上海金融学院计量经济学课件

2,PPT交流学习,1,MultipleRegressionAnalysis,y=b0+b1x1+b2x2+.bkxk+u3.AsymptoticProperties,06.05.2020,Economics20-Prof.Anderson,2,Consistency,UndertheGauss-MarkovassumptionsOLSisBLUE,butinothercasesitwontalwaysbepossibletofindunbiasedestimatorsInthosecases,wemaysettleforestimatorsthatareconsistent,meaningasn,thedistributionoftheestimatorcollapsestotheparametervalue,06.05.2020,Economics20-Prof.Anderson,3,SamplingDistributionsasn,b1,n1,n2,n3,n1n2n3,06.05.2020,Economics20-Prof.Anderson,4,ConsistencyofOLS,UndertheGauss-Markovassumptions,theOLSestimatorisconsistent(andunbiased)ConsistencycanbeprovedforthesimpleregressioncaseinamannersimilartotheproofofunbiasednessWillneedtotakeprobabilitylimit(plim)toestablishconsistency,06.05.2020,Economics20-Prof.Anderson,5,ProvingConsistency,06.05.2020,Economics20-Prof.Anderson,6,AWeakerAssumption,Forunbiasedness,weassumedazeroconditionalmeanE(u|x1,x2,xk)=0Forconsistency,wecanhavetheweakerassumptionofzeromeanandzerocorrelationE(u)=0andCov(xj,u)=0,forj=1,2,kWithoutthisassumption,OLSwillbebiasedandinconsistent!,06.05.2020,Economics20-Prof.Anderson,7,DerivingtheInconsistency,Justaswecouldderivetheomittedvariablebiasearlier,nowwewanttothinkabouttheinconsistency,orasymptoticbias,inthiscase,06.05.2020,Economics20-Prof.Anderson,8,AsymptoticBias(cont),So,thinkingaboutthedirectionoftheasymptoticbiasisjustlikethinkingaboutthedirectionofbiasforanomittedvariableMaindifferenceisthatasymptoticbiasusesthepopulationvarianceandcovariance,whilebiasusesthesamplecounterpartsRemember,inconsistencyisalargesampleproblemitdoesntgoawayasadddata,06.05.2020,Economics20-Prof.Anderson,9,LargeSampleInference,RecallthatundertheCLMassumptions,thesamplingdistributionsarenormal,sowecouldderivetandFdistributionsfortestingThisexactnormalitywasduetoassumingthepopulationerrordistributionwasnormalThisassumptionofnormalerrorsimpliedthatthedistributionofy,giventhexs,wasnormalaswell,06.05.2020,Economics20-Prof.Anderson,10,LargeSampleInference(cont),EasytocomeupwithexamplesforwhichthisexactnormalityassumptionwillfailAnyclearlyskewedvariable,likewages,arrests,savings,etc.cantbenormal,sinceanormaldistributionissymmetricNormalityassumptionnotneededtoconcludeOLSisBLUE,onlyforinference,06.05.2020,Economics20-Prof.Anderson,11,CentralLimitTheorem,Basedonthecentrallimittheorem,wecanshowthatOLSestimatorsareasymptoticallynormalAsymptoticNormalityimpliesthatP(Zz)F(z)asn,orP(Zz)F(z)Thecentrallimittheoremstatesthatthestandardizedaverageofanypopulationwithmeanmandvariances2isasymptoticallyN(0,1),or,06.05.2020,Economics20-Prof.Anderson,12,AsymptoticNormality,06.05.2020,Economics20-Prof.Anderson,13,AsymptoticNormality(cont),Becausethetdistributionapproachesthenormaldistributionforlargedf,wecanalsosaythat,Notethatwhilewenolongerneedtoassumenormalitywithalargesample,wedostillneedhomoskedasticity,06.05.2020,Economics20-Prof.Anderson,14,AsymptoticStandardErrors,Ifuisnotnormallydistributed,wesometimeswillrefertothestandarderrorasanasymptoticstandarderror,since,So,wecanexpectstandarderrorstoshrinkatarateproportionaltotheinverseofn,06.05.2020,Economics20-Prof.Anderson,15,LagrangeMultiplierstatistic,Onceweareusinglargesamplesandrelyingonasymptoticnormalityforinference,wecanusemorethattandFstatsTheLagrangemultiplierorLMstatisticisanalternativefortestingmultipleexclusionrestrictionsBecausetheLMstatisticusesanauxiliaryregressionitssometimescalledannR2stat,06.05.2020,Economics20-Prof.Anderson,16,LMStatistic(cont),Supposewehaveastandardmodel,y=b0+b1x1+b2x2+.bkxk+uandournullhypothesisisH0:bk-q+1=0,.,bk=0First,wejustruntherestrictedmodel,06.05.2020,Economics20-Prof.Anderson,17,LMStatistic(cont),Withalargesample,theresultfromanFtestandfromanLMtestshouldbesimilarUnliketheFtestandttestforoneexclusion,theLMtestandFtestwillnotbeidentical,06.05.2020,Economics20-Prof.Anderson,18,AsymptoticEfficiency,EstimatorsbesidesOLSwillbeconsistentHoweve