计量经济学导论(伍德里奇第三版)课后习题答案

CHAPTER1

SOLUTIONSTOPROBLEMS

1.1(i)Ideally,wecouldrandomlyassignstudentstoclassesofdifferentsizes.

Thatis,eachstudentisassignedadifferentclasssizewithoutregardtoanystudent

characteristicssuchasabilityandfamilybackground.Forreasonswewillseein

Chapter2,wewouldlikesubstantialvariationinclasssizes(subject,ofcourse,to

ethicalconsiderationsandresourceconstraints).

(ii)Anegativecorrelationmeansthatlargerclasssizeisassociatedwithlower

performance.Wemightfindanegativecorrelationbecauselargerclasssizeactually

hurtsperformance.

However,withobservationaldata,thereareotherreasonswemightfindanegative

relationship.Forexample,childrenfrommoreaffluentfamiliesmightbemore

likelytoattendschoolswithsmallerclasssizes,andaffluentchildrengenerallyscore

betteronstandardizedtests.Anotherpossibilityisthat,withinaschool,aprincipal

mightassignthebetterstudentstosmallerclasses.Or,someparentsmightinsisttheir

childrenareinthesmallerclasses,andthesesameparentstendtobemoreinvolvedin

theirchildren'seducation.

(iii)Giventhepotentialforconfoundingfactors-someofwhicharelistedin(ii)一

findinganegativecorrelationwouldnotbestrongevidencethatsmallerclasssizes

actuallyleadtobetterperformance.Somewayofcontrollingfortheconfounding

factorsisneeded,andthisisthesubjectofmultipleregressionanalysis.

1.2(i)Hereisonewaytoposethequestion:Iftwofirms,sayAandB,areidentical

inall

respectsexceptthatfirmAsuppliesjobtrainingonehourperworkermorethanfirm

B,byhowmuchwouldfirmA'soutputdifferfromfirmB's?

(ii)Firmsarelikelytochoosejobtrainingdependingonthecharacteristicsof

workers.Someobservedcharacteristicsareyearsofschooling,yearsintheworkforce,

andexperienceinaparticularjob.Firmsmightevendiscriminatebasedonage,

gender,orrace.Perhapsfirmschoosetooffertrainingtomoreorlessableworkers,

where—abilityIImightbedifficultto

quantifybutwhereamanagerhassomeideaabouttherelativeabilitiesofdifferent

employees.Moreover,differentkindsofworkersmightbeattractedtofirmsthat

offermorejobtrainingonaverage,andthismightnotbeevidenttoemployers.

(iii)Theamountofcapitalandtechnologyavailabletoworkerswouldalsoaffect

output.So,twofirmswithexactlythesamekindsofemployeeswouldgenerally

havedifferentoutputsiftheyusedifferentamountsofcapitalortechnology.The

qualityofmanagerswouldalsohaveaneffect.

(iv)No,unlesstheamountoftrainingisrandomlyassigned.Themanyfactors

listedinparts(ii)and(iii)cancontributetofindingapositivecorrelationbetween

outputandtrainingevenifjobtrainingdoesnotimproveworkerproductivity.

1.3Itdoesnotmakesensetoposethequestionintermsofcausality.Economists

wouldassumethatstudentschooseamixofstudyingandworking(andother

activities,suchasattendingclass,

1

leisure,andsleeping)basedonrationalbehavior,suchasmaximizingutilitysubject

totheconstraintthatthereareonly168hoursinaweek.Wecanthenusestatistical

methodstomeasuretheassociationbetweenstudyingandworking,including

regressionanalysisthatwecoverstartinginChapter2.Butwewouldnotbe

claimingthatonevariable—causesIItheother.Theyarebothchoicevariablesof

thestudent.

CHAPTER2

SOLUTIONSTOPROBLEMS

2.1(i)Income,age,andfamilybackground(suchasnumberofsiblings)arejusta

few

possibilities.Itseemsthateachofthesecouldbecorrelatedwithyearsof

education.(Incomeandeducationareprobablypositivelycorrelated;ageand

educationmaybenegativelycorrelatedbecausewomeninmorerecentcohortshave,

onaverage,moreeducation;andnumberofsiblingsandeducationareprobably

negativelycorrelated.)

