ESSAYSONTHEECONOMICSOFEDUCATIONByAndrewJacobBiblerADISSERTATIONSubmittedtoMichiganStateUniversityinpartialfulÞllmentoftherequirementsforthedegreeofEconomics-DoctorofPhilosophy2016ABSTRACTESSAYSONTHEECONOMICSOFEDUCATIONByAndrewJacobBiblerChapter1estimatestheimpactofduallanguageandimmersioneducationonstudentachieve-ment.DuallanguageclassroomsprovideEnglishLanguageLearners(ELLs)anopportunitytoreceiveinstructionintheirnativelanguageastheytransitiontoEnglishßuency.Thismightal-lowELLstobuildastrongerfoundationincoresubjectsandleadtobetteracademicoutcomes.DuallanguageandimmersioneducationhavealsogrownsubstantiallyinpopularityamongEnglishspeakingfamiliesacrosstheU.S.,astheypresentanoptiontolearncontentin,andpresumablybecomeßuentin,asecondlanguage.Despitethespikeinpractice,thereislittlecausalevidenceonwhate!ectattendingaduallanguageschoolhasonstudentachievement.Iexamineduallanguageandimmersioneducation,andstudentachievementusingschoolchoicelotteriesfromCharlotte-MecklenburgSchoolDistrict,Þndinglocalaveragetreatmente!ectsonmathandreadingexamscoresofmorethan0.06standarddeviationsperyearforparticipantswhowereeligibleforEnglishsecondlanguage(ESL)servicesordesignatedlimitedEnglishproÞcient(LEP).ThereisalsosomeevidencethatattendingaduallanguageschoolledtoalowerprobabilityofhavinglimitedEnglishproÞcientstatusstartinginthirdgrade.ForapplicantswhowerenoteligibleforESLservicesordesignatedasLEP,attendingaduallanguageschoolhasresultedinhigherendofgradeexamscoresofabout0.09and0.05standarddeviationsperyearinmathandreading,respectively.Chapter2buildsuponrecentresearchdiscussingapparentgenderdi!erencesinreturnstoparentalinvestments.Parentaltimeinvestmentsareonepotentialmechanism,ifcorrelatedwithhouseholdstructuredi!erentiallybygender,thatcouldhelpexplaindocumentedgenderdi!erencesinnon-cognitiveskills,aswellasthesensitivityofoutcomesandbehaviortofamilystructureforboys.Thispaperinvestigatesgenderdi!erencesinparentaltimeinvestmentsaroundchangesinhouseholdcomposition.IÞndthat,althoughbothboysandgirlsexperiencedecreasesinparentaltimeinvestmentsfollowingachangeinfamilystructurefromatwo-parenttoasingle-parenthouse-hold,thelossforboysisrelativelylarge.Thedi!erenceisstrongestinpaternalweekdayinvestments,forwhichboysloseanadditional24minutesperday,equivalenttoroughly35%ofaveragepaternalweekdayinvestments.ThereisnosigniÞcantevidencethatmotherscompensateforthelossbyincreasingtheirinvestmentstoboysrelativetogirls.Chapter3discussestheconstructionofconÞdenceintervalsinteachervalue-added(VA).Theuseofteachervalue-addedmodelstomeasureteachere!ectivenessisexpandingrapidly,withteachervalue-addedestimatesbeingincorporatedintoteacherevaluationsystemsandpotentiallyhigh-stakesdecisions.However,westillknowlittleabouttheprecisionofvalue-addedestimates,ortheperformanceoftheresultingconÞdenceintervals,whichcouldplayanimportantroleinthedecision-makingofdistrictsandpolicymakers.OurstudyaimstoÞllagapintheliteraturebyprovidingcomparisonsofconÞdenceintervalperformancefortheOLS-Lag(DOLS)value-addedestimator.WeusesimulatedstudentachievementdatatostudythebehaviorofstandarderrorsandconÞdenceintervalsforteachervalue-addedestimatesunderseveralstudentgroupingandclassroomassignmentscenarios.WeproposeasimplemethodforcalculatingconÞdenceintervals,whichincludesacriticalvalueadjustment,andcompareitÕsperformancetothatoftheconÞdenceintervalsyougetfromusingatypicalvarianceestimatorwithstandardnormalcriticalvalues.WeÞndthatthismethodgenerallyleadsto95%conÞdenceintervalcoverageratesnear95%,whichweconsideradesirablefeature.Inparticular,themethodhasadvantagesoverstandardconÞdenceintervalcalculationswhenvalue-addedestimatesarebasedonasmallnumberofclassroomsperteacherandstudentsaregroupedonunobservableheterogeneity.Wethenusestudent-leveladministrativedatatocomparestandarderrorsandconÞdenceintervalsacrossthedi!erentmethods.Ourproposedmethodassigns18.5%ofteachersinthetopdecileofthevalue-addeddistributionconÞdenceintervalswithalowerboundabovethe75thpercentileofthedistribution,amuchlowerpercentagethanthatfromignor-ingcorrelationinunobservables,60%,orusingcohort-schoolclustering,75.8%.Thesedi!erencesindicatethatthepolicyconclusionsdrawnfromusingthevalue-addedestimatesmaydependonthechoiceofconÞdenceinterval.Hence,knowledgeofthereliabilityofvalue-addedmodelscouldbeacriticalpartofthedecisionmakingprocessofadministratorsandpolicymakers.ACKNOWLEDGEMENTSIwouldliketothankToddElderforhisguidanceandhelpthroughout.IamalsogratefultoScottImberman,StacyDickert-Conlin,andMadelineMavrogordatoforservingonmydisserta-tioncommitteeandalwayso!eringhelpfuladvice,aswellasKellyVosters,andMichelleMaxÞeldforprovidingfeedbackonthedi!erentchapters.IhavealsobeneÞtedfromusefuldiscussionsandcommentsfrommanyotherclassmates,faculty,andseminarparticipantsthroughoutthisprocess.Iwouldliketonotethatthethirdchapterisco-authoredwithCassandraGuarino,KellyVosters,andJe!reyWooldridge,andthankthemforworkingwithme.IamgratefultotheNorthCarolinaEducationResearchDataCenter(NCERDC),andCharlotte-MecklenburgSchoolDistrictforprovidingthedatausedintheÞrstchapter,andinparticulartoKaraBonneau,SusanFrieje,andLindsayMessingerfortheirassistanceinthatprocess.ThisresearchwassupportedbyaPre-DoctoralTrainingGrantfromtheIES,U.S.DepartmentofEducation(AwardR305B090011)toMichiganStateUniversity,andbyIESStatisticalResearchandMethodologygrantR305D10028.ivTABLEOFCONTENTSLISTOFTABLES.........................................................................viiLISTOFFIGURES........................................................................ixChapter1DualLanguageEducationandStudentAchievement.....................11.1Introduction............................................................................11.1.1Literature..........................................................................31.2Lottery..................................................................................81.2.1MagnetProgramsandPriorityGroups.............................................91.2.1.1PriorityGroups.............................................................101.2.1.2CreatingLotteryFixedE!ects...............................................111.3Data...................................................................................121.4EmpiricalStrategy.....................................................................201.5Results.................................................................................231.6Conclusion.............................................................................29Chapter2HouseholdCompositionandGenderDi!erencesinParentalTimeIn-vestments...................................................................................312.1Introduction...........................................................................312.1.1Literature........................................................................332.2Data...................................................................................342.3Estimation.............................................................................392.4Results.................................................................................422.4.1HeterogeneityinGenderGaps....................................................482.4.2CompositionoftheInvestmentGaps..............................................492.4.3HouseholdStructureandChildBehavior..........................................502.5Conclusion.............................................................................51Chapter3PrecisionforPolicy:CalculatingStandardErrorsinValue-AddedModels............................................................................................543.1Introduction...........................................................................543.2Discussion.............................................................................563.3Simulations............................................................................593.3.1SimulationDesign................................................................593.3.2AnalyticMethods.................................................................613.3.3SimulationResults................................................................663.4ConÞdenceIntervalsinPractice........................................................783.4.1Data..............................................................................783.4.2Methods..........................................................................793.4.3Results...........................................................................793.5Conclusion.............................................................................82vAPPENDICES.............................................................................85AppendixAFiguresforChapter1.........................................................86AppendixBTablesforChapter1..........................................................89AppendixCFiguresforChapter2........................................................101AppendixDTablesforChapter2.........................................................105AppendixETablesforChapter3.........................................................116AppendixFFiguresforChapter3........................................................123AppendixGSupplementalTablesforChapter1...........................................126AppendixHSupplementalTablesforChapter2...........................................132AppendixISupplementalTablesforChapter3............................................135AppendixJSupplementalFiguresforChapter3..........................................137REFERENCES............................................................................142viLISTOFTABLESTableB.1:ApplicationNumbersandNeighborhoodSchoolCharacterstics....................90TableB.2:SecondandThirdChoices........................................................91TableB.3:DualLanguageandNeighborhoodSchoolCharacteristics..........................91TableB.4:SummaryandBalance-EnglishProÞcientSample................................92TableB.5:SummaryandBalance-ESL/LEPSample........................................93TableB.6:ImpactofAttendingaDualLanguageSchoolonAchievement.....................94TableB.7:ImpactofAttendingaDualLanguageSchoolonAchievement.....................95TableB.8:AttritionandWeighting..........................................................96TableB.9:ImpactofAttendingaDualLanguageSchoolonAchievement-Weighted.........97TableB.10:HeterogeneousE!ects............................................................98TableB.11:E!ectsbyGrade.................................................................99TableB.12:ImpactofDualLanguageSchoolingonLEPStatus.............................100TableD.1:WaveISummarybyGenderandHouseholdStructure...........................106TableD.2:OLSEstimatesofGenderGapsinTimeInvestments.............................107TableD.3:FixedE!ectsEstimatesofGenderGapsinTimeInvestments....................108TableD.4:GenderGapsinTimeInvestmentsbyInitialHHStructure.......................109TableD.5:FEGenderGapsinProbabilityofPTI>0........................................110TableD.6:GenderGapsinProbabilityofPTI>0byInitialHHStructure...................111TableD.7:GenderGapsbyAge............................................................112TableD.8:GenderGapsbyRace...........................................................113TableD.9:GenderGapsbyActivityType..................................................114viiTableD.10:ChildBehaviorandHHStructure..............................................115TableE.1:SummaryofSimulationDesign..................................................117TableE.2:AverageStandardErrorsandCoverageRates(StudentFE-N(0,1),noCohort-SchoolFE)..........................................................................................118TableE.3:AverageStandardErrorsandCoverageRates(StudentFE-N(0,0.25),Cohort-SchoolFEN(0,0.25))................................................................................119TableE.4:AverageStandardErrorsandCoverageRates,Cohort-by-cohortEstimation......120TableE.5:AverageStandardErrorsandCoverageRates,(StudentFE-N(0,1),noCohort-SchoolFE)..........................................................................................121TableE.6:AverageStandardErrorsandCoverageRates,Cohort-by-cohortEstimation......121TableE.7:StudentCharacteristics,byDistrict..............................................122TableE.8:Percentof95%ConÞdenceIntervalsAbove/BelowCuto!s,ByRegionofValue-AddedDistribution..................................................................................122TableG.1:ImpactofDualLanguageEducation-ConstantE!ect...........................127TableG.2:ImpactofDualLanguageEducation-3rdGradeAttendanceMeasure...........128TableG.3:GradesThreeThroughFiveOnly................................................129TableG.4:CohortInteractions..............................................................130TableG.5:AttritionandWeighting(Panel).................................................131TableH.1:FEEstimatesofGenderGapswithDayofWeekFEs............................133TableH.2:GenderGapsinInvestmentsbyInitialHHStructure(DayofWeekFEs).........134TableI.1:EstimationSampleSizesandGraphSampleSizes.................................136viiiLISTOFFIGURESFigureA.1:AverageTestScoresbyDLAttendance............................................87FigureA.2:LEPAverageTestScoresbyDLAttendance.......................................87FigureA.3:AverageTestScoresbyLEPStatus...............................................88FigureA.4:ProportioninCMSDLSchool....................................................88FigureC.1:MTI-Weekday..................................................................102FigureC.2:MTI-Weekdend.................................................................102FigureC.3:FTI-Weekday...................................................................103FigureC.4:FTI-Weekend...................................................................103FigureC.5:InvestmentsfromMother/Father-Boys..........................................104FigureC.6:InvestmentsfromMother/Father-Girls..........................................104FigureF.1:AverageStandardErrors,byDistrict.............................................124FigureF.2:AverageConÞdenceIntervalWidths,byDistrict..................................125FigureJ.1:AverageStandardErrors,byDistrict.............................................138FigureJ.2:AverageConÞdenceIntervalWidths,byDistrict..................................139FigureJ.3:AverageStandardErrors,byDistrict.............................................140FigureJ.4AverageConÞdenceIntervalWidths,byDistrict...................................141ixChapter1DualLanguageEducationandStudentAchievement1.1IntroductionDuallanguageclassroomsuseanon-EnglishlanguageofinstructionforasigniÞcantamountofthecurriculum.TheyareprimarilyusedtoprovideinstructiontoEnglishlanguagelearners(ELLs)intheirÞrstlanguageandtopromotebilingualismandbiculturalismamongnativeEnglishspeakers.Therearetwotypesofduallanguageclassrooms.1Two-wayclassroomstypicallyenrollstudentsfromtwodi!erentlanguagebackgroundsandteachcurriculuminbothlanguages.TherewereonlyabouttensuchprogramsintheU.S.in1980,butthatnumberwasalmost250by2000HowardandSugarman[2001].Incontrast,moststudentsinaone-way(immersion)classroomshareasimilarlanguagebackground,butreceiveinstructioninasecondlanguage.Thenumberofone-wayclassroomsregisteredwiththeCenterforAppliedLingusticsincreasedfromfewerthan50toalmost450overthelastfewdecades(CenterforAppliedLinguistics,2011).Recentexpansionsinseveralstateshavedriventhesenumbersevenhigher.2Despitethegrowth,thereislittlecausalevidenceonthee!ectofduallanguageeducationonstudentachievement.Inthispaper,IuseschoolchoicelotteriesfromCharlotte-MecklenburgSchoolDistrict(CMS)toestimatethee!ectofattendingaduallanguageschoolonstudentachievement.Districtstargetduallanguageeducationtotwotypesofstudents:Englishlanguagelearners(ELLs)whomightbeneÞtfromreceivinginstructionintheirhomelanguage,andotherstudentswhowanttohaveinstructioninasecondlanguage.ForELLs,duallanguageeducationmight1Althoughtherearetwodi!erenttypesofclassroomsstudiedhere,two-wayduallanguageandlanguageimmersion,Iwillrefertobothasduallanguageforsimplicity.2Utahpassedlegislationforfundingofduallanguageprogramsin2008,andsincethenhasimplementedprogramsin118schoolsin22districts.NewYorkCityaddedorexpanded40programsin2015.1allowforaneasiertransitiontofullEnglishinstruction,providingapotentialrouteforimprovingoutcomesofthegrowingandstrugglingELLpopulation.ThealternativeisoftenplacementinanEnglish-onlyclassroomcoupledwithEnglishsecondlanguage(ESL)services,whichcouldmeanmissingimportantclassroominstructiontimeanddisruptiontothestudentandhisorherpeers,andultimatelymakingstudentsmorelikelytofallbehind.Ontheotherhand,placementinanEnglish-onlyclassroommightexpeditethedevelopmentofEnglishskills,leadingtofasterre-classiÞcationoutofELLstatusandhigherscoresonstandardizedexamsthatarewritteninEnglish.DistrictsalsotargetduallanguageeducationtoEnglishspeakingstudentswhodesiretolearninasecondlanguage,andtheinßuxofduallanguageprogramsseemstobedriveninlargepartbydemandfromEnglishspeakingfamiliesWatanabe[2011].3Theprimarygoalfordistrictsino!eringduallanguageeducationtoEnglishspeakersistoprovideanoptionthatallowsthemtobecomebilingual,biliterate,andbicultural.Inaddition,districtscanuseduallanguageschools,aswellasotherspecializedprograms,too!eramorediversesetofschoolingoptionsandcompetewithcharterandprivateschoolstoretainstudentsresidinginthedistrictboundaries.DuallanguageprogramsareoftenpromotedusinghightestperformanceofparticipantsasevidenceofincreasedcognitivedevelopmentMaxwell[2012,2014].However,lackofformaltraininginEnglishcouldslowprogressasmeasuredbyscoresonstandardizedexams.Itisunclearwhetherduallanguageprogramswouldincreaseordecreasetestscoresforthisgroupofstudents.Whetherduallanguageeducationhasanye!ectonstudentachievementandthedirectionofthosee!ectsareempiricalquestions,butthereisverylittlecausalevidenceduetoendogeneityfromselfselectionintotheprograms.Inthispaper,Iestimatethecausale!ectofattendingaduallan-guageschoolonachievementonstandardizedmathandreadingexamsbyexploitingquasi-randomassignmentfromoversubscribedadmissionslotteries.IfocusonstudentswhoappliedthroughtheCharlotte-Mecklenburgschoolchoicelotteryfortheirkindergartenyear,andspeciÞedaduallan-guageschoolastheirÞrstchoice.Iusequasi-randomassignmenttoaduallanguageschoolthroughthelotteryasaninstrumentforduallanguageschoolattendancetoidentifythelocalaveragetreat-mente!ectofduallanguageschoolingonachievement.Thetreatmentdi!ersbywhetherornotthe3About70percentoftheestimationsampleinthisstudyarestudentswhowereneveridentiÞedasEnglishlanguagelearnersorlimitedEnglishproÞcienctinthedata.2studentusesEnglishasahomelanguage.ForanativeEnglishspeaker,thetreatmentistoreceiveinstructioninasecondlanguageandthealternativetoattendingaduallanguageschoolisreceivinginstructionintheirhomelanguage.ForELLs,thetypicalalternativetoattendingaduallanguageschoolistoreceiveinstructioninEnglish(nottheirhomelanguage)accompaniedbyotherESLservices.Becauseofthisdivideintreatment,Iestimatee!ectsseparatelyfortwosubgroupsusingaproxyforwhetherornotthestudentwasproÞcientinEnglishwhentheyenteredschool.TheÞrstgroupismadeupofstudentswhowereeligibleforEnglishsecondlanguage(ESL)servicesorweredesignatedaslimitedEnglishproÞcient(LEP)4atanypointinthedata.IwillrefertothisgroupofstudentsastheÒESL/LEPÓsample.IntheESL/LEPsample,IÞndthatattendingaduallanguageschoolleadstoincreasedscoresonmathandreadingexamsofmorethan0.06standarddeviationsperyearofparticipation.Furthermore,IÞndsomeevidencethatattendingaduallanguageschoolhasledtoalowerprobabilityofbeingdesignatedaslimitedEnglishproÞcientingradesthreethroughsixforstudentsintheESL/LEPsample.ThesecondsubgroupismadeupofstudentswhowerenevereligibleforESLservicesordesignatedLEP.IwillrefertothisgroupastheÒEnglishÓortheÒnon-ESL/LEPÓsample.Amongthisgroup,Iestimatethatattendingaduallanguageschoolleadsto0.09standarddeviationshigherachievementinmathperyearofparticipation,and0.05standarddeviationshigherachievementinreadingperyear.TheestimatesarestatisticallysigniÞcantandgenerallyrobusttoanumberofalternatespeciÞcations.1.1.1LiteratureBilingualeducationbroadlyreferstoeducationalprogramsthataretargetedtowardELLsandincludesomeamountofhomelanguageinstruction.Drawingconclusionsfrompreviousliteratureiscomplicatedbythefactthatbilingualeducationcantakeseveralforms,andthedegreetowhichhomelanguageinstructionisusedvarieswithinandacrossprogramtypes.Two-wayduallanguage4NorthCarolinausesthetermlimitedEnglishproÞcient(LEP)torefertostudentswhodonotuseEnglishasaprimarylanguageintheirhome,andscorebelowaspeciÞedcuto!onanEnglishskillstest.Thereissomevariationamongresearchersandschooldistrictsinhowtheyrefertothisgroupofstudents.ThetermEnglishlanguagelearner(ELL)isoftenusedinplaceof,orinterchangeablywithLEP.IusethetermLEPwhenreferringtothedesignationgiventostudentsinNorthCarolinabecausethatisthetermthestateuses,butusethetermsELLandLEPinterchangeablywhenreferringtothisgroupofstudentsgenerally.3classroomstendtousethenon-Englishlanguageforalargeproportionofinstruction(50%ormore)throughoutelementaryschool.InstructionisnotgenerallybasedoncurrentEnglishabilityofELLstudents,astwo-wayclassroomsenrolldominantspeakersofbothlanguagesandprovideinstructioninbothlanguages.Otherformsofbilingualeducation,suchastransitionalbilingualeducationandstructuredEnglishimmersion,aremorefocusedonexpeditingEnglishßuencyanddonotnecessarilygroupELLsandnon-ELLsinthesameclassroom.5SomepriorresearchfocusesontheachievementgapforELLstudentsinduallanguage(DL)programs,showingthatELLsparticipatinginDLprogramshavehighertestscoresthanELLsinnon-duallanguageclassrooms.TwodetailedreportsonsixdistrictsinNorthCarolina,includingCMS,ÞndthatELLsintwo-wayprogramsscorehigherthanstudentsinEnglish-onlyclassroomsonend-of-gradeexamsThomasandCollier[2009],Thomasetal.[2010].CollierandThomas[2004]summarize18yearsofresultsonone-andtwo-wayprogramsfrom23di!erentschooldistricts.Studentsinbothprogramtypescloseatleast70%oftheELLtestscoregapbytheendofÞfthgrade,butthiscouldbedrivenbyselfselection.Onetechniquesometimesemployedinanattempttoovercometheselfselectionissueismatchingonpretestscoresorotherobservablecharacterisitcs.Cazabon,Lambert,andHall[1999]studiedatwo-wayprograminCambridge,MA.TheyusedapretesttomatcheachDLstudentwithacontrol,andshowthatELLsassignedtoaduallanguageclassroomoutperformedthecontrolgrouponEnglishbasedmathandreadingexams.Cobb,Vega,andKronauge[2009]alsomatchstudentsonobservablecharacteristicstoconsidertheimpactofDLeducationonachievementandÞndpositivee!ectsinwritingandmathfornativeSpanishspeakers,withthee!ectsbeingmorepronouncedoneyearaftercompletionoftheprogram.Severalattemptshavebeenmadebyresearcherstosummarizeestimatesfromthemostmethod-ologicallyrigorousstudies.Inameta-analysisontransitionalbilingualeducationprograms,RossellandBaker[1996]deemedonly25percentofthestudiestheyconsideredtobemethodologicallyacceptable.Atransitionalbilingualeducationmodelteachesreadinginthenativelanguageinearlygrades,butmovestocompleteEnglishinstructionasearlyassecondgrade.Theydeterminedthat5SeeValentinoandReardon[2015]foragooddescriptionofsomeofthedi!erencesbetweenduallanguageclass-roomsandotherformsofbilingualeducation.4arelativelysmallpercentageofthemostrigorousstudiesestimatepositivee!ectsoftransitionalbilingualprograms.Whileothermeta-analysesagreethatmuchofthepriorresearchisßawed,theysuggestthatamongacceptablestudiestherearepositivee!ectsofbilingualprogramsacrosssubjectsandindi!erenttypesofprogramsGreene[1998],SlavinandCheung[2005],Willig[1985].Mostoftheliteraturetheyexamineddoesnotaccountforselectionbias,butthereareahandfulofstudiesthatusedrandomassignmentinanattempttoestimatecausale!ects.However,thesestudiesweregenerallybasedonsmallsamples(e.g.lessthan175students)andfromnearly30yearsagoGreene[1998].Morerecently,ValentinoandReardon[2015]usedataonstudentpreferencesfromalargeurbandistricttocomparethetestscoresofstudentsinbilingualeducationandEnglish-onlyprograms.Theystudystudentperformanceonexamsacrossprogramtypesconditionalonthetypeofprogramthatthestudentpreferred.Theassignmentisquasi-random,buttheydonotuseknowledgeoftheassignmentmechanismtocompletelyexploittherandomnessintheassignment.TheyÞndthatduallanguagestudentsprogressfasterinmathandEnglishlanguageartsperformanceaftersecondgrade,leadingtobetterlong-runperformancethananyoftheotherthreeprograms(includingEnglishim-mersion)ValentinoandReardon[2015].Similartothisstudy,Steeleetal.[2016]exploitrandomassignmentfromoversubscribedadmissionslotteriesintoduallanguageprogramsinPortland,Ore-gon.Theyreportmostlypositive,insigniÞcante!ectsforduallanguagestudentsonreadingandmathexamscores.However,therearetwoimportantdi!erencesinthisstudy.First,theypooltwosubgroupsofstudents-ELLsandnon-ELLs-together,despitethefactthatthetreatmente!ectsforthesetwogroupscouldbeofdi!erentsignsandmagnitudes,whichwouldhaveimportantpolicyimplications.6Anotherimportantdi!erencebetweentheSteeleetal.studyandthispaperisthattheduallanguageprogramsinPortlandPublicSchools(PPS)arestrandprograms,meaningthattheyonlymakeupaportionoftheschool.AlloftheduallanguageprogramsinCMSarehousedinthreeschools,whereeveryclassroomintheschoolisaduallanguageclassroom.Otherrecentstudieshaveexaminedcausale!ectsofbilingualeducationprograms,buttheclass-roomsinthesestudiesarenotnecessarilytwo-wayduallanguageprograms.Slavinetal.[2011]6AboutninepercentofstudentsareELLsatthetimeofapplicationandÞfteenpercenthaveanon-EnglishhomelanguageSteeleetal.[2016].5userandomassignmentinkindergartentoeitheranEnglishimmersionortransitionalbilingualclassroomtostudydi!erencesinEnglishandSpanishreadingscoresforseveralyearsfollowingtheassignment.TheyfoundthatstudentsassignedtoatransitionalclassroomscoredloweronEnglishreadingexamsinearlygrades,buttherewerenostatisticallysigniÞcantdi!erencesbyfourthgrade.GuoandKoretz[2013]useadi!erence-in-di!erencesframeworktostudythee!ectofaMassachusettspolicythatshiftedtheearlyelementaryeducationforELLsfromaseveralyeartransitionalbilingualmodeltoaone-yearsheltered(orstructured)Englishimmersionmodel.StructuredmodelstargetinstructiontothecurrentEnglishabilityofthestudentswhileexpeditingEnglishßuencyrelativetotransitionalmodels,sothisrepresentsaclearshiftawayfrominstructioninthehomelanguageofthestudents.SimilartotheÞndingsofSlavinetal.[2011],GuoandKoretzÞndthatthepolicyhadnoe!ect(orasmallpositivee!ect)onfourthgradeEnglishreadingscores.Chinetal.[2013]usedistrictlevelvariationinthenumberofLEPstudentsinTexastostudywhetherhavingabilingualeducationoptionimprovesachievementforLEPsandtheirnon-LEPpeers.Theyidentifythetreat-mente!ectsusingthediscontinuitygeneratedbyaTexasrulethatdistrictswithatleasttwentyLEPstudentswhoshareacommonlanguageinaspeciÞcgrademusto!erabilingualeducationoptiontothosestudents.TheydonotÞndsigniÞcantincreasesinthetestscoresofLEPstudentsfromdistrictsthato!erbilingualeducation,butdoÞndanincreaseinthescoresofnon-LEPsindistrictsthato!erbilingualprograms.TheirÞndingssuggestthato!eringbilingualeducationresultedinpositivespillovere!ectstonon-LEPstudents.Estimatingpeere!ectsdirectlyhasrarelybeendoneinthissettingandwithmixedevidenceCho[2012],Geayetal.[2013].Whiletestscoresareoneoutcomeofinterest,districtsmayalsocareaboutthedurationofLEPclassiÞcationoflanguageminoritystudents.WhenastudentenrollsinadistrictinNorthCarolina,theirparenttakesasurveythatasksaboutthelanguagesthestudentusesathome.Thedistrictusesthatsurvey,andpossiblyinterviewswiththeparentsand/orstudent,todeterminethehomelanguageofthestudent.IfthehomelanguageofthestudentisnotEnglish,thenthestudentmusttakeatestthatdeterminesLEPstatusandeligibilityforESLservices.WhenastudentisidentiÞedasLEPbasedonthescoreoftheplacementtest,theyarerequiredtocontinuetestingannuallyuntiltheyarere-classiÞedoutofLEPstatus.LEPclassiÞcationisimportantforseveralreasons.6Itisanothermeasureofstudentprogressthatdi!ersfromthemathandreadingexams.Second,studentswithLEPstatusmaybeeligiblefortestingaccommodations.Lastly,o!eringESLservicesiscostly,sodistrictsbeneÞtfromprogramsthatexpeditereclassiÞcation,allelseequal.UmanskyandReardon[2014]showthatduallanguageparticipantsinalargeurbandistrictarereclassiÞedoutofLEPstatusataslowerrateinearlygrades,buthavehighertotalreclassiÞcationandEnglishproÞciencythanstudentsfromEnglishimmersionclassroomsbytheendofhighschool.Similarly,Steeleetal.[2016]reportslowerreclassiÞcationoutofELLstatusforduallanguageparticipantsthroughoutelementaryandmiddleschool.Whenestimatingtreatmente!ectsthough,theyÞndthatattendingaduallanguageclassroomledtoahigherprobabilityofexitingELLstatusstartinginÞfthgrade.Inadditiontoestimatingthee!ectofduallanguageschoolingonachievement,Iestimatethee!ectofduallanguageschoolingonLEPclassiÞcationamongstudentswhowereeveridentiÞedasLEPoreligibleforESLservices.FornativeEnglishspeakers,thereisconcernthatattendingduallanguageschoolsmaypromotebilingualismattheexpenseofachievementasmeasuredbystandardizedtests,whicharewritteninEnglish.Thequestionformanyparentsiswhethertheirchildcanattendaduallanguageschoolandbecomebilingualwithoutfallingbehindinothersubjects.Learninginasecondlanguagecouldcreateconfusionorfrustrationthatwouldnegativelyimpactachievement,especiallyintheshort-run.Ontheotherhand,thementaljugglinginvolvedwiththinkingintwolanguagesmightpromotecognitivedevelopment.Thistheoreticalconnectionhaspreviouslybeenmadeinresearchrelatedtoworkingmemory,whichisusedtostoreandprocessinformationandexecuterelatedtasksBaddeleyandHitch[1974],Baddeley[2003],Alloway[2010].Workingmemorycanbeconsideredameasureofabilitytolearn.ItisstronglycorrelatedwithacademicoutcomesandmuchofthegrowthinworkingmemorycapacitytakesplacebeforeadolescenceAlloway[2010].WorkingmemoryiscloselyassociatedwithsecondlanguageacquisitionBaddeley[2003],nativelanguagevocabularyDufvaetal.[2001],andlisteningandreadingcomprehensionChrysochoouetal.[2011],Dufvaetal.[2001],butempiricalevidencedirectlysupportingacausallinkbetweensecondlanguageacquisitionandcognitivedevelopmentthroughworkingmemoryissparse.InthisspeciÞcsetting,studentsapplyforentryintoaDLschoolfortheirkindergartenyearbutdonÕttaketheirÞrsthighstakes7examuntilthirdgrade,sostudentsandteachershavesometimetoovercomeanyinitialdi"cultiesinadjustingtothenewlanguage.ThegapintimebetweenschoolassignmentandtestingallowsteachersandadministratorstocommittoteachinginthesecondlanguageintheÞrstfewyearsofschoolwhentherearenohighstakesexamslooming.Ifworkingintwolanguagescanboostcognitivedevelopment,thenonemightexpectittoshowupinthisenvironment.SomepriorliteraturehaspointedoutthepositiveachievementgapforEnglishdominantstudentsinduallanguageprograms.Englishspeakingparticipantsoftwo-wayprogramsinNorthCarolinascorehigherthantheirpeersonend-of-gradeexamsandhavebetterattendanceThomasandCollier[2009],Thomasetal.[2010].Otherresearchusesmatchingonpretestsandobservables,Þndingthatlearningaportionofcurriculuminasecondlanguagedoesnothinderprogressandmightbeassociatedwithpositivee!ectsonachievementinreadingCobbetal.[2009],Cazabonetal.