beenstudiedextensivelyintheliteraturewithregardstheirtoprimaryobjectivetheclasssizeeectSomeofthemorewellcitedpapersincludeKrueger1999Hanushek1999andFinnandAchilles1990Wearguethattheeffec ID: 469666
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1IntroductionThequestionoftheexistenceandthequantitativeimportanceofpeereectsinuencingindividualbehaviorhaslongeludedcredibleempiricalstudy.Theessentialproblemisthatwhethertheresearcherisinterestedinhowindividualbehaviorisaectedbygroupcharacteristics(termedexogenousorcontextualects)orgroupbehavior(termedendogenouseects),dataarerarelyavail-ableinwhichtherelevantgroupsortheirassociatedtraitsareexogenouslyassigned.Whilethiscriticismappliestoanyempiricalstudywhenweexaminehowindividualtraitsareassociatedwithindividualoutcomes,theproblemisparticularlyvexinginthestudyofpeereects.Theconceptualproblemsarenumerous,andwellelucidatedintheliterature(seeespeciallythewritingsofManski(1993,1995,2000)intheeconomicsliterature,andHauser(1970)inthesociologyliterature)andindicatethenumerouspitfallswherebyaresearchermayerroneouslyinferthepresenceofpeereects,wheninfacttheestimatesmayonlybeindicativeoftherespondentandherassociatedgroupsharingacommonenvironment.Astheconceptualidearelatedtothestudyofpeereectsplacesthesameindividualinavarietyofalternativegroupsettings(basedeitheron(exogenous)inputsoroutcomes,dependingonwhatisofinteresttotheresearcher),theidealdatarequiredbytheempiricalresearcherneedstosamplealargenumberofnearlyidenticalindividualsplacedinamultiplicityofalternativegroupsettings.Theproblemishowtomimicthisconceptualidealwithobservationaldata,wherebyalternativegroupsettingsalmostsurelycarrywiththemdierencesbasedonunobservedcharacteristicsaswell.Hereagain,theproblemoftheunobservablesconfoundinginferenceisclearlynotuniquetothestudyofpeereects.Butasoneofthecanonicalmodesofdetectingandquantifyingtheimportanceofpeereectsplacessomemeasureofgroupoutcomesasoneofthekeyexplanatoryfactorsinaregressionforindividualbehavior,thepresenceoftheseunobservablesbecomesparticularlyacute.Inparticular,evenifwecanarguethattheothercovariatesinsucharegressionareplausiblyexogenous,totheextentthattheunobservablesaresharedbysomeoralloftheothergroupoutcomes,thenthesummarymeasureofthegroupoutcomesthatservesasthepeereectmeasurewillappearspuriouslyimportantforthatreason.Thus,thecriteriathatmustbeimposedontheunobservablesinorderfortheresearchertoclaimthattheestimatedpeereectsrepresentsomethingofbehavioralsignicance(asopposedtosimplyrepresentingaquantiedversionofthestatementthattheyallshareacommonenvironment)arefarmorestringentthanforasimpleregressionwhichisusedtounderstandindividualattributesandindividualoutcomes.WetakeupthischallengeinthispaperbyutilizingdataonanexperimentconductedinTennesseeintheearly1980sdesignedostensiblytostudytheef-fectsofclasssizeonstudentachievementingradesKindergartenthroughthirdgrade.ThesedataarecommonlycalledtheProjectSTARdata,andtheyhave beenstudiedextensivelyintheliteraturewithregardstheirtoprimaryobjec-tive,theclasssizeeect.Someofthemorewell-citedpapersincludeKrueger(1999),Hanushek(1999),andFinnandAchilles(1990).Wearguethattheef-fectsfoundbytheseauthorsrepresentareduced-formimpactofclasssize,butthattheydonottrytobreaktheseeectsdownintotheirconstituentcom-ponents.Inparticular,wetaketheviewthatHeckman(1992)hasoeredonsocialexperimentsgenerally,inthattheyconstituteablackboxofunderlyingcomponents.Heckmanhaspointedoutthatitisessentialtounderstandthesemorestructuralcomponentsofsocialexperimentssoastoproperlyextrapo-latetheknowledgegainedfromthemtolarge-scalepolicyimplementation.Inourworkhere,wefocusonthecrucialaspectofProjectSTARinthatitwasconductedoverseveralgrades.Astheexperimentprogressedovertime,fromKindergartentothirdgrade,itispossiblethattheexperimentaleectscapturelessapureclasssizeeectandpotentiallymoreafeedbackeect(orsocialmultiplier),operatingthroughtheexperimentallyinducedpeerqualitydiencesacrossclasses.ItisimportanttonotethatwedonotdisagreewiththeauthorswhohavewrittenontheProjectSTARresultsasregardsthereducedformresultstheyndandreport,butwedooeranalternativeinterpretationoftheseresultsinsuchawaythatallowforquitedierentpolicyproposals(i.e.notbasedentirelyonchangingclasssizes)whichmayoerthesameslateofacademicoutcomes.AtthecoreofourreinterpretationoftheProjectSTARresultsisthemainpurposeofthispaper,whichistoestimatepeereectsusingdatawhereinsomefractionofthevariationinreferencegroupcharacteristicsisexogenouslydetermined.Weareinterestedinthispaperinendogenouspeereects(astermedbyManski)wherebyindividualoutcomesarealteredbysomeaspectofthedistributionofthereferencegroupoutcomes.Suchpeergroupeectshavethefeaturethattheygenerateafeedbackeect,sothattheintensitytowhichsocialprogramsoperatewithinandbetweengroupsaectsthetotalaggregateoutcome.Positivefeedback,forexample,wouldimplythatsocialprogramswhicharehighlyconcentratedongroupsofindividualswillbemoreecientthanprogramswhicharesprinkledacrossthelandscape.WhiletheProjectSTARdesigninprinciplekeptstudentsassignedtoSmallclassesinSmallclassesforthedurationoftheexperiment(andthesameforthestudentsinRegularsizedclasses),theexitandsubsequentreplacementofstudentsfromandintotheProjectSTARschoolsmeantthatthepopulationofstudentsparticipatingintheexperimenthaddierentialexposurestotheSmallandRegularclasssizetreatments.Thisfactisthekeytoouridenticationstrategyfortheestimationofthepeergroupseects.Thesimultaneousdeterminationofanindividualstudentoutcomeandhercorrespondingclassgroupoutcomes,aswellastheircommonexposuretoaclasssizeofagiventype(SmallorRegular),bothnecessitatethatweneedameansbywhichwecanusetheexperimentaldesigntodeliveraninstrumentalvari-able(s)bywhichsomefractionofthevarianceingroupoutcomesisexogenously determined.Werestudentsexogenouslyassignedtonotjustclasstypesschools,andweretestscoresavailableforthenewlyenteringstudentsbeforetheyenrolledintheProjectSTARschools,wecouldsimplyutilizeordinaryleastsquares,usingameasuresuchasthesamplemeanofthelaggedtestscoresofastudentscurrentclassmatesasthepeergroupmeasure.Whilethisap-proachisnotpossibleowingtothelackoftestscoresforthenewentrants,thisideadoesemphasizethevalueofthelongitudinalnatureoftheexperiment.Inparticular,asuitableversionofpreviousexposuretotheSmallclasstreatmentisagoodcandidateforaninstrument.Attheindividuallevel,thispriorexposuretothetreatmentisacomponentoflaggedtestscoresthatwecanobserve,andsousingthefractionoftheclasspreviouslyexposedtotheSmallclasstreatmentisasuitablecandidateinstrumentforthestudentscurrentpeergroupaveragetestscores.Thefactthattheinstrumentislaggediswhatallowsustoavoidthesimultaneousdeterminationoftheindividualstudentsoutcome,aswellastheoutcomesofherpeers.Thisideautilizestheexperimentaldesigntoextractthevariationinstudentperformanceduetotheimpactoftheexperimentinanearliergrade,becauseoftheboostinperformanceowingtotheSmallclasstreatmentversusboththeRegularclasstreatmentandtheentiregroupofnewlyenteringstudentswhohadnopriorexposuretotheexperiment.Thisiswheretheexit,andsubsequentreplenishment,ofstudentsoutofandintotheProjectSTARschoolsiscrucialforourpurposes.Intheextremecasewherenoexitandentrytakesplace,thenourinstrumentforpeergroupqualitywouldbeperfectlycollinearwiththeclasstypeindicator,andwewouldbeunabletoinferwhatisapeergroupeectfromwhatisaclasstypeeFortunately,theentryandexitpatternsofstudentsacrossclassesaswellasacrossschoolswasquitediverse,andsowehaverathergoodpowerinexplain-inggroupoutcomes,evenconditionalonaclasstypeindicatorincludedasaregressor.Weinterpretthecoecientontheclasstyperegressorasapureclasstype(orsize)eect,netofthefeedbackeectsduetoalterationsinpeergroupqualityfromtheimpactoftheexperimentintheearliergrades.Notsur-prisingly,owingtothelagnatureofourstrategytosplitthesetwoeectsapartgiventheoverallreducedformeect,wehavenopowertotelltheseapartforKindergarten,andextremelylittlepowertodosoasoftherstgrade.However,forthesecondandthirdgrades,wehaverelativelygoodpower,andwendthataftercontrollingfortheexperimentallydeterminedpeergroupeect,thepureclasssizeeectisrenderedmuchsmallerthanthereducedformeectsfoundintheearlierstudiesonProjectSTAR,andinmanycases,thesepureclasssizeeectsareinsignicantlydierentfromzero.Thebulkofthereducedformectsasofthesecondandthirdgradesappearstobeduetothefeedbackof Infact,thisisalsoaversionofthereectionproblem(aslabeledbyManski(1993))wherebyitisunclearwhatfractionofstudentsperformingwellinaSmallclassisduetotheclasssizeeectasopposedtothepeergroupeect.AbsententryandexitofstudentsfromtheProjectSTARschools,wewouldbeunabletoapportionwhatfractionofaclasstypeectisduetoapureresourceeect,andwhatfractionisduetoapeereect. thepeergroupeWealsocommentonthemethodsusedtoestimatetheimportanceofpeergroupeectscommonlyusedintheliterature,andlinkthesetomethodsusedtostudyphenomenawhichmaybequitedistinctfromthestudyofpeergroupects.Fundamentally,peergroupeectsarespillovereectswherebygroupoutputexceedsindividualeectssummedtothegrouplevel.Thedegreetowhichtheper-persongroupoutputexceedstheindividualoutputisthepeerect.Weshowthatthisispreciselywhatisestimatedbythecanonicalap-proachintheliteraturewhichestimatesvariantsofregressionsofindividualoutcomesontypicallytheaverageoftheoutcomesoftheothermembersofthepeergroup.Wealsodiscussthespecicationproblemswhichleadtomean-inglesscoecientsof1inextremecircumstances,butpossiblylessthan1(butwithnomoremeaning)inmoretypicalsettings,therebyobscuringthespuriousregressionproblemsplaguingtheresearchexercise.Wethenconsideravarietyofalternativemeansbywhichpeergroupeectsmaybeestimatedfromthedata,aswellasspecicationchecksthatcanbeperformed.ThenextsectionofthepaperdiscussestheProjectSTARexperimentalde-signandtheaspectsofthedatawhicharecrucialforourresearchquestion.WethenprovideabriefconceptualdiscussioninSectionthreeoftheidenticationissuesinvolvedinextractingthepeergroupeectsfromtheProjectSTARdata.InSectionfourwediscussourcoreempiricalresults.Sectionvethenconsidersthemoreconceptualissuesinvolvedintheestimationofpeereectsgenerally,andSectionsixconcludes.2TheProjectSTARExperimentalDesignandDataProjectSTARwasfundedbytheTennesseeStateLegislatureandconductedbytheTennesseeDepartmentofEducationwiththegoalofobtainingconclusiveresultsregardingtheecacyofclasssizereductions.Theambiguityoftheexistingempiricalliterature,whichusedobservationaldata,compelledtheLeg-islaturetoappropriatefundinginordertodesign,implement,andinterpretanexperimentalstudybeforeinvestinginacross-the-boardslashingofclasssizes.The79schoolsthatparticipatedintherstyearofthestudy,the1985-86schoolyear,wereselectedtoprovidevariationinbothgeographiclocationacrossthestateandinthesizeandeconomicstatusoftheschoollocations(schoolsweredesignatedasinnercity,suburban,urban,orrural).Importantly,theexperi-mentalrandomizationtookplacewithinschools,sothatparticipatingschoolswererequiredtobelargeenoughtohaveatleastoneclassofeachtypeun-derstudy.Attheoutsetoftheexperiment,kindergartenstudentsandtheir FormorecomprehensivedescriptionsoftheexperimentseeFolger(1989),Wordetal.(1990),FinnandAchilles(1990),andKrueger(1999). teacherswererandomlyassignedtooneofthreeclasstypes:Smallclasses(13-17students),Regularclasses(22-25students),orRegular/aideclasses(22-25students)whichincludedafull-timeteachersaide.Theexperimentaldesigncalledforstudentstoremaininthesameclasstypethroughtheendofthirdgrade,atwhichtimeallchildrenreturnedtoRegularsizeclasses.Studentsen-teringSTARschoolsafterkindergartenwereaddedtotheexperiment.Alltold,therewerebetween6,000and7,000studentsintheexperimentineachyear,andtheexperimentinvolvedatotalof11,600childrenoverallfouryears.Thevalidityofanyexperimentalstudymaybecompromisediftherandomassignmentisnotcredible.Assuch,theschoolsparticipatingintheSTARexperimentwereauditedtoenforcecompliancewiththerandomassignmentprocedures.AcriticalpieceofouridenticationofpeergroupeectslieswiththenewstudentswhoenteredtheparticipatingschoolsduringthecourseoftheSTARexperiment.Fortunately,theprotocolwasforallenteringchildrentoberandomlyassignedtoaclasstype.Allavailableindicationsarethattheinitialrandomassignmenttoclassesofstudents,boththoseattendingkinder-gartenaswellasthoseenteringinlatergrades,andteacherswasdonesoundly.SincetheSTARdataonlycontainsinformationontheactualclasstypeastu-dentattendedinagivenyear,andnotthetypeofclasstowhichthestudentwasrandomlyassigned,Krueger(1999)exploresthepossibilitythatstudentsswitchedclasstypesimmediatelyaftertheirrandomassignment.Inhissub-sampleof1581studentsin18schools,hendsthatfor99.7%ofstudents,theclasstypeattendedinkindergartenwastheclasstypetowhichthestudentswererandomlyassigned.Thisindicatesthattheinitialrandomassignmentofstudentswastakenveryseriouslybytheparticipatingschools.Notealsothatiftherandomizationweredonecorrectly,wewouldexpecttheaveragecharacteristicsofstudentsacrossthetreatmentandcontrolgroupstolookidenticalpriortothestartoftheexperiment.Unfortunately,studentswerenotgivenabaselinetestbeforeattendingclass,soitsnotpossibletocomparetestscoresacrossclasstypetoaddresscrediblerandomization.Butwecanofcoursecomparetheobservablecharacteristicsofstudents(aswellasteachers)andseeifonaveragetheylooksimilarinSmall,Regular,andRegular/aideclasses.KruegerandWhitmore(2001)performedthisexerciseforbothstudentsandteachers.Forstudents,class-typeassignmentwasmodeledasafunctionofdemographiccharacteristics(freelunch,race,andgender)andschool-by-entry-wavexedeectstoaccountforthefactthatrandomizationoccurredwithinschoolsandatthetimeinwhichastudententeredtheexperiment.Theresultsindicatethatstudentcharacteristicsarenotcorrelatedwithassignmentstatus,aswewouldexpectunderrandomassignmenttoclasstype.Ananalogousmodelwasestimatedfortheassignmentofteachers,withtherelevantdemographic Theaverageclasssizeoverthecourseoftheexperimentwas15.3fortheSmallclasses,22.8fortheRegularclasses,and23.2fortheRegular/aideclasses.Inthe1985-86schoolyear,thestatewidepupil-teacherratioinTennesseewas22.3.Freelunchisintendedasameasureofparentseconomicstatus. characteristicsbeingrace,gender,mastersdegree,andtotalexperience.Again,thesecharacteristicsarenotjointlysignicantinexplainingassignmentstatus,aresultconsistentwiththerandomplacementofteachersinclasstypes.Asiscommoninsocialexperiments,particularlythoseofanextendedlon-gitudinalnature,ProjectSTARdeviatedbothinitsadministrationandduetobehavioralresponsesoftheparticipantsinawaythatwasnotidealgiventheintentionsoftheoriginalexperimentaldesign.Ratherthanweakeningthemeritoftheexperiment,wearguethatinthiscaseparticularexogenouschangesinthecompositionofclassesallowustoaddressabroadersetofissuesthansolelytheeectivenessofclasssizereductions.Therstdeviation,andofonlylim-itedinterestinouranalysis,isattheendofkindergartenstudentsinRegularandRegular/aideclasseswerere-randomizedbetweenthesetwoclasstypes.Inapracticalsense,thedistinctionbetweenRegularandRegular/aideclassesisinconsequentialsincemanyoftheRegularclassesemployedapart-timeaide.Empirically,theresultsoftheProjectSTARexperimentindicatethatthedif-ferenceinstudentachievementbetweenRegularandRegular/aideclassesisinsignicant.Nonetheless,inouranalysisthatfollowsweoftendistinguishbe-tweenRegularandRegular/aideclasseswhenmodelingstudentoutcomes,butourprincipalinstrumentforpeerqualitygroupsRegularandRegular/aidestu-dentstogether.Aseconddeparturefromtheoriginalexperimentalprotocolisthatanumberofstudents,ontheorderof10%peryear,switchedbetweenSmallandRegularclasses.Krueger(1999)attributesthisprimarilytobehavioralproblemsandparentalcomplaints.Ifthestudentswhoswitchedclasstypessystematicallyeredfromthosewhoremainedwiththeirinitialassignments,thenacompar-isonofoutcomesofthetreatmentandcontrolgroupsmaynolongerestimateaparameterofinterest.Finally,studentmobilitysubstantiallyaectedtheexperimentaldesign.Stu-dentsattritedoutoftheexperiment,dueinparttofamilieshavingmovedtoerentschooldistrictsandstudentshavingattendedprivateschools,andstu-dentsenteredSTARschoolsafterkindergarten.SincekindergartenwasnotmandatoryinTennesseeatthetimeoftheexperiment,aparticularlylargein-uxofstudentsisseenenteringinrstgrade(2313newstudentsenteredinrstgradecomparedwith4516ofthekindergartenstudentsremainingintheexper-imentatthattime).Asubstantialnumberofnewentrantsalsoappearedlaterintheexperiment;1679studentsenteredinsecondgradeand1281studentsenteredinthirdgrade.Wearguethatitisprimarilythisinowofnewstudentsthatrendersasimplecomparisonoftreatmentandcontrolgroupsineectiveinisolatingthepureclasssizeeect.Tocrediblyestimatetheclasssizeeect,itisalsonecessarytoconsiderthedierenceinpeergroupcompositioninducedbythenewentrantsand,toalesserextent,thestudentsswitchingbetweenclasstypes.Morespecically,thenewentrantsgeneratevariationinpeerqualityviatwodistinctroutes.First,anewentrantdoesnothavetheboostinachieve-mentprovidedbyattendanceinaSmallclass,soifthestudentisrandomly