1IntroductionNexttodrinkingalcohol,sexisthemostcommonriskybehaviorofte - PDF document

1IntroductionNexttodrinkingalcohol,sexisthemostcommonriskybehaviorofteens.1Whilenumerousstudiestrytomeasuretheimpactofpolicyortechnologicalchangesonsexualbehaviorsofteens2andotherstrytomeasuretheeectofsexualactivitiesoneducationalandhealthoutcomes,3toourknowledgetherearenoeconomicstudiesofafundamentaltradeofacedbyteens:thetradeobetweenbeinginaromanticrelationshipatallandtheinclusionofsexinarelationship.Wemodelandestimatethistradeowithintheconnesofthebestavailabledataonteenrelationshipsandsexualbehaviors.Weshowthatinequilibriumsomewomenhavesexoutofmatchingconcerns.Further,asmenbecomemorescare,thiseectisaccentuated.Touncoverdierencesingender-specicpreferences,wespecifyamodeloftwo-sideddirectedsearch.Utilitiesinarelationshipdependonpartnercharacteristics,thetermsoftherelationship,andonpastsexualexperience.4Individualsofadis-cretetypeononesideofthemarkettargettheirsearchtowardpotentialpartnerswithparticularcharacteristicsandalsotargettheirsearchbasedonthetermsoftherelationship(sexornosexinourcase).5Giventhesuppliesofindividualsofeachtypeonbothsidesofthemarket,theexanteyieldfromtargetingparticularpartnercharacteristicsandtermsdependsontheassociatedprobabilityofmatching.Theprobabilityofmatchinginturnisendogenouslydeterminedbythechoicesofindi-vidualsonone'sownsideofthemarket(rivals)andbyallindividualsontheotherside(prospects).Changesinthesupplyofmenandwomenofdierentcharacteristics 1TheYouthRiskBehaviorSurveillanceSystem(YRBS)oftheCentersforDiseaseCon-troltracksthemajorrisktakingbehaviorsofanationalrepresentativesampleofninththroughtwelfthgraders.Summarystatisticsfor2011showthatmoreteenshavehadsex(47%)thaneverusedmarijuana(40%),otherillicitdrugs,tobacco(44%),orat-temptedsuicide.Onlythebehaviorofeverhavingadrinkofalcohol(71%)outrankedsex.http://www.cdc.gov/healthyyouth/yrbs/pdf/us overview yrbs.pdf2Akerlof,YellenandKatz(1996)discusshowabortionchangedsexualparticiaption;GirmaandPaton(2011)studyemergencycontraceptionavailabilityandunprotectedsex.3SeeSabiaandRees(2008)andSabiaandRees(2009).4Arcidiacono,KhwajaandOuyang(2012)provideevidencethatonceadolescentfemalesbecomesexuallyactive,theygenerallyremainso.5ThisformulationofcharacteristicsandtermsbuildsonworkbyDagsvik(2000)whodevelopsatheoreticalmatchingmodelwithterms,whereagentshavepreferencesovertermsandobservedmatchcharacteristics,andthepreferencedistributioncaninprinciplebebackedoutfromobservedaggregatematchingpatterns.Willis(1999)alsopresentsanequilibriummodelwherethetermsarejointandsingleparenthood,showingthatsexratiochangescangenerateequilibriawherewomenraisechildrenassinglemothers.RecentworkbySalaniandGalichon(2012)addressesmatchingbasedonunobservedcharacteristicsundertransferableutility.2 ticityofsubstitutionmatchingfunctions,weshowthat,asthegenderratiobecomesmoreunfavorable,theindividualbecomesmorelikelytosacricerelationshiptermsforahighermatchprobability.Theadvantagesofourmodelingapproacharethree-fold.First,bylinkingchoicesoverpartnerswithoutcomes,inequilibriumweareabletoweightthedierentgenderratiosappropriately.Standardpracticeistouseonlyonesex-ratiowhenlookingattherelationshipsbetweengenderratiosandoutcomes.Buttherelevantgenderratiosaredeterminedbyequilibriumforces.Second,byallowingpreferencesoverbothpartnercharacteristicsandwhathappensintherelationship(inthiscasegradeandraceofthepartnerpluswhethersexoccurs),weareabletocapturethetradeosmadeacrossthetwowhenthematchingenvironmentchanges.Forexample,increasingthenumberofseniorboysfavorswomen.Thismay,however,stillresultinmoresexualrelationshipsbecausethefemalepreferenceforseniorboysovercomesfemales'preferringnottohavesex.Finally,byworkinginanon-transferableframeworkwithpartnerselection,weareabletoidentifypreferencesforoutcomes.13Inthetransferablecase,iftheoccurrenceofaparticularoutcomeisaectedbythegenderratio,itisunclearhowutilitiesareaectedbythegenderratiobecauseindividualsmaybemakingtransfersnotobservedbytheresearcher,particularlyinthecasewheretheoutcomesofrelevancearediscrete.WeestimatethemodelusingdatafromtheAddHealthdata.ThesedatacontaininformationontheuniverseofstudentsatparticularU.S.highschoolsin1995aswellasanswerstodetailedquestionsaboutrelationshipsforasubsetofthestudents.Themodelisestimatedassumingthatindividualsareabletotargettheirsearchtowardsopposite-sexpartnersofaparticulargradeandraceaswellastospecifywhetherornotsexwilloccurintherelationship.Notsurprisingly,estimatesofthestructuralmodelshowthatmenvaluesexualrelationshipsrelativelymorethanwomen.Bysimulatingchoicesintheabsenceofmatchingconcerns,wendthat31.6%ofwomenand61.8%ofmenwouldprefertobeinasexual,asopposedtoanonsexual,relationship.Thesecounterfactualchoices 13BothHitsch,HortacsuandAriely(2010)andFisman,Iyengar,KamenicaandSimonson(2006)useobservedchoicesofbothmenandwomentobackoutpreferencesforpartnercharacteristics.Fisman,Iyengar,KamenicaandSimonson(2006)observechoiceinaspeed-datingexperimentandHitsch,HortacsuandAriely(2010)observetheminanonline-datingcontext.Chiappori,Oref-ceandQuintana-Domeque(2009)recoversseparatepreferencesformalesandfemalesforpartnercharacteristicsbutnotforrelationshipterms.4 