伍德里奇《计量经济学导论(第四版)》课后习题答案和讲解

STUDENTSOLUTIONSMANUALJeffreyM.Wooldridge

IntroductoryEconometrics:AModernApproach,4e

CONTENTS

PrefaceivChapter1Introduction1Chapter2TheSimpleRegressionModel3

Chapter3MultipleRegressionAnalysis:Estimation9Chapter4MultipleRegression

Analysis:Inference17Chapter5MultipleRegressionAnalysis:OLSAsymptotics24

Chapter6MultipleRegressionAnalysis:FurtherIssues27Chapter7Multiple

RegressionAnalysisWithQualitative34Information:Binary(orDummyVariables

Chapter8Heteroskedasticity42Chapter9MoreonSpecificationandData

Problems47Chapter10BasicRegressionAnalysisWithTimeSeriesData52Chapter11

FurtherIssuesinUsingOLSWithTimeSeriesData58Chapter12SerialCorrelationand

Heteroskedasticityin65TimeSeriesRegressions

Chapter13PoolingCrossSectionsAcrossTime.Simple71PanelDataMethods

Chapter14AdvancedPanelDataMethods78Chapter15InstrumentalVariables

EstimationandTwoStage85LeastSquares

Chapter16SimultaneousEquationsModels92Chapter17LimitedDependent

VariableModelsandSample99SelectionCorrections

Chapter18AdvancedTimeSeriesTopics110

ii

AppendixABasicMathematicalTools117AppendixBFundamentalsof

Probability119AppendixCFundamentalsofMathematicalStatistics120AppendixD

SummaryofMatrixAlgebra122AppendixETheLinearRegressionModelinMatrix

Form123

iii

PREFACE

Thismanualcontainssolutionstotheodd-numberedproblemsandcomputer

exercisesinIntroductoryEconometrics:AModernApproach,4e.Hopefully,youwill

findthatthesolutionsaredetailedenoughtoactasastudysupplementtothetext.Rather

thanjustpresentingthefinalanswer,Iusuallyprovidedetailedsteps,emphasizingwhere

thechaptermaterialisusedinsolvingtheproblems.

Someoftheanswersgivenherearesubjective,andyouoryourinstructormayhave

perfectlyacceptablealternativeanswersoropinions.

IobtainedthesolutionstothecomputerexercisesusingStata,startingwithversion

4.0

andendingwithversion9.0.Nevertheless,almostalloftheestimationmethods

coveredinthetexthavebeenstandardized,anddifferenteconometricsorstatistical

packagesshouldgivethesameanswerstothereporteddegreeofaccuracy.Therecanbe

differenceswhenapplyingmoreadvancedtechniques,asconventionssometimesdiffer

onhowtochooseorestimateauxiliaryparameters.(Examplesincludeheteroskedasticity-

robuststandarderrors,estimatesofarandomeffectsmodel,andcorrectionsforsample

selectionbias.Anydifferencesinestimatesorteststatisticsshouldbepractically

unimportant,providedyouareusingareasonablylargesamplesize.WhileIhave

endeavoredtomakethesolutionsfreeofmistakes,someerrorsmayhavecreptin.I

wouldappreciatehearingfromstudentswhofindmistakes.Iwillkeepalistwouldalso

liketohearfromstudentswhohavesuggestionsforimprovingeitherthesolutionsorthe

problemsthemselves.Icanbereachedviae-mailatwooldril@..Ihopethatyou

findthissolutionsmanualhelpfulwhenusedinconjunctionwiththetext.Ilookforward

tohearingfromyou.

JeffreyM.Wooldridge

DepartmentofEconomics

MichiganStateUniversity

110Marshall-AdamsHall

EastLansing,MI48824-1038

iv

CHAPTER1

SOLUTIONSTOPROBLEMS

1.1Itdoesnotmakesensetoposethequestionintermsofcausality.Economists

wouldassumethatstudentschooseamixofstudyingandworking(andotheractivities,

suchasattendingclass,leisure,andsleepingbasedonrationalbehavior,suchas

maximizingutilitysubjecttotheconstraintthatthereareonly168hoursinaweek.We

canthenusestatisticalmethodstomeasuretheassociationbetweenstudyingandworking,

includingregressionanalysisthatwecoverstartinginChapter2.Butwewouldnotbe

claimingthatonevariable“causes“theother.Theyarebothchoicevariablesofthe

student.

