ESSAYSONASYMMETRICEMPLOYERLEARNINGANDTHEECONOMICSOF EDUCATION By MichaelDavidBates ADISSERTATION Submittedto MichiganStateUniversity inpartialful˝llmentoftherequirements forthedegreeof Economics-DoctorofPhilosophy 2015 ABSTRACT ESSAYSONASYMMETRICEMPLOYERLEARNINGANDTHE ECONOMICSOFEDUCATION By MichaelDavidBates Chapter1adaptsmodelsofpublicandprivateemployerlearningtothemarketfor teachers.Itthenusestatewide,micro-level,administrativedatafromNorthCarolinato formulatevalue-addedmeasures(VAMs)ofteacherproductivity.Itexploitstheadoptionof VAMsofteacherperformancebytwoofthelargestschooldistrictsinthestate,ashockto theavailableinformationforsome,butnotall,employers,toprovideaninitialdirecttestof asymmetricemployerlearning.Consistentwithashocktopublicinformation,forjobmoves withinthedistrict,thiswork˝ndsthattheadoptionofvalue-addedmeasuresincreasesthe probabilitythathigh-VAMteachersmovetohigher-performingschools.Formovesoutof thedistrict,theimpactsofpolicyaremitigatedandevenreversedbyteacherswithlower value-addedmeasuresbecomingmorelikelytomovetohigher-performingschools.This adverseselectiontoplausiblylessinformedprincipalsisconsistentwithasymmetricemployer learning.Further,thischapterprovidesevidencethatthesemovesleadtoanincreasethe inequalityinaccesstohighqualityteaching. Chapter2examinesworkermobility,andempiricallytestswhetherall˝rmslearnabout workers'abilitiesatthesamerate(symmetriclearning)orwhethercurrentemployersaccu- mulateanduseprivateinformationabouttheirworkers(asymmetriclearning).Theemployer learningmodelallowsforbothpublicandprivatelearning,andthus,nestssymmetriclearn- ingasaspecialcase.Themodelpredictsthatconditionalonemployees'easilyobservable referencegroups,workersareadverselyselectedintojobswitchesandlayo˙sonthebasis ofdi˚culttoobservecharacteristics,suchasintellectualability.Inversely,conditionalon ability,themodelpredictsthatasthemeanabilityofaworker'sreferencegroupincreases, thelikelihoodofjobseparationincreases.Underasymmetricprivatelearning,thesee˙ects shouldbecomemorepronouncedoverthelengthofcontinuousworkingspells.Thesame e˙ectsshoulddiminishwithexperience,inthepresenceofpubliclearning.Thisstudyuses datafromthe1979cohortoftheNationalLongitudinalSurveyofYouthtotestthemodel. I˝ndadverseselectiononAFQTofworkerswhobecomeunemployed,andconditionalon AFQTscore,workerswithhighereducationfrommoreselectiveinstitutionsarearepositively selectedintojobswitchesandmovesfromemploymenttounemploymentduringrecessions. Theevidencelargelyrejectssymmetriclearninginfavorasymmetriclearning. Chapter3discussesestimationofmultilevel/hierarchicallinearmodelsthatincludecluster- levelrandominterceptsandrandomslopes.Therandominterceptsandslopesrepresentthe e˙ectsofomittedcluster-levelcovariatesthatmaybecorrelatedwithincludedcovariates. Theresultingcorrelationsbetweenrandome˙ects(interceptsandslopes)andincludedco- variatesleadtobiaswhenusingstandardrandom-e˙ectsestimators.Whenappliedtomod- elswithrandomslopes,thestandard˝xed-e˙ects(FE)estimatordoesnotrelyonstandard cluster-levelexogeneityassumptions,butrequiresanatedvariance thatthevariancesofunit-levelcovariatesareuncorrelatedwiththeirrandomslopes.This workproposesaer-clusterregressi(PC)estimatorthatisstraightforwardtoimplement instandardsoftware,andshowsanalyticallythatitisunbiasedforallregressioncoe˚cients undercluster-levelendogeneityandviolationoftheuncorrelatedvarianceassumption.The PC,RE,andanaugmentedFEestimatorareappliedtoarealdatasetandevaluatedina simulationstudythatdemonstratesthatthePCestimatorperformswellinpractice. ACKNOWLEDGMENTS Iwould˝rstliketoexpressmysinceregratitudetoToddElderforhisguidanceandsupport throughouttheprocessofthisproject.IamalsoverygratefultoDavidArsen,MikeConlin, ScottImberman,andJe˙reyWooldridge.Eachhastaughtmeanincredibleamountand hasgivenmewonderfulandthoughtfuladvice.ThanksalsogoestoSorenAndersonand MichaelWaldman,fortheirhelpfulcommentsandfeedback.Ialsowanttoacknowledge KatherineCastellano,SophiaRabe-Hesketh,andAndersSkrondalwithwhomthethird charterisco-authored. IreceivedgreatassistancefrommanyclassmatesincludingAndrewBibler,PaulBurkan- der,HassanEnayati,ChrisKhawand,DanielLitwok,MichelleMax˝eld,MikeNaretta,Brian Stacy,AmandaStype,PaulThompson,ValentinVerdier,KellyVosters,andGregWalsworth, andIthankthemfortheirmanycontributions.IalsothankBarbaraSchneider,KaraBon- neau,theNorthCarolinaEducationResearchDataCenter,aswellasrepresentativesofGuil- fordCountySchools,WinstonSalem/ForsythCommunitySchools,Charlotte-Mecklenburg Schools,andCumberlandCountySchoolsforassistinginaccessingvaluabledata.Ialsowant toacknowledgethatthisresearchwassupported˝nanciallybytheInstituteofEducation SciencesGrantR305B090011toMichiganStateUniversity. Lastly,Iwishtothankmyfamily.Speci˝cally,Iwanttothankmyparents,Peterand Teri,fortheircontinuedbeliefinmeandsupportthroughoutmyeducation.Also,Iwish tothankmysiblings,Alex,Emily,andLiz,whoeachinspiremetogetthemostfromlife. Mostimportantly,Iwanttothankmywife,Amanda,forherunwaveringloveandsupport, andwithmydaughter,Iris,andson,Adam,forbringingunfathomablejoy. iv TABLEOFCONTENTS LISTOFTABLES .................................... vii LISTOFFIGURES ................................... x Chapter1PublicandPrivateLearningintheMarketforTeachers:Evi- dencefromtheAdoptionofValue-AddedMeasures ...... 1 1.1Introduction....................................1 1.2Setting.......................................6 1.3EmployerLearning,VAMs,andTeacherMobility...............9 1.4Model.......................................14 1.4.1ModelStructure..............................14 1.4.2Bidding..................................17 1.4.3MobilityunderAsymmetricInformation................22 1.5DataandEstimation...............................26 1.5.1Value-AddedMeasures..........................27 1.5.2EstimationSample............................29 1.5.3EstimationStrategy...........................30 1.6Results......................................34 1.6.1MobilityandSorting...........................34 1.6.2Observables................................39 1.6.3Di˙erentialE˙ectsWithRespecttoExperienceandTenure.....41 1.7Robustness....................................42 1.7.1SensitivitytoVAMConstruction....................43 1.7.2StrategicSta˚ng.............................44 1.8Conclusion.....................................47 Chapter2JobSeparationUnderAsymmetricEmployerLearning .... 49 2.1Introduction...................................49 2.2EmployerLearningModelsandEvidence....................51 2.3Model......................................57 2.3.1Framework................................57 2.3.2JobSwitches...............................61 2.3.3Layo˙s..................................64 2.4Estimation....................................69 2.5Data........................................71 2.6EmpiricalAnalysis................................76 2.6.1PrimaryResults.............................76 2.6.2RobustnessResults...........................82 v 2.7Conclusion.....................................86 Chapter3HandlingCorrelationsbetweenCovariatesandRandomSlopes inMultilevelModels .......................... 87 3.1Introduction....................................87 3.2MotivationandMultilevelModel........................90 3.2.1MotivatingExampleandSpeci˝cModel................90 3.2.2GeneralMultilevelModel........................92 3.3StandardEstimatorsandConditionsforUnbiasedness.............94 3.3.1ExogeneityandEndogeneity.......................94 3.3.2Random-e˙ectsestimators........................95 3.3.3Fixed-e˙ectsestimators..........................97 3.4NewEstimatorsandConditionsforUnbiasedness...............100 3.4.1Augmented˝xed-e˙ectsestimation...................100 3.4.2Per-clusterregressionestimation.....................102 3.5EmpiricalExample................................107 3.6SimulationStudy.................................108 3.6.1DataGenerationProcess.........................108 3.6.2Results...................................110 3.6.2.1Bias...............................110 3.6.2.2PrecisionandRMSE......................113 3.6.2.3StandardErrorEstimation..................114 3.7Conclusion.....................................115 APPENDICES ...................................... 117 AppendixATablesforChapter1...........................118 AppendixBTablesforChapter2...........................125 AppendixCFiguresforChapter3..........................137 AppendixDTablesforChapter3...........................142 AppendixEAdditionsforChapter1.........................144 AppendixFSupplementalTablesforChapter1...................159 AppendixGSupplementalFiguresforChapter1..................168 AppendixHAdditionsforChapter2.........................174 AppendixIAdditionsforChapter3..........................199 BIBLIOGRAPHY .................................... 210 vi LISTOFTABLES TableA.1:AverageVAMofTeachersmovingwithinandoutofWinston-Salem andGuilford..............................119 TableA.2:SampleSummaryStatistics.......................119 TableA.3:ProbabilityofMovingSchoolsWithinandOutofDistrict......120 TableA.4:E˙ectsonSorting............................120 TableA.5:E˙ectsofteacherqualityindexontheprobabilityofmoving....121 TableA.6:Di˙erentialE˙ectsWithRespecttoExperienceandTenure.....122 TableA.7:Probabilityofmovingschoolswithin-districtusingrestricteddataVAM123 TableA.8:E˙ectofVAMsconstructedusingvariousnumberofyearsonthe probabilityofmovingtoa"better"school...............123 TableA.9:ProbabilityofMovingtoNon-Strategic-Sta˚ngSchools.......124 TableA.10:E˙ectsonSortingWithinDistrictExcludingStrategic-Sta˚ngSchools124 TableB.1:Jobseparationsbytype,referencegroup,andAFQTrelativetoref- erencegroup...............................126 TableB.2:AFQTPercentiles(StandardizedbyAge)byRaceandEducation..127 TableB.3:WorkHistory..............................127 TableB.4:TerminalTenure.............................128 TableB.5:JobSeparations.............................128 TableB.6:NominalWageChangewithSeparation................129 TableB.7:Changesinthee˙ectsofeasytoobservecharacteristicsontheprob- abilityofmoving.............................129 vii TableB.8:E˙ectsofeasyanddi˚culttoobservecharacteristicsontheproba- bilityofmoving.............................130 TableB.9:Dynamicswithregardtoworkingspelldurationandexperience...131 TableB.10:HeteroskedasticprobitAPEsande˙ectsontheconditionalvariance132 TableB.11:Heteroskedasticprobitscaledcoe˚cients...............133 TableB.12:APEsfromheteroskedasticprobitcontrolfunction..........134 TableB.13:Controlfunctione˙ectsontheconditionalvariance.........135 TableB.14:Scaledcoe˚cientsfromheteroskedasticprobitcontrolfunction...136 TableD.1:Estimatesfromdi˙erentmethodsusingHighSchoolandBeyondData.143 TableD.2:Comparingmethodsforestimatingthecoe˚cients..........143 TableF.1:Thee˙ectsofVAMontheprobabilityofmovingschoolswithin- districtbyyear..............................160 TableF.2:Thee˙ectofVAMontheprobabilityofmovingschoolsout-of-district byyear...................................161 TableF.3:Thee˙ectofVAMonteachersortingwithin-districtbyyear.....162 TableF.4:Probabilityofmovingtohigherorlowergrowthschools.......163 TableF.5:Probabilityofmovingschoolsusingnormalmaximumlikelihoodes- timation..................................163 TableF.6:Changesinthemarginalprobabilityofeachtypeoftransferbetween schools..................................164 TableF.7:Probabilityofmovingschoolsusingalternatestandarderrors....165 TableF.8:Probabilityofmovingincludingalternateindexofteacherquality..166 TableF.9:ProbabilityofmovingschoolsusingEmpiricalBayesVAM......166 TableF.10:Probabilityofmovingschoolsusingrestricted-data,EmpiricalBayes VAM...................................167 viii TableI.1:Comparingmethodsforestimatingthecoe˚cient 1 of x ij ( 1 =1 ).206 TableI.2:Comparingmethodsforestimatingthecoe˚cient 2 of x ij w j ( 2 =2 )207 TableI.3:Comparingmethodsforestimatingthecoe˚cient 1 of w j ( 1 =3 )208 ix LISTOFFIGURES FigureC.1:Kerneldensityplotsofestimationerrors, b 1 1 ,forcoe˚cientof x ij acrossreplicationsforallmethodswhentheuncorrelatedvariance assumptionholds(leftpanels)andwhenitisviolated(rightpanels). Note. FE+=AugmentedFixed-E˙ects;PC=Per-Cluster;RE= Random-E˙ects.............................138 FigureC.2:Kerneldensityplotsofestimationerrors, b 1 1 ,forcoe˚cientof w j acrossreplicationsforallmethodswhentheuncorrelatedvariance assumptionholds(leftpanels)andwhenitisviolated(rightpanels). Note. FE+=AugmentedFixed-E˙ects;PC=Per-Cluster;RE= Random-E˙ects..............................139 FigureC.3:Estimatedbiasforcoe˚cient 1 of x ij versusclustersize. Note. FE+ =AugmentedFixed-E˙ects;PC=Per-Cluster;RE=Random-E˙ects.140 FigureC.4:Estimatedrootmeansquareerror(RMSE)forcoe˚cient 1 of x ij versusclustersize. Note. FE+=AugmentedFixed-E˙ects;PC= Per-Cluster;RE=Random-E˙ects...................140 FigureC.5:Ratioofmeanstandarderrors(MeanSE)dividedbystandarddevia- tions(SD)ofestimatesversusclustersize. Note. FE+=Augmented Fixed-E˙ects;PC=Per-Cluster;RE=Random-E˙ects.......141 FigureG.1:Thee˙ectsofVAMontheprobabilityofmovingschoolswithin- districtbyyear..............................169 FigureG.2:Thee˙ectsofVAMontheprobabilityofmovingtoaabetterschool within-districtbyyear..........................170 FigureG.3:Thee˙ectofVAMontheprobabilityofmovingschoolsout-of-district byyear...................................171 FigureG.4:Thee˙ectsofVAMontheprobabilityofmovingtoabetterschool out-of-districtbyyear..........................172 FigureG.5:Thee˙ectofVAMonteachersortingwithin-districtbyyear.....173 x FigureI.1:Kerneldensityplotsofestimationerrors, b 2 2 ,forcoe˚cientof x ij w j acrossreplicationsforallmethodswhentheuncorrelated varianceassumptionholds(leftpanels)andwhenitisviolated(right panels). Note. FE+=AugmentedFixed-E˙ects;PC=Per-Cluster; RE=Random-E˙ects..........................209 xi Chapter1 PublicandPrivateLearninginthe MarketforTeachers:Evidencefromthe AdoptionofValue-AddedMeasures 1.1Introduction Gapsininformationhinderthee˚cientallocationofworkersacrossemployersSpence(1973); Jovanovic(1979);GibbonsandKatz(1991a);FarberandGibbons(1996);AltonjiandPierret (2001).Whilealargeliteraturefocusesoninformationalasymmetriesbetweenworkersand employers,amorerecentliteraturefocusesonasymmetricinformationbetweencurrentand prospectiveemployers.Empiricalworkusesthesemodelsofasymmetricemployerlearningto explainempiricalfacts,suchaswagedynamicswithrespecttojobtenureversusexperience, variabilityofwagesafterajobloss,andselectionofmobileorpromotedworkersoneasyor di˚culttoobservecharacteristicsSchönberg(2007);Pinkston(2009);DeVaroandWaldman (2012);Kahn(2013).Ifthecurrentemployerenjoysaninformationaladvantageoverother prospectiveemployers,itbecomesamonoposonistofthatinformation,permittingpersistent gapsbetweenworkers'wagesandtheirmarginalproductsoflabor.Furthermore,workers maynot˛owtotheemployersatwhichtheywouldbemostproductive.Despitethese 1 importantimplicationsandtheintuitiveappealofthetheory,thereislittledirectevidence ofasymmetricemployerlearning.Thisisinpartduetotheabsenceofdirectmeasuresof productivity,andmoreimportantly,duetoalackofexogenousvariationintheinformational landscapeinwhichemployersoperate. Inthispaper,Iadaptmodelsofpublicandprivateemployerlearningtothemarketfor middleandelementaryschoolteachers.Ithenusestatewide,micro-level,administrative datafromNorthCarolinatoformulatevalue-addedmeasures(VAMs)ofteacherproduc- tivity.VAMscalculatehowmuchateachers'studentslearnincomparisontohowmuch thosestudentsareexpectedtolearn.ThereareseveralmethodsforestimatingVAMs.In econometricterms,Iestimateteacher˝xede˙ectsintheregressionofstudenttestscores onstudentcovariatesincludingpasttestscores.Lastly,IexploittheadoptionofVAMs ofteacherperformancebytwoofthelargestschooldistrictsinthestate,ashocktothe availableinformationforsome,butnotall,potentialemployers,toprovideaninitialdirect testofasymmetricemployerlearning. TheadoptionofVAMsinNorthCarolinaprovidesarichcontextforexaminingemployer learning.EachofthetwolargedistrictsthatadoptedVAMsdidsoindi˙erentwaysand separatelyfromtherestofthestate.Thisprovidesthreedi˙erentinformationallandscapes: oneinGuilfordCountySchools(tobereferredtoasGuilford),wheretheteacher,thecurrent (orretaining)principal,andanyhiringprincipalwithinthedistrictweregivendirectaccess totheteacher'sVAMs;oneinWinstonSalem/ForsythCommunitySchools(tobereferred toasWinston-Salem),inwhichonlyteachersandtheircurrentprincipalsreceivedvalue- addedreports;andlastly,intherestofthestate,wheretheinformationstructureremained relativelyconstant.Thesereleasesofstatisticalmeasuresofteachere˙ectivenessbysome, butnotallemployers,provideuniquetestsofpublicandprivatelearninghypotheses. 2 Thisstudyexamineshowtherelationshipbetweenofteacherqualityandtheproba- bilityofmovingschoolschangeswiththeadoptionofVAMsofteachere˙ectiveness.If VAMsareinformative,theyprovideteacherswithapublicsignaloftheirability.Thus, themodelpredictsthatVAMsincreasethelikelihoodthate˙ectiveteachersmovefromone schooltoanotherwithinthedistrict.Iftheinformationspreadseasilythroughthemarket thereshouldbenodi˙erencebetweentheimpactsofVAMsformoveswithin-districtand teachertransitionsoutofGuilfordandWinston-Salem.However,ifretainingprincipalskeep teachers'VAMsprivate,ine˙ectiveteachersmaybecomemorelikelytomoveout-of-district. Thus,theasymmetricemployerlearningmodelpredictsadverseselectionofteachersout-of- district.Lastly,Iinvestigatewhetherprivateorpubliclearningpreviouslyprevailed.Prior publiclearningimpliessmallere˙ectsformoreexperiencedteachersaboutwhomemployers alreadyknowrelativelymore.PriorprivatelearningimpliesthatthereleaseofVAMswould eventhebalanceofinformationmoresoforteacherswithrelativelymoreyearsinagiven school,allelsebeingequal. Usingdi˙erences-in-di˙erencesanalysis,I˝ndthatbyreleasingVAMstoteachersand principals,bothdistrictsincreasetheprobabilitythathigh-VAMteacherswillmovewithin districttoahigher-performingschool.IestimatethatthereleaseofVAMsincreasesthe probabilitythatateacherwithaonestandarddeviationhigherVAMmoveswithin-districtto higher-performingschoolsbyabout10%.I˝ndthattheselectionofmobileteachersbecomes signi˝cantlymorenegativeforteachersmovingtoanotherschooloutsideofGuilfordand Winston-SalemaftertheyadoptVAMs.Thepolicyleadsteacherswhoareafullstandard deviationbelowaveragetobecome15%morelikelytomovefromGuilfordtoahigher- performingschoolintherestofthestate.InWinston-Salem,thee˙ectofthepolicyonthe probabilitythatahigh-VAMteachermovestoahigher-performingschoolis60%smallerfor 3 teachersmovingout-of-districtthanitisforteachersmovingwithin-district.Thefactthat weseepositiveselectiontoprincipalswithaccesstotheinformationandmuchsmallere˙ects andevennegativeselectionformovestothosewithoutaccesstotheVAMsisconsistentwith asymmetricemployerlearning. Intheprimaryeducationcontext,questionsofe˚ciencyandequityareofparticular importance.Previousresearch˝ndswidevariationinthequalityofteachersRivkinetal. (2005);Chettyetal.(2011,2014).Yet,atthepointofhire,detectinggoodteachersis di˚cult,sinceeasilyobservableteachercharacteristics,suchaseducationalattainmentand collegeselectivity,arenothighlycorrelatedwithteachere˙ectivenessRivkinetal.(2005); StaigerandRocko˙(2010).Informationalgapsmayleadschoolsanddistrictstohirerela- tivelyine˙ectiveteachers,whilepassingonmorecapableones.Thus,asymmetricinformation canhavesigni˝cantrami˝cationsforthestudentstheyserveChettyetal.(2011,2014). Afterthedateofhire,whileprincipalstypicallydonotobserveadirectmeasureofa teachers'e˙ectiveness,theycanobservetheirteachersinactionandinspectstudentout- comes.However,thequalityofateachermayremaindi˚cultfortheemployingschoolto uncover,andharderstillforotherschoolstolearn.Theamountofuncertaintyinthemarket, andwithwhomtheuncertaintylies,candi˙erentiallya˙ectnotonlytheinitialsorting,but alsotheresortingofteachersacrossschools. Persistentinformationalgapsbetweenteachers'truee˙ectivenessandemployers'percep- tionsofitmayleadschoolstoundervaluee˙ectiveteachersandallowine˙ectiveteachers toimpedetheprogressoftheirpupils.Incontrast,completeandpublicinformationallows betterteachersmorechoiceoverwheretoteach.WhenteachersaregivenVAMreports,the VAMsprovidethemanewcrediblewaytosignaltheirability. Intheteacherlabormarket,wagesaretypicallysetrigidlyandarenottiedtoper- 4 formance. 1 Thus,theimplicationsofemployerlearningarefeltprimarilythroughteacher mobilityfromoneschooltoanother.Thereisalargebodyofwork,whichexaminesteacher preferencesBoydetal.(2008);Jackson(2009);Boydetal.(2013).They˝ndthatteachersin generalprefertoteachinschoolsthatarecloserinproximitytotheirhomes,higherperform- ing,andforwhiteteachers,schoolswithalowerpercentageofblackstudents.Consequently, whileprovidinggoodteachersmorechoice,betterinformationmayalsoexacerbatethedi- videinaccesstohighqualityeducation.Thedegreetowhichinformationstaysexclusively withcurrentprincipalstheoreticallymaymitigatethesee˙ects.Thisworkprovidesthe˝rst examinationofwhetherthereleaseofVAMsleadstofurthersortingofteacherstoschools. Risinginequitymaybeanimportantconsequenceofthepolicythathasbeenpreviously overlooked. Thepossibilityofgrowinginequityinaccesstoe˙ectiveteachingisparticularlyimpor- tantgiventhespeedatwhichstatesandschooldistrictsareadoptingVAMs.Theentire stateofNorthCarolinaadoptedteacher-levelVAMsinthe2013schoolyear.AsofMay, 2014,38stateshaverequiredteacherevaluationstoincorporateteachers'impactsonstudent achievementonstandardizedexams.Evenamongtheremainingstates,manylargeschool districtshavealreadyincorporatedVAMsintoevaluationsoftheirteachers.Whilethese policieshavebeencontroversial,thedebatehaspreviouslyignoredthesignalingimpactof VAMsonthedistributionofe˙ectiveteachersacrossschools.Byexaminingchangesinthe sortingofteachers,Ievaluatetheimpactoftheinformationonthedistributionofteacher qualityacrossschools.Therisingmobilityofe˙ectiveteacherstohigh-performingschools andtheriseinthecorrelationbetweenteacherVAMsandschool-widestudentperformance 1 Thereareexceptionstothis.InSection ?? ,Idiscusstwopolicies(ABCgrowthandStrategicSta˚ng) thatdeviatefromthisstandardwagerigidity.TheABCgrowthprogramprovidesincentivestoeveryteacher inschoolsthatmaketheirgrowthtargets.Strategicsta˚ngpolicieso˙erincentivestoteachathard-to-sta˙ schools. 5 inWinston-Saleminparticular,evidencesrisinginequityinaccesstohighqualityeducation asaresultofVAMadoption. 1.2Setting Shockstotheinformationavailableonworkers'productivityarerare.Shockstotheinfor- mationofsome,butnotall,employersinamarketarerarerstill.Thereleaseofteacher performancemeasurestoprincipalsworkingwithintheschooldistrict,butnottothosein therestofthestate,o˙ersanopportunitytoexaminewhetherplausiblyvaluablepersonnel informationspreadsthroughoutthemarket. GuilfordCountySchools(Guilford)contractedwithSAS(originallycalled AnalysistoreceiveteacherEVAAS(EducationValue-AddedAssessmentSystem) measuresofteachere˙ectivenessin2000.Thesemeasuresarebasedonthemodeldevel- opedbySandersetal.(1997)underthenameennesseeValue-AddedAssessment (TVAAS).Infact,theadoptionofVAMsbyGuilfordaccompaniedthetransitionofTVAAS toEVAAS,asthesystemcameunderthemanagementofSAS,whichoriginatedatNorth CarolinaStateUniversity.Thedistrictgaveteachers,theircurrentprincipals,andhiring principalswithinthedistrictdirectaccesstotheseteachervalue-addedmeasures(VAMs). BecauseallhiringprincipalsinGuilfordcandirectlyaccessateacher'sVAM,theintroduction ofVAMsprovidesashocktotheavailableinformationtoallprincipalswithinthedistrict. TherestofthestateofNorthCarolinaadoptedEVAASmeasuresofschoole˙ectiveness in2008,buttherewasnonewteacher-speci˝cinformationprovided.Winston-Salem/Forsyth CommunitySchools(Winston-Salem)tookanadditionalstep,providingSASwithstudent- teachermatchesnecessarytoreceivethesameteacherspeci˝cmeasureofe˙ectivenessalready 6 presentinGuilford.InWinston-Salem,onlytheteacherandhisprincipaldirectlyreceived theVAMreports.TheVAMreportswerenotgivendirectlytoanyotherprincipalswithin thedistrictorotherwise. ThoughtheWinston-SalemdispersedtheVAMsinamorerestrictedway,theintroduc- tionofVAMsinWinston-Salemistheoreticallyalsopublic.AsinGrossman(1981)and Milgrom(1981),eachteachercontemplatingmovingwithinthedistricthasasincentiveto voluntarilydisclosehisscore.BecauseallprincipalsinthedistrictknowthattheVAMexists, ifateacherchoosesnottorevealhisscore,hiringprincipalswithin-districtmaywellassume thatheisasgoodastheaverageteacherwhochoosesnottorevealhisscore.Thus,all teacherswithscoresabovethataveragehaveanincentiverevealtheirscores.Consequently, theaveragescoreofthosewhodonotdisclosedropsuntilonlyteacherswithscoresatthe minimumareindi˙erentbetweenrevealingandkeepingtheinformationprivate.Ifteachers actaspredicted,allteachersvoluntarilydisclosetheirEVAASreports,andtheVAMsalter theinformationavailabletobothcurrentandhiringprincipalswithinWinston-Salem,justas theydoinGuilford.ThisshocktothepublicinformationallowsteacherswithhigherVAMs thantheirresumésmayotherwisesuggesttosignaltheirabilitytoprospectiveemployers. Thesettingandincentivesteachersfacedi˙erwhenmovingoutoftheGuilfordand Winston-Salemdistricts.Perhapsmostimportantly,itispossiblethathiringprincipalsin therestofthestateareunawareoftheexistenceofanapplyingteacher'sEVAASreport. Inwhichcase,ateachermaywithholdhissignalandleavetheprincipal'sexpectationofhis abilityunchanged. 2 Thisinformationalasymmetrymaybeavoidedbyprincipalsthoroughly 2 Inwhichcase,onlythosewhoseVAMsarehigherthanwouldotherwisebeexpectedwouldchooseto reveal,andonlyout-of-districtprincipalshiringthoseteacherswouldbeawareoftheirVAMs'presence. Furthermore,forteacherswhoseVAMisworsethanwouldbeexpectedbytheirresumés,movingoutof districtmaybeanattractivechoice,leadingtomorenegativeselectionofteachersmovingfromdistricts thatadoptVAMs. 7 researchingfromwheretheirapplicantsarecoming.Inwhichcase,thesamepredictions aswereformulatedforwithin-districtmoveswouldapply.However,suchacquisitionof informationiscostly,andprincipalsmayforgoit.Thus,thetestbetweensymmetricand asymmetriclearninghingesonwhethertheadoptionofVAMsleadstheselectionofout-of- districtmobileteacherstobesigni˝cantlymorenegativethanitse˙ectsontheselectionof within-districtmovers. SinceprincipalsinbothGuilfordandWinston-Salemreceivedtrainingaboutthemea- sures,VAMslikelyservedasamoresalientsignalforprincipalswithintheadoptingdistricts thanforthoseintherestofthestate.Out-of-districtprincipalsmayhaveputparticularly lowweightonthemeasuresin2000,whenGuilfordinitiallycontractedwithSAS.Atthat point,onlytwoyearsafterthecreationofEVAAS,NoChildLeftBehindwasstillayear awayfrompassage,andVAMswerelargelyabsentfromeducationpolicydiscussions.The salienceofthesignalwaslikelylessofanissueforteachersmovingfromWinston-Salem, consideringschool-levelEVAASmeasureswereimplementedacrosstheentirestatethesame year.Thismayleadthelearningresultsforout-of-districtmovestobemorepronouncedfor GuilfordthantheyareforteachersleavingWinston-Salem. TosummarizethebasicintuitionofthemodelinSection1.4,ifVAMsprovidemeaningful informationtoallprincipalsinthedistrict,andteachersingeneralprefertoteachatbetter schools,afterdistrictsreleaseVAMs,goodteacherswillbemorelikelytomovetohigher- performingschools.Itisalsopossiblethatcurrentprincipalsbecomelessabletokeepquiet whichteachersarereallygood,whilepassingo˙theworseteacherstounwittingemployers. TableA.1showsexactlythisgeneralpatternformoveswithinGuilfordandWinston-Salem. Inbothdistricts,theaverageVAMofteacherswhomovewithinthedistrictincreasessharply afterreleasingVAMs.Formovesoutofthesedistricts,theaverageVAMofmovingteachers 8 dropsfollowingtheadoptionofthepolicy.Thesemeansarenotconditionalonanyeasily observablecharacteristics,andsoitisdi˚culttosaywhetherthechangesininformation aredrivingthesepatterns.However,theincreasesof0.259and0.119standarddeviations ofaverageVAMsofmoverswithinGuilfordandWinston-Salemrespectivelysuggeststhat releasingVAMswithinthedistrictallowshigh-VAMteacherstomovemoreeasilytoother schools.The0.290and0.143dropinaverageVAMsofmovingoutofGuilfordandWinston- Salemisindicativeoflow-VAMteachersmovingtoplausiblylessinformedprincipalsoutside ofthedistrict. 1.3EmployerLearning,VAMs,andTeacherMobility Thisisthe˝rststudydirectlytestingageneralmodelofpublicandprivatelearningby exploitinginformationshockstoalarge,relevantlabormarket.However,Thereisarobust extantliteraturebuildingmodelsofemployerlearningand˝ttingthemtostylizedempirical facts. FarberandGibbons(1996)providestheseminalmodelandtestforemployerlearning. Theyassumethatemployerscannotdirectlyobservetheabilityofpotentialworkersand mustrelyoncorrelatestoinferworkers'expectedvaluetothe˝rm.Theytreatasubsetof workercharacteristicsaseasilyobservabletoall,anotheraseasilyobservabletothemarket (andnottoresearchers),andyetanothersubsetofpotentialcorrelateswithproductivity aseasilyobservabletotheeconometricians(butnotthemarket).Thisliteraturetypically usesthepercentilefromacognitiveabilityassessment,theArmedForcesQuali˝cationTest (AFQT)fromtheNationalLongitudinalSurveyofYouthof1979(NLSY79),asthisrelatively strongcorrelatewithproductivitythatisveiledtothethemarketatthetimeofhire,but 9 isvisibletoresearchers.Byassumingacompetitivemarketplaceandthatemployersall learnatthesamerate,intheFarberandGibbons(1996)modelwagesperfectlytrack theemployers'learningprocess.AltonjiandPierret(2001)adoptasimilarfoundationin theirexaminationofstatisticaldiscriminationasdoesLange(2007)inhisstudyofthe speedatwhichemployerslearn.Each˝ndsthatthecorrelationbetweenwagesandAFQT scoreincreaseswithexperience,whilethecorrelationbetweenwagesandeasilyobservable characteristicsfallsovertime. Recentworkintheeconomicsofeducationpresentsevidencethatprincipalsalsolearn aboutteacherqualityovertime.WhileStaigerandRocko˙(2010)andRivkinetal.(2005) pointtothedi˚cultyinidentifyinge˙ectiveteachersatthepointofhire,JacobandLefgren (2008)presentsevidencethatprincipals'evaluationsarepositivelycorrelatedwithVAMsof teachere˙ectiveness,butnotperfectly.They˝ndthatprincipalsarebetteratidentifying themostandleaste˙ectiveteacher.Thefactthattheyobserveslightlyhighercorrelations forprincipalswhohaveknowntheirteachersforlongerisfurthersuggestiveofagradual learningprocess. 3 Thestrongestevidenceofprincipalslearningaboutteacherqualitycomes fromRocko˙etal.(2012).TheypresentexperimentalevidencethatteacherVAMsprovide signi˝cantinformationonwhichprincipalsupdatetheirpriorbeliefs.Itisimportantto notethatinthisexperiment,onlyteachers'currentprincipalsreceiveVAMreports,notthe teachersthemselvesorprincipalsofotherschoolswithinthedistrict.Surveysofparticipating principalsshowthatthosewhorandomlyreceivedmorepreciseVAMreportsweremore responsivetotheinformation,thanwereprincipalsreceivingnoisierVAMreports. 4 These 3 ChingosandWest(2011)providefurtherevidencethatprincipalshoneinonthee˙ectivenessoftheir teachers.They˝ndthatprincipalsclassifytheirteachersonthebasisofe˙ectiveness,andmovethem accordingly.Principalsofschoolsunderaccountabilitypressurearemorelikelytomovee˙ectiveteachers intoandlesse˙ectiveteachersoutofhigh-stakesteachingassignments. 4 Rocko˙etal.(2012)also˝ndsthatprovidingVAMstoprincipalscauselesse˙ectiveteacherstoleave atahigherrate.Whiletheauthorsdonotdirectlylinktheseresultstoeitherlearninghypothesis,these 10 resultsareconsistentwiththeBayesianupdatingmodelusedinFarberandGibbons(1996); AltonjiandPierret(2001),andLange(2007). Schönberg(2007);Pinkston(2009);Kahn(2013),andBates(2015)eachrelaxthesym- metriclearningassumptionandallowforprivateemployerlearning.Also,eachusethe NLSY79totesttheirmodelsagainstempiricalfeaturesofthedata.Theircumulative evidenceregardingasymmetriclearningismixed.Whereas,Schönberg(2007)˝ndsthat learningislargelysymmetric,Pinkston(2009)˝ndsthatlearningislargelyasymmetric. Theirdisagreementhingesonwhetherinformationpassesthroughjob-to-jobtransitions,with Pinkston(2009)˝ndingthatthecorrelationbetweenwagesandabilitymovesmoreclosely withrespecttocontinuousworkingspellsthanwithexperience.BothSchönberg(2007)and Bates(2015)thatworkersareonlyadverselyselectedintomobilityinjob-to-unemployment transitions,whereasasymmetriclearningalsopredictssuchselectionforjob-to-jobmoves aswell.However,Bates(2015)alsodemonstratespositiveselectionintomobilityonthe basisofeducation,notingthatconsistentwithasymmetriclearning,thosewhoattendmore competitivecollegesaremorelikelytobothswitchemployersandbelaido˙.Consistent withasymmetricemployerlearning,Kahn(2013)˝ndsthatmovers'wagesaremorevolatile intheimmediateaftermathofatransitionthanarethewagesofthosewhoremaininplace. 