《计量经济学》ch_04_wooldridg

Chapter4MultipleRegressionAnalysis:InferenceWooldridge:IntroductoryEconometrics:AModernApproach,5eInstructedbyprofessorYuan,HuipingChapter4MultipleRegressionAnalysis:Inference4.2TestingHypothesesaboutaSinglePopulationParameter:ThetTest4.3ConfidenceIntervals4.4TestingHypothesesaboutaSingleLinearCombinationoftheParameters4.5TestingMultipleLinearRestrictions:TheFTest4.1SamplingDistributionsoftheOLSEstimators4.6Anapplication—estimationoftheweightsofCPIcomponentsinChinaAssignments:Promblems1,2,4,5,7,8,10ComputerExercisesC1,C2,C3,C8,C9C8:smplifmarr=1andfsize=2(401ksubs.wf1)TheEndStatisticalinferenceintheregressionmodelHypothesistestsaboutpopulationparametersConstructionofconfidenceintervalsSamplingdistributionsoftheOLSestimatorsTheOLSestimatorsarerandomvariablesWealreadyknowtheirexpectedvaluesandtheirvariancesHowever,forhypothesistestsweneedtoknowtheirdistributionInordertoderivetheirdistributionweneedadditionalassumptionsAssumptionaboutdistributionoferrors:normaldistributionChapter4MultipleRegressionAnalysis:Inference4.1SamplingDistributionsoftheOLSEstimators(1/5)ChapterEndAssumptionMLR.6(Normalityoferrorterms)independentlyofItisassumedthattheunobservedfactorsarenormallydistributedaroundthepopulationregressionfunction.Theformandthevarianceofthedistributiondoesnotdependonanyoftheexplanatoryvariables.Itfollowsthat:Chapter4MultipleRegressionAnalysis:Inference4.1SamplingDistributionsoftheOLSEstimators(2/5)ChapterEndDiscussionofthenormalityassumptionTheerrortermisthesumof„many“differentunobservedfactorsSumsofindependentfactorsarenormallydistributed(CLT)Problems:Howmanydifferentfactors?Numberlargeenough?PossiblyveryheterogenuousdistributionsofindividualfactorsHowindependentarethedifferentfactors?ThenormalityoftheerrortermisanempiricalquestionAtleasttheerrordistributionshouldbe„close“tonormalChapter4MultipleRegressionAnalysis:Inference4.1SamplingDistributionsoftheOLSEstimators(3/5)ChapterEndDiscussionofthenormalityassumption(cont.)Exampleswherenormalitycannothold:Wages(nonnegative;also:minimumwage)Numberofarrests(takesonasmallnumberofintegervalues)Unemployment(indicatorvariable,takesononly1or0)Insomecases,normalitycanbeachievedthroughtransformationsofthedependentvariable(e.g.uselog(wage)insteadofwage)Important:Forthepurposesofstatisticalinference,theassumptionofnormalitycanbereplacedbyalargesamplesizeChapter4MultipleRegressionAnalysis:Inference4.1SamplingDistributionsoftheOLSEstimators(4/5)ChapterEndTerminologyTheorem4.1(Normalsamplingdistributions)UnderassumptionsMLR.1–MLR.6:TheestimatorsarenormallydistributedaroundthetrueparameterswiththevariancethatwasderivedearlierThestandardizedestimatorsfollowastandardnormaldistribution„Gauss-Markovassumptions“„Classicallinearmodel(CLM)assumptions“Chapter4MultipleRegressionAnalysis:Inference4.1SamplingDistributionsoftheOLSEstimators(5/5)ChapterEnd4.2.1Theorem4.2tDistributionfortheStandardizedEstimatorsChapter4MultipleRegressionAnalysis:Inference4.2TestingHypothesesaboutaSinglePopulationParameter:ThetTest4.2.3Two-SidedAlternatives4.2.4TestingOtherHypothesesaboutbj4.2.2TestingagainstOne-SidedAlternatives4.2.5Computingp-ValuesfortTests4.2.6AReminderontheLanguageofClassicalHypothesisTesting4.2.7Economic,orPractical,versusStatisticalSignificanceChapterEndUnderassumptionsMLR.1–MLR.6:Ifthestandardizationisdoneusingtheestimatedstandarddeviation(=standarderror),thenormaldistributionisreplacedbyat-distributionNote:Thet-distributionisclosetothestandardnormaldistributionifn-k-1islarge.Chapter4MultipleRegressionAnalysis:Inference4.2.1Theorem4.2tDistributionfortheStandardizedEstimators(1/3)Proof:SectionChapterEndNullhypothesis(formoregeneralhypotheses,seebelow)t-statistic(ort-ratio)Distributionofthet-statisticifthenullhypothesisistrueThet-statisticwillbeusedtotesttheabovenullhypothesis.Thefarthertheestimatedcoefficientisawayfromzero,thelesslikelyitisthatthenullhypothesisholdstrue.Butwhatdoes„far“awayfromzeromean?Thisdependsonthevariabilityoftheestimatedcoefficient,i.e.itsstandarddeviation.Thet-statisticmeasureshowmanyestimatedstandarddeviationstheestimatedcoefficientisawayfromzero.Thepopulationparameterisequaltozero,i.e.aftercontrollingfortheotherindependentvariables,thereisnoeffectofxjonyChapter4MultipleRegressionAnalysis:Inference4.2.1Theorem4.2tDistributionfortheStandardizedEstimators(2/3)SectionChapterEndGoal:Definearejectionrulesothat,ifitistrue,H0isrejectedonlywithasmallprobability(=significancelevel,e.g.5%)Thepreciserejectionruledependsonthealternativehypothesisandthechosensignificancelevelofthetest.Asignificancelevel:theprobabilityofrejectingH0whenitistrue.Chapter4MultipleRegressionAnalysis:Inference4.2.1Theorem4.2tDistributionfortheStandardizedEstimators(3/3)SectionChapterEndTestagainst.Testingagainstone-sidedalternatives(greaterthanzero)4.2.2TestingagainstOne-SidedAlternatives(1/8)Rejectthenullhypothesisinfavourofthealternativehypothesisiftheestimatedcoefficientis„toolarge“(i.e.largerthanacriticalvalue).Constructthecriticalvaluesothat,ifthenullhypothesisistrue,itisrejectedin,forexample,5%ofthecases.Inthegivenexample,thisisthepointofthet-distributionwith28degreesoffreedomthatisexceededin5%ofthecases.!Rejectift-statisticgreaterthan1.701Chapter4MultipleRegressionAnalysis:InferenceSectionChapterEndExample:WageequationTestwhether,aftercontrollingforeducationandtenure,higherworkexperienceleadstohigherhourlywages(1)Testagainst.Onewouldeitherexpectapositiveeffectofexperienceonhourlywageornoeffectatall.Standarderrors4.2.2TestingagainstOne-SidedAlternatives(2/8)Chapter4MultipleRegressionAnalysis:InferenceSectionChapterEndExample:Wageequation(cont.)„Theeffectofexperienceonhourlywageisstatisticallygreaterthanzeroatthe5%(andevenatthe1%)significancelevel.“Thoughttheestimatedreturnforanotheryearofexperience,holdingtenureandeducationfixed,isnotespeciallylarge,wehavepersuasivelyshownthatthepartialeffectofexperienceispositiveinthepopulation.t-statisticCriticalvaluesforthe5%andthe1%significancelevel(theseareconventionalsignificancelevels).Thenullhypothesisisrejectedbecausethet-statisticexceedsthecriticalvalue.(2)Degreesoffreedom;herethestandardnormalapproximationapplies(3)(4)4.2.2TestingagainstOne-SidedAlternatives(3/8)Chapter4MultipleRegressionAnalysis:InferenceSectionChapterEndTestagainst.Testingagainstone-sidedalternatives(lessthanzero)Rejectthenullhypothesisinfavourofthealternativehypothesisiftheestimatedcoefficientis„toosmall“(i.e.smallerthanacriticalvalue).Constructthecriticalvaluesothat,ifthenullhypothesisistrue,itisrejectedin,forexample,5%ofthecases.Inthegivenexample,thisisthepointofthet-distributionwith18degreesoffreedomsothat5%ofthecasesarebelowthepoint.!Rejectift-statisticlessthan-1.7344.2.2TestingagainstOne-SidedAlternatives(4/8)Chapter4MultipleRegressionAnalysis:InferenceSectionChapterEndExample:StudentperformanceandschoolsizeTestwhethersmallerschoolsizeleadstobetterstudentperformanceTestagainst.Dolargerschoolshamperstudentperformanceoristherenosucheffect?PercentageofstudentspassingmathstestAverageannualtea-chercompensationSchoolenrollment(=schoolsize)Staffperonethousandstudents4.2.2TestingagainstOne-SidedAlternatives(5/8)Chapter4MultipleRegressionAnalysis:InferenceSectionChapterEndExample:Studentperformanceandschoolsize(cont.)Onecannotrejectthehypothesisthatthereisnoeffectofschoolsizeonstudentperformance(notevenforalaxsignificancelevelof15%).t-statisticCriticalvaluesforthe5%andthe15%significancelevel.Thenullhypothesisisnotrejectedbecausethet-statisticisnotsmallerthanthecriticalvalue.Degreesoffreedom;herethestandardnormalapproximationapplies4.2.2TestingagainstOne-SidedAlternatives(6/8)Chapter4MultipleRegressionAnalysis:InferenceSectionChapterEndExample:Studentperformanceandschoolsize(cont.)Alternativespecificationoffunctionalform:Testagainst.R-squaredslightlyhigher4.2.2TestingagainstOne-SidedAlternatives(7/8)Chapter4MultipleRegressionAnalysis:InferenceSectionChapterEndExample:Studentperformanceandschoolsize(cont.)Thehypothesisthatthereisnoeffectofschoolsizeonstudentperformancecanberejectedinfavorofthehypothesisthattheeffectisnegative.t-statisticCriticalvalueforthe5%significancelevel!rejectnullhypothesisHowlargeistheeffect?(smalleffect)+10%enrollment!-0.129percentagepointsstudentspasstest4.2.2TestingagainstOne-SidedAlternatives(8/8)Chapter4MultipleRegressionAnalysis:InferenceSectionChapterEndTestingagainsttwo-sidedalternativesTestagainst.Rejectthenullhypothesisinfavourofthealternativehypothesisiftheabsolutevalueoftheestimatedcoefficientistoolarge.Constructthecriticalvaluesothat,ifthenullhypothesisistrue,itisrejectedin,forexample,5%ofthecases.Inthegivenexample,thesearethepointsofthet-distributionsothat5%ofthecaseslieinthetwotails.!Rejectifabsolutevalueoft-statisticislessthan-2.06orgreaterthan2.064.2.3Two-SidedAlternatives(1/3)Chapter4MultipleRegressionAnalysis:InferenceSectionChapterEndExample:DeterminantsofcollegeGPALecturesmissedperweekTheeffectsofhsGPAandskippedaresignificantlydifferentfromzeroatthe1%significancelevel.TheeffectofACTisnotsignificantlydifferentfromzero,notevenatthe10%significancelevel.Forcriticalvalues,usestandardnormaldistribution4.2.3Two-SidedAlternatives(2/3)Chapter4MultipleRegressionAnalysis:InferenceSectionChapterEnd„Statisticallysignificant“variablesinaregressionIfaregressioncoefficientisdifferentfromzeroinatwo-sidedtest,thecorrespondingvariableissaidtobe„statisticallysignificant“Ifthenumberofdegreesoffreedomislargeenoughsothatthenormalapproximationapplies,thefollowingrulesofthumbapply:„statisticallysignificantat10%level“„statisticallysignificantat5%level“„statisticallysignificantat1%level“4.2.3Two-SidedAlternatives(3/3)Chapter4MultipleRegressionAnalysis:InferenceSectionChapterEndTestingmoregeneralhypothesesaboutaregressioncoefficientNullhypothesist-statisticThetestworksexactlyasbefore,exceptthatthehypothesizedvalueissubstractedfromtheestimatewhenformingthestatisticHypothesizedvalueofthecoefficient4.2.4TestingOtherHypothesesaboutbj(1/3)Chapter4MultipleRegressionAnalysis:InferenceSectionChapterEndExample:CampuscrimeandenrollmentAninterestinghypothesisiswhethercrimeincreasesbyonepercentifenrollmentisincreasedbyonepercentThehypothesisisrejectedatthe5%levelEstimateisdifferentfromonebutisthisdifferencestatisticallysignificant?4.2.4TestingOtherHypothesesaboutbj(2/3)Chapter4MultipleRegressionAnalysis:InferenceSectionChapterEnd4.2.4TestingOtherHypothesesaboutbj(3/3)Chapter4MultipleRegressionAnalysis:InferenceSectionChapterEnd4.2.5Computingp-ValuesfortTests(1/2)Computingp-valuesfort-testsIfthesignificancelevelismadesmallerandsmaller,therewillbeapointwherethenullhypothesiscannotberejectedanymoreThereasonisthat,byloweringthesignificancelevel,onewantstoavoidmoreandmoretomaketheerrorofrejectingacorrectH0Thesmallestsignificancelevelatwhichthenullhypothesisisstillrejected,iscalledthep-valueofthehypothesistestAsmallp-valueisevidenceagainstthenullhypothesisbecauseonewouldrejectthenullhypothesisevenatsmallsignificancelevelsAlargep-valueisevidenceinfavorofthenullhypothesisP-valuesaremoreinformativethantestsatfixedsignificancelevelsChapter4MultipleRegressionAnalysis:InferenceSectionChapterEndHowthep-valueiscomputed(here:two-sidedtest)Thep-valueisthesignificancelevelatwhichoneisindifferentbetweenrejectingandnotrejectingthenullhypothesis.Inthetwo-sidedcase,thep-valueisthustheprobabilitythatthet-distributedvariabletakesonalargerabsolutevaluethantherealizedvalueoftheteststatistic,e.g.:Fromthis,itisclearthatanullhypothesisisrejectedifandonlyifthecorrespondingp-valueissmallerthanthesignificancelevel.Forexample,forasignificancelevelof5%thet-statisticwouldnotlieintherejectionregion.valueofteststatisticThesewouldbethecriticalvaluesfora5%significancelevel4.2.5Computingp-ValuesfortTests(2/2)Chapter4MultipleRegressionAnalysis:InferenceSectionChapterEnd4.2.6AReminderontheLanguageofClassicalHypothesisTestingExample4.5[HousingPricesandAirPollution]Wedonotwanttotestthatbnox=0.Instead,H0:bnox=-1t=(-.954+1)/.117=.393Thereislittleevidencethattheelasticityisdifferentfrom-1.wefailtorejectH0atthex%level.H0isacceptedatthex%level.H0:bnox=-.9t=(-.954+.9)/.117=-.462Chapter4MultipleRegressionAnalysis:InferenceSectionChapterEnd4.2.7Economic,orPractical,versusStatisticalSignificance(1/2)economicsignificance:statisticalsignificance:Example4.6[ParticipationRatesin401(k)Plans]Considerbtotemp.Example4.7[EffectofJobTrainingonFirmScrapRates]Considerbhrsemp.Someresearchersinsistonusingsmallersignificancelevelsasthesamplesizeincreases.Mostresearchersarealsowillingtoentertainlargersignificancelevelsinapplicationswithsmallsamplesizes.Chapter4MultipleRegressionAnalysis:InferenceSectionChapterEndGuidelines:Ifthevariableisstatisticallysignificantattheusuallevels,discussthemagnitudeofthecoefficienttogetanideaofitseconomicimportance.Thefactthatacoefficientisstatisticallysignificantdoesnotnecessarilymeanitiseconomicallyorpracticallysignificant!Ifavariableisstatisticallyandeconomicallyimportantbuthasthe„wrong“sign,theregressionmodelmightbemisspecified.Ifavariableisstatisticallyinsignificantattheusuallevels(10%,5%,1%),onemaythinkofdroppingitfromtheregression.Ifthesamplesizeissmall,effectsmightbeimpreciselyestimatedsothatthecasefordroppinginsignificantvariablesislessstrong.variableswithsmalltstatisticsthathavethe“wrong”sign.