计量经济学课后答案伍德里奇CHAPTER 1-12 answers

CHAPTER11.1(i)Ideally,wecouldrandomlyassignstudentstoclassesofdifferentsizes.Thatis,eachstudentisassignedadifferentclasssizewithoutregardtoanystudentcharacteristicssuchasabilityandfamilybackground.ForreasonswewillseeinChapter2,wewouldlikesubstantialvariationinclasssizes(subject,ofcourse,toethicalconsiderationsandresourceconstraints).(ii)Anegativecorrelationmeansthatlargerclasssizeisassociatedwithlowerperformance.Wemightfindanegativecorrelationbecauselargerclasssizeactuallyhurtsperformance.However,withobservationaldata,thereareotherreasonswemightfindanegativerelationship.Forexample,childrenfrommoreaffluentfamiliesmightbemorelikelytoattendschoolswithsmallerclasssizes,andaffluentchildrengenerallyscorebetteronstandardizedtests.Anotherpossibilityisthat,withinaschool,aprincipalmightassignthebetterstudentstosmallerclasses.Or,someparentsmightinsisttheirchildrenareinthesmallerclasses,andthesesameparentstendtobemoreinvolvedintheirchildren’seducation.(iii)Giventhepotentialforconfoundingfactors–someofwhicharelistedin(ii)–findinganegativecorrelationwouldnotbestrongevidencethatsmallerclasssizesactuallyleadtobetterperformance.Somewayofcontrollingfortheconfoundingfactorsisneeded,andthisisthesubjectofmultipleregressionanalysis.1.2(i)Hereisonewaytoposethequestion:Iftwofirms,sayAandB,areidenticalinallrespectsexceptthatfirmAsuppliesjobtrainingonehourperworkermorethanfirmB,byhowmuchwouldfirmA’soutputdifferfromfirmB’s?(ii)Firmsarelikelytochoosejobtrainingdependingonthecharacteristicsofworkers.Someobservedcharacteristicsareyearsofschooling,yearsintheworkforce,andexperienceinaparticularjob.Firmsmightevendiscriminatebasedonage,gender,orrace.Perhapsfirmschoosetooffertrainingtomoreorlessableworkers,where“ability”mightbedifficulttoquantifybutwhereamanagerhassomeideaabouttherelativeabilitiesofdifferentemployees.Moreover,differentkindsofworkersmightbeattractedtofirmsthatoffermorejobtrainingonaverage,andthismightnotbeevidenttoemployers.(iii)Theamountofcapitalandtechnologyavailabletoworkerswouldalsoaffectoutput.So,twofirmswithexactlythesamekindsofemployeeswouldgenerallyhavedifferentoutputsiftheyusedifferentamountsofcapitalortechnology.Thequalityofmanagerswouldalsohaveaneffect.(iv)No,unlesstheamountoftrainingisrandomlyassigned.Themanyfactorslistedinparts(ii)and(iii)cancontributetofindingapositivecorrelationbetweenoutputandtrainingevenifjobtrainingdoesnotimproveworkerproductivity.1.3Itdoesnotmakesensetoposethequestionintermsofcausality.Economistswouldassumethatstudentschooseamixofstudyingandworking(andotheractivities,suchasattendingclass,leisure,andsleeping)basedonrationalbehavior,suchasmaximizingutilitysubjecttotheconstraintthatthereareonly168hoursinaweek.Wecanthenusestatisticalmethodstomeasuretheassociationbetweenstudyingandworking,includingregressionanalysisthatwecoverstartinginChapter2.Butwewouldnotbeclaimingthatonevariable“causes”theother.Theyarebothchoicevariablesofthestudent.CHAPTER22.1(i)Income,age,andfamilybackground(suchasnumberofsiblings)arejustafewpossibilities.Itseemsthateachofthesecouldbecorrelatedwithyearsofeducation.