Econometric-Theory-and-Methods---Russell-Davidson-and-James-G.-MacKinnon.pdf

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Chapter1
RegressionModels
1.1Introduction
Regressionmodelsformthecoreofthedisciplineofeconometrics.Although
econometriciansroutinelyestimateawidevarietyofstatisticalmodels,using
manydi®erenttypesofdata,thevastmajorityoftheseareeitherregression
modelsorcloserelativesofthem.Inthischapter,weintroducetheconceptof
aregressionmodel,discussseveralvarietiesofthem,andintroducetheestima-
tionmethodthatismostcommonlyusedwithregressionmodels,namely,least
squares.Thisestimationmethodisderivedbyusingthemethodofmoments,
whichisaverygeneralprincipleofestimationthathasmanyapplicationsin
econometrics.
Themostelementarytypeofregressionmodelisthesimplelinearregression
model,whichcanbeexpressedbythefollowingequation:
y t = ¯ 1 + ¯ 2 X t + u t : (1 : 01)
Thesubscript t isusedtoindextheobservationsofasample.Thetotalnum-
berofobservations,alsocalledthesamplesize,willbedenotedby n .Thus,
forasampleofsize n ,thesubscript t runsfrom1to n .Eachobservation
comprisesanobservationonadependentvariable,writtenas y t forobserva-
tion t ,andanobservationonasingleexplanatoryvariable,orindependent
variable,writtenas X t .
Therelation(1.01)linkstheobservationsonthedependentandtheexplana-
toryvariablesforeachobservationintermsoftwounknownparameters, ¯ 1
and ¯ 2 ,andanunobservederrorterm, u t .Thus,ofthefivequantitiesthat
appearin(1.01),two, y t and X t ,areobserved,andthree, ¯ 1 , ¯ 2 ,and u t ,are
not.Threeofthem, y t , X t ,and u t ,arespecifictoobservation t ,whilethe
othertwo,theparameters,arecommontoall n observations.
Hereisasimpleexampleofhowaregressionmodellike(1.01)couldarisein
economics.Supposethattheindex t isatimeindex,asthenotationsuggests.
Eachvalueof t couldrepresentayear,forinstance.Then y t couldbehouse-
holdconsumptionasmeasuredinyear t ,and X t couldbemeasureddisposable
incomeofhouseholdsinthesameyear.Inthatcase,(1.01)wouldrepresent
whatinelementarymacroeconomicsiscalledaconsumptionfunction.
Copyrightc ° 1999,RussellDavidsonandJamesG.MacKinnon 3
4 RegressionModels
Ifforthemomentweignorethepresenceoftheerrorterms, ¯ 2 isthemarginal
propensitytoconsumeoutofdisposableincome,and ¯ 1 iswhatissometimes
calledautonomousconsumption.Asistrueofagreatmanyeconometricmod-
els,theparametersinthisexamplecanbeseentohaveadirectinterpretation
intermsofeconomictheory.Thevariables,incomeandconsumption,doin-
deedvaryinvaluefromyeartoyear,astheterm“variables”suggests.In
contrast,theparametersreflectaspectsoftheeconomythatdonotvary,but
takeonthesamevalueseachyear.
Thepurposeofformulatingthemodel(1.01)istotrytoexplaintheobserved
valuesofthedependentvariableintermsofthoseoftheexplanatoryvariable.
Accordingto(1.01),foreach t ,thevalueof y t isgivenbyalinearfunction
of X t ,pluswhatwehavecalledtheerrorterm, u t .Thelinear(strictlyspeak-
ing,a±ne 1 )function,whichinthiscaseis ¯ 1 + ¯ 2 X t ,iscalledtheregression
function.Atthisstageweshouldnotethat,aslongaswesaynothingabout
theunobservedquantity u t ,(1.01)doesnottellusanything.Infact,wecan
allowtheparameters ¯ 1 and ¯ 2 tobequitearbitrary,since,foranygiven ¯ 1
and ¯ 2 ,(1.01)canalwaysbemadetobetruebydefining u t suitably.