(multicollinearity)4.2.7Economic,orPractical,versusStatisticalSignificance(2/2)Chapter4MultipleRegressionAnalysis:InferenceSectionChapterEndCriticalvalueoftwo-sidedtestConfidenceintervalsSimplemanipulationoftheresultinTheorem4.2impliesthatInterpretationoftheconfidenceintervalTheboundsoftheintervalarerandomInrepeatedsamples,theintervalthatisconstructedintheabovewaywillcoverthepopulationregressioncoefficientin95%ofthecasesLowerboundoftheConfidenceintervalUpperboundoftheConfidenceintervalConfidencelevelChapter4MultipleRegressionAnalysis:Inference4.3ConfidenceIntervals(1/3)ChapterEndConfidenceintervalsfortypicalconfidencelevelsRelationshipbetweenconfidenceintervalsandhypothesestestsrejectinfavorofUserulesofthumbChapter4MultipleRegressionAnalysis:Inference4.3ConfidenceIntervals(2/3)ChapterEndExample:Modeloffirms‘R&DexpendituresSpendingonR&DAnnualsalesProfitsaspercentageofsalesTheeffectofsalesonR&Disrelativelypreciselyestimatedastheintervalisnarrow.Moreover,theeffectissignificantlydifferentfromzerobecausezeroisoutsidetheinterval.Thiseffectisimpreciselyestimatedasthein-tervalisverywide.Itisnotevenstatisticallysignificantbecausezeroliesintheinterval.Chapter4MultipleRegressionAnalysis:Inference4.3ConfidenceIntervals(3/3)ChapterEndTestinghypothesesaboutalinearcombinationofparametersExample:Returntoeducationat2yearvs.at4yearcollegesYearsofeducationat2yearcollegesYearsofeducationat4yearcollegesTestagainst.Apossibleteststatisticwouldbe:Thedifferencebetweentheestimatesisnormalizedbytheestimatedstandarddeviationofthedifference.Thenullhypothesiswouldhavetoberejectedifthestatisticis„toonegative“tobelievethatthetruedifferencebetweentheparametersisequaltozero.4.4TestingHypothesesaboutaSingleLinearCombinationoftheParameters(1/4)Chapter4MultipleRegressionAnalysis:InferenceChapterEndInsertintooriginalregressionImpossibletocomputewithstandardregressionoutputbecauseAlternativemethodUsuallynotavailableinregressionoutputDefineandtestagainst.anewregressor(=totalyearsofcollege)4.4TestingHypothesesaboutaSingleLinearCombinationoftheParameters(2/4)Chapter4MultipleRegressionAnalysis:InferenceChapterEndEstimationresultsThismethodworksalwaysforsinglelinearhypothesesTotalyearsofcollegeHypothesisisrejectedat10%levelbutnotat5%level4.4TestingHypothesesaboutaSingleLinearCombinationoftheParameters(3/4)Chapter4MultipleRegressionAnalysis:InferenceChapterEndtwoyear.wf1lslwagecjcjc+univexperseriestotcoll=jc+univlslwagecjctotcollexper4.4TestingHypothesesaboutaSingleLinearCombinationoftheParameters(4/4)Chapter4MultipleRegressionAnalysis:InferenceChapterEnd4.5TestingMultipleLinearRestrictions:TheFTest4.5.2RelationshipbetweenFandtStatistics4.5.3TheR-SquaredFormoftheFStatistic4.5.1TestingExclusionRestrictions4.5.4Computingp-ValuesforFTests4.5.5TheFStatisticforOverallSignificanceofaRegression4.5.6TestingGeneralLinearRestrictionsChapter4MultipleRegressionAnalysis:InferenceChapterEndTestingmultiplelinearrestrictions:TheF-testTestingexclusionrestrictionsYearsintheleagueAveragenumberofgamesperyearSalaryofmajorlea-guebaseballplayerBattingaverageHomerunsperyearRunsbattedinperyearagainstTestwhetherperformancemeasureshavenoeffect/canbeexludedfromregression.4.5.1TestingExclusionRestrictions(1/5)Chapter4MultipleRegressionAnalysis:InferenceSectionChapterEndEstimationoftheunrestrictedmodel(mlb1.wf1)lslog(salary)cyearsgamesyrbavghrunsyrrbisyrNoneofthesevariabelsisstatisticallysignificantwhentestedindividuallyIdea:Howwouldthemodelfitbeifthesevariablesweredroppedfromtheregression?4.5.1TestingExclusionRestrictions(