assignedtoaSmallclasshelowerstheaveragequalityofstudentsinthatclass.Second,theSTARdataindicatesthatstudentswhoenteredtheexperimentafterkindergartenarelowerachieversthanthosewhoattendedSTARschoolsattheoutsetoftheexperiment.Thismayoccurbecausethelateentrantsdidnotattendkindergarten,andmayalsoreectunobservedfamilybackgroundcharacteristicsandparentstastesfortheirchildrenseducation.Thenewen-trantsarethenrandomlyassignedtoaclasstype,andwater-downthequalityofboththeSmallandRegularclasses.Table1summarizesthemeancharacteristicsofstudentsinthesamplebytheirtransitionstatusbetweengrades;studentseitherswitchclasstype,remaininthesameclasstype,orarenewentrantsintotheexperiment.Acomparisonoftheswitcherswiththestayersindicatesthatthemovementofstudentsbetweenclasstypesislikelynonrandom.Comparingtheswitcherstothosewhoremainintheirinitiallyassignedclasstype,weseethattheswitcherstendtohaveaslightlyhighertendencytobeonfreelunch.Butthecomparisonsbe-tweengenderandracerevealessentiallynosystematicdierences.Onaverage,studentswhoswitchedfromaSmallclasstoaRegularclassbetweengradeshadlowertestscorespriortoswitchingthanthosestudentsremaininginaSmallclass.TheaveragesinTable1alsoillustratethedisparitiesbetweenthegroupofnewentrantsandthestudentspreviouslyintheexperiment.Inadditiontolowertestscoreaverages,newentrantsaremorelikelytobenonwhite,male,andonfreelunchthanstudentsalreadyattendingSTARschools.Giventheprobablenonrandomselectionofstudentswhoswitchclasstype,weemphasizethatweprimarilyidentifythepeergroupeectsoofthevaria-tioninducedbythenewentrants.Table2liststhenumberofstudentsineachgradeandclasstypebythestudentsplaceoforigin:randomlyassignedtotherelevantclasstype,switchedfromtheotherclasstype,ornewentrant.Thenumberofstudentspreviouslyrandomlyassignedtotheircurrentclasstypedominatetheswitchers,consistentwiththeexperimentalprotocolforstudentstoremaininthesameclasstypethroughtheendofthirdgrade.Thenewen-trantssubstantiallyoutnumbertheswitchersinanygivenyear,lendingcredencetoouridenticationstrategy.ThisstudyusestheProjectSTARPublicAccessData,whichfollowstheinitialcohortofparticipatingstudents,plusnewentrants,throughthirdgrade.Thedatacontainsstudentlevelobservationsandincludesthewholeuniverseofstudentsintheexperimentinagivenyear,notjustasubsample.Thekeyvariablesincludedforeachobservationarestudentcharacteristics(race,gender,freelunchstatus),teachercharacteristics(race,holdmastersdegree,totalex-perience),classtype,schoolidentiers,andtestscores.ThePublicAccessDatacontainstwotestscores:theStanfordAchievementTest(SAT)inreadingandtheSATinmath,whichwereadministeredtostudentsattheendofeachschool Netofthevariationacrossschools.Becausetheschoolsthemselveswerenotselectedatrandom,allanalysesinthispaperconditiononschooleects. year.FollowingKrueger(1999),werescaledtherawtestscoresintopercentiles.Foreachgradeandtestmeasure,theRegularandRegular/aidestudentsweregroupedtogetherandgivenpercentilescoresrangingfrom0to100.Thestu-dentsinSmallclasseswerethenassignedapercentilescoreforeachtestbasedonwheretheirrawscoresfellinthedistributionofRegular-classstudents.Toobtainthepercentiletestscoremeasureusedinouranalysis,wetooktheav-erageofthepercentilemathscoreandthepercentilereadingscore.Ifoneofthesescoreswasmissing,weusedtheoneavailablescoreasthepercentiletestscore.Ouranalysisforestimatingpeergroupeectsrequiresknowingwhichstu-dentsweretaughtinthesameclass.ThePublicAccessDataonlyidenticlasstype,soif,forexample,therewasmorethanoneSmallclassinaschool,wehadtoinferwhichstudentsweregroupedtogetherandphysicallylocatedinthesameclassroom.Wedidthisbyusingtheteachercharacteristicsvariablescollectedforeachstudent.Ifstudentsinthesameschoolandclasstypehadbeentaughtby,say,awhiteteacherwithamastersdegreeand25yearsoftotalexperience,wecouldsafelyassumethatthesestudentswereclassmates.3TheIdenticationofPeerGroupEectsWiththeProjectSTARDataBeforemovingtoamoregeneraldiscussionofissuesandalternativemethodsoftheestimationofpeereects,webeginwithasimplieddiscussionofhowweusetheProjectSTARdatatoestimatestandardpeergroupeects.Thecanonicalregressionmodelthathasbeenusedintheliteraturetostudypeergroupeects(ofthetypedcoinedendogenousbyManski)isusuallyavariantof:i,j(1)whereistheoutcomeofinterestforindividualwhohasgroupaliation.Asistypicalinthisliterature,westartbyassumingthatthepeergroupliationisknownaprioribytheresearcher,andinourcase,weassumeitisthestudentsclassroom.Thekeyregressorofinterestisthesamplemeanof Krueger(1999)hasaccesstoseveraladditionaltests:theSATwordrecognitiontest,andtheTennesseeBasicSkillsFirst(BSF)testsinreadingandmath.HisprimaryanalysisusestheSATwordrecognitiontestinadditiontotheSATreadingandmathtests.OurabilitytoreplicatehisresultsindicatesthattheabsenceoftheSATwordrecognitiontestinourdataisoflittleconsequence.Inafewcases,itappearsthattwoteachersinthesameschoolandteachinginthesameclasstypedidhaveidenticalcharacteristics.Fortheirstudents,wecouldnotdeterminewhichonesweregroupedtogether,sothesestudentsweredroppedfromouranalysisintherelevantgrade.Thisresultedinourlosing77studentsinkindergartenand47studentsinthethirdgrade.Anextremelysmallminorityofworkonthistopictriestoconfrontthisissueseriously,asopposedtoreplacingourresidualignoranceofpeergroupaliationwithbluntforceas- thegroupoutcomes,netofindividualsoutcome,aquantitycommonlyreferredtoastheleave-outmeandenotedas¯i,ji,j N"1N!1Xk6iy1 )(2)Foreaseofexposition,wehaveassumedthatthegroupsizesarethesameacrossgroupsanditisdesignatedby.Indeed,intheProjectSTARdata,withinaclasstypesubgrouping,theclasssizeisideallyhomogeneous,butinfactitdoesvary.Weletdenotethenumberofgroups,andsothesamplesizeinthissimpliedsetup(ignoringthedierencesinclasssizes)is.Also,thefactthatthedataincludeeveryindividualinagivenclassimpliesthatwecanusetheleave-outmeanasthepeergroupmeasure.IntypicalobservationaldatasetssuchastheHighSchoolandBeyond,ortheNationalEducationLongitudinalStudy(NELS),onlyasmallfractionofastudentspeersinaschoolareincludedinthesurvey,andsoresearcherswouldoftenusethegroupmeaninclusiveindividual,asthatwasmorerepresentativeofthepopulation-levelmeanoutcomefortheschool.ThenatureoftheProjectSTARdataaordsustheluxurythatwedonothavetodealwithsomeoftheissuesthatarisewhenusingthegroupmeaninclusiveofindividualsoutcomewhenstudyingthedeterminantsofWhilethecanonicalapproachhastakenthemeanofreferencegroupbe-haviorastherelevantpeergroupmeasure,hereagainthisisdoneforlackofinformationastowhatfeaturesofthedistributionofpeergroupoutcomesarerelevantforindividualbehavior.Itcouldbethe90thpercentile,orthe10thpercentile,orpossiblynotjustthemean,butperhapsalsolowervarianceaidsinenhancingindividualachievementceterisparibus.Weagreetheseareunsolvedandinterestingissues,butagainignorethemforthemoment,andfocusonidenticationissueswiththesetofcanonicalassumptions.Thepointisthatevenwiththelitanyofstrongassumptionswehavealreadyimposed,theproblemofidentifyingfromtheaboveequationisstillnotnearlysolved.Theessentialproblemsaretwo-fold:(i)Theindividualswhocompriseeachpeergrouparenotgenerallyexogenously(asregardsindividualoutcomes)determinedand(ii)evenwhengroupsareexogenouslyformed(byalotteryorsomerandomizationdevice),individualandgroupoutcomesaresimultaneouslyformed,aproblemtermedthereectionproblembyManskiasananalogytoamirrorimagethoughttobecausingitscorrespondingobjecttomove,asopposedtobesimplyreectingit.Asweindicatedabove,thereectionproblem sumptionsneededtomaketheresearchventureprogress.WoittezandKapteyn(1998)usesurveyresponsesastowhoconstitutespeersastherelevantpeergroup,asopposedtosimplyassigninggenericgroupdesignationsaswehavedone.ConleyandUdry(2000)usesurveyresponsesonconversationsaboutfarmingmethodstodealwithlearningmodelsindevelop-ment.Manski(2000)pointsouttheformidableidenticationproblemswhengroupaliationisnotknownapriori impliesthatsimplyestimatingequation(1)withoutregardtothisissueimpliesnothingmorethanaquantitativestatementthattheindividualandthepeergroupshareacommonenvironment.Tomovebeyondsuchstatementsandtotrytocapturethebehavioralimpactsofapeergrouponindividualbehavior,weneedanempiricalstrategywhichwillabstractfromthetwoprominentsourcesofendogeneityjustdiscussed.Thequestionofpeergroupformationisacommonissueinempiricaleconomicsasitisjustaformofsortingorendogenousmigration.PerhapsoneofitsbestknownformsisthatofTieboutsortingwhereinthedemandforpublicgoodsacrosscommunitiesneedstorstaddresswhythosecommunitiesformedintherstplace.Thegeneralstrategyinsuchsituationsistoeithertrytondsomefractionofthevarianceingroupcompositionwhichisexogenouslydetermined,ortoexploitvariationinthepublicgooddemandwhichisnotdeterminedbythepreferencesofcommunities.Alternatively,onecouldtrytofullymodeltheprocessbywhichgroupsareformed,andtherebyusesourcesofvariationfromthatmodelwhichareunrelatedtotheoutcomeprocess.Unfortunately,thislatterapproachrequiresveryrichdataonpreferencesaswellasdetaileddataongroupmembersandpotentialgroupmembers,oritrunstheriskofbeingatautologicalexerciseinthatitfaceslittledisciplinefromthedata.ip-sideofthisconcernovertheendogeneityinthepeergroupmeasurei,jisalsoensuringthatasuitableinstrumentisalsocorrelatedwiththepeergroupmeasure,netoftheothercovariates.Thisistherankconditionnecessaryforidentication,andthekeyissuehereisthatithastoholdinthepresenceofthecovariates.Thisisnottrivial,asoneofthekeyregressorsistheindicatorforwhetherthechildisassignedtoaSmallclassinhercurrentgrade,whichwelabel.Welet=1whenthechildisassignedtotheSmallclasstreatment,andclearly,foragivenclass,thisdoesnotvaryattheindividualstudentlevel.Therefore,anypeergroupmeasure,oranycandidateinstrumentforthepeergroupmeasure,mustvarywithinclassesinordertosatisfytherankcondition.Naturally,thiswouldruleout,forexample,dierencesinpeergroupmeasuresbetweenthetreatmentandcontrolgroupingsoftheSmallandRegularclasses.Theproblemwithsuchanidenticationstrategyisthatwewouldbeunabletodistinguishbetweenwhatisapureclasssizeeectversuswhatisapeergroupectasthetwomeasuresmovecompletelyintandemwithinschools.Inordertodriveawedgebetweenthecurrentclass-sizedesignationcategoryandsomefactorwhichusestheexperimenttogenerateexogenouschangesinpeergroupcomposition,weturninsteadtothetimingoftheexperimentalimpactsandtheessenceofthefeedbackeect.AswediscussedinSection2,theexitofchildrenfromtheProjectSTARschoolsandthesubsequentrandomassignmentofchildrentoSmallandRegularclassestolltheirplaceimply Thereadershouldalsobearinmindtheexperimentdidnotutilizearandomselectionofschools,asdiscussedinSection2above.Assuch,alleconometricmethodsimplicitlycontainasetofschoolxed-eects.Forthatreason,onlyinstrumentsthatcontainsomewithin-schoolvariationarevalidcandidatestouseasinstrumentalvariables. thatachildwhoisrandomlyassignedtoaSmallclassinhercurrentgradewasnotnecessarilyintheSmallclassinthepreviousyearifshewasnewtotheProjectSTARschools.Inordertoavoidclutteringthenotationwithanadditionalsubscriptdenotingthetimingofvariables,letussticktoourcurrentnotationscheme(oflabelingthingsforthecurrentgradeonly),butdeneanewvariableforthechildrenofclasstoindicatetheirrandomassignmentstatusforthepreviousclassyear.Therefore,=1ifstudentwaspreviouslyrandomlyassignedtoaSmallclass,andduetotheexitandentryofstudents,itisnotnecessarilythecasethatinSmallclasses(i.e.=1)thatis1foreachstudent.Asausefulpieceofadditionalnotation,denethenumberofstudentsineachclasswhowerepreviouslyrandomlyassignedtoaSmallclass,andtheassociatedfractionofstudentswhowerepreviouslyrandomlyassignedtoaSmallclass Nowifallstudentsinthecurrentclasshadvalidtestscoremeasurestakenbeforetheybegantheyearinclass,thenwecouldusethisaverageasonemeasureofthepeergroupqualityandstudytheimpactofthismeasureonindividualtestscoresattheendoftheschoolyear.However,evenapartfromthefactthatweonlyhavesuchdataforincumbentparticipantsintheProjectSTARstudy,thissimplebutdirectapproachwouldhavepotentialpitfalls.First,whileitistruethatstudentswererandomlyassignedtoclasstypes,itisnotcleartheywererandomlyassignedtospecicclasseswithintheclasstypecategorieswithinschools.Second,theOLSapproachofusingthelaggedaverageoftestscoresonthestudentscurrentyearpeersassumestheotherinputstothetestscoreoutcomethatarecommontotheentiregrouparecontrolledforintheregressors.Infact,evenwiththemeasureunderstudy,classsize,thereweresmallbutdetectabledierencesinclasssizeswithinagivenclasstypecategory.Thus,evenwiththeuseofthelaggedmeasure,wemayhavetobecarefultoavoidanomittedvariablesproblemwhenlookingacrossyears.Finally,wecomebacktotherealityofthedatathatwelacktestscoresforthepreviousyearfortheNewEntrants,andsotheywouldhavetodroppedinorderforsuchananalysistobefeasible.Instead,wemakeuseofthehypothesisthattheclasssizetreatmentassignmenthadanimpactonthesubsequentyearstestscoretosolvethesethreeproblems.Inparticular,bygroupingtheNewEntrantswiththeRegularclassstudentsandcontrastingthemwiththeboostintestscoresreceivedbythechildrenplacedinSmallclassesinthepreviousyear,wecanconceptuallyextractthecom-ponentofthelaggedtestscorethatwasinducedbytheexperimentbyusingthevariationincurrentscoresexplainedbylaggedtreatmentstatus.Further- Weareignoringtherathersmallfractionofstudentswhoswitchclasstypeassignmentsinviolationoftheexperimentalprotocol.Theyarenotessentialtoouridenticationstrategy,andtheyonlyaddinessentialcomplexitytoincorporatethemintoourcurrentdiscussion.Asweshalldiscuss,itisnotessential,althoughitisextremelyhelpful,fortheclasssizetreatmentpersetohaveanimpactontestscoresonaverageinorderfortheidenticationstrategytowork. more,asregardsthepossiblefailureoftheexogenousassignmentofstudentstoindividualclasseswithinclasstypes,wecanreplacethiswiththesomewhatweakerassumptionthattheclassgroupingswerenotdeterminedbythefractionofchildrenpreviouslyrandomlyassignedtoSmallclasses.Finally,asregardsthepossibleomittedvariablescommonbothtothestudentandherpeergroup,nowweneedtoonlyworryaboutomittedvariablesthatarecorrelatedwiththefractionofchildrenineachclassthatwerepreviouslyassignedtotheSmallclasstypes.Ofcourse,aswedonothaveanyexplicitrandomizationdevicecreatingtheclasses,wecannotbepositivesometypeofexogeneityfailureispresent,butthisinstrumentalvariablesstrategyofusingthepreviousrandomassignmentindicatorsasaforcingvariableforthelatent(orunobserved)laggedtestscoresislesssusceptibletothesespecicationproblemsthanifthelaggedtestscoreswereobserved,inwhichcasemorestringentidentifyingassumptionswouldhavetobemade.Thestrategythenistousethecontemporaneousaverageofthepeergrouptestscores¯i,jasthepeergroupmeasure.Theinstrumentforthismeasure,whichtackleslitanyofendogeneityproblemsdiscussedabove,isthefractionoftheclassnetofstudentwhowerepreviouslyrandomlyassignedtoaSmallclass:i,j (3)withtheanalogousleaveoutquantityas:i,j(4)NotethatthisinstrumenthandlestheproblemthatthetestscoresfortheNewEntrantsarenotobservedpriortotheirexposuretothetreatment.Inewepickoutthecomponentofthepost-exposuretestoutcomethatisduetohavingbeenexposedtotheSmallclasstreatmentinthepreviousgradeornot,andsouseonlythatvariationinthepredictedpeergroupmeasure.Theuseofthelaggedinstrumentalsodealswiththereectionproblem,asonlythecomponentofthepeergroupmeasurethatvarieswiththelaggedtreatmentisusedinthepredictedpeergroupmeasure.Thepresenceofthecurrentgradeclasstypeindicatorintheregressorset,however,mightrenderthisnothingmorethanaconceptualdiscussion.Inorderfortheinstrumenttohavepower,itmustbethatbecorrelatedwith¯netof.BytheFrisch-WaughTheorem,thismeansthati,j,thefractionofstudentsclassmateswhowerepreviouslyinSmallclasses,musthavesucientvariationafteritslineardependenceonisfactoredout.ThisisclearlywherethedegreeofNewEntrants,andinparticular,theextenttowhichtheNewEntrantsarespreadacrossclassesiskeytogivetheinstrumentanychanceofpowerinourdata.AsweshowinFigures1and2,fortunatelyforourpurposes,theFractionofNewEntrantsdoesindeedhavesignicant