JackmanModelsforOrderedOutcomesp4brl 1unconstrainedfixedoneljfixedegr1egl102dropinterceptfixedunconstrainedegr13unconstrainedunconstrainedtwoljfixedegseeKrehbielandRivers1988orBarte ID: 129930
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Jackman,ModelsforOrderedOutcomes,p2Thecentralideaisthatthereisalatentcontinuousmetricunderlyingtheordinalresponsesobservedbytheanalyst.Thresholdspartitionthereallineintoaseriesofregionscorrespondingtothevariousordinalcategories.Thelatentcontinuousvariable,y*isalinearcombinationofsomepredictors,x,plusadisturbancetermthathasastandardNormaldistribution:y*i=xib+ei,eiN(0,1),8i=1,...,N.(1)yi,theobservedordinalvariable,takesonvalues0throughmaccordingtothefollowingscheme:yi=j()lj-1y*ilj,wherej=0,...,m,andbyslightabuseofnotationinthepursuitofcomplete-nessIdefinel-1=-1,andlm=+1.Likethemodelsforbinarydata,weareconcernedwithhowchangesinthepredictorstranslateintotheprobabilityofobservingaparticularordinaloutcome.Considertheprobabilitiesofeachordinaloutcome:P[yi=0]=P[l-1y*il0],=P[-1y*il0],=P[y*il0],substitutingfrom(1),=P[xib+eil0],=P[eil0-xib],=U(l0-xib);P[yi=1]=P[l0y*il1],=P[l0xib+eil1],=P[l0-xibeil1-xib],=U(l1-xib)-U(l0-xib).ItisstraightforwardtoseethatP[yi=2]=U(l2-xib)-U(l1-xib), Jackman,ModelsforOrderedOutcomes,p4brl 1unconstrainedfixedoneljfixede.g.,r=1e.g.,l1=02dropinterceptfixedunconstrainede.g.,r=13unconstrainedunconstrainedtwoljfixede.g.,seeKrehbielandRivers(1988)orBartels(1991) Table1:OrderedProbitModel,IdentificationConstraints.varianceparameter,r2,ormoretechnically,thestandarddeviation,r.Ifthevarianceofy*iwerealsosomethingtobeestimatedthenthemodel'sparametersareunidentified;evenwithl0``anchoring''themappingofxibtoy*i,allowingr2andbtobothbe``freeparameters''wouldalsoresultinaninfinitecollectionofestimatesthatfitthedataequallywell.Foranycandidateˆbthereisnouniquescalingofy*iviaaˆr2maximizingfittothedata.Settingthevariancetoaknownconstantaprioricircumventsthisproblem.Standardpracticeistosetr2to1ratherthananarbitraryknownconstant,sincethissimplifiestheUijtermsinevaluatingthelog-likelihoodfunction.Likewise,wecouldidentifythelandbparametersavarietyofways,andasImakeclearabove,differentimplementationsofthismodelusedifferentapproaches.Settingl0=0issomerespectshighlyarbitrary,anddonelargelyforprogrammingconvenienceonly,sinceitisthefirstthresholdencounteredinanorderedprobitmodelnomatterhowmanyordinalcategoriestheusermaypasstoacomputerprogramdesignedtoestimatethesemodels.2.2ExploitingIdentificationConstraintsItisimportanttorememberthattheseidentificationconstraintsarenonethelessarbitrary,andinthehandsofaskillfulanalystthiscanbeausefulwaywithwhichtoextractsubstantivemileagefromtheresultsofanorderedprobitmodel.Itissometimespossibletore-definethelatentvariabley*iassubstantivelymeaningfulquantity,suchasmoney,votes,numbersof