﻿
Chapter27. CointegrationandVectorErrorCorrectionModels
240
Considernowanotherprocessy
t
,deﬁnedby
y
t
kx
t
u
t
where,again,kisarealnumberandu
t
isawhitenoiseprocess.Sinceu
t
isstationarybydeﬁnition,
x
t
andy
t
cointegrate:thatis,theirdiﬀerence
z
t
y
t
x
t
ku
t
isastationaryprocess.Fork=0,z
t
issimplezero-meanwhitenoise,whereasfork60theprocess
z
t
iswhitenoisewithanon-zeromean.
Aftersomesimplesubstitutions,thetwo equations abovecanberepresentedjointlyasaVAR(1)
system
"
y
t
x
t
#
"
km
m
#
"
0 1
0 1
#"
y
1
x
1
#
"
u
t
"
t
"
t
#
orinVECMform
"
Ñy
t
Ñx
t
#
"
km
m
#
"
1 1
0 0
#"
y
1
x
1
#
"
u
t
"
t
"
t
#
"
km
m
#
"
1
0
#
h
1
i
"
y
1
x
1
#
"
u
t
"
t
"
t
#
0

0
"
y
1
x
1
#
t
0
z
1
t
;
1. m60:Inthiscasex
t
istrended,aswejustsaw;itfollowsthaty
t
alsofollowsalineartrend
becauseonaverageitkeepsataﬁxeddistancekfromx
t
.Thevector
0
isunrestricted.
2. m=0andk60:Inthiscase,x
t
isnottrendedandasaconsequenceneitherisy
t
.However,
themeandistancebetweeny
t
andx
t
isnon-zero.Thevector
0
isgivenby
0
"
k
0
#
which is s not t null and therefore the VECM shown above does s have a constant t term. . The
constant, however, is s subject to the e restriction that its second element must be0. . More
generally,
0
isamultipleofthevector.NotethattheVECMcouldalsobewrittenas
"
Ñy
t
Ñx
t
#
"
1
0
#
h
1  k
i
2
6
6
4
y
1
x
1
1
3
7
7
5
"
u
t
"
t
"
t
#
whichincorporatestheinterceptintothecointegrationvector.Thisisknownasthe“restricted
constant”case.
3. m=0andk=0: : Thiscaseisthemostrestrictive:clearly,neitherx
t
nory
t
aretrended,and
themeandistancebetweenthemiszero.Thevector
0
isalso0,whichexplainswhythiscase
isreferredtoas“noconstant.”
Inmostcases, thechoicebetweenthesethreepossibilities isbasedonamixofempiricalobser-
vationandeconomicreasoning. Ifthevariablesunderconsiderationseemtofollowalineartrend
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Chapter27. CointegrationandVectorErrorCorrectionModels
241
thenweshouldnotplaceanyrestrictionontheintercept.Otherwise,thequestionarisesofwhether
itmakessensetospecifyacointegrationrelationshipwhichincludesanon-zerointercept.Oneex-
amplewherethisisappropriateistherelationshipbetweentwointerestrates:generallytheseare
nottrended,buttheVARmightstillhaveaninterceptbecausethediﬀerencebetweenthetwo(the
Thepreviousexamplecanbegeneralizedinthreedirections:
1. If f aVAR oforder r greater r than n 1is considered, , thealgebra a gets s moreconvoluted but the
conclusionsareidentical.
2. If f theVAR R includes s morethan two endogenous variables s the cointegration n rank can n be
greaterthan1.Inthiscase,isamatrixwithcolumns,andthecasewithrestrictedconstant
entailstherestrictionthat
0
shouldbesomelinearcombinationofthecolumnsof.
3. Ifalineartrendisincludedinthemodel,thedeterministicpartoftheVARbecomes
0
1
t.