Ifwewishtomakesenseoftheregressionmodel(1.01),then,wemustmake
someassumptionsaboutthepropertiesoftheerrorterm u t .Preciselywhat
thoseassumptionsarewillvaryfromcasetocase.Inallcases,though,itis
assumedthat u t isarandomvariable.Mostcommonly,itisassumedthat,
whateverthevalueof X t ,theexpectationoftherandomvariable u t iszero.
Thisassumptionusuallyservestoidentifytheunknownparameters ¯ 1 and
¯ 2 ,inthesensethat,undertheassumption,(1.01)canbetrueonlyforspecific
valuesofthoseparameters.
Thepresenceoferrortermsinregressionmodelsmeansthattheexplanations
thesemodelsprovideareatbestpartial.Thiswouldnotbesoiftheerror
termscouldbedirectlyobservedaseconomicvariables,forthen u t couldbe
treatedasafurtherexplanatoryvariable.Inthatcase,(1.01)wouldbea
relationlinking y t to X t and u t inacompletelyunambiguousfashion.Given
X t and u t , y t wouldbecompletelyexplainedwithouterror.
Ofcourse,errortermsarenotobservedintherealworld.Theyareincluded
inregressionmodelsbecausewearenotabletospecifyallofthereal-world
factorsthatdetermine y t .Whenwesetupourmodelswith u t asaran-
domvariable,whatwearereallydoingisusingthemathematicalconceptof
randomnesstomodelour ignorance ofthedetailsofeconomicmechanisms.
Whatwearedoingwhenwesupposethatthemeanofanerrortermiszerois
supposingthatthefactorsdetermining y t thatweignorearejustaslikelyto
make y t biggerthanitwouldhavebeenifthosefactorswereabsentasthey
aretomake y t smaller.Thusweareassumingthat,onaverage,thee®ects
oftheneglecteddeterminantstendtocancelout.Thisdoesnotmeanthat
1 Afunction g ( x )issaidtobea±neifittakestheform g ( x )= a + bx fortwo
realnumbers a and b .
Copyrightc ° 1999,RussellDavidsonandJamesG.MacKinnon
1.2Distributions,Densities,andMoments 5
thosee®ectsarenecessarilysmall.Theproportionofthevariationin y t that
isaccountedforbytheerrortermwilldependonthenatureofthedataand
theextentofourignorance.Evenifthisproportionislarge,asitwillbein
somecases,regressionmodelslike(1.01)canbeusefuliftheyallowustosee
how y t isrelatedtothevariables,like X t ,thatwecanactuallyobserve.
Muchoftheliteratureineconometrics,andthereforemuchofthisbook,is
concernedwithhowtoestimate,andtesthypothesesabout,theparameters
ofregressionmodels.Inthecaseof(1.01),theseparametersaretheconstant
term,orintercept, ¯ 1 ,andtheslopecoe±cient, ¯ 2 .Althoughwewillbegin
ourdiscussionofestimationinthischapter,mostofitwillbepostponeduntil
laterchapters.Inthischapter,weareprimarilyconcernedwithunderstanding
regressionmodelsasstatisticalmodels,ratherthanwithestimatingthemor
testinghypothesesaboutthem.
Inthenextsection,wereviewsomeelementaryconceptsfromprobability
theory,includingrandomvariablesandtheirexpectations.Manyreaderswill
alreadybefamiliarwiththeseconcepts.TheywillbeusefulinSection1.3,
wherewediscussthemeaningofregressionmodelsandsomeoftheforms
thatsuchmodelscantake.InSection1.4,wereviewsometopicsfrommatrix
algebraandshowhowmultipleregressionmodelscanbewrittenusingmatrix
notation.Finally,inSection1.5,weintroducethemethodofmomentsand
showhowitleadstoordinaryleastsquaresasawayofestimatingregression
models.