5 OnlyDeVaroandWaldman(2012)departfromtheuseoftheNLSY.Theyuseadmin- istrativepersonnel˝lesfromalarge˝rmtoexaminepromotiondecisionsbasedonprivate andpublicinformation.Insupportofasymmetricemployerlearning,they˝ndthatcon- ditionalonprivateperformancereviews,thosewithmoreeducationaremorelikelytobe resultsintheexperimentalcontextareconsistentwithasymmetricemployerlearning. 5 Kahn(2013)alsoconsidersdi˙erencesbetweenworkerswhoenterapositionduringrecessionsasopposed toeconomicexpansions,withtheideathatthereislessvariationintheabilityofentrantsduringrecessions. Shealsousesvariationintheamountofexposureanoccupationhasoutsidethe˝rm,assumingthatlearning ismoresymmetricinmoreexposedoccupations.Also,thee˙ectsarelargerforthosewhoenterajobduring aneconomicexpansionandforthoseinmoreinsularoccupations. 11 promotedthanarethosewithlesseducation.Theyalsopresentevidencethatlargerwage increasesaccompanypromotionsoflesseducatedworkersthanaccompanypromotionsof higher-educatedworkers.This,theyargue,isduetothefactthatpromotionsareastronger publicsignalforthosewithlower,easilyobservablecharacteristics. AcommoncriticismofmuchoftheearlierliteratureaskswhatAFQTscoresarereally tellingus.ThereislittleevidencethatAFQTscoresarerelatedtoproductivityinmanyjobs heldbythelargelylow-skilledrespondentsoftheNLSY.Similarly,ifemployerscaregreatly aboutAFQTscores,theywouldsimplyadministerthetestthemselves.Byusingamore directmeasureofproductivitythantheassumedcorrelates,thisstudyavoidssuchcriticism. Moreimportantly,thestylizedempiricalfactsgivenasevidenceofasymmetriclearningare consistentwiththetheoreticalmodel,butaresusceptibletoalternativeexplanations.For instance,post-movewagevolatilitymaybeexplainedbydi˙erencesinjobmatchquality, educationmayprovidemorehigherlevelskillsleadingtofasterpromotion,andsymmet- riclearningmayexplainwhylargewageincreasesaccompanypromotionsofless-educated workers.Theabsenceofdirectasymmetricinformationshockshaspreventedtheprevious literaturefromexaminingwhethertheinformationaladvantagespersistandin˛uenceworker mobilitypatternsinequilibrium.Thisworkusesthereleaseofworker-levelperformancedata tosome,butnotall,employersasauniquenaturalexperiment,totestthedegreetowhich theinformationspreadsamongemployers,whethermobilityrespondsinaccordancewith theory,andthetypeoflearningthathadpreviouslyprevailed. Furthermore,whilethereisalargeliteratureexaminingthemobilitypattersofhigher- orlower-VAMteachers,nonehavepreviouslyconsideredthesignalinge˙ectsofVAMson teachermobilityandthedistributionofteacherqualitywithinthemarket.Studentsinpoor, low-achievingschoolsfaceteacherswhoareingenerallessexperienced,lesseducated,and 12 lesse˙ectivethantheircounterpartsinmorea˜uentandhigherachievingschoolsLankford etal.(2002);Clotfelteretal.(2005);Sassetal.(2012). 6 Thoughtheirissigni˝cantchurn withintheteacherlabormarket,Hanusheketal.(2005);Krieg(2006);Goldhaberetal.(2007) andBoydetal.(2008)eachnotethathigherVAMteacherstendtostayintheprofession longerthandotheirlesse˙ectivecounterparts,andhigh-VAMteachersarenomorelikely totransferbetweenschoolsthantheircounterparts. 7 Thereismoredisagreementabout distributionale˙ectsofthisturnover.Boydetal.(2008)˝ndsthat,conditionalonmoving, high-VAMteachersaremorelikelytomovetohigh-performingschoolsthanarelow-VAM teachers,whereasHanusheketal.(2005)andGoldhaberetal.(2007)˝ndnoevidenceof thisresortingofteachers.While,descriptionsofwheree˙ectiveteachershavetraditionally movedfromandtohaveimportantimplicationsforeducationinequity,theyhavelittlepower topredicthowtheadoptionofVAMswillaltertheallocationofteachersacrossschools. Workcloselyexaminingteachers'preferencesoverworkenvironmento˙ersinsightinto howteachermobilitypatternsmaychangewiththeintroductionofVAMs.Jackson(2009) andBoydetal.(2013)analyzesteacherseach˝ndthatonaveragewhiteteachersprefernot toteachinschoolswithalargeproportionofblackstudents.Boydetal.(2013)also˝nd thatteacherspreferschoolsthatarecloser,aresuburban,andhaveasmallerproportionof studentsinpoverty. IfVAMsprovidenewandcredibleinformationtoprincipals,thisnewsignalmayexpand thenumberofschoolswillingtohirehigh-VAMteachers.Takingtheestimatedpreferences fromJackson(2009)andBoydetal.(2013)asgiven,thisexpandedchoicesetmaylead high-VAMteacherstomovetoschoolsthathavelowerproportionsofminorities,aremore 6 Sassetal.(2012)alsonotesthatthereishugevariationinteacherqualitywithinhighpovertyschools. 7 Boydetal.(2008)˝ndsthatine˙ectiveteachersaremorelikelytoleavetheprofessiononlyintheir˝rst yearofteaching. 13 a˜uent,andarehigherachieving.Whilethisearlierliteraturepointsatthepossibility,it hasnotdirectlyexaminedthequestionofrisinginequalityintheallocationofteacherquality asaresultofVAMadoption.GuilfordandWinston-Salem'searlyreleaseofVAMs,allows thisworktoexplorethispreviouslyignoredconsequenceoftheactivelydebatedpolicy. 1.4Model Thissectiondevelopsamodeltoprovidepredictionsforwhichworkersmove,andwherethey howeachmaychangeinresponsetoaninformationshock.PleaseseeAppendixE forproofsofthesepredictions.Themodelbuildsonthemodelofasymmetricemployer learningpresentedinPinkston(2009),whichinturnbuildsuponthecanonicalmodelsof symmetriclearningpresentedinFarberandGibbons(1996)andextendedinAltonjiand Pierret(2001). 1.4.1ModelStructure Teachersreceivetwojobo˙ersinthe˝rstperiodandtakethehighesto˙er.Eachsubsequent period,teachersreceiveoneoutsideo˙erfromeitheraprincipalwithinoroutsideofthe currentdistrictwithagivenprobability.Principalsfacerigidbudgetconstraints,which translatetoa˝xednumberofpositions.Principalswithavacancywhoencounterateacher presenttheteacherwithano˙erre˛ectingtheirexpectationsaboutthee˙ectivenessofthe teacher,whichisbasedupontheinformationavailable.Iitemizetheinformationstructure below: 1.Truee˙ectivenessisgivenby, = m + ,where m isthepopulationmeanofproductivity 14 amongaworker'sreferencegroupand ˘ N (0 ;˙ ) . 8 2.Thepublicsignalisgivenby R x = + ˘ x ; where ˘ ˘ N (0 ;˙ ˘ ( x )) ,and @˙ ˘ ( x ) @x < 0 . 3.Privatesignal: (a)Forhiringprincipals(denotedbythesuperscript h ),theprivatesignalisgiven by P h = + ˝ h where ˝ h ˘ N (0 ;˙ ˝ (0)) . ˙ ˝ (0) is˝xedovertime. (b)Foraretainingprincipal(denotedbythesuperscript r ),theprivatesignalisgiven by P r t = + ˝ r t where ˝ r t ˘ N (0 ;˙ ˝ ( t )) and @˙ ˝ ( t ) @t < 0 . 4.TheVAMserveasanadditionalpieceofinformationthatmayalterboththemean andprecisionofthepublicorprivatesignaldependingonwhetheritisavailableto bothbiddingprincipals.Ithastheform V = + ,where ˘ N (0 ;˙ ) . (a)WhenbothprincipalsareinformedbyVAMs,thepublicsignalbecomes R = ˙ R x + ˙ ˘ ( x ) V ˙ + ˙ ˘ ( x ) .Thevarianceof R isdenotedas ˙ ˘ ( xV ) . (b)WhenonlytheretainingprincipalisinformedbyVAMs,herprivatesignalbe- comes P r = ˙ P r t + ˙ ˝ ( t ) V ˙ + ˙ ˝ ( t ) .Thevarianceof P r isdenotedas ˙ ˝ ( tV ) .Thehiring principal'ssignalremainsunchanged. 5.Thenoiseofeachsignalisorthogonaltothenoiseoftheothersignals. 9 Itisimportanttounderstandthecontextofthislabormarketforteachers.Informulating themodel,Iwillhighlightareasinwhichthismarketispeculiarandthemodelstructuresthat 8 Thenormalityassumptionsarenotnecessary,butareusefulinderivingthecomparativestatics. 9 Theorthogonalityassumptionsarealsonotnecessarytoderivethefollowingpredictions.However, relaxingtheserequirealessrestrictive,thoughmorecomplicatedsetofassumptions,outliningthedirection andmagnitudeofcorrelationsbetweentheerrorsofthesignals. 15 accompanythem.However,theinformationstructureisstandard,baseduponaBayesian updatingmodelwiththemodi˝cationthatemployersreceivetwosignalsratherthanone.I assumethatteachersknowtheire˙ectiveness ( ) ,butcannotcrediblyrevealit.Thereare twobroadclassi˝cationsofprincipals:thosewhoarehiring(denotedbythesuperscript h ); andthosewhoareretainingteachers(denotedbythesuperscript r ).Ifurtherdistinguish betweenwithin-districthiringprincipalswhocanaccesstheincomingVAMs,andout-of- districthiringprincipals,whoseinformationdoesnotchange.Asateacherbeginshercareer, allprincipalsbeginwiththepriorbeliefthatsheisasgoodastheaverageteacherwithher samecharacteristics ( m ) : Theteacherencounterstwoprincipals,bothofwhomarehiring principalsinthis˝rstperiod,towhomshemayprivatelysignalherabilityakintoan interview,(denotedby P h 0 where0indicatesnoadditionalprivateinformation). Overtime,teachersmaydrawontheirexperiencetobolstertheirpublicsignaldenoted by R x (forexamplesconsiderresumésandnetworksofreferences).Anyinformation ( x ) thatiscrediblyrevealedtobothprospectiveemployersproducesmoreprecisepublicsignals. Experienceservesasaproxyforadditionalinformation,asistypicalintheliterature.If thereispubliclearning,generallythevarianceofthepublicsignal ( ˙ ˘ ( x )) willshrinkwith teacherexperience @˙ ˘ ( x ) @x < 0 .Howeveranynewpublicinformationdirectlyproduces thise˙ect. Throughinteractions,observations,and/orattentiontostudentoutcomes,principalsmay obtainprivateinformationunavailabletorivalemployers ( t ) .Retainingprincipals'signals ( P r t ) arecomposedofinformationthatisunavailabletotheotherprospectiveemployer. Yearsoftenurewiththecurrentemployerserveasproxyforthisaccumulated,private information,asistypicalintheliterature.Ifsuchprivatelearningoccurs,whilehiring principals'privatesignalsfrominterviewingtheteacherhaveaconstantlyhighvariance 16 ( ˙ ˝ (0)) ,theprecisionofthecurrentprincipal'ssignal( ˙ ˝ ( t ) )increasesthelongerateacher worksintheschool.Withanyaccumulationofprivateinformation, ˙ ˝ ( t ) <˙ ˝ (0) forall t> 0 : Inordertonestsymmetriclearningwithinthemore˛exiblemodel,Imaintainthat thateveninthisspecialcase,employersreceiveaprivatesignaleachperiod,butthevariance ofthesignalisconstantovertenure ( ˙ ˝ ( t )= ˙ ˝ (0) forall t> 0) . VAMsenterthelearningmodelasanadditionalpieceofinformationthatmayentereither thepublicorprivatesignal.WhetherVAMsin˛uencepublicorprivatesignalsdepends onwhetherVAMsareaccessibletobothprincipals(ascertainlyoccursformoveswithin theunrestrictedGuilfordCountyschooldistrictandtheoreticallyoccursintherestricted Winston-Salemdistrict)orareaccessibletoonlycurrentprincipals(asismorelikelytooccur whencompetingprincipalsarefromdi˙erentdistricts).IfVAMsenterretainingprincipals' privatesignal, P r = ˙ P r t + ˙ ˝ ( t ) V ˙ + ˙ ˝ ( t ) replaces P r t .IfVAMsenterbothprincipals'publicsignal, R = ˙ R x + ˙ ˘ ( x ) V ˙ + ˙ ˘ ( x ) replaces R x .TheintroductionofVAMsaltertheseexpectationsby changingboththecontentofthesignalandthesignal'sprecision,andthustheweightthat principalsascribetoit. 1.4.2Bidding Inmanypubliceducationsystems,strictsalaryschedulesdeterminesteachers'pay.InNorth Carolina,thestatesetsabasesalaryschedulethatdependsexclusivelyuponeasilyobserv- ablecharacteristics,suchaseducationandexperience. 10 Districtstypicallysupplementthis baseamountwithapercentageofthebaseschedule.Ingeneral,thismeansthatagiven teacherwillearnthesamesalaryregardlessofwhereandwhatheisteachingwithinthe 10 Asof2014,NorthCarolinawillmovetopayingteachersinpartbaseduponteachers'VAMs. 17 district. 11 Further,cumbersomedismissalprocessesresultinteachersinitiatingmuchof themobility.Whileprincipalscannotadjustsalariestoin˛uencewhetherateacherstays, principalsmayin˛uenceschoolsta˚ngthroughnon-pecuniarypositionattributes,suchas planningtime,teachingassignments,oradditionalrequirements.Boydetal.(2008,2013), andJackson(2009)eachprovideevidencethatteachershavestrongpreferencesovernon- wagejobattributes. Initially,teacherstakethepositionthato˙ersthehighesttotalcompensation ( C isd ) , whichiscomprisedofsalary ( w d ) setbydistrict d ,characteristicsofschool s ( S sd ) ,and characteristicsofposition i ( J isd ) .Thus, C isd = w d + S sd + J isd . Forsimplicity,Iassumethateachprincipalpresentsasealedbidfortheteacherand paystheminimumofthetwobids.Insuchsealed-bid,second-priceauctions,principals' optimalstrategyistoo˙erthetheirexpectationoftheteacher'se˙ectiveness(assuming thatprincipalsseektomaximizeteachere˙ectivenesswithintheirschools). 1213 Principals formulatetheseexpectationsbyaveragingovertheirpriorbeliefofquality( m ),thepublic signal( R x ),andtheirprivatesignal( P h 0 ).Theyweighteachsignalbyitsprecisionrelative totheothersignals,similartoastandardBayesianupdatingmodel.Aspublicinformation becomesmorecomplete,hiringprincipalsgivelessweighttotheirpriorbeliefsandprivate noisysignalsfrominterviews,andmoreweighttothepublicsignal.Thus,letting Z h NV = 11 InSection1.7,IdiscussboththeABCgrowthandstrategicsta˚ngpolicies,whichdeviatefromthis generalcase.TheABCgrowthprogramprovidesincentivestoeveryteacherinschoolsthatmaketheir growthtargets.Strategicsta˚ngpolicieso˙erincentivestoteachathard-to-sta˙schools.Thebonuses attachedtosuchpositionsvariedformulaicallyandoutsideprincipals'discretion. 12 Previousversionsmodeledopencontinuousbidding,whichpermitstheadoptionofoptimalbidding strategiesfromMilgromandWeber(1982).Thisallowseachschooltoupdatetheoptimalbidconditioning ontherival'sbiddingbehavior.However,bothbiddingprocessesresultinthesamepredictions. 13 Priorworkshowsprincipalscareaboutteachere˙ectiveness,particularlyinschoolsunderaccountability pressure.Otherworkshowsthathigh-VAMteachersalsoleadtoawidearrayofbetterfutureoutcomes fortheirstudents,givingfurtherreasontosuggestprincipalsmaymaximizetheseshort-runmeasuresof e˙ectiveness. 18 ˙ ˝ (0) ˙ ˘ ( x )+ ˙ ˝ (0) ˙ + ˙ ˙ ˘ ( x ) ,ifuninformedofateacher'sVAM(subscriptNV),ahiring principal'soptimalmaximumbid ( b h isdNV ) isgivenbyequation1.1. b h isdNV = ˙ ˝ (0) ˙ ˘ ( x ) Z h NV m + ˙ ˝ (0) ˙ Z h NV R x + ˙ ˙ ˘ ( x ) Z h NV P h 0 : (1.1) Ifthereispubliclearning,asexperienceincreases,morepublicinformationleadstoamore precisepublicsignal.As ˙ ˘ ( x ) declines,hiringprincipalsplacelessweightontheirprior beliefsandnoisyprivateinformation,andmoreweightonthepublicsignal. Aprincipalseekingtoretainherteacher,whoisuninformedofhisVAM,hasanoptimal bid ( b r isdNV ) withveryasimilarformtothatshownisequation1.1.Equation1.2showsher optimalbid,letting Z r NV = ˙ ˝ ( t ) ˙ ˘ ( x )+ ˙ ˝ ( t ) ˙ + ˙ ˙ ˘ ( x ) . b r isdNV = ˙ ˝ ( t ) ˙ ˘ ( x ) Z r NV m + ˙ ˝ ( t ) ˙ Z r NV R x + ˙ ˙ ˘ ( x ) Z r NV P r t : (1.2) Retainingprincipalsprovidemoreweighttotheirprivateinformation( P r t ),iftheyobtain moreusefulinformationthanispubliclyavailable.Thisisre˛ectedby ˙ ˝ ( t ) whichshrinks withadditionalprivateinformationasopposedto ˙ ˝ (0) fromequation1.1,whichremains constantforhiringprincipals. TheintroductionofVAMsalterstheinformationavailabletoprincipals,buttheop- timalbidsthatincorporateVAMshavesimilarformtothoseshowninequations1and2. WhethertheVAMsarepublicorprivateareparticularlyimportantfordeterminingretaining principals'expectationsofagiventeacherintheadoptingdistricts. Ifaprincipal'srivalisfromoutsideofthedistrictanduninformedofthemeasure,the retainingprincipalincorporatestheVAMintoherprivatesignal.Thenewprivatesignal 19 ( P r )becomestheprecision-weightedaverageofthepriorprivateinformationandthenew VAM.Thus,theoptimalbidofaretainingprincipal,whohasaccesstoherteacher'sVAM andwhoserivaldoesnothaveaccesstotheVAM(denotedbythesubscriptRV)isshown inequation3where Z r RV = ˙ ˝ ( tV ) ˙ ˘ ( x )+ ˙ ˝ ( tV ) ˙ + ˙ ˙ ˘ ( x ) . b r isdRV = ˙ ˝ ( tV ) ˙ ˘ ( x ) Z r RV m + ˙ ˝ ( tV ) ˙ Z r RV R x + ˙ ˙ ˘ ( x ) Z r RV P r : (1.3) Equation3issimilartoequation2exceptforthereplacementof P r t by P r andof ˙ ˝ ( t ) by ˙ ˝ ( tV ) .Inexpectation,themagnitudeoftheprivatesignalwillnotchangewiththe introductionofVAMs.However,theprecisionofthecumulativeprivateinformationmust increase. Lemma1:TheprecisionoftheprivatesignalincreaseswiththeincorporationofVAMs intotheprivatesignal( ˙ ˝ ( tV ) <˙ ˝ ( t ) ). Proof:Undertheorthogonalityassumptions, var ( P ) ˙ ˝ ( tV )= ˙ 2 ˙ ˝ ( t )+ ˙ ˙ ˝ ( t ) 2 ( ˙ + ˙ ˝ ( t )) 2 = ˙ ˙ ˝ ( t ) ˙ + ˙ ˝ ( t ) . ˙ ˝ ( t )( ˙ + ˙ ˝ ( t )) ˙ + ˙ ˝ ( t ) ˙ ˙ ˝ ( t ) ˙ + ˙ ˝ ( t ) = ˙ 2 ˝ ( t ) ˙ + ˙ ˝ ( t ) ,and ˙ 2 ˝ ( t ) ˙ + ˙ ˝ ( t ) > 0 ,bypropertyofvariances. Thisdecreaseinthevarianceoftheprivatesignaldecreasestheweightretainingprincipals placeontheirpriorbeliefsandthepublicinformation,whileincreasingtherelativeweight theyplaceontheirnowfullerprivateinformation. Turningbacktohiringprincipals'expectationsofteacherquality,ifahiringprincipal isuninformedofVAMs(ortheirexistence),herexpectationoftheteacher'squalitywould remainunchangedfromthosepresentedinequation1.Thus,theintroductionofVAMs exacerbateinformationalasymmetriesbetweenprospectiveemployers. 20 Incontrast,ifbothbiddingprincipalsareinformedofateacher'sVAM,asislikely thecasewhenbothprincipalsarefromoneoftheadoptingdistrictsafterthepolicytakes e˙ect,theVAMenterstheprincipals'publicsignalofteacherquality.Letting Z r HV = ˙ ˝ ( t ) ˙ ˘ ( xV )+ ˙ ˝ ( t ) ˙ + ˙ ˙ ˘ ( xV ) ,equation4providestheretainingprincipal'soptimal bidwhenthehiringprincipalmayalsoaccessateacher'sVAM(denotedwiththesubscript HV). b r isdHV = ˙ ˝ ( t ) ˙ ˘ ( xV ) Z r HV m + ˙ ˝ ( t ) ˙ Z r HV R + ˙ ˙ ˘ ( xV ) Z r HV P r t : (1.4) Equation4issimilartoequation2withtheexceptionthat R x isreplacedby R ,asVAMs enterthepublicsignal.Whileinexpectationthemagnitudeofthepublicsignalisthesame withorwithoutVAMs,thevarianceofthepublicsignalmustchangeasaresult. Lemma2:TheprecisionofthepublicsignalincreaseswiththeincorporationofVAMs intothepublicsignal( ˙ ˘ ( xV ) <˙ ˘ ( x ) ). Proof:Undertheorthogonalityassumptions, var ( R ) ˙ ˘ ( xV )= ˙ 2 ˙ ˘ ( x )+ ˙ ˙ ˘ ( x ) 2 ( ˙ + ˙ ˘ ( x )) 2 = ˙ ˙ ˘ ( x ) ˙ + ˙ ˘ ( x ) . ˙ ˘ ( x )( ˙ + ˙ ˘ ( x )) ˙ + ˙ ˘ ( x ) ˙ ˙ ˘ ( x ) ˙ + ˙ ˘ ( x ) = ˙ 2 ˘ ( x ) ˙ + ˙ ˘ ( x ) . ˙ 2 ˘ ( x ) ˙ + ˙ ˘ ( x ) > 0 ,bypropertyofvariances. Forequation4,thismeansthatretainingprincipalswillshiftweightthattheyhad previouslyplacedontheprivatesignalontothenewmorecomplete'publically'available information. IfaccesstotheVAMsissharedbetweenemployers,theVAMsenterthepublicsignal ofhiringprincipals,justastheyenterthepublicsignalofretainingprincipals.Letting Z h HV = ˙ ˝ (0) ˙ ˘ ( xV )+ ˙ ˝ (0) ˙ + ˙ ˙ ˘ ( xV ) ,equation5providesthehiringprincipal's 21 optimalbidwhenshemayalsoaccessateacher'sVAM(subscriptedHV). b h isdHV = ˙ ˝ (0) ˙ ˘ ( xV ) Z r HV m + ˙ ˝ (0) ˙ Z r HV R + ˙ ˙ ˘ ( xV ) Z r HV P h 0 : (1.5) Thedi˙erencebetweenequations1and5areinthepublicsignalanditsvariance.Using the˝ndingfromlemma2,thatthevarianceofthepublicsignaldropswiththeintroduction ofVAMs,oncehiringprincipalsmayaccessateacher'sVAM,theyplacelessweightupon theirpriorbeliefsandlessweightupontheirnoisyprivateinformationtheygleanfromthe applicationprocess,andplacemoreweightonthepublicinformationthatnowincludesa teacher'sVAM.Forbidsinwhichbothprincipalsbecomeinformedofateacher'sVAM,the informationbetweenprospectiveemployersbecomesmoresymmetric,andtheirexpectations converge,asbothhiringandretainingprincipalsshiftweightontotheinformationthatthey share. 1.4.3MobilityunderAsymmetricInformation Theteacherlabormarketgenerallymovesinthesummerbetweenschoolyears.Atthat time,teachersmaysampletwoo˙ers,anupdatefromtheircurrentschoolandoneoutside o˙er.Teachersmovetotheschoolthato˙ersthehighestbid. 14 Accordingly,theprobability ofamoveis: P ( M )= P h b h isd b r isd > 0 i : (1.6) Suchschool-to-schooltransfersaremotivatedingeneralbyahiringprincipalvaluingthe teachermoresothandoestheretainingprincipal.Letting standforthecompositeerror 14 Forsimplicity,Imodelmobilitydecisionsasaspotmarket.A˝xedtransitioncostoridiosyncratic teacherpreferencesoverschoolsmaybeaddedwithoutadditionalassumptions. 22 termandsubstitutinginthebidsfrompresentedinequations1and2allowsequation6to bewrittenintheformpresentedinequation1.7. 15 P ( M )= P >˙ ˘ ( x )( ˙ ˝ (0) ˙ ˝ ( t ))( m ) : (1.7) WhiletheVAMsandwhohasaccesstothemalterstheinformationsonwhichprincipals operate,thegeneralformofequation1.7remainsthesame,makingitusefulforillustration. Suchtransitionsmayoccurduetoextremeprivatesignals.However,thismayhappeneven ifbothprincipalsreceivethesameprivatesignalduetodi˙erencesinhoweachprincipal weighsthesignalsshereceives. Forsuchmobility,itisapparentfromequation1.7thatallelseequal,theprobabilityofa moveisinverselyrelatedtotruee˙ectiveness.Intuitively,duetotheiradditionalknowledge ofteachere˙ectiveness,thecurrentschoolshouldvaluethetruee˙ectivenessoftheteacher morethantheoutsidemarket.Becausetheoutsidemarkethaslessinformationabouttrue e˙ectiveness,theoutsideschoolsshouldplacemoreweightontheeasilyobservedcorrelates withteachere˙ectivenessthanthecurrentschool,whichinformthepriorbelief( m ). Theprimaryinvestigationinthisstudyexploreshowmobilitychangeswiththeadoption ofVAMs.TheavailabilityofVAMstosomeprospectiveemployers,butnotothers,provides araretestforthemodellaidoutabove.AsdescribedinSection1.2,bothdistricts'adoption ofVAMs,theoreticallyprovideashocktotheinformationofallprincipalswithinthedistrict. TherearetwoprimarywaysofthinkingabouttheimpactofVAMsinthemodel.The˝rst ismoreinkeepingwiththeprioremployerlearningliterature.VAMsserveasdi˚cult-to- observemeasuresofteacherquality.ResearchersmayuseVAMstoproxydirectlyfor about 15 SeeSubsectionEintheAppendixforalgebraictransformations. 23 whichemployersarelearning.Inthisframework,themodelo˙erspredictionsofwhether betterorworseteachersmoveasresponsetoadoptingtheseVAMs.Equation1.8takesthis broadview. 16 @E h b h HV b r HV ( b h NV b r NV ) j m i @ = ˙ ( ˙ ˝ (0) ˙ ˝ ( t ))(( ˙ ˘ ( x ) ˙ ˘ ( xV ))( ˙ ˝ ( t ) ˙ ˙ ˝ (0) ˙ ˘ ( x ) ˙ ˘ ( xV ) + ˙ ˘ ( xV ) ˙ 2 ˙ ˘ ( x ) ˙ ˝ (0)+ ˙ ˘ ( xV ) ˙ ˝ ( t ) ˙ 2 ˙ ˘ ( x )+( ˙ ˘ ( xV )+ ˙ ˘ ( x )) ˙ ˝ ( t ) ˙ 2 ˙ ˝ (0)) : (1.8) Undertheassumptionthat ˙ ˝ (0) >˙ ˝ ( t ) ,whichisfundamentaltoasymmetricemployer learningandby ˙ ˘ ( x ) >˙ ˘ ( xV ) ,whichwasshowninlemma2, @E h b h HV b r HV ( b h NV b r NV ) j m i @ > 0 .Therefore,themodelpredictsthatprovidingVAMstobothprincipals,asoccurred withinbothdistricts,shouldraisetheprobabilitythatgoodteachersmove,allelseequal. Underthesecondinterpretation,EVAASVAMsenterthetwodistrictsdirectlyasnew signals.Accordingly,themodelo˙erspredictionsonthedi˙erentiale˙ectsofthepolicyon theprobabilityofmovingforteachersreceivingdi˙erentsignals,allelseequal.Aftersome algebra,equation1.9takesthismorenarrowview. 17 @E h b h HV b r HV ( b h NV b r NV ) j mV i @V = 1 Z h HV Z r HV ˙ ˘ ( x ) ˙ + ˙ ˘ ( x ) > 0 (1.9) Therefore,whiletheinterpretationsaresubtlydi˙erent,thecomparativestaticswithrespect toVAMsafterthepolicytakese˙ectarethesame.Withinthedistricts,wherebothprincipals areawareofthesignalsoncetheyareimplemented,themodelpredictsteacherswhoreceive ahigh-VAMsignalbecomemorelikelytotransferschools. 16 SeeAppendixEprovidestherelevantalgebraictransformations. 17 SeeAppendixEfortherelevantalgebraictransformations. 24 RecallfromSection1.2,thatifprincipalsinotherdistrictsknowoftheexistenceof VAMsforteachersfromWinston-SalemandGuilford,thepolicywouldtheoreticallyalter theirinformationaswell.Thepreviouspredictionwouldapplytoout-of-districtmovesas well.However,itisplausiblethatprincipalsinotherdistrictswereuninformedaboutthe policy.Inwhichcase,theadoptionofVAMsinGuilfordandWinston-Salemwouldmakethe balanceofinformationmoreasymmetric,intheeventthatateachercontemplatesmoving toanotherschooloutsideWinston-SalemorGuilford.Ifthehiringprincipalisuninformed oftheVAM,VAMsenterretainingprincipals'privatesignals. ThesametwointerpretationsofVAMs'roleapplyhere.Againbeginningwiththebroader viewofVAMsasameasureof ,equationEdemonstratesthemodel'spredictionswith respecttoteachers'underlyingabilitiesontheprobabilityofmovingtouninformedprinci- pals. 18 @E h b h RV b r RV ( b h NV b r NV ) j m i @ = ˙ ˘ ( x ) 2 ˙ Z h RV Z r RV Z h NV Z r NV ( ˙ ˝ ( t ) ˙ ˝ ( tV ))( ˙ ˝ (0) 2 ˙ 2 + ˙ ˝ (0) 2 ˙ ˘ ( x ) 2 + ˙ ˝ (0) 2 ˙ ˙ ˘ ( x )+ ˙ ˘ ( x ) 2 ˙ 2 : (1.10) Underlemma1, ˙ ˝ ( t ) >˙ ˝ ( tV ) ,whichimpliesthat @E h b h RV b r RV ( b h NV b r NV ) j m i @ < 0 . Therefore,themodelpredictsthatafterthereleaseofVAMstoretainingprincipals,the likelihoodofmovingout-of-districtwilldecreasewithincreasesinteacherquality,andvice versa. UnderthemorenarrowviewofVAMsasonlypertainingtothesignalitself,againthe 18 SeeAppendixEfortherelevantalgebraictransformations. 25 predictionsremainconsistent.Equation1.11presentsthepartialderivativeoftheexpected di˙erenceinthedi˙erencesbetweenemployersbidswithrespecttotheVAMsignalitself. 19 @E h b h HV b r HV ( b h NV b r NV ) j mV i @V = ˙ ˘ ( x ) ˙ ˙ ˝ ( t ) Z r RV ( ˙ + ˙ ˝ ( t )) < 0 (1.11) Herethemodelpredictsadverseselectionofout-of-districtmovesonthebasisofVAMs,all elseequal.Itisimportanttonotethatgood(orhigh-VAM)teachersmaychoosetoreveal theirEVAASreporttoprincipalsinotherdistrictsinane˙orttomoveout-of-district.Ac- cordingly,thefurtheringofinformationasymmetriesbetweenemployersmaynotuniversally applytoout-of-districtmoves.However,aslongassomelow-VAMteachersareabletomove out-of-districtwithoutbeingpenalizedbytheirEVAASreport(ortheirunwillingnesstore- vealit),themodelpredictsmorenegative(smallerinmagnitudeornegative)e˙ectsofVAM ontheprobabilityofmovingout-of-districtafterpolicyimplementationthanareproduced formoveswithin-district.Thus,thetestbetweensymmetricandasymmetriclearningis whethere˙ectsofthepolicyontheselectionofout-ofdistrictmoversaresigni˝cantlymore negativethanthee˙ectsofadoptingVAMsontheselectionofwithin-districtmovers. 1.5DataandEstimation Inthissection,IdescribeboththedataandmethodsusedtogenerateVAMsofteacher e˙ectiveness,andthee˙ectsofthedistrictpoliciesontheteachermobility.Subsection1.5.1 describesthegenerationofVAMs.Subsection1.5.2describestheestimationsample.Sub- section1.5.3describesthedi˙erence-in-di˙erencesestimationapproachusedtoidentifythe e˙ectsofthenewinformationonthemobilitydecisionsofteachersandprincipals. 19 SeeAppendixEfortherelevantalgebraictransformations. 26 1.5.1Value-AddedMeasures Whilethereareothervaluabledimensionsofteaching,manyschoolsanddistrictscarea greatdealaboutteachers'abilitiestoraisetheirstudents'performanceonstandardized assessments.Thisstudyreliesonadministrative,longitudinaldata,whichlinksstudentsto theirteachersandwasgenerouslyprovidedbytheNorthCarolinaEducationResearchData Center(NCERDC)toestimateteachers'abilitiestodojustthat.Thougharobustsourceof data,unfortunately,theNCERDCdoesnotcontaintheexactVAMsissuedtoeachteacher withinthetreatmentdistricts,andneitherdistrictagreedtoreleasethem.Consequently,this studywillmeasurethestudentgainsontheNorthCarolinaEndofGradeexamsattributable toeachteacher. Therearetwoprimarywaystogoaboutthis.The˝rstistoattempttomodeltheexact measuresthatteachersandprincipalsreceive.Thisisprimarilyusefulinexplainingthe teachers'andprincipals'observedbehavior.Thesecondistomodelteachere˙ectivenessas accuratelyaspossible.Thisisprimarilyusefulinevaluatingtheconsequencesofthepolicy. Toillustratethisdistinction,supposethattheEVAASscoreweretotallyuninformative.Ob- servingmobilitybasedonthemwouldclearlyillustratetheimpactoftheadditionalsignal, butwouldo˙ernoinsightintothee˙ectoneducationalequity.Incontrast,usingamea- sureoftruee˙ectivenessprovidesdirectpolicyimplicationsandisalsousefulintestingthe learninghypotheses.Accordingly,Ipreferthissecond,broaderapproach,whichistiedmore closelytotheemployerlearningframework,whichreliesontheerrorinvariablesthatproxy forunderlyingability.Thisstudyfollowsearlierstudiesofemployerlearninginsupposing thattheresearchermayaccessinformationoriginallyunavailabletomarketparticipants. 20 20 WhereasFarberandGibbons(1996);AltonjiandPierret(2001);Lange(2007);Schönberg(2007),and Pinkston(2009)useAFQTscoreasastrongcorrelatewithproductivityaboutwhichemployersmustlearn, 27 Inmypreferredspeci˝cation,Imodelteachere˙ectivenessratherthanattemptingtorepli- catetheEVAASmeasure.Anelementoffeasibilityalsoentersthispreference.TheEVAAS systemisproprietary,andtheexactdataandmethodsusedarenotdisclosed.Furthermore, SASusestwodi˙erentproprietarymodels,andforlargeschooldistrictsitisunclearwhich isused.Ofcourse,inactuality,theresultingmeasuresfromeitherapproacharelikelybe highlycorrelated,andinSection1.7,Ichecktherobustnessofmyresultsagainstotherspec- i˝cations. 21 Inthiscontext,theVAMsneednottotallyencompassateacher'se˙ectiveness. Here,VAMsonlyneedtobestrongercorrelateswithteachere˙ectivenessthanareothercor- relateswithproductivity,suchaseducationalattainmentandlevelofcerti˝cation. 22 While VAMslikelydonotmeasurealltraitsthatprincipalsmayseekintheirteachers,theydodi- rectlymeasureonecomponentofteachingqualitythatisimportanttoprincipalsandpolicy makers. MypreferredmeasureofVAMsiswhatGuarinoetal.(2012)calltheDynamicOLS (DOLS)estimatorpresentedinequation1.12.AccordingtoGuarinoetal.(2012),this DOLSestimatorismorerobusttononrandomstudentassignment,afrequentcriticismof theoftenusedEmpiricalBayesestimator,whichassumesrandomassignmentofstudentsto teachers. 23 IusetheVAMdescribedaboveinthiscapacity. 21 Roseetal.(2012)˝ndsa.91correlationbetweenoneEVAASmeasureandDynamicOLS. 22 Theextantliteraturesupportsthisclaim.AsRivkinetal.(2005)show,easilyobservedteacherchar- acteristicsarenothighlycorrelatedwithteachere˙ectiveness.ExperimentalevidencefromHinrichs(2013) suggeststhatGPAmatterslittletoschoolsinhiringdecisions,andthatthestrongestdeterminantofreceiv- ingapositiveresponsefromaschooliswhethertheteacherholdsanin-statecerti˝cate.However,Jacob andLefgren(2008)˝ndlargeagreementbetweenprincipalevaluationsofteachersandVAMs,atleastinthe tailsofthedistributionsofbothmeasures.Furthermore,recentworkshowssigni˝cantcorrelationbetween teachers'VAMsandmanyimportantfutureoutcomesfortheirstudents,includingeducationalattainment, earnings,andprobabilityofincarcerationChettyetal.(2011,2014). 23 Giventeachers'preferencesfoundinJackson(2009)andBoydetal.(2013),itseemsunlikelythatteacher e˙ectswouldbeuncorrelatedwithstudent-levelcovariates. 28 A ijt = t t + A ijt 1 0 + X it 1 + VAM j + e it (1.12) Here, A ijt representsstudenti'smathematicsachievementinteacherj'sclassinyeart. Including A it 1 allowsforthecorrelationofpreviousmathandreadingtestperformances withcurrentperformance.Additionally, X it isavectorincludingdemographicattributesof individualstudents,suchasgrade,race,gender,specialneeds,andgiftedstatus.Itis VAM j , avectorofteacherindicators,whichisofprimaryinterestforthisstudy.Acknowledging thatVAMscanbesomewhatunstableinanysingleyear,mypreferredestimatesusedata fromeachyearateacheristeaching 4 th through 8 th gradeduringmysampleperiod.This allowsmetogainthemostpreciseestimateofteachers'trueunderlyingability, . 1.5.2EstimationSample Thisstudyrestrictsattentiontothe5,986,132elementaryandmiddleschoolstudent,year observationsfrom1997through2011toconstructtheVAMsfor134,219teacherswhoteach 4 th through 8 th grade.Ilinkthesedatatoeducation,licensing,andworkhistorydataof 67,062leadteacherswithoutteachingassistantsforwhomtherecordsarecomplete.These teachersaredispersedacrossthe2,966schoolsin117schooldistricts.Ifurtherrestrictthe sampletoonlythoseteachersteaching 4 th through 8 th gradeatthetimeofobservation,since theyaretheonlyelementaryandmiddle-schoolteacherstoreceiveVAMs.Thisrestriction paresdownmysamplefrom416,135teacher-yearobservationsto236,018.Attheteacher level,thedataincludestheteachers'race,gender,institutionofhighereducation,degrees earned,experience,andtenureatagivenschool.Eachoftheseareeasilyobservabletoall schoolsandmanyarelikelyusedto˝lterjobcandidates.Iuseperformanceattheschool 29 inwhichtheteachercurrentlyworksasanadditional,easilyobservable,possiblecorrelate withe˙ectiveness.TableA.2providessummarystatisticsformyestimationsample. ThedistrictsthatadoptVAMsdonotdi˙ersubstantiallyfromstateaveragesinachieve- mentorpercentofstudentreceivingpro˝ciencyonthestatestandardizedexams.Given thatbothdistrictsincludeurbancenters,theydohaveahigherproportionofBlackstudents andteachersthandoesanaveragedistrictinthestate.Whileteacherscomefromcolleges ofcomparableselectivity,acrossdistricts,inWinston-Salem,alargershareoftheteaching- forceholdsanadvanceddegree.However,onthebasisofVAMs,teachingqualityinboth districtsisveryclosetothestateaverage. 