2ModelWeanalyzethetradeosamongthreefundamentalsourcesofexpectedutilityfromsearchingforapartner:thetypeofpartner(race,grade),thetermsoftherelationship(sex/nosex)andtheprobabilityofsuccess(matching).Individualsknowinadvancetheirpayosfromdierentpartnertypesandrelationshipterms.And,theymaytargetless-preferredcombinationsoftypesandtermsinordertohavehigherproba-bilitiesofmatching.Itisthisfundamentaltradeothatdistinguishesourmodelfromothers.Atitscore,ourmodelembedssearchandtheattendantriskofnotmatchingintoastaticmodel.Inthatsense,itisanalogoustothewagepostingmodelsinwhichworkerschoosebetweenahighwagermwithalowprobabilityofmatching,oralowwagermwithahighprobabilityofmatching.Inordertodisentanglemaleandfemalepreferences,weproposeatwo-sidedsearchmodelwithnon-transferableutilityandconsideronlyopposite-sex,one-to-onematch-ing.14Wecategorizeeachmaleasatypemwherem2f1;2;:::;Mg.Similarly,eachwomanisgivenatypewwherew2f1;2;:::;Wgelements.Anindividual'stypecandenotesomecollectionofobservedcharacteristicssuchasage,grade,race,orattractiveness.Formales(females)therearethenW(M)typesofmates.Letimindicatethei-thmemberoftypem.Weindexthepossibletermsoftherelationshipbyr2f1;:::;Rg.Thepossibletermscouldincludenothavingsex,havingsexwithprotection,etc.Wemodelsearchasbeingcompletelydirected:menandwomenareabletotargettheirsearchonboththecharacteristicsofthepartneraswellasthetermsoftherelationship.Eachman(woman)thenmakesadiscretechoicetosearchinoneofMR(WR)markets,resultinginMWRtypesofmatches.Searchisthenmodeledasaone-shotgame:therearenodynamicsinthemodel.Individualsrstdecideinwhichmarkettosearch.Couplesarematchedwiththeprobabilityofmatchingdependingonthenumberofsearchersonbothsidesofthemarket. 14Only2%ofthesamplereportedconcurrentsexualmatchesand1%reportedconcurrentrelation-ships,thoughclearlysomereportingproblemsexist.Weproceedinmodelingone-to-onematchinggiventhecomplexityofmodelingmultiplematchesandtherstorderimportanceofthemainreportedmatchforpreferences.6 Wetreatthewrim'sasobservedonlytotheindividual:onlythedistributionisknowntotheotherparticipantsinthemarket.Weassumethatthewrim'sarei.i.d.TypeIextremevalueerrorsandareunknowntotheeconometrician.Inthiscase,wecanestimatetheerrorvariance,,asacoecientonthelogprobabilityterm,capturinghowtheprobabilityofmatchingin uencesutility.Theprobabilityofam-typemansearchingforaw-typewomaninanr-typerelationship,wrmthenfollowsamultinomiallogitform:Pr(w;rjm)=wrm=expwrm+ln(Pwrm)  Pw0Pr0expw0r0m+ln(Pw0r0m) (3)2.2MatchingWenowspecifythematchingprocess.Thematchingprocessisessentiallyaproduc-tionfunction,takingasinputsthenumbersearchingmenandthenumberofsearchingwomenineachmarketandgivingthenumberofmatchesineachmarketasanout-put.Weparameterizethenumberofmatches,X,inmarketfm;w;rgasdependinguponthenumberofm-typemenandw-typewomensearchinginthemarket.LetNmandNwindicatethenumberofm-typemenandnumberofw-typewomenoverall.Recallthatwrmandmrwgivetheprobabilityofm-typemenandw-typewomenwhosearchinmarketfm;w;rgwhicharealsothemarketsharesofsearchingmenandwomen.ThuswrmNmisthenumberofmenoftypemsearchingwomenoftypewonrelationshiptermsr.Thenumberofmatchesinmarketfm;w;rgisthengivenby:16Xmwr=A(wrmNm) 2+(mrwNw) 21 =A[(wrmNm)+(mrwNw)]1 (5) 16Foreaseofexpositionweareassuminganinteriorsolutionsuchthatthenumberofmatchesproducedislessthanboththenumberofmenandthenumberofwomeninthefw;m;rgmarket.Inpractice,wenesttheCESmatchingfunctionintoaLeontieffunctiontoconstrainthenumberofmatchestobelessthanthenumberofsearchingmenandwomen:Xmwr=minnA[(wrmNm)+(mrwNw)]1 ;wrmNm;mrwNwo(4)8 cannotbeaParetomove.182.4ImplicationsofChangingtheGenderRatioGivenourutilityspecicationandmatchingprocess,wenowturntohowchangingthegenderratioleadstochangesintheprobabilitiesofchoosingparticularmarkets.Tobegin,considertwomarketsthatincludewtypewomenandmtypemenbutwheretherelationshiptermsinthetwomarketsaregivenbyrandr0respectively.Now,xthesearchprobabilities,the's,andincreasethenumberofm-typemen.Wecanthenseewhichofthetworelationshipmarketsbecomerelativelymoreattractiveformenandwomenrespectively.Wethenshowhowthesearchprobabilitiesmustadjustinequilibrium.DenotingGmwastheratioNm=Nw,Proposition1showstherelationshipbetweenthegenderratioandsearchbehavior,withtheproofinAppendixA.Proposition1.If0andmr0w�mrw&#x]TJ/;༗ ;.9;Ւ ;&#xTf 2;
.78; 0 ;&#xTd [;wr0m�wrmthenthefollowinghold:(a)mrw wrmmr0w wr0m(b)Pmrw�Pmr0wandPwrmPwr0m(c)Both@mr0w=mrw @Gmw�0and@wr0m=wrm @Gmw�0InProposition1theaveragepreferenceofafm;wgpairissuchthatthewomeninthepairhaveastrongerpreferencefortermsr0overrthantheirmalecounterparts.Thersttwoclaimsareintuitive.Claim(a)statesthatthisrelativepreferencebywomenforr0overrtranslatesintosearchbehavior:inequilibrium,theratioofsearchprobabilitiesforr0versusrmustresultinwomensearchingmoreinr0relativetomen.Thesedierentialsearchprobabilitiesthentranslateintomatchprobabilities.Sincewomenarerelativelymorelikelythanmentosearchinr0,femalematchprobabilitiesmustbelowerinr0thaninr,withthereverseholdingformen,claim(b).Thekeyresultforourempiricalworkisclaim(c).Asthegenderratiomovessuchthatmenbecomerelativelymoreabundant,bothmenandwomenincreasetheirrelativesearchprobabilitiesinthemarketwherewomenhavearelativepreference,inthiscaser0.Theresultfallsdirectlyoutoftheelasticityofsubstitution.Namely, 