[1999].Again,Steeleetal.[2016]Þndthatduallanguageinstructionledtopositive,althoughofteninsigniÞcant,e!ectsonmathandreadingscores.Theire!ectsarenotdirectlycomparablebecausetheypoolallstudents,buttheirsampleiscomprisedofmostlynativeEnglishspeakers.Themostimportantcontributionofthisstudyistoprovidecausalestimatesofthee!ectofat-tendingaduallanguageschoolonachievementusingschoolchoicelotteriesinCharlotte-MecklenburgSchoolDistrictand,inparticular,estimatingseparatetreatmente!ectsfortwogroupsregardedverydi!erentlyforpolicypurposes.Thisisavaluableadditiontotheexistingresearchonbilingualed-ucation,becausemostpriorliteraturedoesnotcrediblyidentifyanytreatmente!ectanddoesnotdisentanglee!ectsforEnglishlanguagelearnerscomparedtonativeEnglishspeakers.Thenextsec-tionprovidesdetailsonthelotteryusedinCMS.Section3discussesthedataandsomedescriptives.Insection4theempiricalstrategyusedforthemainresultsisdiscussed.Section5presentstheempiricalresultsandsection6concludes.1.2LotteryEverystudentenrolledinCharlotte-MecklenburgSchoolDistrictisassignedtoaneighborhoodschoolbasedongeographiczones.Thedistrictusesaschoolchoicelotterytoallocateseatsforstudents8whowishtooptoutoftheirneighborhoodschool.Theempiricalstrategyusedinthispapermakesuseofexogenousvariationcreatedfromoversubscribedlotteries,soitisusefultodescribehowthelotteryoperatesandwhyitfacilitatestheidentiÞcationoftreatmente!ects.Thissectionprovidesdetailsonthelottery.1.2.1MagnetProgramsandPriorityGroupsAllCMSstudentscansubmituptothreeprogramsinorderofpreferencethroughacentralizedlottery.AllstudentswithanoldersiblinginaschoolareguaranteedaseatinthatschoolbymakingittheirÞrstchoice.7Thennon-guaranteedseatsareassignedinthreerounds.IntheÞrstround,onlyÞrstchoicesareconsidered.Iftherearefewerapplicantsthanseatsavailabletoagivenprogram,thenalloftheapplicantstothatprogramwillbeassignedtotheirÞrstchoice.IdentiÞcationcomesfromcomparingwinnersandlosersfromthesamelottery,soestimatesaredrivenbyoversubscribedlotteries.Whenthenumberofapplicantsisgreaterthanthenumberofavailableseats(thechoiceisoversubscribed),seatsareawardedquasi-randomly.Seatassignmentisnotcompletelyrandom,becausetheprobabilityofwinningforaparticularstudentdependsontheprioritygroupthatthestudentisassignedto.Prioritygroupsrefertosetsofstudentsthatmeet(ordonotmeet)somepre-speciÞedcriteria.InCMS,overthesampleperiodtheyarebasedongeographiclocationandwhetherthestudentÕsneighborhoodschoolisaTitleIchoiceschool.Withthatinmind,thedistrictgivesprioritywithacoupleofapparentgoals.First,theycareabouttransportationcostsandallowingstudentstoattendschoolsthatareclosetohome.Studentswholivewithincloseproximitytoafullmagnetschoolaregivenpriority.Inaddition,thedistrictissplitintofourgeographiczones.Magnetschoolso!ertransportationtoatleastone,anduptofourofthezones,leadingthezonestobereferredtoastransportationzones.Studentswholiveinazoneservedbyamagnetaregivenpriorityforadmissiontothatschooloverstudentswholiveinazonethatisnotservedbythatschool.Studentsoutsideofthezonecanstillapply,butlivingoutsideof7Studentswhomeetadmissioncriteriaandhaveatwinoroldersiblingassignedtoamagnetprogramreceiveguaranteedadmissiontothatprogram.TheapplyingstudentmustspeciÞyitastheirÞrstchoiceinordertobeguaranteedadmissionthroughsiblingpreference.Thesiblingguaranteerequiresthatthestudentshavethesameresidenceandatleastonecommonparentorguardian.9theschoolÕstransportationzonemeanstheyhavealowerprobabilityofwinning,allelseequal.Theyarealsorequiredtoprovidetheirowntransportation.Thedistrictalsocaresaboutequity.Theyshowthisbyo!eringprioritytostudentswhoareassignedtoTitleIchoiceschools.TitleIschoolsarethosewithahighpercentageofstudentseligibleforfreeandreducedpricelunch(FRPL).ATitleIschoolbecomesaTitleIchoiceschooliftheyfailtomeetadequateyearlyprogressinthesamesubjectfortwoconsecutiveyears.NoChildLeftBehind(NCLB)requiresthatthedistrictallowstudentsassignedtoTitleIchoiceschoolstheopportunitytoattendanon-TitleIchoiceschool,butitdoesnotrequirethedistricttoallowstudentstochoosetheschooltheyareo!ered.Infact,theycouldbeo!eredaschoolthattheydidnotapplyforinthelottery.Assigningstudentstoprioritygroupsalterstheprobabilitiesofwinning,andmeansthatassign-mentisnotunconditionallyrandom.Iuselottery(programofapplicationbyyearbyprioritygroup)Þxede!ectstoexploitthefactthatwinnersshouldberandomlychosenwithinthesegroups.Inad-ditiontoprioritygroups,allapplicantsareorderedbasedonrandomlyassignednumbers.Whenachoiceisoversubscribed,thecombinationofprioritygroupsandrandomlyassignednumbersdeter-minewhowinsthelottery.Thenextsubsectiondiscussestheprioritygroups,andgivesmoredetailonhowlotterywinnersaredeterminedduringandaftertheÞrstround.1.2.1.1PriorityGroupsSeatsareallocatedbasedonprioritygroupandlotterynumber.ThetoppriorityforapplicantstofullmagnetschoolsinCMSisgiventostudentswholivewithinone-thirdmileoftheschool,butonlytwentypercentofseatscanbeassignedthroughthatpriority.8Forexample,iftherearetenseatsavailabletoaspeciÞcfullmagnetschoolandmorethantwoapplicantslivewithinone-thirdmileoftheschool,thenthestudentswiththeÞrsttwonumberswinunderthatpriority.Thentheymovetothesecondprioritygroup,studentswithTitleIchoiceneighborhoodschools.9Onlytenpercent8Studentswiththispriorityarestillsubjecttothelotteryifdemandfromthispriorityexceeds20%ofseatsavailable.Theareacanbeextendedbeyond1/3ofamilebythesuperintendentifthenumberofstudentsenrolledmeetingthiscriteriaforaspeciÞcgradeislessthan15.9TheÞrstseatsawardedareforstudentswhoqualifyforFRPLandthosebelowtheirgradelevelinreading.Forkindergartenstudents,belowgradelevelinreadingisdeÞnedashavingapersonalizededucationplan.10ofavailableseatscanbeassignedthroughthispriority.Continuingwiththeexample,thestudentwiththeÞrstnumberwhomeetsthesecondpriorityisassignedaseat,buttherestofthestudentsassignedtoTitleIchoiceschoolsremainunassigned.Finally,theymovetothethirdpriority,allstudentswholiveintransportationzonesservedbythemagnetschool.10Thereisnolimitonthenumberofseatsassignedthroughthispriority,sointhisexample,studentswiththenextsevennumberswholiveinthetransportationzoneareadmitted.Thelastpriorityisforstudentsfromtransportationzonesnotservedbythemagnetschool.11Inthisexample,ifmorethantwostudentsmeetpriorityone,thenthatprioritygroupisoversubscribed.TheidentiÞcationstrategyreliesoncomparingstudentswhometaspeciÞcpriorityandwonwithstudentswhometthatpriorityanddidnotwin.Similarly,ifmorethanonestudentmeetsthesecondpriority,thenthatlotteryisoversubscribedaswell.Finally,ifmorethansevenstudentsmeetprioritythree,thenthatlotteryisoversubscribed.Insuchacase,studentsfromallthreeofthoseprioritygroupscontributetotheestimates.Incontrast,considerwhathappenstothestudentsinthelastprioritygroup,thosefromoutsideofthetransportationzone.Sincenostudentsinthelastprioritygroupwonaseat,thosestudentsdonotdirectlycontributetotheestimates.AftergoingthroughallÞrstchoices,secondchoicesareconsidered.IfastudentÕssecondchoiceisalreadyfullfromtheÞrstroundofassignments,thentheyremainunassignedinthesecondround.Thenthirdchoicesareconsidered.Allstudentsareassignedtoadefaultneighborhoodschoolbasedonpre-determinedgeographiczonesifnototherwiseassignedinthelottery.Sincethelotteryconsidersstudentchoicesinorder,studentsaremostlikelytowinachoicebypickingitÞrst,andmoreseatsareawardedintheÞrstroundthaninthesecondorthird.InthefollowinganalysisIrestricttostudentswhomadeaduallanguageschooltheirÞrstchoice.ThetreatmentassignmentvariableisadummyvariableforwinningtheirÞrstchoice,whichshouldberandomwithinlottery.1.2.1.2CreatingLotteryFixedE!ects10ThispriorityÞrstlimitsthenumberofseatsfromanyparticularneighborhoodschoolassignmentzonetobeproportionaltothepotentialnumberofapplicantstotheschool.Thenpriorityopensuptoallstudentsinthetransportationzone.Thisrestrictiondoesnotseemtobepracticallyimportant.11Applicantsfromoutsideofthetransportationzonemustprovidetheirowntransportation.11AlthoughlotteryÞxede!ectsarenotexplicitlygiveninthedata,IuseavailableinformationtoconstructÞxede!ects.Thedatacontainuptothreechoicesforeverystudentinorderofpreference,aswellassiblingplacement,TitleIchoiceplacement,FRPLstatus,andtransportationzone.12IstartwiththesampleofallapplicantswithoutaguaranteedseatandproceedinthefollowingwaytogeneratelotteryÞxede!ects.1.ProxyTitleIchoiceschoolusingwhetherornotanystudentfromtheirneighborhoodschoolwasplacedundertheTitleIchoiceoptionthatyear.2.GenerateprioritygroupsusingFRPL,transportationzone,andTitleIchoiceproxy.3.LotteryÞxede!ectsarepriority-year-programofapplicationcombinations.SincethelotteryÞxede!ectsaregenerated,theyareaproxytothetruelotteryÞxede!ects.Theassignment,conditionalonlottery,providestheexogenousvariationusedtoestimatecausale!ects.1.3DataTherearethreeduallanguageschoolsinCMS.Allthreearefullmagnetschools,meaningthatadmissionrequiresalotteryapplicationandeverystudentintheschoolparticipatesintheduallanguageprogram.CollinswoodLanguageAcademyandOaklawnLanguageAcademyo!ertwo-wayEnglish-Spanishclassrooms,andWaddellLanguageAcademy(formerlySmithLanguageAcademy)o!ersfullimmersionstrandsinMandarin,French,GermanandJapanese.Collinswoodstartedin1997andnowhousesgradesK-8.Inkindergarten,90%ofinstructionaltimeisinSpanishThomasandCollier[2009].IngradesonethroughÞve,halfofthecontentistaughtineachlanguage.Oaklawnisanewerprogram,startedin2004,butfollowsasimilarmodeltothatofCollinswood.Thecurriculumistaught90%inSpanishinkindergarten,75%inÞrstgrade,and50%ingradestwo12CMSstoppedreportingFRPLafter2010.For2011IproxyforFRPLatthetimeofapplicationusingFRPLfromtheNCERDCdata.12throughÞve.13Spanishisbyfarthemostcommonnon-EnglishlanguageamongstudentsinCMS,andthetwo-wayprogramsaretargetedtowardnativespeakersofbothlanguages.TheGermanandFrenchone-wayimmersionclasseso!eredatWaddellhavecompleteforeignlanguageinstructioningradesK-2,whereastheMandarinandJapaneseclassesteachonehourinEnglishperdayingradesK-2ThomasandCollier[2009].AllfourprogramsatWaddelltarget90%ofinstructionaltimeinthenon-Englishlanguageingrades3Ð5.Theone-wayprogramsprimarilytargetEnglishspeakingstudents,buttheyadmitELLswhospeakthepartnerlanguageoranotherlanguagealtogether.CMSandtheNorthCarolinaEducationResearchDataCenter(NCERDC)providedthedataforthisstudy.CMSprovidedeightyears(2006Ð2013)oflotteryresultswithassignmentintothethreeduallanguageschools.NCERDClinkedthelotterydatafromCMSwithstatewidedata.Thefollowinganalysiswillfocusonend-of-gradeexamscoresinmathandreading,whichbegininthirdgrade.Linkingthelotterydatawithstatewidedataprovidesinformationonend-of-gradeexamscores,andallowsforthetrackingofstudentswholeavethedistrictbutstayintheNorthCarolinapublicschoolsystem.Sincelotteryresultscouldimpactschoolattendancedecisions,thishelpstomitigateattritionissuesRouse[1998],Steeleetal.[2016].Myanalysissampleincludesstudentsenteringkindergartenfromthe2006-2007through2010-2011schoolyearswhosubmittedanapplicationforaduallanguageschoolintheCMSschoolchoicelottery,andwerelinkedfromtheCMSdatatotheNCERDCdata.14Istartwiththe2006-2007schoolyearbecauseofchangesimplementedthatyeartothelotterysystem,includinghowtheprioritygroupsweredetermined.End-of-gradeexamsstartinthirdgrade,sothelastyearofentryusedwhenestimatinge!ectsonexamscoresisthe2010-2011schoolyear.15Sinceestimationreliesonapplicantswithnon-guaranteedseatsinoversubscribedlotteries,thereareacoupleofthingsworthnoting.FromtheÞrstrowofTableB.1,between20and30percentoftheseatsineachschoolwereawardedtostudentswithsiblingguarantee.Thosestudentsaredroppedfromtheestimationsample.ThesecondrowofTableB.1showsthepercentageofapplicantsto13Bothschoolsuseteamteaching,dividedbylanguageofinstruction.14NCERDCwasabletolinkbetween93%and97%ofallobservationsfromtheCMSdataineachyear.AmongobservationsofrisingkindergartenersintheCMSdatawhochoseaduallanguageschoolÞrstinthelotteryoverthesampleperiod,93.5%werematchedwiththeNCERDCdata.15Thelatestexamscoresarefromthe2013-2014schoolyear.ThiswillbeupdatedasNCERDCreleasesnewexamscoreseachyear.13eachschoolthatwontheirÞrstchoice.Only56percentofapplicantswholistedCollinswoodastheirÞrstchoicewontheirÞrstchoice,and78percentofÞrstchoiceapplicantstoWaddellwontheirÞrstchoiceinthelottery.16TheCMSassignmentmechanismonlyconsidersÞrstchoicesintheÞrstroundofseatallocation,soifaschoolÞllsupintheÞrstround,thensecondandthirdchoiceapplicantstothatschoolwillnotwinaseat.TableB.2showsapplicationnumbersforstudentswhochoseoneoftheduallanguageschoolsastheirsecondorthirdchoice,butnotastheirÞrstchoice.Fromcolumn1,49percentofstudentswhomadeaduallanguageschooltheirsecondchoice,wontheirÞrstchoicetoanon-duallanguageschool.About14percentofstudentswhochoseaduallanguageschoolwiththeirsecondchoiceandnottheirÞrst,wonthatchoice,butonly10percentattendedaduallanguageschool.TheCMSdataalsocontaintheneighborhoodschoolthateachstudentisassignedto,whichhelpstodescribetheoutsideoptionsthatstudentsareforegoingtoenteraduallanguageschool.Characteristicsoftheneighborhoodschoolsoftheapplicantsareinformativeforthinkingaboutthecounterfactual.Languageofinstructionisnottheonlythingthatchangesforthestudentwhentheyoptoutoftheirneighborhoodschoolandintoaduallanguageschool.SpeciÞcally,therecouldbechangesinpeerqualityandcompositionofthestudentbody.MeancharacteristicsoftheschoolsthatapplicantsareoptingoutofaredisplayedinTableB.1.ApplicantstoOaklawncomefromschoolsthathavearelativelyhighproportionofminorities(12percentwhite),76percentofstudentsonfreeandreducedpricelunch,andscore0.3standarddeviationsbelowthestateaverageonend-of-grademathandreadingexams.Theycomefromneighborhoodschoolsthatscoreworsethantheaverageforallapplicants.Ontheotherhand,applicantstoWaddellandCollinswoodcomefromneighborhoodschoolswithasmallerpercentageofFRPLstudents(57percentand65percent,respectively),butstillscorebelowthestateaverageonend-of-grademathandreadingexams.Thethreeduallanguageschoolsaregenerallyhigherperformingthantheotherschoolsintheirrespectiveneighborhoods.TableB.3showsthatover75percentofstudentsatOaklawnareatgradelevelinreading,butonly50percentofthestudentsatschoolsintheareanearOaklawnare16Thesepercentagesincludestudentswithguaranteedseats,sotheyareoverestimatesofthepercentageofwinnersamongthosewithnon-guaranteedseats.About43percentofnon-guaranteedapplicantstoCollinswoodwonand69percentofnon-guaranteedapplicantstoWaddellwon.14atgradelevelinreading.17Selfselectionandpeere!ectscouldplayasigniÞcantroleinthehighperformanceofDLschools,butthereareotherfeaturesthatmighthurttheirperformancerelativetoneighborhoodschools.SpeciÞcally,DLschoolsexperiencehigherteacherturnoverandbeginwithlargerclassesinkindergarten.Duallanguageclassroomsneedteacherswhoareßuentinthelanguageofinstruction,sotheschoolsinCMSoftenrecruitteachersfromabroad.TheteachersarepermittedtoworkintheU.S.foralimitedamountoftime,leadingtohigherturnover.ThisisparticularlytrueinCollinswoodandOaklawn.TableB.3showsthatover50percentoftheteachersineachofthoseschoolshaszeroto3yearsofexperience,comparedtoabout30percentintheneighboringschoolsandothermagnets.Notallteachersandsta!membersinduallanguageschoolscomefromabroad,noraretheynecessarilyßuentinasecondlanguage.Sincetheyoftenimplementteamteaching,inmostgradesthereisatleastoneEnglishspeakingteacher.TableB.3showsthattheduallanguageschoolsdohavehighlyexperiencedteachers,althoughtheyhaveasmallerproportionthantheneighboringschoolsandothermagnets.Fromcolumnseven,37percentofteachersatothermagnetshave11ormoreyearsofexperience,butthatnumberisonly25percentatCollinswood(column4)and27percentatWaddell(column2).Sincestudentscannotenrollinaduallanguageschoolafterkindergarten(orÞrstgrade)withoutmeetingaminimumlanguagerequirement,theschoolsstartwithlargerclasssizes,anticipatingsomeattritionthroughoutelementaryschool.Theaveragekindergartenclasshas21.4studentsatCollinswoodand22.3atWaddell,asseenincolumns2and4ofTableB.3.Thatis3morestudentsthantheotherschoolsintheirrespectiveareas.Fromcolumn7ofTableB.3,othermagnetschoolshave18.7studentsinakindergartenclassonaverage.Althoughthereareseveraldi!erencesbetweenduallanguageschoolsandthetypicalneighborhoodschool,greaterteacherturnoverandlargerearlyelementaryschoolclasssizearetwocharactersticsofDLschoolsthatcouldleadtolowerachievement.FiguresA.1-A.3providedescriptivecomparisonsofaveragestandardizedmathandreadingscoresbyLEPandDLstatus.FiguresA.1-A.2compareaveragestandardizedmathandreadingscoresfor17Irefertoschoolsintheirareaastheneighborhoodschoolzonethattheduallanguageschoolisinaswellasalloftheschoolzonescontiguoustothatzone.Sincealloftheduallanguageschoolsarefullmagnetschools,nostudentsareautomaticallyassignedtothem.Insteadeachstudenthasaneighborhoodschoolassignmentthattheyattendunlesstheyoptoutthroughthelottery,changeaddress,orenrollinacharterorprivateschool.15duallanguageandnon-duallanguagestudents.18ThesearedescriptivecomparisionsofDLandnon-DLstudentssimilartowhathasgenerallybeenexaminedinpriorstudies.Theyrepresentagoodstartingpoint,butignoreusefulinformationonlotteryÞxede!ectsandsiblingplacement.FigureA.1graphsthecomparisonfornon-LEPstudents.Non-LEP,duallanguagestudentsscorewellabovethestateaverageinreadingineverygrade,andthereisadivergencebetweenduallanguagestudentsandtherestofthedistrictfromgradesthreethrougheight.Inseventhandeighthgradetheduallanguagestudentsscoremorethan0.3standarddeviationsabovethestateaverageinreading.Non-LEP,duallanguagestudentsalsoscorewellabovethestateaverageinmath,butthegapbetweenDLstudentsandtherestofthedistrictdisplaysaslightdownwardtrendwithgrade.Theduallanguage,non-LEPstudentsscoreabout0.3standarddeviationsabovethemeaningrades4and5,butabout0.2standarddeviationsabovethemeanineighthgrade.ThecounterfactualsinFigureA.1,linesforthenon-DLstudents,includeallnon-DLstudentsinthedistrict,mostofwhomhadnointerestinattendingaduallanguageschool.Sincetherearelikelysystematicdi!erencesbetweenDLapplicantsandnon-applicants,estimatesgeneratedfromthissortofanalysisshouldnotbeconsideredcausal.FigureA.2displaystheanalogouscomparisonforstudentswhowereidentiÞedasLEPinatleastonegrade,threethrougheight.Non-duallanguage,LEPstudentsscorebelowthestateaverageinmathandreadingineverygrade.Ontheotherhand,LEPstudentswhoattendduallanguageschoolsscoreabout0.2standarddeviationsabovethestateaverageinmathinthirdgrade,andmorethan0.2abovetheaverageineverygradeafterthat.Theyalsoscoreatthestateaverageinreadinginthirdgrade,andaboveitineverygradeafterthird.Onceagain,FigureA.2providesevidencethatLEP,DLstudentsscoreabovetheirnon-DLpeersinmathandreadingonaverage,butthedi!erencesshouldnotbeinterpretedascausale!ects.WhilethesegraphsprovideusefuldescriptionsofthegapsintestscoresforDLstudents,theydonotprovidecausalevidenceonthedi!erencesinscores.Forcausalevidence,Iturntotherandomizationcreatedbytheoversubscribedlotteries.Onlylotteryapplicantswithnon-guaranteed18FiguresA.1-A.2graphaveragestandardized(byyearandgradeacrossthestate)residualizedscores.Theyareresidualsfromlinearregressionsofstandardizedexamscoresongradedummies,yeardummies,sex,FRPLstatus,andexceptionality.ForthepurposesofFiguresA.1-A.2,duallanguagestudentsareallstudentswhoattendedaduallanguageschoolinanygrade,3-8.FigureA.1usesallstudentswhowerenotidentiÞedaslimitedEnglishproÞcientinanygrade,3-8,andFigureA.2usesallstudentswhowereidentiÞedaslimitedEnglishproÞcientinatleastoneofthosegrades.16seatsareusedtoestimatecausale!ectsbecausetheestimationstrategyreliesoncomparingwinnersandlosersofthesamelottery.TablesB.4andB.5describethelotterywinnersandlosers.Columns1-3describetheapplicationsample,whichincludesallapplicantsregardlessofwhethertheyhavevalidtestscoresinthedata.Columns4-6describetheestimationsamples,whicharerestrictedtoapplicantswhoremainedinthesamplelongenoughtohavetestscoresavailable.AveragemathandreadingscoresontheirÞrstexamaredisplayedforlotterywinnersincolumn4andthosewholostthelotteryincolumn5.Thedi!erencesinthesescoresgivetherawtestscoregapsafterrestrictingtotheestimationsample.FromTableB.4,amongthenon-ESL/LEPstudents,lotterywinnersscoredabout0.24standarddeviations(0.52-0.28)higherthanlotterylosersontheirÞrstend-of-grademathexam.19Thesedi!erencesstilldonotwarrantacausalinterpretation,becausetheyignorelotteryÞxede!ects.Theanalogousdi!erencesbetweenwinnersandlosersintheESL/LEPsampleareshowninTableB.5.Lotterywinnersscored0.08standarddeviationsabovethestatemeanontheirÞrstmathexam,andlotterylosersscoredabout0.2standarddeviationsbelowthestatemean.Thatisadi!erenceofabout0.28standarddeviationsinfavoroflotterywinnersontheirÞrstend-of-grademathexam.Similarly,lotterywinnersintheESL/LEPsamplescoredabout0.24standarddeviationshigherthanlotterylosersontheirÞrstend-of-gradereadingexam,althoughbothgroupsscoredbelowthestateaverage.Ifstudentsperfectlycompliedwiththelotteryassignment,thenassignmentwouldbesynonomouswithattendanceandthecausale!ectcouldbeestimatedusingOLSregressionsofachievementonassignment/attendance.However,studentsdonotperfectlycomplywithinitialassignmentfromthelottery.FromTableB.4,89.9percentofÞrstchoicelotterywinnersand37.5percentofÞrstchoicelotterylosersfromthenon-ESL/LEPestimationsubsampleattendaduallanguageschool,meaningthatthereisnon-complianceamongwinnersandlosers.TheÞrstrowofTableB.5givestheanalogousÞguresfortheESL/LEPsubsample.Ninety-threepercentoflotterywinnersfromtheESL/LEPestimationsampleattendaduallanguageschooland29percentoflotterylosersattendaduallanguageschool.Winnersarenotboundtoattendtheschooltheywonthelotteryfor,andlotteryloserscan19ThescoresincludedarefromtheÞrstexamscoreavailableforeachstudent,whichistypicallythethirdgradescore.17endupinaduallanguageschooldespitelosingtheinitiallottery.Thereareseveralwaysthiscanhappen.First,theycouldwinaseattoadi!erentduallanguageprogramwiththeirsecondorthirdchoiceinthelottery.ThisissomewhatunlikelysincetheyaretypicallyÞlledupbystudentsmakingthemtheirÞrstchoice,butitdoeshappen.FromTableB.4,about31(21)percentoflotterylosersinthenon-ESL/LEPestimationsamplechoseaduallanguageschoolwiththeirsecond(third)choice.Morethan11percentofthelotterylosersinthatsamplewonaseatinaduallanguageprogramwiththeirsecondorthirdchoice.TableB.5showsthatlotterylosersfromtheESL/LEPsamplewerelesslikelytochooseaduallanguageschoolfortheirsecond(third)choice,asonly23(12)percentdid,andonly2percentofthemwonaseatinaduallanguageschool.Second,studentswhodonotwintheirÞrstchoiceareplacedonawaitinglistforthatschool,whichisaccessedifseatsbecomeavailable.Ifalotterywinnerchoosesnottotaketheseato!eredtothem,theseatiso!eredtothenextstudentonthewaitinglist.Thisislikelyamajorsourceofnon-compliancefromlotterylosers.Fromthenon-ESL/LEPestimationsampleinTableB.4,10.1percentofwinnersdonotendupattendingaduallanguageschool.FromTableB.5,7.1percentoflotterywinnersintheESL/LEPestimationsampledonotattendaduallanguageschool.Evenifawinningstudentenrollsintheduallanguageschoolandattendsthatschool,buteventuallyexits,thatseatcanbeo!eredtoanotherstudent.ThewaitinglistcanbeaccessedallthewaythroughtheÞrstacademicquarteroftheschoolyear.Lastly,studentscanreapplyintheschoolchoicelotteryforthesubsequentyearandwinaseat.20Asdiscussedinfurtherdetailinthenextsection,non-compliancedoesnotinvalidatetheempiricalstrategyusedinthispaper.Forcausalinference,assignmentmustbeasigniÞcantpredictorofattendanceandmustbeexogenousconditionalonlotteryÞxede!ects.IÞrstexaminewhetherassignmentisasigniÞcantpredictorofattendingaDLprogram.Itestfordi!erencesinDLattendancebetweenwinnersandlosers,conditionalonlotteryÞxede!ects,byregressingthedummyvariableforattendingaDLschoolonadummyforwinningandlotteryÞxede!ects.TheÞrstrowofTablesB.4andB.5displaystheestimatedcoe"cientsonthedummyforwinningthelottery,whichindicatewhetherwinningthelotteryactuallypredictsDLattendance.Rejectingthenullhypothesisofnoe!ect20ThereisalsoasecondlotterymainlyforstudentswhoenrolledinCMSafterthedeadlinefortheÞrstlottery,butsecondlotteryapplicantsareplacedattheendofthewaitlistforoversubscribedprograms.18indicatesthatwinningiscorrelatedwithattendance.Thetestincolumn6ofTableB.4suggeststhatlotterywinnersinthenon-ESL/LEPestimationsampleareabout52percentmorelikelytoattendaduallanguageschoolthanlotterylosers.Column6ofTableB.5showsthatintheESL/LEPestimationsamplewinnersarealmost67percentmorelikelytoattendaDLschool.BothestimatesarestatisticallysigniÞcant,suggestingthatwinningthelotteryisagoodpredictorforattendingaDLschool,whichisnecessarytoimplementtheidentiÞcationstrategyusedinthispaper.Theremainingtests,foundincolumnsthreeandsixofTablesB.4andB.5,givesomeindicationwhetherthelotteryresultsaretrulyrandom.Assignmentisrandomwithinlotterygroups,sothegeneratedlotteryÞxede!ectsareincludedineachtest.Sincelotterygroupsdependongeographiclocationandfreeandreducedpricelunchstatus,weshouldnÕtexpectÞxedcharacteristicsofap-plicantstobeunrelatedtowinningthelotteryunconditionally.Arejectionofthenullhypothesissuggeststhatwinningthelotterymightberelatedtothatcharacteristicinsomenon-randomwayandgenerallygivescauseforconcernabouttheidentiÞcationstrategyproposedbelow.Testsincolumn3ofTablesB.4andB.5arefortheapplicationsample,whichisthesamplethattheran-domizationactuallytookplacein.Noneofthetestsinthenon-ESL/LEPapplicationsamplerejectthenullhypothesis.Afterrestrictingtotheestimationsample,theonlyrejectionincolumn6ofTableB.4isonthecoe"cientintheregressionofadummyvariableforblackonwinningthelottery.FromTableB.5,thereisarejectionofthenullhypothesisforthedummyvariableforHispanicinboththeapplicationandestimationsamples.Thereareatleasttworeasonswhyatestmightrejecteveniftheinitialassignmentisrandom.TheÞrstcouldbefromnon-randomattiritionfromthesample.SinceIamestimatinge!ectsonmathandreadingscores,studentswhodonotremaininthesamplelongenoughtoobservetestoutcomesmustbedroppedforestimation.Eventhoughassignmentisrandomatthetimeofapplication,itisnotnecessarilyrandomwhenrestrictingtotheapplicantsthatremaininthesamplethroughthirdgrade.StayinginthedistrictcouldberelatedtowinningthelotteryandtheresultingattritionwouldleadtoselectionbiasRouse[1998],Steeleetal.[2016].21WhilethiswouldnotexplaintherejectionintheESL/LEPapplicationsample,itcould21Attritionislikelyhigherbecauseofthelagbetweenapplicationandtestingandthefocusonstudentsenteringkindergarten.Ihaveatleastonesetofexamscoresforabouteighty-Þvepercentoftheapplicantsample(seetherowslabeledÒNon-missingTestScoresÓinTablesB.4andB.5).19explaintherejectioninthenon-ESL/LEPestimationsample.IincludeinitialtestsinTablesB.4andB.5fornon-randomattrition,whicharefromOLSregressionsofhavingatleastoneavailablesetoftestscoresonwinningthelottery.Bothtestsfailtorejectthenullhypothesis,suggestingthatnon-randomattritionisnotanissue.22Anotherpossibleexplanationisthatassigmentisac-tuallyrandom,andtherejectionofthenullhypothesisisanartifactofmeasurementerrorintheconstructedproxiesusedforlotteryÞxede!ects.SinceprioritygroupsdependonfreeandreducedpricelunchstatusandcharacteristicsofthestudentÕsneighborhoodschool,Icontrolforthisßexi-blybyincludingfreeandreducedlunchbycohortdummyvariablesandneighborhoodschoolÞxede!ects,inadditiontothelotteryÞxede!ects.Tofurtheralleviateconcernsofendogeneityandnon-randomattrition,Iincludeanumberofrobustnesschecksincludingusingweightsbasedontheprobabilityofremaininginthesample.1.4EmpiricalStrategyThee!ectofattendingaduallanguageschoolonachievementcanbeestimateddirectlyusingOLStoestimateequation1.1.Yi,j,g,s=!á1[DualLanguage]i,j+"áXi,j,g,s+!j+Ns+#i,j,g,s(1.1)WhereYi,j,g,srepresentsanend-of-grademathorreadingexamscoreofstudentiingradegwhoappliedtolotteryjfromneighborhoodschools.Thekeyvariable,1[DualLanguage]i,j,isadummyvariablethatisequaltooneifthestudentattendedaduallanguageschool.23LotteryÞxede!ects,!j,areincludedbecausewinningthelotteryisnotunconditionallyrandom,butstudentsaredrawn22Iincludeamoreformaldiscussionofnon-randomattritionbelow,aswellasadiscussionofestimatesweightedfornon-randomattrition.23Usingenrollmentintheyearoftheexamisonewaytomeasureparticipation.Thatleavesalotoftimebetweenapplicationandwhenenrollmentismeasured.Onemightworrythatthiscouldbiasestimatessincestudentshavetimetoapplytootherschoolsorsimplywithdrawfromtheduallanguageprogram,bothofwhicharelikelynon-random.Forthisreason,Ipreferusingenrollmentinkindergartenastheparticipationmeasure.Inoneyearofthedata(2007),theschoolofattendanceinkindergartenismissingforanon-trivialportionofapplicants,manyofwhomshowupinaduallanguageschoolinÞrstgrade.Forthisreason,IactuallymeasureattendanceasshowingupinaduallanguageschoolineitherkindergartenorÞrstgrade.20randomlywithinlottery.Fixede!ectsfortheneighborhoodschool24thatthestudentwasassignedtoatthetimeofapplication,Ns,andstudentlevelcovariates,Xi,j,g,s,arealsoincluded.Gradesarepooledforestimation,sogradeofexamdummyvariablesareincludedinXi,j,g,s.Oneconcernwiththisapproachisthat,althoughtheassignmentisrandomconditionalonlotteryÞxede!ects,compliancewithinitialassignmentmaynotberandom,leadingtoabiasedandinconsistentestimatorfortheaveragetreatmente!ect.Compliancemightbenon-randomforacoupleofreasons.Inparticular,over30percentoflotterylosersendupattendingaduallanguageschool.Studentswhoattendaduallanguageschool,despitelosingthelotteryfortheirÞrstchoice,mightbesystematicallydi!erentfromthestudentswholostanddidnotendupattendingaduallanguageschool.Forexample,studentswhochoseaduallanguageprogramwiththeirsecondand/orthirdchoicearemorelikelytoattendaduallanguageschoolrelativetothosewhodidnotspecifyaduallanguageschoolwiththeirsecondand/orthirdchoice.Non-compliancecouldrepresentstrengthofpreferencesforduallanguageschoolingorfortheirneighborhoodschool,ortheabilityofparentstomaneuvertheirwayintotheirÞrstchoiceschool.SinceOLSestimatorsarebiasedandinconsistentifattendingaduallanguageschoolisnon-random,Ifocusonestimatingtheintention-to-treatandlocalaveragetreatmente!ectswhichareconsistentwhenassignmentisrandomconditionalonlotteryÞxede!ects.Ifollowastandardapproachforestimatingtreatmente!ectsusingapplicantsforoversubscribedlotteriesDemingetal.