Thereasoningispracticallythesameasaboveexceptthatthefocusnowcenterson
1
rather
than
0
. Thecounterpartto o the“restrictedconstant”casediscussedaboveis a“restricted
trend”case,suchthatthecointegrationrelationshipsincludeatrendbuttheﬁrstdiﬀerences
ofthevariablesinquestiondo not. . Inthecaseofanunrestrictedtrend,thetrendappears
inboth thecointegrationrelationships andtheﬁrstdiﬀerences, which corresponds to o the
Inorder toaccommodatetheﬁvecases, gretlprovidesthefollowingoptionsto thecoint2and
vecmcommands:
t
optionﬂag
description
0
--nc
noconstant
0
;
0
?
0
0
--rc
restrictedconstant
0
--uc
unrestrictedconstant
0
1
t;
0
?
1
0
--crt
constant+restrictedtrend
0
1
t
--ct
constant+unrestrictedtrend
optionofusingthe--seasonaloptions, for augmenting
t
withcenteredseasonaldummies. In
eachcase,p-valuesarecomputedviatheapproximationsdevisedbyDoornik(1998).
27.4 TheJohansencointegrationtests
Thetwo Johansentestsfor cointegrationareused toestablishtherank of , or in n other r words
thenumber of f cointegrating vectors. . Theseare e the“-max” test, , for hypotheses s on n individual
eigenvalues,andthe“trace”test,forjointhypotheses. Supposethattheeigenvalues
i
aresorted
fromlargesttosmallest.Thenullhypothesisforthe-maxtestonthei-theigenvalueisthat
i
0.
j
0forallji.
“Model,TimeSeries,CointegrationTest,Johansen”.
t
oneincludesintheVAR(seesection27.3above).Thefollowingcodeusesthedenmarkdataﬁle,
suppliedwithgretl,toreplicateJohansen’sexamplefoundinhis1995book.
open denmark
coint2 2 2 LRM M LRY IBO IDE --rc c --seasonal
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Chapter27. CointegrationandVectorErrorCorrectionModels
242
Inthiscase,thevectory
t
inequation(27.2)comprisesthefourvariablesLRM,LRY,IBO,IDE.The
numberoflagsequalspin(27.2)(thatis,thenumberoflagsofthemodelwritteninVARform).
Partoftheoutputisreportedbelow:
Johansen test:
Number of equations s = = 4
Lag order = 2
Estimation period: : 1974:3 3 - 1987:3 (T = 53)
Case 2: : Restricted constant
Rank Eigenvalue Trace test p-value
Lmax test
p-value
0
0.43317
49.144 [0.1284]
30.087 [0.0286]
1
0.17758
19.057 [0.7833]
10.362 [0.8017]
2
0.11279
8.6950 [0.7645]
6.3427 [0.7483]
3
0.043411
2.3522 [0.7088]
2.3522 [0.7076]
Boththetraceand-maxtestsacceptthenullhypothesisthatthesmallesteigenvalueis0(seethe
lastrowofthetable),sowemayconcludethattheseriesareinfactnon-stationary.However,some
linearcombinationmaybeI(0),sincethe-maxtestrejectsthehypothesisthattherankofÕis0
(thoughthetracetestgiveslessclear-cutevidenceforthis,withap-valueof0:1284).
27.5 Identiﬁcationofthecointegrationvectors
Thecoreproblemintheestimationofequation(27.2)istoﬁndanestimateofÕthathasbycon-
structionrankr,soitcanbewrittenasÕ
0
,whereisthematrixcontainingthecointegration
ablesrespondtodeviationfromequilibriuminthepreviousperiod.