(ii)Notifthefactorswelistedinpart(i)arecorrelatedwitheduc.Becausewe

wouldliketoholdthesefactorsfixed,theyarepartoftheerrortenn.Butifuis

correlatedwitheducthenE(u|educ)0,andsoSLR.4fails.

2.2Intheequationy=0+lx+u,addandsubtract0fromtherighthand

sidetogety=(0+0)+lx+(u0),Callthenewerrore=u0,so

thatE(e)=0.Thenewinterceptis0+0,buttheslopeisstill1.

2.3(i)Letyi=GPAi,xi=ACTi,andn=8.Then=25.875,=3.2125,(xi

-)(yi-)=iIn

八=5.8125,and(xi-)2=56.875.Fromequation(2.9),weobtaintheslopeas

liIn

八=一5.8125/56.875.1022,roundedtofourplacesafterthedecimal.From

(2.17),0

-3.2125-(.1022)25.875.5681.Sowecanwrite1

.5681+.1022ACTGPAn=8.

TheinterceptdoesnothaveausefulinterpretationbecauseACTisnotclosetozero

forthe

increasesby.1022(5)=.511.populationofinterest.IfACTis5pointshigher,

GPA

(ii)Thefittedvaluesandresiduals——roundedtofourdecimalplaces——aregiven

alongwiththeobservationnumberiandGPAinthefollowingtable:

2

Youcanverifythattheresiduals,asreportedinthetable,sumto.0002,whichis

prettyclosetozerogiventheinherentroundingerror.

=.5681+.1022(20)2.61.(iii)WhenACT=20,GPA

\2,isabout.4347(roundedtofourdecimalplaces),(iv)Thesumofsquared

residuals,u

i1

nn

andthetotalsumofsquares,(yi-)2,isabout1.0288.SotheR-squaredfrom

the

i1

regressionis

R2=1-SSR/SST1-(.4347/1.0288).577.

Therefore,about57.7%ofthevariationinGPAisexplainedbyACTinthissmall

sampleofstudents.

2.4(i)Whencigs=0,predictedbirthweightis119.77ounces.Whencigs=20,

bwght=109.49.

Thisisaboutan8.6%drop.

(ii)Notnecessarily.Therearemanyotherfactorsthatcanaffectbirthweight,

particularlyoverallhealthofthemotherandqualityofprenatalcare.Thesecouldbe

correlatedwith

cigarettesmokingduringbirth.Also,somethingsuchascaffeineconsumptioncan

affectbirthweight,andmightalsobecorrelatedwithcigarettesmoking.

(iii)Ifwewantapredictedbwghtof125,thencigs=(125-119.77)/(-.524)

-10.18,orabout-10cigarettes!Thisisnonsense,ofcourse,anditshowswhat

happenswhenwearetryingtopredictsomethingascomplicatedasbirthweightwith

onlyasingleexplanatoryvariable.Thelargestpredictedbirthweightisnecessarily

119.77.Yetalmost700ofthebirthsinthesamplehadabirthweighthigherthan

119.77.

3

(iv)1,176outof1,388womendidnotsmokewhilepregnant,orabout84.7%.

Becauseweareusingonlycigstoexplainbirthweight,wehaveonlyonepredicted

birthweightatcigs=0.Thepredictedbirthweightisnecessarilyroughlyinthe

middleoftheobservedbirthweightsatcigs=0,andsowewillunderpredicthigh

birthrates.

2.5(i)Theinterceptimpliesthatwheninc=0,consispredictedtobenegative

$124.84.This,ofcourse,cannotbetrue,andreflectsthatfactthatthisconsumption

functionmightbeapoorpredictorofconsumptionatverylow-incomelevels.On

theotherhand,onanannualbasis,$124.84isnotsofarfromzero.

=-124.84+.853(30,000)=25,465.16dollars,(ii)Justplug30,000intothe

equation:cons

(iii)TheMPCandtheAPCareshowninthefollowinggraph.Eventhoughthe

interceptisnegative,thesmallestAPCinthesampleispositive.Thegraphstartsat

anannualincomelevel

increaseshousingprices.

