《计量经济学英》PPT课件

1,MultipleRegressionAnalysis,y=b0+b1x1+b2x2+.bkxk+u2.Inference,2,AssumptionsoftheClassicalLinearModel(CLM),Sofar,weknowthatgiventheGauss-Markovassumptions,OLSisBLUE,Inordertodoclassicalhypothesistesting,weneedtoaddanotherassumption(beyondtheGauss-Markovassumptions)Assumethatuisindependentofx1,x2,xkanduisnormallydistributedwithzeromeanandvariances2:uNormal(0,s2),3,CLMAssumptions(cont),UnderCLM,OLSisnotonlyBLUE,butistheminimumvarianceunbiasedestimatorWecansummarizethepopulationassumptionsofCLMasfollowsy|xNormal(b0+b1x1+bkxk,s2)Whilefornowwejustassumenormality,clearthatsometimesnotthecaseLargesampleswillletusdropnormality,4,.,.,x1,x2,Thehomoskedasticnormaldistributionwithasingleexplanatoryvariable,E(y|x)=b0+b1x,y,f(y|x),Normaldistributions,5,NormalSamplingDistributions,6,ThetTest,7,ThetTest(cont),KnowingthesamplingdistributionforthestandardizedestimatorallowsustocarryouthypothesistestsStartwithanullhypothesisForexample,H0:bj=0Ifacceptnull,thenacceptthatxjhasnoeffectony,controllingforotherxs,8,ThetTest(cont),9,tTest:One-SidedAlternatives,Besidesournull,H0,weneedanalternativehypothesis,H1,andasignificancelevelH1maybeone-sided,ortwo-sidedH1:bj0andH1:bj0,c,0,a,(1-a),One-SidedAlternatives(cont),Failtoreject,reject,12,One-sidedvsTwo-sided,Becausethetdistributionissymmetric,testingH1:bjthancthenwefailtorejectthenullForatwo-sidedtest,wesetthecriticalvaluebasedona/2andrejectH1:bj0iftheabsolutevalueofthetstatisticc,13,yi=b0+b1Xi1+bkXik+uiH0:bj=0H1:bj0,c,0,a/2,(1-a),-c,a/2,Two-SidedAlternatives,reject,reject,failtoreject,14,SummaryforH0:bj=0,Unlessotherwisestated,thealternativeisassumedtobetwo-sidedIfwerejectthenull,wetypicallysay“xjisstatisticallysignificantatthea%level”Ifwefailtorejectthenull,wetypicallysay“xjisstatisticallyinsignificantatthea%level”,15,Testingotherhypotheses,AmoregeneralformofthetstatisticrecognizesthatwemaywanttotestsomethinglikeH0:bj=ajInthiscase,theappropriatetstatisticis,16,ConfidenceIntervals,Anotherwaytouseclassicalstatisticaltestingistoconstructaconfidenceintervalusingthesamecriticalvalueaswasusedforatwo-sidedtestA(1-a)%confidenceintervalisdefinedas,17,Computingp-valuesforttests,Analternativetotheclassicalapproachistoask,“whatisthesmallestsignificancelevelatwhichthenullwouldberejected?”So,computethetstatistic,andthenlookupwhatpercentileitisintheappropriatetdistributionthisisthep-valuep-valueistheprobabilitywewouldobservethetstatisticwedid,ifthenullweretrue,18,Stataandp-values,ttests,etc.,Mostcomputerpackageswillcomputethep-valueforyou,assumingatwo-sidedtestIfyoureallywantaone-sidedalternative,justdividethetwo-sidedp-valueby2Stataprovidesthetstatistic,p-value,and95%confidenceintervalforH0:bj=0foryou,incolumnslabeled“t”,“P|t|”and“95%Conf.Interval”,respectively,19,TestingaLinearCombination,Supposeinsteadoftestingwhetherb1isequaltoaconstant,youwanttotestifitisequaltoanotherparameter,thatisH0:b1=b2Usesamebasicprocedureforformingatstatistic,20,TestingLinearCombo(cont),21,TestingaLinearCombo(cont),So,touseformula,needs12,whichstandardoutputdoesnothaveManypackageswillhaveanoptiontogetit,orwilljustperformthetestforyouInStata,afterregyx1x2xkyouwouldtypetestx1=x2togetap-valueforthetestMoregenerally,youcanalwaysrestatetheproblemtogetthetestyouwant,22,Example:,SupposeyouareinterestedintheeffectofcampaignexpendituresonoutcomesModelisvoteA=b0+b1log(expendA)+b2log(expendB)+b3prtystrA+uH0:b1=-b2,orH0:q1=b1+b2=0b1=q1b2,sosubstituteinandrearrangevoteA=b0+q1log(expendA)+b2log(expendB-expendA)+b3prtystrA+u,23,Example(cont):,Thisisthesamemodelasoriginally,butnowyougetastandarderrorforb1b2=q1directlyfromthebasicregressionAnylinearcombinationofparameterscouldbetestedinasimilarmannerOtherexamplesofhypothesesaboutasinglelinearcombinationofparameters:b1=1+b2;b1=5b2;b1=-1/2b2;etc,24,MultipleLinearRestrictions,Everythingwevedonesofarhasinvolvedtestingasinglelinearrestriction,(e.g.b1=0orb1=b2)However,wemaywanttojointlytestmultiplehypothesesaboutourparametersAtypicalexampleistesting“exclusionrestrictions”wewanttoknowifagroupofparametersareallequaltozero,25,TestingExclusionRestrictions,NowthenullhypothesismightbesomethinglikeH0:bk-q+1=0,.,bk=0ThealternativeisjustH1:H0isnottrueCantjustcheckeachtstatisticseparately,becausewewanttoknowiftheqparametersarejointlysignificantatagivenlevelitispossiblefornonetobeindividuallysignificantatthatlevel,26,ExclusionRestrictions(cont),Todothetestweneedtoestimatethe“restrictedmodel”withoutxk-q+1,xkincluded,aswellasthe“unrestrictedmodel”withallxsincludedIntuitively,wewanttoknowifthechangeinSSRisbigenoughtowarrantinclusionofxk-q+1,xk,27,TheFstatistic,TheFstatisticisalwayspositive,sincetheSSRfromtherestrictedmodelcantbelessthantheSSRfromtheunrestrictedEssentiallytheFstatisticismeasuringtherelativeincreaseinSSRwhenmovingfromtheunrestrictedtorestrictedmodelq=numberofrestrictions,ordfrdfurnk1=dfur,28,TheFstatistic(cont),TodecideiftheincreaseinSSRwhenwemovetoarestrictedmodelis“bigenough”torejecttheexclusions,weneedtoknowaboutthesamplingdistributionofourFstatNotsurprisingly,FFq,n-k-1,whereqisreferredtoasthenumeratordegreesoffreedomandnk1asthedenominatordegreesoffreedom,29,0,c,a,(1-a),f(F),F,TheFstatistic(cont),reject,failtoreject,RejectH0atasignificancelevelifFc,30,TheR2formoftheFstatistic,BecausetheSSRsmaybelargeandunwieldy,analternativeformoftheformulaisusefulWeusethefactthatSSR=SST(1R2)foranyregression,socansubstituteinforSSRuandSSRur,31,OverallSignificance,AspecialcaseofexclusionrestrictionsistotestH0:b1=b2=bk=0SincetheR2fromamodelwithonlyaninterceptwillbezero,theFstatisticissimply,32,GeneralLinearRestrictions,ThebasicformoftheFstatisticwillworkforanysetoflinearrestrictionsFirstestimatetheunrestrictedmodelandthenestimatetherestrictedmode