1.2(iIdeally,wecouldrandomlyassignstudentstoclassesofdifferentsizes.Thatis,

eachstudentisassignedadifferentclasssizewithoutregardtoanystudentcharacteristics

suchasabilityandfamilybackground.ForreasonswewillseeinChapter2,wewould

likesubstantialvariationinclasssizes(subject,ofcourse,toethicalconsiderationsand

resourceconstraints,(iiAnegativecorrelationmeansthatlargerclasssizeisassociated

withlowerperformance.Wemightfindanegativecorrelationbecauselargerclasssize

actuallyhurtsperformance.However,withobservationaldata,thereareotherreasonswe

mightfindanegativerelationship.Forexample,childrenfrommoreaffluentfamilies

mightbemorelikelytoattendschoolswithsmallerclasssizes,andaffluentchildren

generallyscorebetteronstandardizedtests.Anotherpossibilityisthat,withinaschool,a

principalmightassignthebetterstudentstosmallerclasses.Or,someparentsmightinsist

theirchildrenareinthesmallerclasses,andthesesameparentstendtobemoreinvolved

intheirchildren'seducation.

(iiiGiventhepotentialfbrconfoundingfactors-someofwhicharelistedin(ii-

findinganegativecorrelationwouldnotbestrongevidencethatsmallerclasssizes

actuallyleadtobetterperformance.Somewayofcontrollingfortheconfoundingfactors

isneeded,andthisisthesubjectofmultipleregressionanalysis.

SOLUTIONSTOCOMPUTEREXERCISES

Cl.l(iTheaverageofeducisabout12.6years.Therearetwopeoplereportingzero

yearsofeducation,and19peoplereporting18yearsofeducation.

(iiTheaverageofwageisabout$5.90,whichseemslowintheyear2008.

(iiiUsingTableB-60inthe2004EconomicReportofthePresident,theCPIwas

56.9in1976and184.0in2003.

(ivToconvert1976dollarsinto2003dollars,weusetheratiooftheCPIs,whichis

184/56.93.23

Therefore,theaveragehourlywagein2003dollarsisroughly

3.23($5.90$19.06

whichisareasonablefigure.

1

(vThesamplecontains252women(thenumberofobservationswithfemale=1and

274men.

Cl.3(iThelargestis100,thesmallestis0.

(ii38outof1,823,orabout2.1percentofthesample.

(iii17

(ivTheaverageofmath4isabout71.9andtheaverageofread4isabout60.1.So,at

leastin2001,thereadingtestwashardertopass.

(vThesamplecorrelationbetweenmath4andread4isabout.843,whichisavery

highdegreeof(linearassociation.Notsurprisingly,schoolsthathavehighpassrateson

onetesthaveastrongtendencytohavehighpassratesontheothertest.

(viTheaverageofexpppisabout$5,194.87.Thestandarddeviationis$1,091.89,

whichshowsratherwidevariationinspendingperpupil.[Theminimumis$1,206.88and

themaximumis$11,957.64.]

2

CHAPTER2

SOLUTIONSTOPROBLEMS

2.2(iLetyi=GPAi,xi=ACTi,andn=8.Then=25.875,=3.2125,1

n

i(xi-(yi-=

5.8125,and1

n

i(xi-2=56.875.Fromequation(2.9,weobtaintheslopeas1

八=5.8125/56.875~.1022,roundedtofourplacesafterthedecimal.From(2.17,0

八=_1

=3.2125-(.102225.875..5681.SowecanwriteGPA

=.5681+.1022ACT

n=8.

TheinterceptdoesnothaveausefulinterpretationbecauseACTisnotclosetozero

forthe

populationofinterest.IfACTis5pointshigher,GPA

increasesby.1022(5=.511.