Withoutfurther speciﬁcation, theproblem m hasmultiplesolutions (infact, inﬁnitelymany). . The
parametersandareunder-identiﬁed: ifallcolumnsofarecointegrationvectors, , thenany
arbitrarylinearcombinationsofthosecolumnsisacointegrationvectortoo. Toputitdiﬀerently,
ifÕ
0
0
0
forspeciﬁcmatrices
0
and
0
,thenÕalsoequals
0
QQ
1
0
0
foranyconformable
non-singular matrix Q. In n order r to o ﬁnd d a unique solution, , it t is therefore necessaryto impose
somerestrictionsonand/or. Itcanbeshownthattheminimumnumberofrestrictionsthat
isnecessarytoguaranteeidentiﬁcationisr
2
. Normalizingonecoeﬃcientpercolumnto1(or 1,
accordingtotaste)isatrivialﬁrststep,whichalsohelpsinthattheremainingcoeﬃcientscanbe
interpretedastheparametersintheequilibriumrelations,butthisonlysuﬃceswhen1.
The method that gretl uses by default is s known as s the“Phillips s normalization”, , or “triangular
representation”.Thestartingpointiswritinginpartitionedformasin
"
1
2
#
where
1
is an matrix and
2
is n rr. Assuming g that
1
has full rank,  can n be
post-multipliedby
1
1
,giving
ˆ
"
I
2
1
1
#
"
I
B
#
:
Thecoeﬃcientsthatgretlproducesare
ˆ
,withBknownasthematrixofunrestrictedcoeﬃcients.
Intermsoftheunderlyingequilibriumrelationship,thePhillipsnormalizationexpressesthesystem
1
For comparison with other r studies, , you u may y wish h to normalize  diﬀerently. . Using g the set command you
can do o set t vecm_norm diag g to o select a normalization that t simply scales the e columns of f the e original  such h that
ij
 1 for  and  r,asused in the empiricalsection ofBoswijk and Doornik (2004). . Another r alternative is
set vecm_norm first,which scales  such that the elements on the ﬁrst row equal1. . To o suppress normalization
altogether,useset vecm_norm none.(Toreturntothedefault:set vecm_norm m phillips.)
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Chapter27. CointegrationandVectorErrorCorrectionModels
243
ofequilibriumrelationsas
y
1;t
b
1;r1
y
r1;t
:::b
1;n
y
n;t
y
2;t
b
2;r1
y
r1;t
:::b
2;n
y
n;t
.
.
.
y
r;t
b
r;r1
y
r1;t
:::b
r;n
y
r;t
wheretheﬁrstvariablesareexpressedasfunctionsoftheremainingn r.
Although the triangular representation n ensures that t the e statistical problem of estimating  is
solved,theresultingequilibriumrelationshipsmaybediﬃculttointerpret. Inthiscase, , theuser
maywanttoachieveidentiﬁcationbyspecifyingmanuallythesystemofr
2
constraintsthatgretl
willusetoproduceanestimateof.
Asanexample,considerthemoneydemandsystempresentedinsection9.6ofVerbeek(2004).The
variablesusedarem(thelogofrealmoneystockM1),infl(inﬂation),cpr(thecommercialpaper
rate),y(logofrealGDP)andtbr(theTreasurybillrate).
2
Estimationofcanbeperformedviathecommands
open money.gdt
smpl 1954:1 1994:4
vecm 6 2 2 m m infl cpr r y y tbr --rc
Maximum likelihood estimates, observations s 1954:1-1994:4 (T = 164)
Cointegration rank k = = 2
Case 2: : Restricted constant
beta (cointegrating g vectors, , standard errors in parentheses)
m
1.0000
0.0000
(0.0000)
(0.0000)
infl
0.0000
1.0000
(0.0000)
(0.0000)
cpr
0.56108
-24.367
(0.10638)
(4.2113)
y
-0.40446
-0.91166
(0.10277)
(4.0683)
tbr
-0.54293
24.786
(0.10962)
(4.3394)
const
-3.7483
16.751
(0.78082)
(30.909)
Interpretationofthecoeﬃcientsofthecointegrationmatrixwouldbeeasierifameaningcould
beattachedtoeachofitscolumns.Thisispossiblebyhypothesizingtheexistenceoftwolong-run
relationships:amoneydemandequation
mc
1
1
infl
2
y
3
tbr
cprc
2
4
infl
5
y
6
tbr
2
Thisdatasetisavailableintheverbeekdatapackage;seehttp://gretl.sourceforge.net/gretl_data.html.
