odd15-17

CHAPTER1515.1.a.Sincetheregressorsareallorthogonalbyconstruction--dkWdm=ii0fork$m,andalli--thecoefficientondmisobtainedfromtheregressionyondm,i=1,...,N.Butthisiseasilyseentobethefractioniiofyinthesamplefallingintocategorym.Therefore,thefittedvaluesareijustthecellfrequencies,andthesearenecessarilyin[0,1].b.Thefittedvaluesforeachcategorywillbethesame.Ifwedropd1butaddanoverallintercept,theoverallinterceptisthecellfrequencyforthefirstcategory,andthecoefficientondmbecomesthedifferenceincellfrequencybetweencategorymandcategoryone,m=2,...,M.215.3.a.IfP(y=1|z,z)=F(zD+gz+gz)then12111222dP(y=1|z,z)122-------------------------------------------------------------------------=(g+2gz)Wf(zD+gz+gz);122111222dz2forgivenz,thisisestimatedas^^^^^2(g+2gz)Wf(zD+gz+gz),122111222where,ofcourse,theestimatesaretheprobitestimates.b.InthemodelP(y=1|z,z,d)=F(zD+gz+gd+gzd),121111221321thepartialeffectofzis2dP(y=1|z,z,d)121---------------------------------------------------------------------------------------=(g+gd)Wf(zD+gz+gd+gzd).131111221321dz2Theeffectofdismeasuredasthedifferenceintheprobabilitiesatd=111andd=0:1P(y=1|z,d=1)-P(y=1|z,d=0)11=F[zD+(g+g)z+g]-F(zD+gz).1113221112Again,toestimatetheseeffectsatgivenzand--inthefirstcase,d--we189justreplacetheparameterswiththeirprobitestimates,anduseaverageorotherinterestingvaluesofz.c.WewouldapplythedeltamethodfromChapter3.Thus,wewouldrequirethefullvariancematrixoftheprobitestimatesaswellasthegradientoftheexpressionofinterest,suchas(g+2gz)Wf(zD+gz+12211122gz),withrespecttoallprobitparameters.(Notwithrespecttothez.)22j15.5.a.IfP(y=1|z,q)=F(zD+gzq)then1112dP(y=1|z,q)-----------------------------------------------------------------=gqWf(zD+gzq),11112dz2assumingthatzisnotfunctionallyrelatedtoz.21*b.Writey=zD+r,wherer=gzq+e,andeisindependentof1112(z,q)withastandardnormaldistribution.Becauseqisassumedindependent22ofz,q|z~Normal(0,gz+1);thisfollowsbecauseE(r|z)=gzE(q|z)+1212E(e|z)=0.Also,2222Var(r|z)=gzVar(q|z)+Var(e|z)+2gzCov(q,e|z)=gz+1121212becauseCov(q,e|z)=0byindependencebetweeneand(z,q).Thus,5======================================22r/rgz+1hasastandardnormaldistributionindependentofz.Itfollows12that5======================================&22*P(y=1|z)=FzD/rgz+1.(15.90)1112782c.BecauseP(y=1|z)dependsonlyong,thisiswhatwecanestimate1alongwithD.(Forexample,g=-2andg=2giveexactlythesamemodel1112forP(y=1|z).)Thisiswhywedefiner=g.TestingH:r=0ismost1101easilydoneusingthescoreorLMtestbecause,underH,wehaveastandard0^probitmodel.LetDdenotetheprobitestimatesunderthenullthatr=0.115================================================^^^^^^~^^^DefineF=F(zD),f=f(zD),u=y-----F,andu_u/rF(1-F)ii11ii11iiiiiii90(thestandardizedresiduals).Thegradientofthemeanfunctionin(15.90)^withrespecttoD,evaluatedunderthenullestimates,issimplyfz.The1ii1onlyotherquantityneededisthegradientwithrespecttorevaluatedatthe1nullestimates.Butthepartialderivativeof(15.90)withrespecttoris,1foreachi,-3/25==========================================2&2*(22)-(zD)(z/2)rz+1fzD/rgz+1.i11i21i2i111i27890^^2^Whenweevaluatethisatr=0andDweget-(zD)(z/2)f.Then,the11i11i2i2scorestatisticcanbeobtainedasNRfromtheregressionu5================================================5================================================~^^^^2^^^uonfz/rF(1-F),(zD)zf/rF(1-F);iii1iii11i2iii2a2underH,NR~c.0u1d.ThemodelcanbeestimatedbyMLEusingtheformulationwithrin12placeofg.Butthisisnotastandardprobitestimation.115.7.a.ThefollowingStataoutputisforparta:.regarr86pcnvavgsentottimeptime86inc86blackhispanborn60Source|SSdfMSNumberofobs=2725-------------+------------------------------F(8,2716)=30.48Model|44.972091685.62151145ProbF=0.0000Residual|500.8444222716.184405163R-squared=0.0824-------------+------------------------------AdjR-squared=0.0797Total|545.8165142724.20037317RootMSE=.42942------------------------------------------------------------------------------arr86|Coef.Std.Err.tP|t|[95%Conf.Interval]-------------+----------------------------------------------------------------pcnv|-.1543802.0209336-7.370.000-.1954275-.1133329avgsen|.0035024.00634170.550.581-.0089326.0159374tottime|-.0020613.0048884-0.420.673-.0116466.007524ptime86|-.0215953.0044679-4.830.000-.0303561-.0128344inc86|-.0012248.000127-9.650.000-.0014738-.0009759black|.1617183.02350446.880.000.1156299.2078066hispan|.0892586.02055924.340.000.0489454.1295718born60|.0028698.01719860.170.867-.0308539.0365936_cons|.3609831.016092722.430.000.329428.392538291------------------------------------------------------------------------------.regarr86pcnvavgsentottimeptime86inc86blackhispanborn60,robustRegressionwithrobuststandarderrorsNumberofobs=2725F(8,2716)=37.59ProbF=0.0000R-squared=0.0824RootMSE=.42942------------------------------------------------------------------------------|Robustarr86|Coef.Std.Err.tP|t|[95%Conf.Interval]-------------+----------------------------------------------------------------pcnv|-.1543802.018964-8.140.000-.1915656-.1171948avgsen|.0035024.00588760.590.552-.0080423.0150471tottime|-.0020613.0042256-0.49