上海金融学院计量经济学课件ch05.ppt
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上海金融学院计量经济学课件ch05.ppt
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MultipleRegressionAnalysiswy=b0+b1x1+b2x2+.bkxk+uw3.AsymptoticProperties1Economics20-Prof.AndersonConsistencyUndertheGauss-MarkovassumptionsOLSisBLUE,butinothercasesitwontalwaysbepossibletofindunbiasedestimatorsInthosecases,wemaysettleforestimatorsthatareconsistent,meaningasn,thedistributionoftheestimatorcollapsestotheparametervalue2Economics20-Prof.AndersonSamplingDistributionsasnb1n1n2n3n1n2n33Economics20-Prof.AndersonConsistencyofOLSUndertheGauss-Markovassumptions,theOLSestimatorisconsistent(andunbiased)ConsistencycanbeprovedforthesimpleregressioncaseinamannersimilartotheproofofunbiasednessWillneedtotakeprobabilitylimit(plim)toestablishconsistency4Economics20-Prof.AndersonProvingConsistency5Economics20-Prof.AndersonAWeakerAssumptionForunbiasedness,weassumedazeroconditionalmeanE(u|x1,x2,xk)=0Forconsistency,wecanhavetheweakerassumptionofzeromeanandzerocorrelationE(u)=0andCov(xj,u)=0,forj=1,2,kWithoutthisassumption,OLSwillbebiasedandinconsistent!
6Economics20-Prof.AndersonDerivingtheInconsistencyJustaswecouldderivetheomittedvariablebiasearlier,nowwewanttothinkabouttheinconsistency,orasymptoticbias,inthiscase7Economics20-Prof.AndersonAsymptoticBias(cont)So,thinkingaboutthedirectionoftheasymptoticbiasisjustlikethinkingaboutthedirectionofbiasforanomittedvariableMaindifferenceisthatasymptoticbiasusesthepopulationvarianceandcovariance,whilebiasusesthesamplecounterpartsRemember,inconsistencyisalargesampleproblemitdoesntgoawayasadddata8Economics20-Prof.AndersonLargeSampleInferenceRecallthatundertheCLMassumptions,thesamplingdistributionsarenormal,sowecouldderivetandFdistributionsfortestingThisexactnormalitywasduetoassumingthepopulationerrordistributionwasnormalThisassumptionofnormalerrorsimpliedthatthedistributionofy,giventhexs,wasnormalaswell9Economics20-Prof.AndersonLargeSampleInference(cont)EasytocomeupwithexamplesforwhichthisexactnormalityassumptionwillfailAnyclearlyskewedvariable,likewages,arrests,savings,etc.cantbenormal,sinceanormaldistributionissymmetricNormalityassumptionnotneededtoconcludeOLSisBLUE,onlyforinference10Economics20-Prof.AndersonCentralLimitTheoremBasedonthecentrallimittheorem,wecanshowthatOLSestimatorsareasymptoticallynormalAsymptoticNormalityimpliesthatP(Zz)F(z)asn,orP(Zz)F(z)Thecentrallimittheoremstatesthatthestandardizedaverageofanypopulationwithmeanmandvariances2isasymptoticallyN(0,1),or11Economics20-Prof.AndersonAsymptoticNormality12Economics20-Prof.AndersonAsymptoticNormality(cont)Becausethetdistributionapproachesthenormaldistributionforlargedf,wecanalsosaythatNotethatwhilewenolongerneedtoassumenormalitywithalargesample,wedostillneedhomoskedasticity13Economics20-Prof.AndersonAsymptoticStandardErrorsIfuisnotnormallydistributed,wesometimeswillrefertothestandarderrorasanasymptoticstandarderror,sinceSo,wecanexpectstandarderrorstoshrinkatarateproportionaltotheinverseofn14Economics20-Prof.AndersonLagrangeMultiplierstatisticOnceweareusinglargesamplesandrelyingonasymptoticnormalityforinference,wecanusemorethattandFstatsTheLagrangemultiplierorLMstatisticisanalternativefortestingmultipleexclusionrestrictionsBecausetheLMstatisticusesanauxiliaryregressionitssometimescalledannR2stat15Economics20-Prof.AndersonLMStatistic(cont)Supposewehaveastandardmodel,y=b0+b1x1+b2x2+.bkxk+uandournullhypothesisisH0:
bk-q+1=0,.,bk=0First,wejustruntherestrictedmodel16Economics20-Prof.AndersonLMStatistic(cont)Withalargesample,theresultfromanFtestandfromanLMtestshouldbesimilarUnliketheFtestandttestforoneexclusion,theLMtestandFtestwillnotbeidentical17Economics20-Prof.AndersonAsymptoticEfficiencyEstimatorsbesidesOLSwillbeconsistentHowever,undertheGauss-Markovassumptions,theOLSestimatorswillhavethesmallestasymptoticvariancesWesaythatOLSisasymptoticallyefficientImportanttorememberourassumptionsthough,ifnothomoskedastic,nottrue18Economics20-Prof.Anderson
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