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Chapter27. CointegrationandVectorErrorCorrectionModels
244
whichimplythatthecointegrationmatrixcanbenormalizedas
2
6
6
6
6
6
6
6
6
6
6
4
1
0
1
4
1
2
5
3
6
c
1
c
2
3
7
7
7
7
7
7
7
7
7
7
5
Thisrenormalizationcanbeaccomplishedbymeansoftherestrictcommand,tobegivenafter
entry.Thesyntaxforenteringtherestrictionsshouldbefairlyobvious:3
restrict
b[1,1] = = -1
b[1,3] = = 0
b[2,1] = = 0
b[2,3] = = -1
end restrict
whichproduces
Cointegrating vectors (standard errors in n parentheses)
m
-1.0000
0.0000
(0.0000)
(0.0000)
infl
-0.023026
0.041039
(0.0054666)
(0.027790)
cpr
0.0000
-1.0000
(0.0000)
(0.0000)
y
0.42545
-0.037414
(0.033718)
(0.17140)
tbr
-0.027790
1.0172
(0.0045445)
(0.023102)
const
3.3625
0.68744
(0.25318)
(1.2870)
27.6 Over-identifyingrestrictions
OnepurposeofimposingrestrictionsonaVECMsystemissimplytoachieveidentiﬁcation.Ifthese
restrictionsaresimplynormalizations,theyarenottestableandshouldhavenoeﬀectonthemax-
derivefromtheeconomictheoryunderlyingtheequilibriumrelationships;substantiverestrictions
ofthissortarethentestableviaalikelihood-ratiostatistic.
Gretliscapableoftestinggenerallinearrestrictionsoftheform
R
b
vecq
(27.5)
and/or
R
a
vec0
(27.6)
Notethattherestrictionmaybenon-homogeneous(q0)buttherestrictionmustbehomoge-
neous. Nonlinearrestrictionsarenotsupported,andneitherarerestrictionsthatcrossbetween
3
thecolumnof(theparticularcointegratingvector).Thisisstandardpracticeintheliterature,anddefensibleinsofaras
itisthecolumnsof(thecointegratingrelationsorequilibriumerrors)thatareofprimaryinterest.
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Chapter27. CointegrationandVectorErrorCorrectionModels
245
and. Whenr >1,suchrestrictionsmaybeincommonacrossallthecolumnsof(or)ormay
bespeciﬁctocertaincolumnsofthesematrices. ForusefuldiscussionsofthispointseeBoswijk
(1995)andBoswijkandDoornik(2004),section4.4.
Therestrictions(27.5)and(27.6)maybewritteninexplicitformas
vecHh
0
(27.7)
and
vec
0
(27.8)
respectively,whereand arethefreeparametervectorsassociatedwithandrespectively.
Wemayrefertothefreeparameterscollectivelyas(thecolumnvectorformedbyconcatenating
and ).Gretlusesthisrepresentationinternallywhentestingtherestrictions.
Ifthelistofrestrictionsthatispassedtotherestrictcommandcontainsmoreconstraintsthan
mandcanbegiventhe--full switch, inwhichcasefullestimatesfortherestrictedsystemare
printed(includingthe—
i
terms)andthesystemthus restrictedbecomesthe“currentmodel”for
thepurposesoffurthertests. Thusyouareabletocarryoutcumulativetests,asinChapter7of
Johansen(1995).
