《计量经济学》ch-04-wooldridg

Chapter4,MultipleRegressionAnalysis:Inference,Wooldridge:IntroductoryEconometrics:AModernApproach,5eInstructedbyprofessorYuan,Huiping,Chapter4MultipleRegressionAnalysis:Inference,4.2TestingHypothesesaboutaSinglePopulationParameter:ThetTest,4.3ConfidenceIntervals,4.4TestingHypothesesaboutaSingleLinearCombinationoftheParameters,4.5TestingMultipleLinearRestrictions:TheFTest,4.1SamplingDistributionsoftheOLSEstimators,4.6AnapplicationestimationoftheweightsofCPIcomponentsinChina,Assignments:Promblems1,2,4,5,7,8,10ComputerExercisesC1,C2,C3,C8,C9C8:smplifmarr=1andfsize=2(401ksubs.wf1),TheEnd,StatisticalinferenceintheregressionmodelHypothesistestsaboutpopulationparametersConstructionofconfidenceintervalsSamplingdistributionsoftheOLSestimatorsTheOLSestimatorsarerandomvariablesWealreadyknowtheirexpectedvaluesandtheirvariancesHowever,forhypothesistestsweneedtoknowtheirdistributionInordertoderivetheirdistributionweneedadditionalassumptionsAssumptionaboutdistributionoferrors:normaldistribution,Chapter4MultipleRegressionAnalysis:Inference,4.1SamplingDistributionsoftheOLSEstimators(1/5),Chapter,End,AssumptionMLR.6(Normalityoferrorterms),independentlyof,Itisassumedthattheunobservedfactorsarenormallydistributedaroundthepopulationregressionfunction.Theformandthevarianceofthedistributiondoesnotdependonanyoftheexplanatoryvariables.Itfollowsthat:,Chapter4MultipleRegressionAnalysis:Inference,4.1SamplingDistributionsoftheOLSEstimators(2/5),Chapter,End,DiscussionofthenormalityassumptionTheerrortermisthesumofmany“differentunobservedfactorsSumsofindependentfactorsarenormallydistributed(CLT)Problems:Howmanydifferentfactors?Numberlargeenough?PossiblyveryheterogenuousdistributionsofindividualfactorsHowindependentarethedifferentfactors?ThenormalityoftheerrortermisanempiricalquestionAtleasttheerrordistributionshouldbeclose“tonormal,Chapter4MultipleRegressionAnalysis:Inference,4.1SamplingDistributionsoftheOLSEstimators(3/5),Chapter,End,Discussionofthenormalityassumption(cont.)Exampleswherenormalitycannothold:Wages(nonnegative;also:minimumwage)Numberofarrests(takesonasmallnumberofintegervalues)Unemployment(indicatorvariable,takesononly1or0)Insomecases,normalitycanbeachievedthroughtransformationsofthedependentvariable(e.g.uselog(wage)insteadofwage)Important:Forthepurposesofstatisticalinference,theassumptionofnormalitycanbereplacedbyalargesamplesize,Chapter4MultipleRegressionAnalysis:Inference,4.1SamplingDistributionsoftheOLSEstimators(4/5),Chapter,End,TerminologyTheorem4.1(Normalsamplingdistributions),UnderassumptionsMLR.1MLR.6:,Theestimatorsarenormallydistributedaroundthetrueparameterswiththevariancethatwasderivedearlier,Thestandardizedestimatorsfollowastandardnormaldistribution,Gauss-Markovassumptions“,Classicallinearmodel(CLM)assumptions“,Chapter4MultipleRegressionAnalysis:Inference,4.1SamplingDistributionsoftheOLSEstimators(5/5),Chapter,End,4.2.1Theorem4.2tDistributionfortheStandardizedEstimators,Chapter4MultipleRegressionAnalysis:Inference,4.2TestingHypothesesaboutaSinglePopulationParameter:ThetTest,4.2.3Two-SidedAlternatives,4.2.4TestingOtherHypothesesaboutbj,4.2.2TestingagainstOne-SidedAlternatives,4.2.5Computingp-ValuesfortTests,4.2.6AReminderontheLanguageofClassicalHypothesisTesting,4.2.7Economic,orPractical,versusStatisticalSignificance,Chapter,End,UnderassumptionsMLR.1MLR.6:,Ifthestandardizationisdoneusingtheestimatedstandarddeviation(=standarderror),thenormaldistributionisreplacedbyat-distribution,Note:Thet-distributionisclosetothestandardnormaldistributionifn-k-1islarge.,Chapter4MultipleRegressionAnalysis:Inference,4.2.1Theorem4.2tDistributionfortheStandardizedEstimators(1/3),Proof:,Section,Chapter,End,Nullhypothesis(formoregeneralhypotheses,seebelow)t-statistic(ort-ratio)Distributionofthet-statisticifthenullhypothesisistrue,Thet-statisticwillbeusedtotesttheabovenullhypothesis.Thefarthertheestimatedcoefficientisawayfromzero,thelesslikelyitisthatthenullhypothesisholdstrue.Butwhatdoesfar“awayfromzeromean?,Thisdependsonthevariabilityoftheestimatedcoefficient,i.e.itsstandarddeviation.Thet-statisticmeasureshowmanyestimatedstandarddeviationstheestimatedcoefficientisawayfromzero.,Thepopulationparameterisequaltozero,i.e.aftercontrollingfortheotherindependentvariables,thereisnoeffectofxjony,Chapter4MultipleRegressionAnalysis:Inference,4.2.1Theorem4.2tDistributionfortheStandardizedEstimators(2/3),Section,Chapter,End,Goal:Definearejectionrulesothat,ifitistrue,H0isrejectedonlywithasmallprobability(=significancelevel,e.g.5%),Thepreciserejectionruledependsonthealternativehypothesisandthechosensignificancelevelofthetest.Asignificancelevel:theprobabilityofrejectingH0whenitistrue.,Chapter4MultipleRegressionAnalysis:Inference,4.2.1Theorem4.2tDistributionfortheStandardizedEstimators(3/3),Section,Chapter,End,Testagainst.,Testingagainstone-sidedalternatives(greaterthanzero),4.2.2TestingagainstOne-SidedAlternatives(1/8),Rejectthenullhypothesisinfavourofthealternativehypothesisiftheestimatedcoefficientistoolarge“(i.e.largerthanacriticalvalue).Constructthecriticalvaluesothat,ifthenullhypothesisistrue,itisrejectedin,forexample,5%ofthecases.Inthegivenexample,thisisthepointofthet-distributionwith28degreesoffreedomthatisexceededin5%ofthecases.!Rejectift-statisticgreaterthan1.701,Chapter4MultipleRegressionAnalysis:Inference,Section,Chapter,End,Example:WageequationTestwhether,aftercontrollingforeducationandtenure,higherworkexperienceleadstohigherhourlywages,(1)Testagainst.,Onewouldeitherexpectapositiveeffectofexperienceonhourlywageornoeffectatall.,Standarderrors,4.2.2TestingagainstOne-SidedAlternatives(2/8),Chapter4MultipleRegressionAnalysis:Inference,Section,Chapter,End,Example:Wageequation(cont.),Theeffectofexperienceonhourlywageisstatisticallygreaterthanzeroatthe5%(andevenatthe1%)significancelevel.