多元线性回归模型：假设检验

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1、MultipleRegressionAnalysisy=b0+b1x1+b2x2+...bkxk+u2.Inference1AssumptionsoftheClassicalLinearModel(CLM)Sofar,weknowthatgiventheGauss-Markovassumptions,OLSisBLUE,Inordertodoclassicalhypothesistesting,weneedtoaddanotherassumption(beyondtheGauss-Markovassumptions)Assu

2、methatuisindependentofx1,x2,…,xkanduisnormallydistributedwithzeromeanandvariances2:u~Normal(0,s2)2CLMAssumptions(cont)UnderCLM,OLSisnotonlyBLUE,butistheminimumvarianceunbiasedestimatorWecansummarizethepopulationassumptionsofCLMasfollowsy

3、x~Normal(b0+b1x1+…+bkxk,s2)

4、Whilefornowwejustassumenormality,clearthatsometimesnotthecaseLargesampleswillletusdropnormality3..x1x2ThehomoskedasticnormaldistributionwithasingleexplanatoryvariableE(y

5、x)=b0+b1xyf(y

6、x)Normaldistributions4NormalSamplingDistributions5ThetTest6ThetTest(cont)Knowingt

7、hesamplingdistributionforthestandardizedestimatorallowsustocarryouthypothesistestsStartwithanullhypothesisForexample,H0:bj=0Ifacceptnull,thenacceptthatxjhasnoeffectony,controllingforotherx’s7ThetTest(cont)8tTest:One-SidedAlternativesBesidesournull,H0,weneedanaltern

8、ativehypothesis,H1,andasignificancelevelH1maybeone-sided,ortwo-sidedH1:bj>0andH1:bj<0areone-sidedH1:bj0isatwo-sidedalternativeIfwewanttohaveonlya5%probabilityofrejectingH0ifitisreallytrue,thenwesayoursignificancelevelis5%9One-SidedAlternatives(cont)Havingpickedasi

9、gnificancelevel,a,welookupthe(1–a)thpercentileinatdistributionwithn–k–1dfandcallthisc,thecriticalvalueWecanrejectthenullhypothesisifthetstatisticisgreaterthanthecriticalvalueIfthetstatisticislessthanthecriticalvaluethenwefailtorejectthenull10yi=b0+b1xi1+…+bkxik+uiH

10、0:bj=0H1:bj>0c0a(1-a)One-SidedAlternatives(cont)Failtorejectreject11Examples1HourlyWageEquationH0:bexper=0H1:bexper>012One-sidedvsTwo-sidedBecausethetdistributionissymmetric,testingH1:bj<0isstraightforward.ThecriticalvalueisjustthenegativeofbeforeWecanrejectthenull

11、ifthetstatistic<–c,andifthetstatistic>than–cthenwefailtorejectthenullForatwo-sidedtest,wesetthecriticalvaluebasedona/2andrejectH1:bj0iftheabsolu