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首页 伍德里奇计量经济学导论教师用书及作业题答案

伍德里奇计量经济学导论教师用书及作业题答案

伍德里奇计量经济学导论教师用书及作业题答案

kukurany
2009-06-12 0人阅读 举报 0 0 暂无简介

简介:本文档为《伍德里奇计量经济学导论教师用书及作业题答案pdf》,可适用于其他资料领域

CHAPTERTEACHINGNOTESYouhavesubstantiallatitudeaboutwhattoemphasizeinChapterIfinditusefultotalkabouttheeconomicsofcrimeexample(Example)andthewageexample(Example)sothatstudentssee,attheoutset,thateconometricsislinkedtoeconomicreasoning,ifnoteconomictheoryIliketofamiliarizestudentswiththeimportantdatastructuresthatempiricaleconomistsuse,focusingprimarilyoncrosssectionalandtimeseriesdatasets,asthesearewhatIcoverinafirstsemestercourseItisprobablyagoodideatomentionthegrowingimportanceofdatasetsthathavebothacrosssectionalandtimedimensionIspendalmostanentirelecturetalkingabouttheproblemsinherentindrawingcausalinferencesinthesocialsciencesIdothismostlythroughtheagriculturalyield,returntoeducation,andcrimeexamplesTheseexamplesalsocontrastexperimentalandnonexperimentaldataStudentsstudyingbusinessandfinancetendtofindthetermstructureofinterestratesexamplemorerelevant,althoughtheissuethereistestingtheimplicationofasimpletheory,asopposedtoinferringcausalityIhavefoundthatspendingtimetalkingabouttheseexamples,inplaceofaformalreviewofprobabilityandstatistics,ismoresuccessful(andmoreenjoyableforthestudentsandme)CHAPTERTEACHINGNOTESThisisthechapterwhereIexpectstudentstofollowmost,ifnotall,ofthealgebraicderivationsInclassIliketoderiveatleasttheunbiasednessoftheOLSslopecoefficient,andusuallyIderivethevarianceAtaminimum,ItalkaboutthefactorsaffectingthevarianceTosimplifythenotation,afterIemphasizetheassumptionsinthepopulationmodel,andassumerandomsampling,IjustconditiononthevaluesoftheexplanatoryvariablesinthesampleTechnically,thisisjustifiedbyrandomsamplingbecause,forexample,E(ui|x,x,…,xn)=E(ui|xi)byindependentsamplingIfindthatstudentsareabletofocusonthekeyassumptionSLRandsubsequentlytakemywordabouthowconditioningontheindependentvariablesinthesampleisharmless(Ifyouprefer,theappendixtoChapterdoestheconditioningargumentcarefully)Becausestatisticalinferenceisnomoredifficultinmultipleregressionthaninsimpleregression,IpostponeinferenceuntilChapter(Thisreducesredundancyandallowsyoutofocusontheinterpretivedifferencesbetweensimpleandmultipleregression)Youmightnoticehow,comparedwithmostothertexts,IuserelativelyfewassumptionstoderivetheunbiasednessoftheOLSslopeestimator,followedbytheformulaforitsvarianceThisisbecauseIdonotintroduceredundantorunnecessaryassumptionsForexample,onceSLRisassumed,nothingfurtherabouttherelationshipbetweenuandxisneededtoobtaintheunbiasednessofOLSunderrandomsamplingSOLUTIONSTOPROBLEMS(i)Income,age,andfamilybackground(suchasnumberofsiblings)arejustafewpossibilitiesItseemsthateachofthesecouldbecorrelatedwithyearsofeducation(Incomeandeducationareprobablypositivelycorrelatedageandeducationmaybenegativelycorrelatedbecausewomeninmorerecentcohortshave,onaverage,moreeducationandnumberofsiblingsandeducationareprobablynegativelycorrelated)(ii)Notifthefactorswelistedinpart(i)arecorrelatedwitheducBecausewewouldliketoholdthesefactorsfixed,theyarepartoftheerrortermButifuiscorrelatedwitheducthenE(u|educ)≠,andsoSLRfailsIntheequationy=ββxu,addandsubtractαfromtherighthandsidetogety=(αβ)βx(u−α)Callthenewerrore=u−α,sothatE(e)=Thenewinterceptisαβ,buttheslopeisstillβ(i)Letyi=GPAi,xi=ACTi,andn=Thenx=,y=,(xni=∑i–x)(yi–y)=,and(xni=∑i–x)=Fromequation(),weobtaintheslopeasˆβ=,roundedtofourplacesafterthedecimalFrom(),≈βˆ=y–ˆβx–()Sowecanwrite≈≈=ACT�GPAn=TheinterceptdoesnothaveausefulinterpretationbecauseACTisnotclosetozeroforthepopulationofinterestIfACTispointshigher,increasesby()=�GPA(ii)ThefittedvaluesandresidualsroundedtofourdecimalplacesaregivenalongwiththeobservationnumberiandGPAinthefollowingtable:iGPA�GPAuˆ–––Youcanverifythattheresiduals,asreportedinthetable,sumto−,whichisprettyclosetozerogiventheinherentroundingerror(iii)WhenACT=,=()ˆGPA≈(iv)Thesumofsquaredresiduals,ˆniiu=∑,isabout(roundedtofourdecimalplaces),andthetotalsumofsquares,(yni=∑i–y),isaboutSotheRsquaredfromtheregressionisR=–SSRSST≈–()≈Therefore,aboutofthevariationinGPAisexplainedbyACTinthissmallsampleofstudents(i)Whencigs=,predictedbirthweightisouncesWhencigs=,=Thisisaboutandrop�bwght(ii)NotnecessarilyTherearemanyotherfactorsthatcanaffectbirthweight,particularlyoverallhealthofthemotherandqualityofprenatalcareThesecouldbecorrelatedwithcigarettesmokingduringbirthAlso,somethingsuchascaffeineconsumptioncanaffectbirthweight,andmightalsobecorrelatedwithcigarettesmoking(iii)Ifwewantapredictedbwghtof,thencigs=(–)(–)–,orabout–cigarettes!Thisisnonsense,ofcourse,anditshowswhathappenswhenwearetryingtopredictsomethingascomplicatedasbirthweightwithonlyasingleexplanatoryvariableThelargestpredictedbirthweightisnecessarilyYetalmostofthebirthsinthesamplehadabirthweighthigherthan≈(iv),outof,womendidnotsmokewhilepregnant,orabout(i)Theinterceptimpliesthatwheninc=,consispredictedtobenegative$This,ofcourse,cannotbetrue,andreflectsthatfactthatthisconsumptionfunctionmightbeapoorpredictorofconsumptionatverylowincomelevelsOntheotherhand,onanannualbasis,$isnotsofarfromzero(ii)Justplug,intotheequation:=–(,)=,dollars�cons(iii)TheMPCandtheAPCareshowninthefollowinggraphEventhoughtheinterceptisnegative,thesmallestAPCinthesampleispositiveThegraphstartsatanannualincomelevelof$,(indollars)incAPCMPCAPCMPC(i)YesIflivingclosertoanincineratordepresseshousingprices,thenbeingfartherawayincreaseshousingprices(ii)Ifthecitychosetolocatetheincineratorinanareaawayfrommoreexpensiveneighborhoods,thenlog(dist)ispositivelycorrelatedwithhousingqualityThiswouldviolateSLR,andOLSestimationisbiased(iii)Sizeofthehouse,numberofbathrooms,sizeofthelot,ageofthehome,andqualityoftheneighborhood(includingschoolquality),arejustahandfuloffactorsAsmentionedinpart(ii),thesecouldcertainlybecorrelatedwithdistandlog(dist)(i)Whenweconditiononincincomputinganexpectation,incbecomesaconstantSoE(u|inc)=E(inc⋅e|inc)=inc⋅E(e|inc)=inc⋅becauseE(e|inc)=E(e)=(ii)Again,whenweconditiononincincomputingavariance,incbecomesaconstantSoVar(u|inc)=Var(inc⋅e|inc)=(inc)Var(e|inc)=eσincbecauseVar(e|inc)=eσ(iii)Familieswithlowincomesdonothavemuchdiscretionaboutspendingtypically,alowincomefamilymustspendonfood,clothing,housing,andothernecessitiesHigherincomepeoplehavemorediscretion,andsomemightchoosemoreconsumptionwhileothersmoresavingThisdiscretionsuggestswidervariabilityinsavingamonghigherincomefamilies(i)Fromequation(),β=niiixy=⎛⎞⎜⎟⎝⎠∑niix=⎛⎞⎜⎟⎝⎠∑Plugginginyi=ββxiuigivesβ=()iniiixxuββ=⎝⎠∑nii⎛⎞⎜⎟x=⎛⎞⎜⎟⎝⎠∑Afterstandardalgebra,thenumeratorcanbewrittenasinnniiiiiixxxββ===u∑∑∑Puttingthisoverthedenominatorshowswecanwriteβasβ=βniix=⎛⎞⎜⎟⎝⎠∑niix=⎛⎜⎝⎠⎞⎟∑βniiixu=⎛⎞⎜⎟⎝⎠∑niix=⎛⎞⎜⎟⎝⎠∑Conditionalonthexi,wehaveE(β)=βniix=⎛⎞⎜⎟⎝⎠∑niix=⎛⎜⎝⎠⎞⎟∑βbecauseE(ui)=foralliTherefore,thebiasinβisgivenbythefirstterminthisequationThisbiasisobviouslyzerowhenβ=Itisalsozerowhenniix=∑=,whichisthesameasx=Inthelattercase,regressionthroughtheoriginisidenticaltoregressionwithanintercept(ii)Fromthelastexpressionforβinpart(i)wehave,conditionalonthexi,Var(β)=Varniix−=⎛⎞⎜⎟⎝⎠∑niiixu=⎛⎞⎜⎟⎝⎠∑=niix−=⎛⎞⎜⎟⎝⎠∑Var()niiixu=⎛⎞⎜⎟⎝⎠∑=niix−=⎛⎞⎜⎟⎝⎠∑niixσ=⎛⎞⎜⎟⎝⎠∑=σniix=⎛⎞⎜⎟⎝⎠∑(iii)From(),Var(ˆβ)=σ()niixx=⎛−⎜⎝⎠∑⎞⎟Fromthehint,niix=∑≥(nii)xx=−∑,andsoVar(β)≤Var(ˆβ)Amoredirectwaytoseethisistowrite()niixx=−∑=()niixnx=−∑,whichislessthanniix=∑unlessx=(iv)Foragivensamplesize,thebiasinβincreasesasxincreases(holdingthesumoftheixfixed)Butasxincreases,thevarianceofˆβincreasesrelativetoVar(β)ThebiasinβisalsosmallwhenβissmallTherefore,whetherwepreferβorˆβonameansquarederrorbasisdependsonthesizesofβ,x,andn(inadditiontothesizeofniix=∑)(i)Wefollowthehint,notingthatcy=cy(thesampleaverageofiscicytimesthesampleaverageofyi)andcx=cxWhenweregresscyioncxi(includinganintercept)weuseequation()toobtaintheslope:��()()()(()()()()()nniiiiiinniiiiniiiniicxcxcycyccxxyycxcxcxxxxyyccccxxββ======−−−−==−−−−=⋅=−∑∑∑∑∑∑)From(),weobtaintheinterceptasβ=(cy)–β(cx)=(cy)–(cc)ˆβ(cx)=c(y–ˆβx)=cβˆ)becausetheinterceptfromregressingyionxiis(y–ˆβx)(ii)Weusethesameapproachfrompart(i)alongwiththefactthat(cy)=cyand(cx)=cxTherefore,()(icycy−)=(cyi)–(cy)=yi–yand(cxi)–(cx)=xi–xSocandcentirelydropoutoftheslopeformulafortheregressionof(cyi)on(cxi),andβ=ˆβTheinterceptisβ=()cy–β(cx)=(cy)–ˆβ(cx)=(ˆyxβ−)c–cˆβ=βˆc–cˆβ,whichiswhatwewantedtoshow(iii)Wecansimplyapplypart(ii)becauselog()log()log()icycyi=Inotherwords,replacecwithlog(c),yiwithlog(yi),andsetc=(iv)Again,wecanapplypart(ii)withc=andreplacingcwithlog(c)andxiwithlog(xi)If垐andββaretheoriginalinterceptandslope,thenˆββ=and垐log()cβββ=−SOLUTIONSTOCOMPUTEREXERCISES(i)Theaverageprateisaboutandtheaveragemrateisabout(ii)Theestimatedequationis�prate=mraten=,,R=(iii)Theinterceptimpliesthat,evenifmrate=,thepredictedparticipationrateispercentThecoefficientonmrateimpliesthataonedollarincreaseinthematchrate–afairlylargeincrease–isestimatedtoincreasepratebypercentagepointsThisassumes,ofcourse,thatthischangeprateispossible(if,say,prateisalreadyat,thisinterpretationmakesnosense)(iv)Ifweplugmrate=intotheequationwegetˆprate=()=Thisisimpossible,aswecanhaveatmostapercentparticipationrateThisillustratesthat,especiallywhendependentvariablesarebounded,asimpleregressionmodelcangivestrangepredictionsforextremevaluesoftheindependentvariable(Inthesampleof,firms,onlyhavemrate≥)(v)mrateexplainsaboutofthevariationinprateThisisnotmuch,andsuggeststhatmanyotherfactorsinfluence(k)planparticipationrates(i)Averagesalaryisabout,whichmeans$,becausesalaryisinthousandsofdollarsAverageceotenisabout(ii)TherearefiveCEOswithceoten=Thelongesttenureisyears(iii)Theestimatedequationis=ceoten�log()salaryn=,R=  WeobtaintheapproximatepercentagechangeinsalarygivenΔceoten=bymultiplyingthecoefficientonceotenby,()=Therefore,onemoreyearasCEOispredictedtoincreasesalarybyalmost(i)Theestimatedequationis=,–totwrk�sleepn=,R=Theinterceptimpliesthattheestimatedamountofsleepperweekforsomeonewhodoesnotworkis,minutes,orabouthoursThiscomestoabouthourspernight(ii)IfsomeoneworkstwomorehoursperweekthenΔtotwrk=(becausetotwrkismeasuredinminutes),andso=–()=–minutesThisisonlyafewminutesanightIfsomeoneweretoworkonemorehouroneachoffiveworkingdays,=�sleepΔ�sleepΔ–()=–minutes,oraboutfiveminutesanight(i)Averagesalaryisabout$andaverageIQisaboutThesamplestandarddeviationofIQisabout,whichisprettyclosetothepopulationvalueof(ii)Thiscallsforalevellevelmodel:=IQ�wagen=,R=AnincreaseinIQofincreasespredictedmonthlysalaryby()=$(indollars)IQscoredoesnotevenexplainofthevariationinwage(iii)Thiscallsforaloglevelmodel:�log()wage=IQn=,R=IfΔIQ=then=()=,whichisthe(approximate)proportionatechangeinpredictedwageThepercentageincreaseisthereforeapproximately�log()wageΔ(i)Theconstantelasticitymodelisaloglogmodel:log(rd)=ββlog(sales)u,whereβistheelasticityofrdwithrespecttosales(ii)Theestimatedequationis=–log(sales)�log()rdn=,R=Theestimatedelasticityofrdwithrespecttosalesis,whichisjustaboveoneAonepercentincreaseinsalesisestimatedtoincreaserdbyaboutCHAPTERTEACHINGNOTESForundergraduates,Idonotdomostofthederivationsinthischapter,atleastnotindetailRather,Ifocusoninterpretingtheassumptions,whichmostlyconcernthepopulationOtherthanrandomsampling,theonlyassumptionthatinvolvesmorethanpopulationconsiderationsistheassumptionaboutnoperfectcollinearity,wherethepossibilityofperfectcollinearityinthesample(evenifitdoesnotoccurinthepopulation)shouldbetouchedonThemoreimportantissueisperfectcollinearityinthepopulation,butthisisfairlyeasytodispensewithviaexamplesThesecomefrommyexperienceswiththekindsofmodelspecificationissuesthatbeginnershavetroublewithThecomparisonofsimpleandmultipleregressionestimates–basedontheparticularsampleathand,asopposedtotheirstatisticalproperties–usuallymakesastrongimpressionSometimesIdonotbotherwiththe“partiallingout”interpretationofmultipleregressionAsfarasstatisticalproperties,noticehowItreattheproblemofincludinganirrelevantvariable:noseparatederivationisneeded,astheresultfollowsformTheoremIdoliketoderivetheomittedvariablebiasinthesimplecaseThisisnotmuchmoredifficultthanshowingunbiasednessofOLSinthesimpleregressioncaseunderthefirstfourGaussMarkovassumptionsItisimportanttogetthestudentsthinkingaboutthisproblemearlyon,andbeforetoomanyadditional(unnecessary)assumptionshavebeenintroducedIhaveintentionallykeptthediscussionofmulticollinearitytoaminimumThispartlyindicatesmybias,butitalsoreflectsrealityItis,ofcourse,veryimportantforstudentstounderstandthepotentialconsequencesofhavinghighlycorrelatedindependentvariablesButthisisoftenbeyondourcontrol,exceptthatwecanasklessofourmultipleregressionanalysisIftwoormoreexplanatoryvariablesarehighlycorrelatedinthesample,weshouldnotexpecttopreciselyestimatetheirceterisparibuseffectsinthepopulationIfindextensivetreatmentsofmulticollinearity,whereone“tests”orsomehow“solves”themulticollinearityproblem,tobemisleading,atbestEventheorganizationofsometextsgivestheimpressionthatimperfectmulticollinearityissomehowaviolationoftheGaussMarkovassumptions:theyincludemulticollinearityinachapterorpartofthebookdevotedto“violationofthebasicassumptions,”orsomethinglikethatIhavenoticedthatmaster’sstudentswhohavehadsomeundergraduateeconometricsareoftenconfusedonthemulticollinearityissueItisveryimportantthatstudentsnotconfusemulticollinearityamongtheincludedexplanatoryvariablesinaregressionmodelwiththebiascausedbyomittinganimportantvariableIdonotprovetheGaussMarkovtheoremInstead,IemphasizeitsimplicationsSometimes,andcertainlyforadvancedbeginners,IputaspecialcaseofProblemonamidtermexam,whereImakeaparticularchoiceforthefunctiong(x)Ratherthanhavethestudentsdirectlycomparethevariances,theyshouldappealtotheGaussMarkovtheoremforthesuperiorityofOLSoveranyotherlinear,unbiasedestimatorSOLUTIONSTOPROBLEMS(i)hspercisdefinedsothatthesmalleritis,thelowerthestudent’sstandinginhighschoolEverythingelseequal,theworsethestudent’sstandinginhighschool,thelowerishisherexpectedcollegeGPA(ii)Justplugthesevaluesintotheequation:colgpa$=−()()=(iii)ThedifferencebetweenAandBissimplytimesthecoefficientonsat,becausehspercisthesameforbothstudentsSoAispredictedtohaveascore()higher≈(iv)Withhspercfixed,Δcol=ΔsatNow,wewanttofindΔsatsuchthatΔcol=,so=(Δsat)orΔsat=()gpa$gpa$≈Perhapsnotsurprisingly,alargeceterisparibusdifferenceinSATscore–almosttwoandonehalfstandarddeviation

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  • omg995 这个是英文版pdf

    2010-09-29 05:09:21

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伍德里奇计量经济学导论教师用书及作业题答案

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