variationacrossclassesforallthreegrades.Figure1isahistogramofthefractionofeachSmallclasswhowerepreviouslyrandomlyassignedtoaSmallclassaswell.Weretherenonewentrants,andnostudentsswitchingclasstype,thehistogramforeachgradewouldbeasinglebarat1.Infact,wecanseewhilethereisapronouncedtendencyforthatfractiontofallbetween0.5and1,thehistogramrevealssubstantialvariabilityinthisfractionacrossclasses.Figure2doesthesameexercisefortheRegularclasses,whereabsentthenewentrantsandswitchers,eachhistogramwouldbeasinglebarat0.Whilethevariationacrossclasseshereislessvisuallyapparent,itisalsoclearwehavesomepower.Finally,asweshallseebelowwhenwepresenttherst-stageregressionresults,thisnetvariation(netoftheSmallclassindicator)intheinstrumentalsohasdecentexplanatorypoweratthethirdgradelevel,andmoderateatthesecondgradelevel,forthepeergroupoutcomes.Theinclusionoftheclasstypeindicatoralsohelpseasetheexogeneityrequirementsforthegroupformation.Forexample,thepresenceoftheclasstypeindicatorintheregressionhastheeectofsweepingoutallobservedandunobservedfactorsthatvarypurelyattheclasstypelevel.Soifweassumethatthe(possiblyendogenous)sortingthattakesplacewithinclasstypesofstudentsandteachersintoparticularclassesisthesamefortheSmallandReg-ularclasses,thenthepresenceofthetreatmentindicatorwillbalancethebias(Heckman,1997)andnetitoutofourestimatedequation.Thepointisthatrandomizationcreatestwogroupingsofstudentsandteachersthatare,inprinciple,identicaloneithersideofthetreatmentandcontrolline.Whilethesortingwithinthetwoclustersofstudentsandteachersintoclassesmaywellbeendogenous,aslongasthatprocessisthesameforbothgroups,thepresenceofthetreatmentindicatorwillguaranteethatitwillbedierencedout.Ofcourse,ifstudentsandteachersareassignednotjusttoclasstypesonthebasisofrandomization,butalsoindividualclasseswithinclasstypes,thenthisentirediscussionismoot.ButwehavebeenunabletoverifywithcertaintythatallschoolsintheProjectSTARexperimentcreatedclassroomgroupingsviaran-domization,andsoweproceedundertheseweakerassumptions.Whiletheideaofidenticalendogenousprocessesleadingtoclassformation(underthescenariowherewedispensewiththepossibilitythatclasseswereformedviaarandom-izationscheme),weshouldmentionitisnotdiculttoconstructbehavioralmodelsinwhichtheseprocesseswouldnotbeidenticalowingpreciselytotheeringclasssizesoneithersideofthetreatmentandcontrollines.Thatisaverynuancedversionoftheendogenoussortingstory,andtospeakmoretoitempiricallywouldrequirefarricherdatathanwehaveaccesstohere.Ourinstrumentalvariablesstrategyyieldsdierencesinthepowertodetectpeereectsacrossgrades.First,itshouldbeobviousbytheverynatureofouridenticationstrategy,inthatitreliesonthelaggedSmallclassassignment ApointweowetoAndyFosterforpushingusthinkbeyondthepurelystatisticalstatementofthisidentifyingassumption. variable,thatpeereectswillnotevenbeestimableviathisstrategyforKinder-garten.GiventhatnotallchildrenattendKindergarteninTennessee,thisisperhapsnotaseriousshortcomingofourstrategy.Bydefault,weassignallofthereducedformeecttothepureclasssizeeectinexaminingtheKinder-gartenclasstypeestimate,althoughwhatwearereallysayingisthat,givenouridenticationstrategy,wecannottellifsomeportionofthiseectisreallybeingdrivenbypeergroupeectsorsomeothersource.Similarly,whilewearenotprohibitedfromempiricallyestimatingapeergroupeectfortheFirstgradewithourstrategy,aswewillsee,wereallyhavequitelowempiricalpower.Thisbringsustotheconceptualpointwewishtomakeonthissubjectinthissection.BecauseouridenticationstrategyliterallyreliesuponthefeedbackthetreatmentassignmenteectonstudentsastheProjectSTARcohortages,weexpecttoseegreaternotionalpowerforthelatergrades.Wewishtostressthatofcoursethefailuretodetectaneectdoesnotimplythereisandsoinourcontextthefailuretodetectpeereectsintheearlygradesmaysimplybesymptomaticoftheverydesignofouridenticationstrategy.Tosummarizethissection,werelymostheavilyontheaspectofProjectSTARthatitrandomlyassignsaSmallclasstreatmenttoindividualsandthenclustersthosechildrendierentlyastheexperimentprogressedacrossgrades.Thisisthekeytoouridenticationstrategyinextractingmeasurementoftheendogenouspeergroupeectsfromthesetypeofdata.WewilldiscussthespeciceconometricpropertiesofourestimationschemeandhowittsinwithamoregeneraldiscussionofpeergroupeectsinSection5below.WedonotarguethatthestudentsinProjectSTARarerandomlyplacedintoindividualclasses,butmerelyclasstypes(SmallorRegular)withineachparticipatingschool.Thetechnicalliteratureonthisaspectisunclear,andinanycase,ourstrategyisoperationalif,asweassume,studentsandteachersareonlyguaranteedtobeassignedrandomlytoclasstypesandnotpurelyclasses.Thebottomlineiswearerelyingonthesocialmultipliereectsoftheclasssizereductionstoidentifythepeereectsandnottherandomassignmentofstudentstodierentpeergroups.Theextraassumptionwemustincurlackingtherandomassignmenttoindividualclassesisthatthepotentialsortingthatdoesoccuralongthelinesofourinstrumentisthesameprocessacrossthetworandomlydeterminedtreatmentgroups.Finally,aswestatedattheoutset,wehavefornowadoptedthecanonicalapproachoftheliteratureinotherrespects,suchasadoptingtheregressionmodelthatislinearinthepeergroupmeanoutcomeaswellastheextremelycriticalassumptionthattherelevantpeergroupisthestudentsclassmatesasregardsthetestscoreoutcomes. 4TheEvidenceontheSocialMultiplierEectsoftheSmallClassSizeTreatmentinProjectSTARInthissectionweusetheProjectSTARdatatogetherwithouridenticationstrategyjustdiscussedintheprevioussectiontoestimatepeereects.Beforewemovetothatestimationframework,werstreplicatetheearlierworkdonewithProjectSTARontheclasssizeeectsasinKrueger(1999),andtheninterprettheseasreduced-form(ortotal)classsizeeectsthatwetrytopullapartintotheirunderlyingcomponentsofthepeergroupeectandtheresidualwhichwecallthepureclasssizeeect.Weconsiderboththeinstrumentalvariablesaswellasthereducedformresults,thelatterofwhichcombinethedirectclasssizeeectstogetherwiththesocialmultiplierorfeedbackeectscreatedbytheexperiment.Thereducedformallowsustobegintoperturbthecanonicalframeworktoalternativespecications.Wealsoconsidertherobustnessofourbaselineinstrumentalvariablesresultstoalternativeinstrumentationstrategies,aswellasassessthesensitivityofourresultstodeparturesoftheProjectSTARdatafromtheexperimentalprotocol(suchasclasstypeswitchers).4.1EstimatesofthePeerEectsandthePureClassSizeects:InsidetheBlackBoxofProjectSTARWebeginourempiricalanalysiswithrstpresentingthereduced-formclasssizeectsusingtheProjectSTARdata.Theresultsarebrokenoutbythefourgradesforwhichtheexperimentran,andaswediscussedabove,allregressionsincludeschoolectsastheSTARdatawerenotarandomsampleofschools.Owingtotherandomizationofstudentsandteacherswithinschoolstothethreeclasstypes-Small,Regular,andRegularwithateachersaide(weuseRegularasouromittedbasegroup)-asimpleOLSregressionestimatesthetreatmenteectsofinterestasthecoecientsontheSmallandRegular/aidedummies.TheseresultsarepresentedinTable3,andourresultsreproducetheanalogousresultspresentedbyKrueger(1999)andHanushek(1999)(withoutregardtotheirsubsequentinterpretationoftheseresults).Inshort,theRegu-lar/aideclassesdomarginallybetter,althoughthedierenceisnotstatisticallydistinguishablefromtheRegularclassbasegroup.TheSmallclassestimates,howeverareallquitesignicantatconventionallevels,andrangefromalowof4.8percentilepointstoahighof7.3percentilepointsrelativetotheRegularclassstudents.ItisnotmuchviolencetotheseresultstosummarizethemassayingthatbeinginaSmallclassappearstohaveroughlya5percentilepoint Inanexperimentalsetting,theinclusionofcovariateshelpsincounteringsmallimperfec-tionsintherandomizationalongobservabledimensions,butprimarilyservestoreducetheresidualuncertaintyandsoreducethesamplingerroroftheeectsofinterest. gainoverstudentsinRegularclasses(ofeithertype)ateachofthefourgradelevels.Whatwewishtodoisessentiallypryapartthis5percentileeectintoitsconstituentcomponentsofapureclasssizeeectandthepeergroupewhichisthefocusofourwork.Analternativestatementofourgoalistosplittheclasssizeeectintoitsdirectandindirecteects,althoughthislanguageisratherimpreciseandleavestheimplicationsforpolicycounterfactualsrathermuddled.Whereasearlierauthors,especiallyKrueger(1999),interpretedtheroughly5percentilepointgainimpliedbythecoecientontheSmallClassindicatoraspertainingtothecausaleectoftheSmallClasssizeascomparedtotheomittedcontrolgroup,RegularClasses,wewishtoremainmoreagnosticatthisstage.WeinterpretthisasthetotaleectofbeingassignedtotheSmallClasstype,butweviewthiscategorizationasabundleofcomponentswhichcomprisetheblackboxoftheclasstype,andwhichmayincludepeereectsandotherelementsofageneralschoolingproductionfunction.Attheinceptionoftheprogram(i.e.KindergartenandpossiblyFirstGrade)itseemsplausiblethatthecohortdesigntothestudywouldmorepreciselyreectapureclasssizeect.Butasthecohortages,itbecomesincreasinglydiculttoarguethatthesimplecontrastbetweentheTreatmentandControlgroupsreectsapureclasssizeeect,withoutallowingforthepossibilitythattheexperimentallyinducedchangesinthepeergroupcompositionsmightalsoplayarole.WhattheearlierliteratureasexempliedbyKrueger(1999)andHanushek(1999)focusedonwasthelackofwideningofthegapbetweenthestudentswhoremainintheSmallclassesastheexperimentprogressed,andwhythe5pointgainappearedtobeaonceandforallgain,asopposedtoanincreaseintheslopeofthetestscore-graderelationshipaswellasintheintercept.Table4presentsthesimplestpossibledeparturefromtheTreatmentandControlindicatorsusedtomeasuretheclasssizeeectsfromTable3.InTable4weincludetheadditionalcharacteristicoftheclassesgivenbytheaverage(leave-outmean)testscoreoftheclass¯i,j-ameasureweintendtocapturethepeergroupeectsasarticulatedinSection3above.Wearenottryingtoascribeanybehavioralsignicancetotheseregressions,butwewanttopresentabenchmarkbywhichtheIVestimateswepresentbelowmightbecompared.Inparticular,owingtothereectionproblemwhichwediscussedinSection2,theindividualoutcomeandthepeergroupoutcome¯i,jaresimultaneouslydeterminedandsothereversecausalitywouldhavetobeconsideredformallytogivethisabehavioralinterpretation.Theremarkablestabilityofthees- AswediscussedinSections2and3,wedonothavetestscoreoutcomesfortheNewEntrantspriortotheirentrytotheProjectSTARschools.Therefore,wecannotresorttoadhocxestothereectionproblembyutilizingalaggedversionofthepeergroupmeasure(i.e.thestudentscurrentpeerstestscoreinthepreviousgrade).However,wediduse,purelyforcomparisonsake,thelaggedmeanpeergroupeectlaggedonegradeforthosestudentswhowereintheProjectSTARschoolsinthepreviousgrade.Thisexercisehas timatedcoecientsacrossgradesonthepeereectmeasurecertainlypresagetheanalyticalresultsweconsiderinthenextsectionandintheAppendixthatderivethesamplepropertiesofthetypeofpeergroupestimatorsconsideredinTable4.Acrossthethreecolumns,weseethattheestimatedcoecientsonthepeergroupmeasuresarevirtuallyidenticalat0.58withstandarderrorsof0.04.ThecoecientsontheSmallclassindicatorsexhibitalittlemoreheterogeneityacrossgrades,andtheyhavefallentoroughlyhalftheiroriginalmagnitudesfromthetotalprogrameectestimatesgiveninTable3.ThepointestimatessuggestasmalldeclineintheSmallclasseectsacrossthethreegrades(asinTable3),althoughthedeclineisnotstatisticallysignicant.Allthreeesti-matesoftheSmallclasseect,however,remainstatisticallydistinctfromzeroevenafterincludingthecontemporaneouspeereectmeasureasanadditionalregressor.AtthebottomofTable4wepresentwhatwecallthenormalizedpeerectwhichplacestheestimatedcoecientonthepeergroupmeasuregiveninrstrowofeachcolumnonthesamescaleasthecoecientontheSmallclassindicator.Conceptually,itcapturesthediscreteeectofmovingfromaSmalltoaRegularsizedclassontheaveragepeergroupmeasure.Fromameasurementperspective,wecanviewthesumoftheeectsontheSmallclassindicatorandonthisnormalizedpeergroupeectasroughlysplittingtheoverall(roughly5percentilepoint)reduced-formexperimentaleectintoitsconstituentcomponentsofthedirectclasssizeeectandthefeedbackeinducedbythepeergroupeect.Aswecanseeinthelastrow,thenormalizedpeereectsreectthehomogeneityofthepeereectcoecientsandtheyvaryfromroughly4to3points.IfwesumtheSmallclasseectinthesecondrowofTable4withthenormalizedpeereect,wegettheestimatedtotalSmallclassectsof6.66forFirstgrade,5.26forSecondgrade,and4.95forThirdgrade. theeectofreplacingthereectionproblemwhichhindersthebehavioralinterpretationoftheresultsinTable4withanotherproblem,whichis,whatdoesthelaggedpeergroupmeasuremeanifitisonlyconstructedoverthosestudentswhowereintheexperimentlastyear?Interpretationproblemsaside,wendthebiggestchangeoccursintherstgrade,wheretheestimatedcoecientonthepeereectdropsfromtheestimated0.58inTable4to0.05withastandarderrorof0.07.Thesecondandthirdgradeestimatesonthepeergroupmeasuredropbyabouthalfto0.21forbothgrades.Forthemostpart,theSmallclassdummycoecientsremainqualitativelythesame,althoughthepointestimatesshowamorepronouncedmonotonicdeclineacrossgrades.ButasbothversionsofTable4suerfrommeasurementorsimultaneityproblems,weonlyusethemtoserveasabenchmarktocontrastourlaterresultsto.HereagainwewouldberemissifwedidnotpointoutthepresenceofthereectionproblemandtheproblemswithinterpretingtheresultsinTable4behaviorally.AsregardstheSmallclasseect,obviouslyonepotentialimpactisthatitenhancestheperformanceofastudentspeers.Therefore,includingitasacovariatewillobviouslydiminishthepotentialeectoftheClasssizemechanism,asitsimplysplitsthetotaleectdisplayedinTable3intoadirectandindirecteect,withthecontemporaneouspeergroupmeasurebeingapotentialoutcomeofthecontemporaneousclasstypeindicator.TheIVestimatorsconsideredbelowdonothavethismechanicalproblemofsimplysplittingtheoveralleectofpurelythecontemporaneousclasssizemeasure. IfwecomparethesetothetotalexperimentaleectsoftheSmallclasstypepresentedinTable3,thesewere7.31,5.94,and4.76.Thus,forthemostpart,theSmallclassdirecteectandthenormalizedpeereectcombinedappeartoaccountfortheaveragetotalexperimentaleectoftheSmallclassassignment.WeturnnowtoourinstrumentalvariablesstrategywhichavoidstheretionproblemandalsoaccountsfortheaspectofthesamplingdesignoftheexperimentinthatwedonothavetestscoresfortheNewEntrantspriortotheirjoiningtheProjectSTARschools.Aswediscussedintheprevioussec-tion,weuseasaninstrumentforthecontemporaneouspeergroupmeasure¯i,jthefractionofthecurrentpeergroupstudentswhowereassignedtotheSmallclasstreatmentinthepreviousgradei,j .TheinstrumentthereforetreatsstudentswhoareeitherNewEntrantstotheexperimentorpre-viouslyrandomlyassignedtooneoftheRegularclasstypesasthesameasfarasexplainingvariationintheclasstoclassvariationinaveragetestscores.Aswenotedinourconceptualdiscussionintheprevioussection,thisstrat-egylookstohavepromisesincethefractionofstudentswhowerepreviouslyrandomlyassignedtoaSmallclasshasgoodvariationacrossclassesfortheSmallclasstypegroup(owingtothesignicantquantityoftheNewEntrants).InTable5wepresenttherststageoftheprojectionof¯i,ji,j.Wedothisbygrade,andasthegradeincreases,obviouslythenumberofpotentialinstrumentsgrows,asstudentsmayhaverstbeenexposedtotheSmallclasstreatmentinanever-increasingnumberofpriorgrades.So,forexample,bythethirdgrade,therearethreesuchpossibleinstruments.BylookingattherstthreerowsofTable5,thereadercanseethatforthemostpart,theinstrumentsareindividuallygenerallynotstatisticallydistinctfromzero.TheexceptionstothisaretheKindergarteneectfortheSecondgraderegression,whichismarginallystatisticallysignicant,andtheratherlargepointestimatefortheThirdgrade,whichishighlysignicantatconventionallevels.Thejointtestonthecombinedsignicanceoftheinstrumentsforeachregressionisgiveninthe4throwfromthebottomofthetable.TherethereadercanseewehavequitelowpowerfortheFirstgrade,weaktomoderatepowerfortheSecondgrade,andquitegoodpowerfortheThirdgradeowinglargelytotheKindergartenpeermeasureeect.Thispatternofpowerforourinstrumentalvariablesframeworkweanticipatedinourpreviousconceptualdiscussionofourstrategy,asitreliesdirectlyonthefeedbacknotionofwhatapeergroupeectis,andsoitonlybe-comesdetectableasthecohortagesandthefeedbackeectspotentiallysurfacefromtheenvironment. TotheextentthattheRegularclasssizerepresentstheaverageclasssizeintheschoolsfromwhichthesestudentscame,thismaynotbesuchabadapproximation.TherandomassignmentoftheNewEntrantstotheSmallandRegularclasstypeshelpsbalancetheerencesbetweentheNewEntrantsandthepreviouslyassignedstudentsalongunobserveddimensionsoncethecontemporaneousclasstypeindicatorisconditionedon.AswenotedinSection2,however,thereisplentyofevidencetosuggestthatunconditionallytheNewEntrantsandthosestudentspreviouslyrandomlyassignedtoevenjustRegularclassesareobservationallydistinct. Noticealsothatbecauseweareinstrumentingforagroupedversionofthedependentvariable,therst-stageregressionisalsoalmostthereducedformforthetwoequationsystemattheindividuallevel.Therefore,wecanalsoexam-inetheeectoftheclasstypeindicatorsafterholdingconstantthedirectpeertreatmenteectsofinterest.Thisapproachhastheadvantageofavoidinganysortofreectiontypeproblems.However,asregardsourprincipleidentifyingassumption,itmaybesubjecttotheendogenoussortingobjectionifthesortingissystematicallydierentbetweentheSmallandRegularclasses.Butkeepingwithourassumptionthatthisbiasisbalancedacrossthetreatmentarmsoftheexperiment,thenthecoecientsontheSmallclassindicatorstellsustowhatextenttheSmallclasseectofTable3isonlyreectiveofthespillovereectsgeneratedbythepastimpactoftheexperiment.Indeed,whiletheSmallclassectfortheFirstgrade,6.39(andstatisticallydistinctfromzero),isstillclosetoitsTable3estimate,thepointestimatefortheGrade2eectishalfitsTable3value,andisstatisticallyindistinguishablefromzero.Finally,theGradethreepointestimateisactuallynegative,butisagainindistinguishablefromzero.Thus,ourconclusionswhichweshalldiscussbelowregardingtheinsignicanceoftheSmallclasseectsatGrade2and3oftheProjectSTARexperimentarenotsubjecttoacriticismthatwemayhavemishandledthetreatmentoftheendogenouspeereects.OncemeasurescapturingthepriorexposuretotheSmallclasstreatmentofanindividualspeersareincluded,thecurrenteectsofhavingbeenassignedtoaSmallclassaresubstantiallyattenuated.ThesecondstageinstrumentalvariablesresultspresentedinTable6rep-resentthecoreresultsofourpaper.Theyshowthatonceweaccountforthesimultaneousdeterminationoftheindividualandcontemporaneouspeergroupoutcomes¯i,jusingthelaggedfractionofthepeergroupexposedtothetreatmentasaninstrument,theestimatedpeereectsswampthedirectSmallclasssizeeectsingrades2and3.Therstgradepointestimateofthepeerectisroughlyone-thirdofthesecondandthirdgradeestimates,andisquiteimpreciselyestimated.Assuch,itisindistinguishablefromnoeect,althoughasweindicatedabove,andwewishtostressagain,thislackofndinganeshouldcertainlynotbeconstruedtoimplythatthereisnoeect,asthepoweroftheempiricaldesignisquiteweakhere.Indeed,thecondenceintervalontherstgradeeectmorethanencompassestheSecondandThirdgradeeandsocouldevenbeconstruedasconsistentwiththosepointestimates. Theuseofthetermalmostheremaybeunclear.Forthemostpart,thedependentvariableintherststageregressionpresentedinTable5,¯i,jvarieslittleacrossstudentswithinclasses,butmoresoacrossclasses.Belowweshallconsiderreducedformspurelyattheclassroomlevel,asthetreatmentsofinterestvaryonlyattheclasslevelratherthantheindividuallevel,andsointhissense,thestandarderrorspresentedinTable5over-countthedegreesoffreedomforthesetreatments.TheclasslevelresultsarepresentedinAppendixTable1,andshowthatourcorrectionforthewithin-classcorrelationoftheerrorsalmostcompletelycompensatesforthepossibleoverstatementofthedegreesoffreedom.Thus,inferencesdrawnfromTable5arenotdeceptiveowingtotheover-countingofthedegreesoffreedom. ThenormalizedpeereectsarepresentedinthelastrowofTable6,androughlyspeaking,theSecondandThirdgradeeectshaveapointestimateofabout4.5.TheSmallclasseectspresentedinthesecondrowarenowextremelysmallrelativetothe5percentilepointestimatesoftheoveralleectpresentedinTable3,andquiteindistinguishablefromzero.Giventheprecisionofthestandarderrorsonthesetwopointestimates,wecanclearlyrejecttheirequalitytotheearlierreduced-formeects.Thispatternisreversed,however,fortheFirstgradeestimates.TheretheSmallclasseectremainslargelyunchangedat4.91,althoughthestandarderroronthisestimateisextremelylarge,soitisalsoindistinguishablefromzero.Theestimatednormalizedpeereectislessthanhalfthegradetwoandthreeeects,atroughly2percentilepoints.Theassociated-statistic,however,islessthan0.30,reectingthelowpower,andaswealreadynoted,thepeereectfortheFirstgradeisindistinguishablefromThisverystarkpatternoftheapparentcompleteovertakingoftheSmallclasssizeeectbythepeereectasofthesecondgrademaystrikethereaderasunusual,andperhapsindicativeofsomespuriousattributeofthesettingdrivingtheseresults.Forthatreasonwenextturntoexaminingthesensitivityofthesebasicresultstoalternativespecicationsandmeasurementschemes.However,itisalsousefultopauseforamomentandpointoutoneexercisethispaperwillnotbeabletoshedmuchlighton.Namely,asameasurementdevice,wehavepositedthatindividualoutcomesvarywiththemeanoutcomesofthereferencegroup.Butwehavenotconsideredthebehavioralmodelbywhichtheseindividualoutcomes,whicharepresumablytheresultofunderlyingchoicesandinputs,cometobeinuencedbythereferencegroup.Manski(2000)amongothershasdelineatedthreebroadchannelsbywhichthepeergroupmechanismmightpropagate:1.Preferenceinteractions2.Expectationinteractionsand3.Constraintinteractions.Whilewecertainlyagreethatfortheevidenceinthispapertoleadtoprecisepolicyprescriptionswewouldneedtoestablishhowthesebehavioralmechanismsleadtothepeergroupinuencesweobserve,weemphasizethattheProjectSTARdatadonotsamplecharacteristicsthatenableustospeaktothesealternativeexplanationsempirically.Itispossibleatthisjuncturetooerstorieswhichmightrationalizethispatternofresultsacrossgradesthatrelydierentlyonsaythepreferenceversustheexpectationsrationalesbehindthepeerinuences,butweshallavoidthisexposttheorizinginthispaper,andleavethisexplorationuntiltherelevantvariablescanbesampled.4.2AssessingtheRobustnessofthePeerEectResultsTable7presentsourrstsetofrobustnesschecksofourbasicspecicationpresentedinTable6.EssentiallythistableisconcernedwiththefactthatsinceeachstudentinProjectSTARcanberepresentedasagivenexperimentallyassignedtype,thenusingonesourceofvariationisequivalenttousingone minusanothersourceofvariation.Forexample,eachstudentcurrentlyinaSmallclasswaseitherpreviouslyrandomlyassignedtoaSmallclasslastyear(PRASC),aNewEntranttotheProjectSTARschools(NE),oroneoftherathersmallfractionofclasstypeSwitchers(S).Ifweleteachofthesevariablesdenotetheirrespectivefractions,thenwehaveforeachSmallclass:PRASC(5)SothenitisidenticallytruethatforjusttheSmallclasses,usingthefractionPRASCasaninstrument,aswedidinTable6,isequivalenttousing(1-NE-S)asaninstrument.FortheRegulartypeclasses,astudentwhowasPRASCwhoisnowinaRegularclassisclearlyaSwitcher,andsowewillreplacethedesignationofswitchertoPRASCfortheRegularclasses,tokeepthenotationforaSwitcher,S,asbeingjustforthosewhoswitchfromaRegulartoaSmallclass.IntroducingthenotationofPRARCforthosestudentswhoareinaRegularclassnowwhowerepreviouslyrandomlyassignedthere,wehave:PRARCPRASC(6)SonowwehavetheidentitythatPRASC=1-NE-PRARC,andsousingPRASCasaninstrumentfortheRegularclassesisidenticaltousing1-NE-PRARCasaninstrument.NoticewehavepurposefullynotusednotationtodistinguishbetweenNewEntrantstoRegularclassesversusNewEntrantstoSmallclasses,astherandomizationshouldequatethosetwogroups.However,PRARCandPRASCarepotentiallydistinctgroupsastheyhavebeenexposedtodierenttreatmentsatanearlierpointintheexperiment.Thebasicpointofspellingouttheseidentitiesisthatusingthevariationexplainedbytheproportionsofstudentsintheclasseswhowere,forexample,previouslyrandomlyassignedtoaSmallclassisidenticallythesameasusingthemirrorimage(andthusthesamerststageprojectionandthesameIVestimate)proportionsoftheothergroupsofstudentsacrossclasses.ThispointisusefultokeepinmindininterpretingTable7.First,wecanexaminethepossibilitythatthepeereectworksdierentlyfortheSmallandRegularclasstypes.Therefore,therstrowofTable7poolstheclasstypesasinTable6andusesPRASCastheinstrument,thusreplicatingtherstrowofTable6.Thenexttworowsallowthepeereecttobepotentiallydierentacrossclasstypes.FortheThirdgrade,theestimatedpeereectcoecientsareroughlythesame,androughlyaveragetothepooledThirdgradeeectpresentedinTable6.FortheSecondgrade,theSmallclasspeereectisroughlythesameasthepooledSecondgradeeectfromTable6.However,whenlookingjustwithintheRegularclasses,theestimatedpeereectishighlyimpreciseandthepointestimateisactuallynegative.Nowhereiswheretheidentitiesjustpresentedbecomeuseful.Aswenotedabove,fortheRegularclasses,thenumberofstudentsPRASCisequaltothenumberofSwitchers(intoRegularclasstypes). Therefore,aregressionwhichusesonlythefractionofclasstypeSwitcherswillproduceanidenticalpointestimate,andbylookingatthelastrowofTable7,thereadercanseethatthe-0.56pointestimatefromthethirdrowisidenticaltothe-0.56estimatefortheSecondgradeinthelastrow.Thus,whenweallowthepeereectcoecienttodierbyclasstype,wecanseethatinthecaseoftheSecondgrade,thepointestimateisquitedierentfortheRegularclassesthanforthepooled(acrossclasstypes)estimategiveninTable6.LikewiseforthepeergroupeectfortheRegularclassesfortheFirstgradeasisshownintherstcolumnofthethirdrowofTable7.IncontrasttotheTable6pooledestimate,thepointestimatehereisroughlythesamemagnitude(andstatisticalsignicance)oftheSecondandThirdgradeestimatesfromTable6.Andofcoursehereagain,theestimateisidenticaltotheFirstgradeestimatefortheRegularclassesinthelastrowofTable7whichusesthefractionofclasstypeSwitchersastheexcludedinstrument.Ourpointindisplayingthisnumericalequalityoftheestimatedeects,aswellasthebriefconceptualdiscussionwejustprovidedonthereverseimageformofidenticationispreciselytohighlighttoaskepticalreaderthatouridenticationstrategyusesdierentgroupstoidentifyeectswhenwepoolacrossclasstypes.Thereader,forpossiblygoodreasons,maybeworriedaboutrelyingentirelyclasstypeswitcherstoidentifyapeergroupeect,asstudentswhoopttoswitchclasstypes(inthiscasethesomewhatmoreunusualchoiceofswitchingfromaSmalltoaRegularsizedclass)isendogenouslydeterminedwithrespecttotheoutcome.SuchreadersmaythereforewishtodiscardthoseaspectsofouranalysisthatincludetheseSwitchersasasourceofidentifyinginformation.Forthisreason,theymaywishtoinsteadfocusontheSmallclassestimatesgiveninthesecondrowofTable7,asopposedtothepooledclasstypeestimatesgiveninTable6.TheSmallclasspeereectestimatesfromthesecondrowofTable7arequalitativelythesameasthepooledpeereectestimatesfromTable6fortheSecondandThirdgrades.TheFirstgradepeereectestimate,whilestillquiteimpreciselyestimated,isnowquitelargeat1.72andisstatisticallydistinctfromzeroatconventionallevels.ThisdiscrepancywithourTable6resultsisinsomesensereectiveofthelowpowerpropertiesofouridenticationdesignwithregardstotheFirstgradesettingthatwehavediscussedpreviously.Acrossthemultiplicityofspecicationswehavepresentedbothinthepaper,aswellasthosenotpresented,wetendtondmuchmoresystematicanduniformpeerectestimatesfortheSecondandThirdgrade,whereastheresultsfortheFirstgradearemuchmoremixedandfarmorespecicationdependent.ThispatternisalsoseeninthespecicationswepresentinthemiddlerowsofTable7wherewenowusethepercentofNewEntrantsintheclassastheinstrumentforthepeergroupmeasure.TheideahereistousethevariationinpeergroupqualityinducedbythosestudentswhowerenotexposedtoeitherthetreatmentorcontrolgroupsofProjectSTAR.AswenotedindiscussingourprimaryidenticationstrategyunderlyingTable6,wemightexpectthatthe NewEntrantsarecomparabletothestudentsalreadyassignedtothecontrolclassesintheProjectSTARgroups,butifthereissometypeofspillover,orsimplythattheNewEntrantsrepresentadistinctgroupapartfromthepre-existingProjectSTARstudents,thenthisstrategymightbeappropriate.Ourprimaryintent,however,isnottooerastrongbehavioraljusticationforthisinstrumentalvariablesstrategy,butsimplyanalternativemeasurementstrat-egyofthepeergroupcoecient.Forthemostpart,ourconclusionsfromtheotherpartsofTables6and7stand.TheSecondandThirdgraderesultstendtobestatisticallydistinctfrom0,althoughtheestimatedeectsarediminishedincomparisontoTable6.ThisisespeciallytruewhenwebreaktheestimatedectsoutbyclasstypeandwelookattheeectsforjusttheRegularclasses-theseeectsareroughlyhalfoftheirTable6counterparts.Partofthisat-tenuationmightarisefromthemixingofthestudentspreviouslyassignedtoeitherSmallorRegularclassesunderthisidenticationstrategy.FortheFirstgrade,wedondastatisticallyandeconomicallysignicantestimatedeectforthepooledclasstypespecication,buttheeectestimatedforjusttheSmallclasstypesishighlyimprecise,andfortheRegularclasstypeitisjustbelowconventionallevelsofstatisticalsignicance.Overall,thisalternativemeasure-mentstrategydoesnotalterourprimaryconclusionsfromTable6,andthisisespeciallysoaswethinkmorecarefullyastowhatsourceofvariationthisalternativestrategypicksoutofthevariationinpeerqualityacrossclasses.Finally,wepresentwhatmightbethoughtofastheperversesourceofvariationinpeerqualityacrossclasses,andthatisusingthefractionofstudentsineachclasswhoopttoswitchawayfromtheirinitiallyrandomlyassignedclasstype.AswediscussedinSectiontwo,andwasalsodiscussedinKrueger(1999),thismaynotbesuchacontaminatedsourceofvariationasthereadermightthinkatrstblush,asmanyofthestudentswhoswitchclasstypesaredocumentedtodosofordisciplinaryreasonsandthelike.Thus,itisnotobviousthatastudentwhoswitchesfromaRegulartoaSmallclassdoessobecausesheismoreacademicallymotivated.Forthatmatter,weshouldmentionthatwhilethenumbersareonlyaboutone-thirdaslarge,wedoseesomedegreeofswitchersintheoppositedirection,fromSmalltoRegularclasses.Fortheaveragestudent,itmaybeplausibletothinkthatthesedierentgroupsofswitchers,fromRegulartoSmallandfromSmalltoRegular,impartdierentbiasesontheestimatedeects,whichiswhywepresentthembrokenoutbyclasstype(andnotpooled)inthelasttworowsofTable7.Forthereasonsjustdiscussed,therefore,itisperhapsnottoosurprisingthatthepointestimatesofthepeereectsbasedonthosewhoswitchintotheSmallclassesfromtheRegularclasses(presentedinthenexttolastrow)areslightlylargerthantheestimatesbasedontheswitchersfromtheSmalltoRegularclasses(presentedinthelastrow).However,thedierencesareextremelyslight(amaximumof0.07)andarenotstatisticallysignicantacrossgrades.Thus,ifthereaderdoespositthattheswitchersendogenouslyselectintoclasssizesbasedonpreferencesforacademicqualitywendnostatisticalevidenceofasystematicbiasinone directionbasedontheseestimates.Forthatreason,wehavenotexcludedtheswitchersfromouroverallanalysisastheseestimatesandfurtherspecicationchecksindicatethattheydonotexertasystematicinuenceonourestimatedects.WeinterpretthisasconrmingtheProjectSTARinformalsurvey-basedevidenceandKruegers(1999)evidencethatthetreatmentofswitchersinalternativespecicationsisessentiallyinconsequentialintheimpactonthenalresults.4.3WhatDothePeerEectsMean?Ourintentsofarhasbeentotakethecanonicalapproachofestimatinganequationsuchas:i,j(7)andtoconstructusefulestimatesofthepeereectmanifestedinbyutilizingtherandomassignmentfeaturesavailableintheProjectSTARdata.Inparticu-lar,ouridentifyingstrategyreliedonthenotionthatsubjectingastudenttotheSmallclasstypetreatmentinducednotjustpotentiallyaboostinthatchildstestscoreoutcome,butanindirectorspillovereectonthechildsclassmatesthroughthepeergroupeect.ThisiswhatwemeanbythefeedbackorsocialmultipliereectoftheSmallclasstypetreatment.However,ashasbeennotedfrequentlyintheliterature,thelinear-in-the-peer-group-meanspecicationjustpresentedimpliesthatgivenapopulationofstudentsofaparticularquality,areallocationofthosestudentsintoalternativegroupingswouldleadtothesameaggregateoutcomeifthisspecicationaccordstotheunderlyingmech-anismgeneratingthedata.Wenowturntothequestionofwhether,giventheProjectSTARtreatment,non-linearitiesinthepeergroupeectexistsothatreallocationsoralternativegroupingsofstudentsexposedtothetreatmentectaggregateoutput.Intermsofpolicyquestions,thiswouldspeaktothepureeciencyimplicationsofabilitytrackinginwhichclassesareformedtohomogenizealongthebasisofinitialtestscoreoutcomes.Toexaminethis,weturndirectlytotheclass-levelreducedforms(asinstru-mentingnon-linearversionsof¯i,jobscuresthebasicpoint)whereweallowtheinstrumentofthepercentageofstudentspreviouslyrandomlyassignedtoaSmallclasstoenterinaratherarbitrarilynon-linearwaybybreakingthepercentageintovedummiesasitvariesfrom0to100percent.Wehavein-cludedtheothercovariatesinthesespecicationsbygrade(including,ofcourse,classtype)buthavesuppressedreportingthosecoecientsforbrevity.Unfor-tunately,astherelevantvariationhereoccursattheclasslevelandwehave