(ii)Ifthecitychosetolocatetheincineratorinanareaawayfrommoreexpensive

neighborhoods,thenlog(dist)ispositivelycorrelatedwithhousingquality.This

wouldviolateSLR.4,andOLSestimationisbiased.

(iii)Sizeofthehouse,numberofbathrooms,sizeofthelot,ageofthehome,and

qualityoftheneighborhood(includingschoolquality),arejustahandfuloffactors.

Asmentionedinpart(ii),thesecouldcertainlybecorrelatedwithdist[andlog(dist)].

4

2.7(i)Whenweconditiononinc

E(u|inc

e|inc

)=E(e|inc

0becauseE(e|inc)=E(e)=0.

(ii)Again,whenweconditiononinc

Var(u|inc

e|inc

2Var(e|inc)=e2incbecauseVar(e|inc)=e2.

(iii)Familieswithlowincomesdonothavemuchdiscretionaboutspending;

typically,alow-incomefamilymustspendonfood,clothing,housing,andother

necessities.Higherincomepeoplehavemorediscretion,andsomemightchoose

moreconsumptionwhileothersmoresaving.Thisdiscretionsuggestswider

variabilityinsavingamonghigherincomefamilies.

2.8(i)Fromequation(2.66),

nn21=xiyi/xi.i1i1

Plugginginyi=0+Ixi+uigives

nn21=xi(0Ixiui)xi.i1i1

Afterstandardalgebra,thenumeratorcanbewrittenas

0xi1xxiui.2nnn

ililii1

asPuttingthisoverthedenominatorshowswecanwrite1

nn2nn21=0xi/xi+1+

xiui/xi.i1i1i1i1

Conditionalonthexi,wehave

nn2E(1)=0xi/xi+1i1i1

isgivenbythefirstterminthisequation,becauseE(ui)=0foralli.

Therefore,thebiasin1

Thisbiasisobviouslyzerowhen0=0.Itisalsozerowhenxi=0,whichis

thesameas

iIn

=0.Inthelattercase,regressionthroughtheoriginisidenticaltoregressionwith

anintercept.

5

inpart(i)wehave,conditionalonthexi,(ii)Fromthelastexpressionfor1

)Var(1

n2nn2n2

=xiVarxiui=xixiVar(ui)

i1i1i1i1

2

2

2

n22n22=xixi=/xi2.

i1i1i1

nnn222八(iii)From(2.57),Var(1)=/(xi).Fromthehint,xi

(xi)2,andso

ili1i1

n

)Var()Amoredirectwaytoseethisistowrite(x)2=x2n()2,

whichVar(iill

i1

i1

islessthanxi2unless=0.

i1

n

increasesasincreases(holdingthesumofthe(iv)Foragivensamplesize,the

biasin1

"increasesrelativetoVar().Thebiasinx2fixed).Butasincreases,

thevarianceof

i

111

or'onameansquarederrorisalsosmallwhen0issmall.Therefore,

whetherweprefer11basisdependsonthesizesof0,,andn(inadditiontothe

sizeofxi2).

iIn

2.9(i)Wefollowthehint,notingthatcly=cl(thesampleaverageofclyiiscl

timesthesampleaverageofyi)andc2x=c2.Whenweregressclyionc2xi

(includinganintercept)weuseequation(2.19)toobtaintheslope:

1

(excx)(cc)cc(x)(y)

2in

2

li

1

12n

i

i

i1

nn

i1

(exc)

2i

2

iIn

n

2

c(x)

2

2

i

i1

2

clilc2

(xi)(yi)

2

i

(x)

i1

eri.c2

=(cl)-(c2)=(cl)-[(cl/c2)A](c2)=From(2.17),weobtainthe

interceptasOil

八)becausetheinterceptfromregressingyionxiis(-A)=cl").cl(-

1

1

(ii)Weusethesameapproachfrompart(i)alongwiththefactthat(cly)=c1+

and

(c2x)=c2+.Therefore,(clyi)(cly)=(cl+yi)一(cl+)=yi-

and(c2+xi)-

6

(c2x)=xi-.Soclandc2entirelydropoutoftheslopeformulaforthe

regressionof(cl+yi)

=(cy)-=1Theinterceptis(cx)=(cl+)-"(c2+)=on

(c2+xi),and1101121

八+cl-c2")+c1-c2'=",whichiswhatwewantedtoshow.(0111

(iii)Wecansimplyapplypart(ii)becauselog(clyi)log(cl)log(yi).Inother

words,replaceclwithlog(cl),yiwithlog(yi),andsetc2=0.

(iv)Again,wecanapplypart(ii)withcl=0andreplacingc2withlog(c2)andxi

withlog(xi).八and八aretheoriginalinterceptandslope,then八and

「log(c)\If01110021

2.10(i)Thisderivationisessentiallydoneinequation(2.52),once(1/SSTx)is

broughtinsidethesummation(whichisvalidbecauseSSTxdoesnotdependoni).

Then,justdefine

widi/SSTx.

八,)E[(八)],weshowthatthelatteriszero.But,frompart(i),(ii)

BecauseCov(111

nnwE(u).Becausetheuarepairwiseuncorrelated人)]

=EE[(wuilliiliiili

(theyareindependent),E(ui)E(ui2/n)2/n(becauseE(uiuh)0,ih).

Therefore,iIwiE(ui)ilwi(2/n)(2/n)iIwi0.

nnnA八and,pluggingin(iii)TheformulafortheOLSintercept

is010"()(人)givesOOHOITandare

uncorrelated,(iv)Because1

[Var()Var(八)22/n(2/SST)22/n22/SST,Var(Olxx

whichiswhatwewantedtoshow.

八)2[SST/n2]/SST(v)UsingthehintandsubstitutiongivesVar(Oxx

12222122nx/SSTnx/SSTx.xiliili

2.11(i)Wewouldwanttorandomlyassignthenumberofhoursinthepreparation

coursesothathoursisindependentofotherfactorsthataffectperformanceonthe

SAT.Then,wewouldcollectinformationonSATscoreforeachstudentinthe

experiment,yieldingadataset{(sati,hoursi):iwherenisthenumberof

studentswecanaffordtohaveinthestudy.Fromequation(2.7),weshouldtryto

getasmuchvariationinhoursiasisfeasible.

nn

7

(ii)Herearethreefactors:innateability,familyincome,andgeneralhealthonthe

dayoftheexam.Ifwethinkstudentswithhighernativeintelligencethinktheydo

notneedtopreparefbrtheSAT,thenabilityandhourswillbenegativelycorrelated.

Familyincomewouldprobablybepositivelycorrelatedwithhours,becausehigher

incomefamiliescanmoreeasilyafford

preparationcourses.Rulingoutchronichealthproblems,healthonthedayofthe

examshouldberoughlyuncorrelatedwithhoursspentinapreparationcourse.

(iii)Ifpreparationcoursesareeffective,1shouldbepositive:otherfactorsequal,

an

increaseinhoursshouldincreasesat.

(iv)Theintercept,0,hasausefulinterpretationinthisexample:becauseE(u)=0,

0istheaverageSATscorefbrstudentsinthepopulationwithhours=0.

CHAPTER3

SOLUTIONSTOPROBLEMS

3.1(i)hspercisdefinedsothatthesmalleritis,thelowerthestudentJsstandingin

highschool.Everythingelseequal,theworsethestudent?sstandinginhighschool,

thelowerishis/herexpectedcollegeGPA.

(ii)Justplugthesevaluesintotheequation:

=1.392.0135(20)+.00148(1050)=2.676.colgpa

(iii)ThedifferencebetweenAandBissimply140timesthecoefficientonsat,

becausehspercisthesamefbrbothstudents.SoAispredictedtohavea

score.00148(140).207higher.

(iv)Withhspercfixed,colgpa=.00148sat.Now,wewanttofindsat

suchthat

=.5,so.5=.00148(sat)orsat=.5/(.00148)338.Perhapsnot

surprisingly,acolgpa

largeceterisparibusdifferenceinSATscore-almosttwoandone-halfstandard

deviations-isneededtoobtainapredicteddifferenceincollegeGPAorahalfapoint.

3.2(i)Yes.Becauseofbudgetconstraints,itmakessensethat,themoresiblings

thereareinafamily,thelesseducationanyonechildinthefamilyhas.Tofindthe

increaseinthenumberofsiblingsthatreducespredictededucationbyoneyear,we

solve1=.094(sibs),sosibs=1/.09410.6.

