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上海财经大学研究生试题命题纸（20--20学年第学期）课程名称：计量经济学（I）（经济学院各专业）（B）命题教师：周亚虹1.Givenarandomvariablesy,andrandomvectorsx,z.Thelinearprojectionofyonxisdefinedas：.1(|)(')(')LyxxExxExy−=(a)Let,showthat(|)uyLyx=−(')0Exu=(b)showthat((|,)|)(|)LLyxzxLyx=(c)showthat((|,)|)(|)LEyxzxLyx=2．Letyandzberandomscalars,andxbea1K×randomvector,whereoneelementofxcanbeunitytoallowforanonzerointercept.Considerthepopulationmodel(|,)Eyxzxzβγ=+,2(|,)Varyxzσ=.whereinterestliesinthevector1K×β.Toruleouttrivialities,assumethat0γ≠.Inaddition,assumethatxandzareorthogonalinthepopulation:(')0Exz=.ConsidertwoestimatorsofβbasedonNindependentandidenticallydistributedobservations:(1)ˆβ(obtainedalongwithˆγ)isfromtheregressionofyonxandz;(2)β%isfromtheregressionofyonx.Bothestimatorsareconsistentforβ.(alongwiththestandardrankconditions)(a)Showthat,withoutandadditionalassumptions(exceptthoseneededtoapplythelawoflargenumbersandcentrallimittheorem),ˆ()(AVarNAVarN)ββββ−−%−)isalwayspositivesemidefinite(andusuallypositivedefinite).Therefore-fromthestandpointofasymptoticanalysis-itisalwaysbettertoincludevariablesinaregressionmodelthatareuncorrelatedwiththevariablesofinterest.(b)Considerthespecialcasewhere2(KKzxμ=−,where()KKExμ≡,andKxissymmetricallydistributed:.Then3()KKExμ−0=KβisthepartialeffectofKxon(|)EyxevaluatedatKKxμ=2)K.Isitbettertoestimatetheaveragepartialeffectwithorwithout(Kzxμ=−1K×includedasaregressor?3.SupposethatbistheleastsquarescoefficientvectorintheregressionofYonXandthatcisanyothervector.Provethatthedifferenceonthetwosumsofsquaredresidualsis()YXc'()()'()()''(YXcYXbYXbcbXXcb−−−−−=−−)Provethatthisdifferenceispositive.4.Supposetheregressionmodelisiiyxiαβε=++,()(1/)exp(/)0ifελελ=−.Thisisratherapeculiarmodelinthatallofthedisturbancesareassumedtobepositive.Notethatthedisturbanceshave()iEελ=.Showthattheleastsquaresconstanttermisunbiasedbuttheinterceptisbiased.5．Arerentratesinfluencedbythestudentpopulationinacollegetown?LetrentbetheaveragemonthlyrentpaidonrentalunitsinacollegetownintheUnitedStates.Letpopdenotethetotalcitypopulation,avginctheaveragecityincome,andpctstuthestudentpopulationasapercentageofthetotalpopulation.Onemodeltotestforarelationshipis01)+log(rentβ23log(pop)+log(avginc)+pctstu+u=βββ(a)Statethenullhypothesisthatsizeofthestudentbodyrelativetothepopulationhasnoceterisparibuseffectonmonthlyrents.Statethealternativethatthereisaneffect.(b)Whatsignsdoyouexpectfor1βand2β?