Butjusttore-emphasize,itistruethatthisimpliesthattheclasssizetreatmentsappliedtoapopulationofstudentsindividuallyproducesthesameaggregateoutputcomparedtothedesignofitbeingappliedtogroupsofstudentseach.ThelatterdesigncontainsthefeedbackorsocialmultipliereectoftheSmallclasstypeassignmentweareattemptingtomeasure.Ifthisisnotclearatpresent,wehopethatitwillbecleartothereaderbytheendofthenextsection. onlyabout330classesinthedata,wehavelittlepowertodetectthesenon-linearities.Thisiscompoundedbythefactthatforeachgrade,onlyalittlemorethan100oftheclasses(ofeithertheSmallorRegulartype)containmorethan20percentofchildrenwhowerepreviouslyrandomlyassignedtoaSmallclass.Theaveragecellsizeoutsideofthisbasegroup,therefore,isonlyabout30classes.Forthemostpart,thelinear-in-the-group-meanmodelappearstobeconsistentwiththedata.ThereisextremelyslightevidenceofalargerpointectoncethefractionofstudentswhowerepreviouslyinaSmallclasspassesa40percentthresholdforallthreegrades.AndthereisalsoslightevidenceintheThirdgradeofalargerbenetasthisthresholdismovedto60percent.Assum-ingaparticularparametricformofthenon-linearitywouldlendgreaterpowertothisexercise,butwewereunabletoquantifyaconvincingnon-linearpatternthatwefeltappropriatelysummarizedthisreducedform.Suchnon-linearitiesmayexist,butitwilllikelytakeasamplemuchlargerthantheProjectSTARdesigninthenumberofclassesdimensiontomeasurethemwithaccuracy.InTable9wetakeontheideathatitmaynotbethequalityofastudentspeersthatmattersforindividualoutcomes,butmoreofthesameness.Thatis,imagineaschoolinwhichanentirerstgradeclassispromotedintacttothesecondgrade,sothatthestudentsclassmatesremainexactlythesame.InTable8aclassinacelllike80to100percentofclassmateswerepreviouslyrandomlyassignedtoaSmallclassmighthavesimplybeenaSmallclassthatwasmovedvirtuallyintactacrossgrades.LookingatTable8wecannottelliftheestimatewascreatedbythesamenessoftheclass,orbecausetheclasswasexposedtotheProjectSTARSmallclasstreatment.InTable9weincludeanadditionalsetofdummycontrols,analogoustothoseusedinTable8,tocontrolforanarbitrarynon-linearproleofclasssameness-i.e.thefractionoftheclassthatwaspreviouslyinthesameclasstogether.Interestingly,evenwiththisadditionalsetofcontrolsforclasssameness,theconclusionsofTable8,withonlyaslightnon-linearityappearingintheThirdgradeatthe60percentthreshold,appeartoholdupquitewell.OnefeaturethatmightbeinterestingforfutureworkonthistopicisthattheclasssamenessestimatestendtobelargerthanthequalityestimatesfortheSecondgradeestimates.However,theoppositeistruefortheThirdgradeestimateswherethequalityorSmallclasstreatmentexposuremeasurestendtohaveestimatedcoecientswhicharelargerthanthesamenesscoecients.5SamplePropertiesofthePeerGroupEectsandAlternativeEstimationSchemesUntilnow,wehaveaskedthereadertobearwiththecanonicalregression-basedestimationframeworkofextractingpeergroupeectestimatesfromasampleofdata.WehavearguedthattheProjectSTARdataprovideasuperiormeans ofestimatingsucheectsbecauseitusesrandomizationtoallocateindividualstotreatmentandcontrolgroups,andtheseindividualsaresampledovertimesothattheresultingfeedback,orsocialmultiplier,eectsofthesocialprogramcanbeextractedfromthedata.Thatframeworkconsistsofthe(appropriatelyinstrumented)on¯regressionfamiliarfromstudiesintheliteraturethattrytogetatquantifyingendogenouspeergroupeects.Weturnnowtoun-wrappingthison¯regressionbyworkingoutitspropertiesinthesampleMuchworkhasbeendoneontheconceptualandpopulationaspectsofthepeerectsmodel,butverylittlehasbeendoneonspellingoutexactlywhatsampleinformationisbeingusedtoproduceanestimatedeect.Weshowthatourin-strumentedpeereectsmodelemployedintheprevioussectioninfactcapturestheveryessenceofanendogenouspeereect,thatbeingthesocialmultiplierorfeedbackeect,ofthesocialprogramusedtocreatetheinstrument.Wethenrelateourapproachtootherinnovationsintheempiricalstudyofexternalities,aswellasrecallrelateddiscussionsfromtheearlyunionwageeectliteratureonthedieringeectsestimatedbyindividual-levelandindustry-leveldatawhichpertaintospillovereTheon¯approachmakessensefromtheusualperspectiveoftryingtoquantifyarelationshipwhereanoutcomeofinterestisregressedonaninputorregresssorofinterest(generallynetofothercovariates,butthisisunimportanttotheideasconsideredhere.)Howeverappealingthoughthatmightbe,thisregressionalsocomesveryclosetorunningaregressionofonitself-thesbeingforotherindividualsinthesamplebeingtheonlyaspectsavingthisfrombeingpurelytautological.Leastsquaresestimatorshavethepropertyofplacingthettedregressionlinethroughthepointofmeansofthedependentandindependentvariablesoftheregression.Therefore,evenwhentheregressionisnotliterallyaregressionofontheforthesameindividual,acoecientof1maystillbeproducedpurelybecauseleastsquaresistheestimatingprocedure-ittellsusnothingabouttheunderlyingtrueparametervaluesgeneratingthedata.Infact,weshowintheAppendixtherelevantalgebrathatestablishesthesamplepropertiesofseveralestimationschemesinwhichtheestimatorequals1withoutconsideringanyunderlyingdatageneratingprocess.TherstoftheseistheOLScasewhenthegroupmeaninclusiveofindividualisusedastheregressor,forexamplebecausethedatasampleonlyafractionofthehypoth-esizedpeergroup(suchastheentireschoolintheHighSchoolandBeyondortheNationalEducationLongitudinalStudy).However,ofmorerelevancetoourworkistheInstrumentalVariablesestimatorwheretheinstrumentisthefullgroupmean(again,inclusiveofindividual),butthepeergroupmeasureistheleaveoutmeanasweareabletousewiththeProjectSTARdata.Thisestimatoralsoprovidesasampleestimateof1regardlessoftheunderlyingdata Altonji(1988)considersalternativeestimationschemesforgroupcharacteristicswhenthesamplecontainsonlyasmallfractionoftherelevantgroupmembers. generatingprocess.Theempiricalliteratureonpeereectshasbeenespeciallypre-occupiedwithtacklingtheendogenouspeergroupaliationproblem.Forthatreason,therecentpapersbyZimmerman(1999)andSacerdote(2001)whichusetherandomassignmentconventionsofafewcollegesindesignatingfreshmenroommateshavedrawnsomeappeal.Aswediscussbelow,however,relyingpurelyonrandomgroupassignmenttostudypeereectsleavestheresearcheranestimatorthatisstillratherfragileinitsproperties.Thepointofthispaper,however,isthataccesstoarandomizedsocialexperiment,wherebyatreatmentalterstheoutcomesofsomeoftheindividualsandthepeergroupformationisthesameprocessacrosstheexperimentalgroups,allowsforestimatorswhicharenotasfragileinextractingmeaningfulpeergroupestimatesfromthedata.Toputthismoresuccinctly,thepresenceofarandomizedsocialexperimentofvaryingintensitiesacrossgroupsallowstheresearchertodirectlyinvestigatethepresenceofspillovereects.Wepresenttherelevantderivationsbehindthisargumentnow,andthenseehowtheytie-intotheinstrumentedpeergroupregressionmethodsweutilizedintheprevioussection.Wethenconcludewithageneraldiscussionoftheestimationofspilloverorexternalityeectsfromotherliteratures.Webeginwithastripped-downversionofourestimatingequation(leavingoutcovariatesforthemoment,droppingconsiderationsoftimingoftheoutcomeandpeergroupmeasure,andassumingthegroupsizesareofhomogeneoussizeN):(8)Ingeneral,evenintheabsenceofcovariates,thisregressionwillnotproduceacoecientof1,unlikethefullmeanspecicationdiscussedintheAppendixwhennocovariateswereincluded.Re-writingtheleaveoutmeanintermsofthefullgroupmeanandtheindividualoutcome,wehave: )(9)Therefore,theOLSestimatorfortheregressionjustgivenis: N!1(N¯yj"yy PJjNi1 (10)Simplifyingthis,wehave: (N3(¯yj)2"2N2(¯yj)2+PNi=1(yij)2](11)Now,sinceissimplytheBetweenSumofSquares(BSS)intheoutcomevariableandistheTotalSumofSquares(TSS),we maywritethisexpressioninthemoreinterpretativeformusingthisnotation:1)[BSSTSS BSSBSSTSS(12)Finally,usingthenotationWSSfortheWithinSumofSquares,andmakinguseoftheequationTSSBSSWSS,wecanrewritethisas:BSSWSS !1 WSS (13)Wecanusethislastexpressiontobegintodevelopsomeintuitionfortheleastsquareson¯regressionbyunwrappinghowitutilizesvariationintheoutcomemeasurewithinandbetweengroups.First,noticethatthisOLSestimatorofthepeergroupeectgoesto1mechanically(i.e.regardlessoftheunderlyingtruevalueofthepeereect)asoneoftwothingshappen:(i)Thereferencegroupsizegoestoinnityand(ii)theWithinSumofSquares(WSS)intheoutcomemeasuregoesto0.Thistellsusimmediatelythatoursamplewillhavenopowertodetect(true)peereectsifthereisnovariationintheoutcomemeasurewithingroupsbutonlyacrossgroups.Thiswouldoccur,forexample,ifgroupswereconstructedbyabilitygroupingusedinschoolswherevariationinstudentabilityoccursmostlyacrossclassesratherthanwithinclasses.Failuretoaccountforinstitutionsandbehavioralmechanismsthatleadtotheformationofhomogeneousgroupingsalongreferencegrouplinescaneasilyleadtheresearchertospuriouslyconcludepeereectsarepresent.Similarly,theidealdatacontainalargenumberofreferencegroupssothatthereferencegroupsizeisnottoolargerelativetotheoverallsamplesize,anddoesnotgrowattoofastarateastheoverallsamplesizeincreases.Byignoringtheterminthedenominatorthatisdown-weightedbyorder,wecanderiveamoreintuitiveexpressionthatapproximatesequation(13):WSS (14)Thisexpressioniskeytoourunwrappingoftheon¯regression.Simplyput,ifreferencegroupsareliterallythesumoftheirpartsthentherearenospilloverorpeergroupeects.Consideralteringindividualsoutcomeinapeergroupofsize1(i.e.netofindividualherself).IftheresultantincreaseintheWSSisexactly(BSS,i.e.theblipinthewithin-groupvariationonlyshowsupinthebetween-groupvariationappropriatelyinatedbythenetgroupsize1,thentheestimatedpeereectwillbezero.If,however,thebetweengroupvariationincreasesbymorethanthe1contributionfrom Powerconsiderations,whichwedonotexaminehere,wouldplaceabrakeondrivingtheoptimalreferencegroupsizetooclosetozero,asdoesthetradeoinreducingtheWithinSumofSquaresasthegroupsizediminishes. individualsimpactonthewithin-groupvariation,thentheestimatedpeerectwillbegreaterthanzero.TheupperboundonthecoecientestimatedviaOLSis1,whichoccurswhenthevariationinindividualoutcomesispurelyacrossgroupsratherthanwithingroups.Equation(14)isthekeytoourfollowinganalysis.Itillustratesthebasicintuitionthatthebetweengroupvariationinoutcomescontainsthespillover(orpeer)eects,whereasthewithingroupvariationgivesacleanshotoftheindividualvariationpurgedofthegroup-levelpeereect.Thesameprinci-plethatgrouplevelversusindividualleveldataonthesamevariablecontainerentspilloverorsortingeectsisalsothebasicprincipleunderlyingtheidenticationstrategiesinBoozer(2001)andSenesky(2000),bothofwhomusecontrastswithinandbetweengroupstopurgeorextracteectswhichmanifestthemselvespurelyatthegrouplevel.Ofcourse,theideaisnotnew,astheworkofLewis(1963,1987)onunionwageeectsarticulatedthispointcarefully.InLewisscase,theearlyindustryleveldataonunionizationpercentagesandaver-agewagesofworkerscontainednotonlythedirectimpactof(individual)unionstatusonwages,butalsothepotentialunionthreatmechanismwherebyhigherunionizationpercentagesinanindustrymeanttheunioncouldextractgreaterdemandsintheformofwages.Thus,Lewisviewedtheunionthreateectasanuisanceandapossiblereasonwhytheearlyestimatesbasedonaggregatedatamightoverstatetheindividualunionwageeectbasedonmicrodata.TheLewisthreateectcorrespondstoourpeergroupeect.Inoursettingitisactuallytheobjectofinterestasopposedtoabiasthatneedstosomehowbeeliminated.Theanalyticsgivenabovelayouthowthetwoformsofestimatingtheunionwageeect-viaaggregate-levelorindividual-level(micro)data-combinetoestimatethefullsetofparameters.Aswejustdiscussed,wereweinterestedsolelyinthedirectect,wecouldutilizethepurelywithin-groupindividual-levelvariationtoestimateaneectpurgedofthespilloverorpeergroupeect.Ofcourse,tomaketheanalogytoLewismoreexact,weneedtointroducetheanalogousvariabletohisunionizationstatuswhichinourcasewouldbeclasssize.Beforecomingtothespecictreatmentofdealingwithclasssize,letusstartbyaddingcovariatestothesimpliedregressiongiveninequation(8).Inthiscase,weamendequation(8)as:i,j(15)Inthiscase,asimpleapplicationoftheFrisch-WaughTheoremallowsustoapplytheintuitiveapproximationwederivedinequation(14)tothevariationintheoutcomenetofitslineardependenceonthecovariatesdenotedas:WSS 1)(BSS(16)wheretheoverbarsdenotethesamplemeansoftherespectivevariables.Thisexpressionhighlightsthesensitivityoftheestimatedpeereectstothetype covariatesincludedintheregression.Forexample,acovariatethatvariessolelyatthegrouporclassroomlevel,suchasteachercharacteristicsorthecurrentclasstype,aectsonlythebetweenvariationinBSS.IthasnoeectontheconditionalWSSasitisorthogonal(byconstruction)totheWSS.Therefore,addingacovariatethatvariessolelyattheclassroomlevelunambiguouslydrivesdowntheestimatedpeereect,themoresoasthecovariateisrelatedtothecrossgroupvariationinoutcomes.Thisisanalternativestatementofthere-ectionprobleminthatallcharacteristicsofthecommonenvironmentsharedbyindividualandherpeersmustbecontrolledfor,ortheestimatedpeerewillbeoverstated.Addingcovariatesthatvarybothwithinandbetweengroupsorclasses,suchasstudentraceorgender,haveanambiguouseectontheestimatedpeereTheireectdependsonwhethertheyexplainrelativelymoreofthewithinorthebetweenclassvariationintestscores.Totheextentthattheylargelysoakupthewithinclassvariation,butlessofthebetweenclassvariation,thiswillleadtoalargerestimatedpeereectthatapproaches1.Acovariatethataectsthewithinandbetweenvariationproportionately(i.e.a1unitchangeinacovariateforthewithinvariationequatestoa unitchangeinthebetweenvariationforagivenindividual)willcontributezerototheestimatedpeereasnospilloverispresent.Studieswhichrelypurelyonexogenous(orrandomlyformed)groupassign-mentmechanisms,suchasZimmerman(2000)orSacerdote(2000),essentiallyfollowtheapproachjustdescribed.Theyincludeinthecovariatesanumberoffactorswhichdescribetheindividualheterogeneity,andrunaregressionoftheindividualoutcomeonalaggedversionoftheoutcomeoftheirrandomlyassignedcollegeroommate.Thediscussionwejustpresentedshowsthattheirestimatedeectreliesentirelyonhowthecovariatesaectthevariationinout-comeswithinandbetweenroommatepairs.Ifthecovariatesdolittletocontrolforthepossiblyheterogeneousenvironmentssharedbyroommatepairsinthebetweenpairvariation,buttheyparseoutindividualvariationquitewell,thensuchstudiesmaybeestimatingspuriouslylargepeergroupeects.AsweshowintheAppendix,thelaggingoftheoutcomevariable(toovercomethesimul-taneityproblem)usedasthekeyright-handsideregressorsimplymodiestheexpressiongiveninequation(16)bymultiplyingitbytheautocorrelationco-cientinthecurrentandlaggedoutcomesbeingusedintheregression.Iftherandomizationoftheroommatesisdonecorrectly,andtheappropriateco-variatesarecontrolledfor,thenourobservationsheredonotindicateaspeciproblemwithsuchstudies.However,wedowishtopointoutthefragilenatureoftheidenticationachievedbyrelyingsolelyonexogenousgroupformation,andthesensitivityofsuchestimatestotheinclusionandexclusionofpotentialcovariates.Inaddition,aswediscussinthenextsubsection,theuseofrandomassignmentforgroupformationhastheproblemthatforlargeenoughgroup,thevariabilityinpeercompositionacrossgroupsgoestozeroasin-creases.Thuswhilerandomizationhelpsensuregroupformationisexogenous, itrunstheriskinlargegroupsettingsthatthepeereectwillnotevenbeidentied.Insmallgroups,thevariabilityacrossgroupswillariseduetothenite-samplingerror.Weturnnext,therefore,totheempiricalstrategywehaveusedinthispaper.ThisdoesnotrelyonrandomizedgroupassignmentasinZimmerman(1999)andSacerdote(2001),butinsteadonthehypothesisthatconditionalonthecurrentclasstypeassignment,thetreatmentstatusintheearliergrade,,ofastudentspeersisexogenous.Theinclusionofthecurrentclasstypedummyinthelistofcovariatesallowsthatifthereendogenousselectionintoindividualclassesbasedonthesoftheclass,itmustbethesameprocessforboththeSmallandRegularclasses,sothatthebiasisthusdierencedoutacrossthetreatmentandcontrollinesbythepresenceof.Thenecessaryexclusionrestrictionisthatanotherstudents(callthem)priortreatmentstatushasnoimpactonstudentsoutcomeexceptviatheendogenouspeerectmechanism.Thus,wetaketheinstrumentfortheendogenous¯i,jtobe: (17)Andasabove,sincethereferencegroupsizeistakenasconstantacrossgroups,denethepartoftheinstrumenti,jthatvariesbyi,j(18)beingsimplythetotalnumberofstudentspreviouslyassignedtotheSmallclasstreatmentinthecurrentclass.Finally,inorderfortheinstrumenttohavepowerconditionalonthecovariates(mostimportantly,conditionalontheclasstypeindicator)weneedtoassumetheassignmentstatustotheSmallclasspreviouslyhasaneectaboveandbeyondthecurrentclasstypestatus.Simplyput,thismeansweneedtheSmallclassassignmenttohavenotpurelyjustaonceandforalleect,butalsoaneectontheslopeofthetestscoreproleacrossgrades.Infact,empiricallywecomedangerouslyclosetonothavinganypower,asKrueger(1999)reportsthatmuchoftheProjectSTAReectsareoftheonce-and-for-allvariety.However,healsopresentspointestimatesthatshowaslopeeectthatisaboutone-fththesizeofthe5percentilepointintercepteect.Thus,whilethepowerisreduceditisstillpresent,anditisworthnoting,thepowerwillalsotendtobegreatertheearlierintheexperimentthestudentwasassignedtotheSmallclasstreatment,forthisreason.Withthisinstrumentinhand,wenowconsiderthesamplepropertiesoftheInstrumentalVariablesestimatorofequation(15),wherethepeergroupmeasure¯istakentobeendogenousandinstrumentedwith¯i,j.Taking