《计量经济学导论》ch课件

Chapter7

MultipleRegressionAnalysiswithQualitativeInformationWooldridge:IntroductoryEconometrics:AModernApproach,5eChapter7MultipleRegressionQualitativeInformationExamples:gender,race,industry,region,ratinggrade,…AwaytoincorporatequalitativeinformationistousedummyvariablesTheymayappearasthedependentorasindependentvariablesAsingledummyindependentvariableDummyvariable:=1ifthepersonisawoman=0ifthepersonisman=thewagegain/lossifthepersonisawomanratherthanaman(holdingotherthingsfixed)MultipleRegressionAnalysis:QualitativeInformationQualitativeInformationDummyvGraphicalIllustrationAlternativeinterpretationofcoefficient:i.e.thedifferenceinmeanwagebetweenmenandwomenwiththesamelevelofeducation.InterceptshiftMultipleRegressionAnalysis:QualitativeInformationGraphicalIllustrationAlternatDummyvariabletrapThismodelcannotbeestimated(perfectcollinearity)Whenusingdummyvariables,onecategoryalwayshastobeomitted:Alternatively,onecouldomittheintercept:ThebasecategoryaremenThebasecategoryarewomenDisadvantages:

1)Moredifficulttotestfordiffe-rencesbetweentheparameters2)R-squaredformulaonlyvalidifregressioncontainsinterceptMultipleRegressionAnalysis:QualitativeInformationDummyvariabletrapThismodelEstimatedwageequationwithinterceptshiftDoesthatmeanthatwomenarediscriminatedagainst?Notnecessarily.Beingfemalemaybecorrelatedwithotherproduc-tivitycharacteristicsthathavenotbeencontrolledfor.Holdingeducation,experience,andtenurefixed,womenearn1.81$lessperhourthanmenMultipleRegressionAnalysis:QualitativeInformationEstimatedwageequationwithiComparingmeansofsubpopulationsdescribedbydummiesDiscussionItcaneasilybetestedwhetherdifferenceinmeansissignificantThewagedifferencebetweenmenandwomenislargerifnootherthingsarecontrolledfor;i.e.partofthedifferenceisduetodiffer-encesineducation,experienceandtenurebetweenmenandwomenNotholdingotherfactorsconstant,womenearn2.51$perhourlessthanmen,i.e.thedifferencebetweenthemeanwageofmenandthatofwomenis2.51$.MultipleRegressionAnalysis:QualitativeInformationComparingmeansofsubpopulatiFurtherexample:EffectsoftraininggrantsonhoursoftrainingThisisanexampleofprogramevaluationTreatmentgroup(=grantreceivers)vs.controlgroup(=nogrant)Istheeffectoftreatmentontheoutcomeofinterestcausal?HourstrainingperemployeeDummyindicatingwhetherfirmreceivedtraininggrantMultipleRegressionAnalysis:QualitativeInformationFurtherexample:EffectsoftrUsingdummyexplanatoryvariablesinequationsforlog(y)DummyindicatingwhetherhouseisofcolonialstyleAsthedummyforcolonialstylechangesfrom0to1,thehousepriceincreasesby5.4percentagepointsMultipleRegressionAnalysis:QualitativeInformationUsingdummyexplanatoryvariabHoldingotherthingsfixed,marriedwomenearn19.8%lessthansinglemen(=thebasecategory)Usingdummyvariablesformultiplecategories1)Definemembershipineachcategorybyadummyvariable2)Leaveoutonecategory(whichbecomesthebasecategory)MultipleRegressionAnalysis:QualitativeInformationHoldingotherthingsfixed,maIncorporatingordinalinformationusingdummyvariablesExample:CitycreditratingsandmunicipalbondinterestratesMunicipalbondrateCreditratingfrom0-4(0=worst,4=best)Thisspecificationwouldprobablynotbeappropriateasthecreditratingonlycontainsordinalinformation.Abetterwaytoincorporatethisinformationistodefinedummies:Dummiesindicatingwhethertheparticularratingapplies,e.g.CR1=1ifCR=1andCR1=0otherwise.Alleffectsaremeasuredincomparisontotheworstrating(=basecategory).MultipleRegressionAnalysis:QualitativeInformationIncorporatingordinalinformatInteractionsinvolvingdummyvariablesAllowingfordifferentslopesInterestinghypotheses=interceptmen=interceptwomen=slopemen=slopewomenInteractiontermThereturntoeducationisthesameformenandwomenThewholewageequationisthesameformenandwomenMultipleRegressionAnalysis:QualitativeInformationInteractionsinvolvingdummyvGraphicalillustrationInteractingboththeinterceptandtheslopewiththefemaledummyenablesonetomodelcompletelyindependentwageequationsformenandwomenMultipleRegressionAnalysis:QualitativeInformationGraphicalillustrationInteractEstimatedwageequationwithinteractiontermNoevidenceagainsthypothesisthatthereturntoeducationisthesameformenandwomenDoesthismeanthatthereisnosignificantevidenceoflowerpayforwomenatthesamelevelsofeduc,exper,andtenure?No:thisisonlytheeffectforeduc=0.Toanswerthequestiononehastorecentertheinteractionterm,e.g.aroundeduc=12.5(=averageeducation).MultipleRegressionAnalysis:QualitativeInformationEstimatedwageequationwithiTestingfordifferencesinregressionfunctionsacrossgroupsUnrestrictedmodel(containsfullsetofinteractions)Restrictedmodel(sameregressionforbothgroups)CollegegradepointaverageStandardizedaptitudetestscoreHighschoolrankpercentileTotalhoursspentincollegecoursesMultipleRegressionAnalysis:QualitativeInformationTestingfordifferencesinregNullhypothesisEstimationoftheunrestrictedmodelAllinteractioneffectsarezero,i.e.thesameregressioncoefficientsapplytomenandwomenTestedindividually,thehypothesisthattheinteractioneffectsarezerocannotberejectedMultipleRegressionAnalysis:QualitativeInformationNullhypothesisAllinteractionJointtestwithF-statisticAlternativewaytocomputeF-statisticinthegivencaseRunseparateregressionsformenandforwomen;theunrestrictedSSRisgivenbythesumoftheSSRofthesetworegressionsRunregressionfortherestrictedmodelandstoreSSRIfthetestiscomputedinthiswayitiscalledtheChow-TestImportant:TestassumesaconstanterrorvarianceaccrossgroupsNullhypothesisisrejectedMultipleRegressionAnalysis:QualitativeInformationJointtestwithF-statisticNulABinarydependentvariable:thelinearprobabilitymodelLinearregressionwhenthedependentvariableisbinaryLinearprobabilitymodel(LPM)Ifthedependentvariableonlytakesonthevalues1and0Inthelinearprobabilitymodel,thecoefficientsdescribetheeffectoftheexplanatoryvariablesontheprobabilitythaty=1MultipleRegressionAnalysis:QualitativeInformationABinarydependentvariable:tDoesnotlooksignificant(butseebelow)Example:Laborforceparticipationofmarriedwomen=1ifinlaborforce,=0otherwiseNon-wifeincome(inthousanddollarsperyear)Ifthenumberofkidsundersixyearsincreasesbyone,thepro-probabilitythatthewomanworksfallsby26.2%MultipleRegressionAnalysis:QualitativeInformationDoesnotlooksignificant(butExample:Femalelaborparticipationofmarriedwomen(cont.)Graphfornwifeinc=50,exper=5,age=30,kindslt6=1,kidsge6=0Negativepredictedprobabilitybutnoproblembecausenowomaninthesamplehaseduc<5.Themaximumlevelofeducationinthesampleiseduc=17.Forthegi-vencase,thisleadstoapredictedprobabilitytobeinthelaborforceofabout50%.MultipleRegressionAnalysis:QualitativeInformationExample:FemalelaborparticipDisadvantagesofthelinearprobabilitymodelPredictedprobabilitiesmaybelargerthanoneorsmallerthanzeroMarginalprobabilityeffectssometimeslogicallyimpossibleThelinearprobabilitymodelisnecessarilyheteroskedasticHeterosceasticityconsistentstandarderrorsneedtobecomputedAdvantangesofthelinearprobabilitymodelEasyestimationandinterpretationEstimatedeffectsandpredictionsoftenreasonablygoodinpracticeVarianceofBer-noullivariableMultipleRegressionAnalysis:QualitativeInformationDisadvantagesofthelinearprMoreonpolicyanalysisandprogramevaluationExample:EffectofjobtraininggrantsonworkerproductivityPercentageofdefectiveitems=1iffirmreceivedtraininggrant,=0otherwiseNoapparenteffectofgrantonproductivityTreatmentgroup:grantreveivers,Controlgroup:firmsthatreceivednograntGrantsweregivenonafirst-come,first-servedbasis.Thisisnotthesameasgivingthemoutrandomly.Itmightbethecasethatfirmswithlessproductiveworkerssawanopportunitytoimproveproductivityandappliedfirst.MultipleRegressionAnalysis:QualitativeInformationMoreonpolicyanalysisandprSelf-selectionintotreatmentasasourceforendogeneityInthegivenandinrelatedexamples,thetreatmentstatusisprobablyrelatedtoothercharacteristicsthatalsoinfluencetheoutcomeThereasonisthatsubjectsself-selectthemselvesintotreatmentdependingontheirindividualcharacteristicsandprospectsExperimentalevaluationInexperiments,assignmenttotreatmentisrandomInthiscase,causaleffectscanbeinferredusingasimpleregressionThedummyindicatingwhetherornottherewastreatmentisunrelatedtootherfactorsaffectingtheoutcome.MultipleRegressionAnalysis:QualitativeInformationSelf-selectionintotreatmentFurtherexampleofanendogenuousdummyregressorArenonwhitecustomersdiscriminatedagainst?Itisimportanttocontrolforothercharacteristicsthatmaybeimportantforloanapproval(fession,unemployment)Omittingimportantcharacteristicsthatarecorrelatedwiththenon-whitedummywillproducespuriousevidencefordiscriminiationDummyindicatingwhetherloanwasapprovedRacedummyCreditratingMultipleRegressionAnalysis:QualitativeInformationFurtherexampleofanendogenuChapter7

MultipleRegressionAnalysiswithQualitativeInformationWooldridge:IntroductoryEconometrics:AModernApproach,5eChapter7MultipleRegressionQualitativeInformationExamples:gender,race,industry,region,ratinggrade,…AwaytoincorporatequalitativeinformationistousedummyvariablesTheymayappearasthedependentorasindependentvariablesAsingledummyindependentvariableDummyvariable:=1ifthepersonisawoman=0ifthepersonisman=thewagegain/lossifthepersonisawomanratherthanaman(holdingotherthingsfixed)MultipleRegressionAnalysis:QualitativeInformationQualitativeInformationDummyvGraphicalIllustrationAlternativeinterpretationofcoefficient:i.e.thedifferenceinmeanwagebetweenmenandwomenwiththesamelevelofeducation.InterceptshiftMultipleRegressionAnalysis:QualitativeInformationGraphicalIllustrationAlternatDummyvariabletrapThismodelcannotbeestimated(perfectcollinearity)Whenusingdummyvariables,onecategoryalwayshastobeomitted:Alternatively,onecouldomittheintercept:ThebasecategoryaremenThebasecategoryarewomenDisadvantages:

1)Moredifficulttotestfordiffe-rencesbetweentheparameters2)R-squaredformulaonlyvalidifregressioncontainsinterceptMultipleRegressionAnalysis:QualitativeInformationDummyvariabletrapThismodelEstimatedwageequationwithinterceptshiftDoesthatmeanthatwomenarediscriminatedagainst?Notnecessarily.Beingfemalemaybecorrelatedwithotherproduc-tivitycharacteristicsthathavenotbeencontrolledfor.Holdingeducation,experience,andtenurefixed,womenearn1.81$lessperhourthanmenMultipleRegressionAnalysis:QualitativeInformationEstimatedwageequationwithiComparingmeansofsubpopulationsdescribedbydummiesDiscussionItcaneasilybetestedwhetherdifferenceinmeansissignificantThewagedifferencebetweenmenandwomenislargerifnootherthingsarecontrolledfor;i.e.partofthedifferenceisduetodiffer-encesineducation,experienceandtenurebetweenmenandwomenNotholdingotherfactorsconstant,womenearn2.51$perhourlessthanmen,i.e.thedifferencebetweenthemeanwageofmenandthatofwomenis2.51$.MultipleRegressionAnalysis:QualitativeInformationComparingmeansofsubpopulatiFurtherexample:EffectsoftraininggrantsonhoursoftrainingThisisanexampleofprogramevaluationTreatmentgroup(=grantreceivers)vs.controlgroup(=nogrant)Istheeffectoftreatmentontheoutcomeofinterestcausal?HourstrainingperemployeeDummyindicatingwhetherfirmreceivedtraininggrantMultipleRegressionAnalysis:QualitativeInformationFurtherexample:EffectsoftrUsingdummyexplanatoryvariablesinequationsforlog(y)DummyindicatingwhetherhouseisofcolonialstyleAsthedummyforcolonialstylechangesfrom0to1,thehousepriceincreasesby5.4percentagepointsMultipleRegressionAnalysis:QualitativeInformationUsingdummyexplanatoryvariabHoldingotherthingsfixed,marriedwomenearn19.8%lessthansinglemen(=thebasecategory)Usingdummyvariablesformultiplecategories1)Definemembershipineachcategorybyadummyvariable2)Leaveoutonecategory(whichbecomesthebasecategory)MultipleRegressionAnalysis:QualitativeInformationHoldingotherthingsfixed,maIncorporatingordinalinformationusingdummyvariablesExample:CitycreditratingsandmunicipalbondinterestratesMunicipalbondrateCreditratingfrom0-4(0=worst,4=best)Thisspecificationwouldprobablynotbeappropriateasthecreditratingonlycontainsordinalinformation.Abetterwaytoincorporatethisinformationistodefinedummies:Dummiesindicatingwhethertheparticularratingapplies,e.g.CR1=1ifCR=1andCR1=0otherwise.Alleffectsaremeasuredincomparisontotheworstrating(=basecategory).MultipleRegressionAnalysis:QualitativeInformationIncorporatingordinalinformatInteractionsinvolvingdummyvariablesAllowingfordifferentslopesInterestinghypotheses=interceptmen=interceptwomen=slopemen=slopewomenInteractiontermThereturntoeducationisthesameformenandwomenThewholewageequationisthesameformenandwomenMultipleRegressionAnalysis:QualitativeInformationInteractionsinvolvingdummyvGraphicalillustrationInteractingboththeinterceptandtheslopewiththefemaledummyenablesonetomodelcompletelyindependentwageequationsformenandwomenMultipleRegressionAnalysis:QualitativeInformationGraphicalillustrationInteractEstimatedwageequationwithinteractiontermNoevidenceagainsthypothesisthatthereturntoeducationisthesameformenandwomenDoesthismeanthatthereisnosignificantevidenceoflowerpayforwomenatthesamelevelsofeduc,exper,andtenure?No:thisisonlytheeffectforeduc=0.Toanswerthequestiononehastorecentertheinteractionterm,e.g.aroundeduc=12.5(=averageeducation).MultipleRegressionAnalysis:QualitativeInformationEstimatedwageequationwithiTestingfordifferencesinregressionfunctionsacrossgroupsUnrestrictedmodel(containsfullsetofinteractions)Restrictedmodel(sameregressionforbothgroups)CollegegradepointaverageStandardizedaptitudetestscoreHighschoolrankpercentileTotalhoursspentincollegecoursesMultipleRegressionAnalysis:QualitativeInformationTestingfordifferencesinregNullhypothesisEstimationoftheunrestrictedmodelAllinteractioneffectsarezero,i.e.thesameregressioncoefficientsapplytomenandwomenTestedindividually,thehypothesisthattheinteractioneffectsarezerocannotberejectedMultipleRegressionAnalysis:QualitativeInformationNullhypothesisAllinteractionJointtestwithF-statisticAlternativewaytocomputeF-statisticinthegivencaseRunseparateregressionsformenandforwomen;theunrestrictedSSRisgivenbythesumoftheSSRofthesetworegressionsRunregressionfortherestrictedmodelandstoreSSRIfthetestiscomputedinthiswayitiscalledtheChow-TestImportant:TestassumesaconstanterrorvarianceaccrossgroupsNullhypothesisisrejectedMultipleRegressionAnalysis:QualitativeInformationJointtestwithF-statisticNulABinarydependentvariable:thelinearprobabilitymodelLinearregressionwhenthedependentvariableisbinaryLinearprobabilitymodel(LPM)Ifthedependentvariableonlytakesonthevalues1and0Inthelinearprobabilitymodel,thecoefficientsdescribetheeffectoftheexplanatoryvariablesontheprobabilitythaty=1MultipleRegressionAnalysis:QualitativeInformationABinarydependentvariable:tDoesnotlooksignificant(butseebelow)Example:Laborforceparticipationofmarriedwomen=1ifinlaborforce,=0otherwiseNon-wifeincome(inthousanddollarsperyear)Ifthenumberofkidsundersixyearsincreasesbyone,thepro-probabilitythatthewomanworksfallsby26.2%MultipleRegressionAnalysis:QualitativeInformationDoesnotlooksignificant(butExample:Femalelaborparticipationofmarriedwomen(cont.)Graphfornwifeinc=50,exper=5,age=30,kindslt6=1,kidsge6=0Negativepredictedprobabilitybutnoproblembecausenowomaninthesamplehaseduc<5.Themaximumlevelofeducationinthesampleiseduc=17.Forthegi-vencase,thisleadstoapredictedprobabilitytobeinthelaborforceofabout50%.MultipleRegressionAnalysis:QualitativeInformationExample:FemalelaborparticipDisadvantagesofthelinearprobabilitymodelPredictedprobabilitiesmaybelargerthanoneorsmallerthanzeroMarginalprobabilityeffectssometimeslogicallyimpossibleThelinearprobabilitymodelisnecessarilyheteroskedasticHeterosceasticityconsistentstandarderrorsneedtobecomputedAdvantangesofthelinearprobabilitymodelEasyestimationan