Syntax
Thefullsyntax for r specifying therestrictionisanextensionof thatexempliﬁedintheprevious
section. Insidearestrict...end d restrictblock,validstatementsareoftheform
parameter linear combination=scalar
whereaparameterlinearcombinationinvolvesaweightedsumofindividualelementsof or
(butnotbothinthesamecombination);thescalarontheright-handsidemustbe0forcombina-
tionsinvolving,butcanbeanyrealnumberforcombinationsinvolving. Below,wegiveafew
examplesofvalidrestrictions:
b[1,1] = = 1.618
b[1,4] + + 2*b[2,5] ] = = 0
a[1,3] = = 0
a[1,1] - - a[1,2] = = 0
Specialsyntaxisusedwhenacertainconstraintshouldbeappliedtoallcolumnsof:inthiscase,
oneindexis givenfor eachb term, and thesquarebrackets aredropped. . Hence, , thefollowing
syntax
restrict
b1 + b2 = 0
end restrict
correspondsto
2
6
6
6
6
4
11
21
11
21
13
23
14
24
3
7
7
7
7
5
Thesameconventionisusedfor: whenonlyoneindexisgivenforan“a”termtherestrictionis
presumedtoapplytoallcolumnsof,orinotherwordsthevariableassociatedwiththegiven
rowofisweaklyexogenous.Forinstance,theformulation
Chapter27. CointegrationandVectorErrorCorrectionModels
246
restrict
a3 = 0
a4 = 0
end restrict
speciﬁesthatvariables3and4donotrespondtothedeviationfromequilibriumintheprevious
period.
4
A varianton the e single-indexsyntax for r commonrestrictions on and is available: : you u can
replacetheindexnumber withthenameof thecorresponding variable, insquarebrackets. . For
example, instead of a3 3 = = 0 0 onecould write e a[cpr] ] = 0, , if the e third variablein thesystem is
namedcpr.
Finally, ashortcut(oranywayanalternative)is availableforsettingupcomplexrestrictions(but
currentlyonlyinrelationto): youcanspecifyR
b
andq,asinR
b
vecq,bygivingthenames
ofpreviouslydeﬁnedmatrices.Forexample,
matrix I4 = I(4)
matrix vR = I4**(I4~zeros(4,1))
matrix vq = mshape(I4,16,1)
restrict
R = vR
q = vq
end restrict
whichmanuallyimposesPhillipsnormalizationontheestimatesforasystemwithcointegrating
rank4.
Therearetwopointstonoteinrelationtothisoption. First,vecistakentoincludethecoeﬃ-
cientsonalltermswithinthecointegrationspace,includingtherestrictedconstantortrend,ifany,
as wellasanyrestrictedexogenousvariables. . Second, , itisacceptabletogiveanR matrixwitha
numberofcolumnsequaltothenumberofrowsof;thisvariantistakentospecifyarestriction
thatisincommonacrossallthecolumnsof.
Anexample
BrandandCassola(2004)proposeamoneydemandsystemfortheEuroarea,inwhichtheypostu-
latethreelong-runequilibriumrelationships:
moneydemand
m
l
l
y
y
Fisherequation
l
Expectationtheoryof ls
interestrates
wheremisrealmoneydemand,landsarelong-andshort-terminterestrates,isoutputand
 isinﬂation.
5
(Thenames for thesevariablesinthegretldataﬁlearem_p, rl, rs,y andinfl,
respectively.)
Thecointegrationrankassumedbytheauthorsis3andthereare5variables,giving15elements
inthematrix. 339restrictionsarerequiredforidentiﬁcation,andajust-identiﬁedsystem
wouldhave15 9 9 6freeparameters. . However,thepostulatedlong-runrelationships s feature
onlythreefreeparameters,sotheover-identiﬁcationrankis3.
4
Note thatwhentwoindicesaregiven inarestrictionontheindexationisconsistentwiththatforrestrictions:
5
Atraditional formulation ofthe Fisher equation would reverse the roles of the variables in the second equation,
butthisdetailis immaterialinthepresentcontext;moreover,theexpectationtheoryofinterestratesimpliesthatthe
anddisappearsfromz
t
.