againthesimplicationthatthegroupsizeisthesameacrossgroups,theIVestimatoris: i,j JjNi i,ji,j(19)Again,the1factordividesoutofthenumeratoranddenominatorandthissimpliesto: JjNiSj"d (20)andmultiplyingoutandpassingthesumoverindividualsthroughthenumeratoranddenominatoryields: N2Sj¯yj"2NSj¯yj+PNi=1yijdij](21)Nowmakeuseofthesamenotation,BSS,WSS,andTSS(torefertotheBetween,Within,andTotalSumofSquaresrespectively)asabove,buthereappliedtocovariancesbetweentheoutcomeandtreatmentindicators,ratherthanpurevariancesintheoutcomevariablewithinandbetweengroups(justforeconomyofnotationinthisstep).Recallingournotationthatweagainhave:1)[BSSTSS BSSBSSTSS(22)whichisthesameexpression,intermsofsumsofsquares,thatwehadaboveinequation(12).UsingtheoperatorCovtorefertothesamplecovariance,itagainsimpliesdowntobeapproximately:Cov Cov(23)Whatissomewhatmorecomfortingaboutthisexpressionthantheanalogousexpressiongiveninequation(16)fortherandomized-groupspeereectsesti-mator,isthatitreliesnotjustontheunivariatevariationintheoutcome(netofthecovariates)withinandbetweengroups,butinsteadnowreliesontheco-variationintheoutcomewiththeprevioustreatmentassignmentdummywithinandbetweengroups.Then,totheextentthatthecovariationislargerBetweenclassesthanWithinclasses,thesecondtermwillbedriventoaquan-titylessthan1,andapositiveestimateofapeereectwillresult.Whereasthepurerandom-assignmentOLSestimatorin(16)reliescruciallyonboththerandomizationbeingdoneproperlyaswellas(moreimportantly)thetypeandquantityofthecovariateswhichareincluded,theIVestimatorpropertiesjustspelledoutinequation(23)indicatethattheIVestimatorislessfragileto thespecicationandtakesadvantageofarandomlyallocatedprogramattheindividuallevel.However,thespuriousdetectionofpeergroupeectsmaystillariseintheIVcase.If,contrarytoourassumptions,priortreatmentassignmentisusedasafactorinassigningstudentstoclasses,andinparticularsuchthatthereisnowithin-classvariationin,theningeneralwewillestimateaspuriouspeereectof1.Whatwerequire,therefore,isthatstudentsareassignedtoindividualclasses,conditionalontheircurrentclasstype,suchthati,janexogenousvariable.InthecontextofProjectSTAR,thisrequiresthattotheextenttheNewEntrantsareplacedintoindividualclassesinawaythatisrelatedtotheiroutcomevariabledierentlythanthosestudentswhowereinProjectSTARfromthepreviousgrade,thenthisdierentialassignmentmechanismmustbethesamefortheSmallandRegularclasses.Soeither(i)thereisnoendogenoussortingonthebasisofthetreatmentassignmentintoclasses,or(ii)totheextentthereisendogenoussorting,thebiasisbalancedacrosstheTreatmentandControlgroups.Byinspection,equation(23)hintsthattheinstrumentedon¯regressioncoecientmaybeestimatedwithoutplacingtheoutcomesofonespeersasaregressorontheright-handside.Instead,whenasocialprogramisavailable,thenanappropriatecomparisonoftheratiooftheeectsofthatsocialprogramwithinandbetweenclassescanprovideevidenceofendogenoussocialeectswithoutresortingtotheratheruncomfortableon¯device.Wetakeupthisanalysisinthenextsubsection.Thediscussionalsoshowsthetiesoftheendogenouspeereectsliteraturetoothersimilarestimatorsofspilloverorexternalityeects,thelinkagestowhichhavenotbeenentirelyclearintheexistingliterature.5.1AlternativeEstimationSchemestotheCanonicalAp-proachBasedonWithinandBetweenGroupCon-trastsWebeginthissubsectionbycomparingthetradeosbetweentherandompeerassignmentstrategiesutilizedbyZimmerman(1999)andSacerdote(2001)tothesocialprogramstrategyusedinthispaperofidentifyingpeereects.Therstthingtonoteintherandomassignmentcaseisthattheestimateofthepeergroupeectisgenerallyheavilyover-identied.Thereasonisthattotheextentthatindividualoutcomesareinuencedbyobservablecharacteristicssuchasgender,race,familybackground,etc.andthegroupcompositionsvaryalong Clearly,unconditionalonthisiscertainlynotthecase.Owingtotheexperimentaldesign,studentswhoremainedintheProjectSTARschoolsfromgradetograderemainedinthesameclasstype,apartfromthesmallnumberofswitchers.Therefore,overall,studentswhowereinaSmallProjectSTARclasslastyeararemuchmorelikelytobefoundinaSmallProjectSTARclassthisyear.Thequestionofexogeneity,therefore,isifstudentsareclusteredintoindividualclasseswithinclasstypeinamannersystematicallyrelatedto theseobservablelines,thenthepeereectcanbeestimatedothesevaryinggroupcompositions.Foreachobservedcharacteristicoftheindividualaseparatepeereectcanbeestimated,providedthevariationintheindividualcharac-teristicacrossgroupsissucient.Iftheresearchermaintainsthehypothesisthatthepeerinuencesworkthroughtheoutcomes(i.e.theendogenouseectsmodelofManski)thentheempiricalmodelwillbeheavilyoveridentied.Ofcourse,onequirkofrelyingontherandomgroupassignmenthypothesisisthatasthegroupsizetendstowardsinnity,thevariationingroupcharacteristicswilltendtozeroifindeedgroupsareformedviaarandomizationscheme.Innitegroupsizes,therewilltendtobevariationincharacteristicsacrossgroupsduetosamplingerror.Forthisreason,thecollegeroommatecontextconsideredbyZimmerman(1999)andSacerdote(2001)whereisquitesmall(generally2or3)isideal.Butoneshouldbecarefulinconsideringasymptoticpropertiesofestimatorsundertherandomgroupformationhypothesis,inthatonlythenumberofgroupsbeallowedtoapproachinnityandnotthegroupsize.Inthelattercase,themodelwouldbeasymptoticallyunidentied.WealsoconsiderinthissubsectionaweakeridentifyingassumptionthatpertainstoourProjectSTARdata.Inthatcase,theclassesthemselvesarenotnecessarilyrandomlyformed,butonlytheclasstypes.However,weargueinthispaperthatclassesareexogenouslyformedalongthelinesofthefractionofchildPreviouslyRandomlyAssignedtoaSmallClassconditionalontheclasstypeindicator(aswellastheothercovariates).Inthatcase,wecannolongerrelyonthedemographicorindividualcharacteristicstoprovideasourceofnecessarilyexogenousvariationinpeerqualities,butonlyhavetheexperimentallyinducedvariationarisingfromhavingbeenpreviouslyexposedtotheSmallclasstreatmentintheProjectSTARschools.Thuswelosetheoveridentiednatureoftheempiricalmodelwiththegainofallowingforweakeridentifyingassumptions.Wearegoingtouseequations14and16astheintuitivebasisthatamo-mentsestimatorconstructedfromtheWithinandBetweenclassestimatorsofthePreviouslyRandomlyAssignedtoaSmallclassindicator(PRASC,denotedaboveas)willreplicatetheinstrumentalvariablesestimatoroftheon¯peereectsregression.ThisisthesameideapursuedinBoozer(2000)wherebyIVestimatorsbasedongroup-levelcharacteristicscanbeseenascontrastormomentestimatorsbasedonhowthestochasticprocessesvarywithinversusbetweengroups.Inthepresentcontext,thishasadirectanalogytotheearlyworkofLewis(1963,1987)regardingwhataggregateorindustryleveldataver-susindividualleveldataonunionizationidenties.Thisalsohastheeectoflinkingouranalysistoconceptsrelatingtothepeereect,suchasPhilipsons(2000)externaltreatmenteectwhichmeasuresthespilloverwhichmayariseinmedicalvaccinationtrials,wherebygreaterdensityofvaccinationmayhavelargeraggregatebenets,evenholdingconstantthetotalnumberofvaccina-tionsadministered.Finally,inthecasewheretheanalyst,likeLewisindealingwiththeunionthreatendsthespilloverorexternalityeectanuisance parameter,theestimationschemediscussedbelowallowsforapureestimateofdirectSmallclasssizeeect,netofthepeereectfeedback.Ratherthandothetediousalgebratoshowtheestimatorweproposeisnumericallyidenticalinthesample,wechoosethesimplertaskofshowingthattheyhavethesamelimitingvalueasthesamplesizegrowsduetothenumberofgroupsgrowinglarge,holdingclasssizesxed.Werstposetheendogenouspeereectsdatageneratingprocess(dgp)as:(24)Notationally,indicatesifthechildwaspreviouslyrandomlyassignedtoasmallclass,andindicatesifthecurrentclassisSmallornot,andsoithasnowithin-classvariationforagivenclassindexedby.Sincethesampleaverageof¯totheclasslevelissimply¯,theWithinclassestimatorderivedfromapplyingOLStothefollowingregression(withthedenotingtheclassspecixede(25)canbewrittenintermsofthedgp(droppingtheerrortermsforeaseofexposi-tion)as:(26)Then,usingequation9,theterminvolvingthepeereectcanbesimpliedto: (27)AndsotheWithinclassregressionofindividualtestscoresonthepreviouslyrandomlyassignedtoasmallclasslastyeardummyaswellastheindividual-levelcovariateswillestimate,intermsofthe )+( (28)Asthegroupsizeislarge,thenthewithin-classestimateswillcomeveryclosetodeliveringacleanshotofthedirectectofhavingpreviouslybeenrandomlyassignedtoasmallclass.Asthemagnitudeofthepeereectinourcaseislessthan1,butthegroupsizeisroughly20,wecanalmostsafelyignorethiscorrectiontothewithinestimatesofdeliveringacleanshotofthedirectectofthepriorexperimentalstatuspurgedofthefeedbackspillovereHowever,whenthegroupsizeisroughly2or3,asinthecaseofZimmerman(1999)orSacerdote(2001)whostudycollegeroommates,thiscorrectionislesslikelytobenegligible.Thecorrectionarisesbecause,whenthepeergroupsaresmall,eachindividualscontributiontothepeereectisnon-negligible.Inthatcase,thewithin-classregressionwilltendtounderstatethedirecteectbecausethewithinregressionsubtractsoutpartofthedirecteectbynettingoutthegroupmeanin¯ Similarly,wecanexaminethelimitingpropertiesoftheBetweenclassre-gression,whereOLSisappliedtotheclassaverageddata:(29)Again,ignoringthetrueerrorterm,wecanre-writethisintermsoftheparam-etersofthe 1"!¯dj0j& (30)Therefore,ifwefocusontheWithinandBetweenclassestimatorsofthecoecientsonthePRASCindicator,wehavethattheWithinestimatorhasthelimit(limitsbeingtakenas,thenumberofgroups,tendstoinnity):plim (31)andsimilarly,theeectonintheBetweenestimatorhasthelimit:plim (32)Thus,toarstapproximation,fortheclasssizelarge,wecanformanestimatorforthepeereectplim (33)TheintuitionisthattheWithinestimatorestimatesthedirecteectofPRASCpurgedoftheclass-levelpeereectduetotheinclusionofthedummies.Ontheotherhand,theBetweenestimatorwillestimateanin-atedversionofthedirecteect,whichisinatedthemorethatthepeeretendstowards1.InthecasewheretheWithinandBetweenestimatesofthePRASCeectarethesame,theimpliedpeereectisthereforezero.Butinlargesamples,theBetweenclassestimateofPRASCwilltendtobelargerthantheWithinclassestimate.Inthissetup,however,nothingabouttheconstruc-tionoftheestimatorimpliestheestimatedpeereectfromanitesamplewillbeboundedontheintervalfrom0to1.Ofcourse,sincethegroupsizeisknown(andintheanalyticshere,as-sumedtobeconstantacrossgroups,unlikeintheProjectSTARdatawhereitvariesslightly,thusintroducinganotherformofapproximation)wecanpro-videtheexactminimumdistanceestimatorbasedontheWithinandBetweenestimatorsas:plim *"1 N"" (34) whichisslightlyattenuatedforlargefromtheapproximateformwegaveabove.Alsonoticethatasthefraction goesto0(i.e.theimpliedpeeregoesto1)theapproximationalsobecomesexact.Roughlyspeaking,ifwetaketheratiooftheWithintotheBetweenestimatestobe0.5,and=21,thiscorrectionshowsuponlyintheseconddecimalplace,andisthuswellwithinthesamplingerrorofourestimatesofthepeereectsintheprevioussection.Similarly,theexactestimatorforthedirecteectofisnotsimplytheWithinestimator,butinsteadaslightlylargerversion:plim N" (35)Hereagain,inthecasewherethereisnospillovereectmanifestedintheestimates,theWithinandBetweenestimateswillbethesame,andsoindeedtheWithingroupestimatewillbeanestimateofthedirecteectoftheSmallclasssizeeectpurgedofthegrouplevelfeedbackeect.Andeveninthepresenceofafeedbackeect,forlargeenoughgroupsizes,theWithingroupestimatorofthetreatmenteectprovidesacleanestimateofthedirecteectoftheprogram,netofthesocialmultipliereects.Ofcourse,identicationrequiresthatthefractionofthosetreatedvarywithingroups(andgroupsarenotsegregatedbytreatmentstatus,aswouldbetheProjectSTARdataweretheirnoNewEntrantsandperfectadherencetotheexperimentaldesignprotocol)aswellasthatfractionmustvaryacrossgroupssothereisvariationinthevariableofinterest.Nextweturntotheimportantobservationthatinstudieswherearandom-izationdeviceisusedtoassignpeergroups,theimpliedpeergroupeectwillgenerallybeoveridentied.Thereasonisthatoftentheresearcherhasavail-ableothercharacteristicsoftheindividuals,capturedintheregressors,thatareassociatedwithdierencesinstudentperformance.Assuch,eventhoughthereisnotasocialprogramalteringindividualperformanceasisthecasewiththePRASCindicator,thedieringcompositionsofpeergroupsasreby¯allowforidenticationofthepeereectcoecientbycontrastingtheWithinandBetweencoecientsonthesinequations21and23inthemannerjustdiscussedabovefortheregressor.TakeforexamplethethelementofthecoecientvectorsonthesfromtheWithinregressioninequation18andtheBetweenregressioninequation22,thenweshouldhaveforeachelementintheregressorsetthat:plim +k)ÃN"1 N"$k plim +0k)#$N"1 N"$0k (36)thusshowingtheoveridentiednatureoftherandomgroupformationcasewhentheanalysthasinformationonmorethanoneindividualcharacteristicthat variesinintensityacrossgroups.Thecaveathereisthatasgetslarge,thenifgroupsaretrulyformedrandomly,thevarianceinthecross-groupvariationinaveragegroupcharacteristicswillshrinktozero.Fornite,theregenerallywillbevariationintheaveragesthatarisesduetosamplingerror.Thus,ideallytheanalystwillhaveaccesstodatainwhichtheaveragegroupsizeintheran-domlyformedgroupscaseissmall,asotherwisetheabilitytodetectpeereectswillbeminimized.Inthisrespect,thecollegeroommatesettingofZimmerman(2000)andSacerdote(2000)isideal,asisquitesmall.InProjectSTARthiswouldbemoreofaproblemwereclassroomassignments,ratherthanclasstypeassignments,randomlydeterminedasthiswouldunderminetheidenticationofpeergroupeThepointsthatwewishtoemphasizefromthediscussioninthissectionare:(i)Thelinearpeergroupmodelthatistypicallyusedintheliteraturewhengroupsarerandomlyformedisgenerallyoveridentied,aslongasthereremainscientvariationintheexogenouscharacteristicsacrossgroups.Thiswilltendtooccurwhenthegroupsizeissmall,andthevariationingroupchar-acteristicsthusarisesbysamplingerror-clearly,thesecharacteristicsmustvarycientlyacrossgroups,andmustbecorrelatedwithindividualperformancetobeofvalue.Theoveridenticationarisesfromthenumberofrestrictionstherandomizationofgroupformationimplies.(ii)Evenintheabsenceofrandomlyformedgroups,anexogenouslyassignedsocialprogramoperatingattheindi-viduallevelwillallowforidenticationofendogenouspeereectsaslongastheintensityoftheprogramvarieswithinandbetweengroups.Iftheprogramvariedonlybetweengroups,butgroupswerestratiedbyprogramstatus,thenwecouldnotseparatelyidentifytheindividualeectfromthespilloverecreatedbytheendogenouspeereects.Similarly,asthewithingroupvariationessentiallyonlyidentiesthedirectectoftheprogram,lackofvariationinthefractionofparticipantsinthesocialprogramacrossgroupswouldeliminatetheverysourceofvariationthatiscrucialinidentifyingthefeedbackeects.Thiswouldariseinourcontextifstudentswereplacedinclasses(andnotjustclasstypes)randomlyandclasssizesweresucientlylargesoastoeliminatevari-abilityinclasscharacteristicswhichareneededtocreatedierentialexposurestothepeerqualities.