18Notethatbecausethedecisionsarebasedonexpectedutilityandprobabilisticsearchthesta-bilityoftheresultingmatchingwillnothold.10 dentsatarandomlysampledsetof80communitiesacrosstheUnitedStates.21At-temptsweremadetohaveasmanystudentsaspossiblefromeachschoollloutthesurveyduringaschoolday.Questionsconsistmainlyofindividualdatalikeage,race,andgrade,withlimitedinformationonacademics,extra-curricularactivitiesandriskybehavior.Weusethissampletoconstructschoollevelaggregatesbyobservablecharacteristics,gradeandrace,allowingustocalculategenderratios.TheAddHealthdataalsoincludesarandomsampleofstudentswhoweread-ministeredamoredetailedin-homesurvey.Thein-homesampleincludesdetailedrelationshiphistoriesandsexualbehaviors.Therelationshiphistoriesincludebothwhathappenswithintherelationshipsaswellascharacteristicsofthepartner.Anaturalprobleminthissurveydesignistheissueofwhatconstitutesarelationshiptorespondents,particularlywhenmenandwomenmaydenerelationshipsdier-ently.Thedenitionthata\relationship"referredtofromhereon,consistsofboththefollowing(i)asholdinghandsand(ii)kissing.Thisdenitionresultsinthemostsymmetricdistributionofresponseswithinschoolsandallowsforthemostdatainthesurveytobeaccessed.22Therelationshiphistoryallowsustodeterminewhetherrespondentshadsexwithadierentpartnerpriortothecurrentpartnership.Werestrictattentiontoschoolswhichenrollbothmenandwomen.Asampleofongoingrelationshipsshowed55%ofpartnersmetinthesameschool,thenextclosestavenueofmatchingwasmutualfriends,accountingforonly24%ofmatches.23Sincethefocusherewillbeonacrosssectionofthematchingdistribution,wecountonlycurrentrelationshipsamongpartnerswhoattendthesameschool.Thosematchedwithsomeoneoutsideoftheschoolareinitiallydropped,consistentwithassumingtheoutsidematchingmarketisfrictionless,whichwerelaxinSection5.3.24 21Aschoolpair,consistingofahighschoolandarandomlyselectedfeederschool(middleschoolorjuniorhighschoolfromthesamedistrict)weretakenfromeachcommunity.22Applyingthisdenition48.8%ofongoingin-schoolrelationshipscamefromwomenand51.2%frommen.Withperfectreportingandagreementoverthedenitionwewouldseeparity.23Thedataonwhereindividualsmetisonlyavailableforasubsampleofrelationships.Theotheravenueswere9%priorfriends,andlessthan5%intheirneighborhood,placeofworship,andcasualacquaintanceseach.24From14840studentsbetweengrades9-12,wedrop:schoolsdiscussedabove(2622),schoolswithfewerthan10reportedstudents(146),unweighteddata(996),missingdata(106)andthosematchingoutsidetheschool(3771),giving7141validobservationsforin-schoolsearchers.Ofthosecurrentlymatchedoutside,roughly48%werematchedwithapartnerwhoseageorgrademadethemhighlyunlikelytobeabletoattendthesameschoolatthetimeofsurvey.Apresumablylargeandunknownfractionofthesematchesbeganassameschoolmatchesinthepast.Theschoolisstilltheprimarymatchingmarket,despiteourinabilitytoobservesame-schoolstatusinthepast.12 3.1DirectMeasuresofPreferencesSomedirectinformationongenderdierencesinpreferencesforsexcanbefoundfromquestionsthatwereaskedofthein-homesample.Individualswereaskedaboutwhethertheywouldwantaromanticrelationshipoverthenextyearandwhatphysicaleventswouldoccurbetweenthepartners.Includedinthequestionswerewhethertheidealrelationshipwouldincludehavingsex.29Table2showselicitedpreferencesoversexandrelationshipsoverall,bygradeandbyrace.ComparingTable1toTable2,moreindividualspreferhavingrelationshipsthando,suggestingsignicantsearchfrictions.Whilepreferencesforrelationshipsarethesameforbothmenandwomen(over95%wantarelationshipasdened),preferencesforsexarenot.While60%ofmenwouldprefertohavesex,thefractionofwomenwhoprefertohavesexisonly36%.Preferencesforsexrisewithage.Evenwiththisrise,comparingthesexpreferencesforwomenofaparticulargradewiththesexpreferencesformenofanothergradeshowsstrongermalepreferencesforsex.NotefromTable1thathalfofcurrentrelationshipsentailsex,whichishigherthantheself-reportedpreferencesforwomenaveragedoveranygrade,evenconditionalonwantingarelationship.Thissuggeststhepossibilitythatwomenmaybesacricingwhattheywantinordertoformrelationships.Toinvestigatethisfurther,Table3showstheprobabilityofhavingsexconditionalonwhethertherespondent'sidealrelationshipincludessex.Themeansarepresentedseparatelyformenandwomen,andshowthatwomenwhowanttohavesexaresignicantlymorelikelytohavesexthanmenwhowanttohavesex.Further,womenwhodon'twanttohavesexarealsosignicantlymorelikelytohavesexthenmenwhodon'twanttohavesex.Thesecondrowshowsthatthesemale/femaledierencesholdconditionalonbeingmatched:itisnotjustthatwomenwhowantsexsortintorelationshipsatahigherrate,theyalsoseetheirpreferencesimplementedwithinmatchesmorefrequentlythansimilarmen.Incontrast,womenwhodonotwanttohavesexseetheirpreferencesimplementedwithinmatcheslessfrequentlythansimilarmen.Finallywealsoconditiononhavinghadsexinpast,whereweseethedierenceislargelydrivenbydierencesamongthesexuallyinexperienced:inthisgroupwomenare12pointsmorelikelytohavesexconditionalonwantingtohave 