Chapter 26
Vector Autoregressions
Gretl provides a standard set of procedures for dealing with the multivariate time-series models
knownasVARs(VectorAutoRegression). Moregeneralmodels—suchasVARMAs,nonlinearmodels
or multivariate GARCH models—are not provided as of now, although it is entirely possible to
estimatethembywritingcustomproceduresinthegretlscriptinglanguage. Inthischapter,we will
brieflyreviewgretl’sVAR toolbox.
26.1 Notation
AVAR is astructure whose aim is to modelthe time persistence of avector ofn time series, y
t
,
viaamultivariate autoregression,asin
y
t
A
1
y
t 1
A
2
y
t 2
A
p
y
t p
Bx
t

t
(26.1)
The number of lags p is called the order of the VAR. The vector x
t
, if present, contains a set of
exogenousvariables,oftenincludingaconstant,possiblywithatimetrendandseasonaldummies.
Thevector
t
istypicallyassumedto be avector white noise, withcovariancematrixÖ.
Equation(26.1)canbewrittenmore compactlyas
ALy
t
Bx
t

t
(26.2)
whereAL isamatrixpolynomialinthelagoperator,oras
2
6
6
6
6
4
y
t
y
t 1

y
t p 1
3
7
7
7
7
5
A
2
6
6
6
6
4
y
t 1
y
t 2

y
t p
3
7
7
7
7
5
2
6
6
6
6
4
B
0

0
3
7
7
7
7
5
x
t
2
6
6
6
6
4
t
0

0
3
7
7
7
7
5
(26.3)
ThematrixAisknownasthe “companionmatrix” andequals
A
2
6
6
6
6
6
4
A
1
A
2
 A
p
I
0

0
0
I

0
.
.
.
.
.
.
.
.
.
.
.
.
3
7
7
7
7
7
5
Equation(26.3)isknownasthe“companionform”oftheVAR.
Another representationofinterestis theso-called“VMArepresentation”,whichiswritteninterms
ofaninfiniteseries ofmatrices Ò
i
definedas
Ò
i
@y
t
@
t i
(26.4)
TheÒ
i
matricesmaybederivedbyrecursive substitutioninequation(26.1): forexample,assuming
for simplicitythatB 0 andp1,equation(26.1)wouldbecome
y
t
Ay
t 1

t
230
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Chapter26. VectorAutoregressions
231
whichcouldbe rewrittenas
y
t
A
n1
y
t n 1

t
A
t 1
A
2
t 2
A
n
t n
Inthis case Ò
i
A
i
. Ingeneral, itis possible to compute Ò
i
as the nn north-westblock of the
i-thpower ofthe companionmatrixA(so Ò
0
is alwaysanidentitymatrix).
The VAR is said to be stable if allthe eigenvalues ofthe companion matrix A are smaller than 1
in absolute value, or equivalently, if the matrix polynomial AL in equation (26.2) is such that
jAzj 0 implies jzj >1. Ifthis isthe case, lim
n!1
Ò
n
0 and the vector y
t
is stationary; as a
consequence,the equation
y
t
Ey
t

