetherinsystematicwaysSuchvariablesaresaidtobecollinear,andtheproblemislabeledcollinearity6.4PoorDataQuality,Collinearity,andInsignificanceConsiderthemodel:Thevarianceoftheleastsquaresestimatorforβ2is:6.4.1TheConsequencesofCollinearityEq.6.316.4PoorDataQuality,Collinearity,andInsignificanceExactorextremecollinearityexistswhenx2andx3areperfectlycorrelated,inwhichcaser23=1andvar(b2)goestoinfinitySimilarly,ifx2exhibitsnovariationequalszeroandvar(b2)againgoestoinfinityInthiscasex2iscollinearwiththeconstantterm6.4PoorDataQuality,Collinearity,andInsignificance6.4.1TheConsequencesofCollinearityIngeneral,wheneverthereareoneormoreexactlinearrelationshipsamongtheexplanatoryvariables,thentheconditionofexactcollinearityexistsInthiscasetheleastsquaresestimatorisnotdefinedWecannotobtainestimatesofβk’susingtheleastsquaresprinciple6.4PoorDataQuality,Collinearity,andInsignificance6.4.1TheConsequencesofCollinearityTheeffectsofthisimpreciseinformationare:Whenestimatorstandarderrorsarelarge,itislikelythattheusualt-testswillleadtotheconclusionthatparameterestimatesarenotsignificantlydifferentfromzeroEstimatorsmaybeverysensitivetotheadditionordeletionofafewobservations,ortothedeletionofanapparentlyinsignificantvariableAccurateforecastsmaystillbepossibleifthenatureofthecollinearrelationshipremainsthesamewithintheout-of-sampleobservations6.4PoorDataQuality,Collinearity,andInsignificance6.4.1TheConsequencesofCollinearityAregressionofMPGonCYLyields:NowaddENGandWGT:6.4.2AnExample6.4PoorDataQuality,Collinearity,andInsignificanceOnesimplewaytodetectcollinearrelationshipsistousesamplecorrelationcoefficientsbetweenpairsofexplanatoryvariablesThesesamplecorrelationsaredescriptivemeasuresoflinearassociationHowever,insomecasesinwhichcollinearrelationshipsinvolvemorethantwooftheexplanatoryvariables,thecollinearitymaynotbedetectedbyexaminingpairwisecorrelations6.4.3IdentifyingandMitigatingCollinearity6.4PoorDataQuality,Collinearity,andInsignificanceTryanauxiliarymodel:IfR2fromthisartificialmodelishigh,above0.80,say,theimplicationisthatalargeportionofthevariationinx2isexplainedbyvariationintheotherexplanatoryvariables6.4PoorDataQuality,Collinearity,andInsignificance6.4.3IdentifyingandMitigatingCollinearityThecollinearityproblemisthatthedatadonotcontainenough‘‘information’’abouttheindividualeffectsofexplanatoryvariablestopermitustoestimatealltheparametersofthestatisticalmodelpreciselyConsequently,onesolutionistoobtainmoreinformationandincludeitintheanalysis.Asecondway