多元回歸分析虛擬變量

多元回歸分析虛擬變量1第一頁，共十八頁，2022年，8月28日DummyVariablesAdummyvariableisavariablethattakesonthevalue1or0Examples:male(=1ifaremale,0otherwise),south(=1ifinthesouth,0otherwise),etc.Dummyvariablesarealsocalledbinaryvariables,forobviousreasonspersonwageeducfemalemarried13.1021023.24221133.0020046.00440155.30701……………52511.565115263.5500Table:Apartiallistingofthedatainwage1.raw2第二頁，共十八頁，2022年，8月28日ADummyIndependentVariableConsiderasimplemodelwithonecontinuousvariable(x)andonedummy(d)y=b0+d0d+b1x+uThiscanbeinterpretedasaninterceptshiftIfd=0,theny=b0+b1x+uIfd=1,theny=(b0+d0)+b1x+uThecaseofd=0isthebasegroup,thend0=E(y|x,d=1)-E(y|x,d=0)3第三頁，共十八頁，2022年，8月28日Exampleofd0>0xy}b0y=b0+b1xslope=b1d=0{d0y=(b0+d0)+b1xd=1(y=b0+d0d+b1x+u)4第四頁，共十八頁，2022年，8月28日DummiesforMultipleCategoriesWecanusedummyvariablestocontrolforsomethingwithmultiplecategoriesWagedeterminations:wage=-1.57-1.81female+0.572educ+0.025exper+0.141tenure(0.72)(0.26)(0.049)(0.012)(0.021)

n=526R2=0.364Thecoefficientoffemale(-1.81)meansthewageoffemaleis$1.81lessperhourthanmaleworkersaftercontrollingothervariables.wage=7.10-2.51female

(0.21)(0.30)Thismeansthattheaveragemalewageperhouris$7.10,andfemale’swageis$2.51less,whichis$4.59perhour.Istheresignificantwagedifferencebtwmenandwomen?Yes,itindeedis.Becausethet-valueoffemaleis-2.51/0.30=-8.37Logformlog(wage)=0.501–0.301female+0.087educ+0.005exper+0.017tenure(0.102)(0.037)(0.0069)(0.016)(0.0030)

n=526R2=0.3923Femaleis30.1%lessthanmen.Theexactdifferenceislog(wageF)-log(wageM)=-0.301,(wageF-wageM)/wageM=exp(-0.301)-1=0.3512=35.12%5第五頁，共十八頁，2022年，8月28日Example:EffectsofComputerOwnershiponCollegeGPAWhetherastudentownacomputereffecttheperformanceofthestudent?colGPA=b0+d0

PC+b1hsGPA+b2ACT+ucolGPA=1.26

+0.157PC+0.447hsGPA+0.0087ACT+u

(0.33)(0.057)(0.094)(0.0105)

n=141,R2=0.219ThismeansthatastudentwhoownsacomputerhasapredictedGPAabout0.16pointshigherthanacomparablestudentwithoutacomputer.ThecoefficientofPCisdifferentfromzero,that’s,thedifferencebetweentwodifferenttypestudentsissignificant.6第六頁，共十八頁，2022年，8月28日MultipleCategories(cont)AnycategoricalvariablecanbeturnedintoasetofdummyvariablesBecausethebasegroupisrepresentedbytheintercept,iftherearencategoriesthereshouldben–1dummyvariablesIftherearealotofcategories,itmaymakesensetogroupsometogetherExample:wagedeterminationslog(wage)=0.388+0.292marrmale-0.120marrfem-0.097singfem+(0.102)(0.055)(0.058)(0.057)0.084educ+0.003exper+0.016tenure(0.007)(0.0017)(0.003)n=526R2=0.42387第七頁，共十八頁，2022年，8月28日InteractionsAmongDummiesInteractingdummyvariablesislikesubdividingthegrouplog(wage)=0.388-0.097female+0.292married-0.316female?married+(0.102)(0.057)(0.055)(0.074)0.084educ+0.003exper+0.016tenure(0.0069)(0.0017)(0.003)n=526R2=0.4238Thebasegroupissinglemenwhenfemale=0andmarried=0So,whenfemale=0andmarried=1,theinterceptforthemarriedmenis0.388+.0292=0.680female=1,married=0,singlewomen0.388-0.097=0.291female=1,married=1,marriedwomen0.388-0.097+0.292-0.316=0.2678第八頁，共十八頁，2022年，8月28日MoreonDummyInteractionslog(wage)=b0+b1