1.5.3EstimationStrategy Theregressionbaseddi˙erences-in-di˙erencesapproachallowsmetoisolatemobilitybased onunderlyinge˙ectivenessfrommobilitybasedoncorrelateswithe˙ectiveness.Further- more,easilyobservable,lowercorrelateswithe˙ectivenessmaybecomelesstiedtothe probabilityofmovingaftertheintroductionofVAMs.Iestimatethefollowingspeci˝cation: y jdt = T t + d d + VAM j G 1dt + X jdt G 2dt + ˘ jdt ; (1.13) G hdt = h 1 + TreatDist jd h2 + Post t h3 + TreatDist jd Post t h4 ;h =1 ; 2 ; where y jdt isthelatentprobabilityofajobchangeforteacherjindistrictdandinyeart. Ionlyobservethebinaryoutcomeofwhenamoveoccurs. T t representsyeare˙ects, d d 30 representsdistrict˝xede˙ects,and X jdt isavectorofteacherandschoolcharacteristics includingteacherexperience,tenure, 24 race,highestdegreeearnedandselectivityofbachelor degreegrantinginstitution,aswellaspercentofstudentswhoareBlackandpercentof studentstestingabovepro˝ciencyattheschoollevel. G 1dt and G 2dt capturethedi˙erences inthee˙ectsofVAMsonmobilitybasedonwhetherVAMswereavailableforteacherjin districtd,attimet.Interactionswithtreatmentdistrictindicatorsseparatepermanent di˙erencesintheimpactsofVAMsandothercharacteristicsfromconfoundingthee˙ectof treatment,whileinteractionswithindicatorsforpostyearsdothesameforstatewidechanges inthee˙ectsatthetimesthepoliciestakee˙ect.Thus,theidentifyingvariationcomesfrom thedi˙erencesbetweenadoptingdistrictsandtherestofthestateinthedi˙erencesinthe predictivepowerofVAMsontheprobabilityofmovingschoolsbetweenpre-andpost-policy years. Keepinginmindpreviouslyestimatedteacherpreferencesandmoreimportantlypotential di˙erencesininformationavailable,Iexaminethesixtypesofjobchangesseparately:within districtmoves,withindistrictmovestohigher-performingschools,withindistrictmovesto lower-performingschools,out-of-districtmoves,out-of-districtmovestohigher-performing schools,andout-of-districtmovestolower-performingschools.Giventhatteachersinitiate mostmoves,movestoworseschoolsarelikelydrivenbylargelybyidiosyncraticteacher preferences.DuetotheindirectmechanismbywhichhiringprincipalsinWinston-Salem obtainteachers'VAMsandthepotentialadditionalsalienceofVAMsignalstoprincipals outsidethedistrictduringWinston-Salem'slateradoption,Iseparatetreatmentbydistrict. GivenhowthedistrictsdistributedVAMs,itseemsclearthatthenewinformationwould 24 Becausetenureisgeneratedandcensoredforjobmatchesbeginningpriorto1995,anindicatorofwhether thecurrentmatchexistedin1995isincludedinallregressions. 31 bepublicbetweentwoprincipalsinGuilford.Perhapstoalesserextantthesameholdsfor Winston-Salem.Accordingly,themodelpredicts 14WD > 0 (where 14WD isthee˙ect oftheinteractionofVAMwithreceivingtreatmentontheprobabilityofmovingwithin- district).Furthermore,becausetherewouldbemoreinformationavailableonmoreexperi- encedteachers,iftherepreviouslybeensomedegreeofpubliclearning,themodelpredicts thee˙ectstodiminishwithteacherexperience.Likewise,iftherehadpreviouslybeenprivate learning,thelearningmodelpredictstheshocktopublicinformationtohavelargerrami˝- cationsforteacherswithmoretenureatagivenschoolallelseequal.Inlaterspeci˝cations, IinteractVAMwithexperienceandthedi˙erence-in-di˙erences, G ,interactions. Whencomparingtheexpectationsofaretainingprincipalwithinoneofthetreatment districtstoahiringprincipalinanotherdistrictthereissomeambiguityastowhether VAMsprovideamorepreciseexpectationforbothprincipalsoronlythecurrentone.Thus, thesymmetriclearningmodelforout-of-districtmovespredicts 14OD = 14WD (where 14OD isthee˙ectoftheinteractionofVAMwithreceivingtreatmentontheprobabilityof movingout-of-district).Ifcurrentprincipalscankeepinformationfromemployersinother districts,thesignalimprovestheprecisionofthecurrentprincipal'ssignalaboutthetrue qualityoftheteacher,whiletheexpectationoftheout-of-districtprincipalisuna˙ected.In whichcase,theasymmetriclearningmodelwouldapplypredicting 14WD > 14OD and possibly 14OD < 0 forout-ofdistrictmoves. Thistypeofmovementmayhaveimportantimplicationsforthedistributionofteacher qualityacrossschools.Ifbetterteachersaremoreabletosignaltheirtruequality,anddo soingeneraltomovetobetterschools,thedivideinteacherqualitybetweentheworst andbestschoolsmaywiden.Accordingly,Iestimateequation1.13substitutingpercentof studentspro˝cientintheschooltaughtatthesubsequentyear,forthebinaryvariableof 32 whetherteachersmove.Again,ifVAMsareinformative,andteachersdoingeneralprefer toteachatbetterschools, 14SQ > 0 inthisregressionaswell.( 14SQ isthee˙ectofthe interactionofVAMwithreceivingtreatmentonthepro˝ciencylevelsoftheschoolwhere theteacherworksthesubsequentyear.)Similartotheprobabilityofmovingtoabetter school,wemayexpectthesee˙ectstobesomewhatmutedforteachersmovinglaterintheir careers,inwhichcasehiringprincipalsmayalreadyhavemorecompleteinformation. Therearetwodistinctissuesthatcomplicatetheestimationofstandarderrorsinthis study.First,thepolicyvariationoccursatthedistrictlevel.Asaresult,theerrorsmaybe correlatedforteachersmovingfromorwithinthesamedistrict.Theappropriateresponse tothissingleissueistoclusterthestandarderrorsatthedistrictlevel.Thesecond,issue resultsfromthefactthattheteacherVAMsareestimated.Bysimplyclusteringthestandard errors,theVAMsaretreatedasthoughtheyareknown,andthus,theydonotaccountfor theinherentvariabilityduetoestimationerror.Werethisasingularissue,itwouldbe appropriatetobootstrapthestudentdatatoaccountforthisestimationerror.Itmayseem naturaltothencluster-bootstrapatthedistrictlevel.However,thissamplesallstudents foraeveryteacherinasampleddistrict,andasaresult,doesnotactuallyaddressthe estimationerror.Infact,thestandarderrorsfromtheclusterbootstraparesmallerthan thenon-bootstrapclusteredstandarderrorsbyaboutafactoroften. Accordingly,Iadoptasamplingapproachthataccountsforboththeestimationerror ofVAMandtheclusterednatureofthedata.First,Isampledistrictsrandomlywithre- placementjustaswiththestandardcluster-bootstrap.Ithenconductstrati˝edsamplingat theteacherlevel,suchthatforeveryteacherwhowasoriginallysampled,Irandomlysam- plestudent/yearobservationswithreplacement.Insodoing,thisprovidesgenerallymore conservativestandarderrorsacrossparameters.Thestandarderrorsonthee˙ectsofthe 33 policyontherelationshipbetweenVAMsandtheprobabilityofmovingschoolsarecom- parabletothestandardbootstrappedstandarderrors,andthestandarderrorsonallother estimatedcoe˚cientsarecomparabletothenon-bootstrappeddistrict-clusteredstandarder- rors.TableF.7intheAppendix ?? presentsallstandarderrorsforTableA.3forcomparison. Throughouttheremainderofthispaper,Ipresentthemoreconservativedistrict-clustered- teacher-strati˝ed-bootstrapstandarderrors(CSBSEs). 1.6Results 1.6.1MobilityandSorting TableA.3presentstheestimatedimpactofrevealingEVAASreportsofteachere˙ectiveness ontherelationshipbetweenteachers'VAMsandtheprobabilityateachermovestoanother school.Giventheevidencethatteachersprefertoteachinschoolswithhigher-performing students,TableA.3decomposese˙ectsbywhetherthereceivingschoolhashigherorlower- performingstudents. 25 Thetestbetweensymmetricandasymmetricemployerlearning focusesonhowthee˙ectsofVAMsontheprobabilityofmovingwithin-districtdi˙erfrom thee˙ectsofVAMsontheprobabilityofmovingout-of-districtafterthetreatmentdistricts adoptthemeasuresofteacherquality.PanelArestrictsattentiontowithin-districtmoves, andPanelBpresentsevidencefromout-of-districtmoves. The˝rstrowpresentsthetherelationshipbetweenVAMsandtheprobabilityofeach typeofmoveintherestofthestate,regardlessofanydistrictsadoptingthepolicy.In 25 Primarye˙ectsofVAMsondi˙erenttypesofmovesaswellasonfutureschoolperformancefurther supportsthisdistinction.Ide˝neamovetoahigherperformingschoolasamoveinwhichtheschooltaught atthefollowingyearhasahigherpercentageofstudentswhoachievepro˝ciencythanthecurrentschool. Pro˝ciencyratesaredemeanedbyyearstatewideaverages,whileamovetoalower-performingschoolis de˝nedinthereverseway. 34 general,thereislittlerelationshipbetweenVAMsandtheprobabilityofmovingwithinor outofthedistrict.However,whendiscerningbetweenmovestomoreandlesspro˝cient schoolsafamiliarpatternemerges.Fromcolumns2and3ofPanelA,ateacherwitha standarddeviationhigherVAMisabout0.3percentagepointsmorelikelytomovetoa higher-performingschooland0.2percentagepointslesslikelytomovetoalower-performing schoolwithinthedistrict.PanelBexhibitsthesamepatternregardingmovestoschools outsideofthecurrentdistrict.AonestandarddeviationincreaseinVAMbeforethepolicy takese˙ectraisestheprobabilityofmovingtoahigher-performingschoolbyaboutatenth ofapercentagepointandlowerstheprobabilityofmovingtolower-performingschoolby aboutthesamemagnitude. WithinbothGuilfordandWinston-Salem,thereleaseofVAMsintensi˝esthispattern. Fromthecoe˚cientontheinteractionsbetweenpolicytreatmentandVAMsinbothdistricts, astandarddeviationincreaseinateacher'sVAMleadstoaboutahalfofapercentagepoint increaseintheprobabilityofmovingwithindistrictafterthedistrictreleasedthevalue- addedinformation.Whilethemagnitudesofthee˙ectsareveryclosebetweendistricts, theyareonlystatisticallysigni˝cantbeyondthe95%con˝dencelevelforGuilford.Column 2illustratesthattheseresultsaredrivenbymovestohigher-performingschools,asthemodel predicts.Fromcolumn2,theestimatedcoe˚cientsimplythattheadoptionofVAMsraises theprobabilitythatateacherwithonestandarddeviationhigherVAMwillmovetoahigher- performingschoolbyover14%(p-value.011)inGuilfordandnearly18%(p-value.009)in Winston-Salem.Column3revealslittlechangeinthee˙ectsofVAMsontheprobabilityof movingtoalower-performingschoolwithindistrict.Thesimilarityofthepointestimates ontheimpactofVAMspost-treatmentbetweenGuilfordandWinston-Salemprovidesno evidencethatrelyinguponteacherstovoluntarilydisclosetheirVAMstohiringprincipals 35 mitigatesthee˙ects. FromSection1.4,thee˙ectofthepolicyshouldbenodi˙erentwhetherteachersmoveto schoolswithinoroutsideofthedistrict,underthesymmetriclearninghypothesis.However, asymmetricemployerlearningpredictsthepolicytogiveprincipalsinGuilfordandWinston- Salemaninformationaladvantageoverprincipalsinotherdistricts.Thistranslatesinto smallerselectione˙ectsforteachersmovingtootherdistrictsthanforwithin-districtmoves, andthesee˙ectsmayevenbenegative.ThesecondcolumnofPanelBpresentschangesinthe e˙ectofteacherqualityontheprobabilityofmovingtoabetter,out-of-districtschoolafter theadoptionofVAMs.Again,thesechangesinselectionofmobileworkersareconsistent withtheemployerlearningmodel. ThechangeinselectionofteachersleavingGuilfordprovidesthestrongestevidenceof growinginformationalasymmetriesbetweenemployers.InGuilford,ateacherwhohasa standarddeviationlowerVAM,isafullpercentagepointmorelikelytomoveout-of-district. Thissameteacherisaboutahalfapercentagepointmorelikelytomovetoabetterschool out-of-district(p-value0.001).Thereisalsoastatisticallysigni˝cante˙ectontheprobability ofmovingtolower-performingschoolsoutofGuilford.Whilethemodeldoesnotpredictthis typeofmovement,itisnotsurprising.LowVAMsmayleadcurrentprincipalstodevalue someoftheirteachers,whomayrespondbymovingtolower-performingschoolsthatare notprivytotheirvalue-addedscores. InWinston-Salem,thedi˙erencebetweenwithin-andout-of-districtmovesislesspro- nounced,thoughstillconsistentwithprivateemployerlearning.WhileinWinston-Salem, ateacherwithonestandarddeviationhigherVAMismorelikelytomovetoahigher- performingschoolout-of-districtafterthepolicytakese˙ect,thepointestimateisonly38% ofthatfrommovingwithin-districtandisnolongerstatisticallysigni˝cant.Wereoutside 36 principalsinformedofthesignal,wewouldexpectthesamepositivee˙ectsfoundinthe secondcolumnofPanelAtobepresentininthesecondcolumnofPanelB. Thefactthate˙ectsaremorenegativeinGuilfordthanWinston-Salem,maybeex- plainedbydi˙erencesinthesalienceofthesignalsbetweenteachersmovingfromGuilford asopposedtothosemovingfromWinston-Salem.Guilford'sadoptionoftheEVAASmea- suresofteachere˙ectivenessoccurredin2000.Itisunlikelythatatthattimeprincipalsin otherdistrictshadmuchunderstandingofthemeasures,ortheirreliability.Incontrast,the restofthestateadoptedschool-levelEVAASreportssimultaneouslywithWinston-Salem's adoptionofteacherlevelVAMs.Giventhisdi˙erenceincontexts,highVAMteachersfrom Winston-SalemmayhavebeenbetterabletousetheirVAMstoobtainpositionsoutsideof Winston-Salem,thanwouldacomparableteachermovingearlierfromGuilford.InWinston- Salem,theincreaseinhigh-VAMteachers'abilitytosignaltheire˙ectivenessmaymitigate anye˙ectsfromrelativelylowVAMteachersexploitingtheinformationalasymmetry.The mitigatede˙ectsofVAMforthosemovingoutofWinston-Saleminadditiontothenega- tiveselectionofteachersmovingawayfromGuilfordevidencesinformationalasymmetries betweenpotentialemployerswithinasopposedtooutsideofthedistrict. Turningtotheimplicationsofsuchmobilityforeducationalequityingeneral,TableA.4 presentstheresultsofhowthesortingofteacherstoschoolschangeswiththeimplementation ofthepolicy.Thecoe˚cientonVAMdescribestherelationshipbetweenteachers'VAMs andthepro˝ciencyleveloftheschooltheyteachatthesubsequentyearintherestofthe state.Acrossbothcolumns,aonestandarddeviationincreaseinateacher'sVAMleadsto aboutaquarterofapercentagepointincreaseinthepercentofstudentswhoarepro˝cient intheschoolinwhichheteachesthesubsequentyear.Theresultthatstudentsinbetter schoolsalsogetbetterteachersisconsistentwith˝ndingsinBoydetal.(2005)andBoyd 37 etal.(2008). Column1examinesthee˙ectofthepolicyonsortingforallteachersinthesamplewho remainteachinginNorthCarolinathefollowingyear.Column2restrictsthesampleto thosewhoremainwithintheircurrentdistrict.Thesecondcolumnmaybemoreinforma- tiveforpredictingthee˙ectsintherestofthestateaftertheadoptionofEVAASVAMs becomesstatewide.Theoretically,thee˙ectsmaybemorepronouncedforthestateasa whole,becausethecostsofmovingoutofstateareingeneralhigherthanthoseofmov- ingoutofaschooldistrict.Thedi˙erenceinresultsfromTableA.3betweenwithin-and out-of-districtmovesimplymorepositivecorrelationsbetweenteacherVAMsandschool performanceamongthosewhoremainindistrictthanoverall,asaresultofthepolicy.Ta- bleA.4re˛ectsthosepatterns.Includingteacherswhomovewithinandoutofdistrict,it seemsfromcolumn1thatreleasingVAMsofteachere˙ectivenessdoeslittletochangethe distributionofteacherqualityacrossschools.However,turningtothesampleofteachers whoremaininthesamedistrictincolumn2,whilethereisnoevidenceofsortingingeneral risinginGuilfordasaresultofthepolicy,inWinston-Salem,onaverageI˝ndateacher withonestandarddeviationhigherVAMwillbeataschoolthathas0.2percentagepoints higherpro˝ciencyratesafterthedistrictreleasesVAMs.InWinston-Salem,thistranslates toabouta70%increaseinthecorrelationbetweenteacherqualityandstudentperformance asaresultofthepolicy.Thislargee˙ectforWinston-Salemtakentogetherwiththemobility patternsfromTableA.3evidencerisinginequalityinthedistributionofe˙ectiveteachersas anunintendedconsequenceofVAMadoption. 38 1.6.2Observables InadditiontopredictingmobilitydynamicswithrespecttoteacherVAMs,themodelpre- sentedinSection1.4alsoo˙erspredictionsregardingeasilyobservablecovariateswithteacher e˙ectiveness.IninstanceswheretheVAMsshocktheavailablepublicinformation,themodel predictsprincipalswouldplacelessemphasisoneasilyobservablecovariateswithteacher e˙ectiveness,suchasdegreeattainmentandcollegeselectivity.IncaseswhereVAMsexac- erbateinformationalasymmetriesbetweencurrentandhiringprincipals,thesamecovariates expectedlyreceiveadditionalemphasisontheprobabilityofamove. Toprovideeaseofinterpretation,Igenerateanindexofeasilyobservableteacherquality bytakingthe˝ttedvaluesfromtheOLSregressionofteacherVAMsonteachercovariates.I includeascomponentsofthisindex,anindicatorforhavinganadvanceddegree,avectorof indicatorsforBarron'sCollegeCompetitivenessindex,yearsofexperience,yearsoftenure, anindicatorforwhethertenureiscensored,race,gender,andavectorofyearindicators. 26 Ingeneral,thosewithhighobservablecharacteristicsaremorelikelytomovewithin district.Thatresultisdrivenbymovestohigher-performingschools,whilethosewithlower observablecharacteristicsaremorelikelytomovetolower-performingschools.Formoves out-ofdistrict,thepositiverelationshipbetweentheindexandtheprobabilityofmovingto abetterschoolo˙setsthenegativerelationshipbetweentheindexandtheprobabilityof movingtoalower-performingschool.Theserelationshipsareexpectedgiventhesortingof teachersbasedonobservablecharacteristicsasshowninJackson(2009)amongothers. The˝rsttwocolumnsofTableA.5donotbearoutthepredictionsforwithindistrict moves.Whilenoisy,thepointestimatesofthee˙ectsoftheteacherindexontheprob- 26 TheVAMsusedinthisanalysisaretheresidualsfromtheprojectionofmystandardVAMsonthe componentsoftheindex. 39 abilityofmovingschoolswithin-districtaftertheadoptionofVAMsarepositive,though onlystatisticallysigni˝cantlysoformovestobetterschoolswithinGuilford.Whilenot expected,thisresultmaybeexplainedbytheadditionalchurnthataccompaniestheadop- tionofVAMsparticularlyformovestobetterschoolswithinGuilford.Morepositionsmay becomeavailableasaresultofhigh-VAMteachersmovingtobetterschools,andlow-VAM teachersmovingoutofdistrict.Asaresult,thosewithgoodobservables˝nditeasierto moveinadditiontothosewithhighVAMs.Heterogeneousopennessamongprincipalsto VAMsmayalsocontribute. 27 Inwhichcase,ashigh-VAMteachersmovetoprincipalsthat valueVAMsthosewithotherfavorableattributesmovetotheprincipalswhovaluethose characteristics. Thechangeintherelationshipbetweentheindexandtheprobabilityofmovingout-of- districtwiththeadoptionsofVAMsismoresupportiveofthemodel.Whereasmoversoutof Guilfordareadverselyselectedonthebasisofthehard-to-observeVAM,theyarepositively selectedonthebasisofthisindexofeasilyobservablemeasuresofteacherquality.Thisis trueacrossmovestohigherorlowerperformingschools,andprovidesfurtherevidencethat themovingteacherswithahighindex,butlowVAMwereabletokeeptheirVAMprivate, whileutilizingtheirotherwisestrongresuméstomovetouninformedprincipals.Giventhat itisplausiblethatmoreteachersmovingfromWinston-Salemcouldinformout-of-district principalsoftheirVAMs,resultsineitherdirectionmaymakesense.Accordingly,theresults formovesoutofWinston-Salemarenotveryinformative.Whiletheresultsformovesoutof Guilfordarereassuring,cumulatively,theevidencefromchangesintherelationshipbetween theindexofeasilyobservableteachercharacteristics,andtheprobabilityofmovingschools 27 InformalconversationswithprincipalsinWinston-SalemandGuilfordindicatethismaybethecase,as twocurrentlowerelementaryprincipalsthatIspokewithindicatedthatteachers'VAMsplayedalimited roleintheirhiringdecisions. 40 istoomixedtodrawde˝nitiveconclusions. 28 1.6.3Di˙erentialE˙ectsWithRespecttoExperienceandTenure The˝nalpieceofprimaryanalysisexaminesthee˙ectsofthepolicyonthecorrelation betweenteacherVAMsandtheprobabilityofmovingwithrespecttoyearsofexperience andtenure.Ifteachersareabletodrawuponeachyearofexperiencetobetterdemonstrate howgoodtheyarethroughresumés,references,oranyotherdevice,thereleaseofVAMs wouldnotserveasmuchofashockforteachersaboutwhomtherealreadyexistsagreat dealofinformation.Themodelpredictsthatifthereissubstantialpubliclearningprior toVAMadoption,thee˙ectsofthepolicyshouldbelessdramaticformoreexperienced teachers.WhileTableA.6exhibitsthisrelationshipforteachersmovingoutofthedistrict, thesameisnottrueforteachersmovingwithindistrict.Takingthepointestimatesliterally, ateacherwith5moreyearsofexperienceandonestandarddeviationhigherVAMistwice aslikelytomovewithinGuilfordtoabetterschoolafterthereleaseofVAM,thanisaless experienced,butotherwisesimilarteacher.InWinston-Salem,theestimatesonthistriple interactionaretoonoisytodrawreliableinference.Whiletheobservedpatternofstronger e˙ectsformoreexperiencedteachersmayseemstrange,thispatternmayoccurifittakes timetorealizethatmovingisworthwhileorifreleasingVAMsallowabuiltupstockofmore 28 Inunreportedregressions,withtheexceptionofout-of-GuilfordmovestheresultsshowninTableA.5 areverysensitivetothevariablecompositionoftheteacherqualityindex.TableF.8inAppendix ?? demonstratesthattheseresultsarealsosensitivetothecovariatesincludedintheindex.Theregressionsin TableF.8includesmeasuresofqualityintheindexofteacherquality,sinceitislikelythatotherprincipals usesending-schoolqualityasanimportantsignaloftheteacher'squality.Inwhichcase,percentofstudents atcurrentschoolwhoareongradelevelandwhoareBlackarereasonabletoincludeintheindex.In TableF.8,thecoe˚cientestimatesoneachoftheinteractionterms,whichareofprimaryinterest,carrythe predictedsign.However,thecoe˚cientestimatesontheindexfortherestofthestatehavetheopposite signaspredicted.Thisinconsistencyislikelyduetocurrentschoolqualitya˙ectingtheprobabilityboth throughteachers'willingnesstomoveaswellasprincipals'willingnesstohirethem.Itremainsnoteworthy thatteachersingoodschoolwithotherhighobservables,areevenlesslikelytomovewithindistrictafter thedistrictadoptsVAMs. 41 experiencedteacherswhocouldnotpreviouslysignaltheirqualitytomove.Fromcolumns 3and4,inbothdistricts,eachadditionalyearofexperiencemitigatesthenegativeselection ofinexperiencedteachersmovingoutofthedistrict.ForGuilfordandWinston-Salem,5 yearsofadditionalexperiencecutsthee˙ectofVAMontheprobabilityofmovingtoa betterschooloutsidethedistrictby15%and20%,respectively.Thesamegeneralpattern holdswithregardtointeractionswithtenure,thoughthestandarderrorsonthecoe˚cient estimatesforinteractionswithtenurearelarger.Wereprivatelearningalreadyprevalent inthemarket,themodelpredictsthee˙ectsofthepolicytobelargerforthosewhohave taughtatthesameschoolforlonger,allelsebeingequal.Thisisconsistentwiththeresults incolumns1and2.Whiletheseresultslargelysuggestpriorprivatelearning,themixed evidenceonpubliclearningmakesmehesitanttodrawde˝nitiveconclusionsontheprior learningenvironment. 1.7Robustness Inthefollowingsection,Iexaminetherobustnessofthee˙ectsofVAMadoption.Sec- tion1.7.1considerschangesine˙ectswhenusingonlyprioryearsofstudentdatawhen constructingVAMs.Section1.7.2considerswhetherotherdistrictpoliciesthatpaidteach- erstoworkinhard-to-sta˙schoolsimpacttheestimatede˙ects.AppendixEincludessevel additionalrobustnessexercisesincludingconsiderationofteachermobilityinaccordance withthestateABCgrowthbonus-paysystem;within-district,year-by-yearanalysisofthe changinge˙ectsofVAMsonmobilityandsorting;andconsiderationofalternatefunctional formsforthemobilityanalysis,suchasnormalMaximumLikelihoodEstimationaswellas competingrisksregressiontoexaminethepossibilityofcorrelatederrorsbetweentypesof 42 moves. 29 1.7.1SensitivitytoVAMConstruction Thepossibilitythatteachersmayhavedi˙erentVAMsaftermovingtootherschools,may presentissuesforusingVAMsconstructedfromstudentdatafromateacher'sentireca- reer.Thiscouldresultfrommovesleadingtohighermatchqualitybetweenteachersand schools,asJackson(2013)˝nds.Itmayalsoresultfromtransitoryadjustmentcosts,giving atheoreticallyambiguousdirectionofpotentialbias. 30 Consequently,inTableA.7,IallowteachersVAMscorestovaryeachyear,usingonly datafromthecurrentandpreviousyearstoconstructateacher'sVAMinanygivenyear. Themaine˙ectshold,thoughtheyareingeneralsomewhatexaggeratedinWinston-Salem andsmallerinGuilford.Still,theadoptionofVAMsraisestheprobabilitythatgoodteach- ersmovetobetterschools.WhereasinWinston-Salem,thee˙ectgrowstoafullpercentage point,inGuilford,ateacherwithanonestandarddeviationhigherVAMbecomes0.3per- centagepointsmorelikelytomovetobetterschoolpost-policy.Fromthemiddlecolumnof PanelB,thenegativeselectionofteachersmovingoutofGuilfordfallstojust30%ofthe estimategiveninTableA.3.PanelCinTableA.7correspondswithTableA.4.Whilethe e˙ectonteachersortingdoublesinWinston-Salem,theresultsbecomemorenegativeand statisticallyinsigni˝cantinGuilford. WhileitispossiblesubsequentmatchqualityincreasesforteachersfromGuilfordand 29 BecausejobmobilityisoftenlocalizedIalsorestrictedanalysistodistrictswhichshareaborderwith GuilfordandWinston-Salem.Theresultsfromthisrestrictionwerenoisyanduninformative,andare unreportedhere. 30 Morecloselyapproximatingtheinformationthatteachersandprincipalsreceiveisanotherrationale forrestrictingthedatausedingeneratingteacherVAMs.InwhichcaseusingEmpiricalBayesestimation provideswhatisbelievedtobeacloserapproximationtothealgorithmusedincreatingtheEVAASmeasures. TableF.10inAppendix ?? providesresultsusingEmpiricalBayesestimationontherestrictedsampleof studenttestscoresincalculatingteacherVAMs.Theresultsareverysimilar. 43 decreasesforteachersinWinston-Salem,Ibelievemeasurementerrormayprovideamore plausibleexplanation.InGuilford,thee˙ectofVAMpriortothetheirreleaseisidenti˝ed o˙ofjusttwoyearsofdata.Asaresult,theestimatesofteachersVAMsarenoisierforthis periodaswellasintheimmediateaftermathofthepolicy.Measurementerrorintheprimary variableofinterestmayattenuatetheestimatesinGuilfordwherethereislittledataprior totheadoptionofthepolicy,whilethee˙ectsinWinston-Salembecomerelativelystronger. Onewayofgettingaroundthisissueistousea˝xednumberofyearspriortothecurrent periodwhenconstructingVAMs.Unfortunately,theadoptionofVAMsbyGuilfordcomes justthreeyearsintothestudentdatasample.SincetheconstructionofVAMsrequires atleastoneprioryearofstudentdata,thisgivesjusttwoyearsatwhichIcould˝xmy VAMestimate.Notonlywouldthisforceanoisierestimateofeachteacher'sVAMforthe entiresample,italsoprovidesmerelyoneyearofdatapriortotheadoptionofthepolicyin Guilford.Todemonstratethechangesoftheestimateswithvaryingthenumberofyearsof datausedinconstructingVAMs,IdropGuilfordfromtheanalysisandvarythenumberof prioryearsofdataIusetoconstructtheVAMsfrom2to8.TableA.8demonstratesthat thoughtherelationshipbetweenyearsusedandthee˙ectoftheinteractionofthepolicyin Winston-SalemandVAMisnotmonotonicasthesampleusedvaries,theestimatesusing moreyearsofdataareclearlythelargest.Thisfurthersuggestscorrelatedmeasurement errorpresentsaproblemforthisapproach. 1.7.2StrategicSta˚ng Apossiblecomplicationarisesduetoalternateteachercompenstionplans.Districtstrategic sta˚ngpolicies,whichaimtoattractmorecapableteacherstoteachinandstayathard- to-sta˙schoolsmaybeproblematicbecausetheyoccuredintreatmentdistrictsduringthe 44 sampleperiodandcouldpotentiallyalterteacherpreferencesoverschools. 31 Charlotte- MecklenburgSchools(CMS)andWinston-Salemwerebyfartheearliestadoptersofthese initiativeswithCMSbeginningitsEquityPlusprogramin1999andWinston-Salemfollowing suitin2000.By2012eachmajordistrictinNorthCarolinaadoptedsomeprogramto attractteacherstohard-to-sta˙schools.InCMS,teachersreceivedasigningbonustoenter atargetedschoolandteacherswithamastersdegreecouldreceiveupto$2,500peryear toremainintheschool.Asmallerincentivewaso˙eredtoteachersenrolledinmasters programs,thoughthedistrictalsoo˙eredtuitionreimbursement.Winston-Salemawarded 20%ofthedistrictsalarysupplement($500-$1,500)toeachteacherintargetedschools. Furthermoretheentirestateo˙ered$1,800bonusestomath,science,andspecialeducation teacherswhotaughtinhighpovertyorlowachievingschoolsduringthethreeyearperiod 2002-2004.In2007,Guilfordadopteditsownstrategicsta˚ngprogram,inwhichbonuses rangedfrom$5,000-$25,500dependingonsubjecttaught,gradelevel,andVAM.Cumberland CountySchoolsgavestipendsto30teacacrosstheir10mostdi˚cultschool.In 2008,CMSbegantailoringtheirplanmoretowardstargetingbetterteachersandWinston- Salem,followedsuitin2012.Theseprogramsmayreversewhichschoolsaremostdesirable toteachers.Withlargeenoughincentives,high-VAMteachersmayopttoworkatlow performingschool,whichisinfacttheintentofthepolicy. TableA.9reportssimilarinformationasisprovidedinTableA.3,withthedi˙erencethat thebinarydependentvariableinTableA.9isequaltooneifamoveoccursandthereceiving schoolisnotclassi˝edasstrategicsta˚ng.Asmightbeexpected,theresultsarequite similartothoseinTableA.3,asteachersworkinginstrategicsta˚ngschoolscomprisejust 31 istheo˚cialtermforlaterpolicieswiththesameobjectives.Earlierpolicieshada varietyofdi˙erentnames;EquityPlus(1and2),FocusSchool,andMissionPossible. 45 4%ofthesample.However,thepolicyhasamuchlargere˙ectonthecorrelationbetween VAMsandtheprobabilityofmovingwithinWinston-Salem.Column2showsthatreleasing VAMsraisestheprobabilitythatateacherwithonestandarddeviationhigherVAMwill movewithinWinston-Salembyafullpercentagepoint,whichisnearlydoublethee˙ect foundwhenexaminingallschoolstogether.Also,thee˙ectofthepolicyonthecorrelation betweenVAMsandtheprobabilityofmovingoutofWinston-Salemdropsby40%,when restrictinganalysistomovestonon-strategicsta˚ngschools.Bothchangesservetowiden thegapintheestimatesbetweenmoveswithinandoutofWinston-Salem,providingfurther evidenceofprivatelearning. TableA.10presentstheimpactsofthepolicyonteachersortingwithin-districtand within-districtamongnon-strategicsta˚ngschools.Column1inTableA.10isidenticalto column2inTableA.4.Iincludeithereforeaseofcomparison.Thethirdcolumnsrestrictthe samplefurthertonon-strategicsta˚ngschools.Movingfromcolumn1to2,inbothdistricts, thepointestimatede˙ectofthepolicyonthedegreetowhichhigh-VAMteacherssort intohighperformingschoolsbecomesmorepositive.ForGuilford,thecoe˚cientbecomes positive,thoughneitherpracticallynorstatisticallysigni˝cantlyso.InWinston-Salem,the pointestimateofthesortinge˙ectsmorethantriple.TableA.10providesnoevidencethat strategicsta˚ngpoliciesaredrivingtheearlierresults.Ifanything,itseemsthatthesepay policiesmayhavemutedwhatwouldotherwisehavebeenmuchlargerimpactsofreleasing VAMs. 46 1.8Conclusion Ifemployersareunabletolearnaccurateinformationabouttheirteachingforceovertime, theirsubsequentpersonneldecisionsregardingteacherswouldbenobetteratidentifying e˙ectiveteachersthanatthepointofhire.Iflearningisentirelyasymmetric,thatisother schoolsarenobetterabletotellthee˙ectivenessofanexperiencedapplicantthanofanovice applicant,e˙ectiveteachersbecometrappedinschoolsinwhichtheydonotwishtoteach, whileprincipalsshu˜etheirlesscapableteacherstootherschoolsinwhatthedocumentary Waiting for SupermantermsLemonGuggenheim(2011).Thereleaseofvalue- addedmeasuresofteachere˙ectivenessdoesseemtoprovideactionableinformationtothose whoareawareofthem.Theevidenceabovesuggeststhatthenewinformationprovides e˙ectiveteacherswithmoremobility,whileLemonbecomesfocusedonthe uninformed. Additionally,theevidencefromsubsequentteachersortingsuggeststhattheincrease inmobilityleadstoincreasedinequityinthedistributionofteacherqualityacrossschools. Despitethefactthat38stateshaveadoptedVAMsofteachere˙ectiveness,andoftencon- tentiously,thissignalingroleofthemeasureshasavoideddiscussion.Thepolicyimplication ofthis˝ndingisnottouniversallyavoidusingVAMs.However,itwouldbeusefultopro- videpolicymakersanestimateofthecostofretaininghigh-VAMteachersinhard-to-sta˙ schools.Theanalysisexcludingstrategicsta˚ngschoolsimpliesthatthesortingmayhave beenlargerwithouttheincentivestoinduceteacherstoworkinlower-performingschools. AsmentionedinSection1.7.2,severaldistrictsinNorthCarolinaareimplementingarange ofsta˚ngpoliciesdesignedtoinduceteacherstoworkinlow-performingschools.Some incorporateVAMsintotheincentiveschemes. 47 Clotfelteretal.(2011)andGlazermanetal.