29AddHealthresponsestoquestionsregardingeverhavingsexareverysimilartotheNLSY97(cf.Arcidiacono,KhwajaandOuyang(2012)):beginningatatwelfthgradesexparticipationrateinthelow60%range,andfallingroughly10%pergrade.14 Wong(2003b),whoarguesamarriagetaboodramaticallyin uencesthefrequencyofcross-racematchingamongblackmales.Blackmenalsomakeupalargerfractionofmatchesthantheydoapercentageofthepopulation.3.3TheGenderRatioandItsImplicationsforRelationshipTermsGivenevidencethatcertaincharacteristicsin uencewhetherone'spreferenceswillalignwithwhathappensinarelationship,thesupplyofthesecharacteristicsmayalsohaveaneectonthetermsoftherelationship.Whenmen,andinparticular,areinshortsupply,womenmayneedtosacricetheirpreferencesmoreinordertosuccessfullymatch.WeexaminehowgenderratiosvaryacrossschoolsinTable6,payingparticularattentiontothegenderratiosforwhitesandblacksbygrade.EachcellinTable6givestheratiooffemaletomalestudentsforeachgrade-racepairing.32Table6showsthatthereisasubstantialamountofvariationinthegenderratio,particularlyamongblacks.33Breakingoutthegenderratioalongdierentdimensions(race,andgrade-racegroupings)spreadstheinitiallycondenseddistribution.34ThebottompanelofTable6showstheprobabilityofhavingsexconditionalmatchingconditionalonthefractionfemalebeingabovethe75thpercentileorbelowthe25thpercentile.Thersttworowsshowthecaseswhengenderratioismeasuredusingthewholeschoolandthenusingonlythoseofthesamerace.Inbothcases,aratiooffemalestomalesisassociatedwithmoresex,thoughthedierencesarenotlarge.TheevidenceinSection3.2showedthatthemostcommonmatchesarebetweenthoseofthesameraceandgradesowenextconsiderthepercent-femaleofthesamegrade-raceofthepartner.Forawomanmatchingwitha12thgradewhitemale,thisvariableistheratiooffemalestomalesamong12thgradewhites.Giventhehighlikelihoodofindividualsmatchingintheirowngrade-racepair,thisvariableservesasacrudemeasureoftheoutsideoptionsthepartnerfaces.ThenalrowofTable6showsthatifthefractionfemaleinthepartner'srace-gradecellishighertheprobabilityofsexintherelationshipismorelikely:whenwomenfacemorecompetitionforpartners,moresexresults.Tofurtherinvestigatetheroleofcompetitionindeterminingrelationshipterms, 32Aminimumof5observationsfromtheraceorgrade-racepairisrequiredforaschooltoenterTable633ThisdispersionisevenmorepronouncedforHispanicandother-racestudents.34Thepopulationshavebeenscaleddownbyoneminustheestimatedconditionalprobabilityofmatchingoutsidetheschoolforeachage-race-gender-schoolgroup.16 4.1UtilityRatherthanhavingseparate's(utilities)foreverytypeofrelationship,weputsomestructureontheutilityfunction.Denotethegradeassociatedwithanm-typemanasGm2f1;2;3;4g.Whenamansearchesforanw-typewoman,thegradeofthepartnerisPGw.Similarly,Rm2f1;2;3;4ggivestheraceofanm-typemanwiththecorrespondingraceofthepotentialw-typepartnergivenbyPGw.Wespecifytheutilityofanon-sexualrelationshipasafunctionofthepartner'sgradeandraceaswellaswhetherthepartnerisinthesamegradeasthesearchingindividual,SGmw=I(Gm=PGw)whereIistheindicatorfunction,andthesamerace,SRmw=I(Rm=PRm).Denotingsearchingintheno-sexmarketbyr=1,weformulatethedeterministicpartofutilityformenandwomenmatchingintheno-sexmarketas:mw1=1SGmw+2PGw+3SRmw+4Xj=1I(PRw=j)4j(8)wm1=1SGmw+5PGm+3SRmw+4Xj=1I(PRm=j)6j(9)wheretheinterceptofanon-sexualrelationshipisnormalizedtozero.Toeconomizeonparameters,thisspecicationsetstheextrautilityassociatedwithbeinginthesamegradeorbeingofthesameracetobethesameformenandwomen.Theeectofpartnergradeandrace,however,isallowedtovarybygender.Thespecicationissetsuchthatcertainrace/gendercombinationsmaybemoredesirablethanotherrace/gendercombinations.Theutilityofsexualrelationshipstakestheutilityofnon-sexualrelationshipsandaddsaninterceptaswellasallowingwhethertheindividualhashadinsexinthepast,PSiw,toaectthecurrentutilityofsex.Notethatwearenotspecifyingthatpartnershavepreferencesforindividualswhohavehadsexinthepastbutratherthosewhohavehadsexinthepasthavepreferencesforsexnow.Hence,thetypesmandwdonotincludepastsex,anditisthereforenottargeted.Wealsoincludeagradeprolewhichcapturesthetransitionprocessintosexualactivity(e.g.social,biologicalandotherfactorschangingasadolescentsage).Denotingsearchinginthesexmarketbyr=2,wespecifythedeterministicpartofutilityformenandwomen18 theithwomanoftypewisthengivenby:Liw()=I(yiw=1)"XmXrI(diw=fm;rg)(ln[mrw(;N;PSiw)]+ln[Pmrw(;N)])#+I(yiw=0)ln"XmXrmrw(;N;PSiw)(1�Pmrw(;N))#(12)Notethattheprobabilityofmatchingisnotaectedbypastsexexceptthroughthesearchprobabilities.Thetermsaretheequilibriumprobabilitiesofsearchingineachmarketforeachtypeindividual.Thelikelihooddescribedsofarwasforagenericschool.Denotetheschoolsinthedatabys2f1;:::;Sg.Summingtheloglikelihoodsoverallthepossiblemtypesandwtypesateachschoolsimpliesthattheparameterscanbeestimatedusing:^=argmax XsXwNswXi=1Lsiw()!whereaxedpointinthesearchprobabilitiesformenandwomenissolvedateachiteration.5ResultsTheestimatesofthestructuralmodelarepresentedinTable8.37Keytodisentanglingmaleandfemalepreferencesgivenobservedmatchesistheeectofthedierentgenderratiosonthesearchdecisions.Thesegenderratiosmanifestthemselvesthroughtheireectontheprobabilityofmatching.Theparametersofthematchingfunction,andA,areidentiedthroughvariationinmatchesacrossschoolswithdierentgenderratiosandtheoverallmatchraterespectively.38Theestimatesofaresignicantand 