2/5)Chapter4MultipleRegressionAnalysis:InferenceSectionChapterEndEstimationoftherestrictedmodelTeststatisticThesumofsquaredresidualsnecessarilyincreases,butistheincreasestatisticallysignificant?NumberofrestrictionsTherelativeincreaseofthesumofsquaredresidualswhengoingfromH1toH0followsaF-distribution(ifthenullhypothesisH0iscorrect)4.5.1TestingExclusionRestrictions(3/5)Chapter4MultipleRegressionAnalysis:InferenceSectionChapterEndRejectionrule(Figure4.7)AF-distributedvariableonlytakesonpositivevalues.ThiscorrespondstothefactthatthesumofsquaredresidualscanonlyincreaseifonemovesfromH1toH0.Choosethecriticalvaluesothatthenullhypo-thesisisrejectedin,forexample,5%ofthecases,althoughitistrue.4.5.1TestingExclusionRestrictions(4/5)Chapter4MultipleRegressionAnalysis:InferenceSectionChapterEndTestdecisioninexampleDiscussionThethreevariablesare„jointlysignificant“TheywerenotsignificantwhentestedindividuallyThelikelyreasonismulticollinearitybetweenthemNumberofrestrictionstobetestedDegreesoffreedomintheunrestrictedmodelThenullhypothesisisoverwhel-minglyrejected(evenatverysmallsignificancelevels).mlb1.wf1PerformanceinEviews.4.5.1TestingExclusionRestrictions(5/5)Chapter4MultipleRegressionAnalysis:InferenceSectionChapterEnd4.5.2RelationshipbetweenFandtStatisticsTheFstatisticfortestingexclusionofasinglevariableisequaltothesquareofthecorrespondingtstatistic,whosealternativeistwo-sided.Itispossiblethat,inagroupofseveralexplanatoryvariables,onevariablehasasignificanttstatistic,butthegroupofvariablesisjointlyinsignificantattheusualsignificancelevels.Often,whenavariableisverystatisticallysignificantanditistestedjointlywithanothersetofvariables,thesetwillbejointlysignificant.Chapter4MultipleRegressionAnalysis:InferenceSectionChapterEnd4.5.3TheR-SquaredFormoftheFStatisticChapter4MultipleRegressionAnalysis:InferenceSectionChapterEnd4.5.4Computingp-ValuesforFTestsThep-valueistheprobabilityofobservingavalueofFatleastaslargeaswedid,giventhatthenullhypothesisistrue.Chapter4MultipleRegressionAnalysis:InferenceSectionChapterEndTestofoverallsignificanceofaregressionThetestofoverallsignificanceisreportedinmostregressionpackages;thenullhypothesisisusuallyoverwhelminglyrejectedThenullhypothesisstatesthattheexplanatoryvariablesarenotusefulatallinexplainingthedependentvariable4.5.5TheFStatisticforOverallSignificanceofaRegression(1/2)Restrictedmodel(regressiononconstant)Chapter4MultipleRegressionAnalysis:InferenceSectionChapterEndEndSectionChapterExample4.9[Parents’EducationinaBirthWeightEquation](bwght.wf1)lsbwghtccigsparityfamincmotheducfatheduc4.5.5TheFStatisticforOverallSignificanceofaRegression(2/2)Chapter4MultipleRegressionAnalysis:InferenceDependentVariable:BWGHTIncludedobservations:1191VariableCoefficientStd.Errort-StatisticProb.  C114.523.72830.7160.0000CIGS-0.5960.110-5.4010.0000PARITY1.7880.6592.7110.0068FAMINC0.0560.0371.5330.1256MOTHEDUC-0.3700.320-1.1580.2470FATHEDUC0.4720.2831.6710.0949R-squared0.038748    Meandependentvar119.5298AdjustedR-squared0.034692    S.D.dependentvar20.14124F-statistic9.553500    Durbin-Watsonstat1.911657Prob(F-statistic)0.000000TestinggenerallinearrestrictionswiththeF-testExample:TestwhetherhousepriceassessmentsarerationalTheassessedhousingvalue(beforethehousewassold)Sizeoflot(infeet)ActualhousepriceSquarefootageNumberofbedroomsIfhousepriceassessmentsarerational,a1%changeinthea