Multinomial and Ordered Response Models.pdf

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1、MultinomialandOrderedResponseModelsPhDEconometricsWeitanHanqingAdvancedInstituteofEconomicsandFinanceRenminUniversityofChinaWeiTan(RUC)MultinomialandOrderedResponseModels1/28MultinomialLogitLetydenotearandomvariabletakingonthevaluesf0;1;


;JgforJapositiveinteger,an

2、dxa1Kvectorwith…rst-elementunity.Themultinomiallogit(MNL)modelhasresponseprobabilities:"#XJP(y=jjx)=exp(xj)=1+exp(xh);j=1;


;J(1)h=1wherejisK1,j=1;


;J.Becausetheresponseprobabilitymustsumtounity,"#XJP(y=0jx)=1=1+exp(xh)h=1WhenJ=1,1istheK1vectorofunknownparamet

3、ers,andwegetthebinarylogitmodel.WeiTan(RUC)MultinomialandOrderedResponseModels2/28MultinomialLogitThepartiale¤ectsforMNLmodelarecomplicated.Forcontinuousxk,wecanwrite("#)XJ@P(y=jjx)=P(y=jjx)jkhkexp(xh)=g(g;)@xkh=1(2)PJhkisthekthelementofhandg(x;)=1+h=1exp(xh).Equat

4、ion(2)showsthateventhedirectionofthee¤ectisnotdeterminedentirelybyjk.Asimplerinterpretationofjisgivenbypj(x;)=p0(x;)=exp(xj);j=1;2;


;J(3)wherepj(x;)denotestheresponseprobabilityinequation(1).Thus,thechangeinpj(x;)=p0(x;)isapproximatelyjkexp(xj)
xkforroughlycontinu

5、ousxk.Equivalently,thelog-oddsratioislinearinx:log[pj(x;)=p0(x;)]=xj.Thisresultextendstogeneraljandh:log[pj(x;)=ph(x;)]=x(jh).WeiTan(RUC)MultinomialandOrderedResponseModels3/28MultinomialLogitThereisanotherusefulfactaboutthemultinomiallogitmodel.SinceP(y=jory=hjx)=

6、pj(x;)+ph(x;),pj(x;)P(y=jjy=jory=h;x)==x(jh)pj(x;)+ph(x;)where(
)isthelogisticfunction.Inotherwords,conditionalonthechoicebeingeitherjorh,theprobabilitythattheoutcomeisjfollowsastandardlogitmodelwithparametervectorjh.EstimationoftheMNLmodelisbestcariedoutbymax

7、imumlikelihood.ForeachitheconditionalloglikelihoodisXJli()=1[yi=j]log[pj(xi;)]j=0wheretheindicatorfunctionselectsouttheappropriateresponseprobabilityforeachobservationi.Asusual,weestimatebyPNmaximizingi=1li().WeiTan(RUC)MultinomialandOrderedResponseModels4/28Multino

8、mialLogit:ExampleExample(SchoolandEmploymentDecisionsforYoungMen):ThedataKEANE.DTA(asubsetfromKeaneandWolpin,1997)containsemploymentandsch