X1
i0
Ò
i
t i
(26.5)
is alegitimateWoldrepresentation.
Ifthe VARisnotstable,thenthe inferentialproceduresthatarecalledfor become somewhatmore
specialized, except for some simple cases. In particular, if the number of eigenvalues of A with
modulus1 isbetween1 andn 1,thecanonicaltooltodealwiththese modelsisthe cointegrated
VARmodel, discussedinchapter27.
26.2 Estimation
The gretlcommandfor estimating a VAR is var which, inthe command line interface, is invoked
inthe followingmanner:
[ modelname <- ] var p Ylist [; Xlist]
where p is a scalar (the VAR order) and Ylist is a list of variables specifying the content of y
t
.
TheoptionalXlist argumentcanbe usedtospecifyasetofexogenousvariables. Ifthisargument
is omitted, the vector x
t
is taken to contain a constant (only); if present, it must be separated
from Ylist by a semicolon. Note, however, that a few common choices can be obtained in a
simplerway: the options--trendand--seasonalscallfor inclusionofalineartrendandasetof
seasonaldummiesrespectively. Inadditionthe--ncoption(no constant)canbeusedto suppress
thestandardinclusionofaconstant.
The“<-” constructcanbeusedtostorethe modelunder aname (seesection3.2), ifsodesired. To
estimate a VAR using the graphicalinterface, choose “Time Series, Vector Autoregression”, under
theModelmenu.
The parameters in eq. (26.1) are typically free from restrictions, which implies that multivariate
OLS provides a consistent and asymptotically efficient estimator of all the parameters.
1
Given
the simplicity of OLS, this is what every software package, including gretl, uses; example script
26.1 exemplifies the fact t that the var r command gives s you u exactly the output you would have
from abattery of OLS regressions. The advantage of using the dedicatedcommand is that, after
estimationis done, itmakes it mucheasier to access certain quantitiesandmanage certaintasks.
For example, the $coeff accessor returns the estimated coefficients as a matrix withn columns
and$sigma returnsanestimate ofthe matrixÖ,the covariance matrixof
t
.
Moreover,foreachvariableinthesystemanF testisautomaticallyperformed,inwhichthenullhy-
pothesisisthatnolagsofvariablej are significantinthe equationforvariable i. Thisiscommonly
knownas aGrangercausalitytest.
Inaddition,twoaccessors becomeavailable for thecompanionmatrix($compan)andtheVMArep-
resentation($vma). Thelatterdeserves adetaileddescription: sincethe VMArepresentation(26.5)
isofinfiniteorder,gretldefinesahorizonuptowhichtheÒ
i
matricesare computedautomatically.
1
Infact,undernormalityof
t
OLSisindeedtheconditionalMLestimator.Youmaywanttouseothermethodsifyou
needtoestimateaVARinwhichsomeparametersareconstrained.
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Chapter26. VectorAutoregressions
232
Example26.1:EstimationofaVAR viaOLS
Input:
open sw_ch14.gdt
genr infl = 400*sdiff(log(PUNEW))
scalar p = 2
list X = LHUR infl
list Xlag = lags(p,X)
loop foreach i X
ols $i const Xlag
endloop
var p X
Output(selectedportions):
Model 1: OLS, using observations 1960:3-1999:4 (T = 158)
Dependent variable: LHUR
coefficient
std. error
t-ratio
p-value
--------------------------------------------------------
const
0.113673
0.0875210
1.299
0.1960
LHUR_1
1.54297
0.0680518
22.67
8.78e-51 ***
LHUR_2
-0.583104
0.0645879
-9.028
7.00e-16 ***
infl_1
0.0219040
0.00874581
2.505
0.0133
**
infl_2
-0.0148408
0.00920536
-1.612
0.1090
Mean dependent var
6.019198
S.D. dependent var
1.502549
Sum squared resid
8.654176
S.E. of regression
0.237830
...
VAR system, lag order 2
OLS estimates, observations 1960:3-1999:4 (T = 158)
Log-likelihood = -322.73663
Determinant of covariance matrix = 0.20382769
AIC = 4.2119
BIC = 4.4057
HQC = 4.2906
Portmanteau test: LB(39) = 226.984, df = 148 [0.0000]
Equation 1: LHUR
coefficient
std. error
t-ratio
p-value
--------------------------------------------------------
const
0.113673
0.0875210
1.299
0.1960
LHUR_1
1.54297
0.0680518
22.67
8.78e-51 ***
LHUR_2
-0.583104
0.0645879
-9.028
7.00e-16 ***
infl_1
0.0219040
0.00874581
2.505
0.0133
**
infl_2
-0.0148408
0.00920536
-1.612
0.1090
Mean dependent var
6.019198
S.D. dependent var
1.502549
Sum squared resid
8.654176
S.E. of regression
0.237830
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Chapter26. VectorAutoregressions
233
Periodicity
horizon
Quarterly
20 (5years)
Monthly
24 (2years)
Daily
3weeks
Allothercases
10
Table26.1:VMAhorizonas afunctionofthedatasetperiodicity
Bydefault, thisis afunctionof theperiodicityof thedata(see table26.1), but it canbe setbythe
usertoanydesiredvalueviathesetcommandwiththe horizon parameter,asin
set horizon 30
Callingthehorizonh,the$vma accessorreturnsanh1n
2
matrix,inwhichthei1-throw
is thevectorizedformofÒ
i
.
VARlag-orderselection
Inorder to helpthe user choose the most appropriate VAR order, gretl providesa specialvariant
ofthe varcommand:
var p Ylist [; Xlist] --lagselect