(Incomeandeducationareprobablypositivelycorrelated;ageandeducationmaybenegativelycorrelatedbecausewomeninmorerecentcohortshave,onaverage,moreeducation;andnumberofsiblingsandeducationareprobablynegativelycorrelated.)(ii)Notifthefactorswelistedinpart(i)arecorrelatedwitheduc.Becausewewouldliketoholdthesefactorsfixed,theyarepartoftheerrorterm.ButifuiscorrelatedwitheducthenE(u|educ)(0,andsoSLR.4fails.2.2Intheequationy=(0+(1x+u,addandsubtract(0fromtherighthandsidetogety =((0+(0)+(1x+(u((0).Callthenewerrore =u ((0,sothatE(e) =0.Thenewinterceptis(0 +(0,buttheslopeisstill(1.2.3(i)Letyi =GPAi,xi =ACTi,andn =8.Then=25.875, =3.2125,(xi –)(yi –) =5.8125,and(xi –)2 =56.875.Fromequation(2.9),weobtaintheslopeas=5.8125/56.875 .1022,roundedtofourplacesafterthedecimal.From(2.17), = –EMBEDEquation.DSMT4 3.2125 –(.1022)25.875.5681.Sowecanwrite=.5681+.1022ACTn=8.TheinterceptdoesnothaveausefulinterpretationbecauseACTisnotclosetozeroforthepopulationofinterest.IfACTis5pointshigher,increasesby.1022(5) =.511.(ii)Thefittedvaluesandresiduals—roundedtofourdecimalplaces—aregivenalongwiththeobservationnumberiandGPAinthefollowingtable: i GPA 1 2.8 2.7143 .0857 2 3.4 3.0209 .3791 3 3.0 3.2253 –.2253 4 3.5 3.3275 .1725 5 3.6 3.5319 .0681 6 3.0 3.1231 –.1231 7 2.7 3.1231 –.4231 8 3.7 3.6341 .0659Youcanverifythattheresiduals,asreportedinthetable,sumto(.0002,whichisprettyclosetozerogiventheinherentroundingerror.(iii)WhenACT=20,=.5681+.1022(20)2.61.(iv)Thesumofsquaredresiduals,,isabout.4347(roundedtofourdecimalplaces),andthetotalsumofsquares,(yi–)2,isabout1.0288.SotheR-squaredfromtheregressionisR2=1–SSR/SST1–(.4347/1.0288).577.Therefore,about57.7%ofthevariationinGPAisexplainedbyACTinthissmallsampleofstudents.2.4(i)Whencigs=0,predictedbirthweightis119.77ounces.Whencigs =20, =109.49.Thisisaboutan8.6%drop.(ii)Notnecessarily.Therearemanyotherfactorsthatcanaffectbirthweight,particularlyoverallhealthofthemotherandqualityofprenatalcare.Thesecouldbecorrelatedwithcigarettesmokingduringbirth.Also,somethingsuchascaffeineconsumptioncanaffectbirthweight,andmightalsobecorrelatedwithcigarettesmoking.(iii)Ifwewantapredictedbwghtof125,thencigs=(125–119.77)/(–.524)–10.18,orabout–10cigarettes!Thisisnonsense,ofcourse,anditshowswhathappenswhenwearetryingtopredictsomethingascomplicatedasbirthweightwithonlyasingleexplanatoryvariable.Thelargestpredictedbirthweightisnecessarily119.77.Yetalmost700ofthebirthsinthesamplehadabirthweighthigherthan119.77.(iv)1,176outof1,388womendidnotsmokewhilepregnant,orabout84.7%.Becauseweareusingonlycigstoexplainbirthweight,wehaveonlyonepredictedbirthweightatcigs=0.Thepredictedbirthweightisnecessarilyroughlyinthemiddleoftheobservedbirthweightsatcigs=0,andsowewillunderpredicthighbirthrates.2.5(i)Theinterceptimpliesthatwheninc =0,consispredictedtobenegative$124.84.This,ofcourse,cannotbetrue,andreflectsthatfactthatthisconsumptionfunctionmightbeapoorpredictorofconsumptionatverylow-incomelevels.Ontheotherhand,onanannualbasis,$124.84isnotsofarfromzero.(ii)Justplug30,000intotheequation:=–124.84+.853(30,000) =25,465.16dollars.(iii)TheMPCandtheAPCareshowninthefollowinggraph.Eventhoughtheinterceptisnegative,thesmallestAPCinthesampleispositive.Thegraphstartsatanannualincomelevelof$1,000(in1970dollars).2.6(i)Yes.Iflivingclosertoanincineratordepresseshousingprices,thenbeingfartherawayincreaseshousingprices.