大学伍德里奇计量经济学第三版教师手册CHAPTER 16.docx
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大学伍德里奇计量经济学第三版教师手册CHAPTER 16.docx
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大学伍德里奇计量经济学第三版教师手册CHAPTER16
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CHAPTER20XXXX
TEACHINGNOTES
IspendsometimeinSection20XXXX.1tryingtodistinguishbetweengoodandinappropriateusesofSEMs.Naturally,thisispartlydeterminedbymytaste,andmanyapplicationsfallintoagrayarea.ButstudentswhoaregoingtolearnaboutSEMSshouldknowthatjustbecausetwo(ormore)variablesarejointlydetermineddoesnotmeanthatitisappropriatetospecifyandestimateanSEM.IhaveseenmanybadapplicationsofSEMswherenoequationinthesystemcanstandonitsownwithaninterestingceterisparibusinterpretation.Inmostcases,theresearchereitherwantedtoestimateatradeoffbetweentwovariables,controllingforotherfactors–inwhichcaseOLSisappropriate–orshouldhavebeenestimatingwhatis(oftenpejoratively)calledthe“reducedform.”
Theidentificationofatwo-equationSEMinSection20XXXX.3isfairlystandardexceptthatIemphasizethatidentificationisafeatureofthepopulation.(TheearlyworkonSEMsalsohadthisemphasis.)Giventhetreatmentof2SLSinChapter20XXXX,therankconditioniseasytostate(andtest).
Romer’s(20XXXX0XX3)inflationandopennessexampleisaniceexampleofusingaggregatecross-sectionaldata.Puristsmaynotlikethelaborsupplyexample,butithasbecomecommontoviewlaborsupplyasbeingatwo-tierdecision.Whiletherearedifferentwaystomodelthetwotiers,specifyingastandardlaborsupplyfunctionconditionalonworkingisnotoutsidetherealmofreasonablemodels.
Section20XXXX.5beginsbyexpressingdoubtsoftheusefulnessofSEMsforaggregatemodelssuchasthosethatarespecifiedbasedonstandardmacroeconomicmodels.Suchmodelsraiseallkindsofthornyissues;theseareignoredinvirtuallyalltexts,wheresuchmodelsarestillusedtoillustrateSEMapplications.
SEMswithpaneldata,whicharecoveredinSection20XXXX.6,arenotcoveredinanyotherintroductorytext.Presumably,ifyouareteachingthismaterial,itistomoreadvancedstudentsinasecondsemester,perhapseveninamoreappliedcourse.Oncestudentshaveseenfirstdifferencingorthewithintransformation,alongwithIVmethods,theywillfindspecifyingandestimatingmodelsofthesortcontainedinExample20XXXX.8straightforward.Levitt’sexampleconcerningprisonpopulationsisespeciallyconvincingbecausehisinstrumentsseemtobetrulyexogenous.
SOLUTIONSTOPROBLEMS
20XXXX.1(i)If1 =0theny1 =1z1 +u1,andsotheright-hand-sidedependsonlyontheexogenousvariablez1andtheerrortermu1.Thisthenisthereducedformfory1.If1 =0,thereducedformfory1isy1 =2z2 +u2.(Notethathavingboth1and2equalzeroisnotinterestingasitimpliesthebizarreconditionu2 –u1 =1z1 2z2.)
If1 0and2 =0,wecanplugy1 =2z2 +u2intothefirstequationandsolvefory2:
2z2+u2=1y2+1z1+u1
or
1y2=1z1 2z2+u1–u2.
Dividingby1(because10)gives
y2=(1/1)z1–(2/1)z2+(u1–u2)/1
21z1+22z2+v2,
where21 =1/1,22 =2/1,andv2 =(u1 –u2)/1.Notethatthereducedformfory2generallydependsonz1andz2(aswellasonu1andu2).
(ii)Ifwemultiplythesecondstructuralequationby(1/2)andsubtractitfromthefirststructuralequation,weobtain
y1–(1/2)y1=1y21y2+1z1–(1/2)2z2+u1–(1/2)u2
=1z1–(1/2)2z2+u1–(1/2)u2
or
[1–(1/2)]y1=1z1–(1/2)2z2+u1–(1/2)u2.
Because1 2,1 –(1/2) 0,andsowecandividetheequationby1 –(1/2)toobtainthereducedformfory1:
y1=20XXXXz1+20XXXXz2+v1,where20XXXX =1/[1 –(1/2)],20XXXX =(1/2)2/[1 –(1/2)],andv1 =[u1 –(1/2)u2]/[1 –(1/2)].
Areducedformdoesexistfory2,ascanbeseenbysubtractingthesecondequationfromthefirst:
0=(1–2)y2+1z1–2z2+u1–u2;
because1 2,wecanrearrangeanddivideby1 2toobtainthereducedform.
(iii)Insupplyanddemandexamples,1 2isveryreasonable.Ifthefirstequationisthesupplyfunction,wegenerallyexpect1 >0,andifthesecondequationisthedemandfunction,2 <0.Thereducedformscanexistevenincaseswherethesupplyfunctionisnotupwardslopingandthedemandfunctionisnotdownwardsloping,butwemightquestiontheusefulnessofsuchmodels.
20XXXX.2Usingsimpleeconomics,thefirstequationmustbethedemandfunction,asitdependsonincome,whichisacommondeterminantofdemand.Thesecondequationcontainsavariable,rainfall,thataffectscropproductionandthereforecornsupply.