Jackman,ModelsforOrderedOutcomes,p6whereuiisanymonotonedecreasingfunction(aweakerassumptionthanthestandardassumptionaboutquadraticutilityfunctions),hisapolicyoption(h2XR),andxiistheunobservedidealpointoflegislatori,i=1,...,N;2.votingissincere(bothKrehbielandRiversandBartelsarequicktodiscountthepossibilityofsophisticatedvotingintheirrespectivecon-texts,withrecoursetoboththeoreticalargumentsandcloseinspectionofthesequenceandnatureofthevotesunderconsideration).3.1KrehbielandRivers:2votes,knownalternativesUndertheseassumptions,inspectingapatternofvotestellsussomethingabouttheregionsofthepolicyspaceinwhichlegislators'idealpointslie.Forinstance,KrehbielandRiversconsiderasequenceoftwovotes,eachvoteabinarychoicebetweenwell-specifiedpointsinagivenunidimensionalpolicyspace.First,legislatorsvotebetweenh1andh3,andsayh3wins.Thenh2isvotedagainsth3.Withh1,h2,andh3known,itisthenpossibletorankorderthelegislatorsintermsofwheretheirunobservedidealpointslieinXR.Toseethis,notethatalegislatorvotesforh1overh3inthefirstvoteifui(h1)�ui(h3)andotherwisevotesagainsth1(inthecaseoftheutilitiesbeingequalweassumethatthelegislatorprefersthestatusquopolicy,herelabelledh1,withoutlossofgenerality.)Inthesecondvote,alegislatorvotesforh2overh3ifui(h2)�ui(h3)andotherwisevotesagainsth2.Theoreticallytherearefourpossiblepatternsofvotinghere,22combinationsofthe``yea''and``nay''possibilities.Ingeneral,legislatorivotesforoptionhjoverhkifandonlyifui(jhj-xij)�ui(jhk-xij).Sinceuiisstrictlydecreasinginhaboutxi,thisconditioncanberewrittenasjhk-xij�jhj-xiji.e.,proposalhkisfurtherawayfromlegislatori'sidealpointthanproposalhj.Theseconditionsimplythefollowing:xihj+hk 2()hjhk, Jackman,ModelsforOrderedOutcomes,p8 Figure1:KrehbielandRiversanalysisof1977minimumwageamendments. Jackman,ModelsforOrderedOutcomes,p9interpretableintermsofthatmetric.Forinstance,thevotesanalyzedbyKrehbielandRiversrefertoproposedlevelsoftheminimumwage,consideredbytheSenateindebatingtheFairLaborStandardAmendmentsAct(1977,S.1871).Thehquantitiesinthiscaseareactualdollaramountscorrespondingtothe1980minimumwage:h1=$2.90perhour(anamendmentproposedbyDeweyBartlett),h2=$3.05perhour(anamendmentproposedbyJohnTower),andh3=$3.15perhour(thelevelproposedbytheLaborandHumanResourcesCommittee,thenunderthechairmanshipofHarrisonA.Williams,andultimatelyapprovedbytheSenate,76-14).Giventhatwecanassociatethesedollarfigureswiththeoptionswiththevotes,itisalsopossibletoassociatedollarfigureswiththecutpointsbetweenthecategoriesdefiningthedependentvariable.Inthiscase,l0=(h1+h3)/2=$3.025perhour,andl1=(h2+h3)/2=$3.10perhour.WiththeseexactrestrictionsonthelparametersKrehbielandRiversareabletomodifythestandardidentificationrestrictionsfortheorderedprobitmodel.Inparticular,thisexactknowledgeonlmeansthatwecanletr2beafreeparameter.Inpractice,onecouldeitherwriteoutandprogramalog-likelihoodfunctionwiththeseidentifyingconstraintsonthel``hard-wired''(andr2afreeparameter),oruseacannedorderedprobitroutineandre-standardizetheestimatestosuittheconstraintsonthethresholds.Thisinvolvesalineartransformationoftheparameterestimates,notingthatthedifferencebetweenl0=0andˆl1isnowjustthedifferencebetween$3.025perhourand$3.10perhour,or.075