When the --lagselect option is given, estimation is performed for all lags up to p and a table
is printed: it displays, for each order, a Likelihood Ratio test for the order p versus p 1, plus
an array of information criteria (see chapter 23). For each information criterion in the table, a
starindicates whatappearsto be the “best” choice. The same outputcanbe obtainedthroughthe
graphicalinterfaceviathe“Time Series,VARlagselection”entryunderthe Modelmenu.
Warning: infinitesamplesthe choice ofthe maximumlag,p,mayaffectthe outcome ofthe proce-
dure. This isnot abug, butrather anunavoidable side effectofthewaythese comparisonsshould
be made. Ifyour sample contains T observationsandyouinvoke the lagselectionprocedure with
maximum order p, gretlexamines allVARs of order ranging form1 to p,estimatedonauniform
sampleofT p observations. Inotherwords,thecomparisonproceduredoesnotusealltheavail-
able data when estimating VARs of order less than p, so as to ensure that all the models in the
comparisonare estimatedonthe same data range. Choosingadifferentvalue of p may therefore
alterthe results,althoughthisisunlikelyto happenifyour sample size isreasonablylarge.
An example of this unpleasant phenomenon is given in example script26.2. As can be seen, ac-
cordingto the Hannan-Quinncriterion, order2seemspreferable toorder 1ifthemaximumtested
orderis4,butthe situationisreversedifthe maximumtestedorderis6.
26.3 Structural VARs
Gretldoesnotcurrentlyprovide anativeimplementationfor thegeneralclassofmodelsknownas
“Structural VARs”; however, it provides an implementationof the Cholesky decomposition-based
approach,the classicandmostpopularSVAR variant.
IRFandFEVD
Assume thatthe disturbance in equation(26.1)can be thoughtofas a linear function of avector
ofstructuralshocks u
t
,whichareassumedtohaveunitvarianceandto be mutuallyunncorrelated,
soVu
t
I. If
t
Ku
t
,itfollowsthatÖV
t
KK0.
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Chapter26. VectorAutoregressions
234
Example26.2:VAR lagselectionviaInformationCriteria
Input:
open denmark
list Y = 1 2 3 4
var 4 Y --lagselect
var 6 Y --lagselect
Output(selectedportions):
VAR system, maximum lag order 4
The asterisks below indicate the best (that is, minimized) values
of the respective information criteria, AIC = Akaike criterion,
BIC = Schwarz Bayesian criterion and HQC = Hannan-Quinn criterion.
lags
loglik
p(LR)
AIC
BIC
HQC
1
609.15315
-23.104045
-22.346466*
-22.814552
2
631.70153
0.00013
-23.360844*
-21.997203
-22.839757*
3
642.38574
0.16478
-23.152382
-21.182677
-22.399699
4
653.22564
0.15383
-22.950025
-20.374257
-21.965748
VAR system, maximum lag order 6
The asterisks below indicate the best (that is, minimized) values
of the respective information criteria, AIC = Akaike criterion,
BIC = Schwarz Bayesian criterion and HQC = Hannan-Quinn criterion.
lags
loglik
p(LR)
AIC
BIC
HQC
1
594.38410
-23.444249
-22.672078*
-23.151288*
2
615.43480
0.00038
-23.650400*
-22.260491
-23.123070
3
624.97613
0.26440
-23.386781
-21.379135
-22.625083
4
636.03766
0.13926
-23.185210
-20.559827
-22.189144
5
658.36014
0.00016
-23.443271
-20.200150
-22.212836
6
669.88472
0.11243
-23.260601
-19.399743
-21.795797
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Chapter26. VectorAutoregressions
235
Themainobjectofinterestinthis settingis thesequenceofmatrices
C
k
@y
t
@u
t i
Ò
k
K;
(26.6)
knownasthe structuralVMArepresentation. Fromthe C
k
matricesdefinedinequation(26.6)two
quantitiesofinterestmaybe derived: theImpulseResponseFunction(IRF)andtheForecastError
VarianceDecomposition(FEVD).
The IRFof variable i to shock j is simplythe sequence of the elements inrowi and columnj of
theC
k
matrices. Insymbols:
I
i;j;k
@y
i;t
@u
j;t k
As a rule, Impulse Response Functions are plotted as a function of k, and are interpreted as the
effect that a shock has on an observable variable through time. Of course, what we observe are
the estimated IRFs, so it is natural to endow them with confidence intervals: following common
practice, gretlcomputesthe confidence intervalsbyusingthe bootstrap;
2
details are givenlater in
this section.
Another quantityofinterestthatmaybe computed fromthe structuralVMArepresentationis the
ForecastErrorVarianceDecomposition(FEVD).Theforecasterrorvarianceafterhstepsisgivenby
Ú
h
Xh
k0
C
k
C
0
k
hencethe variance for variable iis
!
2
i
Ú
h
i;i
Xh
k0
diagC
k
C
0
k
i
Xh
k0
Xn
l1
k
c
i:l
2
where
k
c
i:l
is,trivially,thei;lelementofC
k
.Asaconsequence,theshareofuncertaintyonvariable
ithatcanbe attributedto the j-thshock after hperiodsequals
VD
i;j;h
P
h
k0
k
c
i:j
2
P
h
k0
P
n
l1
k
c
i:l
2
:
Thismakesitpossibletoquantifywhichshocksaremostimportanttodetermineacertainvariable
inthe shortand/orinthe longrun.
Triangularization
Theformula26.6takesK asknown,while ofcourseithastobeestimated. Theestimationproblem
has been the subject of an enormous body of literature we will not even attempt to summarize
