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ThiseditionisintendedforuseoutsideoftheU.S.only,withcontentthatmaybedifferentfromtheU.S.Edition.Thismaynotberesold,copied,ordistributedwithoutthepriorconsentofthepublisher.154CHAPTER14TEACHINGNOTESMypreferenceistoviewthefixedandrandomeffectsmethodsofestimationasapplyingtothesameunderlyingunobservedeffectsmodel.Thename“unobservedeffect”isneutraltotheissueofwhetherthetime-constanteffectsshouldbetreatedasfixedparametersorrandomvariables.WithlargeNandrelativelysmallT,italmostalwaysmakessensetotreatthemasrandomvariables,sincewecanjustviewtheunobservedaiasbeingdrawnfromthepopulationalongwiththeobservedvariables.Especiallyforundergraduatesandmaster’sstudents,itseemssensibletonotraisethephilosophicalissuesunderlyingtheprofessionaldebate.Inmymind,thekeyissueinmostapplicationsiswhethertheunobservedeffectiscorrelatedwiththeobservedexplanatoryvariables.ThefixedeffectstransformationeliminatestheunobservedeffectentirelywhereastherandomeffectstransformationaccountsfortheserialcorrelationinthecompositeerrorviaGLS.(Alternatively,therandomeffectstransformationonlyeliminatesafractionoftheunobservedeffect.)Asapracticalmatter,thefixedeffectsandrandomeffectsestimatesarecloserwhenTislargeorwhenthevarianceoftheunobservedeffectislargerelativetothevarianceoftheidiosyncraticerror.IthinkExample14.4isrepresentativeofwhatoftenhappensinapplicationsthatapplypooledOLS,randomeffects,andfixedeffects,atleastontheestimatesofthemarriageandunionwagepremiums.TherandomeffectsestimatesarebelowpooledOLSandthefixedeffectsestimatesarebelowtherandomeffectsestimates.Choosingbetweenthefixedeffectstransformationandfirstdifferencingisharder,althoughusefulevidencecanbeobtainedbytestingforserialcorrelationinthefirst-differenceestimation.IftheAR(1)coefficientissignificantandnegative(say,lessthan−.3,topickanotquitearbitraryvalue),perhapsfixedeffectsispreferred.Matchedpairssampleshavebeenprofitablyusedinrecenteconomicapplications,anddifferencingorrandomeffectsmethodscanbeapplied.Inanequationsuchas(14.12),thereisprobablynoneedtoallowadifferentinterceptforeachsisterprovidedthatthelabelingofsistersisrandom.Thedifferentinterceptsmightbeneededifacertainfeatureofasisterthatisnotincludedintheobservedcontrolsisusedtodeterminetheordering.Astatisticallysignificantinterceptinthedifferencedequationwouldbeevidenceofthis.ThiseditionisintendedforuseoutsideoftheU.S.only,withcontentthatmaybedifferentfromtheU.S.Edition.Thismaynotberesold,copied,ordistributedwithoutthepriorconsentofthepublisher.155SOLUTIONSTOPROBLEMS14.1First,foreacht1,Var(Δuit)=Var(uit–ui,t-1)=Var(uit)+Var(ui,t-1)=22uσ,whereweusetheassumptionsofnoserialcorrelationin{ut}andconstantvariance.Next,wefindthecovariancebetweenΔuitandΔui,t+1.Becausetheseeachhaveazeromean,thecovarianceisE(Δuit⋅Δui,t+1)=E[(uit–ui,t-1)(ui,t+1–uit)]=E(uitui,t+1)–E(2itu)–E(ui,t-1ui,t+1)+E(ui,t-1uit)=−E(2itu)=2uσ−becauseofthenoserialcorrelationassumption.Becausethevarianceisconstantacrosst,byProblem11.1,Corr(Δuit,Δui,t+1)=Cov(Δuit,Δui,t+1)/Var(∆uit)=22/(2)uuσσ−=−.5.14.2(i)ThebetweenestimatorisjusttheOLSestimatorfromthecross-sectionalregressionofiyonix(includinganintercept).Becausewejusthaveasingleexplanatoryvariableixandtheerrortermisai+iu,wehave,fromSection5.1,plim1()β=β1+Cov(ix,ai+iu)/Var(ix).ButE(ai+iu)=0soCov(ix,ai+iu)=E(ix(ai+iu)]=E(ixai)+E(ixiu)=E(ixai)becauseE(ixiu)=Cov(ix,iu)=0byassumption.NowE(ixai)=11E()TititTxa−=∑=σxa.Therefore,plim1()β=β1+σxa/Var(ix),whichiswhatwewantedtoshow.(ii)If{xit}isseriallyuncorrelatedwithconstantvariance2xσthenVar(ix)=2xσ/T,andsoplim1β=β1+σxa/(2xσ/T)=β1+T(σxa/2xσ).(iii)Aspart(ii)shows,whenthexitarepairwiseuncorrelatedthemagnitudeoftheinconsistencyactuallyincreaseslinearlywithT.Thesigndependsonthecovariancebetweenxitandai.14.3(i)E(eit)=E(vit−ivλ)=E(vit)−λE(iv)=0becauseE(vit)=0forallt.(ii)Var(vit−ivλ)=Var(vit)+λ2Var(iv)−2λ⋅Cov(vit,iv)=2vσ+λ2E(2iv)−2λ⋅E(vitiv).Now,2222E()vitauvσσσ==+andE(vitiv)=11()TitissTEvv−=∑=1T−[2aσ+2aσ+…+(2aσ+2uσ)+…+2aσ]=2aσ+2uσ/T.Therefore,E(2iv)=11()TititTEvv−=∑=2aσ+2uσ/T.Now,wecancollectterms:Var(vit−ivλ)=2222222()(/)2(/)auauauTTσσλσσλσσ+++−+.Now,itisconvenienttowriteλ=1−/ηγ,whereη≡2uσ/Tandγ≡2aσ+2uσ/T.ThenThiseditionisintendedforuseoutsideoftheU.S.only,withcontentthatmaybedifferentfromtheU.S.Edition.Thismaynotberesold,copied,ordistributedwithoutthepriorconsentofthepublisher.156Var(vit−ivλ)=(2aσ+2uσ)−2λ(2aσ+2uσ/T)+λ2(2aσ+2uσ/T)=(2aσ+2uσ)−2(1−/ηγ)γ+(1−/ηγ)2γ=(2aσ+2uσ)−2γ+2ηγ⋅+(1−2/ηγ+η/γ)γ=(2aσ+2uσ)−2γ+2ηγ⋅+(1−2/ηγ+η/γ)γ=(2aσ+2uσ)−2γ+2ηγ⋅+γ−2ηγ⋅+η=(2aσ+2uσ)+η−γ=2uσ.Thisiswhatwewantedtoshow.(iii)WemustshowthatE(eiteis)=0fort≠s.NowE(eiteis)=E[(vit−ivλ)(vis−ivλ)]=E(vitvis)−λE(ivvis)−λE(vitiv)+λ2E(2iv)=2aσ−2λ(2aσ+2uσ/T)+λ2E(2iv)=2aσ−2λ(2aσ+2uσ/T)+λ2(2aσ+2uσ/T).Therestoftheproofisverysimilartopart(ii):E(eiteis)=2aσ−2λ(2aσ+2uσ/T)+λ2(2aσ+2uσ/T)=2aσ−2(1−/ηγ)γ+(1−/ηγ)2γ=2aσ−2γ+2ηγ⋅+(1−2/ηγ+η/γ)γ=2aσ−2γ+2ηγ⋅+(1−2/ηγ+η/γ)γ=2aσ−2γ+2ηγ⋅+γ−2ηγ⋅+η=2aσ+η−γ=0.14.4(i)Men’sathleticsarestillthemostprominent,althoughwomen’ssports,especiallybasketballbutalsogymnastics,softball,andvolleyball,areverypopularat