SOLUTION MANUAL
,Introductory Econometrics: A Modern Approach 8th edition
Solution Manual
CONTENTS
Chapterk1 ThekNaturekofkEconometricskandkEconomickData 1
Chapterk2 ThekSimplekRegressionkModel 5
Chapterk3 MultiplekRegressionkAnalysis:k Estimation 15
Chapterk4 MultiplekRegressionkAnalysis:k Inference 28
Chapterk5 MultiplekRegressionkAnalysis:k OLSkAsymptotics 41
Chapterk6 MultiplekRegressionkAnalysis:k FurtherkIssues 46
Chapterk7 MultiplekRegressionkAnalysiskwithkQualitativek Inf 62
ormation:k Binaryk(orkDummy)kVariables
Chapterk8 Heteroskedasticity 79
Chapterk9 MorekonkSpecificationkandkDatakProblems 91
Chapterk10 BasickRegressionkAnalysiskwithkTimekSerieskData 102
Chapterk11 FurtherkIssueskinkUsingkOLSkwithkTimekSerieskData 114
Chapterk12 SerialkCorrelationkandkHeteroskedasticitykink T 127
imekSerieskRegressions
Chapterk13 PoolingkCrosskSectionskAcrosskTime.k Simplek Pan 139
elkDatakMethods
Chapterk14 AdvancedkPanelkDatakMethods 154
Chapterk15 InstrumentalkVariableskEstimationkandkTwokStagek Le 167
astkSquares
Chapterk16 SimultaneouskEquationskModels 183
Chapterk17 LimitedkDependentkVariablekModelskandkSamplek S 196
electionkCorrections
,Chapterk18 AdvancedkTimekSerieskTopics 217
Chapterk19 CarryingkOutkankEmpiricalkProject 233
AppendixkA BasickMathematicalkTools 234
AppendixkB FundamentalskofkProbability 236
AppendixkC FundamentalskofkMathematicalkStatistics 238
AppendixkD SummarykofkMatrixkAlgebra 242
AppendixkE ThekLinearkRegressionkModelkinkMatrixkForm 244
, CHAPTER 1 k
TEACHINGkNOTES
YoukhaveksubstantialklatitudekaboutkwhatktokemphasizekinkChapterk1.k Ikfindkitkusefulktoktalkkaboutk thek
economicskofkcrimekexamplek(Examplek1.1)kandkthekwagekexamplek(Examplek1.2)ksokthatk studentskse
e,katkthekoutset,kthatkeconometricskisklinkedktokeconomickreasoning,kevenkifkthek economicskisknotkcomplic
atedktheory.
Iklikektokfamiliarizekstudentskwithkthekimportantkdatakstructureskthatkempiricalkeconomistskuse,k focusing
kprimarilykonkcross-sectionalkandktimekserieskdataksets,kaskthesekarekwhatkIkcoverkinkak first-
semesterkcourse.k Itkiskprobablykakgoodkideaktokmentionkthekgrowingkimportancekofkdataksetsk thatkha
vekbothkakcross-sectionalkandktimekdimension.
Ikspendkalmostkankentireklecturektalkingkaboutkthekproblemskinherentkinkdrawingkcausalkinferencesk inkth
eksocialksciences.k Ikdokthiskmostlykthroughkthekagriculturalkyield,kreturnktokeducation,kandk crimekexamp
les.k Thesekexampleskalsokcontrastkexperimentalkandknonexperimentalk (observational)kdata.k Studen
tskstudyingkbusinesskandkfinancektendktokfindkthektermkstructurekofk interestkrateskexamplekmorekreleva
nt,kalthoughkthekissuektherekisktestingkthekimplicationkofkak simplektheory,kaskopposedktokinferringkcausa
lity.k Ikhavekfoundkthatkspendingktimektalkingkaboutk thesekexamples,kinkplacekofkakformalkreviewkofkpr
obabilitykandkstatistics,kiskmoreksuccessfulk(andk morekenjoyablekforkthekstudentskandkme).
,SOLUTIONSkTOkPROBLEMS
1.1 Itkdoesknotkmakeksensektokposekthekquestionkinktermskofkcausality.kEconomistskwouldkassumek thatkstu
dentskchoosekakmixkofkstudyingkandkworkingk(andkotherkactivities,ksuchkaskattendingkclass,k leisure,kandk
sleeping)kbasedkonkrationalkbehavior,ksuchkaskmaximizingkutilityksubjectktokthek constraintkthatktherekar
ekonlyk168khourskinkakweek.k Wekcankthenkusekstatisticalkmethodsktok measurekthekassociationkbetweenk
studyingkandkworking,kincludingkregressionkanalysiskthatkwek coverkstartingkinkChapterk2.k Butkwekwoul
dknotkbekclaimingkthatkonekvariablek“causes”kthekother.k Theykarekbothkchoicekvariableskofkthekstudent.
1.2 (i)kIdeally,kwekcouldkrandomlykassignkstudentsktokclasseskofkdifferentksizes.k Thatkis,keachk studentki
skassignedkakdifferentkclassksizekwithoutkregardktokanykstudentkcharacteristicsksuchkask abilitykandkfa
milykbackground.k ForkreasonskwekwillkseekinkChapterk2,kwekwouldklikeksubstantialk variationkinkclassks
izesk(subject,kofkcourse,ktokethicalkconsiderationskandkresourcekconstraints).