[2014],Rouse[1998].Theintention-to-treate!ectisestimatedbyregressingend-of-grademathandreadingscoresonadummyforwinningthelotteryandasetofcovariatesinthesampleoflotteryapplicants,asshowninequation1.2.Yi,j,g,s=!ITTá1[LotteryWinner]i,j+"ITTáXi,j,g,s+!ITTj+NITTs+#ITTi,j,g,s(1.2)Where1[LotteryWinner]i,jindicateswhetherstudentiwasawinneroflotteryj.Thecoe"cientofinterest,ö!ITT,isanestimateoftheintention-to-treatImbensandAngrist[1994].Thedi!erencebe-tweenequations1.1and1.2isthatequation1.2replacesthevariableofinterest,1[DualLanguage]i,j,withtheassignmentvariable,1[LotteryWinner]i,j.Theestimatorsfromequations1.1and1.2are24Neighborhoodschoolreferstotheschoolthatthestudentwasassignedtoatthetimeofthelottery.Thisistheschoolthatthestudentwouldbeassignedtoattendinkindergartenunlessthestudenteitheroptsoutduringthelottery,enrollsinacharterorprivateschool,orchangesaddress.21notestimatingthesameparameter,butö!ITTisconsistentundertheassumptionthatassignmentisrandom.Whereas,consistencyofö!requiresthelessplausibleassumptionthatattendingaduallanguageschoolisrandom.Boththeintention-to-treatandlocalaveragetreatmente!ectestimatorssharethisadvantageovertheOLSestimatorfromequation1.1.Equations1.3and1.4describeatwo-stageestimationstrategyusingthedummyforwinningthelotteryasaninstrumentforattendingaduallanguageschool.Nowö!LATEisanestimateofthelocalaveragetreatmente!ect,thee!ectforthosewhoareinducedtoparticipatebywinningthelotteryImbensandAngrist[1994].InthemainspeciÞcation,thee!ectsareestimatedbypoolinggradesandinteractingthetreatmentdummywithyearsoftreatment(gradeofexamplusone).Dummyvariablesareincludedforgradeofexam,leadingtoaper-yearofparticipationintepretation.1[DualLanguage]i,j=!DLá1[LotteryWinner]i,j+"DLáXi,j,g,s+!DLj+NDLs+#DLi,j,g,s(1.3)Yi,j,g,s=!LATEáö1[DualLanguage]i,j+"LATEáXi,j,g,s+!LATEj+NLATEs+#LATEi,j,g,s(1.4)IperformspeciÞcationcheckstoalleviateconcernsaboutexogeneityofthetreatmentornon-randomattrition,includingusingweightsbasedontheestimatedprobabilityofremaininginthesample.25Weightingtheregressionsadjustsfornon-randomattritionrelatedtoobservablecharac-teristics.IncludingneighborhoodschoolÞxede!ectsinallofthemainestimatesfurtherrestrictsthecomparisonstohelpwithconcernsaboutmisspeciÞedlotteryÞxede!ects.NeighborhoodschoolisdeÞnedastheschoolthatthestudentwouldhavebeenassignedtoiftheydidnotwinanyseatinthelottery,changeaddress,orenrollinacharterorprivateschool.Havingthesameneighborhoodschoolmeansthatthestudentsliveinthesamegeographicareaandhavethesameoutsideschoolingoption.Forcomparison,IalsoshowestimatesfromanalternativespeciÞcationthatdoesnotincludeneighborhoodschoolÞxede!ects.Inadditiontoestimatingthee!ectofattendanceonacheivement,Iestimatethee!ectofat-tendingaduallanguageschoolonlimitedEnglishproÞciencystatus.Iinteractthetreatmentand25Remaininginthesamplemeansthatthestudenthasvalidend-of-gradeexamscoresforatleastonegrade.Weightsarebasedonestimatedprobabilitiesfromlogitregressionsofanindicatorforstayinginthesampleonrace,gender,FRPL,andadummyforwinningthelottery.22attendancevariableswitheachgrade(threethroughsix),andestimatethee!ectonhavinglim-itedEnglishproÞcientstatusineachgradeontheESL/LEPsample.Priorresearchsuggeststhatduallanguageparticipantsre-classifyataslowerrateinearlygrades,buteventuallysurpasstheirnon-dual-language-schooledpeersUmanskyandReardon[2014].Thisisagoodpointofreference,althoughweshouldnotnecessarilyexpecttheseresultstobethesame.Thesearetwodi!erentcontexts,andIfocusonestimatinge!ectsinaselectsubsample,unlikethedistrictwideanalysisbyUmanskyandReardon[2014].1.5ResultsIbeginbyprovidingÞrststage(ö!DLfromequation1.3)andtreatmente!ect(ö!ITTfromequation1.2andö!LATEfromequation1.4)estimatesfromthemainspeciÞcationinTableB.6.PanelAofTableB.6showsestimatesforthenon-ESL/LEPsampleofapplicants,studentswhowereneveridentiÞedaseligibleforEnglishsecondlanguageservicesoraslimitedEnglishproÞcient.Theestimatede!ectonmathscoresincolumn5suggeststhatamongcompliers,attendingaduallanguageschoolledtoanincreaseinmathscoresof0.089standarddeviations.Thiscanbeinterpretedasaper-yeargaininachievement.Column7showsane!ectonreadingscoresforthissampleof0.053standarddeviationsperyear.BothestimatesarestatisticallysigniÞcantatthetenpercentlevel.TheseestimatesarepromisingforthegrowingpracticeofduallanguageeducationfornativeEnglishspeakers.AtleastinCMS,theduallanguageschoolshavebeensuccessfulindeliveringinstructioninasecondlanguage,andincreasingmathandreadingexamscoresforEnglishproÞcientstudents.Althoughtheseestimatesdonotseparateoutthemechanismsthroughwhichachievementgainsareoperating,theyshowthatitispossibletosuccessfullypromotebilingualismandincreaseacademicachievement.PanelBofTableB.6showsestimatedtreatmente!ectsfortheESL/LEPsample.Theestimatede!ectofattendingaduallanguageschoolonmathscoresincolumn5is0.078.Fromcolumn7,theestimatede!ectonreadingscoresinthissampleis0.064.BothestimatesarestatisticallysigniÞcantattheÞvepercentlevel.Whiletheseestimatesarelarge,theyareinlinewiththefactthattreatment23ismuti-yearandbeginsatayoungage.Theestimatessuggestthatduallanguageeducationcanbeane!ectiveteachingmethodforELLsandhelptoreduceachievementgapsinmathandreading.ConsidertheachievementgapsbetweenLEPandnon-LEPstudents,whicharedisplayedinFigureA.3.ThedistrictaveragemathandreadingscoresforLEPstudentsarebelowthestateaveragesineverygrade,andbelowthedistrictnon-LEPaveragesineverygrade.Thelargestgapinmathscoresisabout0.2standarddeviations,sotheestimateof0.078standarddeviationsperyearislargeenoughtomorethanclosethatgapbythirdgrade.Thelargestdisparityinreadingscoresisalittlemorethan0.4standarddeviations.Theestimatede!ectonreadingscoresof0.064standarddeviationsisenoughtoclosethegapintestscoresbytheendofelementaryschool.AsshowninTableB.7,theestimatesarenotsensitivetotheomissionofneighborhoodschoolÞxede!ects.Theestimateonreadingscoresforthenon-ESL/LEPsamplewithoutneighborhoodschoolÞxede!ects,shownincolumn7ofTableB.7,is0.057andstatisticallysigniÞcantatthetenpercentlevel.Theestimatedimpactonmathscoresinthatsample,reportedincolumn5,is0.086,andisstatisticallysigniÞcantattheÞvepercentlevel.Estimatede!ectsintheESL/LEPsamplewithoutneighborhoodschoolÞxede!ectsarereportedinPanelBofTableB.7.Theestimatede!ectonmathscoresisreportedincolumn5.ItisalittlesmallerthaninthemainspeciÞcation,now0.063,butstillstatisticallysigniÞcantattheÞvepercentlevel.Theestimatede!ectonreadingscores,reportedincolumn7,increasesfrom0.064to0.069,andisstillstatisticallysigniÞcantattheÞvepercentlevel.TheexclusionofneighborhoodschoolÞxede!ectsmakesverylittledi!erence.Allfouroftheestimatesremainpositive,inthesamerangeastheinitialestimates,andstatisticallysigniÞcant.Sincenon-randomattritionwouldleadtoinconsistentestimatorsandtestscoresaremissingforanon-trivialportionofapplicants,Iincludeestimatesthatareweightedbytheinverseoftheestimatedprobabilityofhavingtestscoresinthedata.Iestimatetheprobabilityofremaininginthesamplelongenoughtohavevalidtestscoresusinglogitregressionsondummyvariablesforrace/ethnicity,gender,FRPL,andwinningthelottery,thenusetheinverseoftheestimatedprobabilitiesasweightsintheestimation.TheestimatedprobabilitiesaresummarizedinPanelAofTableB.8.Fromcolumn3inTableB.8,theestimatedprobabilityofhavingasetoftest24scoresinthedataamongthosewhowoninthenon-ESL/LEPsubsampleisalmost85percent,andtheestimatedprobabilityforthosewholostthelotteryisonlyslightlylowerat82percent.Theestimatedaveragepartiale!ectofwinningonremainingthesampleis0.015,andnotstatisticallydi!erentfromzero.Similarly,intheESL/LEPsubsample,theestimatedaveragepartiale!ectofwinningonremaininginthesampleis0.019withastandarderrorof0.042.Theestimatessuggestthatwinningthelotteryisnotastrongpredictorofremaininginthesample,alleviatingconcernsaboutnon-randomattrition.PanelBshowslineartestsfornon-randomattrition,includingteststhatconditiononlotteryÞxede!ectsandneighborhoodschoolÞxede!ects.Fromcolumn2inPanelBofTableB.8,winningthelotterydoesnotappeartobestronglycorrelatedwithremaininginthesampleofEnglishproÞcientstudents.Theestimatedcoe"cientonwinningis-0.002withastandarderrroof0.038.AmongthesampleofESL/LEPstudents,theestimatedcoe"cientonwinningispositive,0.052,butstatisticallyinsigniÞcantwithastandarderrorof0.049.Theselineartestsprovidefurtherevidencethatnon-randomattritionisnotanissue,becausedespiteattritionratesofaroundÞfteenpercentonaverage,attritionisnotstronglycorrelatedwithwinningthelottery.26Despitetheapparentlackofcorrelationbetweenwinningthelotteryandremaininginthesample,weightedestimatesarereportedinTableB.9toshowthattheestimatesarenotsensitivetoweighting.PanelAshowstheinverseprobabilityweightedestimatesforthenon-ESL/LEPsample.Fromcolmun5ofPanelA,theestimatedtreatmente!ectformathscoresisnow0.089,thesameastheinitialestimate,andsigniÞcantatthetenpercentlevel.Theweightedestimateforreadinginthatsample,fromcolumn7,is0.052,almostidenticaltotheintialestimate.Theweightedestimatesonmathandreadingscoresinthenon-ESL/LEPsamplearethesameastheestimatesfromtheinitialspeciÞcation,suggestingthatnon-randomattritionisnotlikelytobeasigniÞcantfactor.EstimatesfortheESL/LEPsampleareshowninPanelBofTableB.9.Theestimatedaveragetreatmente!ectsonmathandreadingscoresarethesameastheinitialestimates,andbotharestillstatisticallysigniÞcantattheÞvepercentlevel.Weightinghasalmostnoimpactonthepointestimates,whichsuggeststhatnon-randomattritionisprobablynotinßatingtheestimatesmuch,26TheestimatesinTableB.8arebasedonasingleobservationforeachindividualapplicant,andadummyvariableindicatingwhetherthestudenthasvalidtestscoresinanygrade.Anotherwaytoinvestigatenon-randomattritionwouldbetoexpandthatdatatoincludeanobservationforeachindividualforeachgradethattheycouldhavetestedin.Theresultsarenotsensitivetothisalternativemethod.SeeTableG.5formoreevidence.25ifatall.Ireportestimatesofheterogeneoustreatmente!ectsinTableB.10;theseallowe!ectstodi!erbygender(columns1-2),programtype(columns3-4),orrace/ethnicity(columns5-7).Heterogeneoustreatmente!ectsareestimatedbyinteractingdummyvariablesindicatingmutuallyexclusivesetsofstudentswiththeattendancevariable,andusingthesamedummyvariablesinteractedwiththeassignmentvariableasinstruments.Estimatesforthenon-ESL/LEPsamplearereportedinPanelA.Columns1and2ofPanelAshowthate!ectsonmathscoresforfemales,0.106,arestrongerthanformales,0.068.Similary,theestimatede!ectonreadingforfemalesis0.069andstatisticallysigniÞcantattheÞvepercentlevel,andthee!ectformalesisastatisticallyinsigniÞcant0.030.Ontheotherhand,columns1and2inPanelBsuggestthatthee!ectisstrongerformalesintheESL/LEPsubsample.Thedi!erenceinheterogeneouse!ectsbygenderbetweensamplesissomewhatstriking,andmayreßectthedi!erenceintreatments.AbigpartofthetreatmentforstudentsintheESL/LEPsubsampleislikelythattheyreceivesomeinstructionintheirhomelanguageasopposedtoEnglishimmersioncoupledwithESLservices.Ontheotherhand,treatmentinthenon-ESL/LEPsampleistypicallyreceivinginstructioninasecondlanguageasopposedtoEnglishimmersion.Thedi!erencesinheterogeneitycouldresultfromdi!eringtreatmentsandpotentiallydi!erentmechanismsfacilitatingthee!ects.E!ectsforone-wayandtwo-wayprogramsarereportedincolumns3and4.Thedi!erencecomesdowntowhichschoolthestudentappliedtosinceWaddellcontainsalloftheone-wayprogramsandtheothertwoschools,CollinswoodandOaklawn,housetwo-wayprogramsonly.Thesizeoftheestimatede!ectsaresimilarbyprogramtypeforthenon-ESL/LEPsample,buttheestimatesone!ectsforone-wayprogramshavemuchlargerstandarderrors.Forexample,theestimatede!ectonmathscoresforone-wayprogramsinthatsampleis0.081,butthestandarderroris0.127.Theestimateonmathfortwo-wayprogramsis0.090withastandarderrorof0.046.ThereisalsoastatisticallysigniÞcantestimatede!ectfortwo-wayapplicantsof0.054onreadingscores,buttheestimatede!ectforone-wayapplicantsissmallerandstatisticallyinsigniÞcant.PanelBinTableB.10reportsestimatedtreatmente!ectsforstudentsintheESL/LEPsampleforone-wayandtwo-wayprograms.Similartothenon-ESL/LEPsample,estimatesforone-wayprogramsareverynoisy.26Theestimateonmathscoresforone-wayprogramsis-0.018inthissample,butthestandarderroris0.176.Theestimatede!ectsfortwo-wayapplicantsintheESL/LEPsampleare0.079and0.065onmathandreadingscores,respectivley.BothestimatesarestatisticallysigniÞcantattheÞvepercentlevel.Finally,Iestimateheterogeneouse!ectsbyrace/ethnicityinthenon-ESL/LEPsampleincolumns5,6,and7ofPanelAinTableB.10.Theestimatedimpactonmathscoresislargestinthewhitesubsample,butestimatesfortheblackandHispanicsubsamplesarealsopositiveandtheestimatefortheHispanicsubsampleisstatisticallysigniÞcantatthetenpercentlevel.Theestimatedtreat-mente!ectonmathscoresforthewhitesubsampleis0.190,whichislargerelativetomostotherestimatede!ects,andissigniÞcantattheÞvepercentlevel.Theestimatede!ectontheblacksub-sampleis0.046butitisstatisticallyinsigniÞcant.Theestimatedtreatmente!ectonmathscoresintheHispanicsubsampleis0.090andsigniÞcantatthetenpercentlevel.Estimatede!ectsonreadingscoresarerelativelysimilaracrossthewhite,andHispanicsubsamples.FromPanelAofTableB.10,theestimatede!ectonreadingscoresintheblacksubsample,0.034,islessthanhalfthesizeofthatestimateinthewhitesubsample,0.084,butbothestimatesarestatisticallyinsigniÞcant.TheonlysigniÞcante!ectonreadingscoresisontheHispanicsubsample,0.115,anditissigniÞcantattheonepercentlevel.Idonotestimatee!ectsforeachrace/ethnicityintheESL/LEPsample,because85%ofthestudentsinthatsampleareHispanic.Anyestimateforotherraceswouldbeunreliable.However,restrictingtotheHispanicsubsampleusingdummyinteractionsshowsthatthemainÞndingisrobustinthissubsample.Column7inPanelBofTableB.10showstheestimatede!ectsonmathandreadingfortheHispanicstudentsintheESL/LEPsample.Theestimatede!ectsonmathandreadingscoresare0.083and0.062,respectively.BothestimatesarestatisticallysigniÞcant.TableB.11showsestimatesbygrade.TheseareestimatedbyinteractingtheDLattendancevariableand/ortheindicatorforwinningthelotterywitheachexamgrade.Theestimatedimpactonmathexamscoresforthenon-ESL/LEPsampleareshownincolumn5ofPanelA.Theestimatede!ectformathscoresonthethirdgradeinteractionis0.374,andsigniÞcantattheÞvepercentlevel.Theestimatede!ectonmathscoresforsixthgradeis0.721andsigniÞcantattheÞvepercentlevel.27ThisestimateisonlyidentiÞedfromtwoofthecohorts,leadingtoarelativelylargestandarderrorof0.346.Theestimatesforthee!ectonreadingscoresinthissampleareshownincolumn6ofPanelAinTableB.11,andtheyalsoappeartoexhibitanincreasingpatternwithgrade.Theestimatesonthethirdandfourthgradeinteractionsare0.215and0.151,respectively.NeitherofthemarestatisticallysigniÞcant.ThelargestestimateisontheÞfthgradeterm,0.431,anditissigniÞcantattheÞvepercentlevel.Estimatede!ectsarealsoreportedbygradefortheESL/LEPsampleinTableB.11.Theestimatede!ectsarestrongerintheESL/LEPsample,buttheestimatede!ectsformathscoresdonotexhibitquiteasstrongofanincreasingpatternwithgrade.Thee!ectonmathscoresonthethirdgradeinteractionfromcolumn5inPanelBofTableB.11is0.393andsigniÞcantattheÞvepercentlevel.Theestimatedcoe"cientonthesixthgradeinteractionis0.542andisalsosigniÞcantattheÞvepercentlevel.Thelargestofalloftheestimatede!ectsonmathscoresfortheESL/LEPsampleisonthefourthgradeinteraction.Thatestimateis0.657andsigniÞcantattheÞvepercentlevel.AlloftheestimatesonreadingscoresintheESL/LEPsamplearepositiveaswell.Thelargestestimate,0.531,isonthesixthgradeinteractionandsigniÞcantattheonepercentlevel.Inadditiontoestimatingtreatmente!ectsonmathandreadingscores,Iestimatethee!ectofattendingaduallanguageschoolonLEPclassiÞcationamongthesampleofstudentsevereligibleforESLservicesorconsideredLEP.Iestimatethee!ectsbyregressingadummyvariableforbeingconsideredLEPinagivenyearonDLattendancebygradeinteractions.Iinstrumentforattendancebygradeinteractionsusingadummyforwinningthelotteryinteractedwitheachgrade.OLSestimatesbygradeareshownincolumns1and2ofTableB.12.Column1showsestimateswithoutneighborhoodschoolÞxede!ects.Everyestimateincolumn1isnegative,meaningthatstudentswhoattendDLschoolsarelesslikelytobeconsideredlimitedEnglishproÞcientineachgrade.Thelargestinabsolutevalueisthe-0.210estimateonthesixthgradeinteractionanditissigniÞcantattheonepercentlevel.Theanalogoustreatmente!ectsareshownincolumn3.Theyareallnegative,butonlytheestimateonthesixthgradeinteraction,-0.168,isstatisticallysigniÞcant.TheseestimatesareinlinewiththehigherEnglishreadingscores,butseemtocountersomeresultsinthepriorliteratureUmanskyandReardon[2014]yetagreewithothersSteeleetal.[2016].These28resultsarenotnecessarilycomparablewithpriorliteratureonre-classiÞcationsinceestimatesarespeciÞctoasetofstudentswhoappliedforduallanguageschoolsinCMS.Furthermore,alloftheestimatesonLEPclassiÞcationarenoisyandmostofthemarenotsigniÞcantlydi!erentfromzero.Ingeneralthough,theysuggestthatmovementsforcingEnglishimmersiononESL/LEPstudentsmightbemisguided.Inthissetting,studentsattendingDLschoolsnotonlyscorehigheronmathandreadingexams,buttheyarealsolesslikelytobeconsideredLEPingrades3-6.1.6ConclusionDuallanguagemagnetschoolsinCharlotte-Mecklenburgo!eranalternativeoptionforstudentstolearncurriculuminanon-Englishlanguage.IÞndthat,conditionalonsomebaselinecharacterstics,duallanguagestudentsscorehigherthantheirpeersonend-of-grademathandreadingexams.Oneconcernwiththisinitialdescriptiveanalysisandpreviousliteratureisthatdi!erencesmaybedrivenbyself-selection.Iuserandomassignmentfromschoolchoicelotteriestoestimatecausale!ectsofattendingaduallanguageschoolonstudentachievement.InthemainspeciÞcation,Iestimatelocalaveragetreatmente!ectsofmorethan0.06standarddeviationsperyearonmath,andalmost0.08standarddeviationsperyearinreadingexamscoresamongstudentswhowereevereligibleforESLservicesorconsideredLEP.Thee!ectsarerobusttoseveralalternativespeciÞcations,andlargeenoughtoclosetheLEP-non-LEPachievementgapinmathandreadingifappliedtoanaverageLEPstudentinCMS.IÞndfurtherevidencethatamongstudentsinthissample,thosewhoattendaduallanguageschoolarelesslikelytobeconsideredLEPingradesthreethroughsix,althoughthedi!erencesaregenerallystatisticallyinsigniÞcant.TheestimatesonachievementandLEPclassiÞcationsuggestthatduallanguageeducationhasledtolargebeneÞtsforstudentswithlimitedEnglishproÞciencyandappearsane!ectivewaytoservethepopulationofELLsinCMS.AmongEnglishÞrstlanguageapplicants,theestimatedimpactonmathscoresofabout0.09standarddeviationsperyearisrobusttodi!erentspeciÞcationsandrepresentsalargeincreaseinachievement.Thee!ectonmathscoresinthenon-ESL/LEPsubsampleissubstantiallystrongeramongfemalesandwhitestudents.Theestimatede!ectinreadingforthissampleis0.053standard29deviationsperyearinthemainspeciÞcation.Thesizeofthee!ectonreadingscoresisalsorobustacrossspeciÞcations.Thereissomeevidencethate!ectsonreadingscoresmightbestrongeramongfemalestudents,butthereislessevidenceofheterogeneitybyschooltypeorrace.ForEnglishÞrstlanguagestudents,itappearsthattheduallanguageschoolsinCMSprovideagoodopportunityforthemtobecomebilingualandbiliteratewithoutsacriÞcingachievementinotherareas.Notonlyaretheynotlosinggroundinmathorreading,theyareexperiencinglargegainsinbothmathandreadingachievement.Futureresearchshouldaimtodisentanglethemechanismsthatfacilitetheachievementgainsrealizedbytheduallanguageandimmersionstudents.Althoughthelotterywinnersinbothsub-samplesinthisstudyhavebeensuccessfulatlearninginanon-Englishlanguageandoutperformingtheirpeerswholosttheduallanguageschoollotteries,thecurrentstudydoesnotspecifythemech-anismsthroughwhichgainswererealized,andthereforecannotdistinguishane!ectoflearninginasecondlanguageitselffrompotentialdi!erencesinpeerandteacherquality,amongotherthings.Separatingoutthesemechanismscouldpointto,orruleout,speciÞcaspectsoftheCMSschoolsthatarecriticaltotheachievementgains,andshouldbeaprimarygoaloffutureresearchonthetopic.30Chapter2HouseholdCompositionandGenderDi!erencesinParentalTimeInvestments2.1IntroductionRecentresearchdiscussestherelativelypoornon-cognitiveoutcomesforboysraisedinsingle-parenthomes.However,westillknowverylittleaboutthemechanismsfacilitatingthesegenderdi!erences.Learningaboutthesemechanismsisimportantforproposingpoliciesortreatmentsthatcouldassistboysinclosingthegapinnon-cognitiveskills,andpotentiallyimprovingoutcomesforboysonotherdimensionsthatarecorrelatedwithgrowingupinasingle-parenthouseholdandhavingpoornon-cognitiveskills(e.g.cognitiveperformance,andeducationalattainment).Whiledi!erencesininvestmentlevelsanddi!erentialreturnstoinvestmentsbothlikelyplayaroleingeneratinggendergapsinnon-cognitiveperformance,separatingtheimportanceofreturnsandlevelsofeachinputisadi"culttask.Becausemanyinputsarecorrelatedwithhouseholdstructure,measuringthereturnstooneinput,canbeconßatedbythelevelsofand/orreturnstoomittedinputs.Parentaltimeinvestments,i.e.theamountoftimethatparentsspendwiththeirchildren,areapotentiallyimportantmechanismthatcouldhelpexplaingenderdi!erencesinnon-cognitivedevelopment.MorespeciÞcally,ifparentaltimeinvestmentsareimportantintheproductionofnon-cognitiveskills,andtheydependonhouseholdcompositiondi!erentiallyforboysandgirls,thentimeinvestmentscouldhelpexplaingendergapsinnon-cognitiveoutcomes.Inthispaper,Ifocusongendergapsintheleveloftimeinvestmentsandhowtheyrelatetohouseholdstructure,i.e.whetherthechildlivesinatwo-parentorsingle-motherhousehold.BecausefatherstendtospendrelativelymoretimewithboysastheyageBakerandMilligan[2013],andsingle-parenthouseholdsaremoreoftenheadedbythemother,growingupinasingle-parenthouseholdcouldbe31moredetrimentalforboysintermsoftimeinvestments.Thispapercontributestotheliteraturebyanalyzingdi!erentialchangesbygenderinthelevelofparentaltimeinvestmentsaroundtransitionsinhouseholdcomposition.Iusewithin-childvariationinparentaltimeinvestmentstoestimategendergapsininvestmentsandinvestigatetheirimportanceasapotentialmechanismforexplaininggenderdi!erencesinout-comesrelatedtohouseholdcomposition.UsingthePanelStudyofIncomeDynamics(PSID)PanelStudyofIncomeDynamics[2014]andtheaccompanyingChildDevelopmentSupplement(CDS),Iobtaindirectmeasuresofparentaltimeinvestmentsandexplorehowinvestmentlevelsrelatetohouseholdcomposition.Ishowthat,whileinvestmentsdecreaseforbothboysandgirlsaftertran-sitioningtosingle-motherhomes,thedecreaseisrelativelylargeforboys.Di!erencesarestrongestthroughpaternalweekdayinvestments,forwhichboysloseanadditional24minutesperday,whichisabout35%oftheaverageweekdayinvestmentfromfathersduringtheÞrstwaveoftheCDS.IalsoestimatelargeadditionaldecreasesforboysthroughthefatherÕsweekendinvestments,althoughthatestimateisgenerallynotstatisticallysigniÞcantandthemagnitudeismoresensitivetospeci-Þcation.Combiningtheweekdayandweekenddata,Iestimatethatpaternalinvestmentsdecreasebyanadditional2.3hoursperweekforboysinsingle-motherhomes,whichisover20%oftheaverageweeklypaternalinvestmentduringtheÞrstwaveoftheCDS.Theinvestmentgapislargerduringadolescence,duringwhichboysinsingle-motherhomesloseover3.3hoursperweekmorethangirls.Furthermore,thereisnostrongevidencethatmotherscompensatefortheadditionallossbyincreasinginvestmentstoboysrelativetogirls.TheseÞndingsconstituteimportantcontributionstotheliteraturebyproposinganothermecha-nismthroughwhichdi!erencesinthegenerationofnon-cognitiveskillscouldoperate,andshowingthatthechangesininvestmentsaresuchthattheirimportanceinexplainingdi!erencesinnon-cognitiveoutcomesisplausible.Furthermore,byfocusingonchildrenwhounderwentchangesinhouseholdstructure,theÞndingsdonotrelyoncomparisonsofinvestmentsacrossindividuals,butratheroncomparisonsofchangesininvestmentsacrossindividuals.Lastly,akeyfeatureofthispaperistheuseofadirectmeasureofparentaltimeinvestments,calculatedfromtwenty-fourhourtimediariescollectedaspartoftheChildDevelopmentSupplementtothePanelStudyofIncome32Dynamics.Theuseofadirectmeasureofparentaltimeallowsfortransparencyandclearinterpre-tationoftheresults.Abriefoverviewofthecurrentliteratureisprovidedinthenextsubsection.Section2describesandsummarizesthedatathatareused.Insection3Idiscusstheestimationprocedure.Section4outlinesresults,andsection5concludes.2.1.1LiteratureRelatedworkdocumentsgenderdi!erencesintheresponsetogrowingupinasingle-parenthouse-hold,andthee!ectondevelopmentofnon-cognitiveskillsBertrandandPan[2013].BertrandandPandocumentagendergapinnon-cognitivebehaviorthatwidenswithage.TheyÞndthatbehaviorismoresensitivetofamilystructureandparentalinputsforboys,butÞndnosystematicdi!erencesinthehomeenvironmentorinvestmentsthatcouldexplainmuchofthegap.TheyusemeasuresofinputssuchastheHOMEindex1,Warmthindex2,andwhetherthechildwasspankedlastweek,allofwhicharecorrelatedwithfamilystructure.Acloserexaminationoftherelationshipbetweentimeinvestmentsandfamilystructuremightrevealgenderdi!erencesininputsthatcouldhelpexplainboththeapparentdi!erencesinreturnstotheseinputs,aswellasthegapinnon-cognitiveskills.InotherworkJacob(2002)showsthatdi!erencesinnon-cognitiveskillscouldexplainasigniÞ-cantportionofthegapincollegeattendance.OtherresearchshowsthatmoreeducatedandhigherincomeparentsinvestmoretimeintotheirchildrenGuryanetal.[2008],thatonlytimeinputsfromparentswithahighlevelofeducationhaveapositiveimpactoncognitiveachievementDelBocaandMancini[2013],andthattheseinvestmentshavealargere!ectearlierinlifeDelBocaetal.[2012].HeckmanandCunha(2008)Þndthatparentaltimeinvestmentsarebetteratincreasingnon-cognitivethancognitiveskillsingeneral,butalsoÞndthatparentaltimeinvestmentsatyoungerageshaveagreatere!ectoncognitiveskillsthaninvestmentsmadeatlaterages.Theyaddthat1Basedonparentresponsestosixquestionsabouttheactivitiesthatthechildparticipatesinandactivitiestheparentparticipatesinwiththechild.AllofthequestionswereaskedduringthechildÕskindergartenyear.SeeBertrandandPan,2013.2Alsoreferredtoasemotionalsupportiveness.Basedonparentresponsestoaseriesofstatementsabouttheirchild,e.g.ÒchildandIoftenhavewarm,closetimestogetherÓandÒbeingaparentisharderthanIthoughtitwouldbe.ÓResponsesweregivenintheSpringofthechildÕskindergartenyear.33non-cognitiveskillsaremoremalleableatlaterstagesinchilddevelopment(i.e.adolescence)thancognitiveskills,andthatnon-cognitiveskillsappeartopromotethegenerationofcognitiveskillsHeckmanandMosso[2014].Onelimitationofthisresearchisthatitonlyconsidersonedimensionofacomplexrelationship,andnosinglemeasureofparentalinputcapturesallrelevantaspectsoftheproductionprocess.Parentaltimeinvestmentsarenotnecessarilycomparablewith,orpreferredtootherinputmeasures.However,timeinvestmentsprovideadirectandclearlyinterpretablemeasureofparentalinvestment.BertrandandPan(2013)focusontheHOMEindex,Warmthindex,andwhetherthechildwasspankedlastweektomeasureparentalinputs.TheyÞndthatthesemeasuresarerelatedtowhetherornotthechildisinatwo-parenthome,butthatbehaviorofboysismuchmoreresponsivetoinputs.Itmightbethecasethatothermeasuresofparentalinputsactuallychangedi!erentiallyforboysandgirls,andactasamechanismexplainingdi!erencesinbehavior,whichcouldalsocontributetoapparentdi!erentialgenderresponsestoinputs.2.2DataToestimatethedi!erentialchangesintimeinvestmentsaroundchangesinhouseholdstructureIusedatafromthePanelStudyofIncomeDynamics(PSID)andtheChildDevelopmentSupplement(CDS)tothePSID.TheCDSisasurveythatwasadministeredtochildrenofPSIDfamiliesinthreewaves(1997,2002/2003,and2007),andincludestimediaries,andsurveysofthechildrenandtheirparents.ThemostcriticalcomponentoftheCDSforthepurposeofthispaperisthecollectionoftwenty-fourhourtimediariesthatcatalogtheactivitiesofeachchildforoneweekdayandoneweekendday.Thediarydataareattheactivitylevelandincludeinformationonthedurationandparticipantsforeachactivity.Iusethetimediariestoconstructmeasuresofparentalinvestmentsbycountingtimethatthechildspentwitheachparentineachdiary.EverychildintheCDSwasassignedonerandomlyselectedweekenddayandweekdaytorecordtheiractivities.TheÞrstwaveoftheCDSincludeschildrenunderage13,andtheyareeligiblefortheCDSuntiltheyturn18.33TheagelimitsrefertothechildÕsageduringaninitialscreening.ThereareasmallnumberofcasesforwhichthechildÕsagewasoutsideoftheselimitsatthepointthatthetimediarydatawasrecorded.34About2,900participantscompletedatleastonetimediaryforCDS-I.Morethan2,500completedatleastoneforCDS-II,andover1,400forCDS-III.Theseadduptoatotalof6,915child-yearobservations.Atotalof3,330childrencompletedatleastonetimediaryinanyperiod,and1,086completedatleastoneinallthreewaves.Morethan1,400completedatleastoneforexactlytwoofthewaves.Therearetwofeaturesofthedatacriticalforthefollowinganalysis.TheÞrstisthepresenceofthetimediariesusedtocalculateparentalinvestments.Investmentsarecalculatedbysummingtimespentwithmother/father4acrossactivity-leveldataforeachchild.Thisisdoneseparatelyforeachweekendandweekdaydiary.Inaddition,IconstructweeklyinvestmentmeasurestohelpsummarizetotalinvestmentsbysummingtheweekdayinvestmentmultipliedbyÞvewiththeweek-endinvestmentmultipliedbytwo.Second,IuseinformationfromtheCDSandPSIDsurveystoconstructvariablesdescribingthecompositionofeachchildÕshousehold,includingpresenceofthechildÕsbiological/adoptivemotherandfather.Ifocusoncomparingtimeinvestmentsforchildrenintwo-parentandsingle-motherhouseholds.FiguresC.1throughC.4displaycross-sectionaldi!erencesininvestmentsacrossgenderandhouseholdtypefromwaveIoftheCDS.FigureC.1graphslocalpolynomialsofinvestmentsbyage,forboysandgirlswhowereintwo-parenthouseholdsduringtheÞrstwaveandthosewhowereinsingle-motherhouseholds.Weekdaytimespentwithmothersdecreasesdramaticallyasageincreasesacrossbothgendersandhouseholdtypes.TheaveragetimespentacrossthesegroupsarewithinaboutthirtyminutesofeachotherateveryageinFigureC.1.However,atageswherethetimespentdi!ers,itisgenerallytruethatmothersspendmoretimewithdaughtersthansons,andthatmothersintwo-parenthouseholdsspendmoretimewiththeirchildren.FigureC.2displaystheanalogousestimatesformaternalweekendinvestmentsbyage.Theoveralllevelsoftheinvestmentsarehigheronweekenddays,andgendergapsarealsomorepronounced.Forexample,inFigureC.1thepatternsforweekdaymaternalinvestmentstogirlsarealmostidenticalforthoseintwo-parentandsingle-motherhouseholds,butinFigureC.2single-motherÕsweekendinvestmentstogirlsare4Theinvestmentmeasuresonlyincludetimespentwithbiological/adoptiveparents.Similarly,whenreferringtoparentalpresenceinthehousehold,Iamreferringtobiological/adoptiveparentsonly.Forexample,achildwholivesinthesamehouseholdastheirbiological/adoptivemotherandastep-fatherisconsideredtobelivinginasingle-motherhouseholdforthepurposesofthisstudy.35lowerateveryagethantheirtwo-parenthouseholdcounterparts.Thegapisroughlybetweenthirtyandsixtyminutesateveryage,whichisasigniÞcantgaprelativetothatinFigureC.1wherethelinesarealmostindistinguishableatsomeages.Similarly,thereisapersistentgapbetweenhouseholdtypesformaternalinvestmentstoboysacrossallages,withboysintwo-parenthouseholdsreceivinglargerinvestments.Furthermore,thegendergapinmotherÕsweekendinvestmentsappearstowidenwithage,withmothersspendingmoretimewithgirlsthanwithboys.BothFiguresC.1andC.2demonstratetheimportanceofthechildÕsagewhenconsideringtimeinvestments,asinvestmentsdecreasesharplywithage.Forexample,mothersinvestbetweensixandsixandone-halfhoursonweekenddaystotheirinfantandtoddlerdaughters,butthatnumberisroughlyfourhoursfortwelvetothirteenyearolds.FiguresC.3andC.4graphtheanalogousestimatesforpaternalinvestmentsfromtheweekdayandweekenddiaries,respectively.Fathersgenerallyspendlesstimewiththeirkidsthanmothersdoacrossallhouseholdtypesandgenders,butthedecreaseininvestmentswithageislessdrastic,especiallyforboys.Infact,FigureC.3showsthatweekdaypaternalinvestmentsaresimilarlylowforbothboysandgirlsinsingle-motherhouseholdsacrossallages.Aslightincreaseinpaternalweekendinvestmentstoboysinsingle-motherhomes,showninFigureC.4,leadstoasmallgap,infavorofboys,thatappearstoincreasewithage.Theincreasinggendergapinweekendpaternalinvestmentsismoreapparentintwo-parenthomes.FromFigureC.4,bothboysandgirlsintwo-parenthomesreceivemorethanfourhoursinweekendpaternalinvestmentsupuntilaboutageÞve,butasteadydeclineinweekendpaternalinvestmentsforgirlsintwo-parenthomesleadstothatnumberdroppingbelowthreehoursaroundagetwelve.However,weekendpaternalinvestmentsforboysintwo-parenthomesremainsteadyataroundfourhoursforallagesrepresentedinthegraph.Oneimplicationoftheseinvestmentpatterns,particularlyforboysintwo-parenthouseholds,isthattheproportionoftotalinvestmentsthatcomefromfathersisincreasingwithage.ThisisshownmoredirectlyinFiguresC.5andC.6,whichgraphtheproportionofthetotalparentalinvestments5thatcomefromeachparentforthoseintwo-parenthouseholdsintheÞrstwaveforboysandgirls,5Totalinvestments,Totali,werecalculatedbyweightingtheweekday,WDi,andweekend,WEi,investmentstoconstructaweeklyinvestment,suchthatTotali=5áWDi+2áWEi.Theweeklymeasureoftimespentwitheachparentdividedbythetotalweeklymeasureistheproportionoftotalparentalinvestmentfromthatparent.Thetwovaluesaddtoonebyconstruction.36respectively.Noticethattheproportionsareroughlythesameforinfantandtoddlerboysandgirls