(2012)haveexaminedthequestionofat- tractingteacherstoundersta˙edschools.Furtherworkisneededtoestimatethecostsand e˙ectivenessofthesepoliciesinretaininge˙ectiveteachersinlow-performingschools,which maycostsubstantiallyless.AsstatesanddistrictscontinuetoadoptteacherVAMs,policy makersshouldbeawareofthepotentialconsequencesofthesepoliciesoneducationalequity, aswellasthecostsofo˙settingthesee˙ects. 48 Chapter2 JobSeparationUnderAsymmetric EmployerLearning 2.1Introduction Employersgamblewitheachnewhire,perhapsparticularlysoasapplicantsbegintheir careers.Thereissigni˝cantuncertaintyabouthowreliable,hardworking,andcapablenew hireswillbe.However,afterworkershavebeenwithaparticular˝rmforsometime,it makessensethatemployersaccumulateinformationaboutthesetraitsandhowtousethem. Atthesametime,someworkersmaydrawontheiracquiredexperiencetobolstertheir resumés,whilecontinuallylookingforotheropportunities.Despitetheintuitiveappealof thissetting,thequestionofwhetherall˝rmsinthemarketlearnabouttheworkersatthe samerate(symmetriclearning)orwhethercurrentemployersenjoythebene˝tsofprivate informationabouttheirworkers(asymmetriclearning)isempiricallylargelyunsettled. Thereareseveralimplicationsstemmingfromtheprocessbywhichcurrentandprospec- tiveemployerslearn.First,informationgapsbetween˝rmsmaypermitpersistentwagegaps basedoneasilyobservablecharacteristics,suchasraceandeducation,evenwhenworkersare equallyproductiveSchönberg(2007);Pinkston(2009).Secondly,thepreservationofinfor- mationaladvantagemaydistortpromotiondecisionsDeVaroandWaldman(2012).Lastly, 49 thedisparatetypesoflearninghavedi˙erentpredictionsforworkermobility.Thisstudy focusesontheseimplications. Undersymmetriclearning,workers'referencegroupsshouldhavenobearingontheprob- abilitytheyexperiencejobseparations,sinceall˝rmsshouldplacethesameimportanceon theworkers'characteristicsthattheycaneasilyobserve.Likewise,gapsininformation cannotexplainworkermobilityonthebasisofhardtoobservecharacteristics,sinceeach employerisequallywellinformedunderthisframework. Incontrast,underasymmetriclearning,informationaboutaworker'strueproductivity ismorevaluablethefurtherthattrueabilityisfromtheaverageworkerwiththesame easilyobservablecharacteristics.Thisheterogeneityinthevalueofinformationabouta givenworker'sproductivityleadsdi˙erentworkerstohavedi˙erentprobabilitiesofleaving. Workerswhohavehigherabilitythantheaverageoftheirrespectivereferencegroupswillbe lesslikelytobebidawaybyanother˝rmorlaido˙bytheretaining˝rm.Inversely,those withhighabilityreferencegroupsaremoreattractivetooutside˝rms.Byexamininghow abilityandreferencegroupsin˛uencetheprobabilityofjobseparations,thisstudyexplores evidenceofthepresenceofasymmetricemployerlearning. Iextendexistingmodelsofasymmetricemployerlearningtodeveloppredictionsregarding theselectionofworkersintojobswitchesandlayo˙s.Thepredictedselectiondi˙ersdepend- ingonwhetherworkercovariatesareeasyordi˚culttoobserve.Ithendeveloppredictions forhowthisselectionwillchangeoverexperienceasopposedtocontinuousworkingspells.I testthesepredictionsusingtheArmedForcesQuali˝cationTest(AFQT)fromtheNational LongitudinalSurveyofYouthof1979(NLSY79)asahardtoobserveworkercharacteristic andraceandeducationasimportantcharacteristicsthatareeasytoobserve.Finally,Itest therobustnessofresultsunderalternatespeci˝cationsanduseacontrolfunctionapproach 50 tohandlepossibleendogeneityinexperienceandcontinuousworkingspells. I˝ndthatingeneral,consistentwithasymmetricemployerlearning,workersareneg- ativelyselectedintomobilityonthebasisoftheirAFQTscore,whiletheyarepositively selectedonthebasisoftheirreferencegroups.ThisselectiononAFQTisdrivencompletely byjob-to-unemploymenttransitions,whereasasymmetriclearningpredictssuchnegativese- lectionthroughbothjob-to-jobandjob-to-unemploymentmoves.Thoughtheinteractionsof AFQTwithexperienceandworkingspelldurationcarrythesignspredictedbyasymmetric learning,theyarestatisticallyinsigni˝cant.Theresultsregardingeducationaremoreconsis- tentwithasymmetricemployerlearning.Workerswithhigherandmoreselectiveeducation aremorelikelytobothtransitionbetweenjobs,aswellasformovesfromemploymentto unemploymentduringeconomicrecessions.Further,theselectiononthebasisofeducation becomesmorepositivewithrespectexperienceandmorenegativewithrespecttoworking spellduration,asasymmetriclearningpredicts. Therestofthestudyproceedsasfollows.Section2situatesthisworkintheexisting literature.Section3laysoutmyextensionofPinkston's[2009]modelofasymmetricemployer learning.Section4describestheestimationstrategiesused.Section5describesthedata, providesde˝nitionsofvariables,anddescriptivestatistics.Section6providesempirical results,andSection7summariestheevidence. 2.2EmployerLearningModelsandEvidence Thisworkfollowsalonglineofpredecessors,suchasFarberandGibbons(1996);Altonji andPierret(2001);Lange(2007),Schönberg[2007],andPinkston(2009),whichexamines howemployerslearnabouttheiremployeesovertime.StartingwithFarberandGibbons 51 [1996],thesestudiespresumethatworkerabilityisheterogeneous.Employerscannotdirectly observetheabilityofpotentialworkersandmustrelyoncorrelatestoinferworkers'expected valuetothe˝rm.Further,theytreatasubsetofcharacteristicsaseasilyobservabletoall, anotheraseasilyobservabletothemarket(andnottoresearchers),andyetanothersubset ofpotentialcorrelateswithproductivityaseasilyobservabletotheeconometricians(butnot themarket).Thisliteraturetypicallyusesthepercentilefromacognitiveabilityassessment, theAFQT,asthisrelativelystrongcorrelatewithproductivitythatisveiledtothethe marketatthetimeofhire,butisavailabletoresearchersthroughouttheworkers'careers. Thisstudydoessoaswell. Muchofthisearlierworkassumesthatall˝rmswithinthemarketlearnatthesamerate. Inwhichcase,allprospectiveemployerslearnmoreaboutworkers'abilitiesovertimeand wagesbecomemorestronglylinkedtopreviouslyunobserved,strongcorrelateswithproduc- tivityandlesstiedtotheeasilyobservedcharacteristics.FarberandGibbons[1996],Altonji andPierret[2001],andLange[2007]each˝ndthispatternregardingwages,education,and AFQTscores. However,itseemsreasonablethatemployersmaylearnmoreabouttheiremployeesthan dooutside˝rms.Indeed,bothGibbonsandKatz(1991b)andDeVaroandWaldman(2012) provideevidenceofsuchinformationalasymmetriesbetween˝rmsthoughneitheriscouched inaleaningframework.Schönberg(2007);Pinkston(2009),andKahn(2013)developmodels ofemployerlearningthatallowemployerstolearnatdi˙erentrates.Theytestthesemodels forthepresenceofasymmetriclearninginthelabormarketusingtheNLSY79,and˝nd con˛ictingevidence. Schönbergdevelopsaninitialmodelforasymmetricemployerlearning,anddevelopstests foritshypotheses.Hermodelassumesthattheretaining˝rmperfectlyobservestheproduc- 52 tivityofitsworkersinthesecondperiodunderasymmetriclearning,butthisinformation islostwheneverajobmatchterminates.Perfectlearninginthesecondperiodimpliesthat therearenoinformationalbene˝tsfromadditionaltenure.Thisisatestableassertion,and Lange(2007)presentsevidenceofcontinuouslearning,˝ndingthatwhileabouthalfthe learningoccursinthe˝rstthreeyears,20yearslaterthevarianceoftheerrorcontinuesto decline.Further,awinner'scursebefallsoutside˝rmsastheydrawtheleastproductive workersfromtheirretaining˝rms.Allowingdi˙erencesinmatchqualitytomotivatemore productiveworkerstoleaveattenuatestheseverityofthewinner'scurse,andtheoretically leadstovolatilewagesearlyinajobmatch. Inaccordancewithhermodel,Schönberg˝ndslittleevidenceofasymmetriclearning.Her empiricalworkfocusesonlyonwhitemaleworkers,abstractingawayfromtheimplications ofemployerlearningforstatisticaldiscrimination.She˝ndsthatonlywhitecollegeeducated jobleaversarenegativelyselectedonthebasisofAFQTscore,andthatadverseselectionis drivenbyjob-to-unemploymenttransitions,whichshecontendsweakensthecaseforasym- metricemployerlearning.Furthermore,Schönberg˝ndsthattheimpactofschoolingon wagesdecreaseswithexperienceandremainsrelativelyconstantwithtenure.Consequently, sheconcludesthatlearninginthemarketislargelysymmetric. Kahn(2013)extendsSchönberg'sframeworktotestwhetherjobmoversexperiencemore volatilewagepatternsafteratransitionthandothosewhoremaininplace.UsingtheNLSY, sheconsidersdi˙erencesbetweenworkerswhoenterapositionduringrecessionsasopposed toeconomicexpansions,withtheideathatthereislessvariationintheabilityofentrants duringrecessions.Shealsousesvariationintheamountofexposureanoccupationhas outsidethe˝rm,assumingthatlearningismoresymmetricinmoreexposedoccupations. Kahn˝ndsthatmovers'wagesaremorevolatileintheimmediateaftermathofatransition 53 thanarethewagesofthosewhoremaininplace.Also,thee˙ectsarelargerforthosewho enterajobduringaneconomicexpansionandforthoseinmoreinsularoccupations.These features,sheargues,aresupportiveoftheasymmetriclearninghypothesis.Perhapsmore applicabletothiswork,Kahn˝ndsevidenceofadverseselectiononthebasisofAFQTand yearsofeducationofmoversingeneraland˝ndsthesee˙ectsarestrongerforoccupations thatcommunicatelesswiththeoutsideworld.Again,shearguesthisevidencesupportsthe asymmetriclearningmodel. ThealternatemodelPinkston[2009]developsallowstheprecisionofthesignalthatcur- rentemployersreceivetoincreasewithtime,ratherthanimposingadiscreetjump.Healso allowsracetoin˛uencetheimpactofabilityonworkers'wageprogressions.Furthermore, Pinkstonallowsinformationaboutworkers'productivitytobetransmittedfromincumbent employerstonewemployersduringjob-to-jobtransitions.Consequently,heexamineswage dynamicsoverspellsofcontinuousworkingasopposedtotenure,whichSchönberguses. Additionally,Pinkstonnotes,ashasearlierresearch,thatlengthofworkingspellandexpe- riencemaycontaininformationaboutthequalityofthejobmatchandtheproductivityof theworker.Toaddressthisissue,heusestheresidualsfromtheregressionofworkingspell lengthonworkers'career-averagespelllength,totaldurationofthecurrentjob,andindica- torsformissingvaluesofduration,asinstrumentalvariablesforworkingspell,andusesthe residualsfromtheregressionofexperienceonpotentialexperiencetoinstrumentforcurrent experience.Pinkston˝ndsthatAFQTbecomesalargerfactorinwagedeterminationas workingspellincreaseswhiletheimpactofraceandeducationdiminishwithworkingspell. Bothfeaturesaresuggestiveofasymmetricinformation. SincePinkston[2009]considerstheimplicationofasymmetriclearningonwagedevelop- ment,thisstudyprovidesanaturalextensionofhismodeltoexamineevidenceregarding 54 jobseparation.WhereasPinkstonimposesanexogenousrateofjobdestructioninhismodel forasymmetriclearning,thisworkmodelsandteststheimplicationsofasymmetriclearning onthemarginofjobswitchesandlayo˙s. ThoughSchönberg,[2007]alsoaddressesjobseparationinherstudyofasymmetricem- ployerlearning,thisworkpushespastheranalysisinafewkeyareas.Firstly,themodel developedhererespectsthegradualnatureofthelearningprocess,andallowsinformation toaccumulatethroughoutjob-to-jobtransitions.This˛exibilityallowsmetoformulate theoreticalpredictionsandempiricaltestsofthedynamicsofworkerselectiononthebasis ofeasyanddi˚cult-to-observecharacteristicswithrespecttoexperienceandworkingspell length.Secondly,whilewebothexaminetheimplicationsofhardtoobservecharacteristics onmobility,Ialsodevelopandtestpredictionsregardingtheimpactoftheworkers'reference groups,conditionalonindividualAFQTscores,ontheprobabilityofjobseparation.Fur- thermore,FarberandGibbons(1996);AltonjiandPierret(2001);Lange(2007),Schönberg, [2007]andPinkston(2009)eachuseyearsofeducationtoproxyforeasilyobservableinfor- mation.However,thereiswidevariationqualityofthoseyearsofeducation.Whichcollege anindividualattendsmayprovidemoreinformationaboutthatperson'scognitiveability thanthequantityofeducationobtained.Consequently,Iincorporatecollegeselectivityinto myanalysistoconstructmorerobustreferencegroups.Lastly,Iprovideatheoreticalra- tionaleforasymmetricinformationtoimpactselectioninlayo˙s,andexaminewhetherthe impactofAFQTscoreandreferencegroupmembershiponjob-to-unemploymenttransitions di˙ersbetweendi˙erenteconomicconditions. TableB.1illustratesthisbasicstory.Ingeneral,thoseinpanelBwithlowerAFQT scoresthantheaveragepersonwithsameeducationalattainmentatthesamequalityof institution,experienceahigherrateofjobseparationthanthoseinpanelAwhoseAFQT 55 scoresarehigherthantheirreferencegroups'averages.Thisistrueacrossreferencegroups andseparationtypes,withtheexceptionofthelesseducatedinjob-to-jobtransitions. Restrictingattentiontoeducationalattainmentdoesn'ttellaconsistentlearningstory. Collegegraduatesgenerallyenjoylowermobilityratesthancollegeattendees.Also,those withhighschooldiplomasandrelativelylowAFQTscoresaremorelikelytoenterunemploy- ment,thanarecollegeattendeeswithlowerAFQTscoresthantheaveragecollegeattendee. However,therearesigni˝cantdi˙erencesinaverageAFQTacrosslevelsofeducationthat thistabledoesnottakeintoaccount. Despitedi˙erencesinaverageAFQTbetweenthosewhoattendcompetitivecollegeas opposedtononcompetitiveschools,di˙erencesinmobilitybetweenthesetwogroupsiscom- pletelyconsistentwithasymmetricemployerlearning.Acrossallcategories,thosewhoattend competitivecollegesarebothmorelikelytoswitch˝rmsormovefromemploymenttoun- employmentthanthosewhoattendlesscompetitiveinstitutions.Thissurprisingresult˝ts perfectlyinlinewithasymmetricemployerlearning.AsSection3showsinmoredetail, underasymmetricemployerlearning,outside˝rmsplacemoreemphasisonthispublicsig- nalofworkerabilitythandoretaining˝rms.Thisraisestheoutsideo˙ersforworkerswho attendandgraduatefromcompetitivecolleges,thusraisingtheprobabilitythattheywill bebidawaytoanoutside˝rm.Further,higheroutsideo˙ersdriveupthewagesofworkers withrelativelyhighpublicsignalsmakingtheirwagesclosertotheirexpectedproductivities. Thus,therelittleroomtobu˙eragainsteconomicdownturns,andintheeventofrecessions, employersletgooftheseworkers. 56 2.3Model Thissectiondevelopsamodeltodemonstratetheimplicationsofemployerlearningonthe selectionofworkersintomobility.ItbuildslargelyuponthemodelpresentedinPinkston [2009],whichinturnbuildsuponmodelspresentedinFarberandGibbons(1996)andAltonji andPierret(2001).PleaseseetheAppendixforproofsofpredictions. 2.3.1Framework Employerscareabouttheproductivityoftheirworkers,whichiscomposedoftheworker's underlyingability( )andthequalityofthematchbetweenagiven˝rmandworker( ' f ). Eachperiod,employerslearnworkers'˝xedcharacteristics( m ),apublicsignal( R x where x indexesexperience),aprivatesignal( P f where f maystandforeithertheretaining˝rm(r) orthehiring˝rm(h)),andtheexpectedmatchqualityforthatperiod( E ( ' f ) ). Beforereceivingaprivatesignal,all˝rmsshareapriorbeliefthataworker'sexpected abilityequalstheaverageability ( m ) ofotherworkerswiththesameeasilyobservablechar- acteristics.Thepublicsignalisanalogoustoaresumé,whiletheprivatesignalisinformed initiallybyaninterviewandlaterbythedailyactivitiesoftheemployeeduringworkhours. Thus,theprivatesignaltotheoutside˝rm( P h )isnotsubsetoftheinformationtheretain- ing˝rmobserves( P r ),whichmeansthathiring˝rmscanpro˝tablycompeteforworkers againstbetterinformedretaining˝rms.Witheachsignal,employersorperspectiveem- ployersupdatetheirpriorbeliefsaccordingtothevalueandprecisionofthesignalsthey receive. 57 Toallowthenestingofsymmetriclearning,workersand˝rmsmustlearnabouteach possiblematchqualitycomponentequally,whetherornottheworkeriscurrentlyworking foragivenemployer.Whendiscussingbidding,Iwilluse E ( ' f ) ; because ' f maychange overtimewithinamatchaccordingtotheeconomicclimate.Firmsandworkershavean expectationover ' f ,whichisspeci˝ctotheparticularmatch,butafterthebiddingissettled andwagesset,thesurroundingeconomicconditionsarerealized,and ' f adjusts.Thistiming isimportantwhenmodelinglayo˙sunderdownwardwagerigidity. Whenworkers˝rstenterthemarket,theyreceivetwoo˙ersandgotothehighestbidder. Eachsubsequentperiod,theycontinuetoreceivetwoo˙ers,onefromtheirretaining˝rm andonefromanoutside˝rm.The˝rmsthenbidontheworkerasinastandardEnglish auction.The˝rmwiththehighestbidgetstheworkerandpayshimthehighestbidofthe rival.Thefollowingassumptionsprovidemorestructuretothedescribedmodel: 1.Unobservedabilityoftheworker, = m + ,where ˘ N (0 ;˙ ) . 2.Theprivatesignal, P r = + ˝ where ˝ ˘ N (0 ;˙ ˝ ( t )) ; and @˙ ˝ ( t ) @t < 0 . t indexestenure throughouttheperiodofcontinuousemployment.Forhiringemployers t =0 ; andfor currentemployers t> 0 . ˙ ˝ ( t ) <˙ ˝ (0) forall t> 0 . 3.Thepublicsignal, R x = + ˘; where ˘ ˘ N (0 ;˙ ˘ ( x )) ,and @˙ ˘ ( x ) @x < 0 .Here x indexes experience. 4.Unobservedproductivity, ˆ = + '; where ' ˘ N (0 ;˙ ' ) isthematchqualitybetween theworkerandthe˝rm.Firms'andworkers'expectationofmatchqualityduringthe biddingprocess E ( ' ) ˘ N ( ';˙ E' ) . 58 5.Errorsareorthogonaltooneanother. Theoptimalbidsareprecisionweightedaveragesofthesignalsemployersreceiveandthe expectedabilityofworkerswiththesameeducationandofthesamerace.Inthismodel,the di˙erencebetweenthecurrentandoutsideemployeristhatthecurrentemployerreceives amoreprecisesignaloftheirworkers'productivitythandooutside˝rms ( ˙ ˝ ( t ) <˙ ˝ (0) forall t> 0) .Assumingcontinuousnessinthebiddingprocess,theopennessofanEnglish Auctionallowseach˝rmtolearnthattheother˝rmvaluestheworkeratleastasmuch asitdoesduringtheauction.Thus,theprivatesignalreceivesdoubleweight.Letting W = ˙ ˘ ( x ) ˙ ˝ (0)+ ˙ ˝ (0) ˙ +2 ˙ ˘ ( x ) ˙ and W 0 = ˙ ˘ ( x ) ˙ ˝ ( t )+ ˙ ˝ ( t ) ˙ +2 ˙ ˘ ( x ) ˙ ,from MilgromandWeber(1982),theretaining˝rm'soptimalbid, ( b r ) ,andtheoutside˝rm's optimalbid, ( b h ) ,aregivenbelow: b r = E [ j R x ;P r ;P h = P r ]= ˙ ˘ ( x ) ˙ ˝ ( t ) W 0 m + ˙ ˝ ( t ) ˙ W 0 R x + 2 ˙ ˘ ( x ) ˙ W 0 P r + E ( ' r ) (2.1) b h = E [ j R x ;P r = P h ;P h ]= ˙ ˘ ( x ) ˙ ˝ (0) W m + ˙ ˝ (0) ˙ W R x + 2 ˙ ˘ ( x ) ˙ W P h + E ( ' h ) (2.2) Noticethatiftheretaining˝rm'ssignalismoreprecisethanthatofthehiring˝rm(if ˙ ˝ ( t ) < ˙ ˝ (0) ),theretaining˝rmplacesrelativelylessweightonthereferencegroupand publicinformationandrelativelymoreweightontheprivatesignal.Further,asthepublic signalsbecomemoreprecise,thehiring˝rmplaceslessemphasisontheirprivatesignalin 59 favorofthepublicinformation.Thus,ifboth˝rmsreceivethesameprivatesignal,the di˙erencesinweightingwilllikelyleadtodi˙erentoptimalbids. Itisimportanttonotethatbecausethevaluationofthelosing˝rmisrevealedduring theopenauction,intheeventthattheoutside˝rmwinstheauction,italsocapturesthe retaining˝rm'sprivatevaluationoftheworker.Consequently,informationaccumulates throughoutjob-to-jobtransitionsratherthanresettingwitheachnewemploymentspell.In thisway,theinformationactslessasspeci˝chumancapital,asinBecker(1962),andismore analogoustogeneralhumancapital.However,iftheworkerisforcedtoendureaperiodof unemploymentbetweenspells,themarketlosestheaccumulatedprivateinformation. Undersymmetriclearning,theoptimalbidstakeaverysimilarform,thoughinthis specialcaseallsignalsarepublic.Theoptimalbidoftheretaining˝rmistheweighted averageofthepriorbeliefandthepublicsignaloftheworker'sability,plustheexpected qualityofthematch.Again,employersweighthesignalandpriorbeliefinaccordanceto therelativeprecisionofeach.Thus,theoptimalbidsoftheretaining˝rm( b r )andoutside ˝rm( b h )arerespectivelyshowninequations2.3and2.4: b r = E [ j R x ]+ ' r = ˙ ˘ ( x ) ˙ ˘ ( x )+ ˙ m + ˙ ˙ ˘ ( x )+ ˙ R x + E ( ' r ) (2.3) b h = E [ j R x ]+ ' h = ˙ ˘ ( x ) ˙ ˘ ( x )+ ˙ m + ˙ ˙ ˘ ( x )+ ˙ R x + E ( ' h ) (2.4) Noticethatifemployerslearnaboutworkers'trueabilityovertime,thevarianceofthe publicsignal( ˙ ˘ ( x ) )decreases.Thus,employersplacelessweightontheirpriorbeliefand moreweightonthepublicsignal. 60 2.3.2JobSwitches Assumingeach˝rmplaysitsoptimalstrategy,theprobabilitythataworkerswitches ˝rms( P ( J ) )isequaltotheprobabilitythattheoutside˝rmhasahigheroptimalbid thantheretaining˝rm. 1 Thedi˙erencesbetween˝rmsintheprecisionandweightingof privateinformationprovidesclearpredictionsofselectiononthebasisofbothhardand easy-to-observeworkercharacteristics.Aftersomealgebra,andallowing J tostandforthe compositeerrorterm(includingthedi˙erenceinmatchquality),thedi˙erencebetweenthe hiringandretaining˝rms'optimalbidscanbewrittenas: 23 P ( J )= P [ b h b r > 0]= P J <˙ ˘ ( x )[ ˙ ˝ (0) ˙ ˝ ( t )]( m ) (2.5) Becauseacurrentemployerhasaclearerviewofaworker'sunderlyingability,theretaining ˝rmplacesmoreemphasisonitthandoother˝rms.Therefore,evenifthecurrentand prospectiveemployersreceiveequivalentrelativelyhighprivatesignals,theweightingwill leadthecurrentemployertohaveahigheroptimalbid.Incorporatingthenormalityand orthogonalityassumptionsaboveandallowing ˙ j tostandforthevarianceof J ,the derivativeofequation2.5withrespecttoability( )providesthefollowing: 4 1 Jreferencesjob-to-jobtransitions. 2 J E ( ' r ) E ( ' h ) 2 WW 0 (( ˙ ˙ ˝ ( t )+ ˙ ˘ ( x ) ˙ ˝ ( t )+2 ˙ ˙ ˘ ( x )) ˙ ˙ ˘ ( x ) ˝ h ( ˙ ˙ ˝ (0)+ ˙ ˘ ( x ) ˙ ˝ (0)+ 2 ˙ ˙ ˘ ( x )) ˙ ˙ ˘ ( x ) ˝ r + ˙ 2 ˙ ˘ ( x )( ˙ ˝ (0) ˙ ˝ ( t )) ˘ ) : 3 PleaseseeAppendixHforalgebra. 4 ˙ J = var ( J )=2 ˙ E' + 4 W 2 W 0 2 ( W 0 2 ˙ 2 ˙ ˘ ( x ) 2 ˙ ˝ (0)+ W 2 ˙ 2 ˙ ˘ ( x ) 2 ˙ ˝ ( t )+ ˙ 4 ˙ ˘ ( x ) 2 ( ˙ ˝ (0) ˙ ˝ ( t )) 2 ˙ ˘ ( x )) ; and ˙ E' = var ( E ( ' r ))= var ( E ( ' h )) . 61 @P ( J ) @ = ˚ 8 > < > : 2 ˙ ˘ ( x ) ˙ WW 0 q ˙ j ˙ ˘ ( x )[ ˙ ˝ (0) ˙ ˝ ( t )]( m ) 9 > = > ; 2 ˙ ˘ ( x ) ˙ WW 0 q ˙ j ˙ ˘ ( x )[ ˙ ˝ (0) ˙ ˝ ( t )] < 0 (2.6) ˚ f : g ,beingthenormalprobabilitydensityfunction,ispositive,asiseachvariance.Thus, aslongastheprecisionoftheprivatesignalshrinksthelongeraworkeriswiththeretaining ˝rm( ˙ ˝ (0) >˙ ˝ ( t ) ),whichisfundamentaltoasymmetricemployerlearning,equation2.6 showsthatasability( )increasestheprobabilityofamovedecreases,allelseequal. This˛exiblelearningmodelallowsforfurtherpredictionsabouttheevolutionofthis selectionovertime,speci˝callywithregardtoincreasesinlengthofcontinuousworkingspells asopposedtoexperienceinthemarket.Intuitively,itmakessensethatselectiononthebasis ofabilitywouldmostpronouncedwhentherearethegreatestasymmetriesininformation betweenemployers.Thisoccurswhenaworkerhasbeencontinuouslyworkingforalong periodoftime,andinformationhasaccumulatedwithoneemployerand/ortransferredto anotherthroughthebiddingprocessassociatedwithajob-to-jobmove.Moreformally,the cross-partialof 2 ˙ ˘ ( x ) ˙ WW 0 q ˙ j ˙ ˘ ( x )[ ˙ ˝ (0) ˙ ˝ ( t )]( m ) withrespecttoworkingspelllength ( t )andability( )isnegative. 5 Inversely,thesee˙ectsaresmallerwhentherearesmall asymmetries,suchaswhenaworkerhassu˚cientexperienceinthemarketforhisabilityto beapparenttoallprospectiveemployers.Thus,thecross-partialof 2 ˙ ˘ ( x ) ˙ WW 0 q ˙ j ˙ ˘ ( x )[ ˙ ˝ (0) ˙ ˝ ( t )]( m ) withrespecttoexperience( x )andability( )ispositive. 6 WhileSchönberg(2007)notesthepredictedadverseselectiononthebasisofunderlying abilityforjobswitches,thisworkintroducesanexaminationofselectionintomobilityon 5 PleaseseeAppendixAforproof.Notethatthisisnotthecrosspartialoftheprobabilityofajob-to-job move,butisratherthescaledregressioncoe˚cientontheinteractionbetweenabilityandworkingspell length. 6 PleaseseeAppendixAforproof. 62 thebasisofreferencegrouporeasytoobserveworkercharacteristics.Thederivativeof equation2.5withrespecttoaverageabilityofthereferencegroupisunsurprisinglynearly identicaltoequation2.6,itsimplyhastheoppositesign. @P ( J ) @m = ˚ 8 > < > : 2 ˙ ˘ ( x ) ˙ WW 0 q ˙ j ˙ ˘ ( x )[ ˙ ˝ (0) ˙ ˝ ( t )]( m ) 9 > = > ; 2 ˙ ˘ ( x ) ˙ WW 0 q ˙ j ˙ ˘ ( x )[ ˙ ˝ (0) ˙ ˝ ( t )] > 0 (2.7) Fromequation2.7,conditionalonindividualability,astheaverageofabilityamongwork- ersinthesamereferencegroup( m )increases,theprobabilityofajob-to-jobmoveshould increase.Thisisduetoprospectiveemployersapplyingmoreweighttopublicinformation thandoesthecurrentemployer.Iwillnotduplicatetheabovecloseexaminationofthedy- namicsofthisselectiononthebasisofreferencegroupwithrespecttoworkingspellduration andexperience,duetotheclosenessofequations2.6and2.7.Theinverseofthedynamics withregardtoexperienceandworkingspelldurationisalsotrue.Asworkingspelllength increasesworkerswithhighreferencegroupsbecomeevenmorelikelytoswitchemployers. 7 Conversely,withincreasesinexperience,thosewithhighreferencegroupsbecomelesslikely totransfer˝rms. 8 Undersymmetriclearning,theprobabilityofajobswitch( P ( J ) )isagaintheprobability theoutside˝rmhasahigheroptimalbidthantheretaining˝rm.Theprimarydi˙erence hereisthattherearenoprivatesignals.Thusthedi˙erenceinbidsshowninequation2.8 simpli˝estoequation2.9. P ( ˙ ˘ ( x ) ˙ ˘ ( x )+ ˙ m + ˙ ˙ ˘ ( x )+ ˙ R x + E ( ' h ) " ˙ ˘ ( x ) ˙ ˘ ( x )+ ˙ m + ˙ ˙ ˘ ( x )+ ˙ R x + E ( ' r ) # > 0 ) (2.8) 7 SeeAppendixA 8 SeeAppendixAforproof. 63 P ( J )= P [ E ( ' h ' r ) > 0] (2.9) Noticethatbothindividualabilityandtheaverageabilityofthereferencegroupareelimi- natedfromtheequation,sincethemarketuniformlyweightstheeasyanddi˚cult-to-observe information. Inorderforselectionofjobswitcherstopersistundersymmetriclearning,changein matchqualitymustdi˙erwithability. 9 Giventhe˝ndinginKahn(2013)amongothers,that jobswitchingleadstolargewagegainsforyoungworkers,jobswitchingmaybeadesirable outcome,atwhichhighabilityworkersmaybemoreadept.Inwhichcase,positiveselection onthebasisofabilitymaybeexpectedinundersymmetricemployerlearning.Thissame positivecorrelationbetweenworkerabilityanddi˙erenceinmatchqualitywouldproduce ambiguityinthepredictionsregardingabilityandtheprobabilityofjob-to-jobtransitions underasymmetricemployerlearning. 2.3.3Layo˙s Asymmetricemployerinformationmayalsoprovidemeaningfulpredictionsregarding theprobabilityoflayo˙s,asinGibbonsandKatz(1991b).WhereasGibbonsandKatz (1991b)examinewagepenaltiesoflayo˙sasopposedtoplantclosings,thisstudyprovides furtherrationalefortheir˝ndingsbyexaminingtheeasyanddi˚cult-to-observeworker characteristicsofthosewhoarelaido˙. Avoluntarymovefromemploymenttounemploymentisverydi˙erentfromthesame 9 Thiswouldbeaviolationofassumption5above. 64 movewereitunilaterallydecidedbytheemployer.Therearenoimmediatepredictions fromeitheremployerlearningmodelconsideringvoluntarymovesintounemployment,but therearedi˙erentimplicationsofasymmetricandsymmetriclearningregardinglayo˙s. Unfortunatelyforresearchers,itisoftendi˚culttodiscernquitsfromlayo˙s.Itishelpfulto decomposetheprobabilityofajob-to-unemploymenttransition, P ( JU ) ,intotheprobability ofalayo˙, P ( L ) ,andtheprobabilityofaquit P ( Q ) .Duringrecessions,Davisetal.(2006) andElsbyetal.(2009)eachshowanincreasedin˛owofworkersintounemploymentduring recessions,whichisdrivenbyanincreaseinlayo˙slargeenoughtodominatesadecrease inthenumberofquits.Comparingthemagnitudeofselectione˙ectsontheprobabilityof ajob-to-unemploymentseparationsbetweeneconomicrecessionsandexpansions,provides insightintohowselectiondi˙ersforlayo˙sandquits. Thereisalsonobroadlyacceptedtheoreticaljusti˝cationforlayo˙s.Itseemsthat thereexistsarangeoflowerwagesinwhichworkerswouldprefertoworkuntiltheycould movetoanother˝rmatahigherwageratherthanenduringaperiodofunemployment. Firms,itseems,shouldpreferkeepingaworkeraslongasthewageislessthantheworker's productivity.Evenwithrelativelylargeeconomic˛uctuationsitseemsthereislikelytobe overlap. However,weobservenominaldownwardwagerigidityduringeconomicdownturns.Camp- bellandKamlani(1997)reportthathumanresourcespersonnelmostcommonlylistfearof themostproductiveworkersleavingastheirprimarymotivationforusinglayo˙sratherthan wagereductions.Thisbegsthequestion,whywould˝rmscarewhichworkersleft,ifeach workerispaidtheirmarginalproductoflabor?Asymmetricinformationprovidesonesuch rationale. Inthemodelpresentedabove,theexpectedmarginalpro˝tfromagivenworkerishis 65 conditionalexpectedproductivity,netofhiswage.Inthecaseofsymmetriclearning,this isthedi˙erencebetweentheexpectedmatchqualityattheretaining˝rmandtheexpected matchqualityatthe˝rmwiththenexthighestbid. 10 Underasymmetriclearning,˝rmswillnotnecessarilykeeptheirmostproductiveworkers, butrathertheirmostpro˝tableworkers;thosewhooutperformtheirobservablecharacteris- tics.Theexpectedproductivityis E [ j S x; S f; v h ] whereasthewageis, w = min f E [ j R x ;P r ; P h = P r ] ;E [ j R x ;P r = P h ;P h ] g : Consequently,expectedpro˝tsonagivenworkeraregiven below. E [ ˇ j R x ;P r ;P h ]= ˙ ˝ (0) ˙ ˝ ( t ) ˙ ˘ ( x ) Q m + ˙ ˝ ( t ) ˙ ˘ ( x ) ˙ Q P h + ˙ ˝ (0) ˙ ˘ ( x ) ˙ Q P r + ˙ ˝ ( t ) ˙ ˝ (0) ˙ Q R x + E ( ' r ) ˙ ˝ (0) ˙ ˘ ( x ) Q 0 m + ˙ ˝ (0) ˙ Q 0 R x + 2 ˙ ˙ ˘ ( x ) Q 0 P h + E ( ' h ) (2.10) where Q = ˙ ˝ (0) ˙ ˝ ( t ) ˙ ˘ ( x )+ ˙ ˝ ( t ) ˙ ˘ ( x ) ˙ + ˙ ˝ (0) ˙ ˘ ( x ) ˙ + ˙ ˝ ( t ) ˙ ˘ ( x ) ˙ ˝ (0) and Q 0 = ˙ ˝ (0) ˙ ˘ ( x )+ ˙ ˝ (0) ˙ +2 ˙ ˙ ˘ ( x ) .Inexpectation,theerrorsarezeroleavingthesimpler equation2.11,withaverysimilarformtoequation2.5. 11 E f E [ ˇ j R x ;P r ;P h ] g = ˙ ˘ ( x ) ˙ Q 0 Q ˙ ˘ ( x ) ˙ ˝ (0)( ˙ ˝ (0) ˙ ˝ ( t ))( m ) : (2.11) If ˙ ˝ ( t ) <˙ ˝ (0) (thebasicassumptionofasymmetriclearning),and >m ,the˝rmwill enjoypositiveexpectedpro˝tsontheworker.Thissurprisingresultcomesfromthefact thatretaining˝rmsactasamonopsonisticconsumersoftheinformationtheyacquireabout 10 Assumingcontinuousnessinmatchquality,thisdi˙erencegoestozero,thelongeraworkerisinthe market. 11 SeeAppendixA 66 theproductivityoftheirworkers. 12 Itisimportantforderivingpredictionsregardinglayo˙s thattheirworkersdi˙erintheirpro˝tability,andtheirpro˝tabilitydependsnotonlyon theirproductivity,butalsoupontheobservablecharacteristics. Itispossiblethat˝rmsoverbidonaworkerafterreceivingoverlyfavorablesignals. Subsequentsignalshoneinonthetrueproductivityandtheexpectedvalueisexceededby thepreviouswage.Ordinarily,suchoverbiddingseemsrare.However,inthecontextof economic˛uctuations,itismoreunderstandableforwagestoexceedexpectedproductivity. Allowingthematchcomponent( ' r )ofaworkers'productivitytodependontheeconomic climateprovidesamechanismbywhicheconomicconditionsimpactpro˝tability,ifmatch qualityisrealizedafter˝rmsdeterminewages.Therealizationofalowerthanexpected matchmayleadwagestoexceedproductivityduringeconomicdownturns.Thiswouldbe particularlymorelikelyforworkerswhosewageswerealreadyclosetotheirproductivity.In thepresenceofdownwardwagerigidity,theprobabilitythata˝rmlayso˙aworkeristhe probabilitythattheexpectedpro˝tsfromtheworker(giventhesignalsandrevealedcurrent matchquality)arenegative.Moreformally,allowing L tobethecompositeerrorterm,the probabilityofalayo˙ P ( L ) ,isgivenbyequation2.12below: P ( L )= P ˆ ' r E ( ' h )+ L > ˙ ˘ ( x ) ˙ Q 0 Q ˙ ˘ ( x ) ˙ ˝ (0)( ˙ ˝ (0) ˙ ˝ ( t ))( m ) ˙ : (2.12) Similartojobswitches,equation2.12dependsonthedi˙erencebetweenabilityandreference groupquality,thedi˙erenceinprecisionoftheemployers'signals,andisscaledbythe 12 Theoretically,a˝xedcostofdismissingworkersmaypreventin˝nitehiringanddismissalandallow economicpro˝tstobezero,evenifthereisheterogeneityinthedi˙erencebetweenworkers'wageand theirmarginalproductoflabor.Addingsuchacostcomplicatesalgebraicderivations,butdoesnotchange predictions. 67 precisionofthepublicsignal.Againimposingthenormalityandorthogonalityassumptions andtakingthederivativewithrespecttoabilitygivesthefollowing: 13 @P ( L ) @ = ˚ ( ˙ ˘ ( x ) ˙ Q 0 Q p ˙ L' ˙ ˘ ( x ) ˙ ˝ (0)( ˙ ˝ (0) ˙ ˝ ( t ))( m ) ) ˙ ˘ ( x ) ˙ Q 0 Q p ˙ L' ˙ ˘ ( x ) ˙ ˝ (0)( ˙ ˝ (0) ˙ ˝ ( t )) < 0 : (2.13) Equation2.13illustratesthatasability( )increasestheprobabilityoflayo˙shouldfall. ThisisperfectlyinlinewithGibbonsandKatz(1991b).Takingthederivativewithrespect tomeanreferencegroupability( m )givesperhapsamoresurprisingresult: 14 @P ( L ) @m = ˚ ( ˙ ˘ ( x ) ˙ Q 0 Q p ˙ L' ˙ ˘ ( x ) ˙ ˝ (0)( ˙ ˝ (0) ˙ ˝ ( t ))( m ) ) ˙ ˘ ( x ) ˙ Q 0 Q p ˙ L' ˙ ˘ ( x ) ˙ ˝ (0)( ˙ ˝ (0) ˙ ˝ ( t )) > 0 : (2.14) Conditionalonindividualability,asworkers'referencegroups( m )areingeneralmorepro- ductive,themorelikelytheyaretobelaido˙.Thisisbecause,highreferencegroupworkers' wagesarebidhigherbyoutside˝rms,whichplacesigni˝cantweightonthereferencegroup. Again,thelearningframeworkallowsforfurtherpredictionsconcerningtheevolutionof thisselectionoverexperienceandworkingspelllength.Justaswithjob-to-jobtransitions, thecrosspartialof ˙ ˘ ( x ) ˙ Q 0 Q q ˙ L' [ ˙ ˘ ( x ) ˙ ˝ (0)( ˙ ˝ (0) ˙ ˝ ( t ))( m )] withrespecttoworking spelllengthandabilityisnegative,andispositivewithrespecttoworkingspelllengthand referencegroup.Thisimpliesthatwithincreasesinworkingspellduration,theadverse selectionintounemploymentonthebasisofabilityshouldbecomestronger(morenegative), 13 PleaseseeAppendixA 14 PleaseseeAppendixAforproof. 68 andthepositiveselectiononthebasisofreferencegroupshouldalsobecomestronger(more positive). 15 Thecrosspartialof ˙ ˘ ( x ) ˙ Q 0 Q q ˙ L' [ ˙ ˘ ( x ) ˙ ˝ (0)( ˙ ˝ (0) ˙ ˝ ( t ))( m )] withrespect toexperienceandabilityispositive,andisnegativewithrespecttoexperienceandreference group.Thustheselectiononthebasisofbothabilityandreferencegroupweakenswith increasesinexperience. 