“Thoughttheestimatedreturnforanotheryearofexperience,holdingtenureandeducationfixed,isnotespeciallylarge,wehavepersuasivelyshownthatthepartialeffectofexperienceispositiveinthepopulation.,t-statistic,Criticalvaluesforthe5%andthe1%significancelevel(theseareconventionalsignificancelevels).Thenullhypothesisisrejectedbecausethet-statisticexceedsthecriticalvalue.,(2),Degreesoffreedom;herethestandardnormalapproximationapplies,(3),(4),4.2.2TestingagainstOne-SidedAlternatives(3/8),Chapter4MultipleRegressionAnalysis:Inference,Section,Chapter,End,Testagainst.,Testingagainstone-sidedalternatives(lessthanzero),Rejectthenullhypothesisinfavourofthealternativehypothesisiftheestimatedcoefficientistoosmall“(i.e.smallerthanacriticalvalue).Constructthecriticalvaluesothat,ifthenullhypothesisistrue,itisrejectedin,forexample,5%ofthecases.Inthegivenexample,thisisthepointofthet-distributionwith18degreesoffreedomsothat5%ofthecasesarebelowthepoint.!Rejectift-statisticlessthan-1.734,4.2.2TestingagainstOne-SidedAlternatives(4/8),Chapter4MultipleRegressionAnalysis:Inference,Section,Chapter,End,Example:StudentperformanceandschoolsizeTestwhethersmallerschoolsizeleadstobetterstudentperformance,Testagainst.,Dolargerschoolshamperstudentperformanceoristherenosucheffect?,Percentageofstudentspassingmathstest,Averageannualtea-chercompensation,Schoolenrollment(=schoolsize),Staffperonethousandstudents,4.2.2TestingagainstOne-SidedAlternatives(5/8),Chapter4MultipleRegressionAnalysis:Inference,Section,Chapter,End,Example:Studentperformanceandschoolsize(cont.),Onecannotrejectthehypothesisthatthereisnoeffectofschoolsizeonstudentperformance(notevenforalaxsignificancelevelof15%).,t-statistic,Criticalvaluesforthe5%andthe15%significancelevel.Thenullhypothesisisnotrejectedbecausethet-statisticisnotsmallerthanthecriticalvalue.,Degreesoffreedom;herethestandardnormalapproximationapplies,4.2.2TestingagainstOne-SidedAlternatives(6/8),Chapter4MultipleRegressionAnalysis:Inference,Section,Chapter,End,Example:Studentperformanceandschoolsize(cont.)Alternativespecificationoffunctionalform:,Testagainst.,R-squaredslightlyhigher,4.2.2TestingagainstOne-SidedAlternatives(7/8),Chapter4MultipleRegressionAnalysis:Inference,Section,Chapter,End,Example:Studentperformanceandschoolsize(cont.),Thehypothesisthatthereisnoeffectofschoolsizeonstudentperformancecanberejectedinfavorofthehypothesisthattheeffectisnegative.,t-statistic,Criticalvalueforthe5%significancelevel!rejectnullhypothesis,Howlargeistheeffect?,(smalleffect),+10%enrollment!-0.129percentagepointsstudentspasstest,4.2.2TestingagainstOne-SidedAlternatives(8/8),Chapter4MultipleRegressionAnalysis:Inference,Section,Chapter,End,Testingagainsttwo-sidedalternatives,Testagainst.,Rejectthenullhypothesisinfavourofthealternativehypothesisiftheabsolutevalueoftheestimatedcoefficientistoolarge.Constructthecriticalvaluesothat,ifthenullhypothesisistrue,itisrejectedin,forexample,5%ofthecases.Inthegivenexample,thesearethepointsofthet-distributionsothat5%ofthecaseslieinthetwotails.!Rejectifabsolutevalueoft-statisticislessthan-2.06orgreaterthan2.06,4.2.3Two-SidedAlternatives(1/3),Chapter4MultipleRegressionAnalysis:Inference,Section,Chapter,End,Example:DeterminantsofcollegeGPA,Lecturesmissedperweek,TheeffectsofhsGPAandskippedaresignificantlydifferentfromzeroatthe1%significancelevel.TheeffectofACTisnotsignificantlydifferentfromzero,notevenatthe10%significancelevel.,Forcriticalvalues,usestandardnormaldistribution,4.2.3Two-SidedAlternatives(2/3),Chapter4MultipleRegressionAnalysis:Inference,Section,Chapter,End,Statisticallysignificant“variablesinaregressionIfaregressioncoefficientisdifferentfromzeroinatwo-sidedtest,thecorrespondingvariableissaidtobestatisticallysignificant“Ifthenumberofdegreesoffreedomislargeenoughsothatthenormalapproximationapplies,thefollowingrulesofthumbapply:,statisticallysignificantat10%level“,statisticallysignificantat5%level“,statisticallysignificantat1%level“,4.2.3Two-SidedAlternatives(3/3),Chapter4MultipleRegressionAnalysis:Inference,Section,Chapter,End,TestingmoregeneralhypothesesaboutaregressioncoefficientNullhypothesist-statisticThetestworksexactlyasbefore,exceptthatthehypothesizedvalueissubstractedfromtheestimatewhenformingthestatistic,Hypothesizedvalueofthecoefficient,4.2.4TestingOtherHypothesesaboutbj(1/3),Chapter4MultipleRegressionAnalysis:Inference,Section,Chapter,End,Example:CampuscrimeandenrollmentAninterestinghypothesisiswhethercrimeincreasesbyonepercentifenrollmentisincreasedbyonepercent,Thehypothesisisrejectedatthe5%level,Estimateisdifferentfromonebutisthisdifferencestatisticallysignificant?,4.2.4TestingOtherHypothesesaboutbj(2/3),Chapter4MultipleRegressionAnalysis:Inference,Section,Chapter,End,4.2.4TestingOtherHypothesesaboutbj(3/3),Chapter4MultipleRegressionAnalysis:Inference,Section,Chapter,End,4.2.5Computingp-ValuesfortTests(1/2),Computingp-valuesfort-testsIfthesignificancelevelismadesmallerandsmaller,therewillbeapointwherethenullhypothesiscannotberejectedanymoreThereasonisthat,byloweringthesignificancelevel,onewantstoavoidmoreandmoretomaketheerrorofrejectingacorrectH0Thesmallestsignificancelevelatwhichthenullhypothesisisstillrejected,iscalledthep-valueofthehypothesistestAsmallp-valueisevidenceagainstthenullhypothesisbecauseonewouldrejectthenullhypothesisevenatsmallsignificancelevelsAlargep-valueisevidenceinfavorofthenullhypothesisP-valuesaremoreinformativethantestsatfixedsignificancelevels,Chapter4MultipleRegressionAnalysis:Inference,Section,Chapter,End,Howthep-valueiscomputed(here:two-sidedtest),Thep-valueisthesignificancelevelatwhichoneisindifferentbetweenrejectingandnotrejectingthenullhypothesis.Inthetwo-sidedcase,thep-valueisthustheprobabilitythatthet-distributedvariabletakesonalargerabsolutevaluethantherealizedvalueoftheteststatistic,e.g.:Fromthis,itisclearthatanullhypothesisisrejectedifandonlyifthecorrespondingp-valueissmallerthanthesignificancelevel.Forexample,forasignificancelevelof5%thet-statisticwouldnotlieintherejectionregion.,valueofteststatistic,Thesewouldbethecriticalvaluesfora5%significancelevel,4.2.5Computingp-ValuesfortTests(2/2),Chapter4MultipleRegressionAnalysis:Inference,Section,Chapter,End,4.2.6AReminderontheLanguageofClassicalHypothesisTesting,Example4.5HousingPricesandAirPollutionWedonotwanttotestthatbnox=0.Instead,H0:bnox=-1t=(-.954+1)/.117=.393Thereislittleevidencethattheelasticityisdifferentfrom-1.wefailtorejectH0atthex%level.H0isacceptedatthex%level.H0:bnox=-.9t=(-.954+.9)/.117=-.462,Chapter4MultipleRegressionAnalysis:Inference,Section,Chapter,End,4.2.7Economic,orPractical,versusStatisticalSignificance(1/2),economicsignificance:statisticalsignificance:Example4.6ParticipationRatesin401(k)PlansConsiderbtotemp.Example4.7EffectofJobTrainingonFirmScrapRatesConsiderbhrsemp.Someresearchersinsistonusingsmallersignificancelevelsasthesamplesizeincreases.Mostresearchersarealsowillingtoentertainlargersignificancelevelsinapplicationswithsmallsamplesizes.,Chapter4MultipleRegressionAnalysis:Inference,Section,Chapter,End,Guidelines:Ifthevariableisstatisticallysignificantattheusuallevels,discussthemagnitudeofthecoefficienttogetanideaofitseconomicimportance.Thefactthatacoefficientisstatisticallysignificantdoesnotnecessarilymeanitiseconomicallyorpracticallysignificant!Ifavariableisstatisticallyandeconomicallyimportantbuthasthewrong“sign,theregressionmodelmightbemisspecified.Ifavariableisstatisticallyinsignificantattheusuallevels(10%,5%,1%),onemaythinkofdroppingitfromtheregression.Ifthesamplesizeissmall,effectsmightbeimpreciselyestimatedsothatthecasefordroppinginsignificantvariablesislessstrong.variableswithsmalltstatisticsthathavethe“wrong”sign.(multicollinearity),4.2.7Economic,orPractical,versusStatisticalSignificance(2/2),Chapter4MultipleRegressionAnalysis:Inference,Section,Chapter,End,Criticalvalueoftwo-sidedtest,ConfidenceintervalsSimplemanipulationoftheresultinTheorem4.2impliesthatInterpretationoftheconfidenceintervalTheboundsoftheintervalarerandomInrepeatedsamples,theintervalthatisconstructedintheabovewaywillcoverthepopulationregressioncoefficientin95%ofthecases,LowerboundoftheConfidenceinterval,UpperboundoftheConfidenceinterval,Confidencelevel,Chapter4MultipleRegressionAnalysis:Inference,4.3ConfidenceIntervals(1/3),Chapter,End,ConfidenceintervalsfortypicalconfidencelevelsRelationshipbetweenconfidenceintervalsandhypothesestests,rejectinfavorof,Userulesofthumb,Chapter4MultipleRegressionAnalysis:Inference,4.3ConfidenceIntervals(2/3),Chapter,End,Example:ModeloffirmsRthenullhypothesisisusuallyoverwhelminglyrejected,Thenullhypothesisstatesthattheexplanatoryvariablesarenotusefulatallinexplainingthedependentvariable,4.5.5TheFStatisticforOverallSignificanceofaRegression(1/2),Restrictedmodel(regressiononconstant),Chapter4MultipleRegressionAnalysis:Inference,Section,Chapter,End,End,Section,Chapter,Example4.9ParentsEducationinaBirthWeightEquation(bwght.wf1)lsbwghtccigsparityfamincmotheducfatheduc,4.5.5TheFStatisticforOverallSignificanceofaRegression(2/2),Chapter4MultipleRegressionAnalysis:Inference,TestinggenerallinearrestrictionswiththeF-testExample:Testwhetherhousepriceassessmentsarerational,Theassessedhousingvalue(beforethehousewassold),Sizeoflot(infeet),Actualhouseprice,Squarefootage,Numberofbedrooms,Ifhousepriceassessmentsarerational,a1%changeintheassessmentshouldbeassociatedwitha1%changeinprice.,4.5.6TestingGeneralLinearRestrictions(1/4),Inaddition,otherknownfactorsshouldnotinfluencethepriceoncetheassessedvaluehasbeencontrolledfor.,Chapter4MultipleRegressionAnalysis:Inference,Section,Chapter,End,UnrestrictedregressionRestrictedregressionTeststatistic,cannotberejected,4.5.6TestingGeneralLinearRestrictions(2/4),Chapter4MultipleRegressionAnalysis:Inference,Section,Chapter,End,Perfor