多元回归分析OLS的渐进性课件

多元回归分析：渐进性

y=b0+b1x1+b2x2+...bkxk+u1Economics20-Prof.AndersonPte.Ltd.Allrightsreserved.渐进性的含义如果误差并非正态分布，对任何的样本容量而言，t统计量、F统计量并非恰好服从t分布、F分布。幸运的是，即使没有正态性假定，t统计量和F统计量仍然渐进的服从t分布、F分布，，至少在大样本情况下使如此。2Economics20-Prof.Anderson一致性在高斯-马尔科夫假定下，OLS估计是最优线性无偏的，但我们并非总能得到无偏的估计量。一致性是对一个估计量最起码的要求。在无法满足无偏性的情况下，我们可以搜集尽可能多的样本，即使n→

∞,参数估计值的分布将逼近真实参数值。3Economics20-Prof.Anderson一致性的正式定义4Economics20-Prof.Anderson一致性的直观理解5Economics20-Prof.Anderson当样本容量增加时的样本分布6Economics20-Prof.Anderson7Economics20-Prof.AndersonOLS的一致性在高斯-马尔科夫假定下，OLS估计值是一致且无偏的。类似的，我们可以像无偏性一样证明一致性，为此需要引入概率极限。8Economics20-Prof.Anderson简单回归中证明一致性9Economics20-Prof.Anderson一个较弱的假定为了得到无偏性，我们需要零条件均值假设E(u|x1,x2,…,xk)=0为了得到一致性，我们仅需要较弱的假定：零均值和零相关:E(u)=0,Cov(xj,u)=0,forj=1,2,…,k.不满足上述条件，OLS是有偏和不一致的。10Economics20-Prof.AndersonDerivingtheInconsistencyJustaswecouldderivetheomittedvariablebiasearlier,nowwewanttothinkabouttheinconsistency,orasymptoticbias,inthiscase11Economics20-Prof.AndersonAsymptoticBias(cont)So,thinkingaboutthedirectionoftheasymptoticbiasisjustlikethinkingaboutthedirectionofbiasforanomittedvariableMaindifferenceisthatasymptoticbiasusesthepopulationvarianceandcovariance,whilebiasusesthesamplecounterpartsRemember,inconsistencyisalargesampleproblem–itdoesn’tgoawayasadddata12Economics20-Prof.AndersonSummaryofDirectionofAsymptoticbiasCorr(x1,x2)>0Corr(x1,x2)<0b2>0PositiveasymptoticbiasNegative

asymptoticbiasb2<0NegativeasymptoticbiasPositiveasymptoticbias13Economics20-Prof.AndersonLargeSampleInferenceRecallthatundertheCLMassumptions,thesamplingdistributionsarenormal,sowecouldderivetandFdistributionsfortestingThisexactnormalitywasduetoassumingthepopulationerrordistributionwasnormalThisassumptionofnormalerrorsimpliedthatthedistributionofy,giventhex’s,wasnormalaswell14Economics20-Prof.AndersonLargeSampleInference(cont)EasytocomeupwithexamplesforwhichthisexactnormalityassumptionwillfailAnyclearlyskewedvariable,likewages,arrests,savings,etc.can’tbenormal,sinceanormaldistributionissymmetricNormalityassumptionnotneededtoconcludeOLSisBLUE,onlyforinference15Economics20-Prof.AndersonCentralLimitTheoremBasedonthecentrallimittheorem,wecanshowthatOLSestimatorsareasymptoticallynormalAsymptoticNormalityimpliesthatP(Z<z)F(z)asn,orP(Z<z)F(z)（标准正态累积分布函数）Thecentrallimittheoremstatesthatthestandardizedaverageofanypopulationwithmeanmandvariances2isasymptotically~N(0,1),or16Economics20-Prof.AndersonTheorem5.2AsymptoticNormality17Economics20-Prof.Anderson18Economics20-Prof.AndersonLawoflargenumbers19Economics20-Prof.AndersonAsymptoticNormality(cont)

Becausethetdistributionapproachesthenormaldistributionforlargedf,wecanalsosaythat

Notethatwhilewenolongerneedtoassumenormalitywithalargesample,wedostillneedhomoskedasticity20Economics20-Prof.AndersonAsymptoticStandardErrors

Ifuisnotnormallydistributed,wesometimeswillrefertothestandarderrorasanasymptoticstandarderror,since

So,wecanexpectstandarderrorstoshrinkatarateproportionaltotheinverseof√n21Economics20-Prof.AndersonLagrangeMultiplierstatisticOnceweareusinglargesamplesandrelyingonasymptoticnormalityforinference,wecanusemorethattandFstatsTheLagrangemultiplierorLMstatisticisanalternativefortestingmultipleexclusionrestrictionsBecausetheLMstatisticusesanauxiliaryregressionit’ssometimescalledannR2stat22Economics20-Prof.AndersonLMStatistic(cont)

Supposewehaveastandardmodel,y=b0+b1x1+b2x2+...bkxk+uandournullhypothesisisH0:bk-q+1=0,...,bk=0First,wejustruntherestrictedmodel23Economics20-Prof.AndersonTheideaofLMstatisticIftheomittedvariablesxk-q+1throughxktrulyhavezeropopulationcoefficientsthen,atleastapproximately,shouldbeuncorrelatedwitheachofthesevariablesinthesample.RunningaregressionoftheseresidualsonthoseindependentvariablesexcludedunderH0,weshouldgetasmallenoughR2.However,wemustincludealloftheindependentvariablesintheregressionfortechnicalreasons.24Economics20-Prof.AndersonLMStatistic(cont)Withalargesample,theresultfromanFtestandfromanLMtestshouldbesimilar.LM>c,rejectH25Economics20-Prof.Anderson26Economics20-Prof.AndersonExample:EconomicModelofCrimeNarr86为一个人被拘捕的次数；Pcnv为以前被拘捕后被定罪的次数；Avgsen为过去定罪后被判刑的平均时间长度；Tottime为此人在年龄达到18岁后在1986年以前被送进监狱的总次数；Ptime86为1986年坐牢的月数；Qemp86为此人在1986年合法就业的季度数。27Economics20-Prof.AndersonAsymptoticEfficiency

EstimatorsbesidesOLSwillbeconsistentHowever,undertheGauss-Markovassumptions,theOLSestimatorswillhavethesmallestasymptoticvariancesWesaythatOLSisasymptoticallyefficientImportanttoreme