(ii) Aknegativekcorrelationkmeanskthatklargerkclassksizekiskassociatedkwithklowerkperformance.k W
ekmightkfindkaknegativekcorrelationkbecauseklargerkclassksizekactuallykhurtskperformance.k However,k
withkobservationalkdata,ktherekarekotherkreasonskwekmightkfindkaknegativekrelationship.k Forkexampl
e,kchildrenkfromkmorekaffluentkfamilieskmightkbekmoreklikelyktokattendkschoolskwithk smallerkclassksizes,k
andkaffluentkchildrenkgenerallykscorekbetterkonkstandardizedktests.k Anotherk possibilitykiskthat,kwithink
akschool,kakprincipalkmightkassignkthekbetterkstudentsktoksmallerkclasses.k Or,ksomekparentskmightkinsistkt
heirkchildrenkarekinktheksmallerkclasses,kandktheseksamekparentsk tendktokbekmorekinvolvedkinktheirkchild
ren’skeducation.
(iii) Givenkthekpotentialkforkconfoundingkfactorsk–ksomekofkwhichkareklistedkink(ii)k–
findingkak negativekcorrelationkwouldknotkbekstrongkevidencekthatksmallerkclassksizeskactuallykleadktok
k
betterk performance.kSomekwaykofkcontrollingkforkthekconfoundingkfactorskiskneeded,kandkthiskiskthek s
ubjectkofkmultiplekregressionkanalysis.
1.3 (i)kHerekiskonekwayktokposekthekquestion:kIfktwokfirms,ksaykAkandkB,karekidenticalkinkallk respectskex
ceptkthatkfirmkAksupplieskjobktrainingkonekhourkperkworkerkmorekthankfirmkB,kbykhowk muchkwouldkfirm
kA’skoutputkdifferkfrom kfirmkB’s?
(ii) Firmskareklikelyktokchoosekjobktrainingkdependingkonkthekcharacteristicskofkworkers.kSomek obs
ervedkcharacteristicskarekyearskofkschooling,kyearskinkthekworkforce,kandkexperiencekinkak particularkj
ob.kFirmskmightkevenkdiscriminatekbasedkonkage,kgender,korkrace.kPerhapskfirmskk choosektokofferktraini
ngktokmorekorklesskablekworkers,kwherek“ability”kmightkbekdifficultktok quantifykbutkwherekakmanagerkh
asksomekideakaboutkthekrelativekabilitieskofkdifferentkemployees.k Moreover,kdifferentkkindskofkworke
rskmightkbekattractedktokfirmskthatkofferkmorekjobktrainingkonk average,kandkthiskmightknotkbekevidentkt
okemployers.
(iii) Thekamountkofkcapitalkandktechnologykavailablektokworkerskwouldkalsokaffectkoutput.k So,k tw
okfirmskwithkexactlyktheksamekkindskofkemployeeskwouldkgenerallykhavekdifferentkoutputskifk theykusekd
ifferentkamountskofkcapitalkorktechnology.k Thekqualitykofkmanagerskwouldkalsokhavekank effect.
, (iv) No,kunlesskthekamountkofktrainingkiskrandomlykassigned.k Thekmanykfactorsklistedkinkparts
(ii) andk(iii)kcankcontributektokfindingkakpositivekcorrelationkbetweenkoutputkandktrainingkevenkifk jobkt
rainingkdoesknotkimprovekworkerkproductivity.
SOLUTIONSkTOkCOMPUTERkEXERCISES
C1.1k(i)kThekaveragekofkeduckiskaboutk12.6kyears.k Therekarektwokpeoplekreportingkzerokyearskofk ed
ucation,kandk19kpeoplekreportingk18kyearskofkeducation.
(ii) Thekaveragekofkwagekiskaboutk$5.90,kwhichkseemsklowkinkthekyeark2008.
(iii) UsingkTablekB-
60kinkthek2004kEconomickReportkofkthekPresident,kthekCPIkwask56.9kink 1976kandk184.0kink2003.
(iv) Tokconvertk1976kdollarskintok2003kdollars,kwekusekthekratiokofkthekCPIs,kwhichkisk 184k
/k56.9k 3.23k.k Therefore,kthekaveragekhourlykwagekink2003kdollarskiskroughlyk 3.23($5.9
0)k $19.06k,kwhichkiskakreasonablekfigure.
(v) Theksamplekcontainsk252kwomenk(theknumberkofkobservationskwithkfemalek=k1)kandk274k me
n.
C1.2k(i)kTherekarek1,388kobservationskinktheksample.k Tabulatingkthekvariablekcigskshowskthatk212k wo
menkhavekcigsk>k0.
(ii) Thekaveragekofkcigskiskaboutk2.09,kbutkthiskincludeskthek1,176kwomenkwhokdidknotk smoke.k
Reportingkjustkthekaveragekmaskskthekfactkthatkalmostk85kpercentkofkthekwomenkdidknotk smoke.k Itkma
keskmoreksensektoksaykthatkthek“typical”kwomankdoesknotksmokekduringkpregnancy;k indeed,kthekmedi
anknumberkofkcigarettesksmokedkiskzero.
(iii) Thekaveragekofkcigskoverkthekwomenkwithkcigsk>k0kiskaboutk13.7.kOfkcoursekthiskisk muc
hkhigherkthankthekaveragekoverkthekentireksamplekbecausekwekarekexcludingk1,176kzeros.
(iv) Thekaveragekofkfatheduckiskaboutk13.2.kTherekarek196kobservationskwithkakmissingk val
uekforkfatheduc,kandkthosekobservationskareknecessarilykexcludedkinkcomputingkthekaverage.