(c)Theequationestimatedusing1990datafromRENTAL.RAWfor64collegetownsisˆlog(rent)0.043=+0.066log(pop)+0.507log(avginc)+0.0056pctstu(.844)_(.039)log(pop)_(.081)log(avginc)_(.0017)pctstu264,0.458nR==Whatiswrongwiththestatement:“A10%increaseinpopulationisassociatedwithabouta6.6%increaseinrent”?(d)Testthehypothesisstatedinpart(a)atthe1%level.Solution1:(a)Bydefinition,1(')['((|))](')(')(')(')0ExuExyLyxExyExxExxExy−=−=−=(b)let(|,),bypart(a),vyLyxz=−(')0Exv=.So1(|)(')(')0LvxxExxExv−==(|)((|,)|)((|,)|)(|)((|,)|)LyxLLyxzvxLLyxzxLvxLLyxzx=+=+=(c)let,(|,)vyEyxz=−%(|,)0Evxz=%.Hence1(|)(')(')0LvxxExxExv−==%%(|)((|,)|)((|,)|)(|)((|,)|)LyxLEyxzvxLEyxzxLvxLEyxzx=+=+=%%Solution2:(a)letthen(,)wxz=(|)Eywwδ=.Since2(|)Varywσ=,2ˆ()[(')]AVarNEwwδδσ1−−=whereˆˆˆ(',)'δβγ=K.Importantly,because,isblockdiagonal,withtheupper(')0Exz=(')EwwK×blockgives21ˆ()[(')AVarNExxββσ]−−=Next,weneedtofind(AVarN)ββ−%.Itishelpfultowriteyxvβ=+,wherevzuγ=+and.Because(|,)uyEyxz≡−(')0Exz=and,(')Exu0=('Exv)0=.Further,222222(|)(|)(|)2(|)(|)EvxEzxEuxEzuxEzx2,γγγ=++=σ+whereweuseand(|,)(|,)0EzuxzzEuxz==22(|,)(|,)EuxzVaryxzσ==.Unlessisconstant,theequation2(|)Ezxyxvβ=+generallyviolatesthehomoskedasticityassumption.So,withoutfurtherassumption,12()[(')](')[(')]AVarNExxEvxxExxββ1−−−=%Nowwecanshowˆ()(AVarNAVarN)ββ−−−%ββisalwayspositivesemidefinitebywritingˆ()()AVarNAVarNβββ−−−%β1−01212[(')](')[(')][(')]ExxEvxxExxExxσ−−=−22(')Ezxxγ=≥(b)If2()KKzxμ=−,.Further,2(|)(|,)()KKEyxEyxzxxβγμ==+−(|)2()KKKKEyxxxβγμ∂=+−∂Hence(|)|KKxKKEyxxμβ=∂=∂.If,usingtheconclusionofpart(a),itisbettertoestimatetheaveragepartialeffectwith(')0Exz=2(KKzx)μ=−includedasaregressor.Solution3:Writecas.Then,thesumofsquaredresidualsbasedoncis()bcb+−()'()()'()()''()2()''(YXcYXcYXbYXbcbXXcbcbXYXb−−=−−+−−+−−)=ixBut,thethirdtermiszero,as.Therefore,2()''()2()''0cbXYXbcbXe−−=−()'()'()''()YXcYXceecbXXcb−−−=−−Therighthandsideisnecessarilypositive.Thisconfirmswhatweknewattheoutset,leastsquaresisleastsquares.Solution4:Wecouldwritetheregressionas**()()iiiiyxαλβελαβε=+++−=++Then,weknowthe*()iEελ=,andthatitisindependentofix.Therefore,thesecondformofthemodelsatisfiesallofourassumptionsfortheclassicalregression.Ordinaryleastsquareswillgiveunbiasedestimatorsof*αandβ.Aslongasλisnotzero,theconstanttermwilldifferfromα.Solution5:(a)H0:3β=0.H1:3β≠0.(b)Otherthingsequal,alargerpopulationincreasesthedemandforrentalhousing,whichshouldincreaserents.Thedemandforoverallhousingishigherwhenaverageincomeishigher,pushingupthecostofhousing,includingrentalrates.(c)Thecoefficientonlog(pop)isanelasticity.Acorrectstatementisthat“a10%increaseinpopulationincreasesrentby.066(10)=.66%.”(d)Withdf=64–4=60,the1%criticalvalueforatwo-