.Eachgroupreceivedalittlelessthanseventypercentoftotalinvestmentsfromtheirmotherandalittleoverthirtypercentfromtheirfather.Theproportionschangequitedi!erentlywithageforboysandgirls.Forgirls,theproportionofinvestmentsfromtheirmothersneverdropsbelowaboutsixty-Þvepercent.However,byaboutagetwelve,boysreceiveunderÞfty-Þvepercentoftheirinvestmentsfromtheirmother.Anotherimplicationofthegenderdi!erencesintheinvestments-agerelationshipisthatonemightexpectthedi!erentiale!ectofhouseholdcompositiononparentaltimeinvestmentstodi!erbyage.Theincreasingrelativeimportanceofpaternalinvestmentsforboys,apparentinFiguresC.4andC.5,suggeststhatthepotentialforinvestmentlosses,relativetogirls,increaseswithage.TwoimportantpointsofFiguresC.1-C.6arethatinvestmentsgenerallydeclinewithageandthattherelationshipbetweeninvestmentsandagedi!ersbygender.Thesepatternscouldreßectthewayparentsspreadtheirtimewithmultiplechildren(Price,2008)andtheapparentpreferenceoffatherstospendrelativelymoretimewiththeirsons(BakerandMilligan,2013).Inmostcases,acrossallagegroupsandhouseholdstructures,mothersspendalittlemoretotaltimewithgirls,andfathersspendalittlemorewithboys,onaverage.TheÞguresdemonstratehowimportantageiswhenevaluatingtimeinvestmentsandsuggeststhatusingßexiblecontrolsforageisnecessaryintheanalysisthatfollows.TableD.1summarizesthetimeinvestmentvariablesandcovariatesbygenderandhouseholdtype.Inparticular,Iseparateouttheindividualswhounderwentachangeinhouseholdstructure,becausetheyarecriticalforestimation.Columns3and4displayaveragecharacteristicsforboyswhounderwentachangeatsomepoint.ColumnthreeincludesboyswholivedwithbothparentsintheÞrstperiod,meaningthatthechangeinstructureforthemisgoingfromlivingwithbothparentstolivingwithlessthanbothparents.Ontheotherhand,column4includesindividualswhounderwentchanges,butdidnothavebothparentsinthehouseholdintheÞrstperiod.Columns7and8displaytheaveragesforgirlswhounderwentachangeinhouseholdcompositionatsomepoint.TheÞrstrowsummarizestotalweeklymaternalinvestments,whichwasconstructedbysummingweekdayinvestments(row2)multipliedbyÞvewithweekendinvestments(row3)multipliedbytwo.Girls37receivelargermaternalinvestmentsthanboysacrossallhouseholdtypes.Girlswhowerealwaysintwo-parenthouseholdsreceivedabout26.6hoursinmaternalinvestmentsperweek,relativeto24.8hoursperweekforboys.Thegapinmaternalinvestmentsforchildrenintwo-parenthouseholdsthateventuallysplit,comparingcolumn6withcolumn2,isabout3hoursperweek,withgirlsreceivingmoreinvestments.Bothboysandgirlsinhouseholdsthateventuallysplitreceivedlargermaternalinvestmentsthanthosewhowerealwaysinatwo-parenthousehold.Fromcolumn3,boysinfamiliesthateventuallysplitreceived25.5hoursperweekonaverage,andthoseinhouseholdsthatneversplitreceived24.8hoursperweek.Similarly,girlsintwo-parenthouseholdsthateventuallyhadachangeincompositionreceived27.4hoursperweekinmaternalinvestments,butthoseintwo-parenthouseholdsthatneversplitreceieved26.6hoursperweek.Thesecomparisonsmaybemisleading,becausethehouseholdstructurecategoriesarealsocorrelatedwithage.Forexample,girlsintwoparenthouseholdsthatneverexperienceachangearejustundersevenyearsoldonaverage,buttheaverageageofgirlsintwo-parenthouesholdsthateventuallysplitisunderÞveyears.Thedi!erenceissimilarforboys.This,alongwithFiguresC.1-C.4,demonstrateswhyitisimportanttocontrolßexiblyforagewhenestimatinggendergapsintherelationshipbetweeninvestmentsandhouseholdstructure.Notonlyisagecorrelatedwithinvestmentsdi!erentiallybygender,itisalsocorrelatedwithhouseholdstructure.Withthatinmind,itissimilarlytruethatboysintwo-parenthouseholdsthateventuallysplitreceivedhigherpaternalinvestmentsthanthosewhowerealwaysintwo-parenthouseholds,16.7and16.3hoursperweek,respectively.However,theoppositeistrueforgirls,withgirlswhowerealwaysinatwo-parenthouseholdreceivingnearlytwohoursmoreperweekinpaternalinvestments,despitebeingroughlytwoyearsolderonaverage.Fromcolumn3,boyswhoexperienceachangeinhouseholdcompositionbutareinatwo-parenthouseholdduringwaveIareabouttwoyearsyoungerthanboyswhodonotexperienceachangeandareinatwo-parenthouseholdatwaveI,4.6yearsoldand6.5yearsold,respectively.Boysintwo-parenthouseholdswhoeventuallyseeachangeinhouseholdcompositionalsohavealittleoveronesiblinginthehouseholdonaverage,whereasthoseintwo-parenthouseholdswhoexperiencenochangehaveabout1.3siblingsinthehousehold.Thedi!erencesaresimilarforgirls.Girlswhoexperienceachangeinhouseholdstructure,butlivedwithbothparentsintheÞrstperiodwere384.9yearsoldonaverageandhad1.1siblingsinthehouseholdatwaveI,andthosewhoareinatwo-parenthouseholdanddonÕtexperienceachangewereabout6.8yearsoldwithalmost1.3siblingsinthehousehold.Theracialcompositionofboysandgirlsintwo-parenthouseholdsthateventuallysplitarealsosimilar.Inbothcases,thereareroughlyequalpercentagesofblackandwhiteindividualsinthesubsamples,andthepercentageofHispanicindividualsisrelativelysmall.Lastly,thepercentageofchildrenintwo-parenthouseholdsinwhichtheirparentsaremarriedisabouteighty-sixpercentforgirlsandnearlyninetypercentforboys.2.3EstimationThemaincontributionofthispaperistheestimationofthegendergapsintimeinvestmentsinsingle-motherhouseholds.Toestimatethegendergaps,IuseindividualÞxede!ectsregressions,includinginteractionsbetweenadummyforbeinginasingle-motherhouseholdwithmaleandfemaledummyvariables.Thegendergapisthedi!erenceinthecoe"cientsonthemaleandfemaleinteractions.Tit=$+"MáMiáMOit+"FáFiáMOit+"3áOtherit+Xitá"+ci+#it(2.1)Thelefthandsidevariableinequation(2.1),Tit,representssomemeasureofparentaltimeinvestmentsthatchildireceivedinwavet.FormostspeciÞcationstheinvestmentmeasuresaretheamountoftimethatchildispentwithhisorhermother/fatherfromtheweekday/weekendtwenty-fourhourtimediary,measuredinhours.InthemainspeciÞcation,Ireportestimatesfortheweekdayandweekendinvestments,aswellasatotalweeklyinvestmentconstructedasaweightedsumoftheweekdayandweekendinvestments.IconstructthetotalinvestmentbysummingtheweekdayinvestmentmultpliedbyÞvewiththeweekendinvestmentmultipliedbytwo.Foreaseofreportingandbecauseusingtheweeklymeasurebetterreßectsthee!ectsontotalinvestments,IfocusonreportingestimatesfortotalinvestmentsinalternatespeciÞcationsthatusethehoursmeasures.Ialsoincludeestimatesbasedonequation(2.1)thatreplacethehoursmeasureswithdummyvariablesindicatingwhetherthechildhadanypositiveinvestmentfromhisorhermother/father.Inthat39speciÞcation,theoutcomeofinterestisadummythatequalsonewhenthetimeinvestmentisgreaterthanzeroforthegiventimediary.6Forthecorrespondingtotalinvestmentmeasure,theoutcomeisadummyvariablethatisequaltooneifeithertheweekdayorweekendinvestmentispositive.Theindependentvariablesofinterestaretheinteractionterms,whereMiandFirepresentmaleandfemaledummyvariables,andMOitrepresentsadummyvariableindicatingwhetherchildiwasinasingle-motherhouseholdatwavet.Therearetwoothertypesofhouseholdstructurestoconsider.Livinginthesamehouseholdasbothparentsistheomittedcategory,andOtheritindicateswhetherchildiwasinsomeotherhouseholdtypeduringwavet.Thethirdcategory,Otherit,isconstructedtomakethecategoriesmutuallyexclusiveandcomprehensive.7Ifocusonestimatinggendergapsininvestmentsforthoselivinginsingle-motherhouseholds.Single-motherhouseholdsareofparticularinterest,becausechildrenaremorelikelytolivewiththeirmotherifthefamilyisbrokenup.Furthermore,iffathersinvestrelativelymoreinboysastheygetolder,nothavingtheirfatherpresentinthehouseholdcouldhinderdevelopmentforboys,evenifnotforgirls.Xitrepresentsavectoroftime-varyingobservablecharacteristicsincludingchildÕsageandage-squaredinteractedwithgender,thenumberofbiologicalsiblingsinthehousehold,dummyvariablesindicatingtheCDSwave,dummyvariablesindicatingthepresenceofstepparentsinoroutofthehousehold,andadummyvariableformarriageofparentsinthehousehold.8ThegenderspeciÞcagetermsareimportant,becauseinvestmentsarecloselyrelatedtoageanddi!erentiallybygender.9Childlevel,time-constantcharacteristicsareindicatedbyci,and#itindicatesaperiodspeciÞcerrorterm.Iestimateequation(2.1)usingindividuallevelÞxede!ects,sothatö"MisaÞxede!ectsestimatorofinvestmentsthatboysreceiveinsingle-motherhouseholdsrelativetothoseintwo-parenthouse-holds.Similarly,ö"Fisanestimatorforinvestmentsthatgirlsreceiveinsingle-motherhouseholds.6Fortheweekday/weekendestimates,Tit=1whentheweekday/weekendmaternal/paternaltimeinvestmentisgreatthanzero,andTit=0otherwise.Forthetotalinvestmentregressions,Tit=1wheneithertheweekdayortheweekendmaternal/paternalinvestmentispositive.7Otheritisequalto1ifthechildwasinasingle-fatherhoueshold,orinahouseholdwithneitherparent.Thismakestheestimatedcoe"cienthardtointerpretbutthesehouseholdstructuresarenotthefocusofthisstudy.8EstimatesthatalsoconditiononthedayoftheweekthatthediaryreferencesareincludedinTablesH.1andH.2oftheappendix.ResultsarenotsensitivetothisspeciÞcation.9InanalternatespeciÞcation,Iincludeasetofagedummyvariablesinteractedwithgender.TheresultsarerobusttothisspecÞcation.40Theparameterofinterestisthedi!erencebetweenthetwoinvestmentlevels,"Diff="M!"F.Whenö"Diff<0,thatsuggeststhatboysreceiverelativelylowlevelsofinvestmentsinsingle-motherhouseholds,andö"Diff>0suggeststhatboysinsingle-motherhouseholdarerelativelywello!intermsoftimeinvestments.BecauseIestimate"DiffusingÞxede!ects,itisnecessarytoviewsomeboysandsomegirlsinasingle-motherhouseholdduringonewaveandinatwo-parenthouseholdinanother.Theprocedureexplainedabovedoesnotrestrictthedirectionofthechangeinhouseholdstructure.Thosewhotransitionfromasingle-mothertotwo-parenthouseholdcontributetotheestimatesinthesamewayasthosewhogofromatwo-parenttoasingle-motherhousehold.However,wemightexpectthesetwogroupstobedi!erent.Omittedcharacteristicsandbehaviorcandirectlyinßuencetransitions,aswellastheleveloftimeinvestments.Theagesatwhichthechildisineachhouseholdstructureisalsorelatedtothedirectionofthetransition.Ifthesizeofthegendergapdi!erswithage,itcouldleadtoestimatingdi!erentgapsdependingonthedirectionofthetransition.Iincludeafollow-upanalysis,inwhichIsplitthesamplebyinitialhouseholdtype,andreportseparateestimatesforthesampleofthosewholivedwithbothparentsintheÞrstperiodandforthosewhodidnot.Inaddition,Iestimategendergapsbyageandracetoinvestigatewhetherthegapchangesacrossthosevariables.IalsodecomposethegapsintospeciÞcactivitiestodeterminewhattypesofactivitiesarethemaincontributorstothedi!erentialinvestmentlosses.Lastly,Iinvestigatedi!erentialchangesinparentalratingsofnon-cognitivebehavioraroundthechangesinhouseholdcomposition.Learningmoreabouthowparentaltimeinvestmentsfactorintothegenerationofnon-cognitiveskillsisaprimaryconcern.Iexaminedi!erentialchangesbygenderinexternalizingbehavior,internalizingbehavior,andpositivebehavior10aroundchangesinhouseholdstructure.Todothis,Iestimateequation(2.1)usingindividualÞxede!ectsandreplacingthetimeinvestmentmeasureswiththebehavioralmeasures.Whilethesedataarewell-suitedfor10EachofthethreeratingsisbasedonaseriesofquestionsaskedtothechildÕsprimarycaregiver.Externalizingandinternalizingbehaviorquestionsaskhowfrequentlythechildexhibitssomeexternalizing(i.e.actingout)orinternalizing(i.e.inwardnegativebehavior)andhavethreepossibleanswers:nottrue,sometimestrue,oroftentrue.Thedataarecodedsothathigherscoresmeanthatthechildexhibitsmoreproblematicbehavior.ThepositivebehaviorquestionsaskhowÒlikeÓthechildcertainbehaviors/charactersticsare(e.g.cheerful,notimpulsive),andareansweredonaonetoÞvescale,whereonemeansthebehavior/characteristicisÒnotatalllikethechildÓandÞvemeansitisÒtotallylikethechild.ÓAhigherscoreonthepositivebehaviorratingmeansthatthechilddisplayslessproblematicbehavior.41measuringparentaltimeinvestments,usingparent-ratedbehaviormeasuresmaybeproblematicbecauseparentalperceptionsofthechildÕsbehaviorcouldchangedi!erentiallybygenderaroundchangesinhouseholdstructure.2.4ResultsBeforepresentingtheestimatedgendergapsfromtheÞxede!ectsregressions,IwillbrießydiscussOLSestimatesofequation(2.1).TableD.2showsestimatesofthemaleandfemaleinteractionterms,aswellastheestimateddi!erencebetweenthemaleandfemaleinteractionterms,formaternalandpaternaltotal,weekday,andweekendinvestments.EstimatesinTableD.2areconditionalonasetofcovariates,includingmaleandfemaleinteractionswithageandage-squared.Theestimatedcoe"cientonthemaleinteractionwithsingle-motherhouseholdintheequationfortotalmaternalinvestmentsfromcolumn1,suggeststhatboysinsingle-motherhouseholdsreceive1.24hoursperweekmorefromtheirmothers,relativetochildrenintwo-parenthouseholds.Subtractingthecoe"-cientonthefemaleandsingle-motherhouseholdinteractiongivestheestimatedgenderdi!erenceininvestments.Againfromcolumn1,theestimatedgendergapis0.1hoursperweek,suggestingthatboysarerelativelywello!intermsofmaternalinvestments.Wecandecomposethetotaldi!erencebylookingattheweekdayandweekendgaps.Theestimatedweekdayandweekendgapsinmaternalinvestmentshaveopposingsigns.Fromcolumn2,theestimatedgapis-0.049,meaningthatboysinsingle-motherhouseholdsreceivedroughlythreefewerminutesinweekdayinvestmentsthangirls.Ontheotherhand,fromcolumn3,boysinsingle-motherhouseholdsreceivedroughlyelevenminutesmoreperweekenddaythangirls.Inthiscase,theweekendgapoutweighstheweekdaygap,andtheestimatedtotaldi!erenceispositive.Theestimatesforpaternalinvestmentsareshownincolumns4through6.Fromcolumn4weseethatboysinsingle-motherhomeshaverelativelylargedecreasesininvestments,comparedtogirls,estimatedat-7.5and-6.5hoursperweek,respectively,leadingtoanestimatedgendergapofaboutonehourperweek.TheconcernwithestimatingthegendergapbyOLSisthatitreliesheavilyoncross-sectionalvariation,butbothfamilystructureandgenderarelikelycorrelatedwithunobservabledeterminantsoftimeinvestments.TheÞxede!ectsestimator42ispreferablebecauseitrestrictsthecomparisontochangesininvestmentswithinindividualswhounderwentachangeinhouseholdstructurewithchangesininvestmentsforchildrenwhoremainintwo-parenthouseholds.Theremainderofthereportedestimatesofequation(2.1)areÞxede!ectsestimates.TableD.3reportsÞxede!ectsestimatesofthegendergapintimeinvestmentsbasedonequation(2.1)formaternal/paternaltotal,weekday,andweekendinvestments.ThespeciÞcationreportedinpanelAdoesnotincludeanycontrolvariables,andthespeciÞcationinpanelBincludesthefullsetofcontrols.ColumnfourofpanelAshowsestimatesfortotalweeklypaternalinvestmentswithoutcontrols.Theestimatedgapinpaternalinvestmentsis-1.92,meaningthatpaternalinvestmentsdropbynearlytwohoursmoreperweekforboysinsingle-motherhomesthantheydoforgirls.Afteraddingcontrols,whiletheestimatedcoe"cientsonthegender/single-motherinteractionschange,theestimatedtotalgapinpaternalinvestmentsremainssimilar.Forexample,theestimatedcoe"cientonthemaleinteractionwithsingle-motherhouseholdgoesfrom-10.3withoutcontrolsto-7.3afteraddingcontrols.Similarly,theestimatedcoe"cientonthefemaleinteractiongoesfrom-8.4withoutcontrolsto-4.9withcontrols.Inbothcases,theestimatessuggestthatpaternalinvestmentsdropforboysandgirlsaftergoingtosingle-motherhouseholds,butthedecreaseisrelativelylargeforboys.FrompanelB,theestimatedgenderdi!erenceintotalweeklypaternalinvestmentsis-2.36,whichissimilartotheestimatefrompanelA,suggestingthatpaternalinvestmentsdropbynearlytwoandone-halfhoursmoreperweekforboysinsingle-motherhomesthantheydoforgirls.ThatestimateisbotheconomicallysigniÞcant,astheestimatedgapismorethan20%ofaveragepaternalinvestmentsacrossgenderandhouseholdtypesduringwaveI,andstasticallyimportantwithastandarderrorof1.08.Thegenderdi!erenceisstrongestthroughweekdayinvestments,forwhichtheestimatedgap,fromcolumn5ofpanelB,is-0.4withastandarderrorof0.15.Thatequatestoroughly24minutesperweekdayandisabout35%ofaveragepaternalweekdayinvestmentsduringwaveI.Toputthesizeofthatestimateinperspective,considerFigureC.3again,whichgraphspaternalweekdayinvestmentsduringtheÞrstwaveoftheCDS.Averagepaternalinvestmentstoboysandgirlsintwo-parenthouseholdsgenerallyliebetweenoneandtwohours,dependingonage.Ofcourse,thereisaslightdownwardtrend,andthelinedropsbelowoneandhalfhoursby43aboutagesevenforbothboysandgirls.Theestimateddi!erenceof24minutesrepresentsabouttwentypercentoftheaveragepaternalweekdayinvestmentintwo-parenthouseholdsatthelowerendandfortypercentatthehigherend.Althoughtheweekdaygapdrivesabouteightypercentoftheestimatedtotalweeklygap,thegapinpaternalweekendinvestmentsisalsonegative,-0.22hours,butonlydrivesabouttwentypercentofthegapbecauseofitÕssmallermagnitudeandlowerweightinthemakeupofthetotalweeklyinvestmentmeasure.Thedi!erencebetweentheOLSandÞxede!ectsestimatesfortotalpaternaltimecanbeex-plainedbyexamininghoweachindividualcontributestoeachestimator.IntheOLSestimator,thosewhotransitionfromtwo-parenttosingleparenthouseholds,orviceversa,areintheomittedgroup(two-parenthousehold)inoneperiodandinadi!erentgroupinanother.WiththeÞxede!ectsestimator,theyareneverintheomittedcategory.Girlswhoeventuallygothroughatwo-parenttosingle-mothertransitionhavealowerbaselineinputthanboyswhomakethesametransition,becausepaternalinvestmentstogirlsintwo-parenthouseholdsdecreaseastheygetolder.Sincepaternalinvestmentsareverylowforbothgenderswhentheirfatherisnotinthehousehold,thewithinchilddropinpaternalinvestmentsaftergoingfromatwo-parenttosingle-motherhouseholdisrelativelylargeforboys.Fromcolumn3ofTableD.1,boysintwo-parenthouseholdsthateven-tuallysplitareage4.6andget16.7hoursinweeklypaternalinvestments.Ontheotherhand,girlsinthesamehouseholdstructure,fromcolumn7,areofasimilarage,4.9yearsold,butreceiveanaveragepaternalinvestmentofonly13.2hoursperweek.Whentheaveragepaternalinvestmentdecreasesdrasticallytonearzeroforbothgendersafterthetransitiontoasingle-motherhousehold,thereismoreroomforadecreaseinpaternalinvestmentsforboys.Anotherwaytoseethisistocompareaverageinvestmentsforboysintwo-parenthouseholdsthatneversplitwiththosewhoeventuallysplit,16.3and16.7,respectively,whicharequitesimilar,despitetheagedi!erenceinthetwogroups.Ontheotherhand,girlswhoarealwaysinatwo-parenthouseholdreceiveabout15hoursperweekinpaternalinvestments,butthoseinatwo-parenthouseholdthateventuallysplitsonlyreceiveabout13.2.Finally,considerthesimilarityintheestimatedchangeforboys,ö"M,-7.5hourswhenestimatedbyOLSand-7.3whenestimatedbyÞxede!ects.Thedi!erencebetweentheÞxede!ectsandOLSestimatesofö"Diffisthroughadi!erenceintheestimatesonthecoe"cient44forgirls,ö"F,whichare-6.5and-4.9whenestimatedbyOLSandÞxede!ects,respectively.Whiletheboy-girldi!erencesinpaternalinvestmentsarebotheconomicallyandstatisticallyimportant,itispossiblethatsingle-motherscompensatefortheextralossesbyincreasingtheirinvestmentstoboysrelativetogirls.Columns1-3ofTableD.3showtheestimateddi!erencesintotal,weekday,andweekendmaternalinvestments.FromcolumnstwoandthreeofpanelA,estimatedweekdayandweekendmaternalinvestmentsaresmallerforboysandgirlsinsingle-motherhouseholds.However,bothweekdayandweekendmaternalinvestmentsarerelativelylowforboys,leadingtoestimatedgapsof-0.19forweekdaysand-0.03forweekends.Fromcolumn1ofpanelA,theestimatedtotalweeklygenderdi!erenceinmaternalinvestmentsis-1.4,meaningthatboyssu!erlargerinvestmentlossesthangirlsofalmostoneandone-halfhoursperweek.Addingthefullsetofcontrolvariableschangestheestimatedcoe"cientsontheinteractionterms,butdoesnotleadtosigniÞcantchangesintheestimatedgaps.Column2ofpanelBshowsthattheestimatedgendergapinmaternalweekdayinvestmentsgrowsslightlyinmagnitudeto-0.207.Fromcolumn3ofpanelB,theestimatedgapinweekendinvestmentsisnow-0.003.Thetotalgapinmaternalinvestments,fromcolumn1ofpanelB,isabout-1.1,meaningthatboysinsingle-motherhomesreceivefewermaternalinvestmentsthangirlsinsingle-motherhomes,andthemagnitudeissimilartotheestimated-1.4hoursperweekfrompanelA.Thenegativeestimateonthegapintotalmaternalinvestmentssuggeststhatmothersdonotcompensate,butinsteaddecreasetheirinvestmentstoboysrelativetogirls.However,thestandarderrorfortheestimateddi!erenceisrelativelylarge,1.6,soweshouldnotdrawanystrongconclusionsbasedonthatdi!erence.TheanalysisinTableD.3doesnotrestrictthedirectionofthehouseholdtransition,includingchildrenwhogofromatwo-parenthouseholdtoadi!erenthouseholdtype,aswellasthosewhogofromnotlivinginatwo-parenthouseholdtolivingwithbothparents.However,thesetwotypesoftransitionsandfamiliescouldbequitedi!erentfromeachother.Onereasonthatwemightexpectheterogeneitybasedonthedirectionofthetransitionisthattheagethatthechildisineachhouseholdstructureiscorrelatedwiththedirectionofthetransition.Di!erentialgapsbyagecouldleadtodi!erentestimateswhensplittingthesamplebythedirectionofthetransition.TableD.4showsseparateestimatesbyinitialhouseholdstructure.Columns1and2showestimatesfor45totalweeklyinvestmentsforfamiliesthatwereintactintheÞrstwave,andcolumns3and4showestimatesforchildrenwithlessthantwoparentsinthehouseholdintheÞrstperiod.Fromcolumn2,theestimatedgenderdi!erenceintotalpaternalinvestmentsforthosewhowereinatwo-parenthouseholdintheÞrstwaveis-4.2hoursperweekwithastandarderrorof1.54.Restrictingtothesubsampleofindividualswhowereintwo-parenthouseholdsinwaveoneincreasesthemagnitudeoftheestimatedgapinpaternalinvestmentsbyalmosteightypercent.However,theestimatedgapinmaternalinvestmentsbecomespositive,0.688,afterrestrictingtothoseintwo-parenthouseholdsintheÞrstwave.Thepositivesignontotalmaternalinvestmentssuggeststhatmothersmightcompensatefortheextralossesthatboyssu!erinpaternalinvestments,buttheestimateisnoisy,andsmallerinmagnitudethanthelossinpaternalinvestments.Ifwetaketheseestimatesatfacevalue,thenboysloseanadditional4.2hoursperweekinpaternalinvestments,aboutseventypercentofwaveoneaveragepaternalinvestments,andmotherspartiallycompensateforthelosswithanadditionalfortyminutesperweek.Whiletheincreasepartiallyo!setstheloss,aboutseventeenpercentofit,boysstillsu!ersubstantiallylargerinvestmentlossesfromtransitioningtoasingle-motherhome.Columns3and4ofTableD.4displayestimateswhenrestrictingtothesubsampleofindividualswhowerenotinatwo-parenthomeintheÞrstwave.Therearefewermoversinthisdirectionandthestandarderrorsarerelativelylarge,buttheestimatessuggestthatthedirectionofthemovelikelymatters.Forexample,theestimatedgapinmaternalinvestmentsinthesubsamplewhowerenotintwo-parenthomesinwaveoneis-3.5hoursperweek.Thisestimateisoftheoppositesignfromthesubsampleofchildrenwhowereintwo-parenthomesintheÞrstwave,andrelativelylargeinmagnitude.Inotherwords,thissubsampleisdrivingthenegativeestimatesontotalmaternalinvestmentsinthefullsample.Theestimatedgapinpaternalinvestmentsisstillnegative,-.18,butmuchsmallerinmagnitude.Althoughthereappearstobesomeheterogeneitybasedonthedirectionofthetransition,theestimatesinthesampleofchildrennotintwo-parenthouseholdsintheÞrstwavearenoisierbecausetheyarebasedonarelativelysmallnumberoftransitions.Inadditiontoestimatingdi!erencesinparentaltimeinvestments,Iincludesupplementalesti-matesontheprobabilityofhavingapositiveinvestment.InthisspeciÞcation,Ireplacethenumber46ofhoursspentwiththemother/fatherwithadummyvariableforhavinganyinvestment,andesti-matethedi!erencesusinglinearÞxede!ectsregressions.Fromcolumn1ofTableD.5,theestimatedchangeintheprobabilityofreceivinganonzeromaternalinvestmentaftertransitioningtoasingle-motherhouseholdis0.040forboys,and0.068forgirls,leadingtoanestimatedgendergapof-0.028infavorofgirls.Inotherwords,afterthetransition,girlsseearelativelylargebumpintheirproba-bilityofhavinganyinvestmentfromtheirmothers.However,theestimateisnoisywithastandarderrorof0.033.Interestingly,thesignontheweekdayandweekendmaternalinvestmentsestimatesaredi!erent,withtheestimatedgapinreceivinganyweekdayinvestmentfavoringgirls,andtheestimatedgapinweekendinvestmentsfavoringboys.Forpaternalinvestments,theestimatedgapintheprobabilityofreceivinganyweekdayinvestmentis-0.08,meaningthattheyreceiveaboutan8percentagepointmoredrasticchangeintheprobabilityofrecevingapositiveinvestmentfromtheirfathersonweekdays.Theanalogousweekendestimateis-0.047.However,theestimatedgapinthetotalweeklyprobabilityismuchsmallerinabsolutevaluethanboththeweekdayandweek-endestimates,only-0.01.Thatsuggeststhatboysseearelativelylargedecreaseintheprobabilityofreceivinganinvestmentonanygivenday,butthechangeintheirprobabilityofreceivinganinvestmentatsomepointthroughouttheweekissimilar.Splittingthesamplebyinitialhouseholdcompositionrevealsthedi!erenceinprobabilityofreceivingapositivepaternalinvestmentbetweenthosewhowereandwerenotlivingwithbothparentsintheÞrstwave.FromPanelAofTableD.6,thosewholivedwithbothparentsintheÞrstwaveseearelativelylargedecreaseintheprobabilityofreceivingapositivepaternalinvestmentthroughouttheweek,withanestimateddi!erenceof-0.065.Theestimatedgapwhenlookingatweekdayorweekendinvestmentsonlyis-0.11inbothcases,suggestingthatonanygivenweekdayorweekenddayboyshaveanelevenpercentagepointlargerdecreaseintheprobabilityofreceivingapositivepaternalinvestment.Ontheotherhand,theestimatedgendergapsinpaternalinvestmentsarepositivefortotal,weekday,andweekendinvestmentsforthesubsamplethatlivedwithlessthanbothparentsinwaveI,meaningthatboysinsingle-motherhouseholdswererelativelywello!bythismeasure.However,theestimatesarenoisyinthatsubsamplewithanestimatedweeklygapof0.094andastandarderrorof0.089.472.4.1HeterogeneityinGenderGapsNext,Iconsiderpotentialheterogeneouse!ectsbyageandrace.Becauseofthestrongcorrelationbetweeninvestmentsandage,anddi!erentialagetrendsbygenderandhouseholdstructure,onemightexpectthattheboy-girlinvestmentgapinsingle-motherhouseholdsdependsontheageofthechild.Forexample,FiguresC.5andC.6showhowtheproportionoftotalparentalinvestmentsthatcomefromthefatherincreasewithageforboysintwo-parentshouseholds,whichsuggeststhatpaternalinvestmentsbecomeincreasinglyimportantforboysastheygetolder.FiguresC.3andC.4provideinsighttothattrendbyshowingthatgapsariseinboyandgirlinvestmentswithage,withfathersspendingrelativelymoretimewithboys.Furthermore,thesteepdeclineinmaternalinvest-mentsfromFiguresC.1andC.2meansthatpaternalinvestmentsbecomeincreasinglyimportantinthemakeupoftotalinvestmentsforboys,butlesssoforgirls.Tocomparethegendergapsacrossages,Igrouptheindividualsintofouragebins(0-5,6-10,11-15,and16andover)andestimatethegapforeachbin.PanelAofTableD.7showstheestimatesfortotalmaternalinvestmentsbyagebin.Thesmallestgap,-0.6hoursperweek,isinthe6-10yearoldbin.Fromthere,thegapsizeincreasestoabout-1hourperweekinthe11-15yearoldbin,andthento-2.2hoursperweekinthe16andoverbin.Thereissomesuggestiveevidencethatrelativelossesforboysincreaseastheygetolder,buttheestimatesarenoisy,allwithstandarderrorsof1.8orhigher.PanelBofTableD.7showsestimatesforpaternalinvestmentsbyagebin.Amoreclearpatternemergesinthepaternalinvestmentgaps,withboyssu!eringgreaterinvestmentlosseswithage.Thesmallestestimateis-0.4hoursperweekforages0-5,buttheyincreaseinmagnitudewitheachagebin.Therestoftheestimatedgapsare-1.8hoursperweekfor6-10yearolds,-3.3forthe11-15yearolds,and-3.6forthe16andoveragegroupwithp-valuesof0.12,0.006and0.02,respectively.Thepatternintheseestimateddi!erencessupportstheideathatpaternalinvestmentsbecomeincreasinglyimportantforboysastheygetolder,leadingtorelativelylargeinvestmentlossesduringadolescence.Ialsoestimatedi!erentialgapsbyrace,focusingonthewhiteandblacksubsamples,asthosetworacesmakeupmorethanninetypercentofthesampleofchildrenwhomadeatransitionfromatwo-parenttolessthantwo-parenthousehold.FromTableD.8,theestimatedgendergapintotal48maternalinvestmentsforwhitechildreninsingle-motherhouseholdsis-3.1hoursperweekwithastandarderrorof2.5.Theestimateddi!erencesforblackchildreninsingle-motherhouseholdsis-1.4hoursperweekwithastandarderrorofabout2.3.Bothofthepointestimatesarelargerinmagnitudethantheoverallestimate,buttheestimatesarenoisyandnotstatisticallydi!erentfromzeroatanystandardsigniÞcancelevel.Theestimateddi!erenceinpaternalinvestmentsforwhitechildreninsingle-motherhouseholdsisabout-4hoursperweek,andisstronglystatisticallysigniÞcantwithastandarderrorof1.6.Thatestimateissimilarinmagnitudetotheoverallestimatedgapforthesubsampleofchildrenwhowereintwo-parenthouseholdsintheÞrstwave.Theestimatefortheblacksubsampleisabouthalfthesize,-2hoursperweek,whichissimilartotheestimatedgapacrossthefullsample,andislessstatisticallysigniÞcant,withastandarderrorof1.4.Whileallfouroftheestimatedgapsarenegative,thedi!erencesprovidesomeevidencethattheextralossessu!eredbyboyswhoarewhitearelargerintermsofbothmaternalandpaternalinvestments.Parentalinvestmentsarealsolargeronaverageacrossthesampleofwhitechildren,whichcoulddrivepartofthedi!erence.Forexample,theaverageweeklymaternalinvestmentforwhitechildrenintwo-parenthouseholdsisabout3hoursmoreperweekthanthatforblackchildrenintwo-parenthouseholds(21.5hoursperweekto18.2hoursperweek),andthedi!erenceinweeklypaternalinvestmentsissimilar(14.5hoursperweekto10.9hoursperweek).2.4.2CompositionoftheInvestmentGapsThetimediarydataincludedescriptionsoftheactivitiesthatchildrenparticipatein,makingitpossibletodecomposethetotalgapbyactivity.TableD.9displaysÞxede!ectsestimatesofequation(2.1)byactivitycategory.11Column1providesestimatesontotalmaternalinvestments.Estimatedgapsinmaternalinvestmentsarepositiveonpassiveleisure(0.22hoursperweek),e.g.watchingtelevision,tendingtoneeds(0.1hoursperweek),andchildcare(0.18hoursperweek),meaningthat11Thisisnotacomprehensivesetoftheactivities,buttheyaccountforthemajorityofthedi!erences.TheestimatesforthegapsinpaternalinvestmentsdisplayedinTableD.9adduptoalittemorethanthetotaldi!erence,-2.5hoursperweekcomparedto-2.3,meaningthattheomittedactivitiesadduptoaslightlypositivenumber.Thedisplayedestimatesformaternalinvestmentsadduptoabout-0.7,lessthantotalgapof-1.1,sothegapsintheomittedactivitiessumtoabout-0.4.49mothersincreasetimewithboysrelativetogirlsinthoseactivitiesaftertransitioningtoasingle-motherhousehold.Ontheotherhand,maternalinvestmentsofactiveleisuredecreasesubstantiallymoreforboysinsingle-motherhouseholds.Theestimateof-0.85hoursperweekinactiveleisureisthelargestinmagnitudeofanyactivity.Boysalsoexperiencerelativelylargereductionsinpaternalinvestmentsinleisure,-0.72and-0.75hoursperweekinpassiveandactiveleisure,respectively.Ingeneral,boysseeamuchlargerdropo!inleisureactivitywiththeirparentsthangirlsdoaftertransitioningtoasingle-motherhousehold.TherestoftheestimatesforpaternalinvestmentsdisplayedinTableD.9arealsonegative,butgenerallyofsmallermagnitudethantheestimatesforleisureactivity.Thenexttwolargestarethedi!erencesintendingtoneeds,-0.48hoursperweek,andentertainment,-0.21hoursperweek.2.4.3HouseholdStructureandChildBehaviorOnereasonforanalyzinggendergapsintherelationshipbetweenhouseholdstructureandparentalinvestmentsistoassessthepossibilitythatchangesininvestmentlevelscontributetonon-cognitiveskillforumlation,explainingsomeofthedi!erencesweseeinoutcomesbetweenboysandgirls,e.g.behaviorandeducationalattainment.Ifthedocumentedgapsintimeinvestmentlossesfromtransitioningtosingle-motherhouseholdscontributetocontemporaneousbehavioralissues,thenwemightexpecttoseechangesinbehaviorthatmatchthedi!erentialinvestmentlosses.Inotherwords,iftheadditionalinvestmentdecreasesthatboyssu!erisbadfortheirbehavior,thenwewouldexpectthattoshowupinmeasuresoftheirbehavior.Iestimatethegenderdi!erentialchangesinbehaviorfromtransitioningtosingle-motherhouseholdsbyreplacingtheoutcomevariablesinequation(2.1),parentaltimeinvestments,withmeasuresofchildbehavior.Onemajorweeknessofthisexerciseisthatthebehavioralmeasures,takenfromtheCDS,areparent-reported.ThisisproblematicbecauseparentalperceptionsoftheirchildÕsbehaviorcouldchangedi!erentiallybygenderaroundhouseholdstructuretransitions.Nonetheless,Ishowresultsfromtheanalysistodemonstratetheconcept.TableD.10showsresultsfromÞxede!ectsestimationofequation(2.1),replacingtheparentalinvestmentoutcomeswiththreeparent-ratedbehaviormeasures:externalizingbehavior,internaliz-50ingbehavior,andpositivebehavior.Eachoftheratingsaretheresultofasequenceofquestionsthattheparentanswersaboutthechild.Therearesomeminordi!erencesinthequestionsacrossthethreewaves,soIusetherawresponsestostandardizethesetofquestionscontributingtoeachbehavioralmeasure.Eachofthequestionsrelatedtoexternalizing/internalizingbehaviorarean-sweredonaonethroughthreescale,andthequestionsforpositivebehaviorareonaonethroughÞvescale.Isummedtheanswersandstandardizedthetotalscoreswithineachwave.Higherscoresinexternalizingandinternalizingbehaviorrepresentmoreproblematicbehavior,butahigherscoreonthepositivebehaviorsmeasurerepresentslessproblematicbehavior.Interestingly,Theestimatedcoe"cientonthemaleinteractionwithsingle-motherhouseholdintheequationforexternalizingbehaviorisnegative,-0.11,meaningthatparentsratetheirboysasdisplayinglessexternalizingbe-haviorfollowingthetransition.Ontheotherhand,thecoe"cientonthefemaleinteractiontermispositive,0.16,meaningthatparentsstatethattheirgirlsareactingoutmorefollowingthetransition.Theexternalizingbehaviorratingsforboysbecomemuchmorefavorablefollowingthetransitiontoasingle-motherhousehold,relativetogirls,andtheestimatedgapinchangesinexternalizingbehavior,fromcolumn1,is-0.27standarddeviations.TheresultisalsostatisticallysigniÞcant,withap-valueof0.02.Apparently,thechangeinhouseholdstructureisnotleadingtomoreexternalizingbehaviorforboys,accordingtotheparent-ratedmeasure,likewemighthaveexpectedbasedontheadditionalinvestmentlosses.Theestimatedgapsintheothertwomeasuresaremuchsmallerinmagnitude,-0.007forinternalizingbehaviorand0.11forpositivebehavior.Theestimatedgapinexternalizingbehaviorissurprising,butthisgapshouldbeestimatedusingmeasuresthatarenotbasedonparentratingsinfuturework.2.5ConclusionGendergapsinnon-cognitiveskillsamongadolescentshavebeendocumentedinrecentliterature,butdeterminingthemechanismsthatleadtothesegapsisstilllargelyanopenquestionBertrandandPan[2013],Jacob[2002].Thesegapscouldariseforseveralreasons,includinggenderdi!er-encesinreturnstoandlevelsofinputsinsingle-motherandtwo-parenthouseholds.Bertrandand51PanidentiÞeddi!erencesinreturnstoinputsandfamilystructureasmajorcontributingfactorsinexplainingthisgap,butsuggestedthatdi!erencesinquantityofinputsplaysamuchsmallerrole.Becauseinputscancomeinmanyforms,mostofwhicharecorrelatedwithhouseholdstructureanddi"culttomeasureandinterpret,itisdi"culttodisentaglemechanismsleadingtonon-cognitivedi!erences.ThispaperusesdatafromthePanelStudyofIncomeDynamicsandtheChildDe-velopmentSupplementsurveytoobtaindirectmeasuresofparentaltimeinvestmentsandconsidergenderdi!erencesininvestmentsaroundchangesinhouseholdcomposition.Oneadvantagetothisapproachisthattimeinvestmentsarerelativelystraightforwardinthattheyaresimplymeasuresoftheamountoftimeparentsspendwiththeirchildren,basedontwenty-fourhourtimediariescollectedintheCDS.Findingdi!erentialinvestmentlossesbygendersuggeststhatlivinginasingle-motherhouseholdcouldhavealargenegativeimpactonthequantityofinvestmentsthatboysreceiverelativetogirls,andinturncouldcontributetothegendergapinnon-cognitiveskills,speciÞcallyforthoseinsingle-motherhouseholds.Usingchild-levelÞxede!ects,Iestimatethedi!erencesintimeinvestmentsthatarisefromlivinginasingle-motherhousehold.AlthoughbothboysandgirlsseesigniÞcantreductionsinparentalinvestments,boysexperiencelargerdecreases.Theextralossintotalweeklypaternalinvestmentsamountstoabout2.3hoursperweek,andisstrongestthroughweekdayinvestments,24minutesperweekday,whichaccountforroughlyeightypercentofthetotalloss.TheseÞndingsareeconomicallysigniÞcant,equatingtomorethantwentypercentoftheaveragepaternalinvestmentsintheÞrstwaveofthedata.Thereisnostrongevidencethatmotherscompensatefortheextralossbyincreasinginvestmentstoboysfollowingachangeinhouseholdstructure,relativetogirls.IalsoÞndthattheadditionallossesforboysaregenerallyincreasingwithage,withtheestimatedgapintotalpaternalinvestmentsover3.3hoursperweekforboysduringadolescence.Decomposingthegapinpaternalinvestmentsshowsthattheadditionallossesforboysarelargestinleisureactivities,whichaccountforaboutsixtypercentofthedi!erence.Allofthisevidenceconsideredtogetherwithexistingresearchsuggeststhattimeinvestmentsareanotherpotentialmechanismthatcouldhelpexplainthenon-cognitiveskillgendergap.However,usingparent-reportedratingsofexternalizingandpositivebehaviorfromtheCDS,IÞndthatparents52ratetheirgirlsashavinghigherlevelsofexternalizingbehaviorandlowerlevelsofpositivebehaviorwheninsingle-motherhouseholds,relativetoboys.Futureresearchshouldconsideramoredirectlinkbetweentimeinvestmentsandoutcomes,focusingonmeasuresofnon-cognitiveskillsthatarenotbasedonparentratings.53Chapter3PrecisionforPolicy:CalculatingStandardErrorsinValue-AddedModels3.1IntroductionEarlystudiesonvalue-added(VA)focusedsimplyonthepossibilityofmeasuringteachere!ective-nessSandersetal.[1997],Raudenbush[2004],Rocko![2004],Rubinetal.[2004],Rivkinetal.[2005],Aaronsonetal.[2007].Thesepavedthewaytoseveralstudiesinvestigatingthepotentialforbiasincommonlyusedvalue-addedestimators(e.g.Ballouetal.[2004],McCa!reyetal.[2004],KaneandStaiger[2008],Ballou[2009],BriggsandWeeks[2009],Harris[2009],IshiiandRivkin[2009],KoedelandBetts[2010],LockwoodandMcCa!rey[2009],ReardonandRaudenbush[2009],GoldhaberandHansen[2013],Rothstein[2008],Sassetal.[2014],Corcoranetal.[2011],KoedelandBetts[2011],Condieetal.[2014],BallouandCavalluzzo[2012],GoldhaberandChaplin[2015],Goldhaberetal.[2014],Kaneetal.[2013],Chettyetal.[2014],Guarinoetal.[2015].Whilemuchattentionhasbeengiventoobtainingthevalue-addedestimates,andassessingpotentialbiasinthem,therehasbeenlessfocusonobtainingthecorrectconÞdenceintervalsforthevalue-addedestimates.Precisionoftheestimatesshouldbeconsideredwhenusingtheminpractice,especiallyinhighstakessettings.Forinstance,ifteachervalue-addedestimatesareusedtoidentifythebottomÞvepercentofteachersforsanctionsorlossofemployment,howconÞdentarewethattheteachersinourestimated6thpercentileareactuallymoree!ectivethanthoseinthe4thpercentile?Evenwithunbiasedestimates,weneedthecorrectconÞdenceintervalstoaddressthisquestionandtounderstandwhetherteachersinthetailsofthee!ectivenessdistributionÑwhichisoftenthetargetforpolicyÑaresigniÞcantlydi!erentfromthoseinthemiddleofthedistribution.Suchinferencequestionshavebeenconsideredinalimitednumberofstudies.Raudenbush(2009)54discussesinferencespeciÞcallyregardingboththeadaptivecenteringrandome!ectsapproachpro-posedinthepaperandcomparableÞxede!ectsapproaches.Twootherstudieshaveaddressedimprecisionasitrelatestodistinctionfromthemeanteachere!ectaswellastheabilityofestima-torstodistinguishbetweenanyparticularranks(McCa!reyetal.[2004],Lockwoodetal.[2002],respectively).Usingsimulations,Lockwood,Louis,andMcCa!rey(2002)usemeansquarederrorasametric,showingthatdistinguishingbetweenrankscanbedi"cult,especiallyinthetailsofthedistributionofteacherquality.OurpaperÞllsagapinthepriorresearchbycomparingdi!erentmethodsforcomputingstan-darderrorsandconÞdenceintervalsfortheOLS-Lagestimator.Usingsimulateddataonstudentsandteachers,weexaminethesensitivityofthestandarderrorsontheteachervalue-addedestimatestothemethodofcalculationunderseveralstudentgroupingandclassroomassignmentscenarios.Westudyhowdi!erentstandarderrorsbehavebycomparingthevariabilityofestimatesfromthesimulateddatawiththesizeofthecomputedstandarderrors.WealsoevaluatehowwellconÞdenceintervalscalculatedusingtraditionalmethodsperformbycountinghowfrequentlytheimpliedcon-Þdenceintervalsaroundtheestimatedteachere!ectincludethetrueteachere!ect.OursimulationsshowthatrelyingonstandarderrorsandconÞdenceintervalscalculatedusingtraditionalmethodscouldbemisleadingwhenestimatingteacherÞxede!ectsusingOLS-Lag.Inparticular,whenstudentsaregroupedonunobservablecharacteristics,evenunderrandomassign-mentofclassrooms,traditionalstandarderrors,suchasthoseobtainedusingastandardclusterrobustvarianceestimator,tendtounderreporttheestimatorvariabilitywhenvalue-addedestimatesarebasedonasmallnumberofstudentcohortsperteacher.Consequently,thecorrespondingconÞ-denceintervalsover-rejectthetrueteachere!ects.Furthermore,thetendencytoover-rejectthetrueteachere!ectsisworsewithfewercohortsbecauseregularlyreportedconÞdenceintervalsusethestandardnormaldistributiontoobtaincriticalvalues,andthisturnsouttobeapoorapproximationwithsmallnumbersofcohorts.WeproposeanothermethodforobtainingconÞdenceintervalsthatbothallowsforunrestrictedcorrelationofwithinclassroomerrorsandusesappropriatecriticalval-uesfromt-distributions.Thismethodworkswellinthesimulations,producingconÞdenceintervalswithnearlyperfectcoverageinallcaseswithunbiasedestimators.55Toexpandonthesimulationevidence,weanalyzeadministrativedatafromsixdistrictsinalargestate.Foreachdistrict,wecomparestandarderrorsandconÞdenceintervalsfromourproposedmethodwiththosefromtheregularlyusedmethods.Intwodistricts,weÞndthatthestandarderrorscomputedwithourproposedcohort-by-cohortmethodaresystematicallylargerthanthosefromtheothermethods,suggestingpotentialover-rejectionfromusingtypicalmethods.Moreimportantly,weobserveconsistentdivergenceinthecomputedconÞdenceintervals.Onaverage,weÞndthattheconÞdenceintervalsfromtheproposedmethodaremuchwiderwithasmallnumberofcohorts,makingitdi"culttodistinguishanyquantileoftheteachere!ectivenessdistributionfromanother.Consistentwithourtheoreticalpredictionsandsimulationresults,averagewidthsoftheconÞdenceintervalsconvergewithincreasingcohorts,andaresimilartotraditionallyestimatedconÞdenceintervalsonaveragewithsevencohorts(themaximuminourdata).Finally,weillustratetherelevanceofconÞdenceintervalsinpolicysettingsbycomparinghowwellthedi!erentconÞdenceintervalsallowustoseparateteachersfromdi!erentpointsintheVAdistribution.Thepaperproceedsasfollows.ThenextsectionprovidesabriefdiscussionoftheissuesthatarisewhenconductinginferenceonOLS-Lagteachervalue-addedestimates.Thethirdsectiondescribesoursimulationdesignandanalysisandpresentssimulationresults.Section4describestheadministrativedatathatweuse,themethodsusedtoanalyzetheadministrativedata,andpresentsresultsofthatanalysis.Section5concludes.3.2DiscussionThenestednatureofstudentleveladministrativedataandresultingpotentialforerrorcorrelationleadtoambiguityinhowtoconductinferenceonteachervalue-addedestimatesobtainedfromOLSregressions.Therearetwoissuesthatariseinconductinginferenceonteachervalue-addedestimatesasaresultoferrorcorrelation:obtainingstandarderrorsandusingpropercriticalvalues.Thereareseveraloptionsforcalculatingrobuststandarderrorstoaddressprecision.Forinstance,researchersmightwishtoallowerrorcorrelationwithinstudent,teacher,orschoolbyusingacluster-robustvarianceestimator.Eachofthesechoicescangivedi!erentresults,andknowingwhichiscorrectis56noteasy.Thesecondissue,obtainingpropercriticalvalues,isparticularlyrelevantwhenestimatingvalue-addedfromasmallnumberofclassroomsperteacher.IfateacherÞxede!ectisbasedonasmallnumberofcohorts,thenthereiswhatamountstoasmallsampleissueDonaldandLang[2007].WhenconstructingconÞdenceintervalsforvalue-addedestimates,wewouldliketorelaxtheas-sumptionsabouterrorindependencetoallowforunrestrictedcorrelationwithinclassrooms.ThereareseveralreasonsonemightexpecterrordependencewithinateacherÕsclass,butwefocusonthecaseofnonrandomgroupingofstudentsintoclassrooms.Evenifclassroomsarerandomlyassignedtoteachersandtheestimatorsareunbiased,groupingstudentsonunobservablecharacteristicscom-plicatesinferencebyincreasingresidualvarianceacrossclassroomsandreducingtheamountofindependentinformationavailabletoestimateeachteacherÞxede!ect.Intheory,clusteringatthelevelofindependenceproducesconsistentstandarderrorsbyseparatingsetsofdependenterrorsintoindependentpiecesofinformation.Forexample,iferrortermsareindependentacrossclassrooms,thenclusteringattheclassroomorcohort-schoollevelwillresultinconsistentestimators.However,theasymptoticjustiÞcationforconsistencyinthiscaseworksasthenumberofcohortsperteacherincreases,whichislessusefulconsideringthatteachere!ectivenessistypicallyestimatedusingonlyafewyearsofdata.Asinthebroadergroupeddataliterature,constructingconÞdenceintervalsthusiscomplicatedwhenvalue-addedisestimatedfromasmallnumberofclassroomsperteacher.1Onekeyfeatureofteachervalue-addedseparatesitfrommanyothersettingsÑinthevalue-addedcontextweareestimatingapotentiallylargenumberoftreatments,eachapplyingtoarelativelysmallnumberofindividuals.Therefore,addingmoreteachers(orschools)ise!ectivelyaddingnewtreatmentsratherthanaccumulatinginformationoneachtreatment.Thiscontrastsoft-studiedscenariosthatinvolveaggregatetreatmentcases,whereasmallnumberoftreatmentsapplytoalargenumberofindividuals.Theissueinthevalue-addedcaseisthatusingavarianceestimatorthatallowsforarbitraryerrordependencewithingroup(e.g.,withinclassroom)doesnotnecessarilyleadto1Ourdiscussionisaimedattheelementaryschoolsetting,whereeachteacherhasoneclassroomperyear.Howthiscarriesovertothemiddleorhighschoolsetting,whereteachershaveseveralclassroomsperyear,dependsontheindependenceassumptionthattheresearcheriswillingtomake.57correctconÞdenceintervalswhenthenumberofclassroomsperteacherissmall.ConÞdenceintervalperformancecouldbehinderedthroughboththevarianceestimatesandcriticalvalueswhenthenumberofclassroomsperteacherissmall.Thisisproblematicifschooldistrictswanttomakedecisionsbasedonteacherperformanceoverashortperiodoftime,whichisoftenthecase.Oursimulations,whichwedescribeindetailinthenextsection,investigatetheperformanceofstandarderrorsandconÞdenceintervalsobtainedfromseveraldi!erentmethods.Wegeneratestudentleveldata,estimateteachervalue-addedusingOLS-Lag,andconstructmultipleconÞdenceintervalsaroundtheteachervalue-addedestimatesunderseveralscenariosofstudentgroupingandclassroomassignmenttoteachers.WeevaluatethequalityofthestandarderrorsbycomparingaveragestandarderrorswithstandarddeviationsofestimatedteacherÞxede!ects(i.e.,teachervalue-added).WeevaluatethequalityoftheconÞdenceintervalsbycalculatingcoverageratesof95%conÞdenceintervalsimpliedbyeachmethod.2Weuseadministrativedatatocheckwhetherpatternsfoundinthesimulationscarryoutinthedata,andevaluatethedi!erencesinthemethodsforconstructingconÞdenceintervalsbyexaminingtheabilityofthedi!erentconÞdenceintervalstodistinguishteachersfromdi!erentpercentilesofthevalue-addeddistribution.WecomparetheperformanceacrossÞvedi!erentstandarderrorandconÞdenceintervalcalcula-tions.TheÞrstisthenaŁvecalculationwithnocorrectionforheteroskedasticityorwithingrouperrorcorrelation.WealsocalculatethreesetsofstandarderrorsfromatypicalclusterrobustvarianceestimatorandusestandardnormalcriticalvaluestoconstructconÞdenceintervals.Thecluster-inginthesecasesisattheclassroom,cohort-school,andschoollevel.Finally,weproposeanothermethodthatinvolvescalculatingateachervalue-addedestimateforeachclassforeachteacherusingOLS-Lagregressionsforeachyear.Thenweusetheaverageofthepreliminaryestimatesastheteachervalue-addedestimateandconstructconÞdenceinterval,teacherbyteacher,usingcriticalvaluesfromthet-distributionwithdegreesoffreedomequaltothenumberofclassrooms(forthatteacher)minusone.Therearetwoadvantagesoftheproposedmethod;ite!ectivelydividestheerrortermsintoindependentpiecesofinformation(similartoclassroom-levelclustering)andcor-rectlyadjuststhecriticalvaluesaccordingtothenumberofindependentpiecesofinformation,i.e.,2Thecoveragerate,discussedinmoredetailbelow,referstothepercentofthe500simulationreplicationsforwhichthe95%conÞdenceintervaloftheestimatedteachere!ectcontainsthetruee!ect.58thenumberofclassroomstheteacherhastaught.Wealsoincluderesultsfromschoollevelclustering.Theoretically,though,clusteringattheschoollevelisproblematicbecauseincreasingthenumberofschoolsdoesnotincreasetheamountofinformationperteacher,butratherincreasesthenumberofparameterstoestimate.Inotherwords,clusteringattheschoollevelassumesfewerindependentpiecesofinformationareavailablethanparameterstoestimate.3.3Simulations3.3.1SimulationDesignWestartbysimulatingdataonstudentsandteachers,usingthefollowingtestscoredatageneratingprocess(DGP).Ai,t=%Ai,t!1+"i,t+ci+csi+#i,t(3.1)whereAi,tandAi,t!1representthecurrentandlaggedtestscore.Allobservationsarefromthesamegrade,soweobserveeachstudentonlyonetime.Theteachere!ectsarerepresentedby"i,t.AnindividualstudentÞxede!ect,cohort-schoolÞxede!ect,andrandomerrorsarerepresentedbyci,csiand#i,t,respectively.TableE.1summarizesoursimulationparameterchoices.Eachcomponentofthecurrenttestscoreisgeneratedfromanormaldistribution.Thestandarddeviationsofthebaselinescore(i.e.,thelaggedscore),teacherÞxede!ect,andrandomerrorareÞxedat0.25,0.25,and1,respectively.3IntheÞrstcasepresented,thereisnocohort-schoolÞxede!ect,andthestudentÞxede!ectsaredrawnfromastandardnormaldistribution.4Inthesecondcase,boththestudentÞxede!ectandthecohort-schoolÞxede!ectaredrawnfromanormaldistributionwithstandarddeviationof0.25.TheteacherÞxede!ectsaredrawnfromanormaldistributionwithameanof0.5.53ThepercentoftotalvariancefromteacherÞxede!ectsbasedonourchoicesisaroundthelowerendofthatconsideredinGuarinoetal.[2015],andthepercentfromstudentÞxede!ectsislargerthanwhattheyconsiderintheÞrstcaseandlowerinthesecondcase.4WehavedonethesameanalysiswiththestandarddeviationofthestudentÞxede!ectequalto0.5.Thischangessomeresultsquantitatively,butthegeneralpatternsarethesame.5Basedonthesevalues,thevarianceisalittleover1attheleast,andover2atthemost,dependingonthe59Simulationsarerestrictedtoasinglegrade(withcurrentandlaggedstudenttestscores)tosimplifytheprocess,andaredesignedtoreßecttheelementaryschoolsetting.Studentsdonotrepeatgradessowehaveexactlyoneobservationperstudent.ThemainresultsarebasedonsimulateddataincludingÞveschools,withtwoteachersperschool,and20studentsperclass.WefocusonasmallnumberofschoolsandteachersbecausethereisnogainfromaddingschoolstothesimulationsÑrecall,addingschoolsonlyaddsnewteachers,ortreatments.Informationusefulforestimatingteachere!ectsaccumulateswiththenumberofstudentstheteacherhastaught,somorestudentsperclassormorecohorts(years)ofstudents,butnotwithanincreaseinthenumberofschools.Forthisreason,wefocusmoreonhowt-statisticschangewithincreasingthenumberofstudentcohortsthanonhowtheychangeasmoreteachersandschoolsareadded.6Weconsiderthreegroupingscenariosreßectinghowstudentsmightbegroupedintoclassroomswithineachcohortateachschool.Thesearerandomgrouping(RG),dynamicgrouping(DG),andheterogeneitygrouping(HG).DynamicgroupingisbasedonstudentsÕlaggedtestscores,Ai,t!1,whichplaceshighachievingstudentstogether(andsimilarlyforlowachieving,etc.)tosimulatetrackingorabilitygrouping.7Heterogeneitygrouping(HG)isbasedonanunobservablestudentÞxede!ect,ci,whichmightrepresentanabilitylevelorbehavioralproÞlethatisobservableto,say,aprincipal,butisnotappropriatelycapturedinpriortestscores.Wefocusonrandomassignment(RA)ofclassroomstoteachers,butalsoconsiderpositiveassignment(PA)8,inwhichcasetheclassroomswiththehigheraverageAi,t!1fordynamicgrouping,orhigheraverageciforheterogeneitygrouping,areassignedtotheteacherwiththehigherÞxede!ect.Ineveryscenarioweconsider,studentsaregroupedintotwoclassroomsbasedononeofthethreepossiblegroupingmechanisms,RG,DG,orHG,andthenclassroomsareassignedtothetwoteachersintheschoolrandomly(RA)orpositively(PA).9IneachcasepresentedherethereisnocorrelationbetweenthelaggedscoreandstudentÞxede!ect,corr(Ai,t!1,ci)=0.10distributionsusedforthestudentandcohort-schoolÞxede!ects.6Weincluderesultswith50schoolsinonecasetoshowthattheresultsdonotchange.7GroupingonAi,tandciisimperfectinthesensethatitisnotthecasethatthetophalfofthedistributionisplacedinoneclassroomandthebottomhalfintheother.8Weconsiderpositiveassignmenttodemonstratewhathappenswithabiasedestimator.9Guarinoetal.[2015]provideamorethoroughdiscussionoftheseandotherpossiblegroupingmechanisms.10Wehavedonealternateanalysesusingcorr(Ai,t,ci)=0.5,andresultsaresimilar.60Thetwononrandomgroupingmechanismshavedi!erentimplicationsforthevalue-addedesti-matesthemselves,asonemechanisminvolvesgroupingbasedonanincludedcontrolvariable(Ai,t!1),whiletheotherisbasedonanomittedvariable(ci).Groupingonanomittedvariableleadstohighererrorvarianceacrossclassrooms,butitdoesnotbiastheteacherÞxede!ects(value-added)estimatorsifclassroomsarerandomlyassignedtoteachers.However,ifclassroomassignmentisnonrandom,groupingbasedontheomittedstudentÞxede!ectdoesintroducebiasGuarinoetal.[2015].Inthiscase,wedonotexpecttheconÞdenceintervalscomputedusingthebiasedvalueaddedestimatestoexhibitgoodcoveragerates,butcomparingthevariabilityoftheestimateswiththecomputedstandarderrorsisstillinformative.Tosummarize,weconductasimulationforeachoftheparticulargroupingandassignmentmechanismsundereachofthedistributionalcasesdescribedinTableE.1,andforvariousnumbersofcohortsofstudents.Thesimulationinvolves500repetitionsforeachoftheparticularcombinationsofgrouping,assignment,distributionalcase,andcohortchoices.3.3.2AnalyticMethodsIneachsimulation,weestimateteachere!ectsalongwithfourdi!erenttypesofstandarderrors:standarderrorswithoutanycorrectionforclusteringorheteroskedasticity,standarderrorswithclassroomlevelclustering,cohort-schoollevelclustering,andschoollevelclustering.11Toestimatetheteachere!ects,weregressthecurrenttestscoresonteacherdummyvariablesusingordinaryleastsquares(OLS),controllingforonelaggedtestscore,whichwetermOLS-Lag.12Ai,t=%Ai,t!1+(TeacherDummiesi,t)"0+Xi,t!0+#i,t(3.2)WhereXi,trepresentsasetofobservablestudentcharacteristics.Inthesimulationsthisisempty,butwhenestimatingvalue-addedusingadministrativedataweincludeasetofstudent11Clusteredstandarderrorsareobtainedusingthetypicalclusterrobustsandwichvarianceestimator.12Guarinoetal.[2015]foundthattheOLS-Lagmethodwasthemostrobustmethod,thatis,leastsubjecttobiasoverall,amongasetofestimatorsinvestigated.WethereforeexploretheissueofstandarderrorsandconÞdenceintervalsusingthisestimator.61characteristics.OLS-Lagdoesnotrestrictthecoe"cientonthelaggedtestscore(%),allowingforlessthancompletepersistenceinstudentsÕtestknowledgefromoneyeartothenext.Forthesakeofbrevitywepresentsimulationresultsonlyforthecasewhere%=0.5,whichisinlinewiththeÞndingsofAndrabietal.[2011].13OurÞrstperformancemetricisacomparisonbetweeneachofthefourtypesofstandarderrorsandtheactualvariabilityoftheestimates.Wecalculatetheactualvariabilityoftheestimatesbytakingthestandarddeviationoftheestimatesacrossthe500repetitions,asinequation(3.3).SD(ö"p,q)=!""#1499500$r=1(ö"p,r,q!øö"p,q)2(3.3)ö"p,r,qrepresentsthevalue-addedestimateofteacherp,fromtherthreplication,basedonqcohortsofsimulateddata,andøö"p,qrepresentstheaverageestimateacrossall500repetitions.WetakeSD(ö"p,q)asameasureofthetruevariabilityoftheestimator.Tosummarize,foreachof500simulationreplicationsforaparticularscenario,weestimateteachervalue-addedusingOLS-lag,obtainingonevalue-addedestimateforeachteacher.Thenwecalculatethestandarddeviationoftheestimatesforeachteacher.Toassesstheperformanceofthestandarderrors,weusetheaverageoftheparticulartypeofstandarderroracrossthe500repetitions,asshowninequation(3.4).AvgSEp,qk=1500500$r=1sek(ö"p,r,q)(3.4)Theaveragestandarderrorsareobtainedinthefollowingstepsforeachgroupingandassignmentscenario:Foreachof500simulationreplications,weobtainthefourdi!erentstandarderrorsforeachvalue-addedestimate.Letsek(ö"p,r,q)standforthecorrespondingstandarderrorreportedwhenthekthtypeofclusteringisused.Thatis,k&{noclustering,classclustering,cohort!schoolclustering,schoolclustering}.Thenwecalculatetheaverageofeachstandarderrorforeachteacher,asshowninequation(3.4).Wereporttheaverageoftheseaverages,takenacrossteachers,tosummarizetheoverallresultsforaparticularstandarderrorestimate,andrefertothat13Evidencefrompreviousliteraturesuggeststhatthevalueof!islessthanoneJacobetal.[2010],Rothstein[2008],Andrabietal.[2011].Wehavealsodonesimulationsusing!=1,andtheresultsarenotsensitivetothischange.62astheÒAverageoftheTeacherAverageSE.ÓWeconsiderthecomparisonbetweentheaveragesfromequations(3.3)and(3.4)tobeinformative,becausetheytellushowcloselythestandarderrorsreßecttheactualvariabilityoftheestimates.Werefertothesecondindicatorofperformancethatweuseasthecoveragerate.Thecoveragerateforeachteacheristhepercentofreplicationsforwhichthe95%conÞdenceintervaloftheestimatedteachere!ectcontainsthetruee!ect.Inshort,thisiscomputedbycomparingcomputedt-statisticswiththecriticalvalue.WeÞrstconstructthet-statisticforthetwo-sidedtestwiththenullhypothesisthattheestimatede!ectequalstheknowntruee!ectforeachreplicationforeachteacher,asinequation(3.5).tp,r,qk=ö"p,r,q!"ptruesek(ö"p,r,q)(3.5)Again,ö"p,r,qrepresentstheestimatedteachere!ectforthepthteacherbasedonqcohorts,fromtherthsimulationreplication,andsek(ö"p,r,q)isthecorrespondingstandarderrorcomputedwithclusteringmethodk.Let"ptruerepresentthetruee!ectforteacherp,whichdoesnotchangeacrossreplications.ThenullhypothesiscanbewrittenasH0:"p0="ptrue.Wecalculatetheabsolutevalueofthet-statisticinequation(3.5)foreachteacher,typeofstandarderror,andreplication.Wethencomparethist-statistictotherelevantcriticalvaluefromthestandardnormaldistribution(1.96fora95%conÞdenceinterval):CoverageRatep,r,qk=(1500)500$r=1I[abs(tp,r,qk)"1.96](3.6)Thecoveragerateforagiventeacherandtypeofclusteringisthencalculatedasthepercentoft-statisticswithabsolutevaluelessthan1.96,whichisidenticaltoforminga95%conÞdenceintervalaroundeachestimateandcalculatingthepercentageofreplicationsinwhichwefailtorejectH0.Equation(3.6)providestheexpressionforthecoverageratebasedonqcohorts,clusteringtypek,forteacherp,and500replications.I[á]standsfortheindicatorfunction,andabs(á)standsfortheabsolutevalue.CoverageratessummarizetheperformanceoftheconÞdenceintervalsbyindicatingwhether63weareactuallyrejectingthenullataratethatwewouldexpect.Withanunbiasedestimatorweexpectthat95%conÞdenceintervalsformedusingvalidstandarderrorsandcriticalvalueswillincludethetruee!ectapproximately95%ofthetime.However,withabiasedestimatorcoverageratescouldbelowevenwhentheconÞdenceintervalsarethecorrectsize.Aswediscussfurtherbelow,coverageratesalsodependonappropriatecriticalvalues,whicharekeytoinferenceandconclusions.Standarderrorsalonewouldnotrevealthisimportantdeviation.Wearethusbothinterestedinthecoverageratesandwhetherthestandarderrorscloselyreßecttheactualvariabilitythatweseeintheestimates.Coveragerateshighlightthepolicyimplicationsandpracticalimportance,butdependonbiasaswellasthewidthoftheconÞdenceintervals.Comparingthestandarddeviationoftheestimateswiththeaveragestandarderrortellsushowwellthestandarderrorsreßectthetruevariability,whethertheestimatorisbiasedornot.Therefore,itisimportanttoconsidercoverageratesandstandarderrorsinevaluatingtheperformanceoftheconÞdenceintervals.Inadditiontoreportingresultsbasedonregressionsthatpoolallyears,wereportresultsfromamethodwerefertoastheOLS-Lagcohort-by-cohortapproach,orsimplycohort-by-cohort.Inthisapproachweestimatevalue-addedseparatelyforeachcohort,andconsiderthoseestimatesasindependentpiecesofinformationweusetoconstructconÞdenceintervals.Thisapproachsidestepsissuesrelatedtowithinclassroomgrouping,becauseitdoesnotrequireanassumptionaboutthewithinclassroomerrorcorrelation;rather,themethode!ectivelyusestheclassroomsasseparatepiecesofinformation.Weestimatevalue-addedcohort-by-cohortusingOLS-Lag,obtainingoneteachere!ectestimateforeachyeartheteacherisinthedata.ö"p,r,q=1qq$t=1ö"p,r,qt(3.7)Wetaketheestimatedvalue-addedforeachteacherastheaverageoftheirsingle-yearestimates,asdescribedinequation(3.7).Whereö"p,r,qtrepresentstheestimateforthepthteacher,fromtherthreplication,andusingdatafromthetthcohortonly,andö"p,r,qistheaverageofthesingle-yearestimates.64sek(ö"p,r,q)=SD(ö"p,r,qt)#q(3.8)Weuseequation(3.8)toobtainastandarderrorfortheestimate,whereknowrepresentstheOLS-Lagcohort-by-cohortapproach.Calculatingtheestimatedteachere!ectandstandarderrorinthiswayisequivalenttoregressingthesingle-yearteachere!ectestimatesforeachteacheronaconstant,andusingthereportedestimateandstandarderror.Oncewehaveobtainedanestimate,ö"p,r,q,andcorrespondingstandarderror,sek(ö"p,r,q),foreachrepetitionr,wetakethestandarddeviationoftheestimatesasthemeasureoftruevariability,justasinequation(3.3).Similarly,wetaketheaveragestandarderror,tocomparewiththetruevariability,justasinequation(3.4).Thenweconstructt-statisticsforH0:"p0="ptrueusingö"p,r,q,sek(ö"p,r,q),andthetrueteachere!ect,justaswedidinequation(3.5)forthepreviousmethods.Wethencomparetheset-statisticswiththecriticalvaluefromthet-distributionwithdegreesoffreedomequaltothenumberofyearstheteacherisinthedataminusone,i.e.,(q!1).14Thus,thecoveragerateforthecohort-by-cohortmethoddi!ersfromthepreviousmethodsinthatweusetheappropriatecriticalvaluenowfromthet-distributionwith(q!1)degreesoffreedom:CoverageRatep,r,qk=(1500)500$r=1I[abs(tp,r,qk)"t!