(v) Thekaveragekandkstandardkdeviationkofkfaminckarekaboutk29.027kandk18.739,k resp
ectively,kbutkfaminckiskmeasuredkinkthousandskofkdollars.k So,kinkdollars,kthekaveragekandk standar
dkdeviationkarek$29,027kandk$18,739.
C1.3k(i)kTheklargestkisk100,ktheksmallestkisk0.
(ii) 38koutkofk1,823,korkaboutk2.1kpercentkofktheksample.
, (iii) 17
(iv) Thekaveragekofkmath4kiskaboutk71.9kandkthekaveragekofkread4kiskaboutk60.1.k So,katkleastk i
nk2001,kthekreadingktestkwaskharderktokpass.
(v) Theksamplekcorrelationkbetweenkmath4kandkread4kiskaboutk.843,kwhichkiskakverykhighk deg
reekofk(linear)kassociation.k Notksurprisingly,kschoolskthatkhavekhighkpasskrateskonkonektestk havekaks
trongktendencyktokhavekhighkpasskrateskonkthekotherktest.
(vi) Thekaveragekofkexpppkiskaboutk$5,194.87.k Thekstandardkdeviationkisk$1,091.89,kwhichk s
howskratherkwidekvariationkinkspendingkperkpupil.k [Thekminimumkisk$1,206.88kandkthek maximumkisk
$11,957.64.]
C1.4k(i)k185/445k .416kiskthekfractionkofkmenkreceivingkjobktraining,korkaboutk41.6%.
(ii) Forkmenkreceivingkjobktraining,kthekaveragekofkre78kiskaboutk6.35,kork$6,350.k Forkmenknotk re
ceivingkjobktraining,kthekaveragekofkre78kiskaboutk4.55,kork$4,550.k Thekdifferencekisk$1,800,kwhichkisk
veryklarge.k Onkaverage,kthekmenkreceivingkthekjobktrainingkhadkearningskaboutk40%k higherkthankthos
eknotkreceivingktraining.
(iii) Aboutk24.3%kofkthekmenkwhokreceivedktrainingkwerekunemployedkink1978;kthekfigurekisk 35
.4%kforkmenknotkreceivingktraining.k This,ktoo,kiskakbigkdifference.
(iv) Thekdifferenceskinkearningskandkunemploymentkratesksuggestkthektrainingkprogramkhadk str
ong,kpositivekeffects.k Ourkconclusionskaboutkeconomicksignificancekwouldkbekstrongerkifkwek couldka
lsokestablishkstatisticalksignificancek(whichkiskdonekinkComputerkExercisekC9.10kink Chapterk9).
, CHAPTER 2 k
TEACHINGkNOTES
ThiskiskthekchapterkwherekIkexpectkstudentsktokfollowkmost,kifknotkall,kofkthekalgebraickderivations.k Inkcla
sskIklikektokderivekatkleastkthekunbiasednesskofkthekOLSkslopekcoefficient,kandkusuallykIkk derivekthekvaria
nce.k Atkakminimum,kIktalkkaboutkthekfactorskaffectingkthekvariance.k Toksimplifyk theknotation,kafterkIkem
phasizekthekassumptionskinkthekpopulationkmodel,kandkassumekrandomk sampling,kIkjustkconditionkonkthek
valueskofkthekexplanatorykvariableskinktheksample.k Technically,k thiskiskjustifiedkbykrandomksamplingkb
ecause,kforkexample,kE(ui|x1,x2,…,xn)k=kE(ui|xi)kbyk independentksampling.k Ikfindkthatkstudentskareka
blektokfocuskonkthekkeykassumptionkSLR.4kandk subsequentlyktakekmykwordkaboutkhowkconditioningkonkt
hekindependentkvariableskinktheksamplekisk harmless.k (Ifkyoukprefer,kthekappendixktokChapterk3kdoeskt
hekconditioningkargumentkcarefully.)k Becausekstatisticalkinferencekisknokmorekdifficultkinkmultiplekregre
ssionkthankinksimplekregression,k IkpostponekinferencekuntilkChapterk4.k (Thiskreduceskredundancykandka
llowskyouktokfocuskonkthek interpretivekdifferenceskbetweenksimplekandkmultiplekregression.)
Youkmightknoticekhow,kcomparedkwithkmostkotherktexts,kIkusekrelativelykfewkassumptionsktok derivekthek
unbiasednesskofkthekOLSkslopekestimator,kfollowedkbykthekformulakforkitskvariance.k ThiskiskbecausekIkd
oknotkintroducekredundantkorkunnecessarykassumptions.k Forkexample,koncek SLR.4kiskassumed,knothingk
furtherkaboutkthekrelationshipkbetweenkukandkxkiskneededktokobtainkthek unbiasednesskofkOLSkunderkra
ndomksampling.
,SOLUTIONSkTOkPROBLEMS
2.1k Inkthekequationkyk=k 0k +k 1xk+ku,kaddkandksubtractk 0k fromkthekrightkhandksidektokgetkyk=k( 0k +
0)k+k 1xk+k(uk 0).k Callktheknewkerrorkek=kuk 0,ksokthatkE(e)k=k0.k Theknewkinterceptkisk 0k
+
0,kbutkthekslopekiskstillk 1.
n
2.2k (i)kLetkyik =kGPAi,kxik =kACTi,kandknk=k8.k Thenkk xk=k25.875,k yk =k3.2125,k (xik –k xk)(yik –k yk)k=
i 1
n
5.8125,kandk (xik – ˆk
2k
k xk) =k56.875.k Fromkequationk(2.9),kwekobtainkthekslopekas =
1
i 1
ˆ
5.8125/56.875k .1022,kroundedktokfourkplaceskafterkthekdecimal.k Fromk(2.1 0 kk = k yk –
7),
ˆ1kk xkk 3.2125k–k(.1022)25.875k .5681.k Sokwekcankwrite
GPAk =k .5681k+k.1022kACT
nk=k8.