(ii)Ifthecitychosetolocatetheincineratorinanareaawayfrommoreexpensiveneighborhoods,thenlog(dist)ispositivelycorrelatedwithhousingquality.ThiswouldviolateSLR.4,andOLSestimationisbiased.(iii)Sizeofthehouse,numberofbathrooms,sizeofthelot,ageofthehome,andqualityoftheneighborhood(includingschoolquality),arejustahandfuloffactors.Asmentionedinpart(ii),thesecouldcertainlybecorrelatedwithdist[andlog(dist)].2.7(i)Whenweconditiononincincomputinganexpectation,becomesaconstant.SoE(u|inc) =E(EMBEDEquation.DSMT4e|inc)=EMBEDEquation.DSMT4E(e|inc) =EMBEDEquation.DSMT40becauseE(e|inc) =E(e) =0.(ii)Again,whenweconditiononincincomputingavariance,becomesaconstant.SoVar(u|inc) =Var(EMBEDEquation.DSMT4e|inc) =()2Var(e|inc) =incbecauseVar(e|inc) =.(iii)Familieswithlowincomesdonothavemuchdiscretionaboutspending;typically,alow-incomefamilymustspendonfood,clothing,housing,andothernecessities.Higherincomepeoplehavemorediscretion,andsomemightchoosemoreconsumptionwhileothersmoresaving.Thisdiscretionsuggestswidervariabilityinsavingamonghigherincomefamilies.2.8(i)Fromequation(2.66),=/.Plugginginyi=(0+(1xi+uigives=/.Afterstandardalgebra,thenumeratorcanbewrittenas.Puttingthisoverthedenominatorshowswecanwriteas=(0/+(1+/.Conditionalonthexi,wehaveE()=(0/+(1becauseE(ui)=0foralli.Therefore,thebiasinisgivenbythefirstterminthisequation.Thisbiasisobviouslyzerowhen(0 =0.Itisalsozerowhen =0,whichisthesameas =0.Inthelattercase,regressionthroughtheoriginisidenticaltoregressionwithanintercept.(ii)Fromthelastexpressionforinpart(i)wehave,conditionalonthexi,Var()=Var=EMBEDEquation.DSMT4=EMBEDEquation.DSMT4=/.(iii)From(2.57),Var()=2/.Fromthehint, (,andsoVar() (Var().Amoredirectwaytoseethisistowrite =,whichislessthanunless =0.(iv)Foragivensamplesize,thebiasinincreasesasincreases(holdingthesumofthefixed).Butasincreases,thevarianceofincreasesrelativetoVar().Thebiasinisalsosmallwhenissmall.Therefore,whetherwepreferoronameansquarederrorbasisdependsonthesizesof,,andn(inadditiontothesizeof).2.9(i)Wefollowthehint,notingthat=(thesampleaverageofisc1timesthesampleaverageofyi)and =.Whenweregressc1yionc2xi(includinganintercept)weuseequation(2.19)toobtaintheslope:From(2.17),weobtaintheinterceptas =(c1) –(c2) =(c1) –[(c1/c2)](c2) =c1( –EMBEDEquation.DSMT4) =c1)becausetheinterceptfromregressingyionxiis( –EMBEDEquation.DSMT4).(ii)Weusethesameapproachfrompart(i)alongwiththefactthat =c1 +and =c2 +.Therefore, =(c1 +yi) –(c1 +) =yi –and(c2 +xi) – =xi –.Soc1andc2entirelydropoutoftheslopeformulafortheregressionof(c1 +yi)on(c2 +xi),and =.Theinterceptis = –EMBEDEquation.DSMT4 =(c1 +) –(c2 +) =() +c1 –c2 = +c1 –c2,whichiswhatwewantedtoshow.(iii)Wecansimplyapplypart(ii)because.Inotherwords,replacec1withlog(c1),yiwithlog(yi),andsetc2=0.(iv)Again,wecanapplypart(ii)withc1=0andreplacingc2withlog(c2)andxiwithlog(xi).Ifaretheoriginalinterceptandslope,thenand.2.10(i)Thisderivationisessentiallydoneinequation(2.52),onceisbroughtinsidethesummation(whichisvalidbecausedoesnotdependoni).Then,justdefine.(ii)Becauseweshowthatthelatteriszero.But,frompart(i),EMBEDEquation.DSMT4Becausethearepairwiseuncorrelated(theyareindependent),(because).Therefore,(iii)TheformulafortheOLSinterceptisand,pluggingingives(iv)Becauseareuncorrelated,,whichiswhatwewantedtoshow.