1.2Distributions,Densities,andMoments
Thevariablesthatappearinaneconometricmodelaretreatedaswhatstatis-
ticianscallrandomvariables.Inordertocharacterizearandomvariable,we
mustfirstspecifythesetofallthepossiblevaluesthattherandomvariable
cantakeon.Thesimplestcaseisascalarrandomvariable,orscalarr.v.The
setofpossiblevaluesforascalarr.v.maybethereallineorasubsetofthe
realline,suchasthesetofnonnegativerealnumbers.Itmayalsobetheset
ofintegersorasubsetofthesetofintegers,suchasthenumbers1,2,and3.
Sincearandomvariableisacollectionofpossibilities,randomvariablescannot
beobservedassuch.Whatwedoobservearerealizationsofrandomvariables,
arealizationbeingonevalueoutofthesetofpossiblevalues.Forascalar
randomvariable,eachrealizationisthereforeasinglerealvalue.
If X isanyrandomvariable,probabilitiescanbeassignedtosubsetsofthe
fullsetofpossibilitiesofvaluesfor X ,insomecasestoeachpointinthat
set.Suchsubsetsarecalledevents,andtheirprobabilitiesareassignedbya
probabilitydistribution,accordingtoafewgeneralrules.
Copyrightc ° 1999,RussellDavidsonandJamesG.MacKinnon
6 RegressionModels
DiscreteandContinuousRandomVariables
Theeasiestsortofprobabilitydistributiontoconsiderariseswhen X isa
discreterandomvariable,whichcantakeonafinite,orperhapsacountably
infinitenumberofvalues,whichwemaydenoteas x 1 ;x 2 ;::: .Theprobability
distributionsimplyassignsprobabilities,thatis,numbersbetween0and1,
toeachofthesevalues,insuchawaythattheprobabilitiessumto1:
1 X
p ( x i )=1 ;
i =1
where p ( x i )istheprobabilityassignedto x i .Anyassignmentofnonnega-
tiveprobabilitiesthatsumtooneautomaticallyrespectsallthegeneralrules
alludedtoabove.
Inthecontextofeconometrics,themostcommonlyencountereddiscreteran-
domvariablesoccurinthecontextofbinarydata,whichcantakeonthe
values0and1,andinthecontextofcountdata,whichcantakeonthevalues
0,1,2, ::: ;seeChapter11.
Anotherpossibilityisthat X maybeacontinuousrandomvariable,which,for
thecaseofascalarr.v.,cantakeonanyvalueinsomecontinuoussubsetofthe
realline,orpossiblythewholerealline.Thedependentvariableinaregression
modelisnormallyacontinuousr.v.Foracontinuousr.v.,theprobability
distributioncanberepresentedbyacumulativedistributionfunction,orCDF.
Thisfunction,whichisoftendenoted F ( x ),isdefinedontherealline.Its
valueisPr( X·x ),theprobabilityoftheeventthat X isequaltoorless
thansomevalue x .Ingeneral,thenotationPr( A )signifiestheprobability
assignedtotheevent A ,asubsetofthefullsetofpossibilities.Since X is
continuous,itdoesnotreallymatterwhetherwedefinetheCDFasPr( X·x )
orasPr( X<x )here,butitisconventionaltousetheformerdefinition.
Noticethat,intheprecedingparagraph,weused X todenotearandom
variableand x todenotearealizationof X ,thatis,aparticularvaluethatthe
randomvariable X maytakeon.Thisdistinctionisimportantwhendiscussing
themeaningofaprobabilitydistribution,butitwillrarelybenecessaryin
mostofthisbook.
ProbabilityDistributions
Wemaynowmakeexplicitthegeneralrulesthatmustbeobeyedbyproba-
bilitydistributionsinassigningprobabilitiestoevents.Therearejustthree
oftheserules:
(i)Allprobabilitiesliebetween0and1;
(ii)Thenullsetisassignedprobability0,andthefullsetofpossibilitiesis
assignedprobability1;
(iii)Theprobabilityassignedtoaneventthatistheunionoftwodisjoint
eventsisthesumoftheprobabilitiesassignedtothosedisjointevents.
Copyrightc ° 1999,RussellDavidsonandJamesG.MacKinnon

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