ofadollarperhour.Onewouldre-scale(i.e.,multiply)theprobitestimatesbyaconstantmsuchthatthetransformedˆl1equals.075,andthenadd3.025:i.e.,z*=mz+c,(2)wherezisalocationestimatefromtheprobitmodel(athreshold,aslopeestimate,oranestimatedidealpoint),misthere-scalingconstant,andcisthelocationshift.Notethatwhenre-scalingdispersionparametersliker2orthestandarderrorsoftheparameterestimatesoneneedsonlythescaleshiftgivenbythemultiplierm,andnotthelocationshiftgivenherebym=3.025.Table1oftheKrehbielandRiversarticleincludesestimatesofrproducedbythisscaleshift,andsincer=1isthetypicalparameterization,thescalingconstantmissimplytheestimateofrinTable1.Thatis,KrehbielandRiverscouldhaveobtainedtheestimatesinTable1bytakingtheusualprobitoutput,multiplyingtheestimatesby.105,andadding3.025totheresultfor Jackman,ModelsforOrderedOutcomes,p11 Figure2:Bartels'analysisof1981defenseappropriationvotes. Jackman,ModelsforOrderedOutcomes,p12priori.Thefoursequencesofvotesareshownatthebottomofthefigure,withtheleftmostsequence{``Yea'',``Nay'',``Nay''}correspondingtothelowestcategoryintheorderedprobitanalysis.Bartelsreportsthatofthe108legislatorsinhissample(thoselegislatorswhosedistrictswereincludedinthesamplingframeforthe1980NationalElectionStudy),noneexhibitedapatternofvotesinconsistentwithanassumptionoffixedpreferences,symmetric,single-peakedutilityfunctions,andsincerevoting.Inotherwords,thefourpatternsofvotinglistedatthebottomofFigure2aretheonlylogicalpossibilitiesundertheseassumptionsandweretheonlypatternsobservedinthedata.3.3Re-calibratingestimatesfromtheorderedprobitmodelThedependentvariableintheorderedprobittakesonthevalues0through3,correspondingtothefourcategoriesatthebottomofFigure2.Followingstandardpractice,thefirstprobitthresholdissetto0foridentificationpurposes,andr2to1(seeTableA2,atp470oftheBartelsarticle).Bartelsobtainsthefollowingestimatesoftheotherthresholds,againstwhichIlistthecorrespondingvaluesintermsofknowndollaramounts,and/orthereversionlevelQ:ThresholdProbit$bn l0098.72+Q/2ˆl118399.85+Q/2ˆl21148196.61 Convertingthethresholds(andhence,alltheprobitparametersestimatesaswell)todollaramountsisaccomplishedusingthemethodweemployedfortheKrehbielandRiverscase;i.e.,alineartransformationwillre-calibratetheprobitestimatestothedollarscale.Weneedtoknowaslopeandinterceptparametersforthislineartransformation,plusalsosolvefortheunknownreversionamountQ.Wecansolveforalltheseparametersexactly,sincetheaboveinformationcanberepresentedasasystemofthreeequationsinthreeunknowns,98.72=ml0+-.5Q+c,99.85=mˆl1+-.5Q+c,196.61=mˆl2+0Q+c, Jackman,ModelsforOrderedOutcomes,p14Table2:Bartels(1991)Table1:SourcesofSupportforPentagonAppropria-tions. EstimatedEffectsVariableUnits($billions) Intercept-18396(1340)ConstituencyNES7-pointscale1287opinion(582)ConstituencyNESscale0112competitivenessloser's%vote(0497)Taxburden$1,000spercapita-414(372)Pentagonoutlays$1,000spercapita770(368)PartisanshipRepublican=1387Democrat=0(179)Presidentialvotedifferencein100,000s469influence(345) Jackman,ModelsforOrderedOutcomes,p16Figure3:Simulatedeffectsofchangeinconstituencyopinion. Jackman,ModelsforOrderedOutcomes,p17estimatesinthearticle).Thefirstpanelshowsthefamiliarlinearrelationshipbetweenapredictorandthelatentprobitvariable.Thesecondpanel'sparallellinesaretheeffectsofincreasingrelativetothethreethresholds;eachlineisjustlj-x0iˆb,j=0,1,2.RecallthattheprobabilityofbeinginthejthcategoryisestimatedasU()ofthisquantity,lesstheprobabilityofbeingintheimmediatelylowercategory.Stackingthesequantities,asinthethirdpanelofFigure3resultsinthethreecurvedlines.Theareabelowthelowestcurvedlineisjusttheprobabilityofbeingincategory``0''inBartels'analysis,thecategoryoflegislatorswhosevotingpatternsrevealedrelativelylowlevelsofpreferreddefenseappropriations.Noticehowthisprobabilityfallsawayasconstituencyopinionincreases,tobereplacedbyagreaterprobabilitythatalegislatorbelongstooneofthehighercategories.Thelastpanelshowstheeffectsofincreasingconstituencyopiniononlegislators'idealpoints,intermsofdollars.Sincethedollarmetricisalineartransformationoftheprobitmetric,andtheeffectsoftheindependentvariablesarelinearontheprobitmetric,itisnosurprisethattheeffectsoftheindependentvariablesexpressedontheideal-pointmetricarelinearalso.Thedottedhorizontallinesshowthethresholdsbetweenthecategoriesusedintheorderedprobit.Typicallyoneisinterestedinplotsasinpanels(3)and(4).Thesearethetypesofcomparativestaticspresentedinpractice;Iprovidetheotherplotsforcompleteness.TheSpluscodeIusedappearsbelow.#simulatesomepredictedvalsfromLarry's1991piece#meansofdatax0_c(1,1.205,33.499,2.012,.454,.444,-.269)names(x)_c("Intercept","CnstncyOpn","ConstxComp","TaxBurden","Pentagonoutlays","Partisanship","PresInfluence")#partisanship=0Dem,1Repubincasewanttoplaywiththat#othervarsdefndinapsrpiece#parameterestimates(probit)b_c(-.91,2.095,.00183,-.673,1.253,.63,.764)#thresholdestimatesmu_c(0,.183,1.148) Jackman,ModelsforOrderedOutcomes,p19yrange_par()$usryrange_.95*yrange[4]+.05*yrange[3]lines(xsim,pyhat.out[,2])lines(xsim,pyhat.out[,3])text(1.75,yrange,"(3)")plot(xsim,muyhat.out[,1],type="l",ylab="ProbitScale",cex=.85)yrange_par()$usryrange_.95*yrange[4]+.05*yrange[3]lines(xsim,muyhat.out[,2])lines(xsim,muyhat.out[,3])text(1.75,yrange,"(2)")plot(xsim,money.out,type="l",ylab="IdealPoints,$bn",cex=.85)abline(h=bart(mu),lty=2)yrange_par()$usryrange_.95*yrange[4]+.05*yrange[3]text(-1.15,188,"CQ303",cex=.5)text(-1.15,192,"CQ345",cex=.5)text(-1.15,198,"CQ302",cex=.5)text(1.75,yrange,"(4)")5ReferencesAitchison,J.,andS.Silvey.1957.TheGeneralizationofProbitAnalysistotheCaseofMultipleResponses.Biometrika.44:131--40.Aldrich,JohnH.andForrestD.Nelson.1984.LinearProbability,Logit,andProbitModels.SageUniversityPaperseriesonQuantitativeApplicationsintheSocialSciences,seriesnp.07-045.Sage.NewburyPark,California.Amemiya,Takeshi.1985.AdvancedEconometrics.Harvard.Cambridge.Bartels,LarryM.1991.ConstituencyOpinionandCongressionalPolicMaking:TheReaganDefenseBuildup.AmericanPoliticalScienceReview.85:457--74.