Asymptotic Properties of Residual Based Tests for Cointegration
Author(s): P. C. B. Phillips and S. Ouliaris
Source: Econometrica, Vol. 58, No. 1 (Jan., 1990), pp. 165-193
Published by: The Econometric Society
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Econometrica,Vol. 58, No. 1 (January, 1990), 165-193
ASYMPTOTIC PROPERTIES OF RESIDUAL BASED TESTS
FOR COINTEGRATION
BY P. C. B. PHILLIPSAND S. OULIARIS1
This paper developsan asymptotictheoryfor residualbased tests for cointegration.
These tests involveproceduresthataredesignedto detectthe presenceof a unit root in the
regressionsamongthe levelsof economictimeseries.Attention
residualsof (cointegrating)
by Engle-Granger
is givento the augmentedDickey-Fuller(ADF) test thatis recommended
(1987) and the Z,, and Z, unit root tests recentlyproposedby Phillips(1987).Two new
of the cointegrating
tests are also introduced,one of whichis invariantto the normalization
regression.All of thesetests areshownto be asymptoticallysimilarand simplerepresentations of their limitingdistributionsare givenin termsof standardBrownianmotion.The
ADF and Z, tests are asymptoticallyequivalent.Powerpropertiesof the tests are also
studied.The analysisshowsthatall the tests areconsistentif suitablyconstructedbut that
the ADF and Z, tests have slowerratesof divergenceundercointegrationthan the other
tests. This indicatesthat,at least in largesamples,the Z,, test shouldhave superiorpower
properties.
The paper concludesby addressingthe largerissue of test formulation.Some major
pitfallsare discoveredin proceduresthataredesignedto test a null of cointegration(rather
than no cointegration).Thesedefectsprovidestrongargumentsagainstthe indiscriminate
use of such test formulationsand supportthe continuinguse of residualbased unit root
tests.
A full set of criticalvaluesfor residualbasedtestsis included.Theseallowfor demeaned
and detrendeddata and cointegratingregressionswith up to five variables.
Asymptoticallysimilartests, Brownianmotion, cointegration,conceptual
KEYwoRDs:
pitfalls,powerproperties,residualbasedprocedures,Reyni-mixing.
1. INTRODUCTION
THE PURPOSEOF THISPAPERis to provide an asymptotic analysis of residual
based tests for the presenceof cointegrationin multiple time series. Residual
based tests rely on the residualscalculatedfrom regressionsamongthe levels (or
log levels) of economictime series.They are designedto test the null hypothesis
of no cointegrationby testing the null that there is a unit root in the residuals
against the alternativethat the root is less than unity. If the null of a unit root is
rejected, then the null of no cointegrationis also rejected. The tests might
thereforebe more aptly namedresidualbased unit root tests. Some of the tests
we shall study involve standardproceduresapplied to the residuals of the
cointegrating regressionto detect the presence of a unit root. Two of the
procedures we shall examine are new to this paper. They all fall within
the frameworkof residualbased unit root tests.
Approachesotherthanresidualbasedtests for cointegrationare also available.
Some of these have the advantagethat they may be employed to test for the
IWe are gratefulto LarsPeter Hansenand threerefereesfor helpful commentson two earlier
versionsof this paper.Ourthanksalso go to GlenaAmes for her skill and effortin keyboardingthe
manuscriptof this paperand to the NSF for researchsupportunderGrantNumberSES 8519595.
Sam Ouliarisis now at the Universityof Marylandandis presentlyvisitingtheNationalUniversityof
Singapore.The paperwas firstwrittenin July,1987,and the presentversionwas preparedin January,
1989.
165
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166
P. C. B. PHILLIPS AND S. OULIARIS
presenceof r linearlyindependentcointegratingvectorsagainstr - 1 cointegrating vectorsfor r > 1. For instance,a likelihoodratio test has been consideredby
Johansen (1988) in the context of vector autoregressions(VAR's); a common
stochastic trends test has been proposedby Stock and Watson (1986); and a
bounds test has been suggestedby the authorsin earlierwork (1988). None of
these tests rely on the residualsof cointegratingregressions.However,it is the
residual based procedureswhich have attractedthe attention of empiricalresearchers.This is partlybecauseof the recommendationsof Engle and Granger
(1987), partly becausethe tests are so easy and convenientto apply, and partly
because what they set out to test is clearintuitively.