37Wealsoestimatedmodelswithanestedlogitstructureonraceandsexseparately,andaproduct-dierentiationlogit(e.g.anestedlogitwithoverlappingnests),nestingparameterswerenotsignif-icantlydierentfromone,andwecouldnotrejectthenullthatthelogitmodeltthedatabest.Becauseweestimatethevarianceofthelogiterrors,theinterpretationofthecorrelationparametersisnon-standard.38Ina2-marketmodelwithonlymaleandfemalepreferencesforsex,andA,the4parametersarenotidentiedfromonlyoneschool.Thatisbecausewithinaschoolweobserve3moments:the(1)overallgenderratio,(2)theratioofmenwhohavesexrelativetomenwhodonothavesex(whichequalsthatsameratioforwomenwithoutsamplingerror),and(3)thenumberofmenunmatched(onlythenumberofunmatchedmenorunmatchedwomenisindependentsinceweare20 sex.WethencomparethesemodelestimatedpreferencestothestatedpreferencesdiscussedinTable2.Asthestatedpreferenceswereneverusedintheestimation,agoodagreementoftheestimatedandstatedpreferencescanprovidecompellingevidenceforthecredibilityoftheunderlyingparameterestimates.Table9showsthatthemodeldoesagoodjobofmatchingtheelicitedpreferencesforsex,whichwereportintwoways,conditionalonwantingamatchanduncon-ditionally(asreportedinTable9).Theelicitedpreferencesshow34-36%ofwomenprefersexto31.6%ofwomenpredictedbythemodel,whileformentheelicitedpref-erenceforsexisbetween58%and60%comparedtoamodelpredictionof61.8%.Themodelpredictionsandself-reportsbothshowahigherpreferenceforsexamongblacksrelativetwowhites.Thepredictedmale-raceproleisextremelyclosethesubjectivereports,whilethemodelunder-predictsfemalepreferencesrelativetothesubjectives.Thegradeprolesarealsosimilar.Beginningaround50%,malepreferencesforsexincreasewithgradereachingabove70%by12thgrade.Forfemalesweseegrowthfromaround20%toroughly50%between9thand12thgradeinbothstatedandpredictedpreferences.5.2EquilibriumMatchProbabilitiesTable10presentstheestimatedequilibriummatchprobabilitiesforwhites.39Thetableispartitionedinto16cells,oneforeachpossible(femalegrade,malegrade)match.Thecolumnswithineachcellreporttheprobabilitiesformen(Pwm)andwomen(Pmw).Thus,theupperleftcellshowsthatninthgradewomenseesignicantlyhigherprobabilitiesofmatchingthantheirmaleclassmates.Therstcolumnofnumbersgivestheprobabilitiesthataninthgrademalematchesinhiseightpossiblemarkets(sexorno-sexmarketcrossedwithwomeninfourgrades).ThefourcolumnsheadedbyPmwintherstrowofcellsgivethecomparableeightprobabilitiesforninthgradewomen.Thegrade-matchingpatternsaredrivenbytheequilibriumcombinationofutilityparametersforpartnerandowngradeformenandwomenandprobabilityofmatching.Asexpected,ninthgrademales{inboththesexandnosexmarkets{seethelowestmatchprobabilitiesofanygroup.Attheextreme,allfemalescanalwaysmatchwithaninth-grademaleinthesexmarket,butpreferencesforpartnergradeleadthemtosearchelsewhere.Femalesseeahigherprobabilityof 39Theequilibriumincludesindividualsfromall32groups;wepresentonlythesubsetofprobabil-itiesforwhitesinTable1022 Whentheprobabilityofmatchingintheoutsidemarketisnotone,theIIAprop-ertystillholds,butwenolongerhaveagoodmeasureofthenumberofsearchersintheoutsidemarket.Weallowfortheprobabilityofmatchingintheoutsidemarkettobelessthanonebyassumingthatmatchratesarethesameforallthosewhosearchoutsidetheschool.Namely,weknowthecharacteristicsofthosewhomatchedintheoutsidemarket.If10%offreshmengirlsand16%ofseniorgirlsmatchintheoutsidemarket,foracandidatematchrateof80%,theimpliedfreshmengirlsearchratewould12.5%intheoutsidemarketwiththecorrespondingsearchratesforseniorgirlsbeing20%.Wethenscaledownthenumberofnon-matchedindividualswithparticularcharacteristicsgiventheimpliedoutside-marketsearchratestodeterminethenumberofwomensearchingwithintheschool.Forunmatchedindividual'sinthesamplewerandomlyassignstatusasoutside-searchers,basedontheirtype-school-specicprobabilityofsearchingoutside,whichcombinesdataonmatchingoutsidewithanassumptiononthematchrate.Wethendroptheseoutside-searchers.Moreformally,denoteN0masthenumberofm-typemenwhomatchedintheoutsidemarket.Nmwasdenedasthenumberofsearchingmenoftypemintheschoolandwasformedasthenumberofm-typemenminusN0m.WiththeprobabilityofmatchingintheoutsidemarketgivenbyP0m,insteadofformingNmbysubtractingoN0mforthepopulation,wesubtractoN0m=P0m.Notethattheutilitiesofsearchingintheoutsidemarketareallowedtovaryconditionalonschoolandcharacteristics.Theonlyrestrictionisthat,conditionalonsearchingintheoutsidemarket,thematchrateisthesame.Weneedthisinordertocharacterizethenumberofthosesearchinginsidetheschool.40Resultsfortheestimatedmodelunderdierentmatchrateassumptionsarepre-sentedinTable11,withtherstcolumnrepeatingtheresultsoftheoriginalmodel.ThetoprowsofTable11showthemeanchoiceprobabilityforchoosingsexwithoutequilibriumin uences.Forreasonablematchrates,wecontinuetoseethesamepat-ternsofwomenpreferringsexrelativetomen.However,asthematchratefalls,movesclosertozero,whichistheCobb-Douglascase.41 40Thedetailsofhowwescaledownthenumberofindividualssearchingwithintheschoolareasfollows.GivenanestimateofthefractionofindividualsofagiventypeateachschoolF0m;F0w,wein