16 Theinverseisalsotrue.Forworkerswhosereferencegroupsare generallymorecapablethantheyare,increasesinworkingspelllengthmakeitmorelikely theywillbelaido˙.Withrespecttoexperiencetheselectionofhigherreferencegroupsinto unemploymentshouldbecomeweaker(morenegative)withincreasesinexperience. Giventhatsuchbroadeconomicdownturnsareexogenoustoanindividual'sability,the predictionsregardingselectionintojob-to-unemploymentseparationsduringrecessionsmay bemoreinsulatedfrompossiblecorrelationbetweenabilityandmatchqualitythanisthe caseforjob-to-jobtransitions. 2.4Estimation Equation2.5andequation2.12areconvenientlystructuredfornormalmaximumlike- lihood(probit)estimation.Whilethemodelisstructuredtoestimatetheprobabilityof separation,Ionlyobservethebinaryindicatorforwhetheraseparationoccurred( s = t ), andwhetheritwasajob-to-job( s = j )orjob-to-unemploymentseparation.Ifurtherdis- tinguishjob-to-unemploymentseparationsthatoccurduringeconomicrecessions( s = r )as opposedtoeconomicexpansions( s = e )toexploreselectionintolayo˙sasopposedtoquits. Followingearlierresearch,Iuseage-adjusted, AFQT percentilescoresasthehard-to- 15 PleaseseeAppendixAforproof. 16 PleaseseeAppendixAforproof. 69 observe,strongcorrelatewithability( ).Imodelthereferencegroup( m )istwoways.First, Iconstructaverageadjusted AFQT scoresbyeducationalattainment, 17 collegeselectivity, 18 andrace.However,thisassumesthateacheasilyobservablecharacteristicin˛uencesmobility decisionsthroughthesamemechanismandtothesamedegree.Iexplorethevalidityofthis restrictionbyalsoallowingeachcovariatetoenterseparately.Forsimplicity,Isubstitute AFQT for m fortheremainderofthisdiscussion.Allowing y s tobethelatentprobability ofseparation(where s indexestheseparationtype),Iestimatethefollowingincludingex- perience( Exp )andworkingspellduration( WrkSpl ),whichplayacentralroleinlearning andhumancapitalaccumulation: 19 y s = s 0 + s 1 AFQT + s 2 AFQT + s 3 WrkSpl + s 4 Exp : (2.15) Fromabove,forbothjob-to-jobmoves( y j )andjob-to-unemploymentseparationsduring recessions( y r ),themodelpredicts 1 < 0 and 2 > 0 . Sincethemodelpredictseachtochangeasspelllengthandexperienceincrease,Ialso estimateeachvariablewithinteractiontermswithbothlengthofspellandexperiencein estimation.Insertingtheappropriate AFQT scoresfor and m ,leadstothefollowing equationtobeestimated: y s = f ( s + s WrkSpl + s Exp ) X g ; (2.16) where X isavectorcontaining AFQT , AFQT , WrkSpl ,and Exp ; s isavectorofmain 17 Educationalattainmentisgroupedbyhighschoolgraduateswithnocollege,thosewhohavesome college,andthosewithatleastafouryeardegree. 18 IgroupbyBarron's7binsofcollegecompetitiveness,andaddaseparatebinasanindicatorforifthe institutionwasnotlistedonBarron's. 19 Allregressionsalsoincludeanindicatorforurbanicityofthelabormarketandavectorofyearindicators. 70 e˙ectcoe˚cients; s isavectorofcoe˚cientsoninteractionsofeachvariablewithworking spellduration;and s isavectorofcoe˚cientsoninteractionsofeachvariablewithexpe- rience. 20 ReferringbacktoSection3.2and3.3,themodelpredictsforbothjobswitches andlayo˙sallelseequal,adverseselectiononthebasisof AFQT ,andthatselectionshould becomemorenegativewithincreasesinworkingspellduration( 1 j ; 1 r < 0 ),andshouldbe- comemorepositivewithincreasesinexperience( 1 j ; 1 r < 0 ).Further,themodelpredicts positiveselectiononthebasisof AFQT ,whichshouldbecomestrongerwithincreasesin lengthofworkingspell( 2 j ; 2 r > 0 ),andweakerwithincreasesinexperience( 2 j ; 2 r > 0 ). 2.5Data IusetheNationalLongitudinalSurveyofYouthof1979(NLSY79),sinceitcontainsthe datesofhiringandterminationofeachjobrespondentsheldandmoreimportantlycontains workers'AFQTscorestoproxyfortheunderlyingabilityofeachworker.Iendthesample in2000tobringmyestimatesinlinewiththeexistingliteratureandtoreduceissuesrelated tonon-randomattrition,whichbeginstobecomemoreproblematicinsubsequentyears. Womenareexcludedtominimizeinstancesofjobseparationduetochildrearing.The remainingsampleiscomposedofobservationsof6,403malesover22years.Additionally,I exclude452menforwhomtheNLSY79containsnoAFQTscore.Ialsodropallindividuals forwhomthereismissingdataformorethanaquarterofthetimeperiods.Following Pinkston[2009],Ifurtherrestricttheanalysistomenwhoobtainedatleastahighschool degree,dropping889menwhodidnotcompletetheirsecondaryeducation. 21 20 Naturally, s 3 and s 4 areredundantandonlyoneisincludedinestimation. 21 Thislastrestrictionissigni˝cant.Theinterpretationoftheroleofthereferencegroup'smeanAFQT scoredi˙ersdependingonwhethertheyareincludedinthesample.However,itseemsthatemployerlearning mayhavearelativelysmallerroleinthedeterminationofjobseparationsinthispopulation.Itispossiblethat 71 SinceallindividualscompletedtheAFQTin1979,thesescoresareusefulindemonstrat- ingthecompositionofthepopulationwhochoosedi˙erentlevelsofeducation.However, IshouldnotethattheAFQTtestwasadministeredwhenparticipantswerebetweenthe agesof14and22.Consequently,thoughtheAFQTismeanttomeasuretheaptitudeofthe individual,scoresmayalsore˛ectsomedi˙erencesintheamountandqualityofparticipants' educationreceivedpriortotheadministrationoftheassessment.Inordertocorrectatleast fordevelopmentalin˛uencesandthequantityofeducationavailable,theAFQTscoresused intheanalysisareageadjustedfollowingAltonjiandPierret(2001)andPinkston(2009).I subtracttheaveragepercentilescoreofallthosewhowerethesameagewhentheytookthe testanddividebythestandarddeviation. TableB.2providestheaveragestandardizedAFQTpercentilescoresforworkersofeach racewitheachlevelofeducation,whichwillproxyfortheaverageabilityofeachworker's referencegroupinonespeci˝cation.First,noticethataseducationincreasessodoesthe averageAFQT.Thisisasexpectedsincewegenerallythinkthateducationislesscognitively taxingonthosewithhigherintellectualability.Werethemarginalbene˝ttoadditionaledu- cationequalacrosspeople(anunlikelyassumption),thoseforwhomthecostwaslowerwould choosetoobtainmore.Second,noticethatevenwithinlevelsofeducationalattainment,the meanAFQTpercentilescoredi˙erssigni˝cantlyacrossraces.ThemeanAFQTscorefor aBlack,high-schoolgraduateisapproximately.7standarddeviationslowerthanthatof Whitehigh-schoolgraduates.Amongcollegegraduates,themeanAFQTscoreforBlack menismorethanafullstandarddeviationlowerthanthemeanforWhiteparticipants. IconstructreferenceAFQTusingtheNLSY79.Sincethemodelassumesthatworkers' thepopulationwhochosenottopersistthroughhighschoolmayalsochoosenottopersistinemployment aswell. 72 abilities,forwhichtheAFQTproxies,isdistributednormallyaroundthemeanoftheir referencegroup,usingtheestimatedmeanprovideseaseofinterpretation.Ratherthan guessingthepredictedsignofraceandeducationwithintheanalysisthemodelgivesdirect predictions.However,thisalsoimposestherestrictionthateachdimensionofthereference groupa˙ectstheprobabilityofseparationinthesameway.Consequently,Ialsoconduct theanalysisusingindicatorsforraceandhighestgradecompletedforeducation. Raceis˝xedwithinthedata,whileeducationalattainmentvariesformanyindividuals duringthesurveyperiod.SinceIamattemptingtocreateameasurefortheexpectation ofemployers,anargumentcanbemadeforusingeithera˝xedmeasureofeducational attainmentorallowingeducationtovaryovertime.Iprefertotreateducationas˝xed. First,formanyindividualstheireducationstaysbelow12forseveralyearsandthenjumps to12whenparticipantsaremucholder.Secondly,themodeltreatseducationasadimension ofareferencegroupanditseemslikelythat,employersmaydrawdi˙erentinformationfrom adegreeobtainedlaterinlife.Consequently,educationismeasuredasthehighestgrade completedattheageof25.CollegecompetitivenessismeasuredaccordingtotheBarron's indexofdegreegrantinginstitutionormostrecentschoolattendedattheageof25. Asmentioned,theNLSY79recordsemploymentstatuscoveringthis22yearperiodof observation.Analysisisrestrictedtoperiodsinwhichtheparticipantswereworkingatleast 30hoursinaweekandtojobsinitiatedafterthesurveybegan.Aftermatchingemployers acrossyearsandNLSYjoblines,usingemployerstartandstopdates,thisstudyconstructs measuresofexperience,tenure,workingspell,andjobseparations. Experienceismeasuredasthenumberofquartersanindividualreportsworkingupto thecurrentperiod.Becauseemployersmayinferadditionalinformationabouttheworker fromexperience,followingPinkston(2009),Iusepotentialexperienceinsteadofactual 73 experienceforallsingle-stepestimationandasaninstrumentforactualexperienceinthe controlfunctionestimation.Lengthofworkingspellisde˝nedasthedi˙erencebetween thecurrentquarterandearliestdateofhireoveraperiodinwhichtherespondentworked withoutexperiencingajob-to-unemploymentseparation.Thisavoidsareseteachtimea participantreportsswitchinginandoutofthesameemploymentspellorwithjob-to-job moves.Eachoftheseoccurintheweeklyarrays.TableB.3providessummarystatisticsof theworkhistoriesofthoseincludedinthesample. AsmentionedinBratsbergandTerrell(1998),thetenurevariablerecordedintheNLSY79 isinconsistentinaccountingforthestartandstopweekofjobs.Consequently,Igenerate tenureusingthedi˙erencebetweenthestartdateandbeginningdateofeachquarterin- dividualreportsworkingforaparticularemployer,subtractingperiodstheworkerreports beingtemporarilyoutofworkoronactivecallinthemilitary.Terminaltenureusesthedate therespondentreportsleavingtheemployer.AsnotedinLight(2005),theyoungworkers intheNLSY79arehighlymobile.TableB.4providesaroughdistributionoftheterminal tenurelengthofjobspellswithinthesamplemeasuredinquarters.Noticethatroughly55% ofallemploymentrelationshipsendwithinthe˝rstyearand76%endwithinthe˝rsttwo years.Fromthethirdyearonwardthedrop-o˙islessdramatic. Jobseparationsserveastheprimaryoutcomevariable,andfollowingSchönberg[2007], Idecomposeseparationsintojob-to-jobandjob-to-unemploymenttransitions.Becauseit seemsthatasymmetriclearningmayhavedi˙erentpredictionsforjob-to-unemployment transitionsbetweendi˙erentstatesoftheeconomy,Ifurtherseparateanalysisonmoves tounemploymentbetweeneconomicexpansionsandcontractions.Separationsaretaken directlyfromthequarterinwhichrespondentsreportedtoleavetheirprimaryemployer. Separationsinwhichtherespondentreportedworkingforanewemployerduringthesame 74 ornextquarterwithoutreportingtohavelookedforajoborspentmorethanafullquarter outofthelaborforce,Ide˝neasjob-to-jobmoves.Separationsduringwhichtherespondent reportshavinglookedforajob,Ide˝neasjob-to-unemploymenttransitions.TableB.5 providesthisbreakdowninjobseparations.Overall,itseemsthequarterlytermination rateisnearlytwelvepercentwiththemajorityofmovescomingfromjob-to-unemployment transitions. Becausejobseparationsarepublicevents,employersandworkersrevealsigni˝cantinfor- mationwheneveroneoccurs.Further,thereisevidencethattheinformationcommunicated di˙ersdependingonthetypeofseparationthatoccursGibbonsandKatz(1991b).Accord- ingly,Iprovideabriefexaminationofwagechangesafteraworkerleavesa˝rm.TableB.6 showsthedi˙erencesinworkers'averagewagesoverthedurationofthetwojobsoneither endofajobseparation.Ideally,Iwouldwantthebeginningandendingwage,butthewage informationisrecodedyearlyformostofthesampleandeverytwoyearsfrom1994to2000. Sincetheseparationdataisquarterly,thereissigni˝cantmeasurementerrorandthetable shouldbeinterpretedaccordingly. Themostimmediatepatternseemstobethatthewagegainsinmovingjobsseemsmore bene˝cialifthemovedoesnotincludeaperiodofunemployment.Thisisconsistentwith theyo˙sandstorypresentedinGibbonsandKatz(1991b).Itisalsoconsistent withasymmetricemployerlearning.Asemployerslearnaboutemployeesitisakintothe accumulationofhumancapital.Inajobtounemploymentmove,thisinformationislostand workersareaccordinglypenalizedthroughtheirwages.Secondly,thegainsfrommobility arealmoststrictlyincreasinginAFQTconditionaloneducation.Thefactthatthosewith higherthanaverageAFQTscoreshavehigherwagegrowthissuggestiveofemployerlearning ingeneral.However,lookingatthedi˙erenceinwagechangesbetweenjob-to-jobandjob- 75 to-unemploymenttransitions,thereisnoevidencethattherelativebene˝tsofjobmovesare largestforthosewhohavehigherthanaverageAFQTscores,astheasymmetriclearning modelsuggeststobethecase. 2.6EmpiricalAnalysis 2.6.1PrimaryResults Themostbasicpredictionofasymmetricemployerlearningisthatconditionalonreference group,workerswhoexperiencejobseparationsareadverselyselected.Symmetriclearning o˙ersnorationaleforanindividual'sAFQTtoimpacttheprobabilityofjobseparations,be- causeallemployers(bothcurrentandhiring)equallyweighhardtoobservecharacteristics. Columns3and4ofTableB.7provideresultsfromthemostsimpletestofthishypothesis. Bothinexplicitlyconditioningonreferencegroupsandimplicitlydoingsothroughmean AFQT,I˝ndthathavingonestandarddeviationhigherAFQTscoredecreasestheproba- bilityofjobseparationinagivenquarterby0.61to0.65percentagepoints(p-valuesless than.001).Giventhatthebaseprobabilityofseparatingisabout12%withinsample,this isabouta5%increaseintheprobabilityofterminatinganemploymentmatch. Secondly,asymmetriclearningpredictsthatconditionalontheindividual'sAFQT,asthe meanAFQTofthereferencegroupincreases,theindividualshouldbemorelikelytoleave. Similarly,easilyobservedcorrelateswithproductivityshouldbepositivelyrelatedtothe probabilityofseparation.Thisisbecausetheoutsidemarketplacesmoreweightonworkers' easilyobservedcharacteristicsrelativetothecurrentemployer.TableB.7revealsexactlythis relationship.Fromcolumn3,theestimatede˙ectoftheaverageAFQToftherespondent's referencegroupispositive,andstatisticallysigni˝cantlyso(p-value<0.05).Perhapsmore 76 importantly,withoutconditioningontheindividual'sAFQTinthe˝rstcolumn,theAFQT ofthereferencegroupissigni˝cantlynegativelyrelatedtothelikelihoodofseparation(p- value<0.001).Thissuggeststhatwhileingeneralthosewithmoredesirableobservable characteristicsenjoyjobsecurity,theretaining˝rmallows(orencourages)thosewithlower AFQTwithineachgrouptoleave. Asthereferencegroupdependsonrace,educationalattainment,andcollegeselectivity, itisunsurprisingthattheinclusionofanindividual'sAFQTmakestherelationshipbetween eacheasilyobservablecovariateandtheprobabilityofmovingmorepositive. 22 FromColumn 4,theindicatorforattendingacompetitivecollegeincreasestheprobabilityofseparation by0.8percentagepoints(p-value<0.001),whereaseducationalattainmentisessentialun- relatedtotheprobabilityofjobseparation.Regardingrace,whileWhiterespondentshave higherAFQTscoresonaveragethanBlackrespondentsateachlevelofeducation,Hispanic respondentshavelowerAFQTscoresingeneral.However,indicatorsforWhiteandHispanic arebothpositivelycorrelatedwithjobseparationsconditionalonAFQT.Giventhatthese resultscomefromestimatingequationsthatdonotaccountfordi˙erencesinjobseparations, theyaremoreusefultoprovideasettingratherthandirectevidence. TableB.8issplitintotwopanels.PanelAprovidesestimatesoftheaveragepartial e˙ects(APEs)oftheindividual'sandthereferencesgroup'sAFQTontheprobabilityofjob- to-jobmovesandjob-to-unemploymentseparationsbothduringeconomicexpansionsand recessions.PanelBprovidesAPEsforindividualAFQT,educationalattainment,college competitiveness,race,lengthofworkspell,andtimeinthelabormarketseparatelyfor eachtypeofmove.Themodelprovidesthemostclearpredictionsforjob-to-jobtransitions 22 Hispanicistheonlycovariatethathasitspointestimatefall.However,HispanicshavelowerAFQT scoresthandoBlackrespondents,meaningthatthischangeisalsoinaccordancewiththechangeinthe pointestimateofthesummarymeasure. 77 andjob-to-unemploymentseparationsduringrecessions.Accordingly,themajorityofthe followingdiscussionwillcenteronthesetypesoftransitions. Thethirdandfourthcolumnsofeachpanelrevealthattheadverseselectiononthe basisofanindividual'sAFQTscoreisdrivenbyjob-to-unemploymenttransitions,with thestrongeste˙ectsduringrecessions.I˝ndthataonestandarddeviationincreasein AFQTconditionalonobservablesisassociatedwithafullpercentagepointdecreaseinthe probabilityofexperiencingamovefromemploymenttounemploymentduringarecession. Duringeconomicexpansions,thesameincreaseinanindividual'sAFQTleadstoarounda0.6 percentagepointdropintheprobabilityofseparation.Eachoftheseresultsarestatistically signi˝cantwithp-valueslessthan0.001,thoughtheyarenotstatisticallydi˙erentfromone another.Theseresultsaresuggestiveofasymmetriclearning.AsinGibbonsandKatz (1991b),therationalehereisthat˝rmslayo˙theirleastpro˝tableworkers.Duringjob-to- jobmoves,thoughthetheasymmetriclearninghypothesispredictssimilaradverseselection, inkeepingwithSchönberg[2007],I˝ndnoevidenceofnegativeselectionforthesetypesof moves. NeitherGibbonsandKatz(1991b)norSchönberg(2007)analyzetheselectionwithregard toreferencegroup,conditionalonindividualability,thoughasymmetricemployerlearning clearlypredictspositiveselectionintojobswitchesandlayo˙sonthebasisofeasilyobservable covariates.Consistentwiththeasymmetriclearningmodel,thesecondandthirdcolumnof PanelAindicatepositiveselectionintojob-to-jobandtoalesserdegreejob-to-unemployment transitionsduringrecessions.Beingamemberofareferencegroupwithaonestandard deviationhigherAFQTraisestheprobabilityofajob-to-jobtransitionby0.6percentage pointorabout9percent(p-value<0.001).The˝ndingthatthosewithhigherreference groupsaremorelikelytobebidawayconditionalontheindividual'sAFQTsuggeststhat 78 theoutsidemarketvaluesthereferencegroupmoresothandoesthecurrentemployer. Thepointestimateforreferencegroup'sAFQTisalsopositiveforjob-to-unemployment transitionsduringrecessionsthoughnotstatisticallysigni˝cantlyso.Takenliterally,this positivepointestimateindicatesthatthosewithbetterobservablesaremorelikelytomove fromemploymenttounemploymentduringarecession,conditionalonindividualability. Thissuggeststhatthereislessrentbetweenworkers'wagesandmarginalproducts,presum- ablybecausetheoutsidemarketbidsuptheirwagesonthebasisofthereferencegroup. Theresultingsmallbu˙ercannotinsulatethesehigh-reference-groupworkersfromgeneral productivityshocks. Turningtothecomponentsofreferencegroups,Whiterespondentsareconditionallymore likelythanBlackrespondentstoexperienceaseparationineachenvironmentwiththelargest pointestimatescomingfromjob-to-jobtransitionsandjob-to-unemploymentseparations duringrecession.Theseareexactlywheretheasymmetriclearningmodelpredictsthemto belargest.Whiletheestimatede˙ectofbeingWhiteisstatisticallysigni˝cantforjob-to-job moves,itistoonoisyduringrecessionstodrawmeaningfulinference.Hispanicrespondents ontheotherhandareconditionallylesslikelytobebidawaybyanother˝rmthanareBlack respondentsallelseequal,thoughnotsigni˝cantlyso.GiventhattheaverageAFQTscorefor HispanicrespondentsislowerthanforBlackrespondents,thisisinaccordancewiththeory. Ontheotherhand,duringrecessions,IestimatebeingHispanictoleadtoa3percentage pointhigherprobabilityofexperiencingunemployment.Themagnitudeofthisunpredicted resultraisesquestionsastowhetherasymmetriclearningisdrivingtherelationshipbetween raceandjobmobility.Thismorecomplicatedrelationshipbetweenraceandmobilitylead metopreferthemoretransparentapproachofincludingeachcovariateseparatelyforthe remainderoftheanalysis. 79 Selectiononthebasisofeducationprovidesmoreconsistentevidencesupportingasym- metricemployerlearning.Attendingacompetitivecollegesraisestheprobabilityofsepa- ration,allelseequal.Theseresultsaredrivenbyjobswitchesandjob-to-unemployment separationsduringrecessions.Attendingacompetitive,verycompetitive,highlycompet- itive,ormostcompetitiveundergraduateinstitutionincreasestheprobabilityofswitching jobsbyabout0.4percentagepointsandtheprobabilityofajob-to-unemploymenttransi- tionduringarecessionby1.4percentagepoints.Whilestatisticallyinsigni˝cant,thepoint estimatesindicateeducationispositivelyselectedinjobswitchesandjob-to-unemployment transitionsduringrecessions,wheretheasymmetriclearningmodelhasthestrongestpredic- tions.Duringeconomicexpansionswherethemodelhasweakerpredictions,thisselection isreversed.Inunreportedregressionsexcludingcollegeselectivity,educationalattainment isstatisticallysigni˝cantlypositiveforbothjob-to-jobandjob-to-unemploymenttransitions duringrecessions.These˝ndingsthatmoreeducatedworkersaremorelikelytoswitchjobs andbelaido˙furthersupporttheasymmetriclearninghypothesisandseemtobeaprimary driveroftheresultsregardingreferencegroup. Thetimedynamicsofthee˙ectsofAFQTandeducationprovideadditionalsuggestive evidenceofasymmetriclearning.Asemployerslearn,theyshouldmoreaccuratelyidentify andretainthemostpro˝tableworkersandlayo˙theleastpro˝table.Thus,thesee˙ects shouldgrowstrongerovertime.FromthePinkston's[2009]model,duetothebidding structure,currentemployersrevealtheiraccumulatedinformationtooutside˝rmsinthe eventofajob-to-jobtransfer.Accordingly,thisextensionofthemodelpredictsthatasa continuousworkingspelllengthens,allelseequal,informationasymmetriesgrowbetween retainingandhiring˝rms,andthee˙ectofanindividual'sAFQTscoreshouldbecome morepowerful.Incontrast,informationisrevealedtoallthelongertheparticipantisin 80 themarket(workingornot),andthesmallertheseasymmetriesshouldbecome.Thus,the modelpredictstheselectiontobecomeweakerwithincreasesinpotentialexperience,all elseequal.InTableB.9,Iincludeafullsetofinteractionswithlengthofworkingspelland potentialexperiencetouncoverthesedynamics. I˝ndthatinkeepingwiththeasymmetriclearningmodel'spredictions,thee˙ectsof AFQTseemtogrowmorenegativewithincreasesinthelengthofcontinuousworkingspells andmorepositivewithpotentialexperience.Acrossallcolumns,thescaledcoe˚cients(SCs) ontheinteractionsofAFQTwithspelllengtharenegative,strengtheningtheselectionon AFQTwithincreasesinlengthofworkingspell.However,thepointestimatesarenoisyand farfromstatisticalsigni˝cance.Aspredicted,thecoe˚cientsontheinteractionsbetween potentialexperienceandAFQTarepositiveacrossallcolumns,indicatingthattheselection weakens,thelongertheindividualisinthemarket.Again,thesedynamicsarenotstatis- ticallysigni˝cantlydi˙erentfromzero.Dynamicsinthee˙ectofAFQTontheprobability ofseparationregardingbothincreasesincontinuousworkingspellandincreasesinpotential experienceareconsistentwithasymmetricemployerlearning,buttheresultsaretoonoisy tobeveryinformative. Thedynamicsregardingracearemixed,furtherquestioningwhetheremployerlearning isdrivingtherelationshipbetweenraceandmobility.Regardingeducation,againtheresults aremorestronglysupportiveofasymmetricemployerlearning.Scaledcoe˚cientestimates ontheinteractionbetweencollegecompetitivenessandlengthofworkingspellarestatisti- callysigni˝cantlypositiveinallcolumns,butforjob-to-jobmoves.Theinteractionbetween yearsofeducationandworkingspelldurationisstatisticallysigni˝cantlypositiveacrossall columns.Theseresultsindicatethatthetheselectionintoworkermobilityonthebasisof educationsisgettingmorepositivethelongertheindividualiscontinuouslyworking.These 81 resultsareconsistentwithideathatoutside˝rmsplacemorevalueontheseeasilyobservable covariatesthandocurrentemployers. Incontrast,thescaledcoe˚cientestimatesontheinteractionbetweencollegecompet- itivenessandquartersofpotentialexperiencearestatisticallysigni˝cantlynegativeinall columns,exceptforjob-to-jobmoves,andtheinteractionbetweenyearsofeducationand potentialexperienceisstatisticallysigni˝cantlynegativeacrossallcolumns.Theseresults indicatethattheselectiononthebasisofeducationweakensaspotentialexperienceincreases andthemarketlearnsmoreabouttrueworkerproductivity. 2.6.2RobustnessResults Unlikelinearestimation,normalmaximumlikelihoodestimationisinconsistentinthepres- enceofheteroskedasticity ? .Accordingly,TableB.10andTableB.11presenttheestimated APEsandscaledcoe˚cientsrespectivelytocomparewiththoseusingprobitestimation.In themodellaidoutabove,thecompositeerrorterm, ,dependson ˝ , ˘ ,and v .Itfollows that var ( ) isafunctionofthevarianceoftheincumbentemployer'sprivatesignal, ˙ ˝ ( t ) , andthevarianceofthepublicsignal, ˙ ˘ ( x ) .Since ˙ ˝ ( t ) and ˙ ˘ ( x ) areinturnfunctionsof workingspelllengthandexperience,theasymmetricemployerlearningmodelpredictsthe conditionalvarianceoftheprobabilityofajob-to-jobtransitiontochangewithspelllength andexperienceaswell. Above,Imodeltheerrortermofthebinarymodelofjobseparationsasafunctionofthe noiseofthesignalsthateachemployerreceives.Becausethenoiseofthesignalsdecreases withexperienceandworkingspelllength,Imodelthevarianceofthaterrortermas: ln ( var ( ))= 0 + 1 WrkSpl + 2 Exp: (2.17) 82 Theassumptionsthat @˙ ˘ ( x ) @x < 0 ,whichisbasictoemployerslearninggraduallyovertime, andthat @˙ ˝ ( t ) @t < 0 ,whichisbasictoasymmetricemployerlearning,implythat @˙ @x < 0 and @˙ @t < 0 . 23 Inotherwords,asymmetricemployerlearningpredictsthattheconditional varianceofjob-to-jobtransitionsshoulddecreasewithexperienceandlengthofworkingspell. Iestimatethee˙ectofworkingspelllengthandexperienceonthevarianceimplicitlyusing heteroskedasticprobitestimation.Panel2ofTableB.10providestheestimatede˙ectsof spelllengthandpotentialexperienceontheconditionalvariance.Theestimatede˙ectsof workingspelldurationandpotentialexperienceontheconditionalvariancelargedonotbear outthesepredictions,thoughtheyarestatisticallysigni˝cant.Thus,itremainsimportant toexaminethestabilityoftheearlier˝ndingsunderthisalternatespeci˝cation. TurningtotheselectionintojobseparationfromTableB.10,theestimatedAPEsremain virtuallyunchangedfromthoseinTableB.8usingstandardprobitestimation.Themain resultsandinferencehold,withasmallchangeinthatyearsofeducationalattainment nowreachesthe95%con˝dencethresholdinstatisticalsigni˝canceforjob-to-jobmoves. FromTableB.11,thee˙ectsagainremainlargelystablewithtwokeyexceptions.First, theinteractionbetweenAFQTandworkingspelldurationbecomesstatisticallysigni˝cantly negativeforjob-to-unemploymentseparationsduringrecessions,givingadditionalsupport totheasymmetriclearninghypothesis.Second,thoughtheydonotswitchsign,thescaled coe˚cientestimatesfortheinteractionbetweencompetitivecollegeattendanceandpotential experiencebecomenoisierandlosestatisticalsigni˝canceforseparationsingeneralandjob- to-unemploymentduringrecession.Ingeneral,itisreassuringtoseetheresultsremainthis constant. Pinkston[2009]expressesconcernaboutpossibleendogeneityofexperienceandwork 23 PleaseseeAppendixA,A,A,andAforproofs. 83 spellduration.Theoretically,eachcontainsinformationthatemployersmayusetoformtheir initialexpectations,whichmaybiastheestimatede˙ectsofexperience,workingspell,and theirinteractions.Consequently,hetakesaninstrumentalvariablesapproachakintothat takeninAltonjiandShakotko(1985).However,sincehismodelincorporatesapublicsignal itperhapsdoesnotmakesensetothrowoutallofthisinformation.Itseemsthatthebias intheinteractiontermsarethemainconcern.Failingtocontrolforthepublicinformation containedinexperienceandworkingspelllengthmayleadtobiasintheestimatede˙ects ofAFQTscoresandreferencegroupmembership.Consequently,Iuseacontrolfunction approachtotrytocontrolfortheadditionalpublicinformationcontainedinexperience andspelllength,whileavoidingbiasintheinteractionterms.Toclarify,Iperform˝rst stageregressionsofworkingspellofaverageworkingspellinthesample,numberoftime observationsandcurrentjobdurationandexperienceonpotentialexperience.Itheninteract thoseresidualswitheachvariablethatisinteractedwithspelllengthandexperience.To accountforthefactthattheresidualsareestimated,Ibootstrapallstandarderrorsover 500repetitions.Onebene˝tofthiscontrolfunctionapproachisthattheresidualsaccount forendogeneityintheinteractiontermsandprovideanimmediatestatisticaltestforthe presenceofendogeneity. TableB.12,TableB.13,andTableB.14,providetheaveragepartiale˙ects(APEs),e˙ects ontheconditionalvariance,andscaledcoe˚cientsrespectivelyfromheteroskedasticnormal MLEusingthecontrolfunctionapproachdescribedabove.First,theAPEsoftheresiduals aregenerallysigni˝cantundereitherspeci˝cation,indicatingthepresenceofendogeneity. InbothpanelsofTableB.12,theAPEsofAFQTremainsigni˝cantlynegativeonlyfor job-to-unemploymenttransitionsduringeconomicexpansions,thoughinnocasedothe estimatede˙ectschangesign.Theselectiononeducationismorerobust.Ratherthancollege 84 competitivenesscarryingthepredictivepower,itshiftstoyearsofeducation.Forbothjob switchesandmovestounemploymentduringrecessions,yearsofeducationispositivelyand signi˝cantlyselected,whilecollegecompetitivenessonlyapproachesstatisticalsigni˝cance forjob-to-jobtransitions. Asbefore,TableB.13revealsthatcontrarytoprediction,theconditionalvariancelargely increasesinbothexperienceandworkingspelllength.Itisactuallyincreasesinthecontrol functionresidualsthatcausetheconditionalvariancetoshrink. FromTableB.14,thescaledcoe˚cientestimatesoftheinteractionbetweenAFQTand workingspelllengthisnolongerstrictlynegativeacrosstypesofjobseparations,norarethe scaledcoe˚cientsoftheinteractionbetweenAFQTandexperiencestrictlypositivemoving acrosscolumns.However,nonearestatisticallysigni˝canteither,leavingthedynamicsof AFQTuninformativeregardingworkingspelldurationandexperience.Thedynamicsare moreconsistentregardingeducation.FromTableB.14,asimilarpatternaswasshownin TableB.9andTableB.11persists.Acrossallcolumns,asworkingspelllengthens,educa- tionalattainmentbecomesmorepositivelyselected.Alsoconsistentwiththeory,educational attainmentbecomeslessimportantasexperienceincreases.Regardingcollegeselectivity, whilethecoe˚cientsontheinteractionwithexperiencelosesstatisticalsigni˝cance,the interactionswithworkingspelllengthcontinuetobepositiveingeneralandforjob-to- unemploymenttransitions.Thefactthatthemoreeducatedworkersfrommoreselective schoolarepositivelyselectedintomobility,andthatthise˙ectstrengthensthelongerthe workeriscontinuouslyemployedandweakenswithadditionalexperience,suggeststhatother ˝rmsvaluethesesignalsmoresothandoesthecurrentemployer.ThesensitivityoftheAPE ofAFQT,aswellasthelackofdynamicsregardingAFQTindicatethatperhapsthereare otheraspectsofproductivity,aboutwhichemployersareprimarilylearning. 85 2.7Conclusion Tosummarize,I˝ndthatconsistentwithasymmetriclearning,thosewithhigherlevelsof education,conditionalonindividualabilityaremorelikelytobelaido˙orbidawayby another˝rm.Thisselectionstrengthensascontinuousworkingspellsincreasesandweakens withexperience.Theseresultsarerobusttoalternatespeci˝cations.Tomyknowledge thesearenovelempiricalfactsthataredi˚culttoexplainintheabsenceofprivateemployer learning. Ionly˝ndevidenceofadverseselectiononthebasisofAFQTinjob-to-unemployment separations.Inkeepingwiththeasymmetriclearninghypothesis,themagnitudeofthis e˙ectislargestduringrecessions,thoughinaccountingforendogeneityofexperienceand workingspelldurationthemagnitudeofthise˙ectdropsandinferencebecomestenuous. Theevidenceofonthesee˙ectsstrengtheningwithworkingspelllengthissensitive,andthe dynamicswithregardtoexperiencearemerelysuggestive.Takencumulatively,theevidence forasymmetricemployerlearningisstrongandconsistentregardingworkers'educationand merelysuggestivefromtheanalysisofAFQT.Perhapsthisisindicativethattherearemore dimensionsofproductivityaboutwhichemployersarelearning,andfurtherworkisneeded toexplorewhatthosemightbe. 86 Chapter3 HandlingCorrelationsbetween CovariatesandRandomSlopesin MultilevelModels 3.1Introduction Weconsiderlinearregressionmodelsforclustereddatathatincludecluster-speci˝crandom interceptsandslopes.Suchmodelsarecalledmultilevelmodels,mixedmodels,random- coe˚cientmodels,orhierarchicallinearmodels.Ifthemodelsareviewedas motheperspectivetakeninthispaper,theregressioncoe˚cientsrepresentstructural orcausalparameters,andtheerrortermsrepresentthee˙ectsofomittedcovariates.If thereareomittedconfoundersthatarecorrelatedwithincludedcovariates,thentheerror termsarecorrelatedwiththeincludedcovariates.Thesecorrelationsleadtoomitted-variable bias.Thispaperfocusesonestimationmethodsthatavoidbiasduetoomittedcluster-level confounders,alsoreferredtoaselendogeneityAnalternativeviewofmodels, nottakenhere,isthatregressioncoe˚cientsmerelyrepresentassociationsbetweenincluded variables,orlinearprojectionsinthecaseoflinearmodels,inwhichcasetheerrortermsare orthogonaltothecovariatesbyconstruction(seeSpanos(2006)foradiscussionof 87 versusmodels). Researchonaddressingcluster-levelendogeneityinmultilevelmodelshastraditionally beencon˝nedtocorrelationsbetweenunit-levelcovariates(i.e.,covariatesthatvaryover units)andrandom intercepts thatvaryoverclustersinwhichunitsarenested.Thiscon- stitutesatypeofendogeneityasitinvolvescorrelationwithacluster-level randomerrorterm.Forinstance,inestimatingthee˙ectofCatholicschoolingonstu- dentachievementcontrollingforstudentsocioeconomicstatus(SES),onemayworrythat school-levelomittedvariables,suchasschoolresources,maybecorrelatedwithSES.Left unaddressed,thisendogenitymayleadustomis-attributetheimpactsoftheseomittedvari- ablestothee˙ectofSES.Thisbiasmayinturnspillovertoothercoe˚cients.Toaddress thistypeofendogeneity, ? showsthatconsistentestimatorsofthecoe˚cientsofunit-level covariatescanbeobtainedbya˝xed-e˙ectsapproach.However,withstandard˝xed-e˙ects estimators,coe˚cientsofcluster-levelcovariates(i.e.,covariatesthatonlyvaryoverclus- ters)cannotbeestimated.TheHausmanandTaylor(1981)instrumental-variableestimator resolvesthislimitationandisconsistentforthecoe˚cientsofbothunit-andcluster-level covariatesunderappropriateassumptions(seeCastellanoetal.,2014). Endogeneityintheformofcorrelationsbetweenunit-levelcovariatesandrandom slopes varyingoverclustersmayalsoarise.ReferringbacktotheCatholicschoolingexample, whencontrollingforstudents'SES,theslopeofSES(orSESachievementgradient)may varybetweenschools,duetointeractionsbetweenSESandomittedschool-levelcovariates, suchasschoolresources.IftheomittedvariablesarenegativelycorrelatedwiththeSES achievementgradient,thentherandomslopeswillbenegativelycorrelatedwithSES. Remarkably,suchendogeneityisrarelyconsidered.OneexceptionisFrees(2004)who extendstheMundlakapproachtohandlerandomslopes.AnotherisWooldridge(2005) 88 whoshowsunderseeminglybenignconditionsthattraditional˝xed-e˙ectsestimationof random-interceptmodelsisrobustagainstcorrelationsbetweenunit-levelcovariatesand randomslopes.However,neitheroftheseapproachespermitsestimationofthecoe˚cients ofcluster-levelcovariatesevenifthecovariatesareexogenous(i.e.,notendogenous).This limitationisovercomebyKimandFrees(2007)whousegeneralizedmethodofmoments estimationtoextendtheHausman-Taylorapproachtomultilevelmodelswithrandomslopes. However,theirmethodisdi˚culttoimplement,makingthe˝xed-e˙ectsapproachmore feasibleinpractice. Unfortunately,akeyassumptionrequiredforthe˝xed-e˙ectsapproachmaybeviolated inimportantapplications.Speci˝cally,thewithin-clustervarianceofaunit-levelcovariate mustbeuncorrelatedwiththerandomslopeofthatcovariate,whichwerefertoasthe variancethroughoutthispaper.TurningbacktotheCatholic schoolingexample,itispossiblethatmorediverseschools(schoolswithhighvarianceof SES)maybebetterequippedtomitigatethee˙ectsofSES(lowertheSESachievement gradient)thanschoolsthataremorehomogeneous(schoolswithlowvarianceofSES).Such asituationwoulddirectlyviolatetheuncorrelatedvarianceassumption. Inthispaper,weinvestigateestimationofthecoe˚cientsofunit-levelandcluster-level covariatesinmultilevelmodelsinthepresenceoftwosourcesofendogeneity;(nonzero)cor- relationsbetweenunit-levelcovariatesand both therandominterceptandrandomslopes. Throughout,weassumethatcovariatesareuncorrelatedwiththeunit-levelerrortermand thatcluster-levelcovariatesareuncorrelatedwiththerandominterceptsandrandomslopes. Weproposeasimpleer-cluster(PC)approachthatisunbiasedandconsistent forcoe˚cientsforbothunit-levelandcluster-levelcovariatesunderbothformsofcluster- levelendogeneityandviolationoftheuncorrelatedvarianceassumption.Wecontrastits 89 performancetotherandom-e˙ects(RE)estimatorandwhatwecallthe mented(FE+)approach,whichextendsthe˝xed-e˙ectsapproachtoprovide estimatesofthecoe˚cientsofcluster-levelcovariates.InSection3.2,we˝rstintroduce ourmotivatingempiricalexampleandspeci˝cmodelofinterestandthenpresentourgen- eralmodel.InSection3.3,wediscussthetraditionalrandom-and˝xed-e˙ectsestimators andtheirconditionsforunbiasedness.InSection3.4,weintroducenewestimators,namely theaugmented˝xed-e˙ectsestimatorandtheper-clusterestimator,andshowunderwhat assumptionstheyareunbiased.Weprovideconditionsforconsistencyforallfourestima- torsinAppendixI.AllestimatorsareappliedtoadatasetinSection3.5,andSection3.6 investigatesperformanceoftheestimatorsusingasimulationstudy. 3.2MotivationandMultilevelModel 3.2.1MotivatingExampleandSpeci˝cModel Asamotivatingexample,weconsiderthee˙ectofprivateschoolingonstudentachievement. WeusetheRaudenbushandBryk(2002)datafromthe1982HighSchoolandBeyond (HSB)Surveybecauseitisfamiliarineducationanditisinthepublicdomain,allowingus toprovidedataandcommandsforallestimatorsinAppendixI. Thistwo-leveldatasetprovidesuswithanestimationsampleof7,185students(units) nestedin160schools(clusters),70ofwhichareCatholic(private),andtheremainingof whicharepublic.Thenumberofobservationsperschoolrangesfrom14to67students (Mean=45,SD=12).Weuseamathematicsstandardizedtestscoreforstudent i inschool j asourresponsevariable, y ij ,whichhasameanof12.75andastandarddeviationof6.88.Our 90 primaryvariablesofinterestare w j ,abinaryindicatorforwhetherschool j isCatholic,and x ij ,acontinuousindexofstudents'socioeconomicstatus,composedofparentaleducation, parentaloccupation,andparentalincome.Thisindexhasameanofzeroandastandard deviationof0.78. Wewritethemodelusingatwo-stageformulation,similartothatusedinRaudenbush andBryk(2002).The˝rststageistheLevel-1model: y ij = 0 j + 1 j x ij + ij : (3.1) Thisisasimpleregressionofthestudentmathematicstestscores y ij ontheirsocio-economic status x ij ,wheretheintercept 0 j andslope 1 j canvarybetweenschools,asindicatedby the j subscript.Eachstudent'stestscorecandeviatefromtheschool-speci˝cregressionline byarandomerrorterm ij . Theschool-speci˝cinterceptsandslopesbecome(unobservable)outcomesintheLevel-2 models: 0 j = 0 + 1 w j + u 0 j 1 j = 1 + 2 w j + u 1 j : ThemeaninterceptandslopeofSES,forthepopulationofschools,dependonwhether theschoolsareCatholicorpublic( w j ).Theintercepts 0 and 1 inthesemodelstherefore representthepopulationmeansoftheinterceptsandslopesofSESforpublicschools,whereas theslopes 1 and 2 representthedi˙erencesinpopulationmeansoftheinterceptsandslopes betweenCatholicandprivateschools,respectively.TheLevel-2modelshaveerrors u 0 j and u 1 j toalloweachschool'sinterceptandslopetovarywithinthesub-populationsofCatholic andprivateschools.Assumptionsregardingtheerrortermsarediscussedinsubsequent sections. 91 SubstitutingtheLevel-2modelsintotheLevel-1model,weobtainthereducedformof themodel: y ij = 0 + 1 w j + 1 x ij + 2 w j x ij + u 1 j x ij + u 0 j + ij : (3.2) Weseethat 2 isthecoe˚cientofacross-levelinteractionbetweenthestudent-levelcovariate x ij andtheschool-levelcovariate w j . Inthissetting,itislikelythatthereareomittedschool-levelvariablesthata˙ectstudent achievement,andhenceentertherandomintercept u 0 j ,andarecorrelatedwithstudent SES.Iftheseomittedschool-levelvariablesalsointeractwithstudentSES,thentheyenter therandomslope u 1 j ,andtheslopeisthencorrelatedwithSES.Ignoringsuchendogeneity mayleadtobiaswhenestimatingthecoe˚cientsofthismodel. 3.2.2GeneralMultilevelModel Thegeneralmodelweconsiderinthispaperisfortwo-leveldata,suchastheHSBdata describedabove.Inthecross-sectionalcase,units(Level1)aretypicallyindividualsnested withinclusters(Level2),suchasschools,hospitals,orneighborhoods.Inthelongitudinal case,unitsrefertomeasurementoccasionsnestedwithinindividuals,whoconstitutethe Clustersareindexed j ,with j =1 ;:::;J ,andunitsareindexed ij ,with i = 1 ;:::;n j .Thegeneralmodelincludesunit-levelcovariatesthatvarybetweenunitswithin clusters(andbetweenclusters),aswellascluster-levelcovariatesthatvarybetweenclusters butareconstantwithinthesamecluster.Same-levelandcross-levelinteractionsmayalsobe included,wherecross-levelinteractionsareunit-levelcovariates.Someunit-levelcovariates mayhaverandomslopes. 92 Thegeneralmodelcanbewrittenas y j = W j + X j + Z j u j + j : (3.3) Here y j =( y 1 j ;:::;y n j j ) 0 isthevectorofresponsesforcluster j ,the n j ( P +1) matrix W j includesall P cluster-levelcovariatesandits˝rstcolumnisavectorofones, 1 n j ,forthe intercept;the n j R matrix X j includesall R unit-levelcovariates;the n j ( R 1 +1) matrix Z j includesall R 1 R unit-levelcovariatesin X j thathaverandomslopesandits˝rst columnis 1 n j fortherandomintercept.Finally,the u j arerandome˙ectsorcluster-level errorterms(onerandominterceptand R 1 randomslopes)andthe j areunit-levelerror terms.The u j areassumedtobeindependentofthe j ,andtheclustersareindependent inthesensethaterrortermsaswellascovariatesareindependentacrossclusters.Other assumptionsmaderegarding u j and j dependonwhatestimatorsareusedandarediscussed inSections3.3and3.4.Sometimesallunit-levelcovariateshaverandomslopes,sothat R 1 = R ,buttypically R 1 0 ,Equation3.15isthemodelforthecluster-speci˝cslopeof the r thcolumnof X 1 j .Thevector, w rj ,includesonlythosecluster-levelcovariatesthat interactwiththe r thcolumnof X 1 j .The˝rstelementof r is 1 r andtheotherelements aresubsetsof 2 .NotethatweincludeLevel-2modelsonlyforthecoe˚cientsofthose 102 unit-levelcovariatesthathaverandomslopes. Equations3.14and3.15areageneralizationofthetwo-stageformulationofthemodel fortheHSBexampleinEquations3.1and3.2.Inthatmodel,thereare R 1 =1 covariates X 1 j = x j withrandomslopesand 1 = 1 .Therearealso R 2 =1 cross-levelinteractions betweenunit-levelcovariatesin X 1 j andacluster-levelcovariate w j ,namely X 2 j = w j x j with 2 = 2 .Therearenoremainingcolumnsof X j ,so R 3 =0 ,andthereisno X 3 j or 3 .Further, j =( 0 j ; 1 j ) 0 and w 0 j = w 1 j =(1 ;w j ) 0 , 0 =( 0 ; 1 ) 0 ,and 1 =( 1 ; 2 ) 0 . Step1:Estimationof 3 (Wooldridge,2005,2010,Sec.11.7.2)considersthespecialcaseofthismodelwithoutcluster- levelcovariates,i.e.,with j = 1 + u j anddescribesestimationof 3 (coe˚cientsforallunit- levelcovariateswithoutrandomslopes)byanextensionofthede-meaningtransformation usedinFEestimation.Insteadofpre-multiplyingbythede-meaningoperator Q j ,wepre- multiplytheLevel-1modelbytheprojectionmatrix M j I n j Z j ( Z 0 j Z j ) 1 Z 0 j : De˝ning _ y j = M j y j , _ X 3 j = M j X 3 j ,and _ j = M j j ,andnotingthat _ Z j = M j Z j = 0 , gives _ y j = _ X 3 j 3 + _ j : ThePOLSestimatorof 3 ,denoted b 3CML ,canbeexpressedas b 3CML = 3 + 0 @ J 1 J X j =1 _ X 0 3 j _ X 3 j 1 A 1 0 @ J 1 J X j =1 _ X 0 3 j _ j 1 A ; 103 where P j _ X 0 3 j _ X 3 j isassumedtobenonsingularwithprobability1.Ifthe ij arenormally distributed,thisestimatoralsocorrespondstotheconditionalmaximumlikelihoodestima- tor(CML),conditioningonthesu˚cientstatistics Z 0 j y j fortheuisanceparameters" j Verbekeetal.(2001). Theconditionalexpectationof b 3CML ,givenallcovariates V ,becomes E ( b 3CML j V )= 3 + 0 @ J 1 J X j =1 _ X 0 3 j _ X 3 j 1 A 1 0 @ J 1 J X j =1 _ X 0 3 j E ( _ j j V ) 1 A : Unit-levelexogeneity,whichimpliesthatE ( _ j j V )= 0 ,isasu˚cientconditionforconditional unbiasednessE ( b 3CML j V )= 3 . Step2:Estimationof j Next,formquasi-residualsas r j y j X 3 j b 3CML andthenobtainOLSestimates j fortheregressionsof r j on Z j foreachcluster, j =1 ;:::;J , j =( Z j Z 0 j ) 1 Z 0 j r j = 0 @ n 1 j n j X i =1 z ij z 0 ij 1 A 1 0 @ n 1 j n j X i =1 z ij r ij 1 A ; (3.16) where z 0 ij isthe i throwof Z j ,and P n j i =1 z ij z 0 ij isnonsingularwithprobability1,which requiresthat R 1 +1 n j .Thisstepgivesrisetothenameer-clusterIdentical estimatesof 3 and j areobtainedbytreating j as˝xedparametersinEquation3.14via theinclusionofinteractionsbetweendummyvariablesforclustersandthecolumnsof Z j . 104 TheestimatorinEquation3.16canalternativelybeexpressedas j = j +( Z j Z 0 j ) 1 Z 0 j h X 3 j ( 3 b 3CML )+ j i ; (3.17) andtheconditionalexpectationof j ,given V ,becomes E ( j j V )= j +( Z j Z 0 j ) 1 Z 0 j n X 3 j [ 3 E ( b 3CML j V )]+ E ( j j V ) o ; wherethe Z j Z 0 j areassumedtobenonsingularwithprobability1.BecauseE ( b 3CML j V )= 3 fromStep1,itfollowsthat X 3 j [ 3 E ( b 3CML j V )]= 0 .Itfollowsfromunit-level exogeneitythatE ( j j V )= 0 ,andthereforethe j areconditionallyunbiased;E ( j j V )= j . Step3:Estimationof , 1 ,and 2 Theremainingregressioncoe˚cients (forcluster-levelcovariates), 1 (forunit-levelcovari- ateswithrandomslopes),and 2 (forcross-levelinteractionsinvolvingunit-levelcovariates withrandomslopes)arenowestimated.Thesecoe˚cientsarecontainedinthevectors r , r =0 ;:::;R 1 ,inEquation3.15.WewriteeachLevel-2equationforallclustersusingthe followingvectornotation.Let r =( r 1 ;:::; rJ ) 0 and u r =( u r 1 ;:::;u rJ ) 0 andlet W r have J rows w 0 rj , j =1 ;:::;J .TheLevel-2equationforeach r canthenbewrittenas r = W r r + u r ;r =0 ;:::;R 1 : Denotingthevectorofestimates r ( r 1 ;:::; rJ ) 0 ,themodelcanbewrittenas r = W r r + u r + r r 105 Weestimate r byapplyingOLStotheregressionof r on W r ,giving b r = r +( W r W r ) 1 W r ( u r + r r ) = r + 0 @ J 1 J X j =1 w rj w 0 rj 1 A 1 0 @ J 1 J X j =1 w rj ( u rj + rj rj ) 1 A ; where,foreach r , P j w rj w 0 rj isassumedtobenonsingularwithprobability1. Theconditionalexpectationof b r ,given V ,is E ( b r j V )= r + 0 @ J 1 J X j =1 w rj w 0 rj 1 A 1 0 @ J 1 J X j =1 w rj E ( u rj j V )+ E ( rj j V ) rj 1 A : Itfollowsfromcluster-levelexogeneitythatE ( u rj j V )=0 andfromtheresultsforStep2 thatE ( rj j V )= rj .Hence,E ( b r j V )= r ,andusingthelawofiteratedexpectations, weseethattheestimatorisunbiased;E ( b r )= r . Forthespecialcaseofourmodelwith j = 1 + u j ,theestimatorfor becomesthe samplemeanof r andthatestimatorhasbeenproposedby(Wooldridge,2010,equation (11.80)).Inmodelsinwhich X j =( X 1 j X 2 j ) ,or R 3 =0 ,the˝rststepcanbeskippedand r j = y j .Ourempiricalillustrationisanexampleofthelatterspecialcase.Accordingly,we ˝rstestimate 0 j and 1 j inmodel(3.2)foreachcluster j byregressing y j on x j usingOLS, givingunbiasedestimates 0 j and 1 j .IdenticalestimatesareobtainedbyOLSwithdummy variablesforclustersandinteractionsbetweenthesedummyvariablesand x ij .Next, 0 j and 1 j arebothregressedon w j usingOLS.Intheregressionfor 0 j ,theOLSestimatorfor theinterceptisunbiasedfor 0 andtheOLSestimatorforthecoe˚cientof w j isunbiased for 1 .Intheregressionfor 1 j ,theOLSestimatorfortheinterceptisunbiasedfor 1 and theOLSestimatorforthecoe˚cientof w j isunbiasedfor 2 .Ifwedidnotincludethe 106 cross-levelinteractionterm, x ij w j ,inourmodel,therewouldbeno 2 , R 1 = R andwe wouldregress 1 j onjusttheintercept,i.e.,˝nditssamplemean,toobtaintheunbiased estimateof 1 . 3.5EmpiricalExample Togroundcomparisonsofourestimatorsofinterest,weapplyeachtotheHSBdatain- troducedinSection3.2.1.TableD.1providesestimatesoftheregressioncoe˚cientsfor Equation3.2.AllestimateswereobtainedusingstandardcommandsinStata13StataCorp (2013),suchas mixed and xtreg (seeAppendixI).NotethattheREestimateofthecor- relationbetweentherandominterceptandslopewas1,arelativelyfrequentoccurrencein random-coe˚cientmodelsChungetal.(2014). Castellanoetal.(2014)showthatpositivecorrelationbetweenarandominterceptand astudent-levelcovariateleadstooverestimationofthecoe˚cientofthecovariate.Indeed, fromtheHSBdataresultspresentedinTableD.1,weseethatREproducesthelargest estimateofthecoe˚cientofSES,2.958,approximately6%higherthantheclosestestimate (FE+).Theindicatorvariable w j forCatholicschoolsispositivelycorrelatedwithSESand thereforeover-estimationofthecoe˚cientofSESisaccompaniedbyunderestimationofthe coe˚cientof w j ,withREproducingthesmallestestimateof 1 ,at2.130. Whilethedi˙erencesintheFE+andREestimatesof 1 maybepracticallysigni˝cant, theyarecloseinmagnitudetotheestimatedstandarderrorsofthecoe˚cientestimates.FE+ producesestimatesofboth 1 and 1 thatliebetweentheestimatesproducedbyREand PC,whichisintuitivegiventhatFE+reliesonlyontheuncorrelatedvarianceassumption, whereasREadditionallyrequiresexogeneity,andPCrequiresneitherassumption.PCgives 107 thesmallestestimatede˙ectofSESonmathachievementscores( b 1 =2 : 772 )andthelargest estimatede˙ectofCatholicschooling( b 1 =2 : 253 ).Theseestimatesdi˙erbyabout6%from theREcounterparts,enoughtogivepractitionerspause. Thesmalldi˙erencebetweenestimatesof 1 fromFE+andPCprovidesevidencethat, inthiscase,wemaybeabletoignorethepossibilitythatthewithin-schoolvarianceinSES iscorrelatedwiththerandomslope.Infact,thewithin-schoolstandarddeviationofSES hasacorrelationofonly 0 : 04 withtheestimatedresidualsfromtheregressionof 1 j on w j inthe˝nalstepofthePCapproach. 3.6SimulationStudy WenowconductasimulationstudytoinvestigatetheperformanceoftheRE,FE+,andPC estimators.Inparticular,weareinterestedintheamountofbiasforREandFE+whenthe respectiveassumptionsofcluster-levelexogeneityanduncorrelatedvarianceareviolated.We alsoevaluateallthreeestimators,RE,FE+,andPC,bytheirrootmeansquareerrorsand considerperformanceoftheestimatedstandarderrors.WeuseStata13StataCorp(2013) throughout. 3.6.1DataGenerationProcess WegeneratethedatausingourmodelofinterestinEquation3.2.We˝rstdrawtheschool- levelvariablesforeachof J =100 clusters.Therandomintercepts u 0 j andrandomslopes u 1 j aredrawnfromabivariatenormaldistributionwithzeromeansandvariance-covariance matrixde˝nedbyvariances 0 =0 : 4 2 and 1 =0 : 25 2 andcorrelation ˆ =0 : 5 ,givingthe covariance 10 =0 : 05 .Wespecifythesevariancestore˛ectthosefoundinourempirical 108 example.Theexogenousschool-levelcovariate w j isdrawnindependentlyfromanormal distributionwithmean 1 : 7 andvariance ˙ 2 w =1 . Wethengeneratethestudent-levelcovariate x ij as x ij = b 0 u 0 j + b 1 u 1 j + b 2 w j + ae ij ;e ij ˘ N (0 ;˙ j ) ; (3.18) where a = q 1 0 b 2 0 2 1 b 2 1 ˙ 2 w b 2 2 2 b 0 b 1 10 : Here, b 0 =1 : 33 , b 1 =2 : 13 ,and b 2 =0 : 20 sothat x ij ispositivelycorrelatedwiththerandom intercept,randomslope,andschool-levelcovariate w j .Finally,wegenerate y ij accordingto Equation3.2with 0 =1 , 1 =3 , 1 =1 ,and 2 =2 . Thekeyassumptionunderwhichwewanttoassesstheperformanceofthecompeting estimatorsisthatthesamplewithin-clustervariance s 2 j of x ij isuncorrelatedwiththe randomslope u 1 j .Thus,thepopulationwithin-clusterstandarddeviation, ˙ j ,isofparticular importance.Accordingly,theuncorrelatedvarianceassumptionfactorinthissimulationhas 2levels:whenitholds, ˙ j =1 ,andwhenitisviolated, ˙ j =exp( u 1 j ) . Althoughourempiricalexampleinvolvesschools,whichtendtohavelargenumbersof students,bothREandFEarecommonlyusedwithclassroomsservingasclusters.Fur- thermore,therearenumerousrelevantapplicationswithlongitudinaldatawhereweoften ˝ndevensmallerclustersizes.Thus,wealsovaryclustersize,primarilyconsideringclus- terssizesof 4 and 20 .Forsimplicity,wesetclustersizesequalacrossclusters, n j = n . Wefullycrosstheclustersizeanduncorrelated-variance-assumptionfactors,yieldingfour primarysimulationconditionsde˝nedby:(large/small n ) (uncorrelatedvarianceassump- tionholds/violated).Tofurtherdeterminethee˙ectofclustersizewhentheuncorrelated 109 varianceassumptionisviolated,wealsoconsiderarangeofclusterssizesfrom4to50: n =4 ; 8 ; 14 ; 20 ; 50 . Allconditionsarereplicated500times.Duetooccasionallackofvariationof x ij within somesmallclusters,thePCapproachfailsforsomereplications.Thelowestnumberof successfulreplicationsis489,whichoccurswhenthevarianceof x ij iscorrelatedwiththe randomslopes,andwehaveonly4observationsineachcluster.Forallsimulationconditions withaclustersizeof20,all500replicationsaresuccessful. 3.6.2Results Weevaluatetheperformanceofeachofourthreeestimators(RE,FE+,andPC)ofthe ˝xedregressioncoe˚cientsinourmodelofinterest(Equation3.2)acrossourfoursimulation conditions.Theestimatedbiasandrootmeansquareerror(RMSE)aregiveninTableD.2. AppendixGprovidessupplementaltablesforeachcoe˚cientthatalsoincludethemean standarderrors,standarddeviationsoftheestimates,andtheratiosofthesevalues. 3.6.2.1Bias For 1 ,thecoe˚cientoftheendogenousstudent-levelcovariate x ij ,therearethreemain results.First,thePCestimatorisunbiasedacrossallconditionsevenwhentheuncorrelated varianceassumptionisviolated.FigureC.1clearlyillustratesthis˝ndingastheempirical distributionsoftheerrors(i.e.,estimate parameter)ofthePCestimator(thesolidcurves) arecenteredon0inallfourpanels,whereeachpanelrepresentsoneofthefoursimulation conditions. Second,theREestimatorisbiasedregardlessofwhethertheuncorrelatedvarianceas- sumptionholds,whereastheFE+estimatorisbiasedonlywhenthisassumptionisviolated. 110 ThisresultforREisexpectedgiventhattheREestimatorreliesontheassumptionofboth unit-andcluster-levelexogeneity(seeSection3.2),andcluster-levelexogeneityisviolatedin allfourconditionswiththenonzerocorrelationbetween x ij andboththerandomintercept anditsrandomslope.Wedonote,however,thatviolationoftheuncorrelatedvariance assumptionexacerbatesthemagnitudeoftheREestimator'sbias:forthesmallclustersize condition( n =4 ),theestimatedbiasis1.28timesaslarge,andforthelargerclustersize ( n =20 ),theestimatedbiasmorethandoublesasshowninthe˝rstcolumnofresultsin TableD.2.Incontrast,theFE+approachonlyrequiresunit-levelexogeneityassumptions, andthusproducesunbiasedestimatesundercluster-levelendogeneityaslongasthereisno correlationbetweentherandomslopesandwithin-clustervarianceof x ij .Thisisevident inFigureC.1byobservingthatthecurvesforFE+(dashed)aremoresimilartothosefor PC(solid)intheleft-handplots(foruncorrelatedvariancesimulationconditions)andmore similartothecurvesforRE(dot-dashed)intheright-handplots(forcorrelatedvariance simulationconditions). Thirdly,theestimatedbiasfor 1 islargerthanthatfortheothertworegressioncoe˚- cients,whichisnotsurprisinggiventhat x ij isthesourceoftheendogeneity.Forinstance, asshowninTableD.2,theestimatedbiasof b 1 RE rangesfrom6.2%-21.3%ofthetruevalue. Thenextlargestestimatedbiasis 0 : 053 for b 1 RE underthesmallclustersanduncorrelated variancecondition,whichisonly1.8%ofthecoe˚cient'struevalue( 1 =3 ). Thecoe˚cientoftheinteractionterm, 2 ,istheleasta˙ectedbythesimulationcon- ditions.Weonly˝ndstatisticallysigni˝cantbias(atthe5%level)for b 2 RE forthesmall clustersizeothwhentheuncorrelatedvarianceassumptionholdsandwhenit isviolated.Eveninthesecases,asgiveninTableD.2,theestimatedbiasisrathersmall relativetothemagnitudeofthetruevalue( 2 =2 ):itis0.4%oftheparametervaluewhen 111 theconditionholdsand0.6%whenitisviolated.(Plotsoftheempiricaldistributionsofthe estimationerrorsfor 2 aregiveninFigureI.1inAppendixI.) Theestimatedbiasesoftheestimatorsforthecoe˚cient 1 oftheexogenousschool-level covariate w j followsimilarpatternsasforthecoe˚cient 1 oftheendogenousstudent-level covariate x ij .Justasfor 1 ,thePCestimatorisunbiasedacrossallconditions,theFE+ estimatorisbiasedonlywhenthevarianceof x ij iscorrelatedwiththerandomslope u 1 j (i.e.,uncorrelatedvarianceassumptionviolated),andtheREestimatorisbiasedregardless ofwhethertheuncorrelatedvarianceassumptionisviolated.These˝ndingsareclearly illustratedinFigureC.2bycomparingthecentersoftheempiricaldistributionsoferrorsfor allestimatorsacrossallconditions:thePCcurve(solid)isalwayscenteredon 0 ,whereasthe REcurve(dot-dashed)isalwayscenteredbelow 0 ,andtheFE+curve(dashed)iscentered below 0 onlyforthecorrelatedvarianceconditionsintheright-handpanels.Justaswith 1 ,theFE+estimator'sbiasfor 1 doesnotvarywithclustersizeestimateisabout 0.8%ofthetrueparametervalueforboth n =4 and n =20 asseeninTableD.2.Cluster sizea˙ectstheREestimator'sbiasfor 1 asitdidfor 1 :asclustersizeincreases,thebias decreases.Whentheassumptionholds,thisbiasdecreasesbyabout63%goingfrom n =4 to n =20 ,andbyabout45%whentheassumptionisviolated(seeTableD.2). Giventhat 1 wasmosta˙ectedbytheviolationoftheuncorrelatedvarianceassumption, wefurtherinvestigatedthee˙ectofclustersizeonthisregressioncoe˚cient.FigureC.1 givestheestimatedbiasforeachestimatoracrossclustersizesof4,8,14,20,and50. ThePC(solid)curvehugsthe y =0 line.TheFE+andREcurvescrossat n =20 : asclustersizeincreases,theREestimator'sbiasdecreases(dot-dashedcurve),whereasthe FE+estimator'sbiasisnotasa˙ectedbyclustersize,shownbyitsdot-dashedcurvestaying relativelyconstantacrosstherangeofclustersizes.Thus,clustersizehasadi˙erentiale˙ect 112 onthebiasoftheestimators. Whenusingbiastoevaluatetheestimators,oursimulationstudyprovidesstrongevidence thatourproposedPCestimatoroutperformstheotherestimators. 3.6.2.2PrecisionandRMSE Asisoftenthecase,thereisatrade-o˙betweenbiasandprecision,whichdependsinparton thesizeoftheclusters.Therankorderingoftheestimatorsbytheirstandarddeviation(SD) isapproximatelythesameforthethreeregressioncoe˚cientswithslightdi˙erencesbetween thesmallerandlargerclustersizeconditions.Accordingly,wediscusshowtheprecisionof theestimatorsdependsonclustersize,withoutdistinguishingamongthecoe˚cients. Forthesmallerclustersizesof n =4 ,REproducestheestimateswiththesmallest variances,followedbyFE+,andPCproducesthemostvariableestimates.Thisisclearly illustratedbycomparingthewidthsoftheempiricaldistributionsoferrorsinFigureC.1or C.2foreachestimator:theREcurvesarethenarrowestandthePCcurvesarethewidest. Forinstance,for n =4 andwhentheuncorrelatedvarianceassumptionisviolated,theSD of b 1 PC over500replicationsisabout0.266,whereasforRE,theSDislessthanhalfthat atabout0.114(seeTablesI.1,I.2,andI.3inAppendixIforallSDvalues). Forthelargerclustersizeof n =20 ,REalwaysyieldsthesmallestvariances,butthe variancesarenotmuchsmallerthanthoseforFE+andPC,whichtendtohaveaboutequal variances.Forinstance,for 1 ,forthelargeclustersanduncorrelatedvariancecondition showninthelower,left-handpanelofFigureC.1,itisdi˚culttodiscernanydi˙erencesin thewidthsofthedistributions.Indeed,theSDforRE,inthiscase,isabout0.078andthe SDsforbothFE+andPCare0.080. Withregardtoprecision,REconsistentlyoutperformsFE+andPCforalltheregression 113 coe˚cientsandacrossallthesimulationconditions.However,giventhetradeo˙between biasandprecision,itisusefultoevaluatetheestimatorswithregardtotheirRMSEs,which takesbothbiasandimprecisionintoaccount.Giventhattheestimatesof 1 arethemost a˙ectedbythesimulationconditionsandthatprecisiondependsonclustersize,weconsider theRMSEsasafunctionoftheextendedrangeofclustersizesfor 1 inFigureC.4.Just aswithbiasinFigureC.3,FigurC.4showsthattheFE+andREcurvescrosswithRE outperformingFE+asclustersizeincreases.This˝gurealsoshowsthat,forthesmallest clustersizeof4,theRMSEforPCislargeandsimilartothatofRE.However,withclusters ofatleast8,thePCestimatoroutperformsbothREandFE+withregardstoRMSE, providingstrongevidenceinfavorofthePCestimator. 3.6.2.3StandardErrorEstimation Asa˝nalpoint,weevaluatetheestimatorsintermsofhowwelltheirestimatedstandard errors(SEs)approximatethesamplingSDs.Weagainfocusonthemosta˙ectedregression coe˚cient 1 .FigureC.5displaysthisratioofmeanSEtoSDovertheextendedrangeof clustertoFiguresC.3andC.4.IftheSEestimationworkswell,thisratio shouldequalone.WeseethatboththePC(solidcurve)andRE(dot-dashedcurve)ap- proachesprovidegoodSEestimates.Incontrast,fortheFE+approach,theSEsareseverely underestimatedastheclustersizeincreases.AlthoughboththeFE+andPCapproaches treatestimatedcoe˚cientsfrompreviousstepsasknowninthesubsequentstep,itappears thatunderestimationoftheSEisalargerproblemfortheFE+approach.Accordingly,we recommendusingeitheranalyticallyderivedorbootstrapSEsfortheFE+approach.These couldalsobeusedforthePCapproach,andmaybenecessaryifStep1isrequired. 114 3.7Conclusion Giventhepopularityofmultilevelmodels,studiesthatinvestigatepotentialbiasesforkey parametersandprovidesimplesolutionsareclearlyimportant. Wehaveshownthatcommonlyusedrandom-and˝xed-e˙ectsestimatorsarebiasedin thepresenceofcorrelationbetweenrandom-e˙ectsandthewithin-clustervarianceofunit- levelcovariates.Further,suchbiascanspillovertotheestimationofcoe˚cientsofother covariates.Wehaveproposedanewper-clusterregressionestimatorthatavoidssuchbias, producesgoodestimatesofSEs,andgenerallyhaslowRMSE.Consequently,werecommend broaduseofper-clusterregressionwhenworkingwithlongitudinalornestedcross-sectional datawhentheclustersaresu˚cientlylarge.Statacodeforapplyingthismethodtothe HSBdataisprovidedinAppendixI.Ininstanceswheretheclustersizesaresmallrelative tothenumberofrandome˙ects,orwhereestimatesfortherandompartofthemodelare ofinterest,werecommendusingper-clusterregressionaspartofasensitivityanalysisfor alternativeestimators. Per-clustermethodshavebeenusedinthepastforlinearmultilevelmodels(Burstein etal.,1978,p.369)andmultilevelstructuralequationmodelsChouetal.(2000).Per- clustermethodscanalsobeusedfornonlinearmultilevelmodels,suchasprobitmodelswith randominterceptsBorjasandSueyoshi(1994)andlogitmodelswithrandominterceptsand slopesKornandWhittemore(1979).However,thepurposeofthatworkwastodevelop simpleestimatorsandnottoaddressendogeneityconcerns.ForourproposedPCestimator forlinearmodels,itmightappeartobeine˚cienttouseOLSinthe˝nalstep,nottaking intoaccountthattheinterceptsandslopesareestimatedwithdi˙erentprecisionfordi˙erent clusters.However,FGLSapproaches,suchasthosediscussedbyBerkeyetal.(1998),su˙er 115 fromsimilarbiasesasREestimators,aswecon˝rmedinsimulations(notshown). Analternativeapproachforhandlingendogeneity,proposedforrandom-interceptmodels byAllisonandBollen(1997)andTeachmanetal.(2001)istomodeltheunit-levelcovariates jointlywiththeresponsesusingstructuralequationmodelingandallowthemtobecorrelated withtherandomintercept.Thisapproachcanbegeneralizedtorandom-coe˚cientmodels butbecomesinfeasibleforlargeclustersizes. Insummary,wehavedemonstratedthatourproposed,simple-to-implementper-cluster regressionapproachoutperformsstandardestimatorswhenestimatingregressioncoe˚cients inmultilevelmodelsunderviolationsofboththecluster-levelexogeneityanduncorrelated varianceassumptions.Werecommendthatresearchersconsiderthevalidityoftheuncor- relatedvarianceassumptionandaddthePCmethodtotheirtoolboxwheninvestigating e˙ectsofcovariatesincross-sectionalandlongitudinalanalyses. 116 APPENDICES 117 AppendixA TablesforChapter1 118 TableA.1:AverageVAMofTeachersmovingwithinandoutofWinston-SalemandGuilford Panel A: Within District Movers Panel B: Out of District Movers 1998-19992000-20072008-2010 1998-19992000-20072008-2010 GuilfordMean -0.1660.0930.246 0.116-0.174-0.125 N 101463104 4820634 Winston-SalemMean 0.009-0.0880.031 -0.528-0.100-0.243 N 18827563 2612121 RestofStateMean -0.0690.0200.052 -0.116-0.118-0.109 N 188267931966 9624230833 Note:VAMsaremeasuredinstandarddeviations.Guilford˝rstadoptedVAMsin2000. Winston-Salem˝rstadoptedVAMsin2008. TableA.2:SampleSummaryStatistics Rest of Guilford Winston-Salem North Carolina MeanSDMeanSDMeanSD ScaledScore250.3871.71249.2368.86252.3670.49 PercentPro˝cient0.750.140.740.150.760.13 ShareofBlackStudents0.420.240.360.240.290.24 ShareofBlackTeachers0.250.430.210.410.150.36 ShareofHispanicTeachers0.010.090.000.040.000.06 ShareofTeacherswithAdvancedDegrees0.300.460.360.480.290.45 CollegeSelectivity(Barron's)3.951.433.921.683.931.44 Experience11.599.7613.369.7112.199.85 Tenure3.233.053.593.263.683.35 JobMoves0.090.280.080.280.080.27 Within-DistrictMoves0.060.240.060.240.050.22 Out-of-DistrictMoves0.030.160.020.140.030.16 LeftNCPS0.060.230.040.200.060.24 VAM0.021.010.010.990.001.00 N11,2398,295216,484 Note:VAMismeasuredinstandarddeviationswiththemeancenteredat0. Tenureisgenerated,andiscensoredforthosealreadyworkingatagivenschoolin1995. 119 TableA.3:ProbabilityofMovingSchoolsWithinandOutofDistrict PanalA:Within-DistrictMovesPanalB:Out-Of-DistrictMoves ToahigherToalowerToahigherToalower VARIABLESTotalperformingperformingTotalperformingperforming schoolschoolschoolschool VAM0.00160.0032***-0.0016**0.00020.0014**-0.0012** [0.00129][0.00091][0.00074][0.00096][0.00072][0.00058] VAMxTreatmentGCS0.0058**0.0051**0.0007-0.0103***-0.0054***-0.0049*** [0.00265][0.00199][0.00151][0.00261][0.00195][0.00156] VAMxTreatmentWSF0.0052*0.0060***-0.00080.00090.0023-0.0014 [0.00286][0.00229][0.00194][0.00241][0.00208][0.00129] TreatmentGCS-0.0040-0.00500.0010-0.0162***-0.0232***0.0070*** [0.00851][0.00571][0.00679][0.00374][0.00233][0.00268] TreatmentWSF0.0555***0.0475***0.0080***-0.00200.0147***-0.0167*** [0.00499][0.00372][0.00299][0.00274][0.00224][0.00178] Observations236,018236,018236,018236,018236,018236,018 CSBstandarderrorsfrom500repetitionsappearinbrackets.Allregressionsincludeteacherlevelcovariates andinteractionswithtreatmentindicators,aswellasyearanddistrict˝xede˙ects.