上海財經大學研究生試題命題紙

（20 --20 學年第學期）

課程名稱：計量經濟學（I）（經濟學院各專業）（B） 命題教師：周亞虹

Considertheconditionalexpectation

E(y|x,z)=g(x)+zb

Whereg(.)isageneralfunctionofxandbisaM·1vector.Showthat

E(y%|z%)=z%b

Wherey%=y-E(y|x)andz%=z-E(z|x)

TheTwoVariableRegression.Fortheregressionmodely=α+βx+ε,

Showthattheleastsquaresnormalequationsimplyi ei=0andixiei=0

Showthatthesolutionfortheconstanttermis a=Y－bX

n n

(c）showthatthesolutionforbis b=[(Xi-X)(Yi-Y)]/(Xi-X)2

i=1 i=1

(d)Provethatthesetwovaluesuniquelyminimizethesumofsquaresbyshowingthatthediagonalelementsofthesecondderivativesmatrixofthesumofsquareswithrespecttotheparametersarebothpositveandthatthedeterminantis

n n

4n[(X2)－nX2]=4n[(X-X)2]

i i

i=1 i=1

Whichispositiveunlessallvaluesofxarethesame.

Supposethatyouwishtoestimatetheeffectofclassattendanceonstudentperformance,Amodeltoexplainthestandardizedoutcomeonafinalexam(stndfnl)intermsofpercentageofclassesattended,priorcollegegradepointaverage,andACTscoreis

stndfnl = b0+b1atndrte+b2priGPA+b3ACT+b4priGPAatndrte+u

Letdistbethedistancefromthestudents’livingquarterstothelecturehall.Doyouthinkdistisuncorrelatedwithu?

Assumingthatdistanduareuncorrelated,whatotherassumptionmustdistsatisfyinordertobeavalidIVforatndrte?

Suppose,weaddtheinteractiontermpriGPAatndrte:

stndfnl = b0+b1atndrte+b2priGPA+b3ACT+b4priGPAatndrte+u

Ifatndrteiscorrelatedwithu,then,ingeneral,soispriGPAatndrte.What

mightbeagoodIVforpriGPAatndrte?

K

ProvethatE[b'b]=b'b+s2(1/lk)wherebistheordinaryleastsquaresestimatorand

k=1

lkisacharacteristicrootofX'X.

Themedianstartingsalaryfornewlawschoolgraduatesisdeterminedby

log(salary)=b0+b1LSAT+b2GPA+b3log(libvol)+b4log(cost)+b5rank+u

whereLSATisthemedianLSATscoreforthegraduatingclass,GPAisthemediancollegeGPAfortheclass,libvolisthenumberofvolumesinthelawschoollibrary,costistheannualcostofattendinglawschool,andrankisalawschoolranking(withrank=1beingthebest).

Explainwhyweexpectb5￡0.

Whatsignsdoyouexpectfortheotherslopeparameters?Justifyyouranswers.

UsingthedatainLAWSCH85.RAW,theestimatedequationis

log(sal?ary)=8.34+.0047LSAT+0.248GPA+.095log(libvol)+.038log(cost)-0.0033rank

n=136,R2=0.842

WhatisthepredictedceterisparibusdifferenceinsalaryforschoolswithamedianGPAdifferentbyonepoint?(Reportyouranswerasapercentage.)

Interpretthecoefficientonthevariablelog(libvol).

Wouldyousayitisbettertoattendahigherrankedlawschool?Howmuchisadifferenceinrankingof20worthintermsofpredictedstartingsalary?

Solution1:

Let u=y-E(y|x,z),then

y=g(x)+zb+u whereE(u|x,z)=0

E(y|x)=g(x)+E(z|x)b

y-E(y|x)=[z-E(z|x)]b+ue.g.y%=z%b+uwhereE(u|z%)=0

SoE(y%|z%)=z%b

Solution2;

Theleastsquaresestimationbis

b=(XTX)-1XTy,

andtheresidualvectoreise=y-Xb=(I-X(XTX)-1XT)y=My.

Thenit’seasytoseethat

XTe=XTMy=(XT-XTX(XTX)-1XT)y=0y=0.

ForthetwoVariableRegression,Xhastwocolumns.Thefirstcolumnistheunitaryvectorandthesecondcolumnisx.Wethusgetthat

iei=0andi xiei=0.