atethisfractionbyoneovermatchrateoutsidetheschool.ThenumberofmalesearchersofagiventypeisNm=Nm(1�F0m(1=P0m)).ThroughoutwedonotobserveN0m,butestimateitasN0m=^F0mNm.41Weestimatedmodelswithlowerout-matchingfractions,butbecomesverysmallandinsignif-icanterasingtheidentifyingpowerofthegenderratiosforuncoveringgenderdierencesinutility.24 out:(i)ascomparedtowhites,amongblacksthefractionfemaleismuchhigherandthisfractionrisesfasterwithgradelevel;(ii)thedistributionofpastsexforblackmalesisfarhigherthanforwhitemalesand(iii)thedistributionofpastsexforblackfemalesisalsohigherthanforwhitefemales.Thesedierencesallpushthecompetitiveequilibriumtowardoneinwhichblackwomenhavemoresexthantheirwhitecounterparts.Oursimulatedcounterfactualsmovethematchingmarketsfacedbyblacksclosertothosefacedbywhitesinthreecumulativecounterfactualsteps:(i)rstwechangethegrade-specicgenderratiosamongblackstomatchthoseofwhites,keepingthedistributionofotherindividualcharacteristicsxed,whichisdonebyremovingblackwomenandaddingblackmenwhileholdingthetotalnumberofblacksconstant;wethen(ii)changethedistributionofpastsexamongblackmentomatchthatofwhitemen;andnallywealso(iii)changethedistributionofpastsexamongblackwomentomatchthatofwhitewomen.43Examiningtheresultsateachstagetellsustowhatdegreeeachchannelisresponsibleforthehigherratesofsexualparticipationamongblacks.InTable12wepresenttheaggregateschoolsimulatedmatchingprobabilitiesforblackmen(Pwm)andblackwomen(Pmw)(thesimulationincludesallindividualsofalltypes,butwepresentonlytheprobabilitiesforthesub-setofblack-blackmarkets).Withineachcellthersttworowscontainthebaselineprobabilitiesformatchinginthesexandnosexmarkets,whilesrowsthreeandfourreportonstep(i)(changingthegrade-specicgenderratios),andthelasttworowsreportonstep(iii)(changingalltheconditionsofblackstomatchthoseofwhites).Weomitresultsfrom(ii)becausetheyarequitecloseto(iii).Asseeninthersttworowsintheupperleftcell,inthebaselineaggregateschoolaninthgradeblackfemalehasadramaticallyhigherprobabilityofmatchinginthesexmarket(1.00)thanhermaleclassmate(0.11);andhermaleclassmatecantriplehisprobabilityofmatching(to0.33)bysearchinginthenosexmarket.Thenexttworows(CFOnlyGR)reportonsimulation(i)whereblacksfacethesamegrade-specicgenderratiosaswhites.Thischangeincreases(decreases)theprobabilitiesofmatchinginallmarketsforfemales(males),44butnotuniformly.Consistentwith 43Firstchangingthefemalepast-sexdistribution,andthenthemalepast-sexdistributiongener-atedthesamepatternofresults:themalepast-sexdistributionamongblacksisthemajordriverofthehighersexualparticipation.44Thetableshowsonlyincreasesforblacks,otherraceswillqualitativelyseethesamepattern.26 similarblackandwhitewomencanbeattributedtothedierencesinthematchingmarketstheyface.7ConclusionThecontributionofthispaperistwo-fold.First,weshowhowadirectedsearchmodelcandisentanglemaleandfemalepreferencesfordierentrelationshiptermsusingvariationinthegenderratio.Whentheresearcher'sgoalistounderstanddierencesinmaleandfemalepreferences,directedsearchprovidesanattractivealternativetotransferableutilitymodels:transferableutilitymodelsarediculttouseheresincewerarelyobservetransfers.Second,wehaveappliedthedirectedsearchmodeltotheteenmatchingmarketanduncoveredmaleandfemaledierencesinpreferencesforsex.Thepreferencesfromthestructuralmodelmatchtheself-reportedpreferences,providingacompellingout-of-sampletestforthevalidityfortheapproach.Thatmenandwomenvaluesexdierentlysuggeststhatchangesinsexualbehaviorsmayhavedierentwelfareeectsformenthanforwomen.Further,whengenderratiostiltsuchthatmenbecomeaminority|as,forexample,onmanycollegecampuses|womenaremorelikelytoengageinsexconditionalonformingarelationship,sacricingtheirpreferredrelationshiptermsforahigherprobabilityofmatching.Forhighschoolstudentsourcounterfactualsimulationsshowthat,conditionalonmatching,mostofthegapinsexualengagementbetweenblackandwhitewomenisdrivenbytheunfavorablemarketconditionsthatblackwomenface.Ifconditionsfacedbyblacks(asmeasuredbythegenderratioandsexualexperienceofmales)weresimilartothoseforwhites,theracialgapinsexualparticipationwouldshrinkaremarkable50to75percent.Moregenerally,becausechangesinthesuppliesofvarioustypesofmenandwomenalterstheequilibriummatchdistribution,suchchangeshavethepotentialtodeeplyeectthewellbeingofindividuals{changingwhoentersunionsofmanytypes,whodoesnotenteratall,thecharacteristicsofpartnersforthosewhodomatch,theintra-uniondistributionofthesurplusfromunionsandultimatelythehealthandwellbeingofmosteveryoneincurrentandfuturegenerations.Theseimportantlinksarere ectedinagrowingliteratureontheeectsofimbalancesandchangesinsexratioswithinbothdevelopedanddevelopingcountries.Manyofthesestudiesfocusontheeectsofchangingsexratiosonwhodoesanddoesnotenterunionsand28 Chiappori,Pierre-Andre,\CollectiveLaborSupplyandWelfare,"JournalofPo-liticalEconomy,June1992,100(3),437{67. ,SoniaOrece,andClimentQuintana-Domeque,\FatterAttraction:An-thropometricandSocioeconomicCharacteristicsintheMarriageMarket,"IZADiscussionPapers4594,InstitutefortheStudyofLabor(IZA)November2009.Choo,EugeneandAloysiusSiow,\WhoMarriesWhomandWhy,"JournalofPoliticalEconomy,February2006,114(1),175{201.Dagsvik,JohnK,\AggregationinMatchingMarkets,"InternationalEconomicReview,February2000,41(1),27{57.DelBoca,DanielaandChristopherFlinn,\Endogenoushouseholdinteraction,"JournalofEconometrics,2012,166(1),49{65. and ,\HouseholdBehaviorandtheMarriageMarket,"TechnicalReport2012.Dupuy,ArnaudandAlfredGalichon,\PersonalityTraitsandtheMarriageMar-ket,"IZADiscussionPapers6943,InstitutefortheStudyofLabor(IZA)October2012.Fisman,Raymond,SheenaS.Iyengar,EmirKamenica,andItamarSi-monson,\GenderDierencesinMateSelection:EvidencefromaSpeedDatingExperiment,"TheQuarterlyJournalofEconomics,May2006,121(2),673{697.Fox,Jeremy,\IdenticationinMatchingGames,"QuantitativeEconomics,2010,1(2),203{254.Ge,Suqin,\WomensCollegeDecisions:HowMuchDoesMarriageMatter?,"Jour-nalofLaborEconomics,2011,29(4),pp.773{818.Girma,S.andDavidPaton,\TheImpactofEmergencyBirthControlonTeenPregnancyandSTIs,"JournalofHealthEconomics,2011,Forthcoming.Hitsch,GuenterJ.,A.Hortacsu,andDanAriely,\MatchingandSortinginOnlineDating,"AmericanEconomicReview,March2010,100(1),130{163.30 Rose,Elaina,\EducationandHypergamyinMarriageMarkets,"WorkingPaper353330,UniversityofWashington2004.Sabia,JosephJ.andDanielI.Rees,\Theeectofadolescentvirginitystatusonpsychologicalwell-being,"JournalofHealthEconomics,2008,27(5),1368{1381. and ,\TheEectofSexualAbstinenceonFemales'EducationalAttainment,"Demography,2009,46(4),695715.Salani,BernardandAlfredGalichon,\Cupid'sInvisibleHand:SocialSurplusandIdenticationinMatchingModels,"TechnicalReport2012.Siow,Aloysius,\TestingBecker'sTheoryofPositiveAssortativeMatching,"Uni-versityofToronto,WorkingPaper2009.Willis,RobertJ.,\ATheoryofOut-of-WedlockChildbearing,"JournalofPoliticalEconomy,December1999,107(S6),S33{29.Wilson,WilliamJ.,TheTrulyDisadvantaged,UniversityofChicagoPress,1987.Wong,LindaY.,\StructuralEstimationofMarriageModels,"JournalofLaborEconomics,July2003,21(3),699{728. ,\Whysoonly5.5%ofBlackMenMarryWhiteWomen?,"InternationalEconomicReview,2003,44(3),803{826.AppendixAProofofProposition1Theproofofclaim(a)followsfrommanipulatingthede-nitionofthesearchprobability.Assumingmr0w�mrw�wr0m�wrm,wecanaddthelog-matchprobabilityforeachcombinationtobothsidesinthefollowingway:mr0w+log(Pmr0w)�mrw�log(Pmrw)+log(Pwr0m)�log(Pwrm)�wr0m+log(Pwr0m)�wrm�log(Pwrm)+log(Pmr0w)�log(Pmrw):(15)Exponentiatingbothsidesgivesusaratioofchoiceprobabilitiesandmatchproba-bilitiesbecausethechoiceprobabilitiessharethesamedenominator:emr0w+log(Pmr0w) emrw+log(Pmrw)elog(Pwr0m)�log(Pwrm)�ewr0m+log(Pwr0m) ewrm+log(Pwrm)elog(Pmr0w)�log(Pmrw)(16)32 moreandwehave:1+mrwNw wrmNm1=1+mr0wNw wr0mNm1=(17)whichisthedenitionofPwrmPwr0m.Beginningwiththeinequalitybetweentheratioofchoiceprobabilitieswithfemalechoiceprobabilitiesinthedenominatordeliverstheresultforfemalematchprobabilities.Toevaluateclaim(c),weusetheimplicitfunctiontheoremcoupledwithCramer'srule.Weshowthecasewhenthereisonlyonetypeofmanandonetypeofwomanwithtworelationshiptypes,randr0.Foreaseofnotation,wethendenoteG=Gmr.Ourproof,however,holdsinthegeneralcaseduetotheindependenceofirrelevantalternativesassociatedwiththeTypeIextremevalueerrors.Notethatthedenitionsofsearchprobabilitiesimplythat,inequilibriumthelogoddsratiosforwomensatisfy:ln(mr0w)�ln(1�mr0w)ln(mr0w)�ln(mrw)+ln1+wr0mG mr0w�ln1+(1�wr0m)G 1�mr0w(17)Now,deneF1(mr0w;wr0m;G)andF2(mr0w;wr0m;G)basedontheidentityin18andthecorrespondingexpressionformen,respectively:F1(mr0w;wr0m;G)ln(mr0w)�ln(1�mr0w)�ln(mr0w)+ln(mrw)(18)�ln1+wr0mG mr0w+ln1+(1�wr0m)G 1�mr0wF2(mr0w;wr0m;G)ln(wr0m)�ln(1�wr0m)�ln(wr0m)+ln(wrm)(18)�ln1+mr0w wr0mG+ln1+1�mr0w (1�wr0m)Gwhichcanequivalentlybeexpressedas:F1(mr0w;wr0m;G)2ln(mr0w)�2ln(1�mr0w)�ln(mr0w)+ln(mrw)(18)�lnhmr0w+wr0mGi+lnh(1�mrw)+(1�wr0m)GiF2(mr0w;wr0m;G)2ln(wr0m)�2ln(1�wr0m)�ln(wr0m)+ln(wrm)(18)�lnwr0m+mr0w G+ln(1�wr0m)+1�mr0w GTakingthetotalderivativeoftheidentitiesimplyfromtheimplicitfunctionthe-34 AppealingtoCramer'srule,@mr0w @G=@F1 @G@F2 @wr0m�@F1 @wr0m@F2 @G @F1 @mr0w@F2 @wr0m�@F1 @wr0m@F2 @mr0w(25)@mr0w @G=@F2 @G@F1 @mr0w�@F2 @mr0w@F1 @G @F1 @mr0w@F2 @wr0m�@F1 @wr0m@F2 @mr0w(26)Inbothcases,thenumeratorsarepositive.Bothhaveonenegativeterm,@F1=@wr0mand@F2=@mr0wrespectivelybutthistermismultipliedbynegative1.Thedenominatorsarethesameacrossthetwoexpressions.Thersttermispositivebutthesecondtermisnegative.However,thersttermcanbewrittenasthenegativeofthesecondtermplusadditionalterms.Thesignofthedenominatoristhenpositive,implyingthatbothexpressionsarepositiveaswell.QED.AppendixBInthisappendixwediscussfourissueswiththedata.Thethreeissueswiththedataarei)determiningtheshareofstudentssearchingintheoutsidemarket,ii)determiningthedistributionofpriorsexformales,andiii)caseswherefemalesdonotreportcharacteristicsoftheirpartners,andiv)aggregatingthein-homesampletotheschool.Todealwiththefractionsearchingoutsidetheschoolwebeginwithastrongassumptionandsubsequentlyrelaxit.Weassumeinitiallythateachindividualcouldmatchoutsidetheschoolwithprobabilityone.Thismeansthatweonlyneedthefractionofeachindividualtypematchingoutsidetheschooltocorrecttheaggregategenderratiostore