(ii)Holdingsibsandfeducfixed,onemoreyearofmother?seducationimplies.131

yearsmoreofpredictededucation.Soifamotherhasfourmoreyearsofeducation,

hersonispredictedtohaveaboutahalfayear(.524)moreyearsofeducation.

8

(iii)Sincethenumberofsiblingsisthesame,butmeducandfeducareboth

different,thecoefficientsonmeducandfeducbothneedtobeaccountedfbr.The

predicteddifferenceineducationbetweenBandAis.131(4)+.210(4)=1.364.

3.3(i)Ifadultstradeoffsleepfbrwork,moreworkimplieslesssleep(otherthings

equal),so1<0.

(ii)Thesignsof2and3arenotobvious,atleasttome.Onecouldarguethat

more

educatedpeopleliketogetmoreoutoflife,andso,otherthingsequal,theysleep

less(2<0).Therelationshipbetweensleepingandageismorecomplicated

thanthismodelsuggests,andeconomistsarenotinthebestpositiontojudgesuch

things.

(iii)Sincetotwrkisinminutes,wemustconvertfivehoursintominutes:totwrk

=5(60)=300.Thensleepispredictedtofallby.148(300)=44.4minutes.Fora

week,45minuteslesssleepisnotanoverwhelmingchange.

(iv)Moreeducationimplieslesspredictedtimesleeping,buttheeffectisquite

small.Ifweassumethedifferencebetweencollegeandhighschoolisfouryears,

thecollegegraduatesleepsabout45minuteslessperweek,otherthingsequal.

(v)Notsurprisingly,thethreeexplanatoryvariablesexplainonlyabout11.3%ofthe

variationinsleep.Oneimportantfactorintheerrortermisgeneralhealth.

Anotherismaritalstatus,andwhetherthepersonhaschildren.Health(howeverwe

measurethat),maritalstatus,andnumberandagesofchildrenwouldgenerallybe

correlatedwithtotwrk.(Forexample,lesshealthypeoplewouldtendtoworkless.)

3.4(i)Alargerrankfbralawschoolmeansthattheschoolhaslessprestige;this

lowersstartingsalaries.Forexample,arankof100meansthereare99schools

thoughttobebetter.

(ii)1>0,2>0.BothLSATandGPAaremeasuresofthequalityofthe

enteringclass.Nomatterwherebetterstudentsattendlawschool,weexpectthemto

earnmore,onaverage.3,4>0.Thenumberofvolumesinthelawlibrary

andthetuitioncostarebothmeasuresoftheschoolquality.(Costislessobvious

thanlibraryvolumes,butshouldreflectqualityofthefaculty,physicalplant,andso

on.)

(iii)ThisisjustthecoefficientonGPA,multipliedby100:24.8%.

(iv)Thisisanelasticity:aonepercentincreaseinlibraryvolumesimplies

a.095%increaseinpredictedmedianstartingsalary,otherthingsequal.

(v)Itisdefinitelybettertoattendalawschoolwithalowerrank.IflawschoolA

hasaranking20lessthanlawschoolB,thepredicteddifferenceinstartingsalaryis

100(.0033)(20)=

6.6%higherfbrlawschoolA.

9

3.5(i)No.Bydefinition,study+sleep+work+leisure=168.Therefore,ifwe

changestudy,wemustchangeatleastoneoftheothercategoriessothatthesumis

still168.

(ii)Frompart(i),wecanwrite,say,studyasaperfectlinearfunctionoftheother

independentvariables:study=168sleepworkleisure.Thisholdsfbr

everyobservation,soMLR.3violated.

(iii)Simplydroponeoftheindependentvariables,sayleisure:

GPA=0+1study+2sleep+3work+u.

Now,fbrexample,1isinterpretedasthechangeinGPAwhenstudyincreasesby

onehour,

wheresleep,work,anduareallheldfixed.Ifweareholdingsleepandworkfixed

butincreasingstudybyonehour,thenwemustbereducingleisurebyonehour.The

otherslopeparametershaveasimilarinterpretation.