(iii)ThefactthatourInstrumentalVariablesestimatorofthepeergroupectcanbederivedasapproximately1minustheratiooftheWithinclassesti-matorofthePreviouslyRandomlyAssignedtoaSmallclassindicator(PRASC,)totheBetweenclassestimator,showsthetightrelationshipofthecanon-icalpeergroupestimationschemeandotherproblemsinappliedwork.Lewis(1963,1987)notedinhisworkthetendencyoftheBetweenindustryunionwageectstobelargerthanthemicrodataunionwageeects(Withinindustryornot),andhecarefullyconsideredthepossibilityofaunionthreateectwhichisanalogoustoourpeereectspilloverwhichwasresponsibleforthewedgebe-tweenthesetwosetsofestimates.Morerecently,Philipson(2000)hasproposedaframeworktoconsidertheextrapolationofindividuallybasedclinicaltrials formedicaltreatments,whichhavevaryinglevelsofintensityinthetreatmentpopulationsacrosssites.Hepointsoutthatinthecaseofvaccinations,say,aspilloverorexternalityariseswhenlargerfractionsofchildrenarevaccinatedforanunvaccinatedchild.Heproposesrandomassignmentoftreatmentstatusin-tensitiesnotonlywithinsites,asisclassicallydoneinclinicaltrials,butbetweensitessoastoallowforassessmentofwhathecallstheexternaleects.Suchatwo-stagerandomizationdesign,heargues,allowsforextrapolationofthemicrolevelclinicaltrialstoamacrolevelsettingbyhandlingexplicitlytheimplemen-tationbiasthatarisesbecauseoftheexternaleects.Infact,bycomparinghisproposedestimatorswiththeanalyticswejustpresented-inparticular,theequivalenceoftheIV-endogenouspeergroupeectapproachwiththecon-trastestimatorbasedontheratiooftheWithinandBetweenestimators-thereadercanseethat,conceptuallyatleast,hisproposedestimationschemeisourunwrappedendogenouspeereectestimatorusingtheexogenouslyassignedsocialprogramasthedrivingforcebehindthepeergroupquality.5.2EmpiricalResultsBasedontheWithinandBetweenClassComparisonofPriorTreatmentStatusEInthissubsectionwemakeuseofthewithinandbetweenclassrelationsbetweenthepriorSmallclasstreatmentassignmentvariableandindividualtestscores.Wefocusourempiricalworkhereonillustratingequations(24)to(33)intheprevioussectionusingtheProjectSTARdata.InTable10wepresentintheupperpanelthebetweenclassestimatesofthecurrentclasstype(ect,aswellasthefractionoftheclasspreviouslyrandomlyassignedtoaSmallclass.Asthenumberofclassesisroughlyvepercentofthetotalindividual-levelsample,thestandarderrorsarequitelarge.ThegradeoneSmallclasseectisnowslightlylargerthaninTable6,forexample,at6.48,anditstatisticallysignicantwithawidecondenceinterval.Thegradetwoectisindistinguishablefromzero,andthepointestimateisroughlyhalfthereduced-form5percentilepointgradetwoeect.Thepointestimateforthegradethreeeectisactuallynegative,althoughisstatisticallyindistinguishablefromzero.Nowasequation(32)shows,theestimatesofthecoecientonacrossclasseswillbeaninatedversionofthedirecteectofonstudentperfor-manceaslongasthepeereisgreaterthanzero.Fortherstgrade,thebetweenclassestimateoftheeectofis1.53,andisindistinguishablefromzero.Forgradetwo,theeectissomewhatlargerat4.26,butisagainwellwithinsamplingerrorofzero.Forgradethree,however,weseeaquitelargeestimatedeectof13.77withanassociated-statisticofover3.Thewithinclassestimatesinthebottompanelaremorepreciselyestimatedowingtothelargerdegreesoffreedom.Asweshowedinequation(31),foragroupsizeofroughly=20,thewithinclassestimatesofthecoecientonisessentiallythedirecteectofthisvariableonstudentperformancepurgedof thefeedbackorpeereects.Thedeviatingfromclassmeansofthecovariatesalsoeliminatesthecurrentclasstypeasaregressorasitvariesonlyacrossclasses.Incontrasttotheroleplayedbyinexplainingthecross-classvariationinthetoppanel,inthebottompanel,thelargestestimatedeectofoccursfortherstgrade.Theestimatethereis3.64,whichisstatisticallydistinctfromzero,butstatisticallyindistinguishablefromthereducedformSmallclassectinTable3.Thesecondgradeestimateofthedirecteectofis1.53anditiswellwithinsamplingerrorofzero.Whilethiswithinclassestimateisnotstatisticallydistinctfromthecorrespondingbetween-classestimateof4.26inthetoppanel,itisroughlyone-thirdthesizeofthebetweenclassesuggestingaroleforaspillover(orpeer)eectattheclasslevel.Finally,thegradethreedirecteectestimateis2.33andisstatisticallydistinctfromzeroatconventionallevels.However,asthecross-groupeectintheupperpanelissolargeat13.77,thenthisisratherstrongevidenceofaspillover/feedbackeatthegrouplevel.InthelastrowofTable10wehavecomputedtheimpliedpointestimateofthepeere(inequation(24))usingequation(33).Whilewehavenotyetcomputedthedelta-methodstandarderrors,itshouldbecleartothereaderthepeereectsestimatedthiswaywillhaveamuchwidercondenceintervalthanthecorrespondingpeereectestimatescomputedviaIVinTable6.Theimpliedgradeoneeectisactuallynegative,althoughitisclearlyquiteimpreciseandsowellwithinsamplingerrorofazeroeect,consistentwiththe0.3(andstatisticallyinsignicant)estimateintherstrowofTable6.Thegradetwoestimateof0.64iswellwithinsamplingerrorofthe0.86peereestimatedviaIVinTable6.Weshouldnote,however,thatasthebetweenclassestimateofthegradetwoeectof4.26isstatisticallynon-distinctfromzero,theimpliedpeereectcomputedviaequation(33)islikelynotdistinctfromzeroeither,asitisthebetweenestimatethatcontainstheinformationonthespillovereect.Finally,weseeroughlythesameresultfortheimpliedgradethreeeectof0.83,whichisquitesimilartothecorrespondingeectfromTable6of0.92.InFigure3,wehaveplottedthethirdgradewithinandbetweenclassrela-tionsbetweennetoftheothercovariates(notablythecurrentclasstype)viatheFrisch-WaughTheorem.Wehavesuper-imposedtherelationsontopofthebetweenclassFrisch-Waughresidualsforthe322classes(thewithinclassdatapointsbeingfartoonumeroustodisplaymeaningfully).Thisplotshowsthatthereisnotjustaclusterofclassesorindividualsdrivingtheseestimatedrelationships,buttheeectisspreadthroughoutall322classes.Thettedregressionlinesshowthelargergradientforthebetweenclassrelationshipascomparedtothewithinclassrelationship,thusyieldingvisuallyapparentev-idenceofaspillovereectviaequation(33).Thetwolinescrossatthepointwhereand¯netofthecovariatesis0.Asthesearetted(Frisch-Waugh)residuals,thisistheoverallsamplemeanofbothandbytheconstructionoftheresiduals. 6ConclusionsTherehasbeenarecentspateofexcitingnewempiricalworkdocumentingtheexistenceandmagnitudeofpeereectsineducationalandsocialsettingsgener-ally.Someofthisworkhasmadeinnovativeuseofinstitutionalruleswhichpaircollegefreshmeninarandomizedfashionwithroommates,asinZimmerman(1999)andSacerdote(2001),therebyhurdlingonelargeobstacleinthisliter-ature,thatbeingtheendogenoussortingofindividualsintotheirpeergroups.Ofcourse,therandomassignmentitselfsolvesonlyoneofthemanyproblems,welldelineatedbyManski(1993),thathaveplaguedtheadvancementofthisliterature.Peeraliation,issuesofmodelspecicationsuchastimingandmea-surementissuesgenerally,muststillbepushedtothebackburnerevenwithsuchdata.Furthermore,aswedocumentinthispaper,andSacerdote(2001)notesinhiswork,randomassignmentalonedoesnotallowfordistinguishingwhatmaybeendogenouspeereects-wherebyanindividualisdirectlyaectedbyoutcomesofherpeers,leadingtoasocialmultiplierorfeedbackeect-fromexogenouseects,wherebytheindividualisaectednotbyoutcomesofherpeersperse,butthecharacteristicsofherpeers.Inthispaper,wetakethisliteraturetothenextstepbymakinguseofdatawithasocialprogramadministeredinarandomizedfashionattheindividuallevel.TheProjectSTARdataontheeectsofclasssizereductionsforearly-elementaryschoolstudentsfromTennesseeintheearly1980sisaverynaturaldatasettouseforsuchapurpose.Owingtothecohortdesignoftheexperiment,asthecohortprogressedfromKindergartentothenalgradeoftheproject,ThirdGrade,theexitandreplacementofstudentsoutofandintotheProjectSTARschoolsprovidesasampleofclassroomswithdieringpastexposuretotheSmallclasstreatment.Ifasocialmultiplier,orendogenouspeereisindeedpresent,thenclasseswithhigherintensitiesofstudentsexposedtotheSmallclasstreatmentinthepast,shouldhaveaclassroom-leveleectthatexceedstheindividual-leveleectbyamargingreaterthantheshareofstudentstreated.Inthisway,datawhichcontainarandomlyallocatedsocialprogramcanmeasureaspillovereectofasocialprogramdirectly,therebyassuringndingofanendogenouspeereect.Datawhichconsistofpurelyrandompairingsofstudents,withnosocialprogrampresent,mustrelyonmorestringentidentifyingassumptionstomakesuchaclaim.Furthermore,wealsoshowthatexperimentaldesignssuchasthatproposedbyPhilipson(2000)tostudythespilloverorexternaleectsofmedicinaltrialsareinfactthesamenotionasanendogenouspeereect,ashisconceptualideafocusesonmeasuringthefeedback Inarecentpaper,Mott(2001)corroboratesthisargumentthatmerelydoingrandomassignmentofgroupmembershipsdoesnotguaranteeidenticationofthestructuralpeerectsfromtheestimatedreducedformeectsifexogenouseectsareallowedforinthedgpFurthermore,heveriesourargumentthatarandomlyallocatedsocialprogramidentiendogenousspillovereectsviaaclassicalsimultaneousequationsframeworkforthecase.Seethediscussionsurroundinghisequation(10). ectofaclinicaltrial.Inaddition,heproposesatwo-stagerandomizationscheme,wherebyintensitiesofaclinicaltrialarerandomlyassignednotjusttoindividualswithinasite,butalsowhatfractionofeachsiteiseligibletoreceivethetreatment.Suchadesignwouldbeawelcomeadditiontosocialexperimentsmoregenerally.Thequestionofendogenouspeereectsorexogenouspeereectsishighlyimportant.Evenapartfromconsiderationsonthecostsideofasocialprogram-especiallyfactorssuchasxedsetupcostsperlocale,forexample-thepresenceofendogenouspeereectsonthebenetssideimpliesaneconomyofscale.Inthatcase,socialprogramswhichareclusteredinnaturewillhavegreaterbenethanthoseprogramswhicharesprinkledacrossthelandscape.Inthecontextofeducation,thisliteraturetswellwiththeresearchonthepureresourceeectsasitspeakstotheecientallocationofsuchresourceswithinandbetweenschools.Inthispaper,ourintroductionofthepresenceofthepeereectslurkinginthereducedformSmallclasssizeeectsofProjectSTARturnsouttohaverenderedtheclasssizeresourceepersetoamuchsmallermagnitudebygrades2and3.Ourevidenceimpliesthatespeciallybygrade3,the5percentilepointimpactoftheSmallclasstreatmentisalmostentirelyduetothefeedbackectoftheenhancedpeerqualitiesduetothetreatmentsintheearliergrades.Theevidenceonthegrade2eectislesssharp,althoughitdoesappearthatacrossvariousspecications,abouthalfofthe5percentilereducedformeisattributabletothepeerfeedbackeect.ForGrade1,weshouldre-emphasizethenatureofouridenticationstrategyimplieswehavemuchlesspowertodetectafeedbackeectattheearlystagesoftheexperiment.Withthatinmind,wendnoevidenceofanappreciablefeedbackeectfortherstgrade,andsoattributeallofthe7percentilereducedformeecttotheSmallclassreductionperse.WedothesameforKindergarten,althoughthatderivespurelyfromthedesignofouridenticationstrategy.Insummary,ourresultsimplythatalternativepolicystructures,suchasthetrackingofchildrenfollowingthegrade2and3patternsofProjectSTARwithouttheSmallclassreduction,wouldbeexpectedtoproduceasimilarsetofoutcomesfromthosederivedbyProjectSTAR.Thepeereectsthemselvesappeartohaveovertakenthepureresourceectsinthelatergradesoftheexperiment.Thatsaid,itisimportanttostressthatwerelyontheexperimentalassignmenttotheSmallclasstoproduceaboostinachievementinorderforourpeereectidenticationstrategy.Wedonotreadourresultsasimplyingclasssizereductionshavenoeect,butweeramoreindepthinvestigationofthemechanismsbywhichsuchresourcealterationsdohaveeectsthanhavebeenoeredbypriorexaminationsoftheProjectSTARdata.Suchare-interpretationisaby-productofourinterestinutilizingtheexperimentalSTARdatatoidentifyendogenouspeereWhiletheProjectSTARdatadooersomeimportantadvancementsfortheempiricalstudyofpeereects,itisimportanttonoteseveralofthepitfallscitedbyManski(1993)havebeenheldoutsidethescopeofthispaper.Foremostis ourassumptionthattherelevantpeergroupistheProjectSTARclassroominthecurrentyear.Theproblemisthatlackingsuchastrongassumptionimposedontheempiricalwork,makingheadwaywiththesedataisvirtuallyimpossible.InthecontextofProjectSTAR,however,anecdotalandintrospectiveevidencesuggeststhatearlyelementaryclassroomsdoexertapowerfulinuence,morepowerfulthananyotherreadilyidentiedpeergroupdelineationobservablewithourdata.Inthatsenseweareascomfortableaswecanbeaboutthisassumptionwiththesedata,andareratherfortuitousinhavingtheelementaryschoolsettingasthecontextforourdata.AsManskidiscusses,absentsuchanassumptionofthistype,theidenticationproblemforthepeermodelises-sentiallyinsurmountable.Thesecondhurdlewehaveavoidedaltogetherinthispaperisattemptingtocategorizethepeereectswedondintohowtheyman-ifestthemselves.Manski(1993)oersthreesuchcategorizations:(i)preferenceinteractions(ii)constraintinteractionsand(iii)expectationsinteractions.ThetheoreticalworkbyLazear(2000),forexample,isrelatedtotheconstraintinter-actioncategory.ThemodelproposedbyAkerlof(1997)mightbethoughtofasamixofbothpreferenceandexpectationinteractions.Distinguishingbetweensuchmodelsdoesappeartomattergreatlyforthestructureofpoliciesdesignedtocapturethepeereectspillovers.However,theProjectSTARdata,whilequitegoodatallowingthemeasurementofthespillovereects,sampleslittlethatwouldhelpusempiricallydistinguishbetweenthesealternativemodelsofhowthepeereectsmanifestthemselves.Wehopetheresultsofthispaperpushresearcherstoturntheirattentiontoempiricallydistinguishingbetweenthesealternativemodelsoftheunderlyingmechanisms. 7Appendix-TheAlgebraofInstrumentalVari-ablesEstimationoftheEndogenousPeerEf-fectsModelInthisAppendix,wederivethepropertiesoftheinstrumentalvariablesesti-matorfortheempiricalendogenouspeereectsregressionwheretheresearcherusescharacteristicsofthefullgroupinthesampleaseitheraninstrumentorre-gressor(i.e.boththeIVandOLScases).Forsimplicityofexposition,weignorethepresenceofothercovariates.Conditioningeverythingonasetofexogenouscovariatesdoesnotchangeanythingconceptually,althoughincludingthemmayinfactmasksomeofthemechanicalproblemswitheitherIVorOLSthatweaddresshere.Tostart,considertheempiricalspecicationfortheendogenouseectsmodelas:i,j(37)wherethenotation¯i,jistheleave-outmeanofthetestscoresforclass-room.Itisrelatedtotheusualsamplemeanoftheoftheclasstestscores(denotedas¯)by: (38)sothat: )(39)Inthecasewherethesampleofthepeergroupincludestheentirepeergroup(which,purelybyassumption,weassumetobethestudentsimmediateclassmates),thenitmakessensetorelatestudentsoutcometotheoutcomesofthestudentsintheclassotherthanstudent,hencetheuseoftheleave-outmeanastherelevantpeergroupmeasure.Theinstrumentthatweproposeinthispapertoextracttheexogenousvariationinthepeergroupmeasures¯i,jthefractionoftheclasspreviouslyrandomlyassignedtoaSmallclass.Wedonotincludeanadditionalsubscriptforthetimingofthevariablessimplybecausethatisirrelevanttothisdiscussion.LetbeadummyvariableindicatorforwhetherthestudentwaspreviouslyamemberofaSmallclass.Thenourinstrumentalvariableisgivenby: (40) Incontrast,whenthedatacontainonlyasampleofthepeergroupmembers,thenuseoftheordinarysamplemean¯issensible.Thisisbecauseindividualisrepresentativeofothermembersoftheclasswhomaynothavebeenincludedinthesample. Arewriteoftheexpressionfortheinstrumentalvariable,theusefulnessofwhichwillbeapparentbelow,is: (41)issimplynotationforthenumberofstudentsineachclasspreviouslyrandomlyassignedtoaSmallclass.Thisrewriteisusefulbecausesincethisiscontrolledexperimentaldata,thenumberofstudentsineachclass,,essentiallydoesnotvaryacrossclasses,andsothesubscriptissuperuous.Thevariationintheinstrumentthereforecomesentirelyfromvariationinacrossclasses-i.e. Theinstrumentalvariablesestimatorforintheempiricalmodelgivenaboveis,forasampleofstudentsinclassesisjust: i,j(42)(Thenumberofstudentsineachclass,,simplydividesoutofboththenumeratorandthedenominator.)Nowwecanmakeuseofourrelationoftheleave-outmeantotheusualsamplemeantore-writethisas: JjNij[1 (43)Andnownotethattheonlyquantitiesleftwhichareaectedbythesumoverthesubscriptsareonlythetermsinthenumeratoranddenominator,andsocarryingthosesumsthrough,thissimpliesto: PJjj[1 (44)Thisexpressioniseasilyseentoequal1intheabsenceofothercovariates.Noticethisisnotanasymptoticexpression,butholdsinthesampleFurthermore,thisalgebrafortheIVcaseshowsthatacoecientof1willalsoappearintheOLScasewherethefullgroupmean,¯isusedasthepeergroupmeasure,acoecientof1.Thatthisistruecanreadilybeseenbyinspectionofequation(42),replacingwith¯.Thefactthatbothvariablesvaryonlyatthegrouplevelimpliesthesamealgebraicsimplicationswillhold,andtheequationanalogousto(44)willagainbe1.Ofcourse,sincethesetupjustdiscusseddeliversacoecientofexactly1,itisimprobablearesearcherwouldnotrealizehiserror,andoptforadierentestimationstrategy.Inthissense,theadditionofcovariatesmaymaskthisissuetotheresearcher,asnowthecoecientwillnolongerbeexactly1inthegeneralcase.Assumingthatatleastsomeelementsofthevectorvary attheindividual(aswellasthepeergroup)level,theOLSestimatorisnow(usingmatrixforms,andbeingtheidempotentprojectorintothesubspaceorthogonaltothespacespannedbythecolumnsofthematrix,beingtheidempotentmatrixwhichaveragestothegroup()level):el):y0PBMxPBy]!1y0PBMxy(45)Ifweassume,forexpositiononly,thattheregressorvectorconsistsofonlyasignlenon-constantregressor,thensomestraightforwardbuttediousmanip-ulationallowsustowritethisas:s:(y0PBy)·(x0x)!1"(y0PBx)2]!1(y0PBx)(y0Qx)(46)wheretheidempotentmatrixisthewithin(class)operator.Thus,inorderforthenumeratorofthesecondtermtobenon-zero,theregressormustvarywithinandbetweenclass,aswellasbeingcorrelatedwiththeoutcomebothdimensionsinthesample.Iftheregressorvariesonlyatthegrouplevel(inourcontext,thiscouldbeateachercharacteristic,forexample)thenagain,thesampleestimateofthepeereectwillbepurely1.Notehowever,thatnowthereasonsforwhythecoecientdeviatesfrom1arenotentirelymeaningless.Intuitively,themoretheregressorexplainsthewithingroupvariationintheoutcomeascomparedtothebetweengroupvariation,thecoecientwillbedriventowardszero.Infact,substantiallymoresimplicationontheexpressionabovetellsustheestimatedpeereectwillattainzerowhenthefollowingexpressionholds:(47)Inotherwords,whentheOLScoecientfromaregressionofwithinandbetweengroupsdown-weightedbytheR-squaredfromaregressionofbetweengroups(i.e.thesquaredsamplecorrelationbetweentheBetweengroupvariationinand)equalstheOLScoecientobtainedfromthebetweengroupregressionof.AslongastheBetweenregressioncoecientliesabovethis,however,theestimatedpeereectwillbenon-zero.Aswediscussedinthecontextoftheleaveoutmeanestimatorsusedinthispaper,intuitivelythisisbecausethecovariateuencesthecross-groupvariationintheoutcomethanwouldbeexpectedthaniftherewerenofeedbackeectofthecovariatecreatingaspilloveratthegroup(class)levelascomparedtoitseectattheindividual-level,appropriatelydown-weighted.7.1TheIVEstimatorWhenthePeerMeasureisLaggedSinceand¯aredeterminedsimultaneously,someresearchers(e.g.Zimmer-man(1999)andSacerdote(2001)amongothers)havepositedinsteadthatthe uenceofonespeersdependsontheiroutcomesfromsomeearlierperiod,andthusestimateamodiedregressionoftheonegivenaboveas:ij,tij,t(48)wherethesubscripts1denotetheperiodfortheindividualoutcomeandthepeereectrespectively(thedatingoftheothervariablesisnotessentialtothisdiscussionandsoomittedforsimplicity).Tocutdownontheclutterofnotation,assumethatthesamplecorrelationbetweenij,tandij,tnetoftheregressoristhesameattheindividualandthegrouplevelandrepresented.Ifwelettheestimatorforthelaggedpeereectbedenotedas,thencomparingthisestimatortotheonebasedonthecontemporaneouspeermeasure)wehavethat:(49)Inotherwords,thepeerestimatorwhichisderivedfromaregressionequationusingalaggedpeermeasureusesthesameinformationastheonederivedfromanequationusingthecontemporaneousmeasure,exceptitiscorrectedbytheautocovariancepropertiesintestscores.Butthisestimatorisjustasinherentlyfragileastheonebasedthecontemporaneouspeermeasure,butwillmaskthetendencytoestimateacoecientnear1,duetothedown-weightingbytherst-orderautocorrelationcoecientestimateoftestscores. 