ThekinterceptkdoesknotkhavekakusefulkinterpretationkbecausekACTkisknotkclosektokzerokforkthek populatio
nkofkinterest.kkIfkACTkisk5kpointskhigher,k GPAkincreaseskbyk.1022(5)k=k.511.
(ii)kThekfittedkvalueskandkresidualsk—kroundedktokfourkdecimalkplacesk—
arekgivenkalongkwithk thekobservationknumberkikandkGPAkinkthekfollowingktable:
k
i GPA G û
1 2.8 PA
2.7143 .0857
2 3.4 3.0209 .3791
3 3.0 3.2253 –.2253
4 3.5 3.3275 .1725
5 3.6 3.5319 .0681
6 3.0 3.1231 –.1231
7 2.7 3.1231 –.4231
8 3.7 3.6341 .0659
Youkcankverifykthatkthekresiduals,kaskreportedkinkthektable,ksumktok .0002,kwhichkiskprettykclosektok zer
okgivenkthekinherentkroundingkerror.
(iii)kWhenkACTk=k20,k GPAk=k.5681k+k.1022(20)k 2.61.
, n
(iv)kTheksumkofksquaredkresiduals, ûikk2 ,kiskaboutk.4347k(roundedktokfourkdecimalkplaces),
i
n 1
andkthektotalksumkofksquares,k (yik –k yk)2,kiskaboutk1.0288.k SokthekR-squaredkfromkthe
i 1
regressionkis
R2kk =k 1k–kSSR/SSTk 1k–k(.4347/1.0288)k .577.
Therefore,kaboutk57.7%kofkthekvariationkinkGPAkiskexplainedkbykACTkinkthisksmallksamplekofk students.
2.3 (i)kIncome,kage,kandkfamilykbackgroundk(suchkasknumberkofksiblings)karekjustkakfewk possibilities.k Itks
eemskthatkeachkofkthesekcouldkbekcorrelatedkwithkyearskofkeducation.k (Incomekk andkeducationkarekpro
bablykpositivelykcorrelated;kagekandkeducationkmaykbeknegativelykcorrelatedk becausekwomenkinkmo
rekrecentkcohortskhave,konkaverage,kmorekeducation;kandknumberkofksiblingsk andkeducationkarekproba
blyknegativelykcorrelated.)
(ii)kNotkifkthekfactorskweklistedkinkpartk(i)karekcorrelatedkwithkeduc.k Becausekwekwouldklikektok holdkt
hesekfactorskfixed,ktheykarekpartkofkthekerrorkterm.k Butkifkukiskcorrelatedkwithkeduckthenk E(u|educ)k
0,kandksokSLR.4kfails.
2.4 (i)kWekwouldkwantktokrandomlykassignktheknumberkofkhourskinkthekpreparationkcourseksokthatk hour
skiskindependentkofkotherkfactorskthatkaffectkperformancekonkthekSAT.kThen,kwekwouldk collectkinforma
tionkonkSATkscorekforkeachkstudentkinkthekexperiment,kyieldingkakdatakset
{(satik,khoursik)k:kik 1,...,kn},kwhereknkisktheknumberkofkstudentskwekcankaffordktokhavekinkthekstudy.
Fromkequationk(2.7),kwekshouldktryktokgetkaskmuchkvariationkink hoursi askiskfeasible.
(ii) Herekarekthreekfactors:k innatekability,kfamilykincome,kandkgeneralkhealthkonkthekdaykofkthek ex
am.k Ifkwekthinkkstudentskwithkhigherknativekintelligencekthinkktheykdoknotkneedktokpreparekfork thekSAT,kt
henkabilitykandkhourskwillkbeknegativelykcorrelated.k Familykincomekwouldkprobablykbek positivelykcorr
elatedkwithkhours,kbecausekhigherkincomekfamilieskcankmorekeasilykaffordk preparationkcourses.k Rulin
gkoutkchronickhealthkproblems,khealthkonkthekdaykofkthekexamkshouldkk bekroughlykuncorrelatedkwithkhou
rskspentkinkakpreparationkcourse.
(iii) Ifkpreparationkcourseskarekeffective,k 1kk shouldkbekpositive:kotherkfactorskequal,kank i
ncreasekinkhourskshouldkincreaseksat.
(iv) Thekintercept,k 0k ,khaskakusefulkinterpretationkinkthiskexample:kbecausekE(u)k=k0,k 0
iskthe
averagekSATkscorekforkstudentskinkthekpopulationkwithkhoursk=k0.
2.5 (i)kWhenkwekconditionkonkinckinkcomputingkankexpectation, inc becomeskakconstant.k So
E(u|inc)k=kE(kkkinck e|inc)k inc E(e|inc)k= inc 0kbecausekE(e|inc)k=kE(e)k=k0.