here: seeforexample(Lütkepohl,2005,chapter 9).
Suffice itto say that the mostpopular choice datesback toSims(1980),and consists inassuming
thatK islowertriangular,soitsestimateissimplytheCholeskydecompositionoftheestimateofÖ.
Themainconsequenceofthischoice isthattheorderingofvariableswithinthe vector y
t
becomes
meaningful: since K is also the matrix of Impulse Response Functions at lag 0, the triangularity
assumptionmeans thatthe first variable in the ordering responds instantaneously only to shock
number 1, the second one onlyto shocks 1 and 2, and so forth. For this reason, each variable is
thoughtto“own”one shock: variable 1ownsshock number 1,andsoon.
In this sort of exercise, therefore, the ordering of the y variables is important, and the applied
literature has developed the “most exogenous first” mantra—where, in this setting, “exogenous”
2
Itispossible, in principle, tocompute analyticalconfidenceintervals via an asymptoticapproximation, butthisis
notaverypopularchoice: asymptoticformulaeareknowntooftengiveaverypoor approximationofthefinite-sample
properties.
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Chapter26. VectorAutoregressions
236
reallymeans “instantaneously insensitive to structural shocks”.
3
To put it differently, if variable
foo comes before variable bar in the Y list, it follows that the shock owned by foo affects bar
instantaneously,butnotvice versa.
ImpulseResponseFunctionsandtheFEVDcanbeprintedoutviathecommandlineinterfacebyus-
ing the--impulse-responseand--variance-decompoptions,respectively. If youneed tostore
themintomatrices,youcancomputethe structuralVMAandproceedfromthere. For example,the
followingcodesnippetshowsyouhowto computeamatrixcontainingthe IRFs:
open denmark
list Y = 1 2 3 4
scalar n = nelem(Y)
var 2 Y --quiet --impulse
matrix K = cholesky($sigma)
matrix V = $vma
matrix IRF = V * (K ** I(n))
print IRF
inwhichthe equality
vecC
k
vecÒ
k
KK
0
IvecÒ
k
was used.
FIXME:showallthe nice stuffwe haveunderthe GUI.
IRFbootstrap
FIXME:todo
3
Theword “exogenous” hascaughton in this context, butit’sa ratherunfortunate choice: for a start, each shock
impactson everyvariable after one lag, so nothingis reallyexogenoushere. A better choiceofwordswould probably
havebeensomethinglike“sturdy”,butit’stoolatenow.
Chapter 27
Cointegration and Vector Error Correction Models
27.1 Introduction
The twin concepts of cointegration and error correction have drawn a good deal of attention in
macroeconometricsover recentyears. The attractionofthe Vector ErrorCorrection Model(VECM)
is that it allows the researcher to embed a representation of economic equilibrium relationships
within arelativelyrich time-series specification. This approach overcomes the old dichotomy be-
tween(a)structuralmodelsthatfaithfullyrepresentedmacroeconomictheorybutfailedto fit the
data, and(b)time-seriesmodelsthatwere accuratelytailoredtothedatabutdifficultifnotimpos-
sible tointerpretineconomic terms.
The basicideaofcointegrationrelatescloselyto the conceptofunitroots (see section25.3). Sup-
posewehaveasetofmacroeconomicvariablesofinterest,andwefindwecannotrejectthehypoth-
esisthatsome ofthese variables,consideredindividually,arenon-stationary. Specifically,suppose
wejudge thatasubsetofthe variablesareindividuallyintegrated oforder 1, or I(1). Thatis,while
they are non-stationary in their levels, their first differences are stationary. Given the statistical
problems associatedwith the analysis of non-stationary data(for example, the threatof spurious
regression), the traditional approach in this case was to take first differences of all the variables
before proceedingwiththe analysis.
But this can result in the loss of important information. It may be that while the variables in
question are I(1) when taken individually, there exists a linear combination of the variables that
is stationary without differencing, or I(0). (There could be more than one such linear combina-
tion.) Thatis, while the ensemble ofvariablesmaybe “free to wander” over time,nonetheless the
variables are “tied together” in certain ways. And it may be possible to interpret these ties, or
cointegratingvectors,asrepresentingequilibriumconditions.
For example, suppose we find some or all of the following variables are I(1): moneystock, M, the
price level, P, the nominal interest rate, R, and output, Y. According to standard theories of the
demand for money, we would nonethelessexpectthere tobe anequilibrium relationship between
realbalances,interestrateandoutput;for example
m p
0
1
y
2
r
1
>0;
2
<0
wherelower-casevariable namesdenotelogs. Inequilibrium,then,
m p 
1
2
r 
0
Realistically,we shouldnotexpectthiscondition tobe satisfiedeachperiod. Weneedto allowfor
thepossibilityofshort-rundisequilibrium. Butifthe systemmoves back towardsequilibriumfol-
lowingadisturbance,itfollowsthatthevectorxm;p;y;r
0
isboundbyacointegratingvector
0