1)=E(八)=八+3.6Conditioningontheoutcomesoftheexplanatory

variables,wehaveE(21

)=1+2=.2+E(E(121

3.7Only(ii),omittinganimportantvariable,cancausebias,andthisistrueonly

whentheomittedvariableiscorrelatedwiththeincludedexplanatoryvariables.The

homoskedasticityassumption,MLR.5,playednoroleinshowingthattheOLS

estimatorsareunbiased.

八.)Further,the(Homoskedasticitywasusedtoobtaintheusualvarianceformulas

fbrthej

degreeofcollinearitybetweentheexplanatoryvariablesinthesample,evenifitis

reflectedinacorrelationashighas.95,doesnotaffecttheGauss-Markov

assumptions.Onlyifthereisaperfectlinearrelationshipamongtwoormore

explanatoryvariablesisMLR.3violated.

3.8WecanuseTable3.2.Bydefinition,2>0,andbyassumption,

Corr(xl,x2)<0.

:E()<.Thismeansthat,onaverageacrossTherefore,thereisa

negativebiasin111

differentrandomsamples,thesimpleregressionestimatorunderestimatestheeffect

ofthe

)isnegativeeventhough>0.trainingprogram.Itisevenpossiblethat

E(11

3.9(i)1<0becausemorepollutioncanbeexpectedtolowerhousingvalues;

notethat1istheelasticityofpricewithrespecttonox.2isprobablypositive

becauseroomsroughlymeasuresthesizeofahouse.(However,itdoesnotallowus

todistinguishhomeswhereeachroomislargefromhomeswhereeachroomissmall.)

(ii)Ifweassumethatroomsincreaseswithqualityofthehome,thenlog(nox)and

roomsarenegativelycorrelatedwhenpoorerneighborhoodshavemorepollution,

somethingthatisoftentrue.WecanuseTable3.2todeterminethedirectionofthe

bias.If2>0and

hasadownwardbias.Butbecause<0,Corr(xl,x2)<0,thesimple

regressionestimator11

10

)thismeansthatthesimpleregression,onaverage,overstatestheimportanceof

pollution.[E(1

ismorenegativethan1.](iii)Thisiswhatweexpectfromthetypicalsample

basedonouranalysisinpart(ii).Thesimpleregressionestimate,1.043,ismore

negative(largerinmagnitude)thanthemultipleregressionestimate,.718.As

thoseestimatesareonlyforonesample,wecanneverknowwhichiscloserto1.

Butifthisisa—typicalIIsample,1iscloserto.718.

3.10(i)Becausexlishighlycorrelatedwithx2andx3,andtheselattervariables

havelargepartialeffectsony,thesimpleandmultipleregressioncoefficientsonxl

candifferbylargeamounts.Wehavenotdonethiscaseexplicitly,butgivenequation

(3.46)andthediscussionwithasingleomittedvariable,theintuitionispretty

straightforward.

and八tobesimilar(subject,ofcourse,towhatwemeanby(ii)Herewe

wouldexpect11

—almostuncorrelatedII).Theamountofcorrelationbetweenx2andx3doesnot

directlyeffectthemultipleregressionestimateonxlifxlisessentiallyuncorrelated

withx2andx3.(iii)Inthiscaseweare(unnecessarily)introducing

multicollinearityintotheregression:x2andx3havesmallpartialeffectsonyandyet

x2andx3arehighlycorrelatedwithxl.Adding

")isxandxlikeincreasesthestandarderrorofthecoefficientonxsubstantially,so

se(

2

311

).likelytobemuchlargerthanse(1

(iv)Inthiscase,addingx2andx3willdecreasetheresidualvariancewithout

causing

八)muchcollinearity(becausexlisalmostuncorrelatedwithx2andx3),sowe

shouldseese(1

).Theamountofcorrelationbetweenxandxdoesnotdirectlyaffectsmaller

thanse(

1

2

3

)se(1

3.11Fromequation(3.22)wehave

1

r-y

i1

n

n

ili

r”

i1

2il

'ilaredefinedintheproblem.Asusual,wemustpluginthetruemodelforyi:

wherether

11

1

ili1

n

Ixil2xi23xi3ui

K

i1

n

2

il

'il=0,rAilxi2=0,andr'ilxil=Thenumeratorofthisexpressionsimplifies

becauser

i1

i1

i1

n

n

n

i1

n

2

iTilaretheresidualsfromtheregressionofxilon.Theseallfollowfromthe

factthatther

"ilhavezerosampleaverageandareuncorrelatedinsamplewithxi2.Sothe

numeratorxi2:ther

canbeexpressedasof

1

3Clxi3r'ilui.1r

2

ili1

i1

i1

n

n

n

Puttingthesebackoverthedenominatorgives

11

i1

3n

r"xrA

i1

2

il

n

ili3

Aru

iIn

n

li

i1

2il

Conditionalonallsamplevaluesonxl,x2,andx3,onlythelasttermisrandomdue

toitsdependenceonui.ButE(ui)=0,andso

)=+E(113

rX

i1

n

n

ili3

9

i1

2il

whichiswhatwewantedtoshow.Noticethatthetermmultiplying3isthe

regression

"il.coefficientfromthesimpleregressionofxi3onr

3.12(i)Theshares,bydefinition,addtoone.Ifwedonotomitoneoftheshares

thentheequationwouldsufferfromperfectmulticollinearity.Theparameterswould

nothaveaceterisparibusinterpretation,asitisimpossibletochangeonesharewhile

holdingalloftheothersharesfixed.(ii)Becauseeachshareisaproportion(and

canbeatmostone,whenallothersharesarezero),itmakeslittlesensetoincrease

sharepbyoneunit.Ifsharepincreasesby.01-whichisequivalenttoaone

percentagepointincreaseintheshareofpropertytaxesintotalrevenue-

12

holdingsharel,shareS,andtheotherfactorsfixed,thengrowthincreasesby

1(.01).Withtheothersharesfixed,theexcludedshare,shareF,mustfallby.01

whensharepincreasesby.01.

3.13(i)Fornotationalsimplicity,defineszx=(zi)xi;thisisnotquitethe

sample

iIn

covariancebetweenzandxbecausewedonotdividebyn-1,butweareonly

usingitto

assimplifynotation.Thenwecanwrite1

1

(z)y

i

i1

n

i

szx

Thisisclearlyalinearfunctionoftheyi:taketheweightstobewi=(zi)/szx.

Toshowunbiasedness,asusualweplugyi=0+Ixi+uiintothisequation,and

simplify:

1

(z)(

i

i1

n

n

Ixiui)

n

szx

0(zi)Iszx(zi)ui

i1

i1

szx

1

n

(z)u

i

i1

i

szx

whereweusethefactthat(zi)=0always.Nowszxisafunctionofthezi

andxiandthe

i1

expectedvalueofeachuiiszeroconditionalonallziandxiinthesample.

Therefore,conditional

onthesevalues,

)E(11

(Z)E(u)

i

i

i1

n

szx

1

becauseE(ui)=0foralli.(ii)Fromthefourthequationinpart(i)wehave(again

conditionalontheziandxiinthesample),

13

)VarVar(1(z)u(z)Var(u)2iiiii12szx

n2nni12szx

2(z)ii1

2szx

becauseofthehomoskedasticityassumption[Var(ui)=2foralli].Giventhe

definitionofszx,thisiswhatwewantedtoshow.

")=2/[(x)2].Nowwecanrearrangetheinequalityinthe(iii)Weknow

thatVar(i1

iIn

2hint,dropfromthesamplecovariance,andcancelneverywhere,toget

[(zi)2]/szx>-In

i1

)Var()whichiswhatl/[(xi)2].Whenwemultiplythroughby

2wegetVar(11

iIn

wewantedtoshow.

CHAPTER4

4.1(i)and(iii)generallycausethetstatisticsnottohaveatdistributionunderHO.

HomoskedasticityisoneoftheCLMassumptions.Animportantomittedvariable

violatesAssumptionMLR.3.TheCLMassumptionscontainnomentionofthe

samplecorrelationsamongindependentvariables,excepttoruleoutthecasewhere

thecorrelationisone.

4.2(i)HO:3=0.Hl:3>0.

(ii)Theproportionateeffectonsalaryis.00024(50)=.012.Toobtainthe

percentage

effect,wemultiplythisby100:1.2%.Therefore,a50pointceterisparibus

increaseinrosispredictedtoincrease