***p<0.01,**p<0.05,*p<0.1 TableA.4:E˙ectsonSorting VARIABLES Total Within District VAM0.0028 *** 0.0024 *** [0.00033][0.00033] VAMxTreatmentGCS-0.0005-0.0000 [0.00074][0.0007] VAMxTreatmentWSF0.00070.0017 * [0.00114][0.00102] TreatmentGCS-0.0195 *** -0.0157 *** [0.00211][0.00216] TreatmentWSF0.0290 *** 0.0231 *** [0.00172][0.00168] Observations209,424202,943 CSBstandarderrorsfrom500repetitionsappearinbrackets. Allregressionsusealinearfunctionalform,include teacherlevelcovariates,andtheirinteractions withtreatmentindicators.***p<0.01,**p<0.05,*p<0.1 120 TableA.5:E˙ectsofteacherqualityindexontheprobabilityofmoving Within-DistrictMovesOut-of-DistrictMoves ToahigherToalowerToahigherToalower VariablesTotalperformingperformingTotalperformingperforming schoolsschoolsschoolsschools VAM0.00180.0039***-0.0021***-0.00020.0014**-0.0016*** [0.00111][0.00078][0.00073][0.00091][0.00068][0.00053] TeacherQualityIndex(TQIndex)0.005**0.0071***-0.0021**-0.00050.0031***-0.0035*** [0.00233][0.00173][0.00105][0.00186][0.00115][0.00096] VAMxTreatmentGCS0.0083***0.0069***0.0014-0.0109***-0.0053***-0.0056*** [0.00237][0.00177][0.0014][0.00249][0.00189][0.00145] VAMxTreatmentWSF0.0063**0.0062***0.00000.00010.0018-0.0017 [0.00248][0.00199][0.00193][0.00212][0.00189][0.00115] TQIndexxTreatmentGCS0.00400.0043**-0.00030.0076***0.0061***0.0015* [0.00246][0.00153][0.00145][0.00116][0.00088][0.00088] TQIndexxTreatmentWSF0.00290.00270.0002-0.0011-0.0026***0.0015** [0.00254][0.00192][0.00131][0.00097][0.00078][0.00063] TreatmentGCS0.0142**0.0253***-0.0111***-0.0120***-0.0132***0.0011 [0.00595][0.00449][0.00405][0.00258][0.00167][0.00189] TreatmentWSF-0.00150.0091***-0.0106***0.0118***0.0177***-0.0059*** [0.00383][0.00242][0.00253][0.00251](0.00136][0.00139] Observations236,018236,018236,018236,018236,018236,018 CSBstandarderrorsfrom500repetitionsappearinbrackets.Allregressions usealinearfunctionalform,andincludeteacherlevelcovariatesandinteractionswithtreatmentindicators. TheVAMsusedinthisanalysisaretheresidualsfromtheprojectionofmystandardVAMs onthecomponentsoftheindex.***p<0.01,**p<0.05,*p<0.1 121 TableA.6:Di˙erentialE˙ectsWithRespecttoExperienceandTenure Within District Out of District VARIABLES Total Higher Total Higher Performing Performing VAM-0.00010.0028*-0.00010.0023 [0.0023][0.00161][0.00244][0.00173] ExperiencexVAM-0.00000.0000-0.0000-0.0000 [0.00011][0.00008][0.00011][0.00008] TenurexVAM0.0020**0.00060.00060.0005 [0.0008][0.00059][0.00073][0.00058] VAMxTreatmentGCS0.00330.0050-0.0181***-0.0095* [0.00568][0.00465][0.00693][0.00514] ExperiencexVAMxTreatmentGCS0.0016***0.0010***0.00020.0003 [0.00026][0.0002][0.00032][0.00026] TenurexVAMxTreatmentGCS0.0056***0.00040.00080.0014 [0.00179][0.00146][0.00217][0.00178] VAMxTreatmentWSF-0.0003-0.0010-0.0073-0.0051 [0.00551][0.00431][0.00503][0.00452] ExperiencexVAMxTreatmentWSF0.00030.00050.00020.0002 [0.00043][0.00036][0.00029][0.00025] TenurexVAMxTreatmentWSF0.0028***0.0009*0.00040.0004 [0.00078][0.00055][0.00053][0.00046] Observations236,018236,018236,018236,018 CSBstandarderrorsfrom500repetitionsappearinbrackets.Allregressions usealinearfunctionalform,andincludeteacherlevelcovariatesandinteractionswith treatmentindicators.***p<0.01,**p<0.05,*p<0.1 122 TableA.7:Probabilityofmovingschoolswithin-districtusingrestricteddataVAM PanelA:Within-DistrictMovesB:Out-Of-DistrictMovesC:SchoolQualityGrowth To a higher To a lower To a higher To a lower Within VARIABLES Total performing performing Total performing performing Total District school school school school VAM0.00030.0011-0.0008-0.0013-0.0006-0.00070.00050.0004 [0.00109][0.00097][0.00063][0.00079][0.00056][0.00043][0.00032][0.00033] VAMxTreatmentGCS0.00340.00300.0004-0.0027-0.0016-0.0011-0.0015-0.0010 [0.00249][0.002][0.00152][0.00201][0.00167][0.00102][0.00083][0.00076] VAMxTreatmentWSF0.0061*0.0099***-0.0038*0.00190.0025-0.00050.0025*0.0037*** [0.00312][0.00241][0.00216][0.00247][0.00224][0.00122][0.00131][0.00109] TreatmentGCS-0.0034-0.00420.0008-0.0137***-0.0220***0.0082***-0.0196***-0.0156*** [0.00848][0.00545][0.00717][0.00365][0.00243][0.00275][0.0022][0.00225] TreatmentWSF0.0555***0.0486***0.0068**-0.00170.0151***-0.0168***0.0299***0.0241*** [0.00533][0.00386][0.0033][0.00283][0.00217][0.0019][0.00165][0.00165] Observations236,018236,018236,018236,018236,018236,018209,424202,943 CSBstandarderrorsfrom500repetitionsappearinbrackets. Allregressionsincludeteacherlevelcovariatesandinteractionswithtreatmentindicators. ***p<0.01,**p<0.05,*p<0.1 TableA.8:E˙ectofVAMsconstructedusingvariousnumberofyearsontheprobabilityof movingtoa"better"school VARIABLES 2yr VAM 3yr VAM 4yr VAM 5yr VAM 6yr VAM 7yr VAM 8yr VAM VAM0.0020***0.0023***0.0024***0.0023***0.0025***0.0027***0.0040*** [0.00054][0.0005][0.00051][0.00073][0.00076][0.00072][0.00083] VAMxTreatmentWinston-Salem0.0103***0.0087***0.0076***0.0064**0.0099***0.0118***0.0150*** [0.00241][0.00233][0.00245][0.00287][0.00293][0.003][0.00323] TreatmentWinston-Salem0.0555***0.0540***0.0550***0.0480***0.0427***0.0457***0.0407*** [0.00382][0.00373][0.00362][0.00385][0.00396][0.00427][0.00434] Observations207,673189,531170,598151,067131,567111,78694,884 CSBstandarderrorsfrom500repetitionsappearinbrackets.Allregressionsusealinearfunctionalform, andincludeteacherlevelcovariatesandinteractionswithtreatmentindicators.ObservationsfromGCS areomittedfromtheaboveanalysis.***p<0.01,**p<0.05,*p<0.1 123 TableA.9:ProbabilityofMovingtoNon-Strategic-Sta˚ngSchools PanalA:Within-DistrictMovesPanalB:Out-Of-DistrictMoves ToahigherToalowerToahigherToalower VARIABLESTotalperformingperformingTotalperformingperforming schoolschoolschoolschool VAM0.00140.0031***-0.0018**0.00020.0013*-0.0011* [0.00127][0.00086][0.00076][0.00098][0.00072][0.00059] VAMxTreatmentGCS0.0043*0.0041**0.0002-0.0111***-0.0054***-0.0057*** [0.00244][0.00197][0.00148][0.00248][0.00194][0.0014] VAMxTreatmentWSF0.0100***0.0103***-0.0004-0.00070.0014-0.0021** [0.00233][0.00176][0.00148][0.00208][0.00196][0.00113] TreatmentGCS-0.0118-0.0084-0.0034-0.0158***-0.0238***0.0079*** [0.00848][0.00552][0.00728][0.00362][0.00221][0.00272] TreatmentWSF0.0241***0.0390***-0.0149***-0.00270.0114***-0.0141*** [0.0049][0.00345][0.00287][0.00255][0.00233][0.00142] Observations236,018236,018236,018236,018236,018236,018 CSBstandarderrorsfrom500repetitionsappearinbrackets. Allregressionsincludeteacherlevelcovariatesandinteractionswithtreatmentindicators. ***p<0.01,**p<0.05,*p<0.1 TableA.10:E˙ectsonSortingWithinDistrictExcludingStrategic-Sta˚ngSchools Within Within VARIABLES Total all district non-strategic schools sta˚ng schools VAM0.0028***0.0024***0.0026*** [0.00033][0.00033][0.00034] VAMxTreatmentGCS-0.0005-0.00000.0009 [0.00074][0.0007][0.00072] VAMxTreatmentWSF0.00070.0017*0.0020* [0.00114][0.00102][0.00114] TreatmentGCS-0.0195***-0.0157***0.0029 [0.00211][0.00216][0.00222] TreatmentWSF0.0290***0.0231***0.0196*** [0.00172][0.00168][0.0018] Observations209,424202,943197,364 CSBstandarderrorsfrom500repetitionsappearinbrackets. Allregressionsincludeteacherlevelcovariatesandinteractions withtreatmentindicators.***p<0.01,**p<0.05,*p<0.1 124 AppendixB TablesforChapter2 125 TableB.1:Jobseparationsbytype,referencegroup,andAFQTrelativetoreferencegroup Panel A: Above average reference group AFQT score SeparationsJob-to-JobJob-to-Unemployment FullSampleFullSampleRecessionExpansion HighschoolGraduate 0.1100.0470.1060.056 NoncompetitiveCollegeAttendee 0.1120.0520.0960.053 CompetitiveCollegeAttendee 0.1220.0560.1200.059 NoncompetitiveCollegeGraduate 0.0910.0440.1000.041 CompetitiveCollegeGraduate 0.1020.0480.1190.046 Total 0.1090.0480.1070.053 Panel B: Below average reference group AFQT score SeparationsJob-to-JobJob-to-Unemployment FullSampleFullSampleRecessionExpansion HighschoolGraduate 0.1310.0460.1340.078 NoncompetitiveCollegeAttendee 0.1190.0460.1160.067 CompetitiveCollegeAttendee 0.1270.0530.1140.068 NoncompetitiveCollegeGraduate 0.0980.0480.0980.044 CompetitiveCollegeGraduate 0.1080.0530.1100.048 Total 0.1240.0480.1260.070 126 TableB.2:AFQTPercentiles(StandardizedbyAge)byRaceandEducation Count Mean SD BlackHighschoolGraduate 372-0.3490.718 BlackUncompetitiveCollegeAttendee 122-0.0050.773 BlackCompetitiveCollegeAttendee 650.4540.757 BlackUncompetitiveCollegeGraduate 150.4880.891 BlackCompetitiveCollegeGraduate 540.9320.754 HispanicHighschoolGraduate 722-0.7460.582 HispanicUncompetitiveCollegeAttendee 200-0.3450.645 HispanicCompetitiveCollegeAttendee 89-0.1730.784 HispanicUncompetitiveCollegeGraduate 470.0320.861 HispanicCompetitiveCollegeGraduate 820.5300.842 WhiteHighschoolGraduate 15080.1520.804 WhiteUncompetitiveCollegeAttendee 3910.5850.799 WhiteCompetitiveCollegeAttendee 2530.9630.696 WhiteUncompetitiveCollegeGraduate 1341.1300.708 WhiteCompetitiveCollegeGraduate 5241.3360.607 Total 45780.2030.969 TableB.3:WorkHistory Mean SD Min Max Experience 36.65222.3250.00092.385 PotentialExperience 45.03824.262-4.000103.000 Tenure 14.02715.4930.00385.134 WorkingSpell 21.18820.0070.00392.070 Observations 232388 Allvariablesaremeasuredinquarters. 127 TableB.4:TerminalTenure Year N Share Mean SD Min Max 1 150960.5561.8870.9520.0033.997 2 55380.2045.6611.14347.999 3 23640.0879.7491.119811.997 4 13170.04913.8521.16812.00515.996 5 8270.03017.8321.17116.00319.997 6 5220.01921.8991.15420.00223.986 7 3660.01325.7911.18924.00727.989 8 2930.01129.8851.12828.00831.993 9 2000.00733.7931.16632.01335.997 10 1580.00637.9411.20636.01339.986 >10 4570.01752.59710.40940.01485.134 Total 2713816.7179.2080.00385.134 TableB.5:JobSeparations Mean SD Recessions JobSeparation 0.1780.382 Job-to-UnemploymentMove 0.1160.321 Job-to-JobTransition 0.0620.240 Observations 29557 Expansions JobSeparation 0.1080.310 Job-to-UnemploymentMove 0.0620.241 Job-to-JobTransition 0.0460.210 Observations 202831 FullSample JobSeparation 0.1170.321 Job-to-UnemploymentMove 0.0690.253 Job-to-JobTransition 0.0480.214 Observations 232388 128 TableB.6:NominalWageChangewithSeparation Job Separations Job-to-Job Job-to-Unemployment % Di˙erence in Wage Changes N Mean SD N Mean SD N Mean SD Between JTJ and JTU Moves HighSchoolGraduates AFQT AFQT HSG < 0 : 75 18550.953.56 7421.233.70 11130.773.46 59.6% 0 : 75 0] BaseProbabilityofMoving Forsimplicity,these˝rstderivationsadoptthenotationofbiddingintheabsenceofVAMs. Substitutingthehiringandretainingprincipalsbidsprovidesthefollowing: P ( M )= P [ ˙ ˝ (0) ˙ ˘ ( x ) Z h NV m + ˙ ˝ (0) ˙ Z h NV R x + ˙ ˙ ˘ ( x ) Z h NV P h 0 ( ˙ ˝ ( t ) ˙ ˘ ( x ) Z r NV m + ˙ ˝ ( t ) ˙ Z r NV R x + ˙ ˙ ˘ ( x ) Z r NV P r t ) > 0] ; (E.1) where Z h NV = ˙ ˝ (0) ˙ ˘ ( x )+ ˙ ˝ (0) ˙ + ˙ ˙ ˘ ( x ) and Z r NV = ˙ ˝ ( t ) ˙ ˘ ( x )+ ˙ ˝ ( t ) ˙ + ˙ ˙ ˘ ( x ) Aftersomealgebra,equationE.1becomesthefollowing: = P f ˙ ˘ ( x ) Z h NV Z r NV [( m ) ˙ ˘ ( x )( ˙ ˝ (0) ˙ ˝ ( t ))+( ˙ ˙ ˝ ( t )+ ˙ ˘ ( x ) ˙ ˝ ( t )+ ˙ ˙ ˘ ( x )) ˝ h 0 ( ˙ ˙ ˝ (0)+ ˙ ˘ ( x ) ˙ ˝ (0)+ ˙ ˙ ˘ ( x )) ˝ r t + ˙ ( ˙ ˝ (0) ˙ ˝ ( t )) ˘ ] > 0 g Letting ( ˙ ˙ ˝ ( t )+ ˙ ˘ ( x ) ˙ ˝ ( t )+ ˙ ˙ ˘ ( x )) ˝ h 0 ( ˙ ˙ ˝ (0)+ ˙ ˘ ( x ) ˙ ˝ (0)+ ˙ ˙ ˘ ( x )) ˝ r t + ˙ ( ˙ v ˙ ˝ ( t )) ˘ ,bethecompositeerrorterm,simpli˝estheabove,toequation1.7from 145 withintext,presentedbelow: P ( M )= P >˙ ˘ ( x )[ ˙ ˝ (0) ˙ ˝ ( t )]( m ) : Undertheassumptionsthat ˝ r , ˝ h and ˘ areeachorthogonaltooneanother, ˙ var ( )= var [( ˙ ˙ ˝ ( t )+ ˙ ˘ ( x ) ˙ ˝ ( t )+ ˙ ˙ ˘ ( x )) ˝ h 0 ( ˙ ˙ ˝ (0)+ ˙ ˘ ( x ) ˙ v + ˙ ˙ ˘ ( x )) ˝ r t + ˙ ( ˙ ˝ (0) ˙ ˝ ( t )) ˘ ] = ˙ ˝ ( t )( ˙ ˙ ˝ (0)+ ˙ ˘ ( x ) ˙ ˝ (0)+ ˙ ˙ ˘ ( x )) 2 + ˙ ˝ (0)( ˙ ˙ ˝ ( t )+ ˙ ˘ ( x ) ˙ ˝ ( t )+ ˙ ˙ ˘ ( x )) 2 + ˙ ˘ ( x ) ˙ 2 ( ˙ ˝ (0) ˙ ˝ ( t )) 2 (E.2) Assumingnormalityoftheerrorterms,theprobabilityofaschool-to-schooltransition maybewrittenas: P ( M )= ( 1 p ˙ ˙ ˘ ( x )[ ˙ ˝ (0) ˙ ˝ ( t )]( m ) ) = xt ( m ) g : (E.3) Comparativestaticsforwithin-districtmoveswithrespecttoteacher e˙ectiveness( ) Assumingtheprobabilityofmovingschoolsismonotonicallyincreasinginthedi˙erence between b h and b r ,thesignof @P [ b h HV b r HV > 0 j m ] P [ b h NV b r NV > 0 j m ] @ isimpliedby 146 thesignof @E [ b h HV b r HV ( b h NV b r NV ) j m ] @ .Here,thesubscriptHVdenotesthathiring principalsmayaccessateacher'sVAM,whilethesubscriptNVdenotesthatthereareno VAMsinformingthebidding.Thedi˙erencebetweenhiringandretainingprincipals'bids withoutthepresenceofVAMsisgivenbyequationE.1andisgivenbyequationE.4when bothprincipalsmayaccesstheVAMs. b h HV b r HV = ˙ ˝ (0) ˙ ˘ ( xV ) Z r HV m + ˙ ˝ (0) ˙ Z r HV R + ˙ ˙ ˘ ( xV ) Z r HV P h 0 ˙ ˝ ( t ) ˙ ˘ ( xV ) Z r HV m + ˙ ˝ ( t ) ˙ Z r HV R + ˙ ˙ ˘ ( xV ) Z r HV P r t : (E.4) Theexpectationofthatdi˙erencegivenpriorbeliefsandtheunderlyingabilityinthepres- enceofVAMsisgivenbyequationE.5: E [ b h HV b r HV j m ]= 1 Z h HV Z r HV ( m ) ˙ ˘ ( xV ) 2 ˙ ( ˙ ˝ (0) ˙ ˝ ( t )) : (E.5) Theexpectationofdi˙erencebetweenbidsgivenpriorbeliefsandtheunderlyingability withoutVAMsisgivenbyequationE.6: E [ b h NV b r NV j m ]= 1 Z h HV Z r HV ( m ) ˙ ˘ ( x ) 2 ˙ ( ˙ ˝ (0) ˙ ˝ ( t )) : (E.6) Let A 1 =( m ) ˙ ˘ ( xV ) 2 ˙ ( ˙ ˝ (0) ˙ ˝ ( t )) Let A 0 =( m ) ˙ ˘ ( x ) 2 ˙ ( ˙ ˝ (0) ˙ ˝ ( t )) E [ b h HV b r HV ( b h NV b r NV ) j m ]= A 1 Z h HV Z r HV A 0 Z h NV Z r NV = A 1 Z h NV Z r NV A 0 Z h HV Z r HV Z h HV Z r HV Z h NV Z r NV (E.7) 147 Examiningthenumerator: A 1 Z h NV Z r NV A 0 Z h HV Z r HV =( m ) ˙ ( ˙ ˝ (0) ˙ ˝ ( t ))( ˙ ˘ ( xV ) 2 ( ˙ ˝ ( t ) ˙ ˘ ( x ) 2 ˙ ˝ (0)+ ˙ ˝ ( t ) ˙ ˙ ˝ (0) ˙ ˘ ( x )+ ˙ ˙ ˘ ( x ) 2 ˙ ˝ (0) + ˙ ˝ ( t ) ˙ ˘ ( x ) ˙ ˝ (0) ˙ + ˙ ˝ ( t ) ˙ 2 ˙ ˝ (0)+ ˙ ˘ ( x ) ˙ ˝ (0) ˙ 2 + ˙ ˝ ( t ) ˙ ˘ ( x ) 2 ˙ + ˙ ˝ ( t ) ˙ 2 ˙ ˘ ( x )+ ˙ 2 ˙ ˘ ( x ) 2 ˙ ˘ ( x ) 2 ( ˙ ˝ ( t ) ˙ ˘ ( xV ) ˙ ˝ (0) ˙ ˘ ( xV )+ ˙ ˝ ( t ) ˙ ˙ ˝ (0) ˙ ˘ ( xV ) + ˙ ˙ ˘ ( xV ) ˙ ˝ (0) ˙ ˘ ( xV )+ ˙ ˝ ( t ) ˙ ˘ ( xV ) ˙ ˝ (0) ˙ + ˙ ˝ ( t ) ˙ ˙ ˝ (0) ˙ + ˙ ˙ ˘ ( xV ) ˙ ˝ (0) ˙ + ˙ ˙ ˘ ( xV ) ˙ ˙ ˘ ( xV ) + ˙ ˝ ( t ) ˙ ˘ ( xV ) ˙ ˙ ˘ ( xV )+ ˙ ˝ ( t ) ˙ ˙ ˙ ˘ ( xV )) (E.8) =( m ) ˙ ( ˙ ˝ (0) ˙ ˝ ( t ))( ˙ ˘ ( xV ) 2 ( ˙ ˝ ( t ) ˙ ˙ ˝ (0) ˙ ˘ ( x ) + ˙ ˝ ( t ) ˙ ˘ ( x ) ˙ ˝ (0) ˙ + ˙ ˝ ( t ) ˙ ˙ ˝ (0) ˙ + ˙ ˙ ˘ ( x ) ˙ ˝ (0) ˙ + ˙ ˝ ( t ) ˙ ˙ ˙ ˘ ( x )) ˙ ˘ ( x ) 2 ( ˙ ˝ ( t ) ˙ ˙ ˝ (0) ˙ ˘ ( xV )+ ˙ ˝ ( t ) ˙ ˘ ( xV ) ˙ ˝ (0) ˙ + ˙ ˝ ( t ) ˙ ˙ ˝ (0) ˙ + ˙ ˙ ˘ ( xV ) ˙ ˝ (0) ˙ + ˙ ˝ ( t ) ˙ ˙ ˙ ˘ ( xV )) =( m ) ˙ ( ˙ ˝ (0) ˙ ˝ ( t ))(( ˙ ˘ ( xV ) ˙ ˘ ( x ))( ˙ ˝ ( t ) ˙ ˙ ˝ (0) ˙ ˘ ( x ) ˙ ˘ ( xV ) + ˙ ˘ ( xV ) ˙ 2 ˙ ˘ ( x ) ˙ ˝ (0)+ ˙ ˘ ( xV ) ˙ ˝ ( t ) ˙ 2 ˙ ˘ ( x )+( ˙ ˘ ( xV )+ ˙ ˘ ( x )) ˙ ˝ ( t ) ˙ 2 ˙ ˝ (0)) : 148 @A 1 Z h NV Z r NV A 0 Z h HV Z r HV @ = ˙ ( ˙ ˝ (0) ˙ ˝ ( t )) (( ˙ ˘ ( xV ) ˙ ˘ ( x ))( ˙ ˝ ( t ) ˙ ˙ ˝ (0) ˙ ˘ ( x ) ˙ ˘ ( xV ) + ˙ ˘ ( xV ) ˙ 2 ˙ ˘ ( x ) ˙ ˝ (0)+ ˙ ˘ ( xV ) ˙ ˝ ( t ) ˙ 2 ˙ ˘ ( x ) +( ˙ ˘ ( xV )+ ˙ ˘ ( x )) ˙ ˝ ( t ) ˙ 2 ˙ ˝ (0)) : (E.9) @E [ b h HV b r HV ( b h NV b r NV ) j m ] @ is 1 Z h HV Z r HV Z h NV Z r NV @A 1 Z h NV Z r NV A 0 Z h HV Z r HV @ . 1 Z h HV Z r HV Z h NV Z r NV ispositive,asitispurelyafunctionofvariances.Asafundamental componentofasymmetricemployerlearning,itisassumedthat ˙ ˝ (0) ˙ ˝ ( t ) > 0 .Under lemma2, ˙ ˘ ( xV ) ˙ ˘ ( x ) < 0 .Allothertermsarepositivevariances,whichimpliesthat @E [ b h HV b r HV ( b h NV b r NV ) j m ] @ > 0 ,whichinturnimpliesthattheprobabilityofmoving increaseswithincreasesin . Comparativestaticsforwithin-districtmoveswithrespecttoVAMs ( V ) IndeterminingthecomparativestaticswithregardtotheVAMsignal,Iseektosign @E [ b h HV b r HV ( b h NV b r NV ) j mV ] @V .FromequationE.4: b h HV b r HV = ˙ ˝ (0) ˙ ˘ ( xV ) Z r HV m + ˙ ˝ (0) ˙ Z r HV R + ˙ ˙ ˘ ( xV ) Z r HV P h 0 ˙ ˝ ( t ) ˙ ˘ ( xV ) Z r HV m + ˙ ˝ ( t ) ˙ Z r HV R + ˙ ˙ ˘ ( xV ) Z r HV P r t = 1 Z h HV Z r HV [ ˙ ˘ ( xV ) ˙ ( ˙ ˝ (0) ˙ ˝ ( t ))( ˙ ˘ ( xV )( m )+ ˙ ˙ ˘ + ˙ ˘ ( x ) ˙ + ˙ ˘ ( x ) ) + ˝ h Z r HV ˙ ˘ ( xV ) ˙ ˝ r t Z h HV ˙ ˘ ( xV ) ˙ ] 149 SubstitutingintheVAM ( V ) andpriorpublicsignal ( R x ) separatelyprovidesequationE.10 = 1 Z h HV Z r HV [ ˙ ˘ ( xV ) ˙ ( ˙ ˝ (0) ˙ ˝ ( t ))( ˙ ˘ ( xV )( m (1+ ˙ ) )+ ˙ ˙ R x + ˙ ˘ ( x ) V ˙ + ˙ ˘ ( x ) ) + ˝ h Z r HV ˙ ˘ ( xV ) ˙ ˝ r t Z h HV ˙ ˘ ( xV ) ˙ ] (E.10) TurningbacktotheprobabilityofmovinginabsenceofVAMs, b h NV b r NV = ˙ ˝ (0) ˙ ˘ ( x ) Z h NV m + ˙ ˝ (0) ˙ Z h NV R x + ˙ ˙ ˘ ( x ) Z h NV P H 0 ˙ ˝ ( t ) ˙ ˘ ( x ) Z r NV m + ˙ ˝ ( t ) ˙ Z r NV R x + ˙ ˙ ˘ ( x ) Z r NV P R t = 1 Z h NV Z r NV [ ˙ ˘ ( x ) ˙ ( ˙ ˝ (0) ˙ ˝ ( t ))( ˙ ˘ ( x )( m )+ ˙ ˘ ) + ˝ h Z r NV ˙ ˘ ( x ) ˙ ˝ r t Z h NV ˙ ˘ ( x ) ˙ ] (E.11) CombiningequationE.10withequationE.11andtakingtheexpectationconditionalon priorbeliefsandVAMsprovidesequationE.12: E [ b h HV b r HV ( b h NV b r NV ) j mV ]= 1 Z h HV Z r HV [ ˙ ˘ ( xV ) ˙ ( ˙ ˝ (0) ˙ ˝ ( t ))( ˙ ˘ ( xV ) ( m (1+ ˙ ) )+ ˙ ˙ + ˙ ˘ ( x ) V ˙ + ˙ ˘ ( x ) )] 1 Z h NV Z r NV ˙ ˘ ( x ) ˙ ( ˙ ˝ (0) ˙ ˝ ( t ))( ˙ ˘ ( x )( m ) (E.12) TakingthederivativewithrespecttoVAMs(V)providesequation1.9fromthetext. @E h b h HV b r HV ( b h NV b r NV ) j mV i @V = 1 Z h HV Z r HV ˙ ˘ ( x ) ˙ + ˙ ˘ ( x ) > 0 150 As 1 Z h HV Z r HV ˙ ˘ ( x ) ˙ + ˙ ˘ ( x ) isfunctionofvariances,itmustbepositive.Meaningthatreleas- ingVAMsraisestheprobabilitythathigh-VAMteachersmoveschools. Comparativestaticsforout-of-districtmoveswithrespecttoteacher e˙ectiveness( ) Assumingtheprobabilityofmovingschoolsismonotonicallyincreasinginthedi˙erence between b h and b r ,thesignof @P [ b h RV b r RV > 0 j m ] P [ b h NV b r NV > 0 j m ] @ isimpliedbythe signof @E [ b h RV b r RV ( b h NV b r NV ) j m ] @ .Here,thesubscriptRVdenotesthatonlyretaining principalsmayaccessateacher'sVAM,whilethesubscriptNVdenotesthatthereareno VAMsinformingthebidding.Thedi˙erencebetweenhiringandretainingprincipals'bids withoutthepresenceofVAMsisgivenbyequationE.1,andisgivenbyequationE.13when bothprincipalsmayaccesstheVAMs. b h RV b r RV = ˙ ˝ (0) ˙ ˘ ( x ) Z r RV m + ˙ ˝ (0) ˙ Z r RV R x + ˙ ˙ ˘ ( x ) Z r RV P h 0 ˙ ˝ ( tV ) ˙ ˘ ( x ) Z r RV m + ˙ ˝ ( tV ) ˙ Z r RV R x + ˙ ˙ ˘ ( xV ) Z r RV P r : (E.13) Theexpectationofthatdi˙erencegivenpriorbeliefsandtheunderlyingabilityinthepres- enceofVAMsisgivenbyequationE.14: E [ b h RV b r RV j m ]= 1 Z h RV Z r RV ( m ) ˙ ˘ ( x ) 2 ˙ ( ˙ ˝ (0) ˙ ˝ ( tV )) : (E.14) 151 Theexpectationofdi˙erencebetweenbidsgivenpriorbeliefsandtheunderlyingability withoutVAMsisagaingivenbyequationE.6: E [ b h NV b r NV j m ]= 1 Z h HV Z r HV ( m ) ˙ ˘ ( x ) 2 ˙ ( ˙ ˝ (0) ˙ ˝ ( t )) : CombiningequationE.14withequationE.6givesthefollowing: E [ b h RV b r RV ( b h NV b r NV ) j m ]= ( m ) ˙ ˘ ( x ) 2 ˙ Z h RV Z r RV Z h NV Z r NV [( ˙ ˝ (0) ˙ ˝ ( tV )) ( ˙ ˝ ( t ) ˙ ˘ ( x ) 2 ˙ ˝ (0)+ ˙ ˝ ( t ) ˙ ˙ ˝ (0) ˙ ˘ ( x )+ ˙ ˙ ˘ ( x ) 2 ˙ ˝ (0) + ˙ ˝ ( t ) ˙ ˘ ( x ) ˙ ˝ (0) ˙ + ˙ ˝ ( t ) ˙ 2 ˙ ˝ (0)+ ˙ ˘ ( x ) ˙ ˝ (0) ˙ 2 + ˙ ˝ ( t ) ˙ ˘ ( x ) 2 ˙ + ˙ ˝ ( t ) ˙ 2 ˙ ˘ ( x )+ ˙ 2 ˙ ˘ ( x ) 2 ) ( ˙ ˝ (0) ˙ ˝ ( t ))( ˙ ˝ ( tV ) ˙ ˘ ( x ) ˙ ˝ (0) ˙ ˘ ( x ) + ˙ ˙ ˘ ( x ) ˙ ˝ (0) ˙ ˘ ( x )+ ˙ ˝ ( tV ) ˙ ˘ ( x ) ˙ ˝ (0) ˙ + ˙ ˝ ( tV ) ˙ ˙ ˝ (0) ˙ + ˙ ˙ ˘ ( x ) ˙ ˝ (0) ˙ + ˙ ˝ ( tV ) ˙ ˘ ( x ) ˙ ˙ ˘ ( x )+ ˙ ˝ ( tV ) ˙ ˙ ˙ ˘ ( x ) + ˙ ˝ ( tV ) ˙ ˙ ˝ (0) ˙ ˘ ( x )+ ˙ ˙ ˘ ( x ) ˙ ˙ ˘ ( x ))] = ( m ) ˙ ˘ ( x ) 2 ˙ Z h RV Z r RV Z h NV Z r NV ( ˙ ˝ ( t ) ˙ ˝ ( tV )) ( ˙ ˝ (0) 2 ˙ 2 + ˙ ˝ (0) 2 ˙ ˘ ( x ) 2 + ˙ ˝ (0) 2 ˙ ˙ ˘ ( x )+ ˙ ˘ ( x ) 2 ˙ 2 ) (E.15) TakingthederivativeofequationE.15withrespecttotruee˙ectiveness( ),giveswhat 152 isreferredtointextasequationE. @E h b h RV b r RV ( b h NV b r NV ) j m i @ = ˙ ˘ ( x ) 2 ˙ Z h RV Z r RV Z h NV Z r NV ( ˙ ˝ ( t ) ˙ ˝ ( tV ))( ˙ ˝ (0) 2 ˙ 2 + ˙ ˝ (0) 2 ˙ ˘ ( x ) 2 + ˙ ˝ (0) 2 ˙ ˙ ˘ ( x )+ ˙ ˘ ( x ) 2 ˙ 2 ) : Lemma1demonstratesthat ˙ ˝ ( t ) ˙ ˝ ( tV ) > 0 .Allothertermsarepositivevariances, implyingthat @E [ b h RV b r RV ( b h NV b r NV ) j m ] @ < 0 ,whichinturnimpliesthattheprobability ofout-of-districttransitionsincreaseswithdeclinesinteachere˙ectiveness( ). Comparativestaticsforout-of-districtmoveswithrespecttoVAMs ( V ) IndeterminingthecomparativestaticswithregardtotheVAMsignal,Iseektosign @E [ b h RV b r RV ( b h NV b r NV ) j mV ] @V .Turningbacktotheprobabilityofmovinginabsenceof VAMs,equationE.11provides: b h NV b r NV = 1 Z h NV Z r NV [ ˙ ˘ ( x ) ˙ ( ˙ ˝ (0) ˙ ˝ ( t ))( ˙ ˘ ( x )( m )+ ˙ ˘ )+ ˝ h Z r NV ˙ ˘ ( x ) ˙ ˝ r t Z h NV ˙ ˘ ( x ) ˙ ] Inthecasewhereonlyretainingprincipalsmayaccessateacher'sVAM,asisplausiblefor out-of-districtmoves,thedi˙erencebetweenhiringandretainingprincipalsbidsisgivenby 153 equationE.16: b h RV b r RV = ˙ ˝ (0) ˙ ˘ ( x ) Z r RV m + ˙ ˝ (0) ˙ Z r RV R x + ˙ ˙ ˘ ( x ) Z r RV P h 0 ˙ ˝ ( tV ) ˙ ˘ ( x ) Z r RV m + ˙ ˝ ( tV ) ˙ Z r RV R x + ˙ ˙ ˘ ( xV ) Z r RV P r = 1 Z h RV Z r RV [ ˙ ˘ ( x ) ˙ ( ˙ ˝ (0) ˙ ˝ ( tV ))( ˙ ˘ ( x )( m )+ ˙ ˘ ) + ˝ h Z r HV ˙ ˘ ( x ) ˙ ˙ ˘ ( x ) ˙ Z h RV ˙ ˝ r t + ˙ ˝ ( t ) ˙ + ˙ ˝ ( t ) ] = 1 Z h RV Z r RV [ ˙ ˘ ( x ) ˙ ( ˙ ˝ (0) ˙ ˝ ( tV ))( ˙ ˘ ( x )( m )+ ˙ ˘ ) + ˝ h Z r RV ˙ ˘ ( x ) ˙ ˙ ˘ ( x ) ˙ Z h RV ˙ ˝ r t + ˙ ˝ ( t )( V ) ˙ + ˙ ˝ ( t ) ] (E.16) ThederivativeofequationE.16withrespecttotheVAMsignal( V )isreferredtointext asequation1.11,andispresentedbelow: @E h b h HV b r HV ( b h NV b r NV ) j mV i @V = ˙ ˘ ( x ) ˙ ˙ ˝ ( t ) Z r RV ( ˙ + ˙ ˝ ( t )) < 0 Asequation1.11isthenegativeofafunctionofvariances,itislessthanzero.Thusafter VAMsarereleased,asateacher'sVAMdecreases,theprobabilityofmovingoutofdistrict increases. Robustness:YearinteractionswithVAM Theprimarythreattovalidityfordi˙erence-in-di˙erenceanalysisisdi˙erentialtrends.The tablesbelowprovideyearinteractionswiththeVAMwithinbothtreatmentdistrictsas 154 wellastherestofthestate.Whiletheestimatesaretoonoisytosayanythingconclusive, thepre-policytrendsdonotseemdivergeinawaythatwouldbiasupmyresults.Itis alsonoteworthythatisbothdistrictsthereisaspikeinthecorrelationofVAMwiththe probabilityofmovingwithin-districtsoonafterthepolicytakese˙ect. Robustness:MobilitybasedonABCGrowthPolicies Inthe1996/1997schoolyearthestateofNorthCarolinabeganrewardingteacherswho workedinschoolsinwhichthestudentsmadesubstantialgrowth.Thestateawardedbonuses ofeither$750or$1,500basedonwhethertheschoolachievedgrowthinstudenttestscores beyondpredeterminedtieredthresholds.Thesebonusesweregiventoallteachersinquali- fyingschools.ForadditionaldetailaboutthepolicypleaseseeVigdoretal.(2008)andAhn andVigdor(2012). Asaresult,teachinginhighgrowthschoolsmaybeadditionallyattractivetoteachers sincethebonusesdependeduponschoolperformance.TableF.4iscomparabletoTableA.3 exceptthatthedependentvariablehereiswhethertheteachermovestohigher(lower) growthschoolasopposedtoahigher(lower)performingschoolwithinandoutofdistrict. Thetotalwithinandout-ofdistrictsmobilityestimatesincolumns1and4ofTableA.3are una˙ected,andsotheyareomitted. Whenexaminingthisalternateschoolattributeonwhichteachersmaysort,theprimary ˝ndingsremainintact.Thewithindistrictmobilityisdrivenbymovestomorefavorable schoolsforbothdistricts.Thoughtheresultsareattenuatedhereasateacherwithafull standarddeviationhigherVAMis0.3percentagepointmorelikelytomovewithindistrict 155 toahigherABCgrowthschoolforteacherswhoseVAMsarereleased,theestimatesre- mainstatisticallysigni˝cantlypositiveforbothdistricts.Thoughtheseestimatesarenot statisticallydi˙erentfromtheestimatede˙ectontheprobabilityofmovingtohigherper- formingschools,perhapstheysuggestthatschoolperformancemaybeastrongermotivator forteachermobilitythanstudentgrowth. Theestimatede˙ectsformovesoutsidethedistrictareremarkablyclosebetweenTa- bleA.3andTableF.4.TheadverseselectionofmoversoutofGuilfordCountySchoolsholds formovestobothbetterandworseschools,whilemovesfromWinston-Salemtobetter schoolsremainunrelatedtoteachers'VAMsafterthepolicytakese˙ect. NormalMaximumLikelihoodEstimation TheresultsinTableA.3arefromalinearprobabilitymodel,whicharemorestraightfor- wardbothcomputationallyandininterpretation.Takingthenormalityandorthogonality assumptionsfromSection1.4seriouslywouldsuggestnormalMaximumLikelihoodEstima- tion(probitestimation).AsnotedinAiandNorton(2003),thefunctionalformofprobit estimationincorporatesaninteractionterm,evenwhenoneisnotspeci˝callymodeled.As aresult,iftheresearcherisinterestedinestimatingtheaveragepartiale˙ect(APE)of aninteractionadditionallyprogrammingisnecessary.TableF.5inAppendix ?? provides theAPEsinaccordancewithAiandNorton(2003).ComparisonbetweenTableA.3and TableF.5providesverysimilarresults. 156 CompetingRisksAnalysis Byperformingseparateregressionsforeachtypeofschooltransfer,theaboveanalysistreats eachtypeofmoveasindependentoftheothers.However,itispossiblethatthepropensityof ateachertomovewithin-districttoahigher-performingschoolisrelatedtothepropensityof movingtoahigher-performingschoolinanotherdistrict.Thesamecouldbesaidwithany combinationofoutcomes.Totestthesensitivityofmyearlierresultstothesepossibilities, Iadoptacompetingrisksapproach,asproposedbyFineandGray(1999). Competingriskssurvivalanalysismodelsthesubdistributionhazard( E ( t ) )ofapartic- ulartypeofevent,suchasamovewithinaschooldistrict( E = WD ),asafunctionofan unspeci˝edbaselinehazard( E 0 ( t ) ),aswellasavectoroftime-varyingcovariates( Z ( t ) ). 1 WD ( t j Z )= WD 0 ( t ) exp f Z ( t ) 0 g ; (E.17) Inthecontextofthisstudy,timeatrisk ( t ) isde˝nedasthedi˙erencebetweenthecurrent yearandtheyearatwhichtheteacher˝rstappearsmatchedwiththecurrentschool. 2 Z ( t ) isavectorincludingallcovariatesusedinTableA.3,withtheexceptionoftenure,which isperfectlycorrelatedwith t .Iadditionallyincludedistrictaveragesofallwithin-district- varyingcovariatestocontrolforunobserved,district-widee˙ects,asinMundlak(1978) 3 . TableF.6reportsthecoe˚cientestimatesforeachtypeoftransferbetweenschools. 1 Gray(1988)de˝nesthesubdistributionhazardas, WD ( t )= lim t ! 0 P ( t 0] Substitutinginthebidsandthede˝nitionofeachsignalprovidesthefollowing: = P [ ˙ ˘ ( x ) ˙ ˝ (0) W m + ˙ ˝ (0) ˙ W ( + ˘ )+ 2 ˙ ˘ ( x ) ˙ W ( + ˝ h ) ( ˙ ˘ ( x ) ˙ ˝ ( t ) W 0 m + ˙ ˝ ( t ) ˙ W 0 ( + ˘ )+ 2 ˙ ˘ ( x ) ˙ W 0 ( + ˝ r )) >E ( ' r ) E ( ' h )] ; 175 whichaftersomealgebragives: P f 2 WW 0 [( m ) ˙ ˘ ( x ) 2 ˙ ( ˙ ˝ (0) ˙ ˝ ( t ))+ W˝ h W 0 ˝ r + ˙ 2 ˙ ˘ ( x )( ˙ ˝ (0) ˙ ˝ ( t )) ˘ ] >E ( ' r ) E ( ' h ) g : Letting J E ( ' h E ( ' r ))+ 2 WW 0 ( W˙ ˙ ˘ ( x ) ˝ r W 0 ˙ ˙ ˘ ( x ) ˝ h + ˙ 2 ˙ ˘ ( x )( ˙ ˝ ( t ) ˙ ˝ (0)) ˘ ) bethecompositeerrorterm,providesthesimpli˝cation: P ( J )= P ˆ 2 WW 0 ˙ ˘ ( x ) 2 ˙ [ ˙ ˝ (0) ˙ ˝ ( t )]( m ) > J ˙ : Imposingthenormalandorthogonalityassumptionsprovide: P ( J )= ( 2 p ˙ J WW 0 h ˙ ˘ ( x ) 2 ˙ [ ˙ ˝ (0) ˙ ˝ ( t )]( m ) i ) ; where f : g isthenormalCDF, ˙ = var ( )=2 ˙ ' + 4 W 2 W 0 2 ( W 0 2 ˙ 2 ˙ ˘ ( x ) 2 ˙ ˝ (0) + W 2 ˙ 2 ˙ ˘ ( x ) 2 ˙ ˝ ( t )+ ˙ 4 ˙ ˘ ( x ) 2 ( ˙ ˝ ( t ) ˙ ˝ (0)) 2 ˙ ˘ ( x )) and ˙ ' = var ( E ( ' r ))= var ( E ( ' h )) . Thederivativeoftheprobabilityofjob-to-jobtransitionswithrespecttoability( )is: @P ( J ) @ = ˚ ( 2 ˙ ˘ ( x ) ˙ WW 0 p ˙ j ˙ ˘ ( x )[ ˙ ˝ (0) ˙ ˝ ( t )]( m ) ) 2 ˙ ˘ ( x ) ˙ WW 0 p ˙ j ˙ ˘ ( x )[ ˙ ˝ (0) ˙ ˝ ( t )] ˚ f : g ,beingthenormalpdf,ispositive,asiseachvariance.Thus,aslongastheprecision oftheprivatesignalshrinksthelongeraworkeriswiththeretaining˝rm( ˙ ˝ (0) >˙ ˝ ( t ) ), @P ( J ) @ < 0 . 176 Thederivativeoftheprobabilityofjob-to-jobtransitionswithrespecttoreferencegroup quality( m )is: @P ( J ) @m = ˚ ( 2 ˙ ˘ ( x ) ˙ WW 0 p ˙ j ˙ ˘ ( x )[ ˙ ˝ (0) ˙ ˝ ( t )]( m ) ) 2 ˙ ˘ ( x ) ˙ WW 0 p ˙ j ˙ ˘ ( x )[ ˙ ˝ (0) ˙ ˝ ( t )] Underthesameconditions, @P ( J ) @m > 0 . Job-to-Jobdynamicswithrespecttoworkingspelldura- tion Oneofthemainindicatorsofasymmetricemployerlearningistheevolutionoftheseselection e˙ectsasinformationaccumulatesthelongeraworkeriscontinuouslyemployed.Inorderto examinethedynamicsofthisselectionoverworkingspellduration,Ifocusontheinteractions ofindividualabilitywithworkingspelldurationandreferencegroupwithworkingspell durationrespectively.Below,Iwill˝rst˝ndthederivativeoftheprobabilityofajob-to-job move,P(J),withrespecttoworkingspellduration,inorderto˝ndthepredictedsignofthe scaledcoe˚cientsontheinteractionterms. Thescaledcoe˚cientontheinteractionofindividualabilitywithworkingspellduration, 1 j ,isgivenbelow: 1 j = ˚ f : g @ 2 f : g @t@ = ˚ f : g " @ [ : ] @ ( 1 2 ) ˙ 3 2 @˙ @t 2 WW 0 + @ 2 WW 0 @t ˙ 1 2 ! + @ 2 [ : ] @t@ ˙ 1 2 2 WW 0 # ; where ˚ f : g standsfor ˚ ˆ 2 ˙ ˘ ( x ) ˙ WW 0 p ˙ jj ˙ ˘ ( x )[ ˙ ˝ (0) ˙ ˝ ( t )]( m ) ˙ ; and [ : ] standsfor ˙ ˘ ( x ) 2 ˙ [ ˙ ˝ (0) ˙ ˝ ( t )]( m ) . 177 Thescaledcoe˚cientontheinteractionofaveragereferencegroupabilitywithworking spellduration, 2 j ,isgivenbelow: 2 j = ˚ f : g @ 2 f : g @t@m = ˚ f : g " @ [ : ] @m ( 1 2 ) ˙ 3 2 @˙ @t 2 WW 0 + @ 2 WW 0 @t ˙ 1 2 ! + @ 2 [ : ] @t@m ˙ 1 2 2 WW 0 # : Interactionbetweenworkingspelldurationandability Iwanttoshow: 1 j = ˚ f : g [ ˙ ˘ ( x ) 2 ˙ [ ˙ ˝ (0) ˙ ˝ ( t )]( @ 2 WW 0 @t ˙ 1 2 ˙ 3 2 @˙ @t 1 WW 0 )+ @ 2 [ : ] @t@ ˙ 1 2 2 WW 0 ] < 0 : BelowItakeeachpartofthescaledcoe˚cientontheinteractionbetweenworkingspell durationandabilityseparatelybeforesigningtheentireexpression. Startingwiththe˝rstterm: ˙ ˘ ( x ) 2 ˙ [ ˙ ˝ (0) ˙ ˝ ( t )] @ 2 WW 0 @t ˙ 1 2 = 2 ˙ ˘ ( x ) 2 ˙ [ ˙ ˝ (0) ˙ ˝ ( t )] ˙ 1 2 @˙ ˝ ( t )) @t ( ˙ ˘ ( x )+ ˙ ) W 1 W 2 : Since W , W 0 and ˙ areeachsumsofvariances,theassumptionthat @˙ ˝ ( t )) @t < 0 ,which iskeytoasymmetricemployerlearning,impliesthatthe˝rsttermisalsonegative. Movingtothesecondterm,recallthat: 178 ˙ = 2 ˙ ' + 4 W 2 W 0 2 ( W 0 2 ˙ 2 ˙ ˘ ( x ) 2 ˙ ˝ (0)+ W 2 ˙ 2 ˙ ˘ ( x ) 2 ˙ ˝ ( t )+ ˙ 4 ˙ ˘ ( x ) 2 ( ˙ ˝ ( t ) ˙ ˝ (0)) 2 ˙ ˘ ( x )) . Thus, ˙ ˘ ( x ) 4 ˙ 3 [ ˙ ˝ (0) ˙ ˝ ( t )] ˙ 3 2 @˙ @t 1 WW 0 = ˙ ˘ ( x ) 2 ˙ ( ˙ ˝ (0) ˙ ˝ ( t )) (1) WW 0 ˙ 3 2 @˙ ˝ ( t )) @t f 8 W 2 W 0 2 ˙ 2 ˙ ˘ ( x )( ˙ ˝ (0) ˙ ˝ ( t ))( 1 W 0 ( ˙ ˘ ( x )+ ˙ )( ˙ ˝ (0) ˙ ˝ ( t ))+1) + 4 W 0 3 2( ˙ ˘ ( x )+ ˙ ) ˙ ˝ ( t ) W 0 g : As ˙ ˘ ( x ) 2 ˙ ( ˙ ˝ (0) ˙ ˝ ( t )) WW 0 ˙ 3 2 @˙ ˝ ( t )) @t 4 ˙ ˙ ˘ ( x ) W 0 3 ,cannotbeeliminatedhere,Iwilldiscussa su˚cientconditionunderwhichthissourceofambiguitymaybeeliminatedbelow. Finally,examinethelastterm. @ [ : ] @ = 2 ˙ ˘ ( x ) ˙ WW 0 q ˙ j ˙ ˘ ( x )[ ˙ ˝ (0) ˙ ˝ ( t )] ; whichmeansthat, @ 2 [ : ] @t@ ˙ 1 2 2 ˙ ˘ ( x ) 2 ˙ WW 0 = ˙ 1 2 J 2 WW 0 @˙ ˝ ( t )) @t . Again,theassumptionthat @˙ ˝ ( t )) @t < 0 ,whichiscentraltoasymmetriclearning,implies thattheselectiononabilityshouldgrowstronger(morenegative)withincreasesinworking spellduration. Thus,theinteractiontermbetweenworkingspelldurationandabilityisgivenbythe following: 1 j = ˙ ˘ ( x ) 2 ˙ ˙ 1 2 2 WW 0 @˙ ˝ ( t )) @t f [ ˙ ˝ (0) ˙ ˝ ( t )][( ˙ ˘ ( x )+ ˙ ) W 1 + ˙ 1 4 W 2 W 0 2 ˙ 4 ˙ ˘ ( x ) 3 ( ˙ ˝ (0) ˙ ˝ ( t ))( 1 W 0 ( ˙ ˘ ( x )+ ˙ )( ˙ ˝ (0) ˙ ˝ ( t ))+1) + 2 W 0 2 h 2 W 0 ( ˙ ˘ ( x )+ ˙ ) ˙ 2 ˙ ˘ ( x ) 2 ˙ ˝ ( t ) ˙ 2 ˙ ˘ ( x ) 2 i ]+1 g Withsu˚cientvariabilityinmatchquality,thisentiretermisnegative,meaningthat asworkingspelldurationincreases,selectiononthebasisofabilityshouldbecomemore 179 negative. Interactionbetweenworkingspelldurationandreferencegroup Iwanttoshow: 2 j = ˚ f : g [ ˙ ˘ ( x ) 2 ˙ [ ˙ ˝ (0) ˙ ˝ ( t )]( @ 2 WW 0 @t ˙ 1 2 ˙ 3 2 @˙ @t 1 WW 0 )+ @ 2 [ : ] @t@ ˙ 1 2 2 WW 0 ] > 0 : Thealgebraisequivalenttotheprevioussubsection,justwiththeoppositesign.Thus, thescaledcoe˚cientoftheinteractionbetweenworkingspelldurationandexperienceis givenbyfollowing. 