ectthenumberofmenandwomenofeachtypesearchingintheschool.Weestimate(separatelyformenandwomen)alogitonmatchingoutsidetheschoolwhichisafunctionofindividualgrade,raceandschoolxedeects.Soforinstanceformenwespecifytheprobabilityofmatchingoutsidetheschoolforanmtypemanatschoolsas:P(MatchOutjm;s)=expPgI(Gm=g) g+PrI(Rm=r) r+ s 1+expPgI(Gm=g) g+PrI(Rm=r) r+ s:(26)36 Table1:MeansbyGendera MenWomen CurrentlyMatched(sexorrelationship)0.3370.315InaRelationship0.3150.295HavingSex0.1850.159Priorsex0.3210.237CurrentSexjRaceWhite0.1890.166Black0.2400.190Hispanic0.1670.122Other0.0740.099PriorSexjRaceWhite0.2770.228Black0.5270.314Hispanic0.3650.189Other0.1910.136 N3,6873,418 aSampleincludesonlythosesearchingin-school,undertheassump-tionthatPoutmatch=1.Currentisdenedasongoingatthetimeofthein-homesurvey.Relationshipmeansholdinghandsandkissing;sexreferstosexualintercourse.38 Table3:ConditionalMeansofSexParticipationa Observed:WomenMenDierence P(SexjWantsex=1)0.3400.287-0.053***N11722127P(SexjWantsex=1,Matched)0.7380.682-0.057***N539895P(SexjWantsex=0,Matched)0.2720.201-0.071**N344537WithNoPriorSex P(SexjWantsex=1,Matched)0.5790.458-0.121***N280389P(SexjWantsex=0,Matched)0.1750.122-0.053*N279441WithPriorSex P(SexjWantsex=1,Matched)0.9110.854-0.057N259506P(SexjWantsex=0,Matched)0.7190.538-0.180**N6596 a*,**,***denotesignicanceatthe5,1,and0.01%levelsrespectively.Matchedisdenedhashavingeitherarelationshiporsexin-school.Sampleincludesonlyin-schoolmatches.40 Table6:VariationinGenderRatioa Percentile RatioofFemalestoMales:.25.50.75 Total0.7940.8981.002 White0.7720.8850.994 9th0.7800.9171.07710th0.7690.9141.09511th0.6230.7910.92312th0.6130.8130.957 Black0.7290.9301.096 9th0.4810.8911.17810th0.3300.8731.23211th0.3520.7971.14512th0.1020.7011.013 FractionFemale OverallFractionFemale25th&#x]TJ/;༗ ;.9;Ւ ;&#xTf 1;.00; 0 ;&#xTd [;75th P(SexjMatch)0.5130.530Same-RaceFractionFemale P(SexjMatch)0.4910.552FractionFemaleofPartner'sRace-Grade P(SexjMatch)0.4960.564 aBasedonasampleof73schools.Genderratiosarecalculatedusingonlythosesearchingwithintheschool.Aggregategenderratioreferstotheratioofsearchingfemalestosearchingmales.42 Table8:StructuralModelEstimatesa MatchingParametersEstimatesStandardErrors -0.312(0.090)A0.415(0.002)�10.168(0.003)SexUtility MaleSex(7)-3.788(1.978)FemaleSex(9)-17.662(1.187)Past-SexSex(8)15.276(0.973)MaleGradeSex(12)1.506(0.475)FemaleGradeSex(13)3.738(0.481)MatchUtility Samegrade(1)4.957(0.395)PartnerGradeBoy(2)-0.668(0.338)PartnerGradeGirl(5)4.750(0.243)SameRace(3)10.298(0.506)PartnerBlackBoy(4b)-0.824(1.426)PartnerBlackGirl(6b)4.602(0.960)PartnerHispBoy(4h)-7.276(1.464)PartnerHispGirl(6h)-3.770(1.404)PartnerOtherBoy(4o)-10.768(1.773)PartnerOtherGirl(6o)-7.837(0.881) �log(`)4541.72 aEstimatesarefromsampleofin-schoolsearchers3449feamlesin1083two-sidedmatches.Standarderrorsareinparentheses.44 Table10:EquilibriumProbabilitiesofMatching:Whitesa MaleGrade FemaleGrade9101112 andRelationship:PwmPmw PwmPmw PwmPmw PwmPmw 9 sex0.071.00 0.130.95 0.220.72 0.330.51nosex0.230.69 0.380.45 0.630.26 0.950.14 10 sex0.111.00 0.180.81 0.280.59 0.420.41nosex0.200.75 0.340.50 0.580.29 0.880.15 11 sex0.121.00 0.190.79 0.290.57 0.440.39nosex0.150.90 0.260.62 0.460.37 0.730.21 12 sex0.111.00 0.180.81 0.280.59 0.420.41nosex0.101.00 0.190.77 0.350.48 0.590.28 aEachcellgivestheprobabilityofmatchinginsexornosexmarketsbasedonanindividuals'gradeandpossiblepartnergrade.Pwmistheprobabilityofmatchingforamanlookingforawoman,Pmwistheprobabilityofmatchingforawomanlookingforaman.Table11:VaryingFirst-StageAssumptionsa %NeverMatchedRemoved 02550 PoutmatchPoutmatchPoutmatch Meansex10.6610.6610.66 Male,NoEquilibrium0.6180.6280.5680.5790.4850.498Female,NoEquilibrium0.3160.3020.3620.3440.4250.395MeanStatedPreference Male0.5830.5870.6080.6000.6300.621Female0.3460.3520.3700.3680.3870.384 aNumberofobservationsisdierentforeachmodel.Removingnevermatchedinvolvesshrinkingtheestimationsamplebyrandomremovalonnever-matchedindi-viduals,andshrinkingtheaggregatenumberofsearchingmenandwomenwiththeprobabilityofnever-matchingestimatedwithalogitatthetype-schoollevel,sepa-ratelyformenandwomen.Decreasingtheprobabilityofmatchingoutsidetheschoolalsoshrinkstheestimationsampletheandaggregatenumberofsearchingmenandwomeninasimilarfashion.46 Table13:RacialGapintheProbabilityofSexConditionalMatchingUnderVariousMarketConditions SameTypeP(SexjMatch)a AggregateSchool9101112 White0.2100.3930.5370.648Black0.3120.4970.6440.741Dierence-0.102-0.105-0.106-0.093Changing:OnlyGenderRatios White0.2090.3930.5370.648Black0.2980.4910.6380.738Dierence-0.088-0.099-0.101-0.090GenderRatiosandMalePastSex White0.2090.3930.5380.648Black0.2740.4310.5730.672Dierence-0.065-0.038-0.035-0.024GenderRatiosandAllPastSex White0.2090.3930.5380.648Black0.2390.4010.5440.652Dierence-0.030-0.008-0.006-0.004 aGapismeasuredasP(sexjmatch,whitewithwhite)-P(sexjmatch,blackwithblack).Counter-factualpolicysimulationchangestheblackgenderratiostomatchthoseofwhitesinthreestages,changingthegrade-specicgenderratio,thepast-sexdistributionforblackmales,andthepast-sexdistributionforbothblackfemalesandblackmales.48