Unfortunately, little is known from existing work about the propertiesof
residual based unit root tests. Engle and Granger(1987) provide some experimental evidenceon the basis of whichthey recommendthe use of the augmented
Dickey-Fuller(ADF) t ratio test. They also show that this test and many others
are similartests when the data follow a vectorrandomwalk drivenby iid normal
innovations.They conjecturethat the ADF procedureis asymptoticallysimilarin
more generaltime seriessettings.The presentpaperconfirmsthat conjecture.We
also study the asymptoticbehaviorof variousothertests,includingthe Za and Z,
tests recentlysuggestedin Phillips(1987).Ourasymptotictheorycoversboth the
null of no cointegrationand the alternativeof a cointegratedsystem.It is shown
that the power propertiesof many of the tests dependcriticallyon theirmethod
of construction.In particular,test consistencyrelieson whetherresidualsor first
differencesare used in serialcorrelationcorrectionsthat are designedto eliminate
nuisance parametersunder the null. Our analysisof power also indicatessome
majordifferencesbetweenthe tests. In particular,t ratio proceduressuch as the
ADF and the Z, test divergeunder the alternativeat a slowerrate than direct
coefficient tests such as the Za test and the new varianceratio tests that are
developed in the paper. This indicates that coefficientand varianceratio tests
should have superiorpowerpropertiesover t ratio tests at least in largesamples.
One of the new tests developedin the paperis invariantto the normalizationof
the cointegratingregression.This is in contrastto otherresidualbased tests, such
as the ADF, which are numericallydependenton the preciseformulationof the
cointegratingregression.Invarianceis a usefulproperty,sinceit removesconflicts
that can arise in empiricalworkwherethe test outcomedependson the normalization selected.
The generalquestionof how to formulatetests for the presenceof cointegration is also addressed.In particular,we examinethe potentialof certainprocedures which seek to test a null of cointegrationagainst an alternativeof no
cointegration,ratherthan vice versa.This questionof formulationis important.
It arises frequentlyin seminarand conferencediscussions(for example Engle
(1987)) whereit is often arguedthat a null of cointegrationis the moreappealing.
But the question has not to our knowledgebeen formally addressedin the
literatureuntil now. Our analysispoints to some majorpitfalls in the alternate
approach. The source of the difficultieslies in the failure of conventional
asymptotic theory under a null of cointegration.This is not just a matter of
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167
RESIDUAL BASED TESTS
nonstandardlimit theory.In fact, no generallimit theoryappliesin this case to
certain statistics(like long run varianceestimates)that are most relevantto the
null. Moreover,if tests basedon a specificdistributiontheoryare used, they turn
out to be inconsistent.Thesedifficultiesprovidegood argumentsfor the continuing use of tests that are based on the compositenull of no cointegration.
The plan of the paper is as follows. Section 2 provides some preliminary
theory, including a theorem that is likely to be very useful on invariance
principlesfor processeswhichare linearfiltersof othertime series.This theoryis
needed for an asymptoticanalysisof the ADF. In Section3 we reviewa class of
residualbased tests for cointegrationand develop two new procedures:a variance ratio test and a multivariatetracetest. Both tests have interestinginterpretations and the second has the invariancepropertymentionedearlier.An asymptotic theoryfor the tests is developedin Section4 and it is shownthat the Za, Z,
and ADF tests all have limitingdistributionswhich can be simply expressedas
stochasticintegrals.The ADF and Z, tests are asymptoticallyequivalent.Section
5 studies test consistencyand the asymptoticbehaviorof the tests under the
alternativeof cointegration.Issuesof test formulationare consideredin Section6
and some conclusionsare drawnin Section7. Proofs are given in the Appendix
A. Critical values for the residualbased tests are given in Appendix B. These
allow for up to five variablesin the cointegratingregressionand detrendeddata.
In matters of notation we use the symbol
"="
to signify weak convergence,
the symbol "-" to signifyequalityin distribution,and the inequality" > 0" to
signify positive definite when applied to matrices.Continuousstochasticprocesses such as the Brownianmotion B(r) on [0,11 are writtenas B to achieve
notational economy. Similarly,we write integrals with respect to Lebesgue
measuresuch as JoJB(s)ds more simplyas JoJB.
2. PRELIMINARYTHEORY
Let { zt }0 be an m-vectorintegratedprocesswhose generatingmechanismis
(1)
ztZt=Z'_ + t
(t =1,2 ...)
Our resultsdo not dependon the initializationof (1) and we thereforeallowz0 to
be any randomvariableincluding,of course,a constant.The randomsequence
{ }r