1
;
2
;
3
;
4
, suchthat
0
xisstationary(withameanof
0
). Furthermore,ifequilibriumis
correctlycharacterizedbythe simple model above, we have 
2
 
1
,
3
<0 and
4
>0. These
thingsare testable withinthe contextofcointegrationanalysis.
There are typicallythreestepsinthissortofanalysis:
1. Testtodetermine thenumber ofcointegratingvectors,thecointegratingrankofthe system.
2. EstimateaVECMwiththe appropriate rank,butsubjecttonofurtherrestrictions.
237
Chapter27. CointegrationandVector Error Correction Models
238
3. Probe the interpretation of the cointegrating vectors as equilibrium conditions by means of
restrictionsonthe elementsofthese vectors.
The following sections expand on each of these points, giving further econometric details and
explaininghowto implementtheanalysisusing gretl.
27.2 Vector Error Correction Models as representation of a cointegrated system
Consider a VAR of order p with adeterministic part givenby
t
(typically, a polynomialin time).
One canwrite then-variate processy
t
as
y
t

t
A
1
y
t 1
A
2
y
t 2
A
p
y
t p

t
(27.1)
Butsincey
t i
y
t 1
Ñy
t 1
Ñy
t 2
Ñy
t i1
,we canre-write theabove as
Ñy
t

t
Õy
t 1
p 1
X
i1
i
Ñy
t i

t
;
(27.2)
whereÕ
P
p
i1
A
i
I and—
i
 
P
p
ji1
A
j
.ThisistheVECMrepresentationof(27.1).
Theinterpretationof(27.2)dependscruciallyonr,the rankofthematrixÕ.
 Ifr 0,the processesareallI(1)andnotcointegrated.
 Ifr n,thenÕisinvertibleandthe processes are allI(0).
 Cointegrationoccurs inbetween, when0 <r < n andÕcanbe writtenas 
0
. Inthis case,
y
t
is I(1), but the combination z
t

0
y
t
is I(0). If, for example, r  1 and the first element
of  was 1, then one could write z
t
 y
1;t