20XXXX.3No.Inthisexample,weareinterestedinestimatingthetradeoffbetweensleepingandworking,controllingforsomeotherfactors.OLSisperfectlysuitedforthis,providedwehavebeenabletocontrolforallotherrelevantfactors.Whileitistrueindividualsareassumedtooptimallyallocatetheirtimesubjecttoconstraints,thisdoesnotresultinasystemofsimultaneousequations.Ifwewrotedownsuchasystem,thereisnosenseinwhicheachequationcouldstandonitsown;neitherwouldhaveaninterestingceterisparibusinterpretation.Besides,wecouldnotestimateeitherequationbecauseeconomicreasoninggivesusnowayofexcludingexogenousvariablesfromeitherequation.SeeExample20XXXX.2forasimilardiscussion.
20XXXX.4Wecaneasilyseethattherankconditionforidentifyingthesecondequationdoesnothold:
therearenoexogenousvariablesappearinginthefirstequationthatarenotalsointhesecondequation.Thefirstequationisidentifiedprovided3 0(andwewouldpresume3 <0).Thisgivesusanexogenousvariable,log(price),thatcanbeusedasanIVforalcoholinestimatingthefirstequationby2SLS(whichisjuststandardIVinthiscase).
20XXXX.5(i)Otherthingsequal,ahigherrateofcondomusageshouldreducetherateofsexuallytransmitteddiseases(STDs).So1 <0.
(ii)Ifstudentshavingsexbehaverationally,andcondomusagedoespreventSTDs,thencondomusageshouldincreaseastherateofinfectionincreases.
(iii)Ifweplugthestructuralequationforinfrateintoconuse =0 +1infrate +…,weseethatconusedependson1u1.Because1 >0,conuseispositivelyrelatedtou1.Infact,ifthestructuralerror(u2)intheconuseequationisuncorrelatedwithu1,Cov(conuse,u1) =1Var(u1) >0.Ifweignoretheotherexplanatoryvariablesintheinfrateequation,wecanuseequation(5.4)toobtainthedirectionofbias:
1 >0becauseCov(conuse,u1) >0,where
denotestheOLSestimator.Sincewethink1 <0,OLSisbiasedtowardszero.Inotherwords,ifweuseOLSontheinfrateequation,wearelikelytounderestimatetheimportanceofcondomuseinreducingSTDs.(Remember,themorenegativeis1,themoreeffectiveiscondomusage.)
(iv)Wewouldhavetoassumethatcondisdoesnotappear,inadditiontoconuse,intheinfrateequation.Thisseemsreasonable,asitisusagethatshoulddirectlyaffectSTDs,andnotjusthavingadistributionprogram.Butwemustalsoassumecondisisexogenousintheinfrate:
itcannotbecorrelatedwithunobservedfactors(inu1)thatalsoaffectinfrate.
Wemustalsoassumethatcondishassomepartialeffectonconuse,somethingthatcanbetestedbyestimatingthereducedformforconuse.ItseemslikelythatthisrequirementforanIV–seeequations(20XXXX.30)and(20XXXX.31)–issatisfied.
20XXXX.6(i)Itcouldbethatthedecisiontounionizecertainsegmentsofworkersisrelatedtohowafirmtreatsitsemployees.Whilethetimingmaynotbecontemporaneous,withthesnapshotofasinglecrosssectionwemightaswellassumethatitis.
(ii)Onepossibilityistocollectinformationonwhetherworkers’parentsbelongedtoaunion,andconstructavariablethatisthepercentageofworkerswhohadaparentinaunion(say,perpar).Thismaybe(partially)correlatedwiththepercentofworkersthatbelongtoaunion.
(iii)Wewouldhavetoassumethatpercparisexogenousinthepensionequation.Wecantestwhetherperunionispartiallycorrelatedwithperparbyestimatingthereducedformforperunionanddoingattestonperpar.
20XXXX.7(i)Attendanceatwomen’sbasketballmaygrowinwaysthatareunrelatedtofactorsthatwecanobserveandcontrolfor.Thetasteforwomen’sbasketballmayincreaseovertime,andthiswouldbecapturedbythetimetrend.
(ii)No.Theuniversitysetstheprice,anditmaychangepricebasedonexpectationsofnextyear’sattendance;iftheuniversityusesfactorsthatwecannotobserve,thesearenecessarilyintheerrortermut.Soeventhoughthesupplyisfixed,itdoesnotmeanthatpriceisuncorrelatedwiththeunobservablesaffectingdemand.
(iii)Ifpeopleonlycareabouthowthisyear’steamisdoing,SEASPERCt1canbeexcludedfromtheequationonceWINPERCthasbeencontrolledfor.Ofcourse,thisisnotaverygoodassumptionforallgames,asattendanceearlyintheseasonislikelytoberelatedtohowtheteamdidlastyear.Wewouldalsoneedtocheckthat1PRICEtispartiallycorrelatedwithSEASPERCt1byestimatingthereducedformfor1PRICEt.
(iv)Itdoesmakesensetoincludeameasureofmen’sbasketballticketprices,asattendingawomen’sbasketballgameisasubstituteforattendingamen’sgame.Thecoefficienton1MPRICEtwouldbeexpectedtobepositive:
anincreaseinthepriceofmen’sticketsshouldincreasethedemandforwomen’stickets.Thewinningpercentageofthemen’steamisanothergoodcandidateforanexplanatoryvariableinthewomen’sdemandequation.
(v)Itmightbebettertousefirstdifferencesofthelogs,whicharethengrowthrates.Wewouldthendroptheobservationforthefirstgameineachseason.
(vi)Ifagameis
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