female+b2married+b3

female?married+b4

educ+b5

exper+b6

tenurefemale=0,married=0,log(wage)=b0

+b4

educ+b5

exper+b6

tenurefemale=0,married=1,log(wage)=b0

+b2

+b4

educ+b5

exper+b6

tenurefemale=1,married=0,log(wage)=b0+b1

+b4

educ+b5

exper+b6

tenurefemale=1,married=1,log(wage)=b0+b1+b2

+b3+b4

educ+b5

exper+b6

tenure9第九頁，共十八頁，2022年，8月28日OtherInteractionswithDummiesCanalsoconsiderinteractingadummyvariable,d,withacontinuousvariable,xy

=b0+d1d+b1x+d2d*x+uIfd=0,theny

=b0+b1x+uIfd=1,theny

=(b0+d1)+(b1+d2)x+uThisisinterpretedasachangeintheslopeExample:loghourlywageequation,(p235)log(wage)=0.465-0.210female+0.090educ-0.0072female?educ+0.0046exper+0.017tenure(0.123)(0.174)(0.0087)(0.014)(0.0016)(0.0030)n=526R2=0.3926Returntoeducationformenis0.090,or9%,forwomen,itis0.090-0.0072=0.0828,or8.28%Isthisdifferencesignificant?t=-0.0072/0.014=-0.53,sowecan’trejectthenullhypothesisthatthereisnodifferencebtwthereturntoeducationformenandwomen.10第十頁，共十八頁，2022年，8月28日yxy=b0+b1xy=(b0+d0)+(b1+d1)xExampleofd0>0andd1<0d=1d=011第十一頁，共十八頁，2022年，8月28日TestingforDifferencesAcrossGroupsTherefore,whethertheparametersoftwogroupsarethesameresultinwhetheralltheparametersofthedummyvariableandinteractiontermsarezero.Thatis,H0:d0=0,d1=0,...,dk=012第十二頁，共十八頁，2022年，8月28日TestingforDifferencesAcrossGroupsWagedeterminations:whethermenandwomenhavedifferentinterceptandslopes?Theoriginalmodelislog(wage)=b0

+b1educ+b2

exper++b3tenure+ufemaleisthedummyvariableTheunrestrictedmodelislog(wage)=b0

+d0

female+b1educ+d1female?educ+b2

exper+d2female?exper+b3tenure

+d4

female?tenure+uEstimatedtherestrictedandunrestrictedmodel,wegetlog(wage)=0.284

+0.092

educ+0.0041exper+0.022

tenuren=526SSRr=101.3298log(wage)=0.322+0.034

female+0.096educ-0.016

female?educ+

0.0081

exper-0.0059female?exper+0.018

tenure

–0.0079

female?tenuren=526SSRur=88.5825TheF=[(101.3298-88.5825)/4]/(88.5825/518)=18.82Wewillrejectthenullhypothesisandthereissignificantdifferencebtwmenandwomen.Statacommand:testfemale

female?educfemale?exper

female?tenure13第十三頁，共十八頁，2022年，8月28日TestingforDifferencesAcrossGroupsTestingwhetheraregressionfunctionisdifferentforonegroupversusanothercanbethoughtofassimplytestingforthejointsignificanceofthedummyanditsinteractionswithallotherxvariablesSo,youcanestimatethemodelwithalltheinteractionsandwithoutandformanFstatistic,butthiscouldbeunwieldy14第十四頁，共十八頁，2022年，8月28日TestingforDifferencesAcrossGroups,withoutdummyvariablesWagedeterminations:whethermenandwomenhavedifferentinterceptandslopes?WeestimatethemodelformenandwomenseparatelyMen:log(wage)=0.322

+0.096

educ+0.0081exper+0.0182

tenuren1=274SSR1=49.5472Women:log(wage)=0.356

+0.080

educ+0.0023exper+0.010

tenuren2=252SSR2=39.0353So,wegettheunrestrictedmodel’sSSRur=49.5472+39.0353=88.5825,Thepooledmodellog(wage)=0.284

+0.092

educ+0.0041exper+0.022

tenuren=526SSRr=101.3298ThenewFvalueisF=[(101.3298-(49.5472+39.035))/4]/((49.5472+39.035)/518)=18.82So,wege