=
(ii) Again,kwhenkwekconditionkonkinckinkcomputingkak variance, inc becomeskakconstant.k So
2 2k 2k
Var(u|inc)k=kVar(kkkinck e|inc)k=k(kkkinck) Var(e|inc)k=k inckbecausekVar(e|inc)k=k .
e e
,Introductory Econometrics: A Modern Approach 8th edition
Solution Manual
CONTENTS
Chapterk1 ThekNaturekofkEconometricskandkEconomickData 1
Chapterk2 ThekSimplekRegressionkModel 5
Chapterk3 MultiplekRegressionkAnalysis:k Estimation 15
Chapterk4 MultiplekRegressionkAnalysis:k Inference 28
Chapterk5 MultiplekRegressionkAnalysis:k OLSkAsymptotics 41
Chapterk6 MultiplekRegressionkAnalysis:k FurtherkIssues 46
Chapterk7 MultiplekRegressionkAnalysiskwithkQualitativek Inf 62
ormation:k Binaryk(orkDummy)kVariables
Chapterk8 Heteroskedasticity 79
Chapterk9 MorekonkSpecificationkandkDatakProblems 91
Chapterk10 BasickRegressionkAnalysiskwithkTimekSerieskData 102
Chapterk11 FurtherkIssueskinkUsingkOLSkwithkTimekSerieskData 114
Chapterk12 SerialkCorrelationkandkHeteroskedasticitykink T 127
imekSerieskRegressions
Chapterk13 PoolingkCrosskSectionskAcrosskTime.k Simplek Pan 139
elkDatakMethods
Chapterk14 AdvancedkPanelkDatakMethods 154
Chapterk15 InstrumentalkVariableskEstimationkandkTwokStagek Le 167
astkSquares
Chapterk16 SimultaneouskEquationskModels 183
Chapterk17 LimitedkDependentkVariablekModelskandkSamplek S 196
electionkCorrections
,Chapterk18 AdvancedkTimekSerieskTopics 217
Chapterk19 CarryingkOutkankEmpiricalkProject 233
AppendixkA BasickMathematicalkTools 234
AppendixkB FundamentalskofkProbability 236
AppendixkC FundamentalskofkMathematicalkStatistics 238
AppendixkD SummarykofkMatrixkAlgebra 242
AppendixkE ThekLinearkRegressionkModelkinkMatrixkForm 244
, CHAPTER 1 k
TEACHINGkNOTES
YoukhaveksubstantialklatitudekaboutkwhatktokemphasizekinkChapterk1.k Ikfindkitkusefulktoktalkkaboutk thek
economicskofkcrimekexamplek(Examplek1.1)kandkthekwagekexamplek(Examplek1.2)ksokthatk studentskse
e,katkthekoutset,kthatkeconometricskisklinkedktokeconomickreasoning,kevenkifkthek economicskisknotkcomplic
atedktheory.
Iklikektokfamiliarizekstudentskwithkthekimportantkdatakstructureskthatkempiricalkeconomistskuse,k focusing
kprimarilykonkcross-sectionalkandktimekserieskdataksets,kaskthesekarekwhatkIkcoverkinkak first-
semesterkcourse.k Itkiskprobablykakgoodkideaktokmentionkthekgrowingkimportancekofkdataksetsk thatkha
vekbothkakcross-sectionalkandktimekdimension.
Ikspendkalmostkankentireklecturektalkingkaboutkthekproblemskinherentkinkdrawingkcausalkinferencesk inkth
eksocialksciences.k Ikdokthiskmostlykthroughkthekagriculturalkyield,kreturnktokeducation,kandk crimekexamp
les.k Thesekexampleskalsokcontrastkexperimentalkandknonexperimentalk (observational)kdata.k Studen
tskstudyingkbusinesskandkfinancektendktokfindkthektermkstructurekofk interestkrateskexamplekmorekreleva
nt,kalthoughkthekissuektherekisktestingkthekimplicationkofkak simplektheory,kaskopposedktokinferringkcausa
lity.k Ikhavekfoundkthatkspendingktimektalkingkaboutk thesekexamples,kinkplacekofkakformalkreviewkofkpr
obabilitykandkstatistics,kiskmoreksuccessfulk(andk morekenjoyablekforkthekstudentskandkme).
,SOLUTIONSkTOkPROBLEMS
1.1 Itkdoesknotkmakeksensektokposekthekquestionkinktermskofkcausality.kEconomistskwouldkassumek thatkstu
dentskchoosekakmixkofkstudyingkandkworkingk(andkotherkactivities,ksuchkaskattendingkclass,k leisure,kandk
sleeping)kbasedkonkrationalkbehavior,ksuchkaskmaximizingkutilityksubjectktokthek constraintkthatktherekar
ekonlyk168khourskinkakweek.k Wekcankthenkusekstatisticalkmethodsktok measurekthekassociationkbetweenk
studyingkandkworking,kincludingkregressionkanalysiskthatkwek coverkstartingkinkChapterk2.k Butkwekwoul
dknotkbekclaimingkthatkonekvariablek“causes”kthekother.k Theykarekbothkchoicekvariableskofkthekstudent.
1.2 (i)kIdeally,kwekcouldkrandomlykassignkstudentsktokclasseskofkdifferentksizes.k Thatkis,keachk studentki
skassignedkakdifferentkclassksizekwithoutkregardktokanykstudentkcharacteristicsksuchkask abilitykandkfa
milykbackground.k ForkreasonskwekwillkseekinkChapterk2,kwekwouldklikeksubstantialk variationkinkclassks
izesk(subject,kofkcourse,ktokethicalkconsiderationskandkresourcekconstraints).