Chapter27. CointegrationandVectorErrorCorrectionModels
247
Example27.1:Estimationofamoneydemandsystemwithconstraintson
Input:
open brand_cassola.gdt
# perform m a few transformations
m_p = m_p*100
y = = y*100
infl = infl/4
rs = = rs/4
rl = = rl/4
# replicate e table 4, page 824
vecm 2 3 m_p p infl l rl rs s y y -q
ll0 = \$lnl
restrict --full
b[1,1] = 1
b[1,2] = 0
b[1,4] = 0
b[2,1] = 0
b[2,2] = 1
b[2,4] = 0
b[2,5] = 0
b[3,1] = 0
b[3,2] = 0
b[3,3] = 1
b[3,4] = -1
b[3,5] = 0
end restrict
ll1 = \$rlnl
Partialoutput:
Unrestricted loglikelihood d (lu) = 116.60268
Restricted loglikelihood d (lr) ) = 115.86451
2 * * (lu - lr) ) = = 1.47635
P(Chi-Square(3) > 1.47635) = 0.68774
beta (cointegrating vectors, standard errors in parentheses)
m_p
1.0000
0.0000
0.0000
(0.0000)
(0.0000)
(0.0000)
infl
0.0000
1.0000
0.0000
(0.0000)
(0.0000)
(0.0000)
rl
1.6108
-0.67100
1.0000
(0.62752)
(0.049482)
(0.0000)
rs
0.0000
0.0000
-1.0000
(0.0000)
(0.0000)
(0.0000)
y
-1.3304
0.0000
0.0000
(0.030533)
(0.0000)
(0.0000)
Chapter27. CointegrationandVectorErrorCorrectionModels
248
Example27.1replicatesTable4onpage824oftheBrandandCassolaarticle.
6
Notethatweuse
the\$lnlaccessorafterthevecmcommandtostoretheunrestrictedlog-likelihoodandthe\$rlnl
accessorafterrestrictforitsrestrictedcounterpart.
Theexamplecontinuesinscript27.2,whereweperformfurther testingto check whether(a)the
incomeelasticityinthemoneydemandequationis1(
y
1)and(b)theFisherrelationishomo-
restrictionscanbeappliedwithouthavingtorepeatthepreviousones.(Thesecondscriptcontains
afewprintfcommands,whicharenotstrictlynecessary,toformattheoutputnicely.)Itturnsout
Example27.2:Furthertestingofmoneydemandsystem
Input:
restrict
b[1,5] = -1
end restrict
ll_uie = \$rlnl
restrict
b[2,3] = -1
end restrict
ll_hfh = \$rlnl
# replicate e table 5, page 824
printf "Testing zero restrictions in cointegration space:\n"
printf "
LR-test, rank k = = 3: chi^2(3) = %6.4f f [%6.4f]\n", 2*(ll0-ll1), , \
pvalue(X, 3, 2*(ll0-ll1))
printf "Unit t income e elasticity: LR-test, rank k = = 3:\n"
printf "
chi^2(4) = %g g [%6.4f]\n", 2*(ll0-ll_uie), , \
pvalue(X, 4, 2*(ll0-ll_uie))
printf "Homogeneity in the Fisher hypothesis:\n"
printf "
LR-test, rank k = = 3: chi^2(4) = %6.3f f [%6.4f]\n", 2*(ll0-ll_hfh), , \
pvalue(X, 4, 2*(ll0-ll_hfh))
Output:
Testing zero o restrictions s in cointegration space:
LR-test, rank = 3: chi^2(3) = 1.4763 [0.6877]
Unit income elasticity: : LR-test, , rank = 3:
chi^2(4) = = 17.2071 1 [0.0018]
Homogeneity in the Fisher hypothesis:
LR-test, rank = 3: chi^2(4) = 15.547 [0.0037]
Anothertypeoftestthatiscommonlyperformedisthe“weakexogeneity”test. Inthiscontext,a
variableissaidtobeweaklyexogenousifallcoeﬃcientsonthecorrespondingrowinthematrix