2
y
2;t

n
y
n;t
,whichis equivalentto
sayingthat
y
1
t

2
y
2;t

n
y
n;t
z
t
isalong-runequilibriumrelationship: thedeviationsz
t
maynotbe 0buttheyarestationary.
Inthiscase,(27.2)canbe writtenas
Ñy
t

t

0
y
t 1
p 1
X
i1
i
Ñy
t i

t
:
(27.3)
If  were known, then z
t
would be observable and all the remaining parameters could be
estimatedviaOLS.Inpractice,the procedureestimatesfirstandthenthe rest.
The rank ofÕisinvestigatedbycomputing the eigenvaluesofacloselyrelated matrixwhoserank
isthe sameasÕ: however,thismatrixisbyconstructionsymmetricandpositivesemidefinite. Asa
consequence,allitseigenvaluesarerealandnon-negative,andtestsontherankofÕcantherefore
be carriedoutbytestinghowmanyeigenvaluesare0.
If all the eigenvalues are significantly different from 0, then all the processes are stationary. If,
on the contrary, there is at least one zero eigenvalue, then the y
t
process is integrated, although
some linear combination 
0
y
t
might be stationary. At the other extreme, if no eigenvalues are
significantly different from 0, then not onlyis the process y
t
non-stationary, but the same holds
for anylinearcombination
0
y
t
;inother words,nocointegrationoccurs.
Estimation typically proceeds in two stages: first, a sequence of tests is run to determine r, the
cointegrationrank. Then,foragivenranktheparametersinequation(27.3)areestimated. Thetwo
commands thatgretloffersfor estimatingthese systemsarecoint2 andvecm,respectively.
Thesyntaxfor coint2is
Chapter27. CointegrationandVector Error Correction Models
239
coint2 p ylist [ ; xlist [ ; zlist ] ]
where p is the number of lags in (27.1); ylist is a list containing the y
t
variables; xlist is an
optional list of exogenous variables; and zlist is another optional list of exogenous variables
whoseeffectsareassumedto beconfinedtothe cointegratingrelationships.
Thesyntaxfor vecm is
vecm p r ylist [ ; xlist [ ; zlist ] ]
where pisthe number oflags in(27.1);r isthecointegrationrank;andthe listsylist, xlistand
zlisthave thesame interpretationasincoint2.
Bothcommandscanbegivenspecificoptionsto handlethe treatmentofthedeterministic compo-
nent
t
. Thesearediscussedinthe followingsection.
27.3 Interpretation of the deterministic components
Statistical inference in the context of a cointegrated system depends on the hypotheses one is
willingto make onthedeterministic terms,whichleadsto the famous “fivecases.”
Inequation(27.2), theterm
t
isusuallyunderstoodtotakethe form
t

0

1
t:
Inordertohavethemodelmimicascloselyaspossible thefeaturesoftheobserveddata,thereisa
preliminaryquestiontosettle. Do thedataappear tofollowadeterministictrend? Ifso,isitlinear
or quadratic?
Once this isestablished, one shouldimpose restrictions on
0
and
1
thatareconsistentwiththis
judgement. Forexample,suppose thatthe datado notexhibitadiscernibletrend. Thismeansthat
Ñy
t
is on average zero, so itis reasonable to assume thatits expected value is also zero. Write
equation(27.2)as
—LÑy
t

0

1
tz
t 1

t
;
(27.4)
where z
t
 
0
y
t
is assumed to be stationary and therefore to possess finite moments. Taking
unconditionalexpectations,we get
0
0

1
tm
z
:
Sincetheleft-handsidedoesnotdependont,therestriction
1
0isasafebet.Asfor
0
,thereare
just two ways to make the above expressiontrue: either 
0
0 withm
z
0, or
0
equals  m
z
.
The latterpossibilityislessrestrictiveinthatthe vector
0
maybe non-zero,butis constrainedto
be alinear combinationofthe columnsof. Inthatcase, 
0
canbe writtenas c,andone may
write (27.4)as
—LÑy
t

0
c
"
y
t 1
1
#

t
:
Thelong-runrelationshipthereforecontainsanintercept. Thistypeofrestrictionisusuallywritten
0
?
0
0;
where
?
istheleftnullspace ofthe matrix.
An intuitive understandingof the issue canbe gained by means of asimple example. Consider a
seriesx
t
whichbehavesasfollows
x
t
mx
t 1
"
t
where m isa realnumber and"
t
is awhite noise process: x
t
isthenarandom walk withdriftm.
Inthe specialcasem =0,the driftdisappearsandx
t
is apure randomwalk.
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