上海金融学院计量经济学ch

可编辑1MultipleRegressionAnalysis

y=b0+b1x1+b2x2+...bkxk+u3.AsymptoticProperties可编辑2ConsistencyUndertheGauss-MarkovassumptionsOLSisBLUE,butinothercasesitwon’talwaysbepossibletofindunbiasedestimatorsInthosecases,wemaysettleforestimatorsthatareconsistent,meaningasn

∞,thedistributionoftheestimatorcollapsestotheparametervalue可编辑3SamplingDistributionsasn

b1n1n2n3n1<n2<n3可编辑4ConsistencyofOLSUndertheGauss-Markovassumptions,theOLSestimatorisconsistent(andunbiased)ConsistencycanbeprovedforthesimpleregressioncaseinamannersimilartotheproofofunbiasednessWillneedtotakeprobabilitylimit(plim)toestablishconsistency可编辑5ProvingConsistency可编辑6AWeakerAssumptionForunbiasedness,weassumedazeroconditionalmean–E(u|x1,x2,…,xk)=0Forconsistency,wecanhavetheweakerassumptionofzeromeanandzerocorrelation–E(u)=0andCov(xj,u)=0,forj=1,2,…,kWithoutthisassumption,OLSwillbebiasedandinconsistent!可编辑7DerivingtheInconsistencyJustaswecouldderivetheomittedvariablebiasearlier,nowwewanttothinkabouttheinconsistency,orasymptoticbias,inthiscase可编辑8AsymptoticBias(cont)So,thinkingaboutthedirectionoftheasymptoticbiasisjustlikethinkingaboutthedirectionofbiasforanomittedvariableMaindifferenceisthatasymptoticbiasusesthepopulationvarianceandcovariance,whilebiasusesthesamplecounterpartsRemember,inconsistencyisalargesampleproblem–itdoesn’tgoawayasadddata可编辑9LargeSampleInferenceRecallthatundertheCLMassumptions,thesamplingdistributionsarenormal,sowecouldderivetandFdistributionsfortestingThisexactnormalitywasduetoassumingthepopulationerrordistributionwasnormalThisassumptionofnormalerrorsimpliedthatthedistributionofy,giventhex’s,wasnormalaswell可编辑10LargeSampleInference(cont)EasytocomeupwithexamplesforwhichthisexactnormalityassumptionwillfailAnyclearlyskewedvariable,likewages,arrests,savings,etc.can’tbenormal,sinceanormaldistributionissymmetricNormalityassumptionnotneededtoconcludeOLSisBLUE,onlyforinference2023/12/811可编辑12CentralLimitTheoremBasedonthecentrallimittheorem,wecanshowthatOLSestimatorsareasymptoticallynormalAsymptoticNormalityimpliesthatP(Z<z)F(z)asn,orP(Z<z)F(z)Thecentrallimittheoremstatesthatthestandardizedaverageofanypopulationwithmeanmandvariances2isasymptotically~N(0,1),or可编辑13AsymptoticNormality可编辑14AsymptoticNormality(cont)

Becausethetdistributionapproachesthenormaldistributionforlargedf,wecanalsosaythat

Notethatwhilewenolongerneedtoassumenormalitywithalargesample,wedostillneedhomoskedasticity可编辑15AsymptoticStandardErrors

Ifuisnotnormallydistributed,wesometimeswillrefertothestandarderrorasanasymptoticstandarderror,sinceSo,wecanexpectstandarderrorstoshrinkatarateproportionaltotheinverseof√n可编辑16LagrangeMultiplierstatisticOnceweareusinglargesamplesandrelyingonasymptoticnormalityforinference,wecanusemorethattandFstatsTheLagrangemultiplierorLMstatisticisanalternativefortestingmultipleexclusionrestrictionsBecausetheLMstatisticusesanauxiliaryregressionit’ssometimescalledannR2stat可编辑17LMStatistic(cont)Supposewehaveastandardmodel,y=b0+b1x1+b2x2+...bkxk+uandournullhypothesisisH0:bk-q+1=0,...,bk=0First,wejustruntherestrictedmodel可编辑18LMStatistic(cont)Withalargesample,theresultfromanFtestandfromanLMtestshouldbesimilarUnliketheFtestandttestforoneexclusion,theLMtestandFtestwillnotbeidentical可编辑19AsymptoticEfficiencyEstimatorsbesidesOLSwillbeconsistentHowever,undertheGauss-Mar