(v)Usingthehintandsubstitutiongives2.11(i)WewouldwanttorandomlyassignthenumberofhoursinthepreparationcoursesothathoursisindependentofotherfactorsthataffectperformanceontheSAT.Then,wewouldcollectinformationonSATscoreforeachstudentintheexperiment,yieldingadataset,wherenisthenumberofstudentswecanaffordtohaveinthestudy.Fromequation(2.7),weshouldtrytogetasmuchvariationinasisfeasible.(ii)Herearethreefactors:innateability,familyincome,andgeneralhealthonthedayoftheexam.IfwethinkstudentswithhighernativeintelligencethinktheydonotneedtopreparefortheSAT,thenabilityandhourswillbenegativelycorrelated.Familyincomewouldprobablybepositivelycorrelatedwithhours,becausehigherincomefamiliescanmoreeasilyaffordpreparationcourses.Rulingoutchronichealthproblems,healthonthedayoftheexamshouldberoughlyuncorrelatedwithhoursspentinapreparationcourse.(iii)Ifpreparationcoursesareeffective,shouldbepositive:otherfactorsequal,anincreaseinhoursshouldincreasesat.(iv)Theintercept,,hasausefulinterpretationinthisexample:becauseE(u)=0,istheaverageSATscoreforstudentsinthepopulationwithhours=0.CHAPTER33.1(i)hspercisdefinedsothatthesmalleritis,thelowerthestudent’sstandinginhighschool.Everythingelseequal,theworsethestudent’sstandinginhighschool,thelowerishis/herexpectedcollegeGPA.(ii)Justplugthesevaluesintotheequation:=1.392(.0135(20)+.00148(1050)=2.676.(iii)ThedifferencebetweenAandBissimply140timesthecoefficientonsat,becausehspercisthesameforbothstudents.SoAispredictedtohaveascore.00148(140).207higher.(iv)Withhspercfixed, =.00148(sat.Now,wewanttofind(satsuchthat =.5,so.5 =.00148((sat)or(sat =.5/(.00148)338.Perhapsnotsurprisingly,alargeceterisparibusdifferenceinSATscore –almosttwoandone-halfstandarddeviations –isneededtoobtainapredicteddifferenceincollegeGPAorahalfapoint.3.2(i)Yes.Becauseofbudgetconstraints,itmakessensethat,themoresiblingsthereareinafamily,thelesseducationanyonechildinthefamilyhas.Tofindtheincreaseinthenumberofsiblingsthatreducespredictededucationbyoneyear,wesolve1=.094((sibs),so(sibs =1/.09410.6.(ii)Holdingsibsandfeducfixed,onemoreyearofmother’seducationimplies.131yearsmoreofpredictededucation.Soifamotherhasfourmoreyearsofeducation,hersonispredictedtohaveaboutahalfayear(.524)moreyearsofeducation.(iii)Sincethenumberofsiblingsisthesame,butmeducandfeducarebothdifferent,thecoefficientsonmeducandfeducbothneedtobeaccountedfor.ThepredicteddifferenceineducationbetweenBandAis.131(4) +.210(4) =1.364.3.3(i)Ifadultstradeoffsleepforwork,moreworkimplieslesssleep(otherthingsequal),so <0.(ii)Thesignsofandarenotobvious,atleasttome.Onecouldarguethatmoreeducatedpeopleliketogetmoreoutoflife,andso,otherthingsequal,theysleepless( <0).Therelationshipbetweensleepingandageismorecomplicatedthanthismodelsuggests,andeconomistsarenotinthebestpositiontojudgesuchthings.(iii)Sincetotwrkisinminutes,wemustconvertfivehoursintominutes:(totwrk =5(60) =300.Thensleepispredictedtofallby.148(300) =44.4minutes.Foraweek,45minuteslesssleepisnotanoverwhelmingchange.(iv)Moreeducationimplieslesspredictedtimesleeping,buttheeffectisquitesmall.Ifweassumethedifferencebetweencollegeandhighschoolisfouryears,thecollegegraduatesleepsabout45minuteslessperweek,otherthingsequal.