(iiThefittedvaluesandresiduals—roundedtofourdecimalplaces—aregiven

alongwiththeobservationnumberiandGPAinthefollowingtable:

iGPAGPA

u1

2.82.7143.085723.43.0209.379133.03.2253-.225343.53.3275.172553.6

3.5319.068163.03.1231-.123172.73.1231-.423183.7

3.6341.0659

Youcanverifythattheresiduals,asreportedinthetable,sumto.0002,whichis

prettyclosetozerogiventheinherentroundingerror.

(iiiWhenACT=20,GPA

=.5681+.1022(20=2.61.

(ivThesumofsquaredresiduals,21

X

iiu

,isabout.4347(roundedtofourdecimalplaces,andthetotalsumofsquares,1

n

i(yi-2,isabout1.0288.SotheR-squaredfromthe

regressionis

R2=1-SSR/SST=1-(.4347/1.0288工.577.

Therefore,about57.7%ofthevariationinGPAisexplainedbyACTinthissmall

sampleofstudents.

2.3(iIncome,age,andfamilybackground(suchasnumberofsiblingsarejustafew

possibilities.Itseemsthateachofthesecouldbecorrelatedwithyearsofeducation.

(Incomeandeducationareprobablypositivelycorrelated;ageandeducationmaybe

negativelycorrelatedbecausewomeninmorerecentcohortshave,onaverage,more

education;andnumberofsiblingsandeducationareprobablynegativelycorrelated.

(iiNotifthefactorswelistedinpart(iarecorrelatedwitheduc.Becausewewould

liketoholdthesefactorsfixed,theyarepartoftheerrorterm.Butifuiscorrelatedwith

educthenE(u|educ0,andsoSLR.4fails.

2.4(iWewouldwanttorandomlyassignthenumberofhoursinthepreparation

coursesothathoursisindependentofotherfactorsthataffectperformanceontheSAT.

Then,wewouldcollectinformationonSATscoreforeachstudentintheexperiment,

yieldingadataset

{(,}iisathoursin,wherenisthenumberofstudentswecanaffordtohave

inthestudy.Fromequation(2.7,weshouldtrytogetasmuchvariationinihoursasis

feasible.

(iiHerearethreefactors:innateability,familyincome,andgeneralhealthonthe

dayoftheexam.Ifwethinkstudentswithhighernativeintelligencethinktheydonot

needtopreparefortheSAT,thenabilityandhourswillbenegativelycorrelated.Family

incomewouldprobablybepositivelycorrelatedwithhours,becausehigherincome

familiescanmoreeasilyafford

preparationcourses.Rulingoutchronichealthproblems,healthonthedayofthe

examshouldberoughlyuncorrelatedwithhoursspentinapreparationcourse.

(iiiIfpreparationcoursesareeffective,1shouldbepositive:otherfactorsequal,an

increaseinhoursshouldincreasesat.

(ivTheintercept,0,hasausefulinterpretationinthisexample:becauseE(u=0,0is

theaverageSATscoreforstudentsinthepopulationwithhours=0.

2.5(iWhenweconditiononincE(ulineelineE(e|inc0becauseE(eline=E(e=0.

(iiAgain,whenweconditiononincVar(ulinee|inc2Var(e|inc=2eincbecause

Var(e|inc=2e.

(iiiFamilieswithlowincomesdonothavemuchdiscretionaboutspending;

typically,alow-incomefamilymustspendonfood,clothing,housing,andother

necessities.Higherincomepeoplehavemorediscretion,andsomemightchoosemore

consumptionwhileothersmoresaving.Thisdiscretionsuggestswidervariabilityin

savingamonghigherincomefamilies.

2.8(iWefollowthehint,notingthat1cy=1c(thesampleaverageoflicyisc1

timesthesampleaverageofyiand2=2c.Whenweregressclyionc2xi(including

aninterceptweuseequation(2.19toobtaintheslope:

22

11

121

1

1

2

2

2

22

2

11

11112

2

2

1

(((

((((

八.(

nn

ii

i

i

iii

i

iin

iiin

i

icxcxccccxycxccxxyccccx

From(2.17,weobtaintheinterceptas0=(c1-1(c2=(c1-[(c1/c2I

F(c2=c1(-1

=c10

becausetheinterceptfromregressingyionxiis(-1

(iiWeusethesameapproachfrompart(ialongwiththefactthat1(=c1+and

2(=c2+.Therefore,11((icy=(cl+yi-(cl+=yi-and(c2+xi-2(cx=

xi-.Soc1andc2entirelydropoutoftheslopeformulafortheregressionof(cl+

yion(c2+xi,and1

=1

八.Theinterceptis0

=1

(cy-1

2

(CX=(c1+-1

八(c2+=(r+c1-c2/=(T+c1-c21

八,whichiswhatwewantedtoshow.

(iiiWecansimplyapplypart(iibecause11log(log(log(iicycy.Inotherwords,

replacec1withlog(c1,yiwithlog(yi,andsetc2=0.

(ivAgain,wecanapplypart(iiwithc1=0andreplacingc2withlog(c2andxi

withlog(xi.IfOr"andaretheoriginalinterceptandslope,then1Tand002l"log(c.

2.9(iTheinterceptimpliesthatwheninc=0,consispredictedtobenegative

$124.84.This,ofcourse,cannotbetrue,andreflectsthatfactthatthisconsumption

functionmightbeapoorpredictorofconsumptionatverylow-incomelevels.Onthe

otherhand,onanannualbasis,$124.84isnotsofarfromzero.

(iiJustplug30,000intotheequation:

cons=-124.84+.853(30,000=25,465.16dollars.

(iiiTheMPCandtheAPCareshowninthefollowinggraph.Eventhoughthe

interceptisnegative,thesmallestAPCinthesampleispositive.Thegraphstartsatan

annualincomelevelof$1,000(in1970dollars.

SOLUTIONSTOCOMPUTEREXERCISES

C2.1(iTheaverageprateisabout87.36andtheaveragemrateisabout.732.

(iiTheestimatedequationis

prate=83.05+5.86mrate

n=1,534,R2=.075.

(iiiTheinterceptimpliesthat,evenifmrate=0,thepredictedparticipationrateis

83.05percent.Thecoefficientonmrateimpliesthataone-dollarincreaseinthematch

rate-afairlylargeincrease-isestimatedtoincreaseprateby5.86percentagepoints.

Thisassumes,of

course,thatthischangeprateispossible(if,say,prateisalreadyat98,this

interpretationmakesnosense.

(ivIfweplugmrate=3.5intotheequationweget"prate

=83.05+5.86(3.5=103.59.Thisisimpossible,aswecanhaveatmosta100

percentparticipationrate.Thisillustratesthat,especiallywhendependentvariablesare

bounded,asimpleregressionmodelcangivestrangepredictionsforextremevaluesof

theindependentvariable.(Inthesampleof1,534firms,only34havemrate3.5.

(vmrateexplainsabout7.5%ofthevariationinprate.Thisisnotmuch,and

suggeststhatmanyotherfactorsinfluence401(kplanparticipationrates.

C2.3(iTheestimatedequationis

sleep=3,586.4-.151totwrk

n=706,R2=.l()3.

Theinterceptimpliesthattheestimatedamountofsleepperweekfbrsomeonewho

doesnotworkis3,586.4minutes,orabout59.77hours.Thiscomestoabout8.5hours

pernight.

(iiIfsomeoneworkstwomorehoursperweekthentotwrk=120(becausetotwrkis

measuredinminutes,andsosleep

=-.151(120=-18.12minutes.Thisisonlyafewminutesanight.Ifsomeonewere

toworkonemorehouroneachoffiveworkingdays,sleep

=-.151(300=-45.3minutes,oraboutfiveminutesanight.

C2.5(iTheconstantelasticitymodelisalog-logmodel:

log(rd=0+1log(sales+u,

where1istheelasticityofrdwithrespecttosales.

(iiTheestimatedequationis

log(rd=-4.105+1.076log(sales

n=32,R2=.91O.

Theestimatedelasticityofrdwithrespecttosalesis1.076,whichisjustaboveone.

Aonepercentincreaseinsalesisestimatedtoincreaserdbyabout1.08%.

C2.7(iTheaveragegiftisabout7.44Dutchguilders.Outof4,268respondents,

2,561didnotgiveagift,orabout60percent.

(iiTheaveragemailingsperyearisabout2.05.Theminimumvalueis.25(which

presumablymeansthatsomeonehasbeenonthemailinglistforatleastfouryearsand

themaximumvalueis3.5.

(iiiTheestimatedequationis

2

2.012.654,268,.0138

gift

mailsyearnR

(ivTheslopecoefficientfrompart(iiimeansthateachmailingperyearisassociated

with-perhapseven"causes”-anestimated2.65additionalguilders,onaverage.

Therefore,ifeachmailingcostsoneguilder,theexpectedprofitfromeachmailingis

estimatedtobe1.65guilders.Thisisonlytheaverage,however.Somemailingsgenerate

nocontributions,oracontributionlessthanthemailingcost;othermailingsgenerated

muchmorethanthemailingcost.

(vBecausethesmallestmailsyearinthesampleis.25,thesmallestpredictedvalue

ofgiftsis2.01+2.65(.25=2.67.Evenifwelookattheoverallpopulation,wheresome

peoplehavereceivednomailings,thesmallestpredictedvalueisabouttwo.So,withthis

estimatedequation,weneverpredictzerocharitablegifts.

9

CHAPTER3

SOLUTIONSTOPROBLEMS

3.2(ihspercisdefinedsothatthesmalleritis,thelowerthestudent9sstandingin

highschool.Everythingelseequal,theworsethestudent'sstandinginhighschool,the

lowerishis/herexpectedcollegeGPA.(iiJustplugthesevaluesintotheequation:

colgpa

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

(iiiThedifferencebetweenAandBissimply140timesthecoefficientonsat,

becausehspercisthesameforbothstudents.SoAispredictedtohavea

score.00148(140=.207higher.

(ivWithhspercfixed,colgpa

=.00148sat.Now,wewanttofindsatsuchthatcolgpa

=.5,so.5=.00148(satorsat=.5/(.00148~338.Perhapsnotsurprisingly,alarge

ceterisparibusdifferenceinSATscore-almosttwoandone-halfstandarddeviations-is

neededtoobtainapredicteddifferenceincollegeGPAorahalfapoint.

3.4(iIfadultstradeoffsleepforwork,moreworkimplieslesssleep(otherthings

equal,so1<0.(iiThesignsof2and3arenotobvious,atleasttome.Onecouldargue

thatmoreeducatedpeopleliketogetmoreoutoflife,andso,otherthingsequal,they

sleepless(2<0.Therelationshipbetweensleepingandageismorecomplicatedthanthis

modelsuggests,andeconomistsarenotinthebestpositiontojudgesuchthings,(iii

Sincetotwrkisinminutes,wemustconvertfivehoursintominutes:totwrk=5(60=300.

Thensleepispredictedtofallby.148(300=44.4minutes.Foraweek,45minutesless

sleepisnotanoverwhelmingchange,(ivMoreeducationimplieslesspredictedtime

sleeping,buttheeffectisquitesmall.Ifweassumethedifferencebetweencollegeand

highschoolisfouryears,thecollegegraduatesleepsabout45minuteslessperweek,

otherthingsequal,(vNotsurprisingly,thethreeexplanatoryvariablesexplainonlyabout

11.3%ofthevariationinsleep.Oneimportantfactorintheerrortermisgeneralhealth.

Anotherismaritalstatus,andwhetherthepersonhaschildren.Health(howeverwe

measurethat,maritalstatus,andnumberandagesofchildrenwouldgenerallybe

correlatedwithtotwrk.(Forexample,lesshealthypeoplewouldtendtoworkless.

1()

3.6(iNo.Bydefinition,study+sleep+work+leisure=168.Therefore,ifwe

changestudy,wemustchangeatleastoneoftheothercategoriessothatthesumisstill

168.(iiFrompart(i,wecanwrite,say,studyasaperfectlinearfunctionoftheother

independentvariables:study=168sleepworkleisure.Thisholdsforevery

observation,soMLR.3violated,(iiiSimplydroponeoftheindependentvariables,say

leisure:

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

Now,forexample,1isinterpretedasthechangeinGPAwhenstudyincreasesby

onehour,wheresleep,work,anduareallheldfixed.Ifweareholdingsleepandwork

fixedbutincreasingstudybyonehour,thenwemustbereducingleisurebyonehour.

Theotherslopeparametershaveasimilarinterpretation.

3.8Only(ii,omittinganimportantvariable,cancausebias,andthisistrueonly

whentheomittedvariableiscorrelatedwiththeincludedexplanatoryvariables.The

homoskedasticityassumption,MLR.5,playednoroleinshowingthattheOLSestimators

areunbiased.

(HomoskedasticitywasusedtoobtaintheusualvarianceformulasfortheJ.Further,

thedegreeofcollinearitybetweentheexplanatoryvariablesinthesample,evenifitis

reflectedina

correlationashighas.95,doesnotaffecttheGauss-Markovassumptions.Onlyif

thereisaperfectlinearrelationshipamongtwoormoreexplanatoryvariablesisMLR.3

violated.

3.1()Fromequation(3.22wehave

11

1

21

1

n

ii

iiiry

r

wheretherir

aredefinedintheproblem.Asusual,wemustpluginthetruemodelforyi:

1()

1122331

2

1

1

n

iiiii

in

iirxxxur

11

Thenumeratorofthisexpressionsimplifiesbecause11

"niir

=0,121

Aniiirx=0,and111

"n

iiirx=2

11

n

iir.TheseallfollowfromthefactthatthePir

aretheresidualsfromtheregressionoflixon2ix:the1"ir

havezerosampleaverageandareuncorrelatedinsamplewith2ix.Sothe

numeratorof1

canbeexpressedas

2

11

3131111

….nnn

iiiiiiiir

rxru

Puttingthesebackoverthedenominatorgives

13

11

1113

221

1

1

1

n

n

iii

iin

n

iiiirx

ru

rr

Conditionalonallsamplevaluesonx1,x2,andx3,onlythelasttermisrandom

duetoitsdependenceonui.ButE(ui=0,andso

13

1

113

21

1

AE(=+J

n

iiin

11rx

r

whichiswhatwewantedtoshow.Noticethatthetermmultiplying3isthe

regression

coefficientfromthesimpleregressionofxi3on1'ir

3.11(i1<0becausemorepollutioncanbeexpectedtolowerhousingvalues;note

that1istheelasticityofpricewithrespecttonox.2isprobablypositivebecauserooms

roughlymeasuresthesizeofahouse.(However,itdoesnotallowustodistinguish

homeswhereeachroomislargefromhomeswhereeachroomissmall,(iiIfweassume

thatroomsincreaseswithqualityofthehome,thenlog(noxandroomsarenegatively

correlatedwhenpoorerneighborhoodshavemorepollution,somethingthatisoftentrue.

WecanuseTable3.2todeterminethedirectionofthebias.If2>0and

Corr(x1,x2<0,thesimpleregressionestimator1

hasadownwardbias.Butbecause1

<0,thismeansthatthesimpleregression,onaverage,overstatestheimportanceof

pollution.[E(l

ismorenegativethan1.]

12

(iiiThisiswhatweexpectfromthetypicalsamplebasedonouranalysisinpart(ii.

Thesimpleregressionestimate,1.043,ismorenegative(largerinmagnitudethanthe

multipleregressionestimate,.718.Asthoseestimatesareonlyforonesample,wecan

neverknowwhichisclosertol.Butifthisisa“typical“sample,1iscloserto.718.

3.12(iFornotationalsimplicity,defineszx=1(;n

iiizxthisisnotquitethesample

covariancebetweenzandxbecausewedonotdividebyn-1,butweareonly

usingitto

simplifynotation.Thenwecanwrite1

as

1

1

(・n

i

i

izx

zy

s

Thisisclearlyalinearfunctionoftheyi:taketheweightstobewi=(zi/szx.To

showunbiasedness,asusualweplugyi=0+1xi+uiintothisequation,andsimplify:

11

1

Oil

1

1

1(

(((n

i

iiizx

n

n

izxii

iizx

n

i

i

izx

zxuszszusz

us

whereweusethefactthatI

(n

iiz=0always.Nowszxisafunctionoftheziandxiandthe

expectedvalueofeachuiiszeroconditionalonallziandxiinthesample.

Therefore,conditional

onthesevalues,

1

11

K

E(n

i

i

izx

zus

becauseE(ui=0foralli.(iiFromthefourthequationinpart(iwehave(again

conditionalontheziandxiinthesample,

13

2

11

12

22

2

1

2Var((Var(

Var((nn

iii

iiizx

zx

n

i

izx

zuzusszs

becauseofthehomoskedasticityassumption[Var(ui=2foralli].Giventhe

definitionofszx,thisiswhatwewantedtoshow.

(iiiWeknowthatVar(l

八=2/21](].n

iixNowwecanrearrangetheinequalityinthehint,dropfromthesample

covariance,andcanceln-1

everywhere,toget22

1

[(]/n

izxizs>

21

l/[(].n

iixWhenwemultiplythroughby2wegetVar(lVar(l

八,whichiswhatwewantedtoshow.

SOLUTIONSTOCOMPUTEREXERCISES

C3.1(iProbably2>0,asmoreincometypicallymeansbetternutritionforthe

motherandbetterprenatalcare.(iiOntheonehand,anincreaseinincomegenerally

increasestheconsumptionofagood,andcigsandfaminecouldbepositivelycorrelated.

Ontheother,familyincomesarealsohigherforfamilieswithmoreeducation,andmore

educationandcigarettesmokingtendtobe

negativelycorrelated.Thesamplecorrelationbetweencigsandfamineisabout.173,

indicatinganegativecorrelation,(iiiTheregressionswithoutandwithfamineare

119.77.514bwghtcigs

21,388,.023nR

and116.97.463.093bwght

cigsfamine21,388,.03().nR

Theeffectofcigarettesmokingisslightlysmallerwhenfamineisaddedtothe

regression,butthedifferenceisnotgreat.Thisisduetothefactthatcigsandfamineare

notverycorrelated,andthecoefficientonfamineispracticallysmall.(Thevariable

famineismeasuredinthousands,so$10,000morein1988incomeincreasespredicted

birthweightbyonly.93ounces.

C3.3(iTheconstantelasticityequationis

log(4.62.1621og(.1071og(

salarysalesmktval

2

177,.299.

nR

(iiWecannotincludeprofitsinlogarithmicformbecauseprofitsarenegativefor

nineofthecompaniesinthesample.Whenweadditinlevelsformweget

log(4.69.1611og(.0981og(.000036

salarysalesmktvalprofits

2

177,.299.

nR

Thecoefficientonprofitsisverysmall.Here,profitsaremeasuredinmillions,soif

profitsincreaseby$1billion,whichmeansprofits

=1,000-ahugechange-predictedsalaryincreasesbyaboutonly3.6%,However,

rememberthatweareholdingsalesandmarketvaluefixed.

Together,thesevariables(andwecoulddropprofitswithoutlosinganythingexplain

almost30%ofthesamplevariationinlog(salary.Thisiscertainlynot"most"ofthe

variation,(iiiAddingceotentotheequationgives

log(4.56.1621og(.1021og(.000029.012

salarysalesmktvalprofitsceoten

2

177,.318.

nR

ThismeansthatonemoreyearasCEOincreasespredictedsalarybyabout1.2%.

(ivThesamplecorrelationbetweenlog(mktvalandprofitsisabout.78,whichis

fairlyhigh.Asweknow,thiscausesnobiasintheOLSestimators,althoughitcancause

theirvariancestobelarge.Giventhefairlysubstantialcorrelationbetweenmarketvalue

andfirmprofits,itisnottoosurprisingthatthelatteraddsnothingtoexplainingCEO

salaries.Also,profitsisashorttermmeasureofhowthefirmisdoingwhilemktvalis

basedonpast,current,andexpectedfutureprofitability.

14

C3.5Theregressionofeduconexperand