下册答案(伍德里奇计量经济导论第三版课后答案).doc

CHAPTER13131WITHOUTCHANGESINTHEAVERAGESOFANYEXPLANATORYVARIABLES,THEAVERAGEFERTILITYRATEFELLBY545BETWEEN1972AND1984THISISSIMPLYTHECOEFFICIENTONY84TOACCOUNTFORTHEINCREASEINAVERAGEEDUCATIONLEVELS,WEOBTAINANADDITIONALEFFECT128133122141SOTHEDROPINAVERAGEFERTILITYIFTHEAVERAGEEDUCATIONLEVELINCREASEDBY11IS545141686,ORROUGHLYTWOTHIRDSOFACHILDPERWOMAN132THEFIRSTEQUATIONOMITSTHE1981YEARDUMMYVARIABLE,Y81,ANDSODOESNOTALLOWANYAPPRECIATIONINNOMINALHOUSINGPRICESOVERTHETHREEYEARPERIODINTHEABSENCEOFANINCINERATORTHEINTERACTIONTERMINTHISCASEISSIMPLYPICKINGUPTHEFACTTHATEVENHOMESTHATARENEARTHEINCINERATORSITEHAVEAPPRECIATEDINVALUEOVERTHETHREEYEARSTHISEQUATIONSUFFERSFROMOMITTEDVARIABLEBIASTHESECONDEQUATIONOMITSTHEDUMMYVARIABLEFORBEINGNEARTHEINCINERATORSITE,NEARINC,WHICHMEANSITDOESNOTALLOWFORSYSTEMATICDIFFERENCESINHOMESNEARANDFARFROMTHESITEBEFORETHESITEWASBUILTIF,ASSEEMSTOBETHECASE,THEINCINERATORWASLOCATEDCLOSERTOLESSVALUABLEHOMES,THENOMITTINGNEARINCATTRIBUTESLOWERHOUSINGPRICESTOOMUCHTOTHEINCINERATOREFFECTAGAIN,WEHAVEANOMITTEDVARIABLEPROBLEMTHISISWHYEQUATION139OR,EVENBETTER,THEEQUATIONTHATADDSAFULLSETOFCONTROLS,ISPREFERRED133WEDONOTHAVEREPEATEDOBSERVATIONSONTHESAMECROSSSECTIONALUNITSINEACHTIMEPERIOD,ANDSOITMAKESNOSENSETOLOOKFORPAIRSTODIFFERENCEFOREXAMPLE,INEXAMPLE131,ITISVERYUNLIKELYTHATTHESAMEWOMANAPPEARSINMORETHANONEYEAR,ASNEWRANDOMSAMPLESAREOBTAINEDINEACHYEARINEXAMPLE133,SOMEHOUSESMAYAPPEARINTHESAMPLEFORBOTH1978AND1981,BUTTHEOVERLAPISUSUALLYTOOSMALLTODOATRUEPANELDATAANALYSIS134THESIGNOF1DOESNOTAFFECTTHEDIRECTIONOFBIASINTHEOLSESTIMATOROF,BUT1ONLYWHETHERWEUNDERESTIMATEOROVERESTIMATETHEEFFECTOFINTERESTIFWEWRITECRMRTEI01UNEMIUI,WHEREUIANDUNEMIARENEGATIVELYCORRELATED,THENTHEREISADOWNWARDBIASINTHEOLSESTIMATOROF1BECAUSE10,WEWILLTENDTOUNDERESTIMATETHEEFFECTOFUNEMPLOYMENTONCRIME135NO,WECANNOTINCLUDEAGEASANEXPLANATORYVARIABLEINTHEORIGINALMODELEACHPERSONINTHEPANELDATASETISEXACTLYTWOYEARSOLDERONJANUARY31,1992THANONJANUARY31,1990THISMEANSTHATAGEI2FORALLIBUTTHEEQUATIONWEWOULDESTIMATEISOFTHEFORMSAVINGI01AGEI,WHERE0ISTHECOEFFICIENTTHEYEARDUMMYFOR1992INTHEORIGINALMODELASWEKNOW,WHENWEHAVEANINTERCEPTINTHEMODELWECANNOTINCLUDEANEXPLANATORYVARIABLETHATISCONSTANTACROSSITHISVIOLATESASSUMPTIONMLR3INTUITIVELY,SINCEAGECHANGESBYTHESAMEAMOUNTFOREVERYONE,WECANNOTDISTINGUISHTHEEFFECTOFAGEFROMTHEAGGREGATETIMEEFFECT136ILETFLBEABINARYVARIABLEEQUALTOONEIFAPERSONLIVESINFLORIDA,ANDZEROOTHERWISELETY90BEAYEARDUMMYVARIABLEFOR1990THEN,FROMEQUATION1310,WEHAVETHELINEARPROBABILITYMODELARREST00Y901FL1Y90FLUTHEEFFECTOFTHELAWISMEASUREDBY1,WHICHISTHECHANGEINTHEPROBABILITYOFDRUNKDRIVINGARRESTDUETOTHENEWLAWINFLORIDAINCLUDINGY90ALLOWSFORAGGREGATETRENDSINDRUNKDRIVINGARRESTSTHATWOULDAFFECTBOTHSTATESINCLUDINGFLALLOWSFORSYSTEMATICDIFFERENCESBETWEENFLORIDAANDGEORGIAINEITHERDRUNKDRIVINGBEHAVIORORLAWENFORCEMENTIIITCOULDBETHATTHEPOPULATIONSOFDRIVERSINTHETWOSTATESCHANGEINDIFFERENTWAYSOVERTIMEFOREXAMPLE,AGE,RACE,ORGENDERDISTRIBUTIONSMAYHAVECHANGEDTHELEVELSOFEDUCATIONACROSSTHETWOSTATESMAYHAVECHANGEDASTHESEFACTORSMIGHTAFFECTWHETHERSOMEONEISARRESTEDFORDRUNKDRIVING,ITCOULDBEIMPORTANTTOCONTROLFORTHEMATAMINIMUM,THEREISTHEPOSSIBILITYOFOBTAININGAMOREPRECISEESTIMATOROF1BYREDUCINGTHEERRORVARIANCEESSENTIALLY,ANYEXPLANATORYVARIABLETHATAFFECTSARRESTCANBEUSEDFORTHISPURPOSESEESECTION63FORDISCUSSION137IITISNOTSURPRISINGTHATTHECOEFFICIENTONTHEINTERACTIONTERMCHANGESLITTLEWHENAFCHNGEISDROPPEDFROMTHEEQUATIONBECAUSETHECOEFFICIENTONAFCHNGEIN312ISONLY0077ANDITSTSTATISTICISVERYSMALLTHEINCREASEFROM191TO198ISEASILYEXPLAINEDBYSAMPLINGERRORIIIFHIGHEARNISDROPPEDFROMTHEEQUATIONSOTHATIN310,THENWE10AREASSUMINGTHAT,PRIORTOTHECHANGEINPOLICY,THEREISNODIFFERENCEINAVERAGEDURATIONBETWEENHIGHEARNERSANDLOWEARNERSBUTTHEVERYLARGE256,HIGHLYSTATISTICALLYSIGNIFICANTESTIMATEONHIGHEARNIN312SHOWSTHISPRESUMPTIONTOBEFALSEPRIORTOTHEPOLICYCHANGE,THEHIGHEARNINGGROUPSPENTABOUT292LONGERONUNEMPLOYMENTCOMPENSATIONTHANTHELOWEARNINGEXP25619GROUPBYDROPPINGHIGHEARNFROMTHEREGRESSION,WEATTRIBUTETOTHEPOLICYCHANGETHEDIFFERENCEBETWEENTHETWOGROUPSTHATWOULDBEOBSERVEDWITHOUTANYINTERVENTIONCHAPTER14141FIRST,FOREACHT1,VARUITVARUITUI,T1VARUITVARUI,T1,2UWHEREWEUSETHEASSUMPTIONSOFNOSERIALCORRELATIONINUTANDCONSTANTVARIANCENEXT,WEFINDTHECOVARIANCEBETWEENUITANDUI,T1BECAUSETHESEEACHHAVEAZEROMEAN,THECOVARIANCEISEUITUI,T1EUITUI,T1UI,T1UITEUITUI,T1EEUI,T1UI,T1EUI,T1UITEBECAUSEOFTHENOSERIALCORRELATION2IT2IT2ASSUMPTIONBECAUSETHEVARIANCEISCONSTANTACROSST,BYPROBLEM111,CORRUIT,UI,T1COVUIT,UI,T1/VARUIT52/U142ITHEBETWEENESTIMATORISJUSTTHEOLSESTIMATORFROMTHECROSSSECTIONALREGRESSIONOFONINCLUDINGANINTERCEPTBECAUSEWEJUSTHAVEASINGLEIYIXEXPLANATORYVARIABLEANDTHEERRORTERMISAI,WEHAVE,FROMSECTION51,IIUPLIM1COV,AI/VAR1XIIXBUTEAI0SOCOV,AIEAIEAIEEAIIUXIUIIUXBECAUSEECOV,0BYASSUMPTIONNOWEAIIXIX1ETITTXATHEREFORE,PLIM1XA/VAR,1IWHICHISWHATWEWANTEDTOSHOWIIIFXITISSERIALLYUNCORRELATEDWITHCONSTANTVARIANCETHENVAR2XIX/T,ANDSOPLIM1XA/T1TXA/2X2X2XIIIASPARTIISHOWS,WHENTHEXITAREPAIRWISEUNCORRELATEDTHEMAGNITUDEOFTHEINCONSISTENCYACTUALLYINCREASESLINEARLYWITHTTHESIGNDEPENDSONTHECOVARIANCEBETWEENXITANDAI143IEEITEVITEVITE0BECAUSEEVIT0FORALLTIIIIVARVITVARVIT2VAR2COVVIT,2EIIVIVI2EVITNOW,ANDEVITI2EVITAUI1TITSSE12A/TTHEREFORE,EA2AU2A2U2IV/TNOW,WECANCOLLECTTERMS1TITTEV2AUVARVITI2222/AUAUAUTTNOW,ITISCONVENIENTTOWRITE1,WHERE/TAND/T/2U2AUTHENVARVIT2/T2/TI2AUA2UA2U21122AU/2212/2AU/2212/2AU/2222AU2AUUTHISISWHATWEWANTEDTOSHOWIIIWEMUSTSHOWTHATEEITEIS0FORTSNOWEEITEISEVITVISIEVITVISEVISEVIT2E2/T2EIIIIVAA2UI2/T2/TTHERESTOFTHEPROOFISVERYSIMILARTOAA2UAUPARTIIEEITEIS2/T2/TAA2UA2U2112A/2212/A/2212/A/222A0A144IMENSATHLETICSARESTILLTHEMOSTPROMINENT,ALTHOUGHWOMENSSPORTS,ESPECIALLYBASKETBALLBUTALSOGYMNASTICS,SOFTBALL,ANDVOLLEYBALL,AREVERYPOPULARATSOMEUNIVERSITIESWINNINGPERCENTAGESFORFOOTBALLANDMENSANDWOMENSBASKETBALLAREGOODPOSSIBILITIES,ASWELLASINDICATORSFORWHETHERTEAMSWONCONFERENCECHAMPIONSHIPS,WENTTOAVISIBLEBOWLGAMEFOOTBALL,ORDIDWELLINTHENCAABASKETBALLTOURNAMENTSUCHASMAKINGTHESWEET16WEMUSTBESURETHATWEUSEMEASURESOFATHLETICSUCCESSTHATAREAVAILABLEPRIORTOAPPLICATIONDEADLINESSO,WEWOULDPROBABLYUSEFOOTBALLSUCCESSFROMTHEPREVIOUSSCHOOLYEARBASKETBALLSUCCESSMIGHTHAVETOBELAGGEDONEMOREYEARIITUITIONCOULDBEIMPORTANTCETERISPARIBUS,HIGHERTUITIONSHOULDMEANFEWERAPPLICATIONSMEASURESOFUNIVERSITYQUALITYTHATCHANGEOVERTIME,SUCHASSTUDENT/FACULTYRATIOSORFACULTYGRANTMONEY,COULDBEIMPORTANTIIIANUNOBSERVEDEFFECTSMODELISLOGAPPSIT1D90T2D95T1ATHSUCCIT2LOGTUITIONITAIUIT,T1,2,3THEVARIABLEATHSUCCITISSHORTHANDFORAMEASUREOFATHLETICSUCCESSWEMIGHTINCLUDESEVERALMEASURESIF,FOREXAMPLE,ATHSUCCITISFOOTBALLWINNINGPERCENTAGE,THEN1001ISTHEPERCENTAGECHANGEINAPPLICATIONSGIVENAONEPERCENTAGEPOINTINCREASEINWINNINGPERCENTAGEITISLIKELYTHATAIISCORRELATEDWITHATHLETICSUCCESS,TUITION,ANDSOON,SOFIXEDEFFECTSESTIMATIONISAPPROPRIATEALTERNATIVELY,WECOULDFIRSTDIFFERENCETOREMOVEAI,ASDISCUSSEDINCHAPTER13145IFOREACHSTUDENTWEHAVESEVERALMEASURESOFPERFORMANCE,TYPICALLYTHREEORFOUR,THENUMBEROFCLASSESTAKENBYASTUDENTTHATHAVEFINALEXAMSWHENWESPECIFYANEQUATIONFOREACHSTANDARDIZEDFINALEXAMSCORE,THEERRORSINTHEDIFFERENTEQUATIONSFORTHESAMESTUDENTARECERTAINTOBECORRELATEDSTUDENTSWHOHAVEMOREUNOBSERVEDABILITYTENDTODOBETTERONALLTESTSIIANUNOBSERVEDEFFECTSMODELISSCORESCC1ATNDRTESC2MAJORSC3SATS4CUMGPASASUSC,WHEREASISTHEUNOBSERVEDSTUDENTEFFECTBECAUSESATSCOREANDCUMULATIVEGPADEPENDONLYONTHESTUDENT,ANDNOTONTHEPARTICULARCLASSHE/SHEISTAKING,THESEDONOTHAVEACSUBSCRIPTTHEATTENDANCERATESDOGENERALLYVARYACROSSCLASS,ASDOESTHEINDICATORFORWHETHERACLASSISINTHESTUDENTSMAJORTHETERMCDENOTESDIFFERENTINTERCEPTSFORDIFFERENTCLASSESUNLIKEWITHAPANELDATASET,WHERETIMEISTHENATURALORDERINGOFTHEDATAWITHINEACHCROSSSECTIONALUNIT,ANDTHEAGGREGATETIMEEFFECTSAPPLYTOALLUNITS,INTERCEPTSFORTHEDIFFERENTCLASSESMAYNOTBENEEDEDIFALLSTUDENTSTOOKTHESAMESETOFCLASSESTHENTHISISSIMILARTOAPANELDATASET,ANDWEWOULDWANTTOPUTINDIFFERENTCLASSINTERCEPTSBUTWITHSTUDENTSTAKINGDIFFERENTCOURSES,THECLASSWELABELAS“1”FORSTUDENTANEEDHAVENOTHINGTODOWITHCLASS“1”FORSTUDENTBTHUS,THEDIFFERENTCLASSINTERCEPTSBASEDONARBITRARILYORDERINGTHECLASSESFOREACHSTUDENTPROBABLYARENOTNEEDEDWECANREPLACECWITH0,ANINTERCEPTCONSTANTACROSSCLASSESIIIMAINTAININGTHEASSUMPTIONTHATTHEIDIOSYNCRATICERROR,USC,ISUNCORRELATEDWITHALLEXPLANATORYVARIABLES,WENEEDTHEUNOBSERVEDSTUDENTHETEROGENEITY,AS,TOBEUNCORRELATEDWITHATNDRTESCTHEINCLUSIONOFSATSCOREANDCUMULATIVEGPASHOULDHELPINTHISREGARD,ASAS,ISTHEPARTOFABILITYTHATISNOTCAPTUREDBYSATSANDCUMGPASINOTHERWORDS,CONTROLLINGFORSATSANDCUMGPASCOULDBEENOUGHTOOBTAINTHECETERISPARIBUSEFFECTOFCLASSATTENDANCEIVIFSATSANDCUMGPASARENOTSUFFICIENTCONTROLSFORSTUDENTABILITYANDMOTIVATION,ASISCORRELATEDWITHATNDRTESC,ANDTHISWOULDCAUSEPOOLEDOLSTOBEBIASEDANDINCONSISTENTWECOULDUSEFIXEDEFFECTSINSTEADWITHINEACHSTUDENTWECOMPUTETHEDEMEANEDDATA,WHERE,FOREACHSTUDENT,THEMEANSARECOMPUTEDACROSSCLASSESTHEVARIABLESSATSANDCUMGPASDROPOUTOFTHEANALYSIS146ITHEFULLYROBUSTSTANDARDERRORSARELARGERINEACHCASE,ROUGHLYDOUBLEFORTHETIMECONSTANTREGRESSORSEDUC,BLACK,ANDHISPANONTHETIMEVARYINGEXPLANATORYVARIABLESMARRIEDANDUNION,THEFULLYROBUSTSTANDARDERRORSAREROUGHLY60PERCENTLARGERTHEDIFFERENCESARESMALLERFOREXPERANDEXPER2BUTHARDLYTRIVIALWEEXPECTTHISIFWETHINKTHECOMPOSITEERRORTERM,CONTAINSANUNOBSERVEDEFFECT,ITVTHISINDUCESPOSITIVESERIALCORRELATIONAND,ASWESAWINSECTION121FORTIMEIASERIES,THEUSUALOLSSTANDARDERRORSTENDTOUNDERSTATETHEACTUALSAMPLINGVARIATIONINTHEOLSESTIMATESTHESAMEHOLDSTRUEFORPOOLEDOLSWITHPANELDATAIIONTHETIMECONSTANTEXPLANATORYVARIABLESEDUC,BLACK,ANDHISPAN,THERESTANDARDERRORSANDTHEROBUSTSTANDARDERRORSFORPOOLEDOLSAREROUGHLYTHESAMETHECOEFFICIENTESTIMATESAREVERYSIMILAR,TOOTHEMAINDIFFERENCESARISEINTHESTANDARDERRORSANDCOEFFICIENTSONTHETIMEVARYINGEXPLANATORYVARIABLESFOREXAMPLE,THERESTANDARDERRORSONTHEMARRIEDANDUNIONCOEFFICIENTSARE017AND018,RESPECTIVELY,COMPAREDWITHTHEROBUSTSTANDARDERRORSFORPOOLEDOLSOF026AND027WEEXPECTTHISTOBETRUEBECAUSE,UNDERTHEUNDERTHEREASSUMPTIONS,REISMOREEFFICIENTTHANPOOLEDOLSCHAPTER15151IITHASBEENFAIRLYWELLESTABLISHEDTHATSOCIOECONOMICSTATUSAFFECTSSTUDENTPERFORMANCETHEERRORTERMUCONTAINS,AMONGOTHERTHINGS,FAMILYINCOME,WHICHHASAPOSITIVEEFFECTONGPAANDISALSOVERYLIKELYTOBECORRELATEDWITHPCOWNERSHIPIIFAMILIESWITHHIGHERINCOMESCANAFFORDTOBUYCOMPUTERSFORTHEIRCHILDRENTHEREFORE,FAMILYINCOMECERTAINLYSATISFIESTHESECONDREQUIREMENTFORANINSTRUMENTALVARIABLEITISCORRELATEDWITHTHEENDOGENOUSEXPLANATORYVARIABLESEE155WITHXPCANDZFAMINCBUTASWESUGGESTEDINPARTI,FAMINCHASAPOSITIVEAFFECTONGPA,SOTHEFIRSTREQUIREMENTFORAGOODIV,154,FAILSFORFAMINCIFWEHADFAMINCWEWOULDINCLUDEITASANEXPLANATORYVARIABLEINTHEEQUATIONIFITISTHEONLYIMPORTANTOMITTEDVARIABLECORRELATEDWITHPC,WECOULDTHENESTIMATETHEEXPANDEDEQUATIONBYOLSIIITHISISANATURALEXPERIMENTTHATAFFECTSWHETHERORNOTSOMESTUDENTSOWNCOMPUTERSSOMESTUDENTSWHOBUYCOMPUTERSWHENGIVENTHEGRANTWOULDNOTHAVEWITHOUTTHEGRANTSTUDENTSWHODIDNOTRECEIVETHEGRANTSMIGHTSTILLOWNCOMPUTERSDEFINEADUMMYVARIABLE,GRANT,EQUALTOONEIFTHESTUDENTRECEIVEDAGRANT,ANDZEROOTHERWISETHEN,IFGRANTWASRANDOMLYASSIGNED,ITISUNCORRELATEDWITHUINPARTICULAR,ITISUNCORRELATEDWITHFAMILYINCOMEANDOTHERSOCIOECONOMICFACTORSINUFURTHER,GRANTSHOULDBECORRELATEDWITHPCTHEPROBABILITYOFOWNINGAPCSHOULDBESIGNIFICANTLYHIGHERFORSTUDENTRECEIVINGGRANTSINCIDENTALLY,IFTHEUNIVERSITYGAVEGRANTPRIORITYTOLOWINCOMESTUDENTS,GRANTWOULDBENEGATIVELYCORRELATEDWITHU,ANDIVWOULDBEINCONSISTENT152IITSEEMSREASONABLETOASSUMETHATDISTANDUAREUNCORRELATEDBECAUSECLASSROOMSARENOTUSUALLYASSIGNEDWITHCONVENIENCEFORPARTICULARSTUDENTSINMINDIITHEVARIABLEDISTMUSTBEPARTIALLYCORRELATEDWITHATNDRTEMOREPRECISELY,INTHEREDUCEDFORMATNDRTE01PRIGPA2ACT3DISTV,WEMUSTHAVE30GIVENASAMPLEOFDATAWECANTESTH030AGAINSTH130USINGATTESTIIIWENOWNEEDINSTRUMENTALVARIABLESFORATNDRTEANDTHEINTERACTIONTERM,PRIGPAATNDRTEEVENTHOUGHPRIGPAISEXOGENOUS,ATNDRTEISNOT,ANDSOPRIGPAATNDRTEISGENERALLYCORRELATEDWITHUUNDERTHEEXOGENEITYASSUMPTIONTHATEU|PRIGPA,ACT,DIST0,ANYFUNCTIONOFPRIGPA,ACT,ANDDISTISUNCORRELATEDWITHUINPARTICULAR,THEINTERACTIONPRIGPADISTISUNCORRELATEDWITHUIFDISTISPARTIALLYCORRELATEDWITHATNDRTETHENPRIGPADISTISPARTIALLYCORRELATEDWITHPRIGPAATNDRTESO,WECANESTIMATETHEEQUATIONSTNDFNL01ATNDRTE2PRIGPA3ACT4PRIGPAATNDRTEUBY2SLSUSINGIVSDIST,PRIGPA,ACT,ANDPRIGPADISTITTURNSOUTTHISISNOTGENERALLYOPTIMALITMAYBEBETTERTOADDPRIGPA2ANDPRIGPAACTTOTHEINSTRUMENTLISTTHISWOULDGIVEUSOVERIDENTIFYINGRESTRICTIONSTOTESTSEEWOOLDRIDGE2002,CHAPTERS5AND9FORFURTHERDISCUSSION153ITISEASIESTTOUSE1510BUTWHEREWEDROPREMEMBER,THISISALLOWEDZBECAUSEANDSIMILARLYWHENWEREPLACEXWITHY1NIIZIX1NIZXSOTHENUMERATORINTHEFORMULAFORIS1111,NNNIIIZYZYYNWHEREN1ISTHENUMBEROFOBSERVATIONSWITHZI1,ANDWEHAVEUSEDTHEFACTIZTHAT/N1,THEAVERAGEOFTHEYIOVERTHEIWITHZI1SOFAR,WEHAVE1IYSHOWNTHATTHENUMERATORINISN1NEXT,WRITEASAWEIGHTEDAVERAGEYOFTHEAVERAGESOVERTHETWOSUBGROUPSN0/NN1/N,YYWHEREN0NN1THEREFORE,NN1/NN0/NN0/N1YY1Y0THEREFORE,THENUMERATOROFCANBEWRITTENAS1N0N1/NY0BYSIMPLYREPLACINGYWITHX,THEDENOMINATORINCANBEEXPRESSEDASN0N1/N1XWHENWETAKETHERATIOOFTHESE,THETERMSINVOLVINGN0,N1,ANDN,CANCEL,0XLEAVING/11Y01X0154ITHESTATEMAYSETTHELEVELOFITSMINIMUMWAGEATLEASTPARTLYBASEDONPASTOREXPECTEDCURRENTECONOMICACTIVITY,ANDTHISCOULDCERTAINLYBEPARTOFUTTHENGMINTANDUTARECORRELATED,WHICHCAUSESOLSTOBEBIASEDANDINCONSISTENTIIBECAUSEGGDPTCONTROLSFORTHEOVERALLPERFORMANCEOFTHEUSECONOMY,ITSEEMSREASONABLETHATGUSMINTISUNCORRELATEDWITHTHEDISTURBANCESTOEMPLOYMENTGROWTHFORAPARTICULARSTATEIIIINSOMEYEARS,THEUSMINIMUMWAGEWILLINCREASEINSUCHAWAYSOTHATITEXCEEDSTHESTATEMINIMUMWAGE,ANDTHENTHESTATEMINIMUMWAGEWILLALSOINCREASEEVENIFTHEUSMINIMUMWAGEISNEVERBINDING,ITMAYBETHATTHESTATEINCREASESITSMINIMUMWAGEINRESPONSETOANINCREASEINTHEUSMINIMUMIFTHESTATEMINIMUMISALWAYSTHEUSMINIMUM,THENGMINTISEXOGENOUSINTHISEQUATIONANDWEWOULDJUSTUSEOLS155IFROMEQUATION1519WITHUX,PLIM11/215,WHEREISTHEIVESTIMATORSOTHEASYMPTOTICBIASIS51IIFROMEQUATION1520WITHUX,PLIM1CORRX,U,WHEREIS1THEOLSESTIMATORSOWEWOULDHAVETOHAVECORRX,U5BEFORETHEASYMPTOTICBIASINOLSEXCEEDSTHATOFIVTHISISASIMPLEILLUSTRATIONOFHOWASEEMINGLYSMALLCORRELATION1INTHISCASEBETWEENTHEIVZANDERRORUCANSTILLRESULTINIVBEINGMOREBIASEDTHANOLSIFTHECORRELATIONBETWEENZANDXISWEAK2156IPLUGGING1526INTO1522ANDREARRANGINGGIVESY10101Z12Z2V22Z1U1010112Z112Z2U11V2,ANDSO0010,1112,AND212IIFROMTHEEQUATIONINPARTI,V1U11V2IIIBYASSUMPTION,U1HASZEROMEANANDISUNCORRELATEDWITHZ1ANDZ2,ANDV2HASTHESEPROPERTIESBYDEFINITIONSOV1HASZEROMEANANDISUNCORRELATEDWITHZ1ANDZ2,WHICHMEANSTHATOLSCONSISTENTLYESTIMATESTHEJOLSWOULDONLYBEUNBIASEDIFWEADDTHESTRONGERASSUMPTIONSEU1|Z1,Z2EV2|Z1,Z20157IEVENATAGIVENINCOMELEVEL,SOMESTUDENTSAREMOREMOTIVATEDANDMOREABLETHANOTHERS,ANDTHEIRFAMILIESAREMORESUPPORTIVESAY,INTERMSOFPROVIDINGTRANSPORTATIONANDENTHUSIASTICABOUTEDUCATIONTHEREFORE,THEREISLIKELYTOBEASELFSELECTIONPROBLEMSTUDENTSTHATWOULDDOBETTERANYWAYAREALSOMORELIKELYTOATTENDACHOICESCHOOLIIASSUMINGWEHAVETHEFUNCTIONALFORMFORFAMINCCORRECT,THEANSWERISYESSINCEU1DOESNOTCONTAININCOME,RANDOMASSIGNMENTOFGRANTSWITHININCOMECLASSMEANSTHATGRANTDESIGNATIONISNOTCORRELATEDWITHUNOBSERVABLESSUCHASSTUDENTABILITY,MOTIVATION,ANDFAMILYSUPPORTIIITHEREDUCEDFORMISCHOICE01FAMINC2GRANTV2,ANDWENEED20INOTHERWORDS,AFTERACCOUNTINGFORINCOME,THEGRANTAMOUNTMUSTHAVESOMEAFFECTONCHOICETHISSEEMSREASONABLE,PROVIDEDTHEGRANTAMOUNTSDIFFERWITHINEACHINCOMECLASSIVTHEREDUCEDFORMFORSCOREISJUSTALINEARFUNCTIONOFTHEEXOGENOUSVARIABLESSEEPROBLEM156SCORE01FAMINC2GRANTV1THISEQUATIONALLOWSUSTODIRECTLYESTIMATETHEEFFECTOFINCREASINGTHEGRANTAMOUNTONTHETESTSCORE,HOLDINGFAMILYINCOMEFIXEDFROMAPOLICYPERSPECTIVETHISISITSELFOFSOMEINTEREST158IFAMILYINCOMEANDBACKGROUNDVARIABLES,SUCHASPARENTSEDUCATIONIITHEPOPULATIONMODELISSCORE01GIRLHS2FAMINC3MEDUC4FEDUCU1,WHERETHEVARIABLESARESELFEXPLANATORYIIIPARENTSWHOARESUPPORTIVEANDMOTIVATEDTOHAVETHEIRDAUGHTERSDOWELLINSCHOOLMAYALSOBEMORELIKELYTOENROLLTHEIRDAUGHTERSINAGIRLSHIGHSCHOOLITSEEMSLIKELYTHATGIRLHSANDU1ARECORRELATEDIVLETNUMGHSBETHENUMBEROFGIRLSHIGHSCHOOLSWITHINA20MILERADIUSOFAGIRLSHOMETOBEAVALIDIVFORGIRLHS,NUMGHSMUSTSATISFYTWOREQUIREMENTSITMUSTBEUNCORRELATEDWITHU1ANDITMUSTBEPARTIALLYCORRELATEDWITHGIRLHSTHESECONDREQUIREMENTPROBABLYHOLDS,ANDCANBETESTEDBYESTIMATINGTHEREDUCEDFORMGIRLHS01FAMINC2MEDUC3FEDUC4NUMGHSV2ANDTESTINGNUMGHSFORSTATISTICALSIGNIFICANCETHEFIRSTREQUIREMENTISMOREPROBLEMATICALGIRLSHIGHSCHOOLSTENDTOLOCATEINAREASWHERETHEREISADEMAND,ANDTHISDEMANDCANREFLECTTHESERIOUSNESSWITHWHICHPEOPLEINTHECOMMUNITYVIEWEDUCATIONSOMEAREASOFASTATEHAVEBETTERSTUDENTSONAVERAGEFORREASONSUNRELATEDTOFAMILYINCOMEANDPARENTSEDUCATION,ANDTHESEREASONSMIGHTBECORRELATEDWITHNUMGHSONEPOSSIBILITYISTOINCLUDECOMMUNITYLEVELVARIABLESTHATCANCONTROLFORDIFFERENCESACROSSCOMMUNITIES159JUSTUSEOLSONANEXPANDEDEQUATION,WHERESATANDCUMGPAAREADDEDASPROXYVARIABLESFORSTUDENTABILITYANDMOTIVATIONSEECHAPTER91510IBETTERANDMORESERIOUSSTUDENTSTENDTOGOTOCOLLEGE,ANDTHESESAMEKINDSOFSTUDENTSMAYBEATTRACTEDTOPRIVATEAND,INPARTICULAR,CATHOLICHIGHSCHOOLSTHERESULTINGCORRELATIONBETWEENUANDCATHHSISANOTHEREXAMPLEOFASELFSELECTIONPROBLEMSTUDENTSSELFSELECTTOWARDCATHOLICHIGHSCHOOLS,RATHERTHANBEINGRANDOMLYASSIGNEDTOTHEMIIASTANDARDIZEDSCOREISAMEASUREOFSTUDENTABILITY,SOTHISCANBEUSEDASAPROXYVARIABLEINANOLSREGRESSIONHAVINGTHISMEASUREINANOLSREGRESSIONSHOULDBEANIMPROVEMENTOVERHAVINGNOPROXIESFORSTUDENTABILITYIIITHEFIRSTREQUIREMENTISTHATCATHRE1MUSTBEUNCORRELATEDWITHUNOBSERVEDSTUDENTMOTIVATIONANDABILITYWHATEVERISNOTCAPTUREDBYANYPROXIESANDOTHERFACTORSINTHEERRORTERMTHISHOLDSIFGROWINGUPCATHOLICASOPPOSEDTOATTENDINGACATHOLICHIGHSCHOOLDOESNOTMAKEYOUABETTERSTUDENTITSEEMSREASONABLETOASSUMETHATCATHOLICSDONOTHAVEMOREINNATEABILITYTHANNONCATHOLICSWHETHERBEINGCATHOLICISUNRELATEDTOSTUDENTMOTIVATION,ORPREPARATIONFORHIGHSCHOOL,ISATHORNIERISSUETHESECONDREQUIREMENTISTHATBEINGCATHOLICHASANEFFECTONATTENDINGACATHOLICHIGHSCHOOL,CONTROLLINGFORTHEOTHEREXOGENOUSFACTORSTHATAPPEARINTHESTRUCTURALMODELTHISCANBETESTEDBYESTIMATINGTHEREDUCEDFORMEQUATIONOFTHEFORMCATHHS01CATHRELOTHEREXOGENOUSFACTORSREDUCEDFORMERRORIVEVANSANDSCHWAB1995FI