(v)Notsurprisingly,thethreeexplanatoryvariablesexplainonlyabout11.3%ofthevariationinsleep.Oneimportantfactorintheerrortermisgeneralhealth.Anotherismaritalstatus,andwhetherthepersonhaschildren.Health(howeverwemeasurethat),maritalstatus,andnumberandagesofchildrenwouldgenerallybecorrelatedwithtotwrk.(Forexample,lesshealthypeoplewouldtendtoworkless.)3.4(i)Alargerrankforalawschoolmeansthattheschoolhaslessprestige;thislowersstartingsalaries.Forexample,arankof100meansthereare99schoolsthoughttobebetter.(ii)>0,>0.BothLSATandGPAaremeasuresofthequalityoftheenteringclass.Nomatterwherebetterstudentsattendlawschool,weexpectthemtoearnmore,onaverage.,>0.Thenumberofvolumesinthelawlibraryandthetuitioncostarebothmeasuresoftheschoolquality.(Costislessobviousthanlibraryvolumes,butshouldreflectqualityofthefaculty,physicalplant,andsoon.)(iii)ThisisjustthecoefficientonGPA,multipliedby100:24.8%.(iv)Thisisanelasticity:aonepercentincreaseinlibraryvolumesimpliesa.095%increaseinpredictedmedianstartingsalary,otherthingsequal.(v)Itisdefinitelybettertoattendalawschoolwithalowerrank.IflawschoolAhasaranking20lessthanlawschoolB,thepredicteddifferenceinstartingsalaryis100(.0033)(20) =6.6%higherforlawschoolA.3.5(i)No.Bydefinition,study +sleep +work +leisure =168.Therefore,ifwechangestudy,wemustchangeatleastoneoftheothercategoriessothatthesumisstill168.(ii)Frompart(i),wecanwrite,say,studyasaperfectlinearfunctionoftheotherindependentvariables:study =168(sleep (work (leisure.Thisholdsforeveryobservation,soMLR.3violated.(iii)Simplydroponeoftheindependentvariables,sayleisure:GPA=+study+sleep+work+u.Now,forexample,isinterpretedasthechangeinGPAwhenstudyincreasesbyonehour,wheresleep,work,anduareallheldfixed.Ifweareholdingsleepandworkfixedbutincreasingstudybyonehour,thenwemustbereducingleisurebyonehour.Theotherslopeparametershaveasimilarinterpretation.3.6Conditioningontheoutcomesoftheexplanatoryvariables,wehave =E( +) =E() +E() =(1+(2 =.3.7Only(ii),omittinganimportantvariable,cancausebias,andthisistrueonlywhentheomittedvariableiscorrelatedwiththeincludedexplanatoryvariables.Thehomoskedasticityassumption,MLR.5,playednoroleinshowingthattheOLSestimatorsareunbiased.(Homoskedasticitywasusedtoobtaintheusualvarianceformulasforthe.)Further,thedegreeofcollinearitybetweentheexplanatoryvariablesinthesample,evenifitisreflectedinacorrelationashighas.95,doesnotaffecttheGauss-Markovassumptions.OnlyifthereisaperfectlinearrelationshipamongtwoormoreexplanatoryvariablesisMLR.3violated.3.8WecanuseTable3.2.Bydefinition, >0,andbyassumption,Corr(x1,x2) <0.Therefore,thereisanegativebiasin:E() <.Thismeansthat,onaverageacrossdifferentrandomsamples,thesimpleregressionestimatorunderestimatestheeffectofthetrainingprogram.ItisevenpossiblethatE()isnegativeeventhough >0.3.9(i) <0becausemorepollutioncanbeexpectedtolowerhousingvalues;notethatistheelasticityofpricewithrespecttonox.isprobablypositivebecauseroomsroughlymeasuresthesizeofahouse.(However,itdoesnotallowustodistinguishhomeswhereeachroomislargefromhomeswhereeachroomissmall.)