(ii) Aknegativekcorrelationkmeanskthatklargerkclassksizekiskassociatedkwithklowerkperformance.k W
ekmightkfindkaknegativekcorrelationkbecauseklargerkclassksizekactuallykhurtskperformance.k However,k
withkobservationalkdata,ktherekarekotherkreasonskwekmightkfindkaknegativekrelationship.k Forkexampl
e,kchildrenkfromkmorekaffluentkfamilieskmightkbekmoreklikelyktokattendkschoolskwithk smallerkclassksizes,k
andkaffluentkchildrenkgenerallykscorekbetterkonkstandardizedktests.k Anotherk possibilitykiskthat,kwithink
akschool,kakprincipalkmightkassignkthekbetterkstudentsktoksmallerkclasses.k Or,ksomekparentskmightkinsistkt
heirkchildrenkarekinktheksmallerkclasses,kandktheseksamekparentsk tendktokbekmorekinvolvedkinktheirkchild
ren’skeducation.
(iii) Givenkthekpotentialkforkconfoundingkfactorsk–ksomekofkwhichkareklistedkink(ii)k–
findingkak negativekcorrelationkwouldknotkbekstrongkevidencekthatksmallerkclassksizeskactuallykleadktok
k
betterk performance.kSomekwaykofkcontrollingkforkthekconfoundingkfactorskiskneeded,kandkthiskiskthek s
ubjectkofkmultiplekregressionkanalysis.
1.3 (i)kHerekiskonekwayktokposekthekquestion:kIfktwokfirms,ksaykAkandkB,karekidenticalkinkallk respectskex
ceptkthatkfirmkAksupplieskjobktrainingkonekhourkperkworkerkmorekthankfirmkB,kbykhowk muchkwouldkfirm
kA’skoutputkdifferkfrom kfirmkB’s?
(ii) Firmskareklikelyktokchoosekjobktrainingkdependingkonkthekcharacteristicskofkworkers.kSomek obs
ervedkcharacteristicskarekyearskofkschooling,kyearskinkthekworkforce,kandkexperiencekinkak particularkj
ob.kFirmskmightkevenkdiscriminatekbasedkonkage,kgender,korkrace.kPerhapskfirmskk choosektokofferktraini
ngktokmorekorklesskablekworkers,kwherek“ability”kmightkbekdifficultktok quantifykbutkwherekakmanagerkh
asksomekideakaboutkthekrelativekabilitieskofkdifferentkemployees.k Moreover,kdifferentkkindskofkworke
rskmightkbekattractedktokfirmskthatkofferkmorekjobktrainingkonk average,kandkthiskmightknotkbekevidentkt
okemployers.
(iii) Thekamountkofkcapitalkandktechnologykavailablektokworkerskwouldkalsokaffectkoutput.k So,k tw
okfirmskwithkexactlyktheksamekkindskofkemployeeskwouldkgenerallykhavekdifferentkoutputskifk theykusekd
ifferentkamountskofkcapitalkorktechnology.k Thekqualitykofkmanagerskwouldkalsokhavekank effect.
, (iv) No,kunlesskthekamountkofktrainingkiskrandomlykassigned.k Thekmanykfactorsklistedkinkparts
(ii) andk(iii)kcankcontributektokfindingkakpositivekcorrelationkbetweenkoutputkandktrainingkevenkifk jobkt
rainingkdoesknotkimprovekworkerkproductivity.
SOLUTIONSkTOkCOMPUTERkEXERCISES
C1.1k(i)kThekaveragekofkeduckiskaboutk12.6kyears.k Therekarektwokpeoplekreportingkzerokyearskofk ed
ucation,kandk19kpeoplekreportingk18kyearskofkeducation.
(ii) Thekaveragekofkwagekiskaboutk$5.90,kwhichkseemsklowkinkthekyeark2008.
(iii) UsingkTablekB-
60kinkthek2004kEconomickReportkofkthekPresident,kthekCPIkwask56.9kink 1976kandk184.0kink2003.
(iv) Tokconvertk1976kdollarskintok2003kdollars,kwekusekthekratiokofkthekCPIs,kwhichkisk 184k
/k56.9k 3.23k.k Therefore,kthekaveragekhourlykwagekink2003kdollarskiskroughlyk 3.23($5.9
0)k $19.06k,kwhichkiskakreasonablekfigure.
(v) Theksamplekcontainsk252kwomenk(theknumberkofkobservationskwithkfemalek=k1)kandk274k me
n.
C1.2k(i)kTherekarek1,388kobservationskinktheksample.k Tabulatingkthekvariablekcigskshowskthatk212k wo
menkhavekcigsk>k0.
(ii) Thekaveragekofkcigskiskaboutk2.09,kbutkthiskincludeskthek1,176kwomenkwhokdidknotk smoke.k
Reportingkjustkthekaveragekmaskskthekfactkthatkalmostk85kpercentkofkthekwomenkdidknotk smoke.k Itkma
keskmoreksensektoksaykthatkthek“typical”kwomankdoesknotksmokekduringkpregnancy;k indeed,kthekmedi
anknumberkofkcigarettesksmokedkiskzero.
(iii) Thekaveragekofkcigskoverkthekwomenkwithkcigsk>k0kiskaboutk13.7.kOfkcoursekthiskisk muc
hkhigherkthankthekaveragekoverkthekentireksamplekbecausekwekarekexcludingk1,176kzeros.
(iv) Thekaveragekofkfatheduckiskaboutk13.2.kTherekarek196kobservationskwithkakmissingk val
uekforkfatheduc,kandkthosekobservationskareknecessarilykexcludedkinkcomputingkthekaverage.
(v) Thekaveragekandkstandardkdeviationkofkfaminckarekaboutk29.027kandk18.739,k resp
ectively,kbutkfaminckiskmeasuredkinkthousandskofkdollars.k So,kinkdollars,kthekaveragekandk standar
dkdeviationkarek$29,027kandk$18,739.
C1.3k(i)kTheklargestkisk100,ktheksmallestkisk0.
(ii) 38koutkofk1,823,korkaboutk2.1kpercentkofktheksample.
, (iii) 17
(iv) Thekaveragekofkmath4kiskaboutk71.9kandkthekaveragekofkread4kiskaboutk60.1.k So,katkleastk i
nk2001,kthekreadingktestkwaskharderktokpass.
(v) Theksamplekcorrelationkbetweenkmath4kandkread4kiskaboutk.843,kwhichkiskakverykhighk deg
reekofk(linear)kassociation.k Notksurprisingly,kschoolskthatkhavekhighkpasskrateskonkonektestk havekaks
trongktendencyktokhavekhighkpasskrateskonkthekotherktest.
(vi) Thekaveragekofkexpppkiskaboutk$5,194.87.k Thekstandardkdeviationkisk$1,091.89,kwhichk s
howskratherkwidekvariationkinkspendingkperkpupil.k [Thekminimumkisk$1,206.88kandkthek maximumkisk
$11,957.64.]
C1.4k(i)k185/445k .416kiskthekfractionkofkmenkreceivingkjobktraining,korkaboutk41.6%.
(ii) Forkmenkreceivingkjobktraining,kthekaveragekofkre78kiskaboutk6.35,kork$6,350.k Forkmenknotk re
ceivingkjobktraining,kthekaveragekofkre78kiskaboutk4.55,kork$4,550.k Thekdifferencekisk$1,800,kwhichkisk
veryklarge.k Onkaverage,kthekmenkreceivingkthekjobktrainingkhadkearningskaboutk40%k higherkthankthos
eknotkreceivingktraining.
(iii) Aboutk24.3%kofkthekmenkwhokreceivedktrainingkwerekunemployedkink1978;kthekfigurekisk 35
.4%kforkmenknotkreceivingktraining.k This,ktoo,kiskakbigkdifference.
(iv) Thekdifferenceskinkearningskandkunemploymentkratesksuggestkthektrainingkprogramkhadk str
ong,kpositivekeffects.k Ourkconclusionskaboutkeconomicksignificancekwouldkbekstrongerkifkwek couldka
lsokestablishkstatisticalksignificancek(whichkiskdonekinkComputerkExercisekC9.10kink Chapterk9).
, CHAPTER 2 k
TEACHINGkNOTES
ThiskiskthekchapterkwherekIkexpectkstudentsktokfollowkmost,kifknotkall,kofkthekalgebraickderivations.k Inkcla
sskIklikektokderivekatkleastkthekunbiasednesskofkthekOLSkslopekcoefficient,kandkusuallykIkk derivekthekvaria
nce.k Atkakminimum,kIktalkkaboutkthekfactorskaffectingkthekvariance.k Toksimplifyk theknotation,kafterkIkem
phasizekthekassumptionskinkthekpopulationkmodel,kandkassumekrandomk sampling,kIkjustkconditionkonkthek
valueskofkthekexplanatorykvariableskinktheksample.k Technically,k thiskiskjustifiedkbykrandomksamplingkb
ecause,kforkexample,kE(ui|x1,x2,…,xn)k=kE(ui|xi)kbyk independentksampling.k Ikfindkthatkstudentskareka
blektokfocuskonkthekkeykassumptionkSLR.4kandk subsequentlyktakekmykwordkaboutkhowkconditioningkonkt
hekindependentkvariableskinktheksamplekisk harmless.k (Ifkyoukprefer,kthekappendixktokChapterk3kdoeskt
hekconditioningkargumentkcarefully.)k Becausekstatisticalkinferencekisknokmorekdifficultkinkmultiplekregre
ssionkthankinksimplekregression,k IkpostponekinferencekuntilkChapterk4.k (Thiskreduceskredundancykandka
llowskyouktokfocuskonkthek interpretivekdifferenceskbetweenksimplekandkmultiplekregression.)
Youkmightknoticekhow,kcomparedkwithkmostkotherktexts,kIkusekrelativelykfewkassumptionsktok derivekthek
unbiasednesskofkthekOLSkslopekestimator,kfollowedkbykthekformulakforkitskvariance.k ThiskiskbecausekIkd
oknotkintroducekredundantkorkunnecessarykassumptions.k Forkexample,koncek SLR.4kiskassumed,knothingk
furtherkaboutkthekrelationshipkbetweenkukandkxkiskneededktokobtainkthek unbiasednesskofkOLSkunderkra
ndomksampling.
,SOLUTIONSkTOkPROBLEMS
2.1k Inkthekequationkyk=k 0k +k 1xk+ku,kaddkandksubtractk 0k fromkthekrightkhandksidektokgetkyk=k( 0k +
0)k+k 1xk+k(uk 0).k Callktheknewkerrorkek=kuk 0,ksokthatkE(e)k=k0.k Theknewkinterceptkisk 0k
+
0,kbutkthekslopekiskstillk 1.
n
2.2k (i)kLetkyik =kGPAi,kxik =kACTi,kandknk=k8.k Thenkk xk=k25.875,k yk =k3.2125,k (xik –k xk)(yik –k yk)k=
i 1
n
5.8125,kandk (xik – ˆk
2k
k xk) =k56.875.k Fromkequationk(2.9),kwekobtainkthekslopekas =
1
i 1
ˆ
5.8125/56.875k .1022,kroundedktokfourkplaceskafterkthekdecimal.k Fromk(2.1 0 kk = k yk –
7),
ˆ1kk xkk 3.2125k–k(.1022)25.875k .5681.k Sokwekcankwrite
GPAk =k .5681k+k.1022kACT
nk=k8.