=0.025,df=(q!1)](3.9)Sotoobtainthecohort-by-cohortcoverageratesweswapoutthestandardnormalcriticalvalueforacriticalvaluefromat-distribution,whichdependsonthenumberofcohorts.15AgainthisisequivalenttoconstructingconÞdenceintervalsroundö"p,r,qteacher-by-teacher,usingsek(ö"p,r,q),andcountingthenumberoftimesthatwefailtorejecttrueteachere!ect.Thecoveragerateisparticularlyimportantforevaluatingonevirtueofthecohort-by-cohortmethod-thecriticalvalueadjustment.Ifthet-distributionwithdegreesoffreedomequaltothenumberofcohortsminusoneisagoodapproximationforthedistributionofthethet-statistic14Teachershavethesamenumberofclassroomsinthesimulations,butthenumberofclassroomscandi!erforeachteacherinouranalysisoftheadministrativedata.15Wehavedonetworobustnesschecksnotincludedhere.OnedrawsthestudentÞxede!ectandtherandomerrorcomponentfromatdistributionwith5degreesoffreedom,andtheotherusesachi-squareddistributionwith2degreesoffreedom.65obtainedinthecohort-by-cohortmethod,thenthisadjustmentshouldleadtocoverageratesnearninety-Þvepercent.Whereas,assumingthatthet-statisticfollowsanormaldistributionwouldbetooliberal.3.3.3SimulationResultsIneachofthesimulationresultstables,weshowtheaveragestandarddeviationoftheteacherÞxede!ectsestimates,averageoftheteacheraveragestandarderror,andthecoveragerateforeachmethod.Eachtabledisplaysresultsforthe3,7,and20cohortcases.Wealsoshowresultswithasinglecohortforthenon-clusteredstandarderrorsandconÞdenceintervals.InTableE.2weshowresultsfortheOLS-LagestimatorusingtheDGPwithastudentÞxede!ectdrawnfromastandardnormaldistribution,nocohort-schoolÞxede!ect,and%=0.5(referredtoasCase1inTableE.1).Weusethreedi!erentgroupingscenarios:random(RG),dynamic(DG,basedonAi,t!1),andheterogeneitygrouping(HG,basedonci).WeshowresultsforÞveschoolsinPanelAandforÞftyschoolsinPanelB.16TheÞrstcolumnreferstothenumberofcohortsusedintheestimation.Sofortherowswithonecohort,wesimulateddataforasinglecohort,obtainedthevalued-added(VA)estimates,standarderrorsandt-statisticsusingonlythatcohort,andthenrepeatedthatprocessforeachreplication.Thatmeansthatweuseddatafromonlyoneclassforeachteacher.Estimatesbased3cohortsincludestudentsfromthreeclassroomsforeachteacher,andsoforth.ThesecondcolumnshowstheteacheraveragestandarddeviationoftheVAestimatesaveragedacrosssimulationrepetitions,whichistheaverageofequation(3.3)acrossallteachers.ThistellsustheaveragevariabilityoftheVAestimates.Columnsthreethroughsixprovidetheaverage(acrossteachers)ofequation(3.4)foreachmethod:notclustering,classroomlevelclustering,cohort-schoollevelclustering,andschoollevelclustering,respectively.Weconsiderthecomparisonofcolumn2withcolumns3Ð6informativebecausetheyaretwodi!erentmeasuresofthevariabilityoftheVAestimator.Column2estimatesestimatorvariabilitymoredirectly,whereas,columns3Ð6areaveragesofwhatpractitionersuse.Weexpectthatgoodstandarderrorswillproperlyreßectthe16WeonlyshowresultswithÞftyschoolsinthiscase.Wehaveperformedmoresimulationswith50schools,butaspreviouslydiscussedandshowninTableE.2,addingschoolsdoesnotchangetheresults.66variabilityoftheestimates,sowhenthevaluesincolumnsthreethroughsixareclosetothoseincolumn2,thatmethodisconsideredtobeperformingwellbythismetric.Considercolumn3inPanelA,whichreportstheaveragestandarderrorwithoutclusteringundertherandomgroupingÐrandomassignment(RG-RA)scenario.Becausestudentsaregroupedrandomlyinthiscase,wewouldexpectthestandarderrorstocloselyreßectthestandarddeviationoftheestimates(i.e.,thevariabilityoftheestimator).Thesimilarityofcolumns2and3supportsthis:theaveragestandarderrorusingonecohortandwithoutclustering,reportedincolumn3,is0.435,whichisveryclosetotheaveragestandarddeviationoftheestimatesreportedincolumn2,0.419.Notsurprisingly,theaveragecoveragerateofthe95%conÞdenceintervalinthiscase,0.956reportedincolumn7,isalmostexactly95%.Notclusteringworkswellinthiscase,becausethestudentsaregroupedrandomly,sotreatingthemasindependentobservationsleadstostandarderrorsthatreßectthetruevariabilityoftheestimatorandalmostexactcoveragerates,evenwithasinglecohortofdata.Similarly,notclusteringgivesaveragestandarderrorsthataresimilarinmagnitudetotheaveragestandarddeviationoftheestimateswithanynumberofcohorts.Comparingcolumns2and3whenusingtwentycohorts,theaveragestandarddeviation,0.095,issimilarinmagnitudetotheaveragestandarderrorwithoutclustering,0.097,andthecoveragerate,0.953fromcolumn7,isagainalmostexactly95%.Again,thisworkswell,becausethestudentsarerandomlygroupedinthisscenario,sothereisnoneedtoaccountforwithingroupcorrelationwhenconstructingtheconÞdenceintervals.Ifweaccountforwithinclassroomcorrelationbyclusteringattheclassroomlevelwhengroupingisrandom,wetendtounderestimatethevarianceoftheestimatorandhavelowcoveragerateswithasmallnumberofcohorts.Fromcolumn2ofPanelA,withthreecohortsandrandomgrouping,theaveragestandarddeviationoftheVAestimatesis0.253.However,theaveragestandarderrorunderclassroomclustering,fromcolumn3ofPanelA,is0.202.Theunderestimateofthevarianceleadstoacoveragerateof0.826,whichiswellbelow95%andmeansthatusingthismethodunderestimatesthesizeoftheconÞdenceintervalswithasmallnumberofcohortsandrandomgrouping.However,unnecessaryclassroomclusteringisnotproblematicwhenthenumberofcohortsincreases.Theaveragestandarderrorunderclassroomclusteringwithtwentycohorts,0.095fromcolumn3of67PanelA,isverysimilartotheaveragestandarddeviation,0.095fromcolumn2ofPanelA.Thecorrespondingcoverageratefromcolumn8is0.942,whichismuchbetterthanthe0.826withthreecohorts.Thereasonthatclusteringattheclassroomlevelimproveswithmorecohortsisthatite!ectivelyusesoneindependentpieceofinformationfromeachclassroomtoobtainastandarderroralongwithastandardnormalcriticalvaluetoconstructtheconÞdenceinterval.UndertheRG-RAscenario,theclassroomclusteredvarianceestimatorisasymptoticallyvalidwithanincreasingnumberofcohorts.Howeveritdoesnotperformwellwithasmallnumberofclassroomsperteacher,becauseusingonlythreeobservationstoestimatethestandarderrorforeachÞxede!ecttendstounderestimatethestandarderror.Inthesecondcase,alsoinPanelAofTableE.2,weconsiderdynamicgroupingwithrandomassignment(DG-RA).DG-RAmeansthatthestudentsaregroupedintoclassroomsbasedontheirpriortestscore,Ai,t!1,thenclassroomsarerandomlyassignedtotheteachers.Sincewearecalcu-latingVAusingOLS-Lag,wecontrolforthestudentsÕpriorexamscore.Dynamicgroupingdoesnotincreasethevarianceoftheestimator,becauseweareconditioningonthegroupingvariable.Wecanseethatbycomparingtheaveragestandarddeviationsunderdynamicgroupingwiththeaveragestandarddeviationsunderrandomgrouping.Forexample,fromcolumn2ofPanelA,theaveragestandarddeviationunderdynamicgroupingwithasinglecohortis0.431,andwithtwentycohortsitis0.095.Similarly,theaveragestandarddeviationsunderrandomgroupingwithoneandtwentycohortsare0.431and0.099,respectively.Thissimilaritybetweentheaveragestandarderrorswithoutclusteringforbothdynamicandrandomgroupingleadstoalmostidenticalcoverageratesaswell.Forexample,theaveragecoverageratesunderdynamicgroupingwithoneandtwentyco-hortsare0.954and0.947,respectively.TheperformanceoftheclassroomclusteredstandarderrorsandcorrespondingconÞdenceintervalsarealsosimilarunderrandomanddynamicgrouping.Theclassroomclusteredstandarderrorstendtounderestimatethestandarddeviationoftheestimateswithasmallnumberofcohorts,0.202comparedto0.257withthreecohorts,leadingtocoveragerateswellbelow95%,0.820withthreecohorts.However,theperformanceoftheclassroom-clusteredstandarderrorsimproveswithincreasingcohorts,andtheaveragecoveragerate,0.935fromcolumn8,isnear95%withtwentycohorts.68IncontrasttotheÞrsttwocases,whenstudentsaregroupedbasedonthestudentÞxede!ect,heterogeneitygroupingÐrandomassignment(HG-RA),thennon-clusteredstandarderrorsdonotreßectthetruevariabilityoftheestimatorandthecoverageratesaresubstantiallylower.Theaveragestandarddeviationwithasinglecohortgoesfrom0.419underrandomgroupingto0.888underheterogeneitygrouping.Thishappensbecausegroupingonanunobservablecomponentincreasesunobservedvarianceacrossclassrooms,andnotclusteringfailstoaccountforthatincreasedvariance.Theaveragestandarderrorwithasinglecohortunderheterogeneitygroupingisonly0.399.Thisunobservablecomponentofthestudentlevelequationisnolongerindependentwithin-classroom,andfailingtoaccountforthewithin-classroomcorrelationleadstodrasticallyunderestimatingthevariabilityoftheestimator,evenwhentheclassroomsarerandomlyassignedtoteachers.Whenthestudentsinaclassroomallhavehigh(orlow)averagescores,wemistakenlythinkthatwehavearelativelypreciseestimate,basedonthosetwentyobservations.This,notsurprisingly,leadstoalowcoveragerateof0.546,whichismuchlowerthanwewouldhopetogetfora95%conÞdenceinterval.Moreaccurately,weshouldnotconsiderthosetwentystudent-levelobservationsasindependent,becausetheyweregroupedonanunobservabledeterminantoftestscores,beforethegroupswereassignedrandomlytotheteacher.ClassroomlevelclusteringcanimprovetheperformanceoftheconÞdenceintervals,butonlywithasu"cientnumberofcohorts.Withsevencohortsofdata,eventheclassroom-clusteredstandarderrorsunderreportthestandarddeviationoftheestimates,0.304incolumn4comparedto0.340fromcolumn2.However,thecorrespondingconÞdenceintervalsdocontaintheteachere!ectroughly90%ofthetime,whichismuchbetterthannotclustering.Underheterogeneitygrouping,theclassroomandcohort-schoolclusteredstandarderrorsperformsimilarlytotherandomgroupingcase,meaningthattheyunderreportvariabilityandcoverthetrueparameterlessthan95%ofthetimewithasmallnumberofcohorts,buttheyimprovewithanincreasingnumberofcohorts.ThereasonthatusingclassroomclusteredstandarderrorsworkswithanincreasingnumberofcohortsunderheterogeneitygroupingisthesamereasonitworksintheÞrsttwocases:assumingindependenceacrossclassroomsisvalidinallthreecases,buttheestimatorisasymptoticallyjustiÞedwithanincreasingnumberofclassroomsperteacher,anddoesnÕtnecessarily69workwellwithasmallnumberofcohorts.Unfortunately,districtsoftenwanttomakedecisionsregardingteachere!ectivenessbasedononlyafewyearsofdataandwaitingfortwentyclassroomsworthofobservationsforeachteacherisgenerallynotdesirableorfeasible.TherearetwoÞnalpointstomakeaboutPanelAofTableE.2.InallthreescenariospresentedinPanelAclusteringatthecohort-schoollevelperformssimilarlytoclassroomclustering.Theaveragestandarderrorsaresimilarwitheverynumberofcohorts.Withthreecohorts,theaveragestandarderrorunderRA-RGwithclassroomclusteringis0.202,whichisalmostidenticaltotheaveragestandarderrorwithcohort-schoolclustering,0.204.Similarityincoverageratesfollowsthesimilarityinstandarderrors.TheaveragecoverageratesunderRA-RGforclassroomclusteringforthree,seven,andtwentycohortsare0.826,0.914,and0.942,whicharealmostidenticaltothoseforcohort-schoolclustering,0.824,0.913,and0.943.Theperformanceofclassroomandcohort-schoolclusteringisalsosimilarunderdynamicandheterogeneitygrouping.Forexample,underheterogeneitygroupingwith3cohorts,theaverageclassroomclusteredstandarderroris0.404andtheaveragecohort-schoolclusteredstandarderroris0.423,leadingtocoverageratesof0.822and0.828,respectively.Clusteringattheclassroomlevelandcohort-schoolclusteringgivesuchsimilarresultsbecausethetwocalculationsarenearlyidentical.EachteacherdummyiszeroforallstudentsoutsideofthatteacherÕsclassrooms,sotheinnerproductofeachteacherdummyvariableandtheresidualsisthesamewhetheryoutakeitwithinclassroomorwithincohort.Theonlydi!erenceinthetwovariance-covarianceestimatorsresultsfromtakingtheinnerproductoflaggedscoresandresidualseitherwithinclassroomorwithincohort.17Although,wecontinuetoshowtheresultsforbothclassroomsandcohort-schoolclusteringfromthispointon,werefrainfromgoingintodetailaboutboth,astheyaregenerallythesame.Lastly,clusteringattheschoollevelseverelyunderestimatesthevariationintheestimatorandleadstothelowestcoverageratesofanyoftheconÞdenceintervalsconsidered.Withtwentycohorts,theaveragecoverageratesusingschoollevelclusteringare0.020,0.023,and0.010forRG-RA,DG-RA,andHG-RA,respectively,meaningthattheconÞdenceintervalsonlycoverthetrueteacher17Ofcourse,theirsimilaritycouldbearesultofthesimplicityofthesimulations,asaddingmorecovariateschangesthecalculationofeachvariance-covariancematrix.However,inourevaluationsoftheperformanceoftheclassroomandcohortclusteredstandarderrorsinadministrativedata,weÞndthattherearepracticallynodi!erencesintheiraverageperformance.70e!ectabout1Ð2%ofthetimewithtwentycohorts.Thetheoreticalßawwithusingschoollevelclusteringisthatinformationusedtoestimatethetreatments,i.e.theteacherÞxede!ects,doesnotaccumulatewhenaddingmoreschools.Themechanicalissueisthatteachersarenestedinschools,andweareconductinginferenceonteacherÞxede!ects.ThesameissuewouldariseifweclusteredbyteacherandtriedconductinginferenceonteacherÞxede!ects.BecauseweareestimatingtheÞxede!ectsbyOLSandteachersarenestedinschools,withinteacherresidualsaddtozero,andwithinschoolresidualsalsoaddtozero.Thisisproblematicbecausethecentralmatrixintheclusterrobustvariancesandwichestimatorinvolvestakingwithinclusterinnerproductsoftheright-hand-sidevariablesandtheresiduals,butforteacherÞxede!ectsthisequatestosummingresidualswithinschool.Inanextremecase,inwhichteachersarenestedinschoolsandtherearenoright-hand-sidevariablesotherthantheteacherÞxede!ects,thecentralmatrixofthevariance-covarianceestimate(theÒnumeratorÓ)isamatrixofzeros.Addingright-hand-sidevariablesornothavingteachersnestedinschoolsbreaksthismathematicalresult,buttheproblemwithunderestimationofthestandarderrorsappearstopersist.Althoughwecontinuetoshowresultsfromschoolclustering,wedonotdiscussthemindetailfromthispointon.Toillustratethatincreasingthenumberofschoolshasnoimpactontheresults,wereportresultsforthesamethreecases,RG-RA,DG-RA,andHG-RA,withÞftyschoolsinPanelBofTableE.2.Theresultsarestrikinglysimilartothosefor5schools.SimilartotheÞve-schoolcase,theaveragestandarderrorswithoutclusteringcloselyreßecttheaveragestandarddeviationoftheestimateswhenstudentsaregroupedrandomly.Theaveragestandarderrorwithasinglecohortandnoclustering,0.446fromcolumn3ofPanelB,issimilartotheaveragestandarddeviationoftheestimateswithonecohort,0.454.Theresultingcoveragerate,0.947fromcolumn7,isalmostexactly95%.TheaverageclassroomclusteredstandarderrorsandcorrespondingcoveragerateswithÞftyschoolsandrandomgroupingalsoperformsimilarlytothecasewithÞveschools,withcoverageratesincreasinginthenumberofcohortsfrom0.812to0.944forthreeandtwentycohorts,respectively.ResultsunderheterogeneitygroupingarealsosimilaracrosstheÞveandÞftyschoolcases.TheaveragestandarddeviationoftheestimateswithasinglecohortandÞftyschoolsincreasesto0.894,whichissimilartotheaveragestandarddeviationoftheestimateswithasinglecohortandÞve71schools,0.888.EvenwithÞftyschools,thenon-clusteredstandarderrors,0.410onaverage,stillfailtoaccountfortheincreasedvariancefromheterogeneitygrouping,leadingtoalowcoveragerateof0.586.Again,similartotheÞve-schoolcase,increasingthenumberofcohortsdoesnotimprovethecoveragerateverymuchfortheconÞdenceintervalsthatusenon-clusteredstandarderrors,asevenwithtwentycohortsofstudentsthecoverageratewithoutclusteringis0.663.TheperformanceofthestandarderrorsandconÞdenceintervalsfromclusteringattheclassroomlevelimprovewithmorecohortswhenthereareÞftyschools,thesamewaythattheydowhenthereareÞve.Theaveragestandarderror,fromcolumn4inPanelBunderHG-RA,whenclusteringbyclassroomis0.404,anunderestimateoftheaveragestandarddeviationof0.518.Theresultingcoveragerateis0.812,substantiallylowerthan95%.Withtwentycohorts,theaverageclassroom-clusteredstandarderroris0.194,whichissimilartotheanalogousaveragestandarddeviationoftheestimatesof0.201,andthecoveragerateis0.931.ComparingtheresultswithÞveandÞftyschoolsillustratestheargumentthatthenumberofschoolsisnotimportantforestimatingteachere!ectsinthesimulations.Theinformationonteachere!ectsaccumulatesfromaddingcohorts/classesfortheexistingteachers,andnotfromaddingschools.Forthisreason,weonlypresentanddiscussresultsforsimulationswithÞvesschoolsfromhereon.TableE.3reportstheresultsforCase2,withanunobservablestudentÞxede!ectandanunob-servablecohort-schoolÞxede!ect,eachwithastandarddeviationof0.25,underRG-RA,DG-RA,andHG-RA.Includingasmallcohort-schoolÞxede!ectinßuencestheresultsinasimilarmannerasgroupingbasedonanunobservablecharacteristic,e.g.thestudentÞxede!ect.Anunobservablecohort-schoolÞxede!ectincreasesthevarianceoftheestimates,suchthatstandarderrorswithnoclusteringfailtoaccuratelyreßectthevariationintheestimates.Forexample,underRG-RAwithonecohortofdata,theaveragestandarddeviationofestimatesis0.438andtheaveragestandarderrorwithoutclustering,fromcolumn3,is0.317.Thenon-clusteredstandarderrorsfailtoaccountfortheextravarianceintheestimatorfromtheunobservableÞxede!ect,andasaresult,theaver-agecoverageratewithasinglecohortofdataandrandomgrouping,fromcolumn7ofTableE.3,isonly0.838.Evenwithmorecohorts,thenon-clusteredstandarderrorsfailtoproduceabettercoveragerate,andwithtwentycohortstheaveragecoveragerateisonly0.84.Usingcohort-school72(orclassroom)clustering,however,doestakeintoaccountthecorrelationwithineachcohort.Withasmallnumberofcohortsthough,cohort-schoolclusteringunderestimatestheaveragestandarddeviation,muchlikeclassroomandcohort-schoolclusteringdoinallthreescenariosfromTableE.2.WiththreecohortsunderRG-RAwithacohort-schoolÞxede!ect,theaveragestandarderrorusingcohort-schoolclusteringis0.208,fromcolumn5ofTableE.3,whichisnoticeablylowerthantheaveragestandarddeviationofestimatesof0.258.Thatleadstoanaveragecoveragerateof0.835,muchlowerthanthedesired0.95.However,theaveragecoverageratewithcohort-schoolclusteredstandarderrorsimproveswithanincreasingnumberofcohorts,andwithtwentycohortsthecoveragerateis0.939,asseenincolumn9ofTableE.3.TheresultswithHG-RAandacohort-schoolÞxede!ectarealsoreportedinTableE.3,buttheyaresimilartotheRG-RAresultswithacohort-schoolÞxede!ect.Theyaresimilar,becauseinTableE.3thestudentÞxede!ecthasarelativelylowvariance,andgroupingonalow-variancetermhasamuchsmallerimpactonthevariabilityoftheestimates.Forexample,withthreecohortsofdataunderheterogeneitygrouping,theaveragestandarddeviationoftheestimatesis0.264,fromcolumn2ofTableE.3,butrecallthattheaveragestandarddeviationwiththreecohortsandheterogeneitygroupingonthestandardnormalstudentÞxede!ectsis0.505,fromcolumn2ofPanelAinTableE.2.Thedi!erenceisthatthetotalvariationandvariationintheerrorislargerinTableE.2.Regardless,clusteringatthecohort-schoollevelperformssimilarlyforpracticalpurposesinbothcases,inthatthecoverageratesarelowwithasmallnumberofcohorts,0.814withthreecohorts,butimproveasthenumberofcohortsincrease,0.937withtwenty.Ingeneral,incaseswherewewishtoaccountforwithinclassroomdependence,wecandothatusingclusteringwhenthenumberofclassroomsperteachersislarge.However,clusteringleadstolowcoveragerateswithasmallnumberofcohorts.InPanelAofTableE.4,averagestandarderrorsandcoverageratesfromthecohort-by-cohortmethodarepresentedunderRG-RA,DG-RA,andHG-RAscenarioswiththeunobservablestudentÞxede!ectdrawnfromastandardnormaldistributionandnocohort-schoolÞxede!ect,i.e.Case1fromTableE.1.18AlthoughthemethodforestimatingtheteacherÞxede!ectsischangedslightly18WereplicatedtheanalysisfromPanelAofTableE.4undertwootherscenarios,drawingstudentÞxede!ectsandrandomerrortermsfromat-distributionandachi-squareddistribution.Theresultsaresimilartotheresultsusing73inthismethod,itisnotpracticallyverydi!erentfrompoolingallyearsofdataandusingOLS-lag,soitdoesnotaltertheaveragestandarddeviationsoftheestimatesverymuch.19Fromcolumn2inPanelAofTableE.4,theaveragestandarddeviationoftheestimateswiththreecohortsis0.253,whichisidenticaltotheanalogousnumberfromTableE.2.LikeclusteringattheclassroomlevelwiththeOLS-Lagestimator,theaveragestandarderrorinthecohort-by-cohortmethodunderesti-matestheaveragestandarddeviationoftheestimateswithasmallnumberofcohorts,albeitlessseverely.SinceconstructingtheconÞdenceintervalinthecohort-by-cohortcaseincludesacriticalvalueadjustment,usingthecriticalvaluefromthet-distributionwiththenumberofcohortsmi-nusoneasopposedtothestandardnormalcriticalvalue,thecoverageratewiththreecohortsis0.949,despiteunderestimatingthevariabilityoftheestimator.Asthenumberofcohortsincreases,theaveragestandarderrorandaveragestandarddeviationoftheestimatesbecomeveryclose,withtwentycohortstheyare0.096and0.095,respectively.MuchlikeclassroomclusteringwithOLS-Lag,thismethodallowsforcompletedependenceofthewithinclassroomstudentlevelerrorterms.Thecriticalvalueusedinthismethodapproachesthestandardnormalcriticalvalueasthenumberofcohortsincreases,sothattheadjustmentbecomessmaller,andwithtwentycohortstheconÞdenceintervalscoverthetruee!ectatanaveragerateof0.959,slightlymoreconservativethantheanal-ogouscoveragerateof0.942underrandomgroupingwithtwentycohortsfromcolumn8ofTableE.2.ResultsfortheHG-RAcasewithastudentÞxede!ectdrawnfromastandardnormaldistributionarealsopresentedinPanelAofTableE.4.Theaveragestandarderrorinthecohort-by-cohortmethodreßectssomeoftheincreasedvariabilityintheestimates,evenwithasmallnumberofcohorts.Theaveragestandarddeviationoftheestimateswiththreecohortsis0.506,almostidenticaltotheanalogousnumberfromTableE.2,andtheaveragestandarderroris0.470.Fromcolumn4ofTableE.4,thecoveragerateusingthecohort-by-cohortmethodunderHG-RAwiththreecohortsis0.934,avastimprovementfromanyofthecoverageratespresentedforthesamecaseinTableE.2.Forcomparison,usingOLS-Lagandclassroomclusteredstandarderrorswithstandardnormalstandardnormaldistributions.19BothestimatorsareunbiasedfortheteacherÞxede!ectsunderrandomassignmentofclassroomstoteachersandmechanicallysimilar.74criticalvalues,i.e.column8inTableE.2,onlycoversthetruee!ect82.2%ofthetimeunderHG-RAwiththreecohorts.Withalargernumberofcohorts,thecohort-by-cohortconÞdenceintervalscontinuetoachievealmostexact95%coverage,withaveragecoverageratesof95.1%withsevencohortsand95.3%withtwentycohorts.PanelBshowsresultsusingthecohort-by-cohortmethodforthecasewithasmallstudentÞxede!ectandacohort-schoolÞxede!ect,comparabletoTableE.3.Usingthecohort-by-cohortmethodsdoesnotleadtolowercoveragerateswhenthereisacohort-schoolÞxede!ect,asitdoeswhenusingOLS-Lagwithclassroomorcohort-schoolclusteredstandarderrors.Evenunderheterogeneitygrouping,acaseinwhichwemightexpectlowercoverageratesbasedontheresultsinTableE.3,theaveragecoveragerateusingthecohort-by-cohortmethodis95.1%with3cohorts.Overall,despiteatendencyofthestandarderrorsfromthecohort-by-cohortmethodtounderestimatetheaveragestandarddeviationoftheestimate,theunderestimationislessseverethantheaveragestandarderrorsfromusingclusteringwithOLS-Lag.Thecombinationoftheslightlyhigherstandarderrorsandthecriticalvalueadjustmentleadstocoverageratesnear95%ineverycasepresentedinTableE.4,suggestingthatthecohort-by-cohortmethodoutperformstheotheroptions,especiallywithasmallnumberofcohorts.Next,inTablesE.5andE.6,werelaxtherandomassignmentaspectofthesimulations,andre-evaluatetheperformanceofthedi!erentconÞdenceintervals.AllsimulationresultsinTablesE.5andE.6arebasedontheDGPinCase1ofTableE.1,whichdrawsthestudentÞxede!ectfromastandardnormaldistributionanddoesnotincludeacohort-schoolÞxede!ect.InTableE.5,wereportresultsusingtheOLS-Lagestimatorunderdynamicgroupingwithpositiveassignment(DG-PA)andheterogeneitygroupingwithpositiveassignment(HG-PA).Positiveassignmentmeansthattheclassroomwiththehigheraverageofthegroupingvariable,i.e.laggedscoresfordynamicgroupingandunobservablestudentÞxede!ectforheterogeneitygrouping,isassignedtotheteacherwiththehigherÞxede!ect.Underdynamicgrouping,positiveassignmentdoesnotleadtoabiasedestimatorbecause,onceagain,thegroupingvariableisusedintheconditioningsetwhenestimatingteacherVA.Similartopreviouscases,notclusteringunderDG-PAleadstoaveragestandarderrorsthataresimilartotheaveragestandarddeviationoftheestimates,0.435comparedto0.437witha75singlecohort,andproducesconÞdenceintervalswithcoverageratesaroundthe95%markwithanynumberofcohorts.ThecoveragerateunderDG-PAwithasinglecohortandnoclustering,fromcolumn7ofTableE.5is0.953.AlloftheestimatorswehavediscussedupuntilthispointareunbiasedestimatorsfortheteacherÞxede!ects.However,groupingonunobservableheterogeneityandpositivelyassigning(HG-PA)classroomstoteachersleadstobiasedteachervalue-addedestimators.However,positivelyassigningclassroomswithheterogeneitygroupingdoesnotleadtothesubstantialincreaseintheaveragestandarddeviationoftheestimatesthatwesawpreviously.Infact,theaveragestandarddeviationoftheestimateswithasinglecohortunderheterogeneitygroupingandpositiveassignmentis0.417,whichismuchlowerthanthe0.888underHG-RAfromTableE.2.Thereasonthatheterogeneitygroupingdoesnotinßatetheaveragestandarddeviationoftheestimateswithpositiveassignmentrelativetorandomassignment,isthathigher(lower)VAteachersareconsistentlyassignedtotheclassroomwiththehigher(lower)averagestudentÞxede!ect,ratherthanbeingrandomlyassignedtosomeclassroomswithahighandsomewithalowaveragestudentÞxede!ect.Thefactthattheaveragenon-clusteredstandarderrorsaresimilartotheaveragestandarddeviationoftheestimates,whichare0.399and0.417,respectivelywithonecohort,ismoreacoincidencethanasignofgoodperformance.Becausetheestimatorsarebiasedthough,coverageratesdecreasewithanincreasingnumberofcohorts,andtheaveragecoverageratesforboththenon-clusteredandclassroomclusteringmethodsare0.376withtwentycohortsofdata.Finally,wepresentsimulationresultsforthecohort-by-cohortmethodunderDG-PAandHG-PAinTableE.6.MuchliketheOLS-LagstandarderrorsandconÞdenceintervalsunderpositiveassignment,thecoverageratesarelowanddonotimprovewithincreasingcohortswhenstudentsaregroupedontheunobservablestudentÞxede!ect,becausegroupingonanunobservableandnon-randomassignmentproducesabiasedestimator.CoverageratesunderHG-PAfromTableE.6areonly0.621and0.385withthreeandtwentycohorts,respectively.Withabiasedestimator,suchastheestimatorsweconsiderwithHG-PA,thetighterconÞdenceintervalswithincreasingcohortsaremisleading.Oursimulationresultsshowanumberofkeyresults.Wecomparetheaveragestandarderrors76obtainedfromestimatingteacherVAbyOLS-Lagandnotclustering,clusteringattheclassroomlevel,cohort-schoolclustering,andschoolclusteringwiththeaveragestandarddeviationoftheestimatesacrosssimulationsrepetitions.WeÞndthatunderrandomordynamicgroupingofstudentsandrandomassignmentofclassroomstoteachers,noclusteringgenerallyworkswellwithanynumberofcohorts,whichisnotsurprisingsincethestudentlevelerrorsareindependentwithinclassroominthesecases.Similarly,evenwithasinglecohort,95%conÞdenceintervalsfromthismethodcanobtaincoverageratesaround95%.Clusteringattheclassroomorcohort-schoollevel,however,tendstounderestimatethevariabilityoftheestimatorwithasmallnumberofcohorts,leadingtocoverageratesaround80to83%undermostscenarios.ButtheperformanceoftheconÞdenceintervalsbasedonclassroomandcohort-schoolclusteredstandarderrorsimproveswithincreasingcohorts,andtheyreachcoverageratesaround92Ð94%inmostscenarios.Ontheotherhand,groupingstudentsonanunobservableÞxede!ectandrandomlyassigningclassroomstoteachersincreasestheestimatorvariance,whichnon-clusteredstandarderrorsfailtoaccountforandthusperformspoorly.Whileclusteringattheclassroomlevelproducesstandarderrorsthataccuratelyreßectthevarianceoftheestimatorwithalargenumberofcohortsunderheterogeneitygrouping,theystilltendtounderestimateitwithasmallnumberofcohorts.Consequently,thecoverageratesofconÞdenceintervalsproducedfromclassroomorcohort-schoolclusteringarelowwiththreecohorts,around82%,butimprovewithincreasingcohortsandreachover90%withtwentycohorts.Incomparison,whenweconstructconÞdenceintervalsusingourproposedcohort-by-cohortmethod,weÞndthatevenunderheterogeneitygroupingtheconÞdenceintervalcoverageratesarenear95%.Theimprovedperformanceofthecohort-by-cohortmethodoverclusteringbyclassroomwithOLS-Lagestimationisthroughbothanincreaseintheaveragestandarderrorandacriti-calvalueadjustment.Thecriticalvaluedependsonthenumberofcohortsusedintheestimate,whichcorrespondstothepiecesofindependentinformation.WiththreecohortsunderheterogeneitygroupingandrandomassignmentwithastudentÞxede!ectdrawnfromastandardnormaldistribu-tion,thecohort-by-cohortmethodhasa93.4%coveragerate,whichismuchcloserto95%thanthecoverageratesusingOLS-Lagwithclusteringandonlythreecohortsunderheterogeneitygrouping.However,whenstudentsaregroupednonrandomlyandclassroomsareassignednonrandomly,77theresultingbiasintheteacherVAestimatesisproblematicforallmethods.Inthiscase,thecohort-by-cohortmethoddoesnotimprovetheconÞdenceintervalperformancewhengroupingandclassroomassignmentarebasedonanunobservablestudentÞxede!ect,asthatleadstobiasedestimators.Whilethesimulationsarehelpfulinunderstandingtherelativeperformanceofmethodsunderknownconditions,weturntostudent-leveladministrativedatainthenextsectiontoseehowthemethodscompareinactualdata.Muchlikethesimulations,wecancomparethesizeofthestandarderrorsandconÞdenceintervalsundereachmethod.Furthermore,wedemonstratewhatthedi!erencesintheconÞdenceintervalsmeansforseparatingteachersacrossthevalue-addeddistribution.3.4ConÞdenceIntervalsinPractice3.4.1DataUsingadministrativedataonsixdistrictsfromalargestateforyears2001-2007,wecomparedi!erencesinstandarderrorsandconÞdenceintervalsforvalue-addedestimatesproducedusingthepreviouslydiscussedmethods.Allresultsinthissectionarebasedonestimatingmathvalue-addedforfourthgradeteachers.Werestrictthedatasettofourthgradeonlyforsimplicityandtomakecomparisonswiththesimulationsstraightforward.Wealsorestrictthesampletostudentswithnomissingdataforanyofthevariablesincludedintheregressions,andwhoareinclassroomswith10ormorestudentstotal.20Weestimatevalue-addedseparatelybydistrict,asinequation(3.2),whereXi,trepresentsasetofstudentlevelcharacteristics,includingrace/ethnicityindicators,numberofabsences,andindicatorsforfemale,disability,limitedEnglishproÞcientandfree-andreduced-pricelunch.Thelargestofthesixdistrictshas138,913fourthgradestudentsassignedto2,580teachersintheestimationsample.2120Themainresultsdonotrestricttoteacherswithacertainnumberofclassrooms.Asecondanalysis,whichcanbefoundintheappendix,restrictstoteacherswithallsevenyearsofdataavailabletoavoidchangesinsamplecomposition.21SeeTableE.7foradditionalstudentsamplesizesandaveragecharacteristicsbydistrict.783.4.2MethodsWefocusonconductinginferenceinfourdi!erentways.TheÞrstthreearefromestimatingteachervalue-addedusingOLS-Lagbydistrictforallteachersandpoolingallyears.Foreachestimate,wereportthreestandarderrors:noclustering,cohort-schoolclustering,andschoolclustering.22ThenweconstructconÞdenceintervalsusingstandardnormalcriticalvalues.ThefourthmethodisbasedonthepreviouslydiscussedOLS-Lagcohort-by-cohortmethod.Forthecohort-by-cohortmethod,weÞrstcalculateOLS-Lagvalue-addedestimatesforeachyearinthedata.Thenforeachteacherweusetheaverageofthesingle-yearestimatesastheestimatedteachervalue-added,andthestandarderrorofthatmeantoconstructconÞdenceintervalsusingthecriticalvaluefromthet-distributionwithdegreesoffreedomequaltothenumberofyearsinthedataminusone.Foreachmethod,weprovidetheaveragestandarderrorsandconÞdenceintervalwidth.Lastly,wedemonstratepracticalimplicationsofthedi!erences,bycomparingthepercentagesofteachersindi!erentpartsofthedistributionwithupper(lower)boundsthatareunder(above)di!erentpercentilesofthedistributionundereachmethod.Forexample,weshowthepercentofteacherswithestimatesinthetop10%oftheVAdistributionwhoalsohavealowerboundabovethe90thpercentileforeachconÞdenceinterval.ThisshedslightonhowpolicyprescriptionsdependonthemethodusedtocalculateconÞdenceintervals.3.4.3ResultsFigureF.1comparestheaveragestandarderrorsforeachmethodwith2Ð7cohortsforsixdistricts.Thereissomevariationacrossdistrictsintheorderingofthenon-clustered,cohort-schoolclustered,andcohort-by-cohortstandarderrors.InDistrictA,B,andE,thecohort-by-cohortstandarderrorsarelargerforeverynumberofcohorts,andinsomecasesthedi!erencesaresubstantial.Acrossallofoursimulations,fromTablesE.2,E.3,andE.5,thestandarderrorsobtainedfromcohort-schoolclusteringunderestimatethestandarddeviationoftheestimateswithasmallnumberof22Weomittheclassroomclusteringcaseinourreporting,becauseinmostcasestheyarealmostidenticaltocohort-schoolclusteredstandarderrorsonaverage.79cohorts,andtheaveragecohort-by-cohortstandarderrorsaregenerallylargerwithasmallnumberofcohorts.Thisisperhapsthedi!erencethatweobserveinDistrictsA,B,andEinFigureF.1.ForDistrictAtheaveragecohort-schoolclusteredstandarderrorwithtwocohortsisabout20points,andtheaveragecohort-by-cohortclusteredstandarderrorismorethandoublethat,around50pointswithtwocohorts.23Thedi!erenceinthetwoaveragesshrinkswithincreasingcohorts,andwithsevencohortstheaverageschool-cohortclusteredstandarderrorisalittlelessthan20points,andtheaveragecohort-schoolclusteredstandarderrorisaround35points.TheirconvergenceismorepronouncedinDistrictE,forwhichtheyonlydi!erbyaboutÞvepointswithsevencohorts.Ontheotherhand,inDistrictsC,D,andF,theorderisswitched,withtheaveragecohort-schoolclusteredstandarderrorslargerthanthecohort-by-cohortstandarderrorsformostcohorts.Thelargestgap,inDistrictC,appearstobeincreasingwithcohorts,andwithsevencohortstheaveragecohort-schoolclusteredstandarderrorisalittlelessthan40points,whiletheaveragecohort-by-cohortstandarderrorisalittleover20,leavingagapofabout15points.Thefactthattheaveragecohort-schoolclusteredstandarderrorislargerthantheaveragenon-clusteredstandarderrorisconsistentwithheterogeneitygrouping,aswellasanyscenariowithanobservedcohort-schoolÞxede!ect.However,basedonthesimulations,wemightexpectthecohort-by-cohortstandarderrorstobelarger,relativetothecohort-schoolclusteredstandarderrors.Despitetheaveragecohort-by-cohortstandarderrorbeingsmallerthanthecohort-clusteredstandarderrorinDistrictC,itdoesnotmeanthatthecohort-by-cohortconÞdenceintervalswillbesmaller,astheyusealargercriticalvalue.InFigureF.2wecomparetheaveragewidthofconÞdenceintervalsconstructedfromthedi!erentmethods.Itisencouragingthat,asthenumberofcohortsincreases,thesizesofthecohort-by-cohortandOLS-LagconÞdenceintervalsbecomesimilar.However,withasmallnumberofcohorts,thecohort-by-cohortconÞdenceintervalstendtobemuchmoreconservativethantheclusteringmethods.Ofcourse,partofthedi!erenceisduetothelargercriticalvaluesusedinthecohort-23Toputthesenumbersinperspective,theaveragescorefortheestimationsampleinoneofthedistrictsis1592withastandarddeviationof258.Value-addedestimatesinthatdistrictforteachersinthesampleforatleast2yearsrangefrom-161to306.Theaverageestimatedteachere!ectisabout16andthestandarddeviationofteachere!ectestimatesisalmost56.Theaveragecohortclusteredstandarderroris41.7,whereastheaverageofthenon-clusteredstandarderrorsis29.5.80by-cohortmethod.Insomedistrictsthough,suchasDistrictsA,B,andE,thecohort-by-cohortstandarderrorsarealsolargerthananyothermethod,leadingtoalargergapbetweentheconÞdenceintervals.Forexample,inDistrictE,forwhichtheaveragestandarderrorisover40pointswiththreecohortsunderthecohort-by-cohortmethodbutcloserto30withthreecohortsforthenoclusteringandcohort-schoolclusteringmethods,thecohort-by-cohortconÞdenceintervalsappearespeciallyconservativewithasmallnumberofcohorts.Despitethelargedi!erencewiththreecohorts,thegapintheconÞdenceintervalsismuchsmallerwithsevencohorts,becausethegapsintheaveragestandarderrorsandthedi!erenceinthecriticalvaluesbothdeclinewiththenumberofcohorts.Inotherdistricts,wherethecohort-by-cohortaveragestandarderrorissmallerthanthecohort-schoolclusteredstandarderrors,suchasDistrictsDandF,thecohort-by-cohortconÞdenceintervalsarestillnoticeablylargerwithasmallnumberofcohortsduetotheincreasedcriticalvalue.Bysevencohorts,thedi!erencedisappearsandtheconÞdenceintervalwidthsarevirtuallyidentical.Finally,inTableE.8weconsiderhowfrequentlywecandistinguishteachersfromtheirpeersbasedonVA.Thisisofparticularinterestifdistrictsusevalue-addedestimatestoguideemploymentdecisions.Theteachervalue-addedandstandarderrorsareestimateddistrict-by-district,butthetablereportsaverageresultsacrossallsixdistricts.PanelAshowsthepercentageofteachersfromthebottom10%oftheVAdistributionwhohaveanupperboundontheirconÞdenceintervalbelowthe10th,25th,50th,and90thpercentiles.Fromcolumn2ofPanelA,usingOLS-Lagandnoclustering,7.4%ofteachersinthebottomdecilehaveanupperboundthatisalsobelowthetenthpercentile.Inotherwords,7.4%ofteacherswithVAestimatesinthebottomtenpercenthaveconÞdenceintervalsthatarecompletelycontainedinthebottomdecilewhenwedonotuseanyclustering.Forcomparison,whenusingcohort-schoolclustering,23.5%ofthebottomdecileteachershaveconÞdenceintervalsthatarecompletelycontainedinthebottomdecile.Thefactthatcohort-clusteringleadstomoreteachersinthebottomdecileisinlinewithsomeofthesimulationresults.Aswesawinthesimulations,evenifassumingindependenceacrossclassrooms(orcohorts)iscorrect,simplyclusteringtendstounderestimatethevariancewithasmallnumberofcohorts.Usingthecohort-by-cohortmethod,alsooftenunderestimatedthevariancewithasmallnumberofcohortsinthesimulations,butthecriticalvalueadjustmentleadstohighercoveragerates.Usingthecohort-81by-cohortmethodintheadministrativedata,only0.8%ofbottomdecileteachershaveaconÞdenceintervalwithanupperboundthatisalsobelowthetenthpercentile.Similarly,notclusteringandcohort-schoolclusteringassignconÞdenceintervalsthatarecompletelycontainedbelowthe25thpercentilefor44.1%and77.9%oftheteachers,respectively.However,usingthecohort-by-cohortmethodonlyproducesconÞdenceintervalsthatarecompletelycontainedinthebottomquartileoftheVAdistributionfor1.6%ofteachersinthebottomdecile.Inotherwords,usingthecohort-by-cohortmethodproducesmuchmoreconservativeconclusionsabouttheseparationofteachersacrosstheVAdistribution.Infact,fromcolumn5ofTableE.8,thecohort-by-cohortmethodsuggeststhatonly29%ofbottomdecileteachershavea95%conÞdenceintervalwithanupperboundbelowthe90thpercentile.Ifwetakethatatfacevalue,itsuggeststhatitisdi"culttoexclude71%oftheteachersinthebottomdecilefromhavingatruee!ectinthetopdecile.Fortunately,thecohort-by-cohortconÞdenceintervalsarebetteratdistinguishingteachersintheupperendoftheVAdistribution.InPanelCofTableE.8,weshowthepercentageofteachersinthetopdecilewithlowerboundsabovethe10th,25th,50th,and90thpercentiles.Fromcolumn9,weÞndthat8.1%ofteachersinthetop10%oftheVAdistributionhavelowerboundsthatarealsoabovethe90thpercentilewhenusingthecohort-by-cohortmethod.ThisindicatesmuchbetterseparationthanthelessthanonepercentofthebottomdecileteacherswithconÞdenceintervalsinthebottomdecile.Furthermore,weÞndthatthecohort-by-cohortmethodassignsconÞdenceintervalsto18.5%and45.2%oftopdecileteacherswithlowerboundsabovethe75thand25thpercentiles,respectively.Thebetterseparationoftopdecileteacherscancomefromacombinationofsources.Forexample,conÞdenceintervals,especiallyfromthecohort-by-cohortmethod,shrinkwithmorecohorts.Ingeneral,ifweexpectthatthecohort-by-cohortmethodgeneratessubstantiallybettercoverageratesthantheotheroptions,thensimplyusingOLS-Lagwith,orwithoutclustering,willleadtogreatlyover-estimatingthedegreeofseparationbetweenteachersintheVAdistribution.823.5ConclusionInexploringtheprecisionofteachervalue-addedestimates,weÞrstlookatfoureasilycalculablemethodsofconstructingconÞdenceintervalsforOLS-Lagestimatesofteachervalue-added:noclustering,clusteringattheclassroomlevel,cohort-schoollevel,orschoollevel.Usingsimulations,weÞndthatincertaincasesÑparticularlywhengroupingstudentsbasedonunobservableheterogeneityorunderthepresenceofanunobservablecohortÞxede!ectÑnotclusteringisproblematicasitfailstoaccountforwithinclassroomcorrelation.Furthermore,standardvarianceestimatorsthataremeanttoaccountforwithingroupcorrelation,classroomclusteringandcohort-schoolclusteringinthiscase,canperformwellwithalargenumberofcohorts,butunderestimatethevariancewithasmallnumberofcohorts.Weproposeadi!erentmethodthatwecallOLS-Lagcohort-by-cohort,orsimplycohort-by-cohort,whichincludesacriticalvalueadjustmentand,likeclassroomandcohort-schoolclustering,doesnotrelyonanywithinclassroomindependenceassumption.TheOLS-Lagcohort-by-cohortmethodalsotendstounderestimatethevarianceoftheestimatorwithasmallnumberofcohorts,butproducescoverageratesnear95%underallgroupingscenarioswithrandomassignment,becauseofthecriticalvalueadjustmentusedinconstructingtheconÞdenceintervals.Thereareseveralcaseswherethismethodoutperformstheothers.SpeciÞcally,withasmallnumberofcohortsandwhenstudentsaregroupednon-randomly,theconÞdenceintervalsfromthecohort-by-cohortmethodperformmuchbetterthanthosefromnotclusteringandfromclusteringbyclassroomorcohort-school.Furthermore,withheterogeneitygroupingorcohort-schoolÞxede!ects,usingthecohort-by-cohortmethodoutperformstheconÞdenceintervalsfromnotclusteringevenasthenumberofcohortsincreases,becausenotclusteringcompletelyfailstoaccountforanywithinclasscorrelation.TheconÞdenceintervalsobtainedfromthecohort-by-cohortmethodandtheclassroomandcohort-schoolclusteringmethodsdoconvergewithincreasingcohorts,butusingalargenumberofcohortsisunrealisticintheteacher-valueaddedcontext.Wealsousestudentleveladministrativedatatocomparetheperformanceofthedi!erentmeth-odsinsixdistrictsfromalargestate.WeÞndthatthereissomeheterogeneityinrelativesizeofthecohort-by-cohortstandarderrorsandtheotherstandarderrorsconsidered,butthatinall83casesthecohort-by-cohortconÞdenceintervalsarewiderwithasmallnumberofcohorts,inpartbecauseofthecriticalvalueadjustment.WeÞndthatifweusethecohort-by-cohortmethod,only29%ofteachersinthebottomdecileoftheVAdistributionhavea95%conÞdenceintervalwithanupperboundbelowthe90thpercentileoftheVAdistribution.ThisnumberismuchlowerthanthatproducedusingOLS-Lagestimationwithcohort-schoolclusteringornoclusteringatall.Bothoftheseresultin100%ofbottomdecileteachershavinganupperboundbelowthe75thpercentile.WealsoÞndthat,whilecohort-schoolclusteringassigns75.8%ofthetopdecileteachersalowerboundthatishigherthanthe75thpercentile,placingthemsafelyinthetopquartileoftheVAdistribution,thecohort-by-cohortmethodonlyassigns18.5%ofthetopdecileteachersaconÞdenceintervalthatiscompletelycontainedinthetopquartile.ThisshowsthatthecalculationofconÞ-denceintervalshasimportantimplicationsforidentifyingteachersinthetailsofthedistribution.Thecohort-by-cohortmethodisconservativerelativetotheothermethodsconsidered,suggestingthatusingtheothermethodscouldleaddistrictsandpolicymakerstodrasticallyoverestimatethedegreeofseparationbetweenteachersintheVAdistribution.Inconclusion,wecannotprescribeasinglemethodthatwillnecessarilyprovideaccuratecoverageratesofthetrueteachere!ectsunderallpossibledatascenarios,burrather,weadviseresearchersandpractitionerstoexercisecautionwhendrawinginferencesfromtheseestimates.84APPENDICES85APPENDIXAFiguresforChapter186FigureA.1:AverageTestScoresbyDLAttendanceFigureA.2:LEPAverageTestScoresbyDLAttendance87FigureA.3:AverageTestScoresbyLEPStatusFigureA.4:ProportioninCMSDLSchool88APPENDIXBTablesforChapter189TableB.1:ApplicationNumbersandNeighborhoodSchoolCharacteristicsOne-WaySchoolTwo-WaySchoolsWaddellCollinswoodOaklawnOtherApplicants[1][2][3][4]ApplicantsSiblingPlacement0.2960.2320.2180.230WonFirstChoice0.7820.5610.9510.589DLApplications2.0811.2481.2830.093NeighborhoodSchoolWhite0.3150.2500.1190.197Black0.3830.3890.6070.501Hispanic0.2160.2810.1940.220FRPL0.5740.6520.7600.688LEP0.1800.2320.1750.194EOGExamScoresMath-0.010-0.080-0.318-0.186Reading-0.032-0.136-0.357-0.224N1,1471,11253313,071*Notes:TheÞrstthreerowsdisplayofthetypeandnumberofapplicationssubmittedforallthosesubmittingapplicationsintheCMSschoolchoicelottery.Therestofthetableshowsmeancharacteristicsoftheneighborhoodschoolsthatapplicantsareassignedtoweightedbythenumberofapplicantsfromeachschool.EverythingisbasedonÞrstchoiceschool.90TableB.2:SecondandThirdChoicesSecondChoiceDLThirdChoiceDL[1][2]AttendDLSchool0.1050.087SecondChoiceDL1.0000.219ThirdChoiceDL0.2031.000AssignmentCollinswood0.0000.000Waddell0.0800.072Oaklawn0.0770.075AnyDLChoice0.1570.147ChoiceCollinswood0.3390.245Smith0.3320.475Oaklawn0.3290.279WonFirstChoice0.4900.426SecondChoice0.1430.094ThirdChoice0.0700.117AnyChoice0.7030.638Observations286265TableB.3:DualLanguageandNeighborhoodSchoolCharacteristicsWaddellCollinswoodOaklawnAreaWaddellAreaCollinswoodAreaOaklawnOtherMagnets[1][2][3][4][5][6][7]TeachingExperience0-3Years0.3290.3200.2940.5010.2980.5420.29911+Years0.3000.2720.3670.2550.3590.1950.374FRPL0.8060.3340.7590.5730.9000.6880.568AYPTargets0.8690.9860.8711.0000.7791.0000.873PctatGradeLevelReading0.6120.8220.6120.8800.4960.7580.718Math0.6980.8780.7050.9440.5310.7530.733KGClassSize18.05721.40019.04922.33318.20019.33318.736*Note:Averagecharacteristicsateachduallanguageschool,fortheneighborhoodschoolswithzonescontiguoustoeachduallanguageschool,andallothermagnetschools.91TableB.4:SummaryandBalance-EnglishProÞcientSampleApplicationSampleEstimationSampleWonLostTestWonLostTest[1][2][3][4][5][6]Attended(K/First)0.8700.3580.509***0.8990.3750.521***(0.337)(0.480)(0.042)(0.302)(0.485)(0.050)WonAnyChoice1.0000.3640.645***1.0000.3670.649***(0.000)(0.482)(0.050)(0.000)(0.483)(0.049)WonAnyDLChoice1.0000.1130.880***1.0000.1170.879***(0.000)(0.317)(0.036)(0.000)(0.322)(0.036)Female0.5460.5130.0120.5330.5080.003(0.499)(0.501)(0.045)(0.500)(0.501)(0.050)Black0.3040.354-0.0430.3030.387-0.075*(0.461)(0.479)(0.042)(0.460)(0.488)(0.041)White0.3720.2680.0170.3730.2620.027(0.484)(0.444)(0.043)(0.484)(0.441)(0.047)Hispanic0.1120.215-0.0380.1110.206-0.048(0.316)(0.412)(0.039)(0.315)(0.405)(0.041)SecondChoiceDL0.4070.3280.0740.4080.3150.081(0.492)(0.470)(0.044)(0.492)(0.465)(0.049)ThirdChoiceDL0.2450.2120.0540.2470.2100.037(0.431)(0.409)(0.036)(0.432)(0.408)(0.049)Non-missingTestScores0.8470.8210.0001.0001.000(0.361)(0.384)(0.035)(0.000)(0.000)FRPL0.1810.3730.1740.351(0.374)(0.467)(0.380)(0.478)EOGMathScore0.5240.2810.269**(1.004)(0.979)(0.110)EOGReadingScore0.4520.2490.136(0.941)(0.897)(0.102)LotteryFEXXFRPL-YearDummiesXXNeighborhoodSchoolFEXXObservations339302641287248535NumberofClusters464492TableB.5:SummaryandBalance-ESL/LEPSampleApplicationSampleEstimationSampleWonLostTestWonLostTest[1][2][3][4][5][6]Attended(K/First)0.8910.2750.642***0.9290.2900.669***(0.312)(0.448)(0.067)(0.258)(0.455)(0.068)WonAnyChoice1.0000.2980.735***1.0000.2970.711***(0.000)(0.459)(0.061)(0.000)(0.458)(0.060)WonAnyDLChoice1.0000.0230.976***1.0000.0210.987***(0.000)(0.152)(0.013)(0.000)(0.143)(0.013)Female0.4570.444-0.0460.4420.455-0.052(0.500)(0.498)(0.074)(0.499)(0.500)(0.096)Black0.0540.0530.0160.0620.0550.026(0.227)(0.224)(0.027)(0.242)(0.229)(0.025)White0.0540.0290.0400.0440.0340.030(0.227)(0.169)(0.030)(0.207)(0.183)(0.031)Hispanic0.8290.895-0.099**0.8230.890-0.105**(0.378)(0.308)(0.043)(0.383)(0.314)(0.046)SecondChoiceDL0.2020.205-0.107*0.1950.234-0.136**(0.403)(0.405)(0.058)(0.398)(0.425)(0.065)ThirdChoiceDL0.1470.1110.0030.1420.117-0.028(0.356)(0.315)(0.034)(0.350)(0.323)(0.036)Non-missingTestScores0.8760.8480.0551.0001.000(0.331)(0.360)(0.046)(0.000)(0.000)FRPL0.6790.7260.6900.724(0.462)(0.441)(0.464)(0.448)EOGMathScore0.082-0.1990.154(0.802)(0.888)(0.141)EOGReadingScore-0.074-0.3130.223(0.833)(0.797)(0.141)LotteryFEXXFRPL-YearDummiesXXNeighborhoodSchoolFEXXObservations113145300113145258NumberofClusters393693TableB.6:ImpactofAttendingaDualLanguageSchoolonAchievementPanelA:EnglishSampleOLSMathReadingMathReadingFirstStageITTLATEITTLATE[1][2][3][4][5][6][7]WonFirstChoice0.464***0.042*0.024(0.064)(0.022)(0.015)AttendDLSchool-0.004-0.022*0.089*0.053*(0.018)(0.011)(0.047)(0.032)NeighborhoodSchoolFEXXXXXXXObservations1,4721,4721,4721,4721,4721,4721,472NumberofClusters44444444444444PanelB:ESL/LEPSampleOLSMathReadingMathReadingFirstStageITTLATEITTLATE[1][2][3][4][5][6][7]WonFirstChoice0.667***0.052**0.042*(0.069)(0.024)(0.024)AttendDLSchool0.052***0.065***0.078**0.064**(0.019)(0.019)(0.034)(0.032)NeighborhoodSchoolFEXXXXXXXObservations809809809809809809809NumberofClusters36363636363636Robuststandarderrorsinparentheses***p<0.01,**p<0.05,*p<0.1*Notes:EachregressionincludeslotteryÞxede!ects(priority-year-program)aswellascontrolsforfemale,race,frpl-year,exceptionality,gradeofexam,yearofexam,andneighborhoodschoolÞxede!ects.AttendanceismeasuredbywhetherthestudentattendedaDLschoolinkindergartenorÞrstgradeandinteractedwithyearsoftreatment(gradeplusone).Standarderrorsareclusteredbylottery.94TableB.7:ImpactofAttendingaDualLanguageSchoolonAchievementPanelA:EnglishSampleOLSMathReadingMathReadingFirstStageITTLATEITTLATE[1][2][3][4][5][6][7]WonFirstChoice0.471***0.040*0.027*(0.081)(0.020)(0.014)AttendDLSchool0.011-0.0100.086**0.057*(0.017)(0.014)(0.043)(0.032)Observations1,4721,4721,4721,4721,4721,4721,472NumberofClusters44444444444444PanelB:ESL/LEPSampleOLSMathReadingMathReadingFirstStageITTLATEITTLATE[1][2][3][4][5][6][7]WonFirstChoice0.635***0.040**0.044**(0.084)(0.019)(0.020)AttendDLSchool0.039**0.065***0.063**0.069**(0.019)(0.023)(0.031)(0.030)Observations809809809809809809809NumberofClusters36363636363636Robuststandarderrorsinparentheses***p<0.01,**p<0.05,*p<0.1*Notes:EachregressionincludeslotteryÞxede!ects(priority-year-program)aswellascontrolsforfemale,race,frpl-year,exceptionality,gradeofexam,andyearofexam.AttendanceismeasuredbywhetherthestudentattendedaDLschoolinkindergartenorÞrstgradeandinteractedwithyearsoftreatment(gradeplusone).Standarderrorsareclusteredbylottery.95TableB.8:AttritionandWeightingPanelA:SummaryofProbabilitiesofStayingFullSampleEnglishSampleESL/LEPSampleWinnersLosersWinnersLosersWinnersLosers[1][2][3][4][5][6]AveragePr(Stay)0.8550.8310.8470.8210.8760.848SDPr(Stay)0.0310.0380.0430.0600.0400.047APE0.0240.0150.019(SE)(0.024)(0.030)(0.042)N468473339302129171PanelB:Non-RandomAttritionCoe!cientsonIndicatorforWinningLotteryFullSampleEnglishSampleESL/LEPSample[1][2][3]NoControls0.0240.0260.028(SE)(0.024)(0.030)(0.040)Controls+LotteryFE0.019-0.0030.047(SE)(0.024)(0.029)(0.036)+NeighborhoodSchoolFE0.025-0.0020.052(SE)(0.029)(0.038)(0.049)N941641300*Notes:PanelAsummarizesestimatedprobabilitesofhavingatleastonesetoftestscoresavailable.Theestimatesarebasedonlogitregressionsincludinggender,race,frpl-year,andanindicatorforwinningthelottery.PanelBshowsestimatedcoe"cientsfromOLSregressionsofhavingatleastonesetoftestscoresavailableonwinningthelottery.Thebaselinecontrolsaregender,race,andfrpl-year.ThesecondsetofOLSestimatesalsoconditiononlotteryÞxede!ects,andthethirdsetincludeneighborhoodschoolÞxede!ects.96TableB.9:ImpactofAttendingaDualLanguageSchoolonAchievement-WeightedPanelA:EnglishSampleOLSMathReadingMathReadingFirstStageITTLATEITTLATE[1][2][3][4][5][6][7]WonFirstChoice0.468***0.042*0.024(0.065)(0.022)(0.015)AttendDLSchool-0.004-0.022*0.089*0.052*(0.018)(0.012)(0.046)(0.031)NeighborhoodSchoolFEXXXXXXXObservations1,4721,4721,4721,4721,4721,4721,472NumberofClusters44444444444444PanelB:ESL/LEPSampleOLSMathReadingMathReadingFirstStageITTLATEITTLATE[1][2][3][4][5][6][7]WonFirstChoice0.673***0.052**0.043*(0.071)(0.025)(0.025)AttendDLSchool0.053***0.066***0.078**0.064**(0.019)(0.020)(0.033)(0.032)NeighborhoodSchoolFEXXXXXXXObservations809809809809809809809NumberofClusters36363636363636Robuststandarderrorsinparentheses***p<0.01,**p<0.05,*p<0.1*Notes:EachregressionincludeslotteryÞxede!ects(priority-year-program)aswellascontrolsforfemale,race,frpl-year,exceptionality,gradeofexam,yearofexam,andneighborhoodschoolÞxede!ects.AttendanceismeasuredbywhetherthestudentattendedaDLschoolinkindergartenorÞrstgradeandinteractedwithyearsoftreatment(gradeplusone).Regressionareweightedbytheinverseprobabilityofhavingtestscoresavailableinthedata.Weightsweregeneratedfromlogitregressions.Standarderrorsareclusteredbylottery.97TableB.10:HeterogeneousE!ectsPanelA:EnglishSampleGenderSchoolTypeRace/EthnicityFemaleMaleOne-WayTwo-WayWhiteBlackHispanic[1][2][3][4][5][6][7]Math0.106**0.0680.0810.090*0.190**0.0460.090*(0.054)(0.046)(0.127)(0.046)(0.076)(0.053)(0.048)Reading0.069**0.0300.0320.054**0.0840.0340.115***(0.034)(0.043)(0.104)(0.028)(0.055)(0.034)(0.037)NeighborhoodSchoolFEXXXObservations1,4721,4721,472NumberofClusters444444PanelB:ESL/LEPSampleGenderSchoolTypeRace/EthnicityFemaleMaleOne-WayTwo-WayWhiteBlackHispanic[1][2][3][4][5][6][7]Math0.0510.090**-0.0180.079**--0.083***(0.048)(0.038)(0.176)(0.034)--(0.031)Reading0.0110.087**-0.0550.065**--0.062**(0.045)(0.039)(0.157)(0.031)--(0.031)NeighborhoodSchoolFEXXXObservations809809809NumberofClusters363636Robuststandarderrorsinparentheses***p<0.01,**p<0.05,*p<0.1*Notes:EachregressionincludeslotteryÞxede!ects(priority-year-program)aswellascontrolsforfemale,race,frpl-year,exceptionality,gradeofexam,yearofexam,andneighborhoodschoolÞxede!ects.AttendanceismeasuredbywhetherthestudentattendedaDLschoolinkindergartenorÞrstgradeandinteractedwithyearsoftreatment(gradeplusone).Heterogeneouse!ectsareestimatedusinginteractionswiththeassignmentvariableandinstrumentingforinteractionswiththeattendancevariable.Standarderrorsareclusteredbylottery.98TableB.11:E!ectsbyGradePanelA:EnglishSampleOLSITTLATEMathReadingMathReadingMathReading[1][2][3][4][5][6]GradeThird-0.011-0.152*0.189**0.1080.374**0.215(0.120)(0.082)(0.092)(0.079)(0.171)(0.147)Fourth0.042-0.0860.2110.0700.4460.151(0.112)(0.078)(0.143)(0.093)(0.284)(0.188)Fifth-0.009-0.1150.1390.216**0.2640.431**(0.138)(0.094)(0.150)(0.096)(0.276)(0.205)Sixth-0.107-0.1070.327**0.1360.721**0.298(0.121)(0.091)(0.141)(0.135)(0.346)(0.295)NeighborhoodSchoolFEXXXXXXObservations1,4721,4721,4721,4721,4721,472NumberofClusters444444444444PanelB:ESL/LEPSampleOLSITTLATEMathReadingMathReadingMathReading[1][2][3][4][5][6]GradeThird0.0550.312***0.244**0.244*0.393**0.390**(0.097)(0.101)(0.109)(0.128)(0.174)(0.187)Fourth0.374**0.254*0.413**0.1530.657**0.238(0.138)(0.134)(0.171)(0.167)(0.274)(0.248)Fifth0.397***0.370***0.320*0.2080.471**0.302(0.131)(0.100)(0.163)(0.137)(0.229)(0.188)Sixth0.417***0.408***0.367**0.364**0.542**0.531***(0.134)(0.145)(0.171)(0.140)(0.231)(0.183)NeighborhoodSchoolFEXXXXXXObservations809809809809809809NumberofClusters363636363636Robuststandarderrorsinparentheses***p<0.01,**p<0.05,*p<0.1*Notes:EachregressionincludeslotteryÞxede!ectsandcontrolsforfemale,race,frpl-year,ex-ceptionality,gradeofexam,yearofexam,andneighborhoodschoolÞxede!ects.AttendanceismeasuredbywhetherthestudentattendedaDLschoolinkindergartenorÞrstgradeandintear-actedwitheachexamgrade.Instrumentsareinteractionsbetweengradeofexamandtheindicatorforwinningthelottery.Standarderrorsareclusteredbylottery.99TableB.12:ImpactofDualLanguageSchoolingonLEPStatusLimitedEnglishProÞcientOLSLATE[1][2][3][4]AttendDLSchoolGrade3-0.0250.021-0.0320.045(0.047)(0.055)(0.084)(0.101)Grade4-0.148**-0.120-0.159-0.107(0.071)(0.080)(0.137)(0.160)Grade5-0.192**-0.176*-0.071-0.051(0.079)(0.093)(0.118)(0.144)Grade6-0.210***-0.196**-0.168*-0.141(0.073)(0.076)(0.086)(0.111)NeighborhoodSchoolFEXXObservations809809809809NumberofClusters36363636Robuststandarderrorsinparentheses***p<0.01,**p<0.05,*p<0.1*Notes:EachregressionincludeslotteryÞxede!ects(priority-year-program)aswellascontrolsforfemale,race,frpl-year,exceptionality,gradeofobservation,andyearofobservation.AttendanceismeasuredbywhetherthestudentattendedaDLschoolinkindergartenorÞrstgradeandinteractedwithgradedummyvariables.EstimatesarefromOLSand2SLSinteractionterms.Standarderrorsareclusteredbylottery.100APPENDIXCFiguresforChapter2101FigureC.1:MTI-WeekdayFigureC.2:MTI-Weekend102FigureC.3:FTI-WeekdayFigureC.4:FTI-Weekend103FigureC.5:InvestmentsfromMother/Father-BoysFigureC.6:InvestmentsfromMother/Father-Girls104APPENDIXDTablesforChapter2105TableD.1:WaveISummarybyGenderandHouseholdStructureBoys-NoChangeBoys-ChangeGirls-NoChangeGirls-ChangeBoth<2Both<2Both<2Both<2[1][2][3][4][5][6][7][8]MotherÕsInvestmentsTotal24.80218.84725.56818.11626.60821.69427.45521.946(15.206)(15.226)(15.264)(17.029)(15.486)(19.368)(15.495)(18.872)Weekday3.0732.2893.1002.2383.1992.7053.3012.803(2.415)(2.291)(2.372)(2.554)(2.477)(3.074)(2.566)(2.606)Weekend4.7193.7015.0353.4645.3064.0855.4763.964(3.206)(3.266)(3.156)(3.368)(3.192)(3.504)(3.195)(3.681)FatherÕsInvestmentsTotal16.3713.53116.7715.05615.0612.21813.2465.160(11.397)(7.945)(13.147)(9.518)(11.440)(6.621)(10.919)(8.650)Weekday1.6180.3511.8720.4981.4930.2411.3820.586(1.615)(1.047)(2.125)(1.269)(1.712)(0.927)(1.697)(1.123)Weekend4.1410.8873.7051.2843.7970.5063.1671.116(3.284)(2.139)(3.001)(2.717)(3.158)(1.686)(2.910)(2.112)Age6.5337.2884.6176.2836.8487.3664.9315.540(3.850)(3.626)(3.149)(3.214)(3.847)(3.774)(3.430)(3.429)#BioSibs1.3091.1911.0711.1161.2721.1631.1381.000InHH(1.102)(1.145)(0.968)(1.011)(0.983)(1.074)(1.094)(1.035)White0.6300.2570.4900.2790.6720.2580.4780.292Black0.2190.6460.4710.5810.1640.6530.4340.597Hispanic0.0940.0470.0060.0930.1060.0360.0380.028StepmotherInHH0.0360.0350.0140.028OutofHH0.0900.1050.0930.042StepfatherInHH0.1260.1510.1270.083OutofHH0.0320.0230.0430.083ParentsinHH0.9680.8970.9680.862andMarriedObservations7544441558672041815972*Notes:Thistabledisplaystheaveragesbygenderandhouseholdtypewithstandarddeviationsinparentheses.ColumnslabeledNoChangecontainnumbersforchildrenwhohavenochangeinhouseholdstructureatsomepointinthesample,andthoselabeledChangecontainnumbersforchildrenwhoundergosomechangethroughoutthesampleperiod.ColumnslabeledBothdisplaynumbersforchildrenwhohavebothbiologicalparentsinthehouseholdduringtheÞrstwave,andthoselabeled<2areforthosewithlessthanbothbiologicalparentsinthehouseholdduringtheÞrstwave.106TableD.2:OLSEstimatesofGenderGapsinTimeInvestmentsMotherFatherTotalWeekdayWeekendTotalWeekdayWeekend[1][2][3][4][5][6]SingleMotherHHXMale1.2430.0090.620-7.567-0.774-1.791(0.963)(0.158)(0.214)(0.743)(0.102)(0.209)Female1.1390.0580.430-6.548-0.699-1.489(1.001)(0.163)(0.221)(0.725)(0.099)(0.209)Other-11.513-1.537-1.881-2.069-0.135-0.607(0.999)(0.159)(0.229)(0.936)(0.127)(0.238)Male-Female0.103-0.0490.190-1.019-0.076-0.302(StandardError)(0.699)(0.106)(0.164)(0.441)(0.0596)(0.126)Observations6,6996,7866,7266,6996,7866,726R-squared0.2800.2330.1630.2690.1570.229Robuststandarderrorsinparentheses*Notes:ThistabledisplaysestimatesfromOLSregressionsoftimeinvestmentsonchildgenderinteractedwithadummyforbeinginasingle-motherhousehold.Theomittedcategoryisahouseholdwithbothbiologicalparentspresent.Thefourthrowdisplaysthedi!erenceinthemaleandfemaleinteractions.Eachregressionincludescontrolsforthenumberofbiologicalsiblingsinthehousehold,theCDSwave,amarriageindicatorforparentsinthesamehousehold,dummyvariablesforhavingstepparentsin/outofthehousehold,genderinteractionswithageandage-squared,andamaledummyvariable.Standarderrorsareclusteredbyindividualanddisplayedbelowtheestimates.107TableD.3:FixedE!ectsEstimatesofGenderGapsinTimeInvestmentsPanelA:NoControlsMotherFatherTotalWeekdayWeekendTotalWeekdayWeekend[1][2][3][4][5][6]SingleMotherHHXMale-5.550-0.707-0.905-10.346-1.222-2.077(1.229)(0.186)(0.266)(0.933)(0.131)(0.232)Female-4.146-0.509-0.872-8.425-0.839-2.032(1.388)(0.209)(0.302)(0.767)(0.114)(0.200)Other-15.549-2.023-2.611-6.057-0.711-1.291(1.451)(0.203)(0.332)(1.116)(0.151)(0.301)Male-Female-1.405-0.198-0.033-1.921-0.383-0.044(StandardError)(1.799)(0.270)(0.390)(1.166)(0.167)(0.297)Observations6,6996,7866,7266,6996,7866,726R-squared0.0310.0220.0190.0610.0370.041Individuals3,2833,3083,2873,2833,3083,287PanelB:WithControlsMotherFatherTotalWeekdayWeekendTotalWeekdayWeekend[1][2][3][4][5][6]SingleMotherHHXMale1.2730.0070.558-7.306-0.903-1.332(1.713)(0.274)(0.358)(1.235)(0.173)(0.318)Female2.4420.2140.562-4.941-0.503-1.105(1.863)(0.281)(0.396)(1.135)(0.159)(0.312)Other-7.433-1.036-1.109-2.260-0.342-0.314(1.948)(0.296)(0.409)(1.365)(0.191)(0.354)Male-Female-1.169-0.207-0.003-2.364-0.400-0.227(StandardError)(1.612)(0.243)(0.372)(1.084)(0.158)(0.289)Observations6,6996,7866,7266,6996,7866,726R-squared0.2600.2260.1360.1180.0700.094Individuals3,2833,3083,2873,2833,3083,287Robuststandarderrorsinparentheses*Notes:ThistabledisplaysestimatesfromÞxede!ectsregressionsoftimeinvestmentsonchildgenderinteractedwithadummyforbeinginasingle-motherhousehold.Theomittedcategoryisahouseholdwithbothbiologicalparentspresent.Thefourthrowofeachpaneldisplaysthedi!erenceinthemaleandfemaleinteractions.RegressionsinPanelAdonotuseanycontrolvariables.EachregressioninPanelBincludescontrolsforthenumberofbiologicalsiblingsinthehousehold,theCDSwave,amarriageindicatorforparentsinthesamehousehold,dummyvariablesforhavingstepparentsin/outofthehousehold,andgenderinteractionswithageandage-squared.Standarderrorsareclusteredbyindividualanddisplayedbelowtheestimates.108TableD.4:GenderGapsinInvestmentsbyInitialHHStructureBothinWave1<2inWave1MotherFatherMotherFather[1][2][3][4]SingleMotherHHXMale1.948-9.0343.1970.344(2.137)(1.644)(3.259)(1.737)Female1.259-4.8006.7380.530(2.167)(1.543)(3.682)(1.487)Other-9.022-2.865-4.1973.323(2.761)(2.119)(3.319)(1.728)Male-Female0.688-4.234-3.541-0.186(StandardError)(2.028)(1.540)(2.876)(1.668)Observations4,2844,2842,4152,415R-squared0.2880.1460.2190.068Individuals1,9981,9981,2851,285Robuststandarderrorsinparentheses*Notes:ThistabledisplaysestimatesfromÞxede!ectsregressionsoftotalweeklytimeinvestmentsonchildgenderinteractedwithadummyforbeinginasingle-motherhousehold.Theomittedcategoryisahouseholdwithbothbiologicalparentspresent.Columns1and2restricttothesampleofchildrenwhowereinatwo-parenthouseholdintheÞrstwave,andcolumns3and4restricttothosewhowerenotlivingwithbothparentsintheÞrstwave.Thefourthrowdisplaysthedi!erenceinthemaleandfemaleinteractions.Eachregressioncontrolsforthenumberofbiologicalsiblingsinthehousehold,theCDSwave,amarriageindicatorforparentsinthesamehousehold,dummyvariablesforhavingstepparentsin/outofthehousehold,andgenderinteractionswithageandage-squared.Standarderrorsareclusteredbyindividualanddisplayedbelowtheestimates.109TableD.5:FEGenderGapsinProbabilityofPTI>0MotherFatherTotalWeekdayWeekendTotalWeekdayWeekend[1][2][3][4][5][6]SingleMotherHHXMale0.0400.0470.055-0.419-0.377-0.331(0.032)(0.043)(0.044)(0.054)(0.054)(0.054)Female0.0680.1070.041-0.409-0.297-0.285(0.033)(0.044)(0.044)(0.052)(0.056)(0.050)Other-0.435-0.414-0.364-0.187-0.094-0.071(0.052)(0.060)(0.057)(0.059)(0.065)(0.057)Male-Female-0.028-0.0600.014-0.010-0.080-0.047(StandardError)(0.033)(0.039)(0.044)(0.049)(0.050)(0.049)Observations6,6996,7866,7266,6996,7866,726R-squared0.1520.1550.1330.2020.1170.178Individuals3,2833,3083,2873,2833,3083,287Robuststandarderrorsinparentheses*Notes:ThistabledisplaysestimatesfromÞxede!ectsregressionsofadummyvariableforhavingapositivetimeinvestmentonchildgenderinteractedwithadummyforbeinginasingle-motherhousehold.Theomittedcategoryisahouseholdwithbothbiologicalparentspresent.Thefourthrowdisplaysthedi!erenceinthemaleandfemaleinteractions.Eachregressioncontrolsforthenumberofbiologicalsiblingsinthehousehold,theCDSwave,amarriageindicatorforparentsinthesamehousehold,dummyvariablesforhavingstepparentsin/outofthehousehold,andgenderinteractionswithageandage-squared.Standarderrorsareclusteredbyindividualanddisplayedbelowtheestimates.110TableD.6:GenderGapsinProbabilityofPTI>0byInitialHHStructurePanelA:BothParentsinHHinWave1MotherFatherTotalWeekdayWeekendTotalWeekdayWeekend[1][2][3][4][5][6]SingleMotherHHXMale0.0090.049-0.007-0.595-0.507-0.449(0.035)(0.052)(0.052)(0.058)(0.068)(0.064)Female0.0120.074-0.023-0.530-0.397-0.338(0.034)(0.051)(0.049)(0.060)(0.069)(0.061)Other-0.452-0.417-0.432-0.1380.016-0.042(0.081)(0.092)(0.075)(0.079)(0.096)(0.081)Male-Female-0.003-0.0250.016-0.065-0.110-0.110(StandardError)(0.034)(0.049)(0.053)(0.058)(0.065)(0.061)Observations4,2844,3234,2934,2844,3234,293R-squared0.1420.1470.1460.3040.1490.226Individuals1,9982,0061,9961,9982,0061,996PanelB:LessThanTwoParentsinHHinWave1MotherFatherTotalWeekdayWeekendTotalWeekdayWeekend[1][2][3][4][5][6]SingleMotherHHXMale0.1260.0660.1250.004-0.0220.031(0.070)(0.087)(0.093)(0.101)(0.087)(0.099)Female0.1590.1660.111-0.090-0.025-0.035(0.076)(0.094)(0.091)(0.098)(0.097)(0.088)Other-0.358-0.393-0.2760.0510.0720.140(0.077)(0.092)(0.102)(0.097)(0.093)(0.092)Male-Female-0.033-0.1010.0130.0940.00260.066(StandardError)(0.076)(0.081)(0.084)(0.089)(0.089)(0.084)Observations2,4152,4632,4332,4152,4632,433R-squared0.1720.1730.1290.1020.0800.112Individuals1,2851,3021,2911,2851,3021,291Robuststandarderrorsinparentheses*Notes:ThistabledisplaysestimatesfromÞxede!ectsregressionsofadummyvariableforhavingapositivetimeinvestmentonchildgenderinteractedwithadummyforbeinginasingle-motherhousehold.Theomittedcategoryisahouseholdwithbothbiologicalparentspresent.PanelArestrictstothesampleofchildrenwhowereinatwo-parenthouseholdintheÞrstwave,andPanelBrestrictstothosewhowerenotlivingwithbothparentsintheÞrstwave.Thefourthrowofeachpaneldisplaysthedi!erenceinthemaleandfemaleinteractions.Eachregressioncontrolsforthenumberofbiologicalsiblingsinthehousehold,theCDSwave,amarriageindicatorforparentsinthesamehousehold,dummyvariablesforhavingstepparentsin/outofthehousehold,andgenderinteractionswithageandage-squared.Standarderrorsareclusteredbyindividualanddisplayedbelowtheestimates.111TableD.7:GenderGapsbyAgePanelA:MothersMother-TotalWeeklyInvestmentAges0-5Ages6-10Ages11-1516andOver[1][2][3][4]SingleMotherHHXMale1.0932.1122.235-2.451(2.096)(1.815)(1.833)(2.175)Female2.4862.7253.295-0.205(2.484)(1.975)(1.976)(2.403)Male-Female-1