(ii)Ifweassumethatroomsincreaseswithqualityofthehome,thenlog(nox)androomsarenegativelycorrelatedwhenpoorerneighborhoodshavemorepollution,somethingthatisoftentrue.WecanuseTable3.2todeterminethedirectionofthebias.If >0andCorr(x1,x2) <0,thesimpleregressionestimatorhasadownwardbias.Butbecause <0,thismeansthatthesimpleregression,onaverage,overstatestheimportanceofpollution.[E()ismorenegativethan.](iii)Thisiswhatweexpectfromthetypicalsamplebasedonouranalysisinpart(ii).Thesimpleregressionestimate,(1.043,ismorenegative(largerinmagnitude)thanthemultipleregressionestimate,(.718.Asthoseestimatesareonlyforonesample,wecanneverknowwhichiscloserto.Butifthisisa“typical”sample,iscloserto(.718.3.10(i)Becauseishighlycorrelatedwithand,andtheselattervariableshavelargepartialeffectsony,thesimpleandmultipleregressioncoefficientsoncandifferbylargeamounts.Wehavenotdonethiscaseexplicitly,butgivenequation(3.46)andthediscussionwithasingleomittedvariable,theintuitionisprettystraightforward.(ii)Herewewouldexpectandtobesimilar(subject,ofcourse,towhatwemeanby“almostuncorrelated”).Theamountofcorrelationbetweenanddoesnotdirectlyeffectthemultipleregressionestimateonifisessentiallyuncorrelatedwithand.(iii)Inthiscaseweare(unnecessarily)introducingmulticollinearityintotheregression:andhavesmallpartialeffectsonyandyetandarehighlycorrelatedwith.Addingandlikeincreasesthestandarderrorofthecoefficientonsubstantially,sose()islikelytobemuchlargerthanse().(iv)Inthiscase,addingandwilldecreasetheresidualvariancewithoutcausingmuchcollinearity(becauseisalmostuncorrelatedwithand),soweshouldseese()smallerthanse().Theamountofcorrelationbetweenanddoesnotdirectlyaffectse().3.11Fromequation(3.22)wehavewherethearedefinedintheproblem.Asusual,wemustpluginthetruemodelforyi:Thenumeratorofthisexpressionsimplifiesbecause =0, =0,and =.Theseallfollowfromthefactthatthearetheresidualsfromtheregressionofon:thehavezerosampleaverageandareuncorrelatedinsamplewith.SothenumeratorofcanbeexpressedasPuttingthesebackoverthedenominatorgivesConditionalonallsamplevaluesonx1,x2,andx3,onlythelasttermisrandomduetoitsdependenceonui.ButE(ui) =0,andsowhichiswhatwewantedtoshow.Noticethatthetermmultiplyingistheregressioncoefficientfromthesimpleregressionofxi3on.3.12(i)Theshares,bydefinition,addtoone.Ifwedonotomitoneofthesharesthentheequationwouldsufferfromperfectmulticollinearity.Theparameterswouldnothaveaceterisparibusinterpretation,asitisimpossibletochangeonesharewhileholdingalloftheothersharesfixed.(ii)Becauseeachshareisaproportion(andcanbeatmostone,whenallothersharesarezero),itmakeslittlesensetoincreasesharepbyoneunit.Ifsharepincreasesby.01–whichisequivalenttoaonepercentagepointincreaseintheshareofpropertytaxesintotalrevenue–holdingshareI,shareS,andtheotherfactorsfixed,thengrowthincreasesby(.01).Withtheothersharesfixed,theexcludedshare,shareF,mustfallby.01whensharepincreasesby.01.3.13(i)Fornotationalsimplicity,defineszx =thisisnotquitethesamplecovariancebetweenzandxbecausewedonotdividebyn –1,butweareonlyusingittosimplifynotation.ThenwecanwriteasThisisclearlyalinearfunctionoftheyi:taketheweightstobewi =(zi ()/szx.Toshowunbiasedness,asusualweplugyi = +xi +uiintothisequation,andsimplify:whereweusethefactthat =0always.Nowszxisafunctionoftheziandxiandtheexpectedvalueofeachuiiszeroconditionalonallzi