Fortheequationyi=a+bxi+ei,i=1,L,n,whereaandbaretheestimationsof

a,b.Summarizingalltheseequationstogetherwiththeresultin1)yields

n n n n

yi=na+bxi+ei=na+bxi,

i=1 i=1 i=1 i=1

whichimpliesimmediatelya=Y－bX,with

n n

X=xi/n,Y=yi/n.

i=1 i=1

n

Notethataandbshallminimizethefunction f(a,b)=(y-a-bx)2.Thefirstorderoptimality

i i

conditiongives

i=1

n n n n n

i

yi-an-bxi=0, xiyi-axi-bx2=0,

i=1 i=1 i=1 i=1 i=1

fromwhichoneconcludesimmediately

n n

b=[(Xi-X)(Yi-Y)]/(Xi-X)2

i=1 i=1

ThesecondorderderivativematrixofthesumofsquareswithregardstoaandbisS=XTX,whereX=(i,x),i=(1,L,1)T,x=(x1,L,xn)T.Thenthereis

iTiiTx n

n

xi

S=XTX=iTx xTx= i=1 .

n n

i i

x x2

i=1 i=1

Thatmeans,thediagonalelementsofSareboth positive.ThedeterminantofSis

2

n n 2 n n

Det(S)=nx2-x=n(x-X+X)2-(nX)=n(x-X)2,

i i i i

n n

i=1 i=1 i=1 i=1

fromthefactorthat(xi-X)X=X(xi-X)=0,Whichispositiveunlessallvaluesofxare

i=1 i=1

thesame.

Solution3:

Itseemsreasonabletoassumethatdistanduareuncorrelatedbecauseclassroomsarenotusuallyassignedwithconvenienceforparticularstudentsinmind.

Thevariabledistmustbepartiallycorrelatedwithatndrte. Moreprecisely,inthereducedform

atndrte = p0+p1priGPA+p2ACT+p3dist+v,

wemusthavep3?0. GivenasampleofdatawecantestH0:p3=0againstH1:p3?0usingattest.

Wenowneedinstrumentalvariablesforatndrteandtheinteractionterm,priGPAatndrte. (Even

thoughpriGPAisexogenous,atndrteisnot,andsopriGPAatndrteisgenerallycorrelatedwithu.)UndertheexogeneityassumptionthatE(u|priGPA,ACT,dist)=0,anyfunctionofpriGPA,ACT,anddist

isuncorrelatedwithu. Inparticular,theinteractionpriGPAdistisuncorrelatedwithu. Ifdistis

partiallycorrelatedwithatndrtethenpriGPAdistispartiallycorrelatedwithpriGPAatndrte. So,wecanestimatetheequation

stndfnl = b0+b1atndrte+b2priGPA+b3ACT+b4priGPAatndrte+u

by2SLSusingIVsdist,priGPA,ACT,andpriGPAdist. Itturnsoutthisisnotgenerallyoptimal. Itmay

bebettertoaddpriGPA2andpriGPAACTtotheinstrumentlist.Thiswouldgiveusoveridentifyingrestrictionstotest.

Solution4:

Wewritb=b+(X'X)-1X'e,so

b'b=b'b+e'X(X'X)-1(X'X)-1X'e+2b'(X'X)-1X'e

Theexpectedvalueofthetermiszero,andthefirstisnonstochastic.Tofindtheexpectationofthesecondterm,usethetrace,andpermutee'Xinsidethetraceoperator.Thus,

E(b'b)=b'b+E[e'X(X'X)-1(X'X)-1X'e]

=b'b+E[tr{e'X(X'X)-1(X'X)-1X'e}]

=b'b+E[tr{(X'X)-1X'ee'X(X'X)-1}]

=b'b+tr{(X'X)-1X'E[ee']X(X'X)-1}

=b'b+s2tr{(X'X)-1X'X(X'X)-1}

K

=b'b+s2tr{(X'X)-1}=b'b+s2(1/lk)

k=1

Thetraceoftheinverseequalsthesumofthecharacteristicrootsoftheinverse,whicharethereciprocalsofthecharacteristicrootsofX'X

Solution5:

Alargerrankforalawschoolmeansthattheschoolhaslessprestige;thislowersstartingsalaries. Forexample,arankof100meansthereare99schoolsthoughttobebetter.

b1>0,b2>0. BothLSATandGPAaremeasuresofthequalityofth