.393-0.612-1.060-2.246(StandardError)(2.671)(1.844)(1.843)(2.526)Observations6,6996,6996,6996,699Individuals3,2833,2833,2833,283PanelB:FathersFather-TotalWeeklyInvestmentAges0-5Ages6-10Ages11-1516andOver[1][2][3][4]SingleMotherHHXMale-9.701-7.477-5.321-4.231(1.428)(1.274)(1.311)(1.577)Female-9.290-5.610-1.991-0.633(1.251)(1.209)(1.177)(1.325)Male-Female-0.411-1.867-3.331-3.598(StandardError)(1.399)(1.211)(1.203)(1.590)Observations6,6996,6996,6996,699Individuals3,2833,2833,2833,283Robuststandarderrorsinparentheses*Notes:EachpaneldisplaysestimatesfromaÞxede!ectsregressionoftotalweeklytimeinvestmentsonchildgenderinteractedwithadummyforbeinginasingle-motherhouseholdforfourdi!erentagegroups.Theomittedcategoryisahouseholdwithbothbiologicalparentspresent.Thethirdrowofeachpaneldisplaysthedi!erencesinthemaleandfemaleinteractions.Eachregressioncontrolsforthenumberofbiologicalsiblingsinthehousehold,theCDSwave,amarriageindicatorforparentsinthesamehousehold,dummyvariablesforhavingstepparentsin/outofthehousehold,andgenderinteractionswithageandage-squared.Standarderrorsareclusteredbyindividualanddisplayedbelowtheestimates.112TableD.8:GenderGapsbyRaceMotherFatherWhiteBlackWhiteBlack[1][2][3][4]SingleMotherHHXMale-0.3001.635-9.461-6.390(2.091)(2.018)(1.561)(1.417)Female2.8663.085-5.462-4.373(2.352)(2.244)(1.362)(1.294)Male-Female-3.166-1.450-3.999-2.017(StandardError)(2.500)(2.296)(1.658)(1.426)Observations6,6996,6996,6996,699Individuals3,2833,2833,2833,283Robuststandarderrorsinparentheses*Notes:ThistabledisplaysestimatesfromaÞxede!ectsre-gressionoftotalweeklytimeinvestmentsonchildgenderin-teractedwithadummyforbeinginasingle-motherhouseholdbyrace.Theomittedcategoryisahouseholdwithbothbio-logicalparentspresent.Thethirdrowofeachpaneldisplaysthedi!erencesinthemaleandfemaleinteractions.Eachre-gressioncontrolsforthenumberofbiologicalsiblingsinthehousehold,theCDSwave,amarriageindicatorforparentsinthesamehousehold,dummyvariablesforhavingstepparentsin/outofthehousehold,andgenderinteractionswithageandage-squared.Standarderrorsareclusteredbyindividualanddisplayedbelowtheestimates.113TableD.9:GenderGapsbyActivityTypeMotherFather[1][2]PassiveLeisure0.222-0.723(0.749)(0.461)ActiveLeisure-0.858-0.753(0.565)(0.426)Entertainment-0.060-0.212(0.369)(0.330)TendingtoNeeds0.100-0.483(0.567)(0.365)ObtainingGoodsandServices-0.182-0.195(0.445)(0.265)HouseholdActivity-0.175-0.121(0.222)(0.097)Childcare0.183-0.005(0.171)(0.015)Observations6,6996,699Individuals3,2833,283Robuststandarderrorsinparentheses*Notes:EachestimateisfromaÞxede!ectsregressionofweeklytimeinvestmentsforaspeciÞcactivitycategoryonchildgenderinteractedwithadummyforbeinginasingle-motherhousehold.Theomittedcategoryisahouseholdwithbothbiologicalparentspresent.Eachestimateisfromthedi!erenceinthemaleandfe-maleinteractionterms.Eachregressioncontrolsforthenumberofbiologicalsiblingsinthehousehold,theCDSwave,amarriageindicatorforparentsinthesamehousehold,dummyvariablesforhavingstepparentsin/outofthehousehold,andgenderinterac-tionswithageandage-squared.Standarderrorsareclusteredbyindividualanddisplayedbelowtheestimates.114TableD.10:ChildBehaviorandHHStructureExternalizingInternalizingPositiveBehaviorBehaviorBehavior[1][2][3]SingleMotherHHXMale-0.1130.009-0.008(0.134)(0.136)(0.129)Female0.1600.016-0.117(0.136)(0.145)(0.137)Other-0.163-0.0920.146(0.143)(0.157)(0.148)Male-Female-0.273-0.0070.110(StandardError)(0.117)(0.123)(0.132)Observations6,0606,0586,078R-squared0.0260.0270.010NumberofIndividuals3,2003,1933,197Robuststandarderrorsinparentheses*Notes:ThistabledisplaysestimatesfromÞxede!ectsregressionsofparentratedchildbehaviorsonchildgenderinteractedwithadummyforbeinginasingle-motherhousehold.Theomittedcategoryisahouse-holdwithbothbiologicalparentspresent.Thefourthrowofdisplaysthedi!erenceinthemaleandfemaleinteractions.Eachregressionincludescontrolsforthenumberofbiologicalsiblingsinthehousehold,theCDSwave,amarriageindicatorforparentsinthesamehousehold,dummyvariablesforhavingstepparentsin/outofthehousehold,andgenderin-teractionswithageandage-squared.Standarderrorsareclusteredbyindividualanddisplayedbelowtheestimates.115APPENDIXETablesforChapter3116TableE.1:SummaryofSimulationDesignDistribution(Mean,SD)ElementCase1Case2ScorebaseAit!1N(0,0.25)TeacherÞxede"ect"i,tN(0.5,0.25)StudentÞxede"ectciN(0,1)N(0,0.25)CohortÞxede"ectcsi.N(0,0.25)Randomerror#itN(0,1)117TableE.2:AverageStandardErrorsandCoverageRates(StudentFE-N(0,1),noCohort-SchoolFE)AverageofTeacherAverageSEAverageCoverageRate[1][2][3][4][5][6][7][8][9][10]NumberAvgStDevCohort-Cohort-ofofNoneClassschoolSchoolNoneClassschoolSchoolPANELA-Samplecontains5schoolsRandomGrouping-RandomAssignment10.4190.4350.95630.2530.2510.2020.2040.0080.9460.8260.8240.04770.1600.1640.1490.1500.0030.9560.9140.9130.032200.0950.0970.0950.0950.0010.9530.9420.9430.020DynamicGrouping-RandomAssignment10.4310.4350.95430.2570.2510.2020.2040.0100.9440.8200.8180.06470.1590.1640.1490.1500.0040.9570.9080.9080.045200.0990.0970.0950.0950.0020.9470.9350.9330.023HeterogeneityGrouping-RandomAssignment10.8880.3990.54630.5050.2450.4040.4230.0080.6620.8220.8280.02470.3400.1620.3040.3170.0030.6410.8980.9070.015200.2040.0970.1880.1950.0010.6460.9180.9280.010PANELB-Samplecontains50schoolsRandomGrouping-RandomAssigment10.4540.4460.94730.2600.2580.1980.1980.0030.9460.8120.8120.01570.1680.1690.1540.1540.0010.9520.9080.9080.010200.0970.1000.0970.0970.0000.9570.9440.9440.006DynamicGrouping-RandomAssignment10.4510.4460.94830.2540.2580.1980.1990.0040.9540.8230.8220.02370.1650.1690.1530.1540.0010.9550.9140.9140.014200.1010.1000.0970.0970.0010.9470.9340.9340.008HeterogeneityGrouping-RandomAssignment10.8940.4100.58630.5180.2510.4040.4060.0030.6470.8120.8120.00870.3350.1670.3090.3100.0010.6680.9060.9070.005200.2010.0990.1940.1940.0000.6630.9310.9320.003*Notes:Forallsimulations,lambdaissetto0.5.Theteachere!ectsareestimatedusingpooleddynamicOLS.Columns3-6displaytheaveragestandarderrorsbyclusteringtype.Thetypesofclusteringarenoclustering,classroomlevelclustering,cohort-schoollevelclustering,andschoollevelclustering.Columns7-10displaytheaveragecoveragerateforeachtypeofclustering.118TableE.3:AverageStandardErrorsandCoverageRates(StudentFE-N(0,0.25),Cohort-SchoolFE-N(0,0.25))AverageofTeacherAverageSEAverageCoverageRate[1][2][3][4][5][6][7][8][9][10]NumberAvgStDevCohort-Cohort-ofofNoneClassschoolSchoolNoneClassschoolSchoolRandomGrouping-RandomAssignment10.4380.3170.83830.2580.1860.2130.2080.0060.8460.8350.8210.03270.1730.1230.1600.1560.0020.8380.9050.9000.025200.1000.0730.1000.0970.0010.8400.9420.9390.013DynamicGrouping-RandomAssignment10.4480.3180.83130.2630.1870.2130.2080.0080.8320.8290.8190.04470.1730.1230.1580.1540.0030.8430.9060.8980.033200.1020.0730.1010.0980.0010.8360.9400.9360.019HeterogeneityGrouping-RandomAssignment10.4520.3170.82130.2640.1860.2150.2110.0060.8280.8280.8140.03670.1770.1230.1600.1560.0030.8300.8990.8910.024200.1030.0730.1010.0990.0010.8360.9370.9310.012*Notes:Forallsimulations,lambdaissetto0.5.Theteachere!ectsareestimatedusingpooleddynamicOLS.Columns3-6displaytheaveragestandarderrorsbyclusteringtype.Thetypesofclusteringarenoclustering,classroomlevelclustering,cohort-schoollevelclustering,andschoollevelclustering.Columns7-10displaytheaveragecoveragerateforeachtypeofclustering.119TableE.4:AverageStandardErrorsandCoverageRates,Cohort-by-cohortEstimation[1][2][3][4]NumberofAvgStDevofAvgofTeacherAverageCohortsEstimatesAvgSECoverageRatePanelA:StudentFE-N(0,1),noCohort-SchoolFERandomGrouping-RandomAssignment30.2530.2280.94970.1630.1570.955200.0950.0960.959DynamicGrouping-RandomAssignment30.2580.2270.94470.1650.1570.944200.0990.0970.953HeterogeneityGrouping-RandomAssignment30.5060.4700.93470.3320.3300.951200.2040.1980.953PanelB:StudentFE-N(0,0.25),Cohort-SchoolFE-N(0,0.25)RandomGrouping-RandomAssignment30.2580.2330.95170.1750.1670.952200.1000.0990.957DynamicGrouping-RandomAssignment30.2630.2350.95270.1720.1670.953200.1020.0990.951HeterogeneityGrouping-RandomAssignment30.2640.2370.95170.1760.1680.949200.1030.1000.952*Notes:Forallsimulations,lambdaissetto0.5.Theteachere!ectsareestimatedusingpooledOLS.120TableE.5:AverageStandardErrorsandCoverageRates(StudentFEN(0,1),noCohort-SchoolFE)AverageofTeacherAverageSEAverageCoverageRate[1][2][3][4][5][6][7][8][9][10]NumberAvgStDevCohort-Cohort-ofofNoneClassschoolSchoolNoneClassschoolSchoolDynamicGrouping-PositiveAssignment10.4370.4350.95330.2460.2520.2000.2030.0150.9540.8360.8330.08970.1570.1650.1510.1520.0090.9660.9260.9260.089200.0960.0970.0940.0950.0050.9510.9350.9370.085HeterogeneityGrouping-PositiveAssignment10.4170.3990.54730.2390.2310.1920.1940.0070.3880.3460.3490.01470.1560.1510.1440.1450.0030.3740.3650.3660.014200.0930.0890.0900.0900.0010.3760.3760.3760.006*Notes:Forallsimulations,lambdaissetto0.5.Theteachere!ectsareestimatedusingpooleddynamicOLS.Columns3-6displaytheaveragestandarderrorsbyclusteringtype.Thetypesofclusteringarenoclustering,classroomlevelclustering,cohort-schoollevelclustering,andschoollevelclustering.Columns7-10displaytheaveragecoveragerateforeachtypeofclustering.TableE.6:AverageStandardErrorsandCoverageRates,Cohort-by-cohortEstimation[1][2][3][4]NumberofAvgStDevofAvgofTeacherAverageCohortsEstimatesAvgSECoverageRateDynamicGrouping-PositiveAssignment30.2460.2270.95970.1640.1580.949200.0970.0960.954HeterogeneityGrouping-PositiveAssignment30.2390.2170.62170.1600.1520.382200.0930.0910.385*Notes:Forallsimulations,lambdaissetto0.5.Theteachere!ectsareestimatedusingpooledOLS.121TableE.7:StudentCharacteristics,byDistrictABCDEFMathscore1557.701575.641497.201491.171539.381523.40LaggedMathscore1423.541464.701355.971354.151420.151385.79Daysabsent7.196.726.527.567.186.97Asian(%)0.030.020.010.030.030.04Black(%)0.350.100.270.420.200.28Hispanic(%)0.240.050.600.050.240.24AmericanIndian(%)0.000.000.000.000.000.00Multi-racial(%)0.030.030.010.040.050.03Otherrace/ethnicity(%)0.030.030.020.040.060.04Female(%)0.490.500.500.510.500.49Disability(%)0.140.160.110.110.150.15LEP(%)0.200.010.510.060.190.20FRL(%)0.450.210.710.530.500.53Num.students104,01912,805138,91348,59267,07161,043*Notes:LEPreferstostudentclassiÞedashavingLimitedEnglishProÞciency,andFRLreferstostudentseligibleforfree-orreduced-pricelunch.Thenumberofstudentsisthetotalnumberofstudentsingrade4duringyears2001-2007.TableE.8:Percentof95%ConÞdenceIntervalsAbove/BelowCuto!s,ByRegionofValue-AddedDistributionPercentof95%CIPercentof95%CIUpperBoundsUnderPercentileLowerBoundsOverPercentile[1][2][3][4][5][6][7][8][9]UnderUnderUnderUnderOverOverOverOverMethod10th25th75th90th10th25th75th90thPanelA:Bottom10%PooledOLS-LagNoclustering0.0740.4411.0001.000Cohort-schoolclustering0.2350.7791.0001.000SchoolClustering0.7060.9711.0001.000OLS-Lagcohort-by-cohort0.0080.0160.2100.290PanelB:25th-75thpercentilePooledOLS-LagNoclustering0.2410.7590.8120.264Cohort-schoolclustering0.5070.8960.9020.540SchoolClustering0.8900.9940.9910.889OLS-Lagcohort-by-cohort0.0420.1690.1800.058PanelC:Top10%PooledOLS-LagNoclustering1.0001.0000.6000.200Cohort-schoolclustering0.9920.9830.7580.333SchoolClustering1.0001.0000.9750.792OLS-Lagcohort-by-cohort0.5890.4520.1850.081*Notes:ThesepercentagesareaveragestakenoverthesixdistrictsinfromTableE.7,andinclude432,443fourthgradestudentsand9,102fourthgradeteachersfromsevencohorts.122APPENDIXFFiguresforChapter3123FigureF.1:AverageStandardErrors,byDistrict124FigureF.2:AverageConÞdenceIntervalWidths,byDistrict125APPENDIXGSupplementalTablesforChapter1126TableG.1:ImpactofDualLanguageEducation-ConstantE!ectPanelA:EnglishSampleOLSMathReadingMathReadingFirstStageITTLATEITTLATE[1][2][3][4][5][6][7]WonFirstChoice0.492***0.209*0.120(0.053)(0.115)(0.077)Attended(K/First)-0.016-0.127*0.426*0.245(0.100)(0.063)(0.231)(0.155)NeighborhoodSchoolFEXXXXXXXObservations1,4721,4721,4721,4721,4721,4721,472Numberoflotfe44444444444444PanelB:ESL/LEPSampleOLSMathReadingMathReadingFirstStageITTLATEITTLATE[1][2][3][4][5][6][7]WonFirstChoice0.653***0.296**0.226(0.063)(0.136)(0.138)Attended(K/First)0.266**0.348***0.453**0.346*(0.111)(0.109)(0.199)(0.193)NeighborhoodSchoolFEXXXXXXXObservations809809809809809809809NumberofClusters36363636363636Robuststandarderrorsinparentheses***p<0.01,**p<0.05,*p<0.1*Notes:EachregressionincludeslotteryÞxede!ects(priority-year-program)aswellascontrolsforfemale,race,frpl-year,exceptionality,gradeofexam,yearofexam,andneighborhoodschoolÞxede!ects.AttendanceismeasuredbywhetherthestudentattendedaDLschoolinkindergartenorÞrstgrade.ThetreatmentandattendancevariablesarenotinteractedwithyearsoftreatmentinthisspeciÞcation.Standarderrorsareclusteredbylottery.127TableG.2:ImpactofDualLanguageEducation-3rdGradeAttendanceMeasurePanelA:EnglishSampleOLSMathReadingMathReadingFirstStageITTLATEITTLATE[1][2][3][4][5][6][7]WonFirstChoice0.293***0.209*0.120(0.062)(0.115)(0.077)Attended(3rd)0.043-0.0120.715*0.411(0.142)(0.094)(0.414)(0.274)NeighborhoodSchoolFEXXXXXXXObservations1,4721,4721,4721,4721,4721,4721,472Numberoflotfe44444444444444PanelB:ESL/LEPSampleOLSMathReadingMathReadingFirstStageITTLATEITTLATE[1][2][3][4][5][6][7]WonFirstChoice0.560***0.296**0.226(0.067)(0.136)(0.138)Attended(3rd)0.278**0.347**0.528**0.403*(0.136)(0.163)(0.248)(0.233)NeighborhoodSchoolFEXXXXXXXObservations809809809809809809809Numberofclusters36363636363636Robuststandarderrorsinparentheses***p<0.01,**p<0.05,*p<0.1*Notes:EachregressionincludeslotteryÞxede!ects(priority-year-program)aswellascontrolsforfemale,race,frpl-year,exceptionality,gradeofexam,yearofexam,andneighborhoodschoolÞxede!ects.AttendanceismeasuredbywhetherthestudentattendedaDLschoolinthirdgrade.ThetreatmentandattendancevariablesarenotinteractedwithyearsoftreatmentinthisspeciÞcation.Standarderrorsareclusteredbylottery.128TableG.3:GradesThreeThroughFiveOnlyPanelA:EnglishSampleOLSMathReadingMathReadingFirstStageITTLATEITTLATE[1][2][3][4][5][6][7]WonFirstChoice0.503***0.0340.023*(0.054)(0.024)(0.014)AttendDLSchool-0.001-0.0230.0680.046*(0.022)(0.013)(0.045)(0.027)NeighborhoodSchoolFEXXXXXXXObservations1,1721,1721,1721,1721,1721,1721,172NumberofClusters44444444444444PanelB:ESL/LEPSampleOLSMathReadingMathReadingFirstStageITTLATEITTLATE[1][2][3][4][5][6][7]WonFirstChoice0.652***0.055*0.028(0.061)(0.031)(0.030)AttendDLSchool0.049**0.058**0.084*0.043(0.023)(0.022)(0.047)(0.043)NeighborhoodSchoolFEXXXXXXXObservations623623623623623623623NumberofClusters36363636363636Robuststandarderrorsinparentheses***p<0.01,**p<0.05,*p<0.1*Notes:EachregressionincludeslotteryÞxede!ects(priority-year-program)aswellascontrolsforfemale,race,frpl-year,exceptionality,gradeofexam,yearofexam,andneighborhoodschoolÞxede!ects.AttendanceismeasuredbywhetherthestudentattendedaDLschoolinkindergartenorÞrstgradeandinteractedwithyearsoftreatment(gradeplusone).Standarderrorsareclusteredbylottery.129TableG.4:CohortInteractionsEnglishSampleESL/LEPSampleMathReadingMathReading[1][2][3][4]AttendDLSchool2007Cohort0.401**0.200*0.105***0.133***(0.157)(0.116)(0.038)(0.031)2008Cohort-0.009-0.0140.048-0.012(0.035)(0.030)(0.063)(0.028)2009Cohort0.0740.132**0.0320.045(0.109)(0.058)(0.050)(0.086)2010Cohort0.151**0.153***0.2860.239(0.069)(0.040)(0.197)(0.166)2011Cohort0.131***0.0290.0670.061(0.044)(0.034)(0.086)(0.096)NeighborhoodSchoolFEXXXXObservations1,4711,471809809NumberofLotteryFE44443636Robuststandarderrorsinparentheses***p<0.01,**p<0.05,*p<0.1*Notes:EachregressionincludeslotteryÞxede!ects(priority-year-program)aswellascontrolsforfemale,race,frpl-year,exceptionality,gradeofexam,yearofexam,andneighborhoodschoolÞxede!ects.Re-portedcoe"cientsareoninteractionsbetweentheattendancevariableandcohort.AttendanceismeasuredbywhetherthestudentattendedaDLschoolinkindergartenorÞrstgradeandinteractedwithyearsoftreatment(gradeplusone).Standarderrorsareclusteredbylottery.130TableG.5:AttritionandWeighting(Panel)PanelA:SummaryofProbabilitiesofTestingFullSampleEnglishSampleESL/LEPSampleWinnersLosersWinnersLosersWinnersLosers[1][2][3][4][5][6]AveragePr(Test)0.8530.8320.8430.8250.8780.841SDPr(Test)0.0220.0290.0320.0470.0350.045APE0.0190.0110.027(SE)(0.029)(0.033)(0.038)N153214331105816427617PanelB:Non-RandomAttritionCoe!cientsonIndicatorforWinningLotteryFullSampleEnglishSampleESL/LEPSample[1][2][3]NoControls0.0210.0180.037(SE)(0.022)(0.033)(0.032)Controls+LotteryFE0.021-0.0090.049(SE)(0.031)(0.038)(0.038)+NeighborhoodSchoolFE0.030-0.0030.057(SE)(0.032)(0.046)(0.042)N296519211044*Notes:PanelAsummarizesestimatedprobabilitesofhavingtestscoresavailable.Thistableusesanexpandeddataset,relativetoTable6,whereeachstudenthasanobservationforeachgradethattheycouldhavetestediniftheypassedeachgrade.Theestimatesarebasedonlogitregressionsincludinggender,race,frpl-year,andanindicatorforwinningthelottery.PanelBshowsestimatedcoe"cientsfromOLSregressionsofhavingatleastonesetoftestscoresavailableonwinningthelottery.Thebaselinecontrolsaregender,race,andfrpl-year.ThesecondsetofOLSestimatesalsoconditiononlotteryÞxede!ects,andthethirdsetincludeneighborhoodschoolÞxede!ects.131APPENDIXHSupplementalTablesforChapter2132TableH.1:FEEstimatesofGenderGapswithDayofWeekFEsMotherFatherTotalWeekdayWeekendTotalWeekdayWeekend[1][2][3][4][5][6]Single-MotherHHXMale1.2500.0100.539-7.339-0.901-1.357(1.711)(0.273)(0.359)(1.245)(0.175)(0.317)Female2.3700.2110.535-5.131-0.509-1.139(1.865)(0.279)(0.396)(1.142)(0.160)(0.310)Other-7.493-1.044-1.144-2.311-0.345-0.358(1.941)(0.293)(0.408)(1.381)(0.193)(0.353)Male-Female-1.120-0.2000.004-2.207-0.392-0.218(StandardError)(1.617)(0.244)(0.372)(1.084)(0.159)(0.288)Observations6,6996,7866,7266,6996,7866,726R-squared0.2630.2300.1380.1260.0780.099NumberofIndividuals3,2833,3083,2873,2833,3083,287*Notes:ThisisareplicationofthemainÞxede!ectsestimatesfromTable3,PanelB,butincludesindicatorsfordayoftheweek.Forthetotalweeklytimeregression,thereisadummyincludedforeachweekday-weekenddaycombination,withonecombinationexcluded.Standarderrorsareclusteredbyindividualanddisplayedbelowtheestimates.133TableH.2:GenderGapsinInvestmentsbyInitialHHStructure(DayofWeekFEs)BothinWave1<2inWave1MotherFatherMotherFather[1][2][3][4]Single-MotherHHXMale1.849-9.4173.2460.329(2.145)(1.642)(3.252)(1.725)Female1.192-5.2896.8610.483(2.162)(1.549)(3.743)(1.465)Other-9.097-3.034-4.2313.302(2.700)(2.161)(3.349)(1.724)Male-Female0.657-4.128-3.615-0.154(StandardError)(2.037)(1.534)(2.896)(1.642)Observations4,2844,2842,4152,415R-squared0.2920.1600.2240.080NumberofIndividuals1,9981,9981,2851,285Robuststandarderrorsinparentheses*Notes:ThisisareplicationofthemainÞxede!ectsestimatesbyinitialhouseholdstructurefromTable4,butincludesindicatorsfordayoftheweek.Forthetotalweeklytimeregression,thereisadummyincludedforeachweekday-weekenddaycombination,withonecombinationex-cluded.Standarderrorsareclusteredbyindividualanddisplayedbelowtheestimates.134APPENDIXISupplementalTablesforChapter3135TableI.1:EstimationSampleSizesandGraphSampleSizesEstimationsampleGraphsample-Alt.1Graphsample-Alt.2[1][2][3][4][5][6][7]CohortsTeachersStudentsTeachersStudentsTeachersStudentsDistrictA288428,79443519,1191135,16731,08943,01431520,3611137,60941,37359,10524120,5841139,88751,56274,08319520,54311312,05361,79889,85313717,12611314,21771,955104,01911316,26711316,267DistrictB21053,352512,2611674231334,997372,452161,09641536,652282,402161,40151848,607242,566161,718622110,727192,350162,009725612,805162,344162,344DistrictC21,17041,45358928,6021296,56531,33858,49434626,63412910,25441,62379,14126227,40512913,73651,94799,72720426,07312916,75462,279120,31416023,90412919,43572,580138,91312921,96612921,966DistrictD249715,93626911,373431,846361722,75817410,758432,672473929,47612610,112433,477586936,217918,976434,231698542,841677,744434,95471,11048,592435,645435,645DistrictE259018,17627811,891572,488374327,42819012,162573,706494036,58912710,639574,82451,11046,196929,465575,90861,28856,531759,218576,98571,52367,071577,969577,969DistrictF266017,83532311,883702,572376126,08622812,728703,883493335,26717212,936705,20151,09644,17912011,277706,49661,26352,739849,388707,78771,40861,043708,904708,904136APPENDIXJSupplementalFiguresforChapter3137FigureJ.1:AverageStandardErrors,byDistrictNotes:Foreachdistrict,teachere!ectsandstandarderrorsarecalculatedusinganestimationsampleofall4thgradeteachersandtheirstudentsincohorts"c.Foreachgraphdatapointcorrespondingtoccohorts,thestandarderrorsareaveragedoveronlythesubsampleofteacherswithexactlyccohorts.ThenumberofteachersineachestimationsampleandgraphsamplearelistedinAppendixTableI.1.138FigureJ.2:AverageConÞdenceIntervalWidths,byDistrictNotes:Foreachdistrict,teachere!ectsandstandarderrors(andconÞdenceintervalwidths)arecalculatedusinganestimationsampleofall4thgradeteachersandtheirstudentsincohorts"c.Foreachgraphdatapointcorrespondingtoccohorts,theconÞdenceintervalwidthsareaveragedoveronlythesubsampleofteacherswithexactlyccohorts.ThenumberofteachersineachestimationsampleandgraphsamplearelistedinAppendixTableI.1.139FigureJ.3:AverageStandardErrors,byDistrictNotes:Foreachdistrict,teachere!ectsandstandarderrorsarecalculatedusinganestimationsampleofall4thgradeteachersandtheirstudentsincohorts"c.Foreachgraphdatapointcorrespondingtoccohorts,thestandarderrorsareaveragedoveronlythesubsampleofteacherswithexactly7cohorts.ThenumberofteachersineachestimationsampleandgraphsamplearelistedinAppendixTableI.1.140FigureJ.4:AverageConÞdenceIntervalWidths,byDistrictNotes:Foreachdistrict,teachere!ectsandstandarderrors(andconÞdenceintervalwidths)arecalculatedusinganestimationsampleofall4thgradeteachersandtheirstudentsincohorts"c.Foreachgraphdatapointcorrespondingtoccohorts,theconÞdenceintervalwidthsareaveragedoveronlythesubsampleofteacherswithexactly7cohorts.ThenumberofteachersineachestimationsampleandgraphsamplearelistedinAppendixTableI.1.141REFERENCES142REFERENCESDanielAaronson,LisaBarrow,andWilliamSander.Teachersandstudentachievementinthechicagopublichighschools.JournaloflaborEconomics,25(1):95Ð135,2007.TracyPackiamAlloway.Improvingworkingmemory:SupportingstudentsÕlearning.Sage,2010.TahirAndrabi,JishnuDas,AsimIjazKhwaja,andTristanZajonc.Dovalue-addedestimatesaddvalue?accountingforlearningdynamics.AmericanEconomicJournal:AppliedEconomics,3(3):29Ð54,2011.AlanBaddeley.Workingmemory:lookingbackandlookingforward.Naturereviewsneuroscience,4(10):829Ð839,2003.AlanDBaddeleyandGrahamJHitch.Workingmemory.Thepsychologyoflearningandmotivation,8:47Ð89,1974.MichaelBakerandKevinMilligan.Boy-girldi!erencesinparentaltimeinvestments:Evidencefromthreecountries.NationalBureauofEconomicResearch(No.w18893),2013.DaleBallou.Testscalingandvalue-addedmeasurement.Education,4(4):351Ð383,2009.DaleBallou,WilliamSanders,andPaulWright.Controllingforstudentbackgroundinvalue-addedassessmentofteachers.Journalofeducationalandbehavioralstatistics,29(1):37Ð65,2004.MokherChristineGBallou,DaleandLindaCavalluzzo.Usingvalue-addedassessmentforper-sonneldecisions:HowomittedvariablesandmodelspeciÞcationinßuenceteachersÕoutcomes.UnpublishedManuscript,2012.MarianneBertrandandJessicaPan.Thetroublewithboys:Socialinßuencesandthegendergapindisruptivebehavior.AmericanEconomicJournal:AppliedEconomics,5(1):32Ð64,2013.DerekCBriggsandJonathanPWeeks.Thesensitivityofvalue-addedmodelingtothecreationofaverticalscorescale.Education,4(4):384Ð414,2009.MCazabon,WLambert,andGHall.Two-waybilingualeducation:AreportontheAmigosProgram.Washington,DC:CenterforAppliedLinguistics,1999.RajChetty,JohnNFriedman,andJonahERocko!.Measuringtheimpactsofteachersi:Evaluatingbiasinteachervalue-addedestimates.TheAmericanEconomicReview,104(9):2593Ð2632,2014.RosaMinhyoCho.Aretherepeere!ectsassociatedwithhavingenglishlanguagelearner(ell)classmates?evidencefromtheearlychildhoodlongitudinalstudykindergartencohort(ecls-k).EconomicsofEducationReview,31(5):629Ð643,2012.ElisavetChrysochoou,ZoeBablekou,andNikokaosTsigilis.Workingmemorycontributionstoread-ingcomprehensioncomponentsinmiddlechildhoodchildren.TheAmericanjournalofpsychology,124(3):275Ð289,2011.143BrianCobb,DiegoVega,andCindyKronauge.E!ectsofanelementaryduallanguageimmersionschoolprogramonjuniorhighachievement.MiddleGradesResearchJournal,1(1):27Ð48,2009.ScottCondie,LarsLefgren,andDavidSims.Teacherheterogeneity,value-addedandeducationpolicy.EconomicsofEducationReview,40:76Ð92,2014.SeanPCorcoran,JenniferLJennings,andAndrewABeveridge.Teachere!ectivenessonhigh-andlow-stakestests.SocietyforResearchonEducationalE!ectiveness,2011.DanielaDelBocaandAnnaLauraMancini.Parentaltimeandchildoutcomes:Doesgendermatter?BankofItalyOccasionalPaper,(187),2013.DanielaDelBoca,ChiaraMonfardini,ChetiNicoletti,etal.ChildrenÔsandparentÔstime-usechoicesandcognitivedevelopmentduringadolescence.HumanCapitalandEconomicOpportunityWorkingGroupworkingpaper,6,2012.DavidJDeming,JustineSHastings,ThomasJKane,andDouglasOStaiger.SchoolChoice,SchoolQuality,andPostsecondaryAttainment.104(3):991Ð1013,2014.StephenGDonaldandKevinLang.Inferencewithdi!erence-in-di!erencesandotherpaneldata.ThereviewofEconomicsandStatistics,89(2):221Ð233,2007.MiaDufva,PekkaNiemi,andMarinusJMVoeten.Theroleofphonologicalmemory,wordrecogni-tion,andcomprehensionskillsinreadingdevelopment:Frompreschooltograde2.ReadingandWriting,14(1-2):91Ð117,2001.CharlotteGeay,SandraMcnally,andShqiponjaTelhaj.Non-nativespeakersofEnglishintheclassroom:Whatarethee!ectsonpupilperformance?EconomicJournal,123(November2010):281Ð307,2013.DanGoldhaberandDuncanDunbarChaplin.AssessingtheÔrothsteinfalsiÞcationtestÔ:Doesitreallyshowteachervalue-addedmodelsarebiased?JournalofResearchonEducationalE!ec-tiveness,8(1):8Ð34,2015.DanGoldhaberandMichaelHansen.Isitjustabadclass?assessingthelong-termstabilityofestimatedteacherperformance.Economica,80(319):589Ð612,2013.DanGoldhaber,JoeWalch,andBrianGabele.Doesthemodelmatter?exploringtherelationshipbetweendi!erentstudentachievement-basedteacherassessments.StatisticsandPublicPolicy,1(1):28Ð39,2014.JPGreene.Ameta-analysisofthee!ectivenessofbilingualeducation.Claremont,CA:ThomasRiveraPolicyInstitute,1998.CassandraMGuarino,MarkDReckase,andJe!reyMWooldridge.Canvalue-addedmeasuresofteacherperformancebetrusted?EducationFinanceandPolicy,2015.JonathanGuryan,ErikHurst,andMelissaKearney.Parentaleducationandparentaltimeuse.JournalofEconomicPerspectives,22(3),2008.144DouglasNHarris.Wouldaccountabilitybasedonteachervalueaddedbesmartpolicy?anexami-nationofthestatisticalpropertiesandpolicyalternatives.Education,4(4):319Ð350,2009.JamesJHeckmanandStefanoMosso.Theeconomicsofhumandevelopmentandsocialmobility.NationalBureauofEconomicResearch(No.w19925),2014.ElizabethHowardandJulieSugarman.Twowayimmersionprograms:Featuresandstatistics.OccassionalReportsUCBerkeley,(March):10Ð13,2001.GuidoWImbensandJoshuaDAngrist.IdentiÞcationandestimationoflocalaveragetreatmente!ects.Econometrica,62(2):467Ð475,1994.JunIshiiandStevenGRivkin.Impedimentstotheestimationofteachervalueadded.Education,4(4):520Ð536,2009.BrianAJacob.WheretheboysarenÔt:Non-cognitiveskills,returnstoschoolandthegendergapinhighereducation.EconomicsofEducationReview,21(6):589Ð598,2002.BrianAJacob,LarsLefgren,andDavidPSims.Thepersistenceofteacher-inducedlearning.JournalofHumanresources,45(4):915Ð943,2010.ThomasJKaneandDouglasOStaiger.Estimatingteacherimpactsonstudentachievement:Anexperimentalevaluation.NationalBureauofEconomicResearch(No.w14067),2008.ThomasJKane,DanielFMcCa!rey,TreyMiller,andDouglasOStaiger.HaveweidentiÞede!ectiveteachers?validatingmeasuresofe!ectiveteachingusingrandomassignment.InResearchPaper.METProject.Bill&MelindaGatesFoundation.Citeseer,2013.CoryKoedelandJulianBetts.Valueaddedtowhat?howaceilinginthetestinginstrumentinßuencesvalue-addedestimation.EducationFinanceandPolicy,5(1):54Ð81,2010.CoryKoedelandJulianRBetts.Doesstudentsortinginvalidatevalue-addedmodelsofteachere!ectiveness?anextendedanalysisoftherothsteincritique.Education,6(1):18Ð42,2011.JRLockwoodandDanielFMcCa!rey.Exploringstudent-teacherinteractionsinlongitudinalachievementdata.Education,4(4):439Ð467,2009.JRLockwood,ThomasALouis,andDanielFMcCa!rey.Uncertaintyinrankestimation:Impli-cationsforvalue-addedmodelingaccountabilitysystems.JournalofEducationalandBehavioralStatistics,27(3):255Ð270,2002.LesliAMaxwell.Dualclassesseegrowthinpopularity.EducationWeek,31(26):16Ð17,2012.LesliA.Maxwell.DualLanguageProgramsTakeRootinN.C.EducationWeek,34(8):1,October2014.DanielFMcCa!rey,JRLockwood,DanielKoretz,ThomasALouis,andLauraHamilton.Modelsforvalue-addedmodelingofteachere!ects.Journalofeducationalandbehavioralstatistics,29(1):67Ð101,2004.145publicusedatasetPanelStudyofIncomeDynamics.ProducedanddistributedbytheSurveyResearchCenter,InstituteforSocialResearch,UniversityofMichigan,AnnArbor,MI,2014.StephenWRaudenbush.Whatarevalue-addedmodelsestimatingandwhatdoesthisimplyforstatisticalpractice?JournalofEducationalandBehavioralStatistics,29(1):121Ð129,2004.SeanFReardonandStephenWRaudenbush.Assumptionsofvalue-addedmodelsforestimatingschoole!ects.Education,4(4):492Ð519,2009.StevenGRivkin,EricAHanushek,andJohnFKain.Teachers,schools,andacademicachievement.Econometrica,73(2):417Ð458,2005.JonahERocko!.Theimpactofindividualteachersonstudentachievement:Evidencefrompaneldata.TheAmericanEconomicReview,94(2):247Ð252,2004.JesseRothstein.Teacherqualityineducationalproduction:Tracking,decay,andstudentachieve-ment.TheQuarterlyJournalofEconomics,125(1):175Ð214,2008.CeciliaElenaRouse.Privateschoolvouchersandstudentachievement:Anevaluationofthemil-waukeeparentalchoiceprogram.TheQuarterlyJournalofEconomics,113(2):553Ð602,1998.DonaldBRubin,ElizabethAStuart,andElaineLZanutto.Apotentialoutcomesviewofvalue-addedassessmentineducation.Journalofeducationalandbehavioralstatistics,29(1):103Ð116,2004.WilliamLSanders,SPaulWright,andSandraPHorn.Teacherandclassroomcontexte!ectsonstudentachievement:Implicationsforteacherevaluation.Journalofpersonnelevaluationineducation,11(1):57Ð67,1997.TimRSass,AnastasiaSemykina,andDouglasNHarris.Value-addedmodelsandthemeasurementofteacherproductivity.EconomicsofEducationReview,38:9Ð23,2014.R.E.Slavinanda.Cheung.ASynthesisofResearchonLanguageofReadingInstructionforEnglishLanguageLearners.ReviewofEducationalResearch,75(2):247Ð284,2005.JenniferL.Steele,RobertO.Slater,GemaZamarro,TreyMiller,JenniferLi,andSusanBurkhauser.E!ectsofDual-LanguageImmersiononStudentsÕAcademicPerformance.AmericanEducationalResearchJournal(Forthcoming),2016.WaynePThomasandV.P.Collier.EnglishLearnersinNorthCarolina,2009TheNorthCarolinaContext.FairfaxVA:GeorgeMasonUniversity.AResearchReportProvidedtotheNorthCarolinaDepartmentofPublicInstruction,2009.WPThomas,V.P.Collier,andK.Collier.EnglishlearnersinNorthCarolina,2010.Fairfax,VA:GeorgeMasonUniversity.AResearchReportProvidedtotheNorthCarolinaDepartmentofPublicInstruction,2010.I.M.UmanskyandS.F.Reardon.ReclassiÞcationpatternsamongLatinoEnglishlearnerstudentsinbilingual,dualimmersion,andEnglishimmersionclassrooms.AmericanEducationalResearchJournal,51(5):879Ð912,2014.146RachelAValentinoandSeanFReardon.E!ectivenessoffourinstructionalprogramsdesignedtoserveenglishlearners.EducationalEvaluationandPolicyAnalysis,37(4):612Ð637,2015.TeresaWatanabe.DualLanguageImmersionProgramsGrowinginPopularity.LosAngelesTimes,May8,2011.AnnCWillig.Ameta-analysisofselectedstudiesonthee!ectivenessofbilingualeducation.Reviewofeducationalresearch,55(3):269Ð317,1985.147