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Class Mean Test Score ResidualsFigure 3: Between Class Partial PlotFraction of Class Previously Randomly Assigned to a Small Class-.5 -.25 0 .25 .5 .75 -20 0 20 40 Table 1Mean Characteristics of Switchers, Stayers, and New Entrants,Conditional on School Effects [1867].72.71 357 372 status in current rade .76.69 New Entrant, Small Class[3284][2858].48.45 in Re ular Class 1408 1288 .44.44 in Small Class 45 .48.48 Switch to Re ular Class [ 192 246 .51.47 p revious g rade ) .51.51.52Free Lunch ( status in [ 1917 .45.45 ular Class 366 383 .42.48 New Entrant, Small Class [ 3375 2867 .49.50 in Re ular Class 1435 1293 .50.49 in Small Class 47 .52.53 Switch to Re ular Class [ 192 248 .50.48.48.48Girl [ 1904 .65.65 ular Class 339 380 .63.63 New Entrant, Small Class [ 3570 3375 2867 .65.68 in Re ular Class[1435][1293].66.68 47 .62.74 Switch to Re g ular Class [ 192 248 .63.67.65.67WhiteSecond Grade [ 750 1823 score in current rade 39.9339.65 ular Class 255 357 score in current rade 43.7142.95 New Entrant, Small Class [ 3330 2706 53.3652.26 in Re ular Class 1418 1212 60.0357.92 in Small Class 45 57.5951.63 Switch to Re g ular Class [ 188 230 Notes: Sample sizes of the relevant groups are in brackets. Regular size classes and regular/aide classeshave been collapsed into one group called regular. The sample sizes dont match up within gradesacross variables due to missing observations. For the time-varying characteristics (free lunch andpercentile test score), the switchers and stayers means are computed based on the grade, while Table 2Composition of Class Types in Each Grade Randomly Assigned Notes: Regular and regular/aide students are grouped together. Previously randomly assigned refers tostudents having been randomly assigned in an earlier grade to the class type column under consideration,e.g. in the column for small classes, the previously randomly assigned students were randomly assignedto a class in their grade of entry. Switchers refers to students who were not in the class type, inthe relevant grade, to which they were randomly assigned. In parentheses under the switchers rows arethe number of students who switched class type from the previous year. The total column sumshorizontally across the small and regular class columns. The total rows sum vertically across rows Table 3OLS Estimates of the Experimental Effect on Individual Test Scores by Grade 5816574764375701Number of obs.24.30.31.32R2YesYesYesYesSchool fixed effects(.06)(.07)(.06)(.11).05.07.04.26(1.12)(1.08)(1.13)1.67-.65.34.76(1.79)(1.98)(2.20)1.231.08-4.13-1.02White teacher(.82)(.72)(.87)(.79)-12.21-13.24-13.02-13.03(.59)(.57)(.63)3.213.343.174.46Girl(1.45)(1.25)(1.19)(1.38)7.158.008.399.38White(1.16)(1.07)(.97)(1.14)-.511.641.57.22Regular/aide class(1.26)(1.27)(1.17)(1.25)4.765.947.315.13Small classSecond Grade Notes: Robust standard errors that allow for a correlation of the residuals among members of the same OLS Estimates of Class Size and Peer Group Effects by Grade: ) ( .21 ) ( .28 ) 3.283.084.0057476437Number of obsYesYesYesSchool fixed effects ( .03 ) ( .03 ) ( .03 ) .02.03.02 erience ) ( .47 ) ( .48 ) .61.03.26 ree ) ( .74 ) ( .82 ) .14.53-2.11White teacher ( .81 ) ( .70 ) ( .86 ) -12.28-12.95-12.97 ) ( .60 ) ( .57 ) 3.413.273.14Girl ( 1.43 ) ( 1.24 ) ( 1.17 ) 7.088.098.51White ( .51 ) ( .45 ) ( .43 ) -.30.49.54 ular/aide class ) ( .53 ) ( .58 ) 1.672.182.66Small class ( .04 ) ( .04 ) ( .04 ) .57.58.58Second Grade Notes: Robust standard errors that allow for a correlation of the residuals among members of the sameclass are in parentheses. A constant is included in all regressions. The normalized peer effect isconstructed by considering the thought experiment of moving a student from a regular size class to asmall class allowing the quality of the students peers to change, yet holding class size constant.Formally, it is computed by multiplying the coefficient on peers mean test score by the difference inmean peers test scores for small and regular classes. For example, in third grade, moving from a regularclass to a small class entails an increase in mean peers score from 49.14 to 54.90, yielding a normalizedpeer effect of .57*(54.90 - 49.14) = 3.28 percentile points. This normalized peer effect can be compared Table 5 57476437Number of obs.67.70.73R2YesYesYesSchool fixed effects (p -value ) ( .197 ) ( .509 ) of Peer Variables7.651.640.44F-statistic for Joint Test ( .06 ) ( .07 ) ( .06 ) .05.05.04 erience ) ( 1.18 ) ( 1.09 ) 1.86-1.35.14 ree ) ( 1.84 ) ( 2.04 ) 1.051.00-3.30White teacher ( 1.14 ) ( 1.09 ) ( .99 ) -.501.911.79 ular/aide class ) ( 2.57 ) ( 2.50 ) -1.532.446.39Small class in Second Grade ( 8.00 ) Assi g ned to a Small Class-4.11------ in First Grade ( 9.10 ) ( 8.20 ) Assi g ned to a Small Class3.204.46 in Kinder g arten ) ( 3.84 ) ( 3.56 ) Assi g ned to a Small Class17.376.852.37 Second Grade Notes: Robust standard errors that allow for a correlation of the residuals among members of the sameclass are in parentheses. A constant is included in all regressions, as are student characteristics (white, Table 6Instrumental Variables Estimates of Class Size and Peer Group Effects by Grade:Peers Mean Test Score Instrumented by Random Assignment Status of Peers ) ( .63 ) ( 6.77 ) 4.664.492.0557476437Number of obsYesYesYesSchool fixed effects ( .01 ) ( .01 ) ( .06 ) -.003.02.03 erience ) ( .27 ) ( .78 ) -.02.36.30 ree ) ( .30 ) ( 3.69 ) -.51.26-3.07White teacher ( .82 ) ( .71 ) ( .87 ) -12.32-12.81-12.99 ) ( .60 ) ( .57 ) 3.533.233.16Girl ( 1.44 ) ( 1.25 ) ( 1.19 ) 7.048.138.45White ( .19 ) ( .30 ) ( 1.92 ) -.17-.051.04 ular/aide class ) ( .78 ) ( 7.94 ) -.17.384.91Small class ( .04 ) ( .12 ) ( 1.00 ) .92.86.30Second Grade Notes: Robust standard errors that allow for a correlation of the residuals among members of the sameclass are in parentheses. A constant is included in all regressions. The normalized peer effect isconstructed by considering the thought experiment of moving a student from a regular size class to asmall class allowing the quality of the students peers to change, yet holding class size constant.Formally, it is computed by multiplying the coefficient on peers mean test score by the difference inmean predicted peers test scores for small and regular classes. For example, in third grade, moving froma regular class to a small class entails an increase in mean predicted peers score from 49.44 to 54.51,yielding a normalized peer effect of .92*(54.51 - 49.44) = 4.66 percentile points. This normalized peereffect can be compared directly with the small class coefficient to shed some light on the relative Table 7 ) ( 4.66 ) ( .12 ) 1.00-.56.86 ular Classes ) ( .10 ) ( .13 ) 1.06.81.93 Small Classes in Each ) ( .21 ) ( .27 ) .39.43.52 ular Classes ) ( .13 ) ( 2.05 ) .65.60-.81 Small Classes ( .05 ) ( .08 ) ( .16 ) .72.68.61 yp es in Each ) ( 4.66 ) ( .12 ) 1.00-.56.86 ular Classes ) ( .10 ) ( .71 ) .891.011.72 Small Classes ( .04 ) ( .12 ) ( 1.00 ) .92.86.30 yp es Assi nedSecond Grade Notes: Each cell represents a separate regression. Robust standard errors that allow for a correlationamong members of the same class are in parentheses. A constant is included in all regressions, as are Non-Linearities in Peer Group Effects, Class Level Estimates: 330338Number of obs[40][40] is Between 80% and 100%(3.30)(4.04)(4.23) Assigned to a Small Class10.985.072.23[39][45] is Between 60% and 80%(3.14)(4.06)(4.12) Assigned to a Small Class8.724.753.85[34][24] is Between 40% and 60%(3.20)(4.07)(4.24) Assigned to a Small Class2.195.645.37[10][15] is Between 20% and 40%(3.62)(4.60)(4.22) Assigned to a Small Class1.491.14.57 is Between 0 and 20%[200][207][214]-----------Second Grade Notes: Standard errors are in parentheses. Sample size of each group is in brackets. Additionalcovariates in each regression are a constant, class type, white teacher, teacher has a masters, teachers Non-Linearities in Peer Group Effects, Class Level Estimates Including [53](2.87)(6.24)8.71-11.42[85][25](2.52)(3.25)6.49-.93(2.23)(1.90)3.80-.81[48]----------[40][40] is Between 80% and 100%(3.37)(4.13)(4.72) Assigned to a Small Class9.841.875.64[39][45] is Between 60% and 80%(3.16)(4.07)(4.18) Assigned to a Small Class8.482.834.79[34][24] is Between 40% and 60%(3.22)(4.11)(4.24) Assigned to a Small Class2.113.695.98[10][15] is Between 20% and 40%(3.63)(4.59)(4.24) Assigned to a Small Class1.882.131.01 is Between 0% and 20%[200][207][214]----------Second Grade [38][4] Between 80% and 100%(3.64)(3.11)(7.03) Previously Together is 6.127.09-11.28Average Fraction of ClassNotes: Standard errors are in parentheses. Sample size of each group is in brackets. Additionalcovariates in each regression are the same as in Table 8: class type, white teacher, teacher has a masters, Table 10Between and Within Class Estimates:Dependent Variable is Class Mean (or Individual) Test Score .83.64-1.38 lied Peer Coefficient58296449Number of obs ( .74 ) ( .71 ) ( .66 ) -12.18-12.75-12.88 ) ( .57 ) ( .54 ) 3.422.973.06Girl ( 1.26 ) ( 1.15 ) ( 1.06 ) 6.847.818.25White Small Class ( 1.08 ) ( 1.14 ) ( 1.09 ) Assi ned to a1.533.64 Within Class Estimates:322320336Number of obs ( 6.04 ) ( 5.85 ) ( 4.94 ) -16.96-24.08-12.90Fraction Free lunch ( 6.88 ) ( 7.67 ) ( 7.39 ) .9610.097.08 ) ( 11.09 ) ( 10.31 ) 12.4415.4710.00Fraction White ( 1.49 ) ( 1.43 ) ( 1.33 ) -1.071.281.71 ular/aide Class ) ( 2.97 ) ( 3.01 ) -3.742.876.48Small Small Class ( 3.73 ) ( 4.17 ) ( 4.13 ) Assi ned to a4.261.53 Second Grade Notes: Standard errors are in parentheses. A constant and school fixed effects are included in allregressions. Teacher characteristics are included in the between class regressions. The implied peer Appendix Table 1 330338Number of obs ( .090 ) ( .567 ) ( .620 ) (p -value 0.570.25 of Peer VariablesF-statistic for Joint Test.70.70.73R2YesYesYesSchool fixed effects ( 6.16 ) in Third Grade----- ) ( 5.02 ) in Second Grade-17.53 ) ( 7.87 ) ( 5.22 ) in First Grade-26.29-11.87 ) ( 1.41 ) ( 1.32 ) -1.501.321.55 ular/aide class ) ( 3.03 ) ( 3.01 ) -3.822.307.12Small class in Second Grade ( 10.22 ) Assi g ned to a Small Class6.14----- in First Grade ( 11.29 ) ( 10.83 ) Assi g ned to a Small Class5.899.87 in Kinder g arten ) ( 4.69 ) ( 4.45 ) Assi g ned to a Small Class10.31-2.19-2.21 Second Grade Appendix Table 2Instrumental Variables Estimates of Class Size and Peer Group Effects by Grade:Peers Mean Test Score Instrumented by Random Assignment Status of Peers,Individual PRASC Included as a Covariate ( .40 ) ( 1.12 ) ( 11.33 ) 4.113.57-1.5657476437Number of obsYesYesYesSchool fixed effects ( .01 ) ( .02 ) ( .10 ) .01.03.05 erience ) ( .44 ) ( 1.34 ) .14.24.43 ree ) ( .51 ) ( 6.21 ) -.45.58-4.95White teacher ( .84 ) ( .70 ) ( .89 ) -10.90-11.85-12.43 ) ( .61 ) ( .57 ) 3.072.923.00Girl ( 1.43 ) ( 1.23 ) ( 1.23 ) 6.777.607.97White ( .69 ) ( .72 ) ( .93 ) In a STAR school 6.004.47 arten ) ( .52 ) ( 3.21 ) -.46.131.98 ular/aide class ) ( 1.15 ) ( 13.36 ) -1.34.447.54 ) ( 1.04 ) ( 1.14 ) Assi g ned to a Small Class.81-.371.12 Randoml ) ( .22 ) ( 1.67 ) .83.70-.23Second Grade Notes: Robust standard errors that allow for a correlation of the residuals among members of the same Appendix Table 3 57476437Number of obs ) ( .472 ) ( .604 ) of Peer Variables7.650.750.27F-statistic for Joint Test ( 1.14 ) ( 1.06 ) ( .97 ) -.641.581.59 ular/aide class ) ( 2.57 ) ( 2.38 ) -2.422.076.03Small class in Second Grade ( 7.49 ) Assi g ned to a Small Class-1.85----- in First Grade ( 9.04 ) ( 8.05 ) Assi g ned to a Small Class4.256.86 in Kinder g arten ) ( 3.79 ) ( 3.46 ) Assi g ned to a Small Class13.053.74-1.80 in Second Grade ( 1.68 ) Assi g ned to a Small Class.03------ in First Grade ( 2.13 ) ( 1.73 ) Assi g ned to a Small Class-.94-4.07 in Kinder g arten ) ( 1.12 ) ( 1.01 ) Assi g ned to a Small Class4.232.913.68 Second Grade Notes: Robust standard errors that allow for a correlation of the residuals among members of the sameclass are in parentheses. A constant is included in all regressions, as are student characteristics, teacher Inside the ‘Black Box’ of Project Star: Estimation of Peer EffectsUsing Experimental DataMichael A. Boozermichael.boozer@yale.eduStephen E. Cacciolastephen.cacciola@yale.eduThe credible identification of endogenous peer group effects—i.e. social multiplier or feedbackeffects—has long eluded social scientists. We argue that such effects are most credibly identifiedby a randomly assigned social program which operates at differing intensities within and betweenpeer groups. The data we use are from Project STAR, a class size reduction experimentconducted in Tennessee elementary schools. In these data, classes were comprised of varyingfractions of students who had previously been exposed to the Small class treatment, creating classgroupings of varying experimentally induced quality. We use this variation in class group qualityto estimate the spillover effect. We find that when allowance is made for this ‘feedback’ effect ofprior exposure to the Small class treatment, the peer effects account for much of the totalexperimental effects in the later grades, and the direct class size effects are rendered substantiallysmaller.JEL Classification: Z13, C51, C81, I21, C23Keywords:Peer Effects; Data with a Group Structure; Organization of Schooling; ExperimentalEvidence ECONOMIC GROWTH CENTERYALE UNIVERSITYP.O. Box 208269New Haven, CT 06520-8269CENTER DISCUSSION PAPER NO. 832INSIDE THE ‘BLACK BOX’ OF PROJECT STAR:ESTIMATION OF PEER EFFECTSUSING EXPERIMENTAL DATA1Michael A. BoozerYale UniversityandStephen E. CacciolaYale UniversityJune 2001Note:Center Discussion Papers are preliminary materials circulated to stimulate discussions and criticalcomments.Initial notes (Section Five): May 1997. We thank Dean Hyslop, Ann Stevens, Jenny Hunt, Paul Schultz, Andrew Foster, and seminar audiences atYale, Brown, and CUNY for helpful comments. The identification strategy in this paper was inspired byGeorge Akerlof’s (1997) recounting of Eugene Lang’s scholarship intervention in Harlem.This paper can be downloaded without charge from the Social Science Research Networkelectronic library at: http://papers.ssrn.com/paper.taf?abstract_id=277009 An index to papers in the Economic Growth Center Discussion Paper Series is located at:http://www.econ.yale.edu/~egcenter/research.htm