多元线性回归模型：假设检验

1MultipleRegressionAnalysisy=b0+b1x1+b2x2+...bkxk+u2.Inference2AssumptionsoftheClassicalLinearModel(CLM)Sofar,weknowthatgiventheGauss-Markovassumptions,OLSisBLUE,Inordertodoclassicalhypothesistesting,weneedtoaddanotherassumption(beyondtheGauss-Markovassumptions)Assumethatuisindependentofx1,x2,…,xkanduisnormallydistributedwithzeromeanandvariances2:u~Normal(0,s2)3CLMAssumptions(cont)UnderCLM,OLSisnotonlyBLUE,butistheminimumvarianceunbiasedestimatorWecansummarizethepopulationassumptionsofCLMasfollowsy|x~Normal(b0+b1x1+…+bkxk,s2)Whilefornowwejustassumenormality,clearthatsometimesnotthecaseLargesampleswillletusdropnormality4..x1x2ThehomoskedasticnormaldistributionwithasingleexplanatoryvariableE(y|x)=b0+b1xyf(y|x)Normaldistributions5NormalSamplingDistributionserrorstheofncombinatiolinearaisitbecausenormallyddistributeisˆ0,1Normal~ˆˆthatso,ˆ,Normal~ˆstvariableindependentheofvaluessampletheonlconditionas,assumptionCLMUnderthejbbbbbbbjjjjjjsdVar6ThetTest1:freedomofdegreestheNoteˆbyestimatetohavewebecausenormal)(vsondistributiaisthisNote~ˆˆsassumptionCLMUnderthe221jknttseknjjssbbb7ThetTest(cont)KnowingthesamplingdistributionforthestandardizedestimatorallowsustocarryouthypothesistestsStartwithanullhypothesisForexample,H0:bj=0Ifacceptnull,thenacceptthatxjhasnoeffectony,controllingforotherx’s8ThetTest(cont)0ˆjH,hypothesisnullaccepttheowhethertdeterminetorulerejectionawithalongstatisticourusethenwillWeˆˆ:ˆforstatistictheformtoneedfirsteourtestwperformTotsettjjjbbbb9tTest:One-SidedAlternativesBesidesournull,H0,weneedanalternativehypothesis,H1,andasignificancelevelH1maybeone-sided,ortwo-sidedH1:bj0andH1:bj0areone-sidedH1:bj0isatwo-sidedalternativeIfwewanttohaveonlya5%probabilityofrejectingH0ifitisreallytrue,thenwesayoursignificancelevelis5%10One-SidedAlternatives(cont)Havingpickedasignificancelevel,a,welookupthe(1–a)thpercentileinatdistributionwithn–k–1dfandcallthisc,thecriticalvalueWecanrejectthenullhypothesisifthetstatisticisgreaterthanthecriticalvalueIfthetstatisticislessthanthecriticalvaluethenwefailtorejectthenull11yi=b0+b1xi1+…+bkxik+uiH0:bj=0H1:bj0c0a1aOne-SidedAlternatives(cont)Failtorejectreject12Examples1HourlyWageEquationH0:bexper=0H1:bexper0316.0526)003.0()0017.0()007.0()104.0(022.0exp0041.0092.0284.0)ˆlog(2Rntenureereducgeaw13One-sidedvsTwo-sidedBecausethetdistributionissymmetric,testingH1:bj0isstraightforward.ThecriticalvalueisjustthenegativeofbeforeWecanrejectthenullifthetstatistic–c,andifthetstatisticthan–cthenwefailtorejectthenullForatwo-sidedtest,wesetthecriticalvaluebasedona/2andrejectH1:bj0iftheabsolutevalueofthetstatisticc14yi=b0+b1Xi1+…+bkXik+uiH0:bj=0H1:bj≠0c0a/21a-ca/2Two-SidedAlternativesrejectrejectfailtoreject15SummaryforH0:bj=0Unlessotherwisestated,thealternativeisassumedtobetwo-sidedIfwerejectthenull,wetypicallysay“xjisstatisticallysignificantatthea%level”Ifwefailtorejectthenull,wetypicallysay“xjisstatisticallyinsignificantatthea%level”16Examples2DeterminantsofCollegeGPAcolGPA—collegeGPA(greatpointaverage),hsGPA—highschoolGPAskipped—averagenumbersofleturesmissedperweek.234.0141)026.0()011.0()094.0()33.0(083.0015.0412.039.1ˆ2RnskippedACThsGPAAPcolG17TestingotherhypothesesAmoregeneralformofthetstatisticrecognizesthatwemaywanttotestsomethinglikeH0:bj=ajInthiscase,theappropriatetstatisticisteststandardforthe0where,ˆˆjjjjaseatbb18Examples3CampusCrimeandEnrollmentH0:benroll=1H1:benroll1585.0)11.0()03.1(97)log(27.163.6)ˆlog(2Rnenrollmeicruenrollcrime)log()log(10bb19Examples4HousingPricesandAirPollutionH0:blog(nox)=-1H1:blog(nox)≠-1581.0506)006.0()019.0()043.0()117.0()32.0(052.0255.0)log(134.0)log(954.008.11)log(2Rnstratioroomsdistnoxpriceustratioroomsdistnoxprice43210)log()log()log(bbbbb20ConfidenceIntervalsAnotherwaytouseclassicalstatisticaltestingistoconstructaconfidenceintervalusingthesamecriticalvalueaswasusedforatwo-sidedtestA(1-a)%confidenceintervalisdefinedasondistributiainpercentile2-1theiscwhere,ˆˆ1knjjtsecabb21Computingp-valuesforttestsAnalternativetotheclassicalapproachistoask,“whatisthesmallestsignificancelevelatwhichthenullwouldberejected?”So,computethetstatistic,andthenlookupwhatpercentileitisintheappropriatetdistribution–thisisthep-valuep-valueistheprobabilitywewouldobservethetstatisticwedid,ifthenullweretrue22Mostcomputerpackageswillcomputethep-valueforyou,assumingatwo-sidedtestIfyoureallywantaone-sidedalternative,justdividethetwo-sidedp-valueby2Manysoftware,suchasStataorEviewsprovidesthetstatistic,p-value,and95%confidenceintervalforH0:bj=0foryou23TestingaLinearCombinationSupposeinsteadoftestingwhetherb1isequaltoaconstant,youwanttotestifitisequaltoanotherparameter,thatisH0:b1=b2Usesamebasicprocedureforformingatstatistic2121ˆˆˆˆbbbbset24TestingLinearCombo(cont)211221122221212121212121ˆ,ˆofestimateaniswhere2ˆˆˆˆˆ,ˆ2ˆˆˆˆthen,ˆˆˆˆSincebbbbbbbbbbbbbbbbCovssseseseCovVarVarVarVarse25TestingaLinearCombo(cont)So,touseformula,needs12,whichstandardoutputdoesnothave