2 j = ˙ ˘ ( x ) 2 ˙ ˙ 1 2 2 WW 0 @˙ ˝ ( t )) @t f [ ˙ ˝ (0) ˙ ˝ ( t )][( ˙ ˘ ( x )+ ˙ ) W 1 + ˙ 1 4 W 2 W 0 2 ˙ 4 ˙ ˘ ( x ) 3 ( ˙ ˝ (0) ˙ ˝ ( t ))( 1 W 0 ( ˙ ˘ ( x )+ ˙ )( ˙ ˝ (0) ˙ ˝ ( t ))+1) + 4 W 0 2 h 1 W 0 ( ˙ ˘ ( x )+ ˙ ) ˙ 2 ˙ ˘ ( x ) 2 ˙ ˝ ( t ) ˙ 2 ˙ ˘ ( x ) 2 i ]+1 g Underthesameassumptionsmadepreviously,thistermispositivemeaningthatas workingspelldurationincreases,theselectionofmobileworkersonthebasisoftheirreference groupshouldbecomestronger(morepositive). Job-to-Jobdynamicswithrespecttoexperience Thescaledcoe˚cientontheinteractionofindividualabilitywithexperience, 1 j ,isgiven below: 180 1 j = ˚ f : g @ 2 f : g @x@ = ˚ f : g " @ [ : ] @ ( 1 2 ) ˙ 3 2 @˙ @x 2 WW 0 + @ 2 WW 0 @x ˙ 1 2 ! + @ 2 [ : ] @x@ ˙ 1 2 2 WW 0 # ; where ˚ f : g standsfor ˚ ˆ 2 ˙ ˘ ( x ) ˙ WW 0 p ˙ jj ˙ ˘ ( x )[ ˙ ˝ (0) ˙ ˝ ( t )]( m ) ˙ ; and [ : ] standsfor ˙ ˘ ( x ) 2 ˙ [ ˙ ˝ (0) ˙ ˝ ( t )]( m ) . Thescaledcoe˚cientontheinteractionofaveragereferencegroupabilitywithexperience, 2 j ,isgivenbelow: 2 j = ˚ f : g @ 2 f : g @x@m = ˚ f : g " @ [ : ] @m ( 1 2 ) ˙ 3 2 @˙ @x 2 WW 0 + @ 2 WW 0 @x ˙ 1 2 ! + @ 2 [ : ] @x@m ˙ 1 2 2 WW 0 # : Inthefollowingsubsections,Itakethederivativeof P ( J ) withrespecttoexperience, andthenderivethesescaledcoe˚cientsfortheinteractionswithabilityandreferencegroup respectively. Thederivativeof P ( J ) withrespecttoexperience Thederivativeof P ( J ) withrespecttoexperienceis: @P ( J ) @x = ˚ f : g " @ [ : ] @x ˙ 1 2 2 WW 0 + @ 2 WW 0 @x ˙ 1 2 [ : ] 1 2 ˙ 3 2 @˙ @x 2 WW 0 [ : ] # ; where ˚ f : g standsfor ˚ ( 2 ˙ ˘ ( x ) ˙ WW 0 q ˙ j ˙ ˘ ( x )[ ˙ ˝ (0) ˙ ˝ ( t )]( m ) ) ; and [ : ] standsfor ˙ ˘ ( x ) 2 ˙ [ ˙ ˝ (0) ˙ ˝ ( t )]( m ) .Thefollowingtakeseachcomponentseparatelyand resolvesanycon˛ictingsigns. 181 @ [ : ] @x ˙ 1 2 2 WW 0 , @ [ : ] @x ˙ 1 2 2 WW 0 = ˙ 1 2 4 WW 0 @˙ ˘ ( x ) @x ˙ ˘ ( x ) ˙ [ ˙ ˝ (0) ˙ ˝ ( t )]( m ) : @˙ ˘ ( x ) @x < 0 and ˙ ˝ (0) >˙ ˝ ( t ) byassumptionsintrinsictopublicandprivateemployer learningrespectively.Therefore,if >m , @ [ : ] @x ˙ 1 2 2 WW 0 > 0 ,sinceallothertermsaresums ofvariances,andpositivebyde˝nition. @ 2 WW 0 @x ˙ 1 2 [ : ] @ 2 WW 0 @x ˙ 1 2 [ : ]=( 2) ˙ 1 2 [ ˙ ˘ ( x ) 2 ˙ ( ˙ ˝ (0) ˙ ˝ ( t ))( m )] @˙ ˝ ( t ) @x h ( ˙ ˝ (0)+2 ˙ ) W 2 W 1 +( ˙ ˝ ( t )+2 ˙ ) W 1 W 2 i : Underthesamelearningassumptions,thesignof @ 2 WW 0 @x ˙ 1 2 [ : ] ,alsodependsuponthe whetherability( )ishigherthanreferencegroupquality( m ),althoughhereitisinthe oppositedirection. @ 2 WW 0 @x ˙ 1 2 : @ [ : ] @x ˙ 1 2 2 WW 0 + @ 2 WW 0 @x ˙ 1 2 [ : ] , Thecon˛ictinthesignofthe˝rsttwotermsisresolvedbelow. @ [ : ] @x ˙ 1 2 2 WW 0 + @ 2 WW 0 @x ˙ 1 2 [ : ]= ˙ 1 2 4 WW 0 @˙ ˘ ( x ) @x ˙ ˘ ( x ) ˙ [ ˙ ˝ (0) ˙ ˝ ( t )]( m ) 2 ˙ 1 2 [ ˙ ˘ ( x ) 2 ˙ ( ˙ ˝ (0) ˙ ˝ ( t ))( m )] @˙ ˘ ( x ) @x ( ˙ ˝ (0)+2 ˙ ) W 2 W 1 +( ˙ ˝ ( t )+2 ˙ ) W 1 W 2 = ˙ 1 2 2 WW 0 @˙ ˘ ( x ) @x ˙ ˘ ( x ) ˙ [ ˙ ˝ (0) ˙ ˝ ( t )]( m ) (2 ˙ ˘ ( x )(( ˙ ˝ (0)+2 ˙ ) W 1 +( ˙ ˝ ( t )+2 ˙ ) W 1 ) 182 = ˙ 1 2 2 W 2 W 0 2 @˙ ˘ ( x ) @x ˙ ˘ ( x ) ˙ [ ˙ ˝ (0) ˙ ˝ ( t )]( m ) (2 WW 0 ˙ ˘ ( x )(( ˙ ˝ (0)+2 ˙ ) W 0 +( ˙ ˝ ( t )+2 ˙ ) W = ˙ 1 2 2 W 2 W 0 2 @˙ ˘ ( x ) @x ˙ ˘ ( x ) ˙ [ ˙ ˝ (0) ˙ ˝ ( t )]( m ) ( 2 ˙ ˝ ( t ) ˙ ˘ ( x ) 2 ˙ ˝ (0)+2 ˙ ˝ ( t ) ˙ ˙ ˝ (0) ˙ ˘ ( x )+4 ˙ ˙ ˘ ( x ) 2 ˙ ˝ (0) +2 ˙ ˝ ( t ) ˙ ˘ ( x ) ˙ ˝ (0) ˙ +2 ˙ ˝ ( t ) ˙ 2 ˙ ˝ (0)+4 ˙ ˘ ( x ) ˙ ˝ (0) ˙ 2 +4 ˙ ˝ ( t ) ˙ ˘ ( x ) 2 ˙ +4 ˙ ˝ ( t ) ˙ 2 ˙ ˘ ( x )+8 ˙ 2 ˙ ˘ ( x ) 2 ˙ ˘ ( x )( ˙ ˝ (0)+2 ˙ )( ˙ ˙ ˝ ( t )+ ˙ ˘ ( x ) ˙ ˝ ( t ) +2 ˙ ˙ ˘ ( x )) ˙ ˘ ( x )( ˙ ˝ ( t )+2 ˙ )( ˙ ˙ ˝ ( t )+ ˙ ˘ ( x ) ˙ ˝ ( t )+2 ˙ ˙ ˘ ( x )) = ˙ 1 2 2 W 2 W 0 2 @˙ ˘ ( x ) @x ˙ ˘ ( x ) ˙ [ ˙ ˝ (0) ˙ ˝ ( t )]( m ) ( 2 ˙ ˝ ( t ) ˙ ˘ ( x ) 2 ˙ ˝ (0)+2 ˙ ˝ ( t ) ˙ ˙ ˝ (0) ˙ ˘ ( x )+4 ˙ ˙ ˘ ( x ) 2 ˙ ˝ (0) +2 ˙ ˝ ( t ) ˙ ˘ ( x ) ˙ ˝ (0) ˙ +2 ˙ ˝ ( t ) ˙ 2 ˙ ˝ (0)+4 ˙ ˘ ( x ) ˙ ˝ (0) ˙ 2 +4 ˙ ˝ ( t ) ˙ ˘ ( x ) 2 ˙ +4 ˙ ˝ ( t ) ˙ 2 ˙ ˘ ( x )+8 ˙ 2 ˙ ˘ ( x ) 2 ( ˙ ˘ ( x ) ˙ ˝ (0) ˙ ˙ ˝ ( t )+ ˙ ˝ (0) ˙ ˘ ( x ) 2 ˙ ˝ ( t ) +2 ˙ ˘ ( x ) 2 ˙ ˝ (0) ˙ ) (2 ˙ ˘ ( x ) ˙ 2 ˙ ˝ ( t )+2 ˙ ˙ ˘ ( x ) 2 ˙ ˝ ( t )+4 ˙ 2 ˙ ˘ ( x ) 2 ) ( ˙ ˘ ( x ) ˙ ˝ (0) ˙ ˙ ˝ ( t )+ ˙ ˝ (0) ˙ ˘ ( x ) 2 ˙ ˝ ( t )+2 ˙ ˘ ( x ) 2 ˙ ˝ ( t ) ˙ ) (2 ˙ ˘ ( x ) ˙ 2 ˙ ˝ (0)+2 ˙ ˙ ˘ ( x ) 2 ˙ ˝ (0)+4 ˙ 2 ˙ ˘ ( x ) 2 )) = ˙ 1 2 2 W 2 W 0 2 @˙ ˘ ( x ) @x ˙ ˘ ( x ) ˙ [ ˙ ˝ (0) ˙ ˝ ( t )]( m ) ( 2 ˙ ˝ ( t ) ˙ ˙ ˝ (0) ˙ ˘ ( x )+2 ˙ ˝ ( t ) ˙ 2 ˙ ˝ (0)+2 ˙ ˘ ( x ) ˙ ˝ (0) ˙ 2 +2 ˙ ˝ ( t ) ˙ 2 ˙ ˘ ( x )) > 0 , if >m .Sincethenegativetermislargerinmagnitudethanthepositiveterm, @ [ : ] @x ˙ 1 2 2 WW 0 + @ 2 WW 0 @x ˙ 1 2 [ : ] > 0 ,undertheaboveassumptions. 183 Ambiguityof ( 1 2 ) ˙ 3 2 @˙ @x 2 WW 0 [ : ] ( 1 2 ) ˙ 3 2 @˙ @x 2 WW 0 [ : ]=[ ˙ ˘ ( x ) 2 ˙ ( ˙ ˝ (0) ˙ ˝ ( t ))( m )] ( 1) WW 0 ˙ 3 2 @˙ ˘ ( x ) @x [ ( 8) W 2 W 0 2 h 1 W ( ˙ ˝ (0)+2 ˙ )+ 1 W 0 ( ˙ ˝ ( t )+2 ˙ ) i ( W 0 2 ˙ 2 ˙ ˘ ( x ) 2 ˙ ˝ (0)+ W 2 ( ˙ 2 ˙ ˘ ( x ) 2 ˙ ˝ ( t )+ ˙ 4 ˙ ˘ ( x ) 3 ( ˙ ˝ (0) ˙ ˝ ( t )) 2 ) + 4 W 2 W 0 2 [2 W 0 ( ˙ ˝ (0)+2 ˙ ) ˙ 2 ˙ ˘ ( x ) 2 ˙ ˝ (0)+2 W 0 2 ˙ 2 ˙ ˘ ( x ) ˙ ˝ (0)+2 W ( ˙ ˝ ( t ) +2 ˙ ) ˙ 2 ˙ ˘ ( x ) 2 ˙ ˝ ( t )+2 W 2 ˙ 2 ˙ ˘ ( x ) ˙ ˝ ( t )+3 ˙ 4 ˙ ˘ ( x ) 2 ( ˙ ˝ (0) ˙ ˝ ( t )) 2 )]] =[ ˙ ˘ ( x ) 2 ˙ ( ˙ ˝ (0) ˙ ˝ ( t ))( m )] ( 1) WW 0 ˙ 3 2 @˙ ˘ ( x ) @x [ 8 W 3 ˙ 3 ˙ ˘ ( x ) ˙ ˝ (0) 2 + 8 W 0 3 ˙ 3 ˙ ˘ ( x ) ˙ ˝ ( t ) 2 + 4 W 3 W 0 3 ˙ 4 ˙ ˘ ( x ) 2 ( ˙ ˝ (0) ˙ ˝ ( t )) 2 [2 ˙ ˝ (0) ˙ ˘ ( x ) ˙ ˝ ( t ) ˙ +3 ˙ ˝ (0) ˙ ˝ ( t ) ˙ 2 +2 ˙ ˘ ( x ) ˙ ˝ (0) ˙ 2 +2 ˙ ˘ ( x ) ˙ ˝ ( t ) ˙ 2 ( ˙ ˝ (0)+2 ˙ )( ˙ ˝ ( t )+2 ˙ ) ˙ ˘ ( x ) 2 ]] Theambiguityinthethirdtermisproducedbythefollowing,whichIwillde˝neas A : A ˙ 3 2 1 W 4 W 0 4 h ˙ ˘ ( x ) 2 ˙ [ ˙ ˝ (0) ˙ ˝ ( t )]( m ) i @˙ ˘ ( x ) @x [8 ˙ 3 ˙ ˘ ( x )( W 0 3 ˙ ˝ (0) 2 + W 3 ˙ ˝ ( t ) 2 )+4 ˙ 5 ˙ ˘ ( x ) 2 ( ˙ ˝ (0) ˙ ˝ ( t )) 2 ( 2 ˙ ˝ ( t ) ˙ ˝ (0) ˙ ˘ ( x )+3 ˙ ˝ ( t ) ˙ ˝ (0) ˙ +2 ˙ ˝ ( t ) ˙ ˙ ˘ ( x )+2 ˙ ˝ (0) ˙ ˙ ˘ ( x ))] : 184 Resolutionofambiguityinthethirdterm @ [ : ] @x ˙ 1 2 2 WW 0 + @ 2 WW 0 @x ˙ 1 2 [ : ]+ A> 0 FromAppendixA, @ [ : ] @x ˙ 1 2 2 WW 0 + @ 2 WW 0 @x ˙ 1 2 [ : ]= ˙ 1 2 2 W 2 W 0 2 @˙ ˘ ( x ) @x [ : ]( 2 ˙ ˝ ( t ) ˙ ˙ ˝ (0) ˙ ˘ ( x )+2 ˙ ˝ ( t ) ˙ 2 ˙ ˝ (0)+2 ˙ ˘ ( x ) ˙ ˝ (0) ˙ 2 +2 ˙ ˝ ( t ) ˙ 2 ˙ ˘ ( x )) : Addingthenegative A tothistermgives: @ [ : ] @x ˙ 1 2 2 WW 0 + @ 2 WW 0 @x ˙ 1 2 [ : ]+ A = ˙ 3 2 4 W 4 W 0 4 @˙ ˘ ( x ) @x [ : ][ ˙ W 2 W 0 2 ( ˙ ˝ ( t ) ˙ ˙ ˝ (0) ˙ ˘ ( x )+ ˙ ˝ ( t ) ˙ 2 ˙ ˝ (0)+ ˙ ˘ ( x ) ˙ ˝ (0) ˙ 2 + ˙ ˝ ( t ) ˙ 2 ˙ ˘ ( x )) [2 ˙ 3 ˙ ˘ ( x ) 2 ( W 0 3 ˙ ˝ (0) 2 + W 3 ˙ ˝ ( t ) 2 )+ ˙ 5 ˙ ˘ ( x ) 3 ( ˙ ˝ (0) ˙ ˝ ( t )) 2 ( 2 ˙ ˝ ( t ) ˙ ˝ (0) ˙ ˘ ( x )+3 ˙ ˝ ( t ) ˙ ˝ (0) ˙ +2 ˙ ˝ ( t ) ˙ ˙ ˘ ( x )+2 ˙ ˝ (0) ˙ ˙ ˘ ( x ))] = ˙ 3 2 4 W 4 W 0 4 @˙ ˘ ( x ) @x ˙ ˘ ( x ) ˙ [ ˙ ˝ (0) ˙ ˝ ( t )]( m ) (2 ˙ ' W 2 W 0 2 + 4( W 0 2 ˙ 2 ˙ ˘ ( x ) 2 ˙ ˝ (0)+ W 2 ˙ 2 ˙ ˘ ( x ) 2 ˙ ˝ ( t )+ ˙ 4 ˙ ˘ ( x ) 3 ( ˙ ˝ (0) ˙ ˝ ( t )) 2 ) (2 ˙ ˝ ( t ) ˙ ˙ ˝ (0) ˙ ˘ ( x )+2 ˙ ˝ ( t ) ˙ 2 ˙ ˝ (0)+2 ˙ ˘ ( x ) ˙ ˝ (0) ˙ 2 +2 ˙ ˝ ( t ) ˙ 2 ˙ ˘ ( x )) [2 ˙ 3 ˙ ˘ ( x ) 2 ( W 0 3 ˙ ˝ (0) 2 + W 3 ˙ ˝ ( t ) 2 )+ ˙ 5 ˙ ˘ ( x ) 3 ( ˙ ˝ (0) ˙ ˝ ( t )) 2 ( 2 ˙ ˝ ( t ) ˙ ˝ (0) ˙ ˘ ( x )+3 ˙ ˝ ( t ) ˙ ˝ (0) ˙ +2 ˙ ˝ ( t ) ˙ ˙ ˘ ( x )+2 ˙ ˝ (0) ˙ ˙ ˘ ( x ))] = ˙ 3 2 4 W 4 W 0 4 @˙ ˘ ( x ) @x ˙ ˘ ( x ) ˙ [ ˙ ˝ (0) ˙ ˝ ( t )]( m ) (2 ˙ ' W 2 W 0 2 +8 W 0 3 ˙ 3 ˙ ˘ ( x ) 2 ˙ ˝ (0) 2 +8 W 0 2 ˙ 4 ˙ ˘ ( x ) 3 ˙ ˝ (0) ˙ ˝ ( t ) +8 W 3 ˙ 3 ˙ ˘ ( x ) 2 ˙ ˝ ( t ) 2 +8 W 2 ˙ 4 ˙ ˘ ( x ) 3 ˙ ˝ (0) ˙ ˝ ( t )+4 ˙ 4 ˙ ˘ ( x ) 3 ( ˙ ˝ (0) ˙ ˝ ( t )) 2 ) (2 ˙ ˝ ( t ) ˙ ˙ ˝ (0) ˙ ˘ ( x )+2 ˙ ˝ ( t ) ˙ 2 ˙ ˝ (0)+2 ˙ ˘ ( x ) ˙ ˝ (0) ˙ 2 +2 ˙ ˝ ( t ) ˙ 2 ˙ ˘ ( x )) 185 [2 ˙ 3 ˙ ˘ ( x ) 2 ( W 0 3 ˙ ˝ (0) 2 + W 3 ˙ ˝ ( t ) 2 )+ ˙ 5 ˙ ˘ ( x ) 3 ( ˙ ˝ (0) ˙ ˝ ( t )) 2 ( 2 ˙ ˝ ( t ) ˙ ˝ (0) ˙ ˘ ( x )+3 ˙ ˝ ( t ) ˙ ˝ (0) ˙ +2 ˙ ˝ ( t ) ˙ ˙ ˘ ( x )+2 ˙ ˝ (0) ˙ ˙ ˘ ( x ))] = ˙ 3 2 4 W 4 W 0 4 @˙ ˘ ( x ) @x ˙ ˘ ( x ) ˙ [ ˙ ˝ (0) ˙ ˝ ( t )]( m ) (2 ˙ ' W 2 W 0 2 +6 W 0 3 ˙ 3 ˙ ˘ ( x ) 2 ˙ ˝ (0) 2 +8 W 0 2 ˙ 4 ˙ ˘ ( x ) 3 ˙ ˝ (0) ˙ ˝ ( t )+6 W 3 ˙ 3 ˙ ˘ ( x ) 2 ˙ ˝ ( t ) 2 +8 W 2 ˙ 4 ˙ ˘ ( x ) 3 ˙ ˝ (0) ˙ ˝ ( t )+ ˙ 4 ˙ ˘ ( x ) 3 ( ˙ ˝ (0) ˙ ˝ ( t )) 2 )(6 ˙ ˝ ( t ) ˙ ˙ ˝ (0) ˙ ˘ ( x ) +5 ˙ ˝ ( t ) ˙ 2 ˙ ˝ (0)+6 ˙ ˘ ( x ) ˙ ˝ (0) ˙ 2 +6 ˙ ˝ ( t ) ˙ 2 ˙ ˘ ( x )) . Interactionbetweenexperienceandability Iusethederivativeof P ( J ) withrespecttoexperiencetosignthescaledcoe˚cientforthe interactionofexperienceandability.Takingthederivativeof @ f : g @x withrespectto gives thefollowing. 1 j = ˚ f : g ˙ 3 2 4 W 4 W 0 4 @˙ ˘ ( x ) @x ˙ ˘ ( x ) ˙ [ ˙ ˝ (0) ˙ ˝ ( t )] (2 ˙ ' W 2 W 0 2 +6 W 0 3 ˙ 3 ˙ ˘ ( x ) 2 ˙ ˝ (0) 2 +8 W 0 2 ˙ 4 ˙ ˘ ( x ) 3 ˙ ˝ (0) ˙ ˝ ( t )+6 W 3 ˙ 3 ˙ ˘ ( x ) 2 ˙ ˝ ( t ) 2 +8 W 2 ˙ 4 ˙ ˘ ( x ) 3 ˙ ˝ (0) ˙ ˝ ( t )+ ˙ 4 ˙ ˘ ( x ) 3 ( ˙ ˝ (0) ˙ ˝ ( t )) 2 )(6 ˙ ˝ ( t ) ˙ ˙ ˝ (0) ˙ ˘ ( x ) +5 ˙ ˝ ( t ) ˙ 2 ˙ ˝ (0)+6 ˙ ˘ ( x ) ˙ ˝ (0) ˙ 2 +6 ˙ ˝ ( t ) ˙ 2 ˙ ˘ ( x )) . Undertheassumptionsthat @˙ ˘ ( x ) @x < 0 ,whichisfundamentaltopubicemployerlearning, and ˙ t ( t ) <˙ ˝ (0) ; whichisfundamentaltoprivateemployerlearning, 1 j > 0 .Theselection onthebasisofabilityweakens(becomesmorepositive)withincreasesinexperience. 186 Interactionbetweenexperienceandreferencegroup Ialsousethederivativeof P ( J ) withrespecttoexperiencetosignthescaledcoe˚cientfor theinteractionofexperienceandreferencegroupability.Takingthederivativeof @ f : g @x with respectto m givesthefollowing. 2 j = ˚ f : g ˙ 3 2 4 W 4 W 0 4 @˙ ˘ ( x ) @x ˙ ˘ ( x ) ˙ [ ˙ ˝ (0) ˙ ˝ ( t )] (2 ˙ ' W 2 W 0 2 +6 W 0 3 ˙ 3 ˙ ˘ ( x ) 2 ˙ ˝ (0) 2 +8 W 0 2 ˙ 4 ˙ ˘ ( x ) 3 ˙ ˝ (0) ˙ ˝ ( t )+6 W 3 ˙ 3 ˙ ˘ ( x ) 2 ˙ ˝ ( t ) 2 +8 W 2 ˙ 4 ˙ ˘ ( x ) 3 ˙ ˝ (0) ˙ ˝ ( t )+ ˙ 4 ˙ ˘ ( x ) 3 ( ˙ ˝ (0) ˙ ˝ ( t )) 2 )(6 ˙ ˝ ( t ) ˙ ˙ ˝ (0) ˙ ˘ ( x ) +5 ˙ ˝ ( t ) ˙ 2 ˙ ˝ (0)+6 ˙ ˘ ( x ) ˙ ˝ (0) ˙ 2 +6 ˙ ˝ ( t ) ˙ 2 ˙ ˘ ( x )) . Underthesameassumptions, 2 j < 0 ,whichmeansthattheselectiononthebasisof referencegroupalsoweakens,asinthiscase,theselectione˙ectsbecomemorenegativewith increasesinexperience. ExpectedPro˝ts E [ ˇ j R x ;P r ;P h ]= E [ j R x ;P r ;P h ] w . Forworkerswiththeretaining˝rm, w = E [ j R x ;P r = P h ;P h ] and E [ ˇ j R x ;P r ;P h ]= E [ j R x ;P r ;P h ] E [ j R x ;P r = P h ;P h ] E [ ˇ j R x ;P r ;P h ]= ˙ ˝ (0) ˙ ˝ ( t ) ˙ ˘ ( x ) Q m + ˙ ˝ ( t ) ˙ ˘ ( x ) ˙ Q P h + ˙ ˝ (0) ˙ ˘ ( x ) ˙ Q P r + ˙ ˝ ( t ) ˙ ˝ (0) ˙ Q R x 187 + E ( ' r ) ˙ ˝ (0) ˙ ˘ ( x ) Q 0 m + ˙ ˝ (0) ˙ Q 0 R x + 2 ˙ ˙ ˘ ( x ) Q 0 P h + E ( ' h ) where Q = ˙ ˝ (0) ˙ ˝ ( t ) ˙ ˘ ( x )+ ˙ ˝ ( t ) ˙ ˘ ( x ) ˙ + ˙ ˝ (0) ˙ ˘ ( x ) ˙ + ˙ ˝ ( t ) ˙ ˝ (0) ˙ and Q 0 = ˙ ˝ (0) ˙ ˘ ( x )+ ˙ ˝ (0) ˙ +2 ˙ ˙ ˘ ( x ) .Substitutinginthede˝nitionofeachsignal givesthefollowing: = ˙ ˝ (0) ˙ ˝ ( t ) ˙ ˘ ( x ) Q m + ˙ ˝ ( t ) ˙ ˘ ( x ) ˙ Q ( + v )+ ˙ ˝ (0) ˙ ˘ ( x ) ˙ Q ( + ˝ )+ ˙ ˝ ( t ) ˙ ˝ (0) ˙ Q ( + ˘ ) + E ( ' r ) ˙ ˝ (0) ˙ ˘ ( x ) Q 0 m + ˙ ˝ (0) ˙ Q 0 ( + ˘ )+ 2 ˙ ˙ ˘ ( x ) Q 0 ( + v )+ E ( ' h ) : Thissimpli˝estoanexpressionsimilartothatshownfortheprobabilityofjob-to-job moves. = ˙ ˘ ( x ) ˙ ˙ ˝ (0) Q 0 Q [ ˙ ˘ ( x )( ˙ ˝ (0) ˙ ˝ ( t ))( m )]+ 1 Q 0 Q [ Q 0 ˙ ( ˙ ˝ (0) ˙ ˘ ( x ) ˝ r + ˙ ˝ ( t ) ˙ ˘ ( x ) ˝ h + ˙ ˝ ( t ) ˙ ˝ (0) ˘ ) Q˙ (2 ˙ ˘ ( x ) ˝ h + ˙ ˝ (0) ˘ )]+ E ( ' r ' h ) Notethatinexpectation ˝ h = ˝ r = ˘ = ' =0 ,theexpressionabovesimpli˝esto = ˙ ˘ ( x ) ˙ ˙ ˝ (0) Q 0 Q ˙ ˘ ( x )( ˙ ˝ (0) ˙ ˝ ( t ))( m ) > 0 188 Layo˙selectionwithrespecttoabilityandreferencegroup quality Undernominalwagerigidity,theprobabilityofalayo˙isgivenby: P ( L )= P [ E [ ˇ j R x ;P r ;P h ;' ] < 0] P ( L )= P ˆ L' > ˙ ˘ ( x ) ˙ Q 0 Q ˙ ˘ ( x ) ˙ ˝ (0)( ˙ ˝ (0) ˙ ˝ ( t ))( m ) ˙ where L' = 1 Q 0 Q [( Q 0 ˙ ˝ (0) ˝ r Q˝ h + ˙ ˙ ˝ h ( ˙ ˝ ( t ) ˙ ˝ (0)) ˘ ]+ ' r E ( ' h ) , Q = ˙ ˝ (0) ˙ ˝ ( t ) ˙ ˘ ( x )+ ˙ ˝ ( t ) ˙ ˘ ( x ) ˙ + ˙ ˝ (0) ˙ ˘ ( x ) ˙ + ˙ ˝ ( t ) ˙ ˝ (0) ˙ , and Q 0 = ˙ ˝ (0) ˙ ˘ ( x )+ ˙ ˝ (0) ˙ +2 ˙ ˙ ˘ ( x ) . Imposingthenormalandorthogonalityassumptionsprovide: P ( L )= ( ˙ ˘ ( x ) ˙ Q 0 Q p ˙ L' ˙ ˘ ( x ) ˙ ˝ (0)( ˙ ˝ (0) ˙ ˝ ( t ))( m ) ) where ˙ L' = Q 2 Q 2 [ Q 0 2 ( ˙ ˝ (0) 2 ˙ ˝ ( t )+ Q 2 ˙ ˝ (0)+ ˙ 2 ˙ 2 ˝ h ( ˙ ˝ ( t ) ˙ ˝ (0)) 2 ˙ ˘ ( x )]+ ˙ ' + ˙ E' Takingthederivativewithrespecttoability( )givesthefollowing: @P ( L ) @ = ˚ ( ˙ ˘ ( x ) ˙ Q 0 Q p ˙ L' ˙ ˘ ( x ) ˙ ˝ (0)( ˙ ˝ (0) ˙ ˝ ( t ))( m ) ) 189 ˙ ˘ ( x ) ˙ Q 0 Q p ˙ L' ˙ ˘ ( x ) ˙ ˝ (0)( ˙ ˝ (0) ˙ ˝ ( t )) < 0 : Theequationaboveillustratesthatasability( )increasestheprobabilityoflayo˙should fall,aslongas ˙ ˝ (0) >˙ ˝ ( t ) .Takingthederivativewithrespecttoreferencegroupquality ( m )provides: @P ( L ) @m = ˚ ( ˙ ˘ ( x ) ˙ Q 0 Q p ˙ L ' ˙ ˘ ( x ) ˙ ˝ (0)( ˙ ˝ (0) ˙ ˝ ( t ))( m ) ) ˙ ˘ ( x ) ˙ Q 0 Q p ˙ L' ˙ ˘ ( x ) ˙ ˝ (0)( ˙ ˝ (0) ˙ ˝ ( t )) > 0 : Undertheassumptionthat ˙ ˝ (0) >˙ ˝ ( t ) ,asmeanreferencegroupincreasestheprobability oflayo˙increasesaswell @P ( L ) @m > 0 . Layo˙dynamicswithrespecttoworkingspellduration Again,Iuseinteractionsofworkingspelldurationandexperiencewithabilityandreference grouptoexploretheevolutionoftheseselectione˙ectsovertime.I˝rsttakethederivative of P ( L ) andthenusethisto˝ndandsignthescaledcoe˚cientontheseinteractions. Thederivativeof P ( L ) withrespecttoworkingspelllengthisgivenbythefollowing: @P ( L ) @t = ˚ f : g 2 4 @ 1 QQ 0 @t ( ˙ L' ) 1 2 [ : ]+( 1 2 )( ˙ L' ) 3 2 @˙ L' @t 1 QQ 0 [ : ]+ @ [ : ] @t ( ˙ L' ) 1 2 1 QQ 0 3 5 ; 190 where ˚ f : g standsfor ˚ ( ˙ ˘ ( x ) ˙ Q 0 Q q ˙ L' ˙ ˘ ( x ) ˙ ˝ (0)( ˙ ˝ (0) ˙ ˝ ( t ))( m ) ) ,and [ : ] stands for ˙ ˘ ( x ) ˙ ˝ (0)( ˙ ˝ (0) ˙ ˝ ( t ))( m ) .Below,Iexamineeachtermseparatelytomake thealgebramoremanageable. Firstterm: @ 1 QQ 0 @t ( ˙ L' ) 1 2 [ : ]=( 1)( ˙ L' ) 1 2 [ ˙ ˘ ( x ) 2 ˙ ˙ ˝ (0)( ˙ ˝ (0) ˙ ˝ ( t ))( m )] @˙ ˝ ( t )) @t h ( ˙ ˝ (0) ˙ ˘ ( x )+ ˙ ˘ ( x ) ˙ + ˙ ˘ ( x ) ˙ ) Q 0 1 Q 2 i Turningtothesecondterm,recallthat: ˙ L' = Q 2 Q 2 ( Q 0 2 ˙ ˝ (0) 2 ˙ ˝ ( t )+ Q 2 ˙ ˝ (0)+ ˙ 2 ˙ ˝ (0) 2 ( ˙ ˝ ( t ) ˙ ˝ (0)) 2 ˙ ˘ ( x ))+ ˙ ' + ˙ E' : Q = ˙ ˝ (0) ˙ ˝ ( t ) ˙ ˘ ( x )+ ˙ ˝ ( t ) ˙ ˘ ( x ) ˙ + ˙ ˝ (0) ˙ ˘ ( x ) ˙ + ˙ ˝ ( t ) ˙ ˝ (0) ˙ ; Q 0 = ˙ ˝ (0) ˙ ˘ ( x )+ ˙ ˝ (0) ˙ +2 ˙ ˙ ˘ ( x ) : Thus, ( 1 2 )( ˙ L' ) 3 2 @˙ L' @t 1 QQ 0 [ : ]=( 1 2 )[ ˙ ˘ ( x ) 2 ˙ ˙ ˝ (0)( ˙ ˝ (0) ˙ ˝ ( t ))( m )] 1 QQ 0 ˙ 3 2 @˙ ˝ ( t )) @t [( 1) Q 3 [2( ˙ ˝ (0) ˙ ˘ ( x )+ ˙ ˘ ( x ) ˙ + ˙ ˘ ( x ) ˙ ) ˙ ˝ (0) 2 ˙ ˝ ( t ) Q˙ ˝ (0) 2 ] + 2 ˙ 2 ˙ ˝ (0) 2 ( ˙ ˝ ( t ) ˙ ˝ (0)) ˙ ˘ ( x ) Q 2 Q 0 2 [ Q 1 ( ˙ ˝ (0) ˙ ˘ ( x )+ ˙ ˘ ( x ) ˙ + ˙ ˘ ( x ) ˙ )+( ˙ ˝ (0) ˙ ˝ ( t ))]] 191 Thirdterm: @ [ : ] @t ˙ 1 2 L' 1 QQ 0 =( 1)( ˙ L' ) 1 2 1 QQ 0 @˙ ˝ ( t )) @t h ˙ ˘ ( x ) 2 ˙ ˙ ˝ (0)( m ) i : Recombiningtheterms: @P ( L ) @t = ˚ f : g ˙ 1 2 L' ˙ ˘ ( x ) 2 ˙ ˙ ˝ (0)( m ) QQ 0 @˙ ˝ ( t )) @t [1+( ˙ ˝ (0) ˙ ˝ ( t )) [( ˙ ˝ (0) ˙ ˘ ( x )+ ˙ ˘ ( x ) ˙ + ˙ ˝ (0) ˙ ) Q 1 ] + ˙ 1 [( 1) Q 3 [2( ˙ ˝ (0) ˙ ˘ ( x )+ ˙ ˝ (0) ˙ + ˙ ˘ ( x ) ˙ ) ˙ ˝ (0) 2 ˙ ˝ ( t ) Q˙ ˝ (0) 2 ] + 2 ˙ 2 ˙ ˝ (0) 2 ( ˙ ˝ ( t ) ˙ ˝ (0)) ˙ ˘ ( x ) Q 2 Q 0 2 [ Q 1 ( ˙ ˝ (0) ˙ ˘ ( x )+ ˙ ˝ (0) ˙ + ˙ ˘ ( x ) ˙ )+( ˙ ˝ (0) ˙ ˝ ( t ))]] Interactionbetweenworkingspelldurationandability Thescaledcoe˚cientoftheinteractionbetweenworkingspelllengthandability( 1 L )on theprobabilityoflayo˙isgivenbelow. 1 L = ˚ f : g ˙ 1 2 L' ˙ ˘ ( x ) 2 ˙ ˙ ˝ (0) QQ 0 @˙ ˝ ( t )) @t [1+( ˙ ˝ (0) ˙ ˝ ( t )) [( ˙ ˝ (0) ˙ ˘ ( x )+ ˙ ˝ (0) ˙ + ˙ ˘ ( x ) ˙ ) Q 1 ] + ˙ 1 [( 1) Q 3 [2( ˙ ˝ (0) ˙ ˘ ( x )+ ˙ ˝ (0) ˙ + ˙ ˘ ( x ) ˙ ) ˙ ˝ (0) 2 ˙ ˝ ( t ) Q˙ ˝ (0) 2 ] + 2 ˙ 2 ˙ ˝ (0) 2 ( ˙ ˝ ( t ) ˙ ˝ (0)) ˙ ˘ ( x ) Q 2 Q 0 2 [ Q 1 ( ˙ ˝ (0) ˙ ˝ (0)+ ˙ ˘ ( x ) ˙ + ˙ ˘ ( x ) ˙ )+( ˙ ˝ (0) ˙ ˝ ( t ))]] 192 1 L < 0 followsfromtheassumptionthat @˙ ˝ ( t )) @t < 0 withtheadditionalsu˚cientcondi- tionalthat ˙ ˝ (0) ˙ ˝ ( t ) ˙ ˘ ( x )+ ˙ ˝ (0) ˙ ˝ ( t ) ˙ + ˙ ˝ ( t ) ˙ ˘ ( x ) ˙ >˙ ˝ (0) ˙ ˘ ( x ) ˙ Interactionbetweenworkingspelldurationandreferencegroup 2 L = ˚ f : g ˙ 1 2 L' ˙ ˘ ( x ) 2 ˙ ˙ ˝ (0) QQ 0 @˙ ˝ ( t )) @t [1+( ˙ ˝ (0) ˙ ˝ ( t )) [( ˙ ˝ (0) ˙ ˘ ( x )+ ˙ ˝ (0) ˙ + ˙ ˘ ( x ) ˙ ) Q 1 ] + ˙ 1 [( 1) Q 3 [2( ˙ ˝ (0) ˙ ˘ ( x )+ ˙ ˝ (0) ˙ + ˙ ˘ ( x ) ˙ ) ˙ ˝ (0) 2 ˙ ˝ ( t ) Q˙ ˝ (0) 2 ] + 2 ˙ 2 ˙ ˝ (0) 2 ( ˙ ˝ ( t ) ˙ ˝ (0)) ˙ ˘ ( x ) Q 2 Q 0 2 [ Q 1 ( ˙ ˝ (0) ˙ ˝ (0)+ ˙ ˘ ( x ) ˙ + ˙ ˘ ( x ) ˙ )+( ˙ ˝ (0) ˙ ˝ ( t ))]] 2 L > 0 followsfromthesameassumptions. Layo˙dynamicswithrespecttoexperience Itakethederivativeof P ( L ) withrespecttoexperienceinorderto˝ndthescaledcoe˚cient ontheinteractionbetweenexperienceandandabilityandreferencegroupfortheprobability oflayo˙. Thederivativeof P ( L ) withrespecttoexperienceisgivenbythefollowing: @P ( L ) @x = ˚ f : g 2 4 @ [ : ] @x ( ˙ L' ) 1 2 1 QQ 0 + @ 1 QQ 0 @x ( ˙ L' ) 1 2 [ : ]+( 1 2 )( ˙ L' ) 3 2 @˙ L' @x 1 QQ 0 [ : ] 3 5 ; where ˚ f : g standsfor ˚ ( ˙ ˘ ( x ) ˙ Q 0 Q q ˙ L' ˙ ˘ ( x ) ˙ ˝ (0)( ˙ ˝ (0) ˙ ˝ ( t ))( m ) ) (andispositive 193 byde˝nitionofnormalPDF),and [ : ] standsfor ˙ ˘ ( x ) 2 ˙ ˙ ˝ (0)( ˙ ˝ (0) ˙ ˝ ( t ))( m ) (andundertheasumption ˙ ˝ (0) >˙ ˝ ( t ) ,isnegativeif >m ).Thefollowingtakeseach componentseparatelyandresolvesanycon˛ictingsigns. @ [ : ] @x ( ˙ L' ) 1 2 1 QQ 0 @ [ : ] @x ( ˙ L' ) 1 2 1 QQ 0 =( ˙ L' ) 1 2 1 QQ 0 @˙ ˘ ( x ) @x 2 ˙ ˘ ( x ) ˙ ˙ ˝ (0)( ˙ ˝ (0) ˙ ˝ ( t ))( m ) @ 1 QQ 0 @x ( ˙ L' ) 1 2 [ : ] Recall Q = ˙ ˝ (0) ˙ ˝ ( t ) ˙ ˘ ( x )+ ˙ ˝ ( t ) ˙ ˘ ( x ) ˙ + ˙ ˝ (0) ˙ ˘ ( x ) ˙ + ˙ ˝ ( t ) ˙ ˝ (0) ˙ and Q 0 = ˙ ˝ (0) ˙ ˘ ( x )+ ˙ ˝ (0) ˙ +2 ˙ ˙ ˘ ( x ) @ 1 QQ 0 @x ( ˙ L' ) 1 2 [ : ]=( 1) 1 QQ 0 ( ˙ L' ) 1 2 [ : ] @˙ ˘ ( x ) @x h Q 1 ( ˙ ˝ (0) ˙ ˝ ( t )+ ˙ ˝ ( t ) ˙ + ˙ ˝ (0) ˙ )+ Q 1 ( ˙ ˝ (0)+2 ˙ ) i Resolvingcon˛ictbetween8.7.2and8.7.1 @ [ : ] @x ( ˙ L' ) 1 2 1 QQ 0 + @ 1 QQ 0 @x ( ˙ L' ) 1 2 [ : ]=( ˙ L' ) 1 2 1 QQ 0 @˙ ˘ ( x ) @x [ : ] h 2 ˙ ˘ ( x ) Q 1 ( ˙ ˝ (0) ˙ ˝ ( t )+ ˙ ˝ ( t ) ˙ + ˙ ˝ (0) ˙ )+ Q 1 ( ˙ ˝ (0)+2 ˙ ) =( ˙ L' ) 1 2 1 QQ 0 @˙ ˘ ( x ) @x [ : ] ˙ ˘ ( x ) 1 [ Q 1 ( Q ˙ ˘ ( x )( ˙ ˝ (0) ˙ ˝ ( t )+ ˙ ˝ ( t ) ˙ + ˙ ˝ (0) ˙ )+ Q 1 ( Q 0 ˙ ˘ ( x )(( ˙ ˝ (0)+2 ˙ ))] 194 =( ˙ L' ) 1 2 1 QQ 0 @˙ ˘ ( x ) @x [ : ] ˙ ˘ ( x ) 1 [ Q 1 ˙ ˝ ( t ) ˙ ˝ (0) ˙ + Q 1 ˙ ˝ (0) ˙ ] ( 1 2 )( ˙ L' ) 3 2 @˙ L' @x 1 QQ 0 [ : ] isambiguous Recall ˙ L' = Q 2 Q 2 ( Q 0 2 ˙ ˝ (0) 2 ˙ t ( t )+( ˙ t ( t ) ˙ ˘ ( x )+ ˙ t ( t ) ˙ +2 ˙ ˙ ˘ ( x )) 2 ˙ ˝ (0) 3 + ˙ 2 ˙ ˝ (0) 2 ( ˙ t ( t ) ˙ ˝ (0)) 2 ˙ ˘ ( x ))+ ˙ ' + ˙ E' ,and Q = ˙ ˝ (0) ˙ t ( t ) ˙ ˘ ( x )+ ˙ t ( t ) ˙ ˘ ( x ) ˙ + ˙ ˝ (0) ˙ ˘ ( x ) ˙ + ˙ t ( t ) ˙ ˝ (0) ˙ and Q 0 = ˙ ˝ (0) ˙ ˘ ( x )+ ˙ ˝ (0) ˙ +2 ˙ ˙ ˘ ( x ) ( 1 2 )( ˙ L' ) 3 2 @˙ L' @x 1 QQ 0 [ : ]=( 1 2 )[ ˙ ˘ ( x ) 2 ˙ ˙ ˝ (0)( ˙ t (0) ˙ t ( t ))( m )] 1 QQ 0 3 ˙ 3 2 [2 @ 1 QQ 0 @x ( Q 0 2 ˙ ˝ (0) 2 ˙ t ( t )+( ˙ t ( t ) ˙ ˘ ( x )+ ˙ t ( t ) ˙ +2 ˙ ˙ ˘ ( x )) 2 ˙ ˝ (0) 3 + ˙ 2 ˙ ˝ (0) 2 ( ˙ t ( t ) ˙ ˝ (0)) 2 ˙ ˘ ( x )) +(2 Q 0 ( ˙ ˝ (0)+2 ˙ ) ˙ ˝ (0) 2 ˙ t ( t )+2( ˙ t ( t )+2 ˙ )( ˙ t ( t ) ˙ ˘ ( x )+ ˙ t ( t ) ˙ +2 ˙ ˙ ˘ ( x )) ˙ ˝ (0) 3 + ˙ 2 ˙ ˝ (0) 2 ( ˙ t ( t ) ˙ ˝ (0)) 2 )] =( 1 2 )[ ˙ ˘ ( x ) 2 ˙ ˙ ˝ (0)( ˙ t (0) ˙ t ( t ))( m )] 1 QQ 0 4 ˙ 3 2 @˙ ˘ ( x ) @x [ 2 Q 0 ( ˙ ˝ (0) ˙ t ( t )+ ˙ t ( t ) ˙ + ˙ ˝ (0) ˙ )+ Q ( ˙ ˝ (0)+2 ˙ ) [ Q 0 2 ˙ ˝ (0) 2 ˙ t ( t )+( ˙ t ( t ) ˙ ˘ ( x )+ ˙ t ( t ) ˙ +2 ˙ ˙ ˘ ( x )) 2 ˙ ˝ (0) 3 + ˙ 2 ˙ ˝ (0) 2 ( ˙ t ( t ) ˙ ˝ (0)) 2 ˙ ˘ ( x ))] + QQ 0 (2 Q 0 ( ˙ ˝ (0)+2 ˙ ) ˙ ˝ (0) 2 ˙ t ( t )+2( ˙ t ( t )+2 ˙ )( ˙ t ( t ) ˙ ˘ ( x )+ ˙ t ( t ) ˙ +2 ˙ ˙ ˘ ( x )) ˙ ˝ (0) 3 + ˙ 2 ˙ ˝ (0) 2 ( ˙ t ( t ) ˙ ˝ (0)) 2 )]] =( 1 2 )[ ˙ ˘ ( x ) 2 ˙ ˙ ˝ (0)( ˙ t (0) ˙ t ( t ))( m )] 1 QQ 0 4 ˙ 3 2 @˙ ˘ ( x ) @x [ 2 Q 0 3 ( ˙ ˝ (0) ˙ t ( t )+ ˙ t ( t ) ˙ + ˙ ˝ (0) ˙ ) ˙ ˝ (0) 2 ˙ t ( t ) 195 2 Q 0 ( ˙ ˝ (0) ˙ t ( t )+ ˙ t ( t ) ˙ + ˙ ˝ (0) ˙ )+ Q ( ˙ ˝ (0)+2 ˙ ) ( ˙ t ( t ) ˙ ˘ ( x )+ ˙ t ( t ) ˙ +2 ˙ ˙ ˘ ( x )) 2 ˙ ˝ (0) 3 + ˙ 2 ˙ ˝ (0) 2 ( ˙ t ( t ) ˙ ˝ (0)) 2 ˙ ˘ ( x )) + QQ 0 (2( ˙ t ( t )+2 ˙ )( ˙ t ( t ) ˙ ˘ ( x )+ ˙ t ( t ) ˙ +2 ˙ ˙ ˘ ( x )) ˙ ˝ (0) 3 + ˙ 2 ˙ ˝ (0) 2 ( ˙ t ( t ) ˙ ˝ (0)) 2 )]] =( 1 2 )[ ˙ ˘ ( x ) 2 ˙ ˙ ˝ (0)( ˙ t (0) ˙ t ( t ))( m )] 1 QQ 0 4 ˙ 3 2 @˙ ˘ ( x ) @x [ 2 Q 0 3 ( ˙ ˝ (0) ˙ t ( t )+ ˙ t ( t ) ˙ + ˙ ˝ (0) ˙ ) ˙ ˝ (0) 2 ˙ t ( t ) +2 ˙ ˝ (0) 3 ( ˙ t ( t ) ˙ ˘ ( x )+ ˙ t ( t ) ˙ +2 ˙ ˙ ˘ ( x )) [ QQ 0 ( ˙ t ( t )+2 ˙ ) Q 0 ( ˙ ˝ (0) ˙ t ( t )+ ˙ t ( t ) ˙ + ˙ ˝ (0) ˙ )+ Q ( ˙ ˝ (0)+2 ˙ ) ( ˙ t ( t ) ˙ ˘ ( x )+ ˙ t ( t ) ˙ +2 ˙ ˙ ˘ ( x ))] + ˙ 2 ˙ ˝ (0) 2 ( ˙ t ( t ) ˙ ˝ (0)) 2 [ QQ 0 2 ˙ ˘ ( x )[ Q 0 ( ˙ ˝ (0) ˙ t ( t )+ ˙ t ( t ) ˙ + ˙ ˝ (0) ˙ )+ Q ( ˙ ˝ (0)+2 ˙ )]]] =( 1 2 )[ ˙ ˘ ( x ) 2 ˙ ˙ ˝ (0)( ˙ t (0) ˙ t ( t ))( m )] 1 QQ 0 4 ˙ 3 2 @˙ ˘ ( x ) @x [ 2 Q 0 3 ( ˙ ˝ (0) ˙ t ( t )+ ˙ t ( t ) ˙ + ˙ ˝ (0) ˙ ) ˙ ˝ (0) 2 ˙ t ( t ) 2 ˙ ˝ (0) 3 ( ˙ t ( t ) ˙ ˘ ( x )+ ˙ t ( t ) ˙ +2 ˙ ˙ ˘ ( x )) [ ˙ t ( t ) ˙ 2 ( ˙ ˝ (0) ˙ ˘ ( x ) ˙ t ( t )+ ˙ ˙ ˝ (0) ˙ t ( t )+2 ˙ ˙ ˘ ( x ) ˙ t ( t )+ ˙ ˝ (0) 2 ˙ ˘ ( x )+ ˙ ˙ ˘ ( x ) ˙ ˝ (0)) + Q ( ˙ ˝ (0)+2 ˙ )( ˙ t ( t ) ˙ ˘ ( x )+ ˙ t ( t ) ˙ +2 ˙ ˙ ˘ ( x ))] + ˙ 2 ˙ ˝ (0) 2 ( ˙ t ( t ) ˙ ˝ (0)) 2 [ ˙ ˝ (0) 2 ˙ 2 ˙ t ( t ) ( Q 0 ˙ ˝ (0) ˙ + ˙ ˝ (0) ˙ ˘ ( x )( ˙ ˝ (0) ˙ t ( t )+ ˙ t ( t ) ˙ + ˙ ˝ (0) ˙ )+2 ˙ ˘ ( x ) Q ( ˙ ˝ (0)+2 ˙ ))]] FromAppendixAabove, ( 1 2 )( ˙ L' ) 3 2 @˙ L' @x 1 QQ 0 [ : ] ambiguous. ThistermproducingtheambiguityIde˝neas B . B ( 1 2 )[ ˙ ˘ ( x ) 2 ˙ ˙ ˝ (0)( ˙ t (0) ˙ t ( t ))( m )] 1 QQ 0 4 ˙ 3 2 @˙ ˘ ( x ) @x ˙ 2 ˙ ˝ (0) 2 ( ˙ t ( t ) ˙ ˝ (0)) 2 ˙ ˝ (0) 2 ˙ 2 ˙ t ( t ) 196 Resolvingambiguityin8.7.4 FromAppendixA, @ [ : ] @x ( ˙ L' ) 1 2 1 QQ 0 + @ 1 QQ 0 @x ( ˙ L' ) 1 2 [ : ]= ( ˙ L' ) 1 2 1 QQ 0 @˙ ˘ ( x ) @x [ : ] ˙ ˘ ( x ) 1 [ Q 1 ˙ t ( t ) ˙ ˝ (0) ˙ + Q 1 ˙ ˝ (0) ˙ ] . Thus, @ [ : ] @x ( ˙ L' ) 1 2 1 QQ 0 + @ 1 QQ 0 @x ( ˙ L' ) 1 2 [ : ]+ B = ( ˙ L' ) 1 2 1 QQ 0 @˙ ˘ ( x ) @x [ : ] f ˙ ˘ ( x ) 1 [ Q 1 ˙ t ( t ) ˙ ˝ (0) ˙ + Q 1 ˙ ˝ (0) ˙ ] +( 1 2 )[ ˙ ˘ ( x ) 2 ˙ ˙ ˝ (0)( ˙ t (0) ˙ t ( t ))( m )] 1 QQ 0 4 ˙ 3 2 @˙ ˘ ( x ) @x ˙ 2 ˙ ˝ (0) 2 ( ˙ t ( t ) ˙ ˝ (0)) 2 ˙ ˝ (0) 2 ˙ 2 ˙ t ( t ) = ˙ 3 2 L' 1 QQ 0 @˙ ˘ ( x ) @x [ : ] f ˙ ˘ ( x ) 1 [ Q 1 ˙ t ( t ) ˙ ˝ (0) ˙ + Q 1 ˙ ˝ (0) ˙ ] +( 1 2 )[ ˙ ˘ ( x ) 2 ˙ ˙ ˝ (0)( ˙ t (0) ˙ t ( t ))( m )] 1 QQ 0 4 ˙ 3 2 @˙ ˘ ( x ) @x ˙ 2 ˙ ˝ (0) 2 ( ˙ t ( t ) ˙ ˝ (0)) 2 ˙ ˝ (0) 2 ˙ 2 ˙ t ( t ) =( ˙ L' ) 3 2 1 Q 4 Q 0 4 @˙ ˘ ( x ) @x [ : ] f [2( Q 0 2 ˙ ˝ (0) 2 ˙ t ( t )+( ˙ t ( t ) ˙ ˘ ( x )+ ˙ t ( t ) ˙ +2 ˙ ˙ ˘ ( x )) 2 ˙ ˝ (0) 3 + ˙ 2 ˙ ˝ (0) 2 ( ˙ t ( t ) ˙ ˝ (0)) 2 ˙ ˘ ( x ))+ ˙ ' + ˙ E' ) ˙ 1 ˘ ( Q 0 ˙ t ( t ) ˙ ˝ (0) ˙ + Q˙ ˝ (0) ˙ ] ˙ 2 ˙ ˝ (0) 2 ( ˙ t ( t ) ˙ ˝ (0)) 2 ˙ ˝ (0) 2 ˙ 2 ˙ t ( t ) =( ˙ L' ) 3 2 1 Q 4 Q 0 4 @˙ ˘ ( x ) @x ˙ ˘ ( x ) 2 ˙ ˙ ˝ (0)( ˙ ˝ (0) ˙ t ( t ))( m ) f [2 Q 0 2 ˙ ˝ (0) 2 ˙ t ( t )+( ˙ t ( t ) ˙ ˘ ( x )+ ˙ t ( t ) ˙ +2 ˙ ˙ ˘ ( x )) 2 ˙ ˝ (0) 3 + ˙ ' + ˙ E' ) + ˙ 2 ˙ ˝ (0) 2 ( ˙ t ( t ) ˙ ˝ (0)) 2 h Q 0 ˙ t ( t ) ˙ ˝ (0) ˙ + Q˙ ˝ (0) ˙ ˙ ˝ (0) 2 ˙ 2 ˙ t ( t ) i =( ˙ L' ) 3 2 1 Q 4 Q 0 4 @˙ ˘ ( x ) @x ˙ ˘ ( x ) 2 ˙ ˙ ˝ (0)( ˙ ˝ (0) ˙ t ( t ))( m ) f [2 Q 0 2 ˙ ˝ (0) 2 ˙ t ( t )+( ˙ t ( t ) ˙ ˘ ( x )+ ˙ t ( t ) ˙ +2 ˙ ˙ ˘ ( x )) 2 ˙ ˝ (0) 3 + ˙ ' + ˙ E' ) 197 + ˙ 2 ˙ ˝ (0) 2 ( ˙ t ( t ) ˙ ˝ (0)) 2 ˙ t ( t ) ˙ ˝ (0) ˙ ( ˙ ˝ (0) ˙ ˘ ( x )+2 ˙ ˙ ˘ ( x ))+ Q˙ ˝ (0) ˙ } Interactionbetweenexperienceandability Itakethederivativeofthe @ f : g @x withrespecttoability( )to˝ndthepredictedevolutionof selectionintomobilitywithincreasesinexperience.Theanalyticalscaledcoe˚cient 1 L is givenbelow. 1 L = ˚ f : g ( ˙ L' ) 3 2 1 Q 4 Q 0 4 @˙ ˘ ( x ) @x ˙ ˘ ( x ) 2 ˙ ˙ ˝ (0)( ˙ ˝ (0) ˙ t ( t )) f [2 Q 0 2 ˙ ˝ (0) 2 ˙ t ( t )+( ˙ t ( t ) ˙ ˘ ( x )+ ˙ t ( t ) ˙ +2 ˙ ˙ ˘ ( x )) 2 ˙ ˝ (0) 3 + ˙ ' + ˙ E' ) + ˙ 2 ˙ ˝ (0) 2 ( ˙ t ( t ) ˙ ˝ (0)) 2 ˙ t ( t ) ˙ ˝ (0) ˙ ( ˙ ˝ (0) ˙ ˘ ( x )+2 ˙ ˙ ˘ ( x ))+ Q˙ ˝ (0) ˙ } Undertheassumptionsthat @˙ ˘ ( x ) @x < 0 and ˙ ˝ ( t ) <˙ ˝ (0) , 1 L > 0 .Thismeansthat thenegativeselectiononthebasisofabilityofmobileworkersdecreaseswithincreasesin experience. Interactionbetweenexperienceandreferencegroup Itakethederivativeofthe @ f : g @x withrespecttoreferencegroup( m )to˝ndthepredictedevo- lutionofselectionintomobilitywithincreasesinexperience.Theanalyticalscaledcoe˚cient 2 L isgivenbelow. L = ˚ f : g ( ˙ L' ) 3 2 1 Q 4 Q 0 4 @˙ ˘ ( x ) @x ˙ ˘ ( x ) 2 ˙ ˙ ˝ (0)( ˙ ˝ (0) ˙ t ( t )) f [2 Q 0 2 ˙ ˝ (0) 2 ˙ t ( t )+( ˙ t ( t ) ˙ ˘ ( x )+ ˙ t ( t ) ˙ +2 ˙ ˙ ˘ ( x )) 2 ˙ ˝ (0) 3 + ˙ ' + ˙ E' ) + ˙ 2 ˙ ˝ (0) 2 ( ˙ t ( t ) ˙ ˝ (0)) 2 ˙ t ( t ) ˙ ˝ (0) ˙ ( ˙ ˝ (0) ˙ ˘ ( x )+2 ˙ ˙ ˘ ( x ))+ Q˙ ˝ (0) ˙ } Underthesameassumptionsasabove, 2 L < 0 .Thismeansthatthepositiveselection onthebasisofabilityofmobileworkersbecomesmorenegativewithincreasesinexperience. 198 AppendixI AdditionsforChapter3 199 ConsistencyofEstimators A.1Random-e˙ectsestimator Assumethat p lim b j = j ,where j isoffullrank.Alawoflargenumbersgives(see Equation10), p lim b RE = + A 1 j E [ V 0 j 1 j ( Z j u j + j )] ,where A j E ( V 0 j 1 j V j ) is assumedtobeoffullrank.ConsistencythereforerequiresthatE [ V 0 j 1 j ( Z j u j + j )]= 0 .Underexogeneity,E [ V 0 j 1 j ( Z j u j + j ) j V j ]= V 0 j 1 j [ Z j E ( u j j V j )+ E ( j j V j )]= 0 , becauseE ( u j j V j )= 0 andE ( j j V j )= 0 fromEquations7and6,respectively.Since E [ V 0 j 1 j ( Z j u j + j ) j V j ]= 0 forall V j ,itfollowsthatE [ V 0 j 1 j ( Z j u j + j )]= 0 ,so p lim b RE = . TheconsistencyresultalsoholdsforMLandREMLbecauseforboth,theestimatesofthe regressioncoe˚cientscanbeobtainedbysubstitutingthecorrespondingcovariancematrix estimateintotheFGLSestimator.Consistencyof b RE doesnotrequirethat j = j ,so theestimatorisconsistentevenifthecovariancestructureismisspeci˝ed. A.2Fixed-e˙ectsestimator FromEquation11andalawoflargenumbers, p lim b FE = + A 1 j E [ X 0 j ( Z j u j + j )] , where A j E ( X 0 j X j ) isassumedtobeoffullrank.Therequirementforconsistencyis thatE [ X 0 j ( Z j u j + j )]= 0 .Fromunit-levelexogeneity(Equation6),E ( X 0 j j )= 0 ,sothe remainingrequirementisE ( X 0 j Z j u j )= 0 . 200 Augmented˝xed-e˙ectsestimator FromEquation14andalawoflargenumbers, p lim b = + A 1 j E [ W 0 j ( Z j u j + j )] ,because p lim b FE = undertheuncorrelatedvarianceassumptioninEquation12.Here A j E W 0 j W j isassumedtobeoffullranksothattherequiredexogeneityassumptionfor p lim b = becomesE [ W 0 j ( Z j u j + j )]= 0 .Itfollowsfromunit-levelexogeneity(Equation6) thatE ( W 0 j j )= 0 ,sotheremainingrequirementisthatE ( W 0 j Z j u j )= 0 .Asu˚cient conditionforthisremainingrequirementisthatE ( u j j W j )= 0 ,theassumptionthatcluster- levelcovariatesarecluster-levelexogenousthatisstatedinEquation8. Per-clusterregressionestimation ForStep1, p lim b 3CML = 3 Verbekeetal.(2001).ForStep2,fromEquation18andalaw oflargenumbers, p lim j = j + A 1 j E n z ij [ x 0 3 ij ( 3 b 3CML )+ z 0 ij j + ij ] o .Assume that A j E ( z ij z 0 ij ) isoffullrank.E [ z ij x 0 3 ij ( 3 b 3CML )]= 0 since p lim b 3CML = 3 and E ( z ij ij )= 0 underunit-levelexogeneity.Theremainingrequirementforconsistencyisthat E ( z ij z 0 ij j )= 0 .ForStep3, p lim b r = r + A 1 j E w rj ( u rj + rj rj ) ,fromEquation19 andalawoflargenumbers.Assumethat A j E w rj w 0 rj isoffullrank.Itfollowsfrom cluster-levelexogeneityofthecluster-levelcovariates(Equation8)thatE [ w rj u rj ]= 0 . Moreover,E ( w rj ( rj rj ))= 0 iftheestimationerrorsareuncorrelatedwith w rj orif n j !1 . 201 StataCodeforHSBExample BelowweprovidetheStata13StataCorp(2013)codeweusedtoproducetheresultsreported inTableD.1fortheHighSchoolandBeyond(HSB)dataset,distributedwiththeHLM softwareRaudenbushetal.(2011).Thedataset, hsb.dta ,canalsobedownloadedfromthe websiteforRabe-HeskethandSkrondal(2012): http://www.stata-press.com/data/mlmus3.html . ******************************************************************************** *InitialDataSetup* ******************************************************************************** usehttp://www.stata-press.com/data/mlmus3/hsb,clear keepschoolidmathachsectorses describe /*Thevariablesofinterestare: sector(wj)=cluster-levelindicatorforwhetherschoolisCatholic ses(xij)=unit-levelcovariatethatindexesstudents'socioeconomicstatus mathach(yij)=outcome,students'performanceonamathtest schoolid=cluster(school)identifier*/ *Defineclusteridentifier xtsetschoolid *Generatecross-levelinteractiontermbetweensesandsector(xij*wj) generatesesXsector=ses*sector 202 ******************************************************************************** *RandomEffects(REML)* ******************************************************************************** /*Estimationisdoneinonestepusingcovariatesthatareunit-level, cluster-level,andinteractionsbetweenthetwo.Note:sesisthecovariate witharandomslope*/ mixedmathachsessectorsesXsector||schoolid:ses,/// covariance(unstructured)reml ******************************************************************************** *AugmentedFixedEffects(FE+)* ******************************************************************************** ***STEP1-FE **Estimatecoefficientsofunit-levelcovariatesusingstandardfixedeffects xtregmathachsessesXsector,fe ****STEP2-Regressquasi-residualsoncluster-levelcovariate **Generate"newy"asresidualsfromthefirststageregression generateynew=mathach-_b[ses]*ses-_b[sesXsector]*sesXsector **Regresstheresidualsonthecluster-levelcovariate,withcluster-robustSEs 203 regressynewsector,vce(clusterschoolid) ******************************************************************************** *Per-ClusterRegression(PC)* ******************************************************************************** ***Step1 **Notneededbecausetherearenounit-levelcovariatesthatdonothave **randomslopes(R3=0) ***Step2 **Foreachcluster,regressoutcomeonunit-levelcovariateusingOLS,saving **estimatesoftheintercepts(a1)andcoefficients(a2)instatsby_HSB.dta statsbya1=_b[_cons]a2=_b[ses],by(schoolid)saving(statsby_HSB,replace):/// regressmathachses **Mergeestimatesintodataset(aftersortingdataaccordingtoschoolid) sortschoolid mergem:1schoolidusingstatsby_HSB ***Step3 /*Parta:Regressinterceptestimates(a1)oncluster-levelcovariate, using1observationpercluster-OLSwithrobustSEs*/ **Createindicatorfor1observationpercluster(itdoesn'tmatterwhichone) 204 egenpickone=tag(schoolid) **OLSfor1observationpercluster,withrobustSEs regressa1sectorifpickone==1,vce(robust) *=>estimatedintercept(_cons)=estimatedinterceptofmodel(gamma0) *=>estimatedcoefficientofsector=estimatecoefficientofsector(gamma1) /*Partb:Regresscoefficientestimates(a2)onthecluster-levelcovariates, using1observationpercluster-OLSwithrobustSEs*/ regressa2sectorifpickone==1,vce(robust) *=>estimatedintercept(_cons)=estimatedcoefficientofses(beta1) *=>estimatedcoefficientofsector=estimatedinteractionparameter(beta2) 205 SupplementalTablesandFigures TableI.1:Comparingmethodsforestimatingthecoe˚cient 1 of x ij ( 1 =1 ). Simulation100 100 100 100 MeanSE ConditionMethodBiasRMSEMeanSESDSD SmallClustersRE16.6*21.612.913.80.93 &UncorrelatedFE+0.616.214.916.20.92 VariancePC1.925.524.725.50.97 SmallClusters&RE21.3*24.212.111.41.06 &CorrelatedFE+11.7*19.114.115.20.93 VariancePC-1.826.726.126.60.98 LargeClusters&RE6.2*10.07.57.80.96 &UncorrelatedFE+-0.38.06.08.00.74 VariancePC-0.28.07.98.00.99 LargeClusters&RE12.6*14.47.07.01.00 &CorrelatedFE+12.8*15.25.58.20.67 VariancePC0.77.78.17.71.05 Note. SmallClusters: n j = n =4 ,LargeClusters: n j = n =20 ; UncorrelatedVariance: ˙ 2 j =1 ,CorrelatedVariance: ˙ j =exp( u 1 j ) . RMSE=root-mean-squareerror; MeanSE=meanofthestandarderrorestimatesoverthereplications; MeanSE=meanofthestandarderrorestimatesoverthereplications; SD=standarddeviationofthecoe˚cientestimatesoverthereplications; Estimatedbiasdi˙erssigni˝cantlyfrom0atthe0.05level. 206 TableI.2:Comparingmethodsforestimatingthecoe˚cient 2 of x ij w j ( 2 =2 ) Simulation100 100 100 100 MeanSE ConditionMethodBiasRMSEMeanSESDSD SmallClustersRE0.8*7.06.57.00.94 &UncorrelatedFE+0.08.27.68.20.93 VariancePC-0.613.312.613.30.95 SmallClustersRE1.2*6.16.15.91.03 &CorrelatedFE+0.27.97.27.90.91 VariancePC0.713.413.313.40.99 LargeClustersRE0.23.93.83.90.97 &UncorrelatedFE+0.24.03.04.00.75 VariancePC0.14.14.04.10.99 LargeClustersRE-0.13.53.63.51.01 &CorrelatedFE+-0.34.22.84.20.68 VariancePC-0.34.04.14.01.04 Note. SmallClusters: n j = n =4 ,LargeClusters: n j = n =20 ; UncorrelatedVariance: ˙ 2 j =1 ,CorrelatedVariance: ˙ j =exp( u 1 j ) . RMSE=root-mean-squareerror; MeanSE=meanofthestandarderrorestimatesoverthereplications; SD=standarddeviationofthecoe˚cientestimatesoverthereplications; Estimatedbiasdi˙erssigni˝cantlyfrom0atthe0.05level. 207 TableI.3:Comparingmethodsforestimatingthecoe˚cient 1 of w j ( 1 =3 ) Simulation100 100 100 100 MeanSE ConditionMethodBiasRMSEMeanSESDSD SmallClustersRE-4.5*8.06.66.61.01 &UncorrelatedFE+-0.37.47.17.40.97 VariancePC-0.512.711.712.70.92 SmallClustersRE-5.3*8.26.56.21.04 &CorrelatedFE+-2.4*7.36.86.90.98 VariancePC0.212.211.212.20.92 LargeClustersRE-1.7*4.94.54.60.99 &UncorrelatedFE+0.25.35.45.31.03 VariancePC0.05.25.25.20.99 LargeClustersRE-2.9*5.24.34.31.00 &CorrelatedFE+-2.5*5.24.94.61.05 VariancePC0.05.05.05.01.00 Note. SmallClusters: n j = n =4 ,LargeClusters: n j = n =20 ; UncorrelatedVariance: ˙ 2 j =1 ,CorrelatedVariance: ˙ j =exp( u 1 j ) . 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