ThekinterceptkdoesknotkhavekakusefulkinterpretationkbecausekACTkisknotkclosektokzerokforkthek populatio
nkofkinterest.kkIfkACTkisk5kpointskhigher,k GPAkincreaseskbyk.1022(5)k=k.511.
(ii)kThekfittedkvalueskandkresidualsk—kroundedktokfourkdecimalkplacesk—
arekgivenkalongkwithk thekobservationknumberkikandkGPAkinkthekfollowingktable:
k
i GPA G û
1 2.8 PA
2.7143 .0857
2 3.4 3.0209 .3791
3 3.0 3.2253 –.2253
4 3.5 3.3275 .1725
5 3.6 3.5319 .0681
6 3.0 3.1231 –.1231
7 2.7 3.1231 –.4231
8 3.7 3.6341 .0659
Youkcankverifykthatkthekresiduals,kaskreportedkinkthektable,ksumktok .0002,kwhichkiskprettykclosektok zer
okgivenkthekinherentkroundingkerror.
(iii)kWhenkACTk=k20,k GPAk=k.5681k+k.1022(20)k 2.61.
, n
(iv)kTheksumkofksquaredkresiduals, ûikk2 ,kiskaboutk.4347k(roundedktokfourkdecimalkplaces),
i
n 1
andkthektotalksumkofksquares,k (yik –k yk)2,kiskaboutk1.0288.k SokthekR-squaredkfromkthe
i 1
regressionkis
R2kk =k 1k–kSSR/SSTk 1k–k(.4347/1.0288)k .577.
Therefore,kaboutk57.7%kofkthekvariationkinkGPAkiskexplainedkbykACTkinkthisksmallksamplekofk students.
2.3 (i)kIncome,kage,kandkfamilykbackgroundk(suchkasknumberkofksiblings)karekjustkakfewk possibilities.k Itks
eemskthatkeachkofkthesekcouldkbekcorrelatedkwithkyearskofkeducation.k (Incomekk andkeducationkarekpro
bablykpositivelykcorrelated;kagekandkeducationkmaykbeknegativelykcorrelatedk becausekwomenkinkmo
rekrecentkcohortskhave,konkaverage,kmorekeducation;kandknumberkofksiblingsk andkeducationkarekproba
blyknegativelykcorrelated.)
(ii)kNotkifkthekfactorskweklistedkinkpartk(i)karekcorrelatedkwithkeduc.k Becausekwekwouldklikektok holdkt
hesekfactorskfixed,ktheykarekpartkofkthekerrorkterm.k Butkifkukiskcorrelatedkwithkeduckthenk E(u|educ)k
0,kandksokSLR.4kfails.
2.4 (i)kWekwouldkwantktokrandomlykassignktheknumberkofkhourskinkthekpreparationkcourseksokthatk hour
skiskindependentkofkotherkfactorskthatkaffectkperformancekonkthekSAT.kThen,kwekwouldk collectkinforma
tionkonkSATkscorekforkeachkstudentkinkthekexperiment,kyieldingkakdatakset
{(satik,khoursik)k:kik 1,...,kn},kwhereknkisktheknumberkofkstudentskwekcankaffordktokhavekinkthekstudy.
Fromkequationk(2.7),kwekshouldktryktokgetkaskmuchkvariationkink hoursi askiskfeasible.
(ii) Herekarekthreekfactors:k innatekability,kfamilykincome,kandkgeneralkhealthkonkthekdaykofkthek ex
am.k Ifkwekthinkkstudentskwithkhigherknativekintelligencekthinkktheykdoknotkneedktokpreparekfork thekSAT,kt
henkabilitykandkhourskwillkbeknegativelykcorrelated.k Familykincomekwouldkprobablykbek positivelykcorr
elatedkwithkhours,kbecausekhigherkincomekfamilieskcankmorekeasilykaffordk preparationkcourses.k Rulin
gkoutkchronickhealthkproblems,khealthkonkthekdaykofkthekexamkshouldkk bekroughlykuncorrelatedkwithkhou
rskspentkinkakpreparationkcourse.
(iii) Ifkpreparationkcourseskarekeffective,k 1kk shouldkbekpositive:kotherkfactorskequal,kank i
ncreasekinkhourskshouldkincreaseksat.
(iv) Thekintercept,k 0k ,khaskakusefulkinterpretationkinkthiskexample:kbecausekE(u)k=k0,k 0
iskthe
averagekSATkscorekforkstudentskinkthekpopulationkwithkhoursk=k0.
2.5 (i)kWhenkwekconditionkonkinckinkcomputingkankexpectation, inc becomeskakconstant.k So
E(u|inc)k=kE(kkkinck e|inc)k inc E(e|inc)k= inc 0kbecausekE(e|inc)k=kE(e)k=k0.
=
(ii) Again,kwhenkwekconditionkonkinckinkcomputingkak variance, inc becomeskakconstant.k So
2 2k 2k
Var(u|inc)k=kVar(kkkinck e|inc)k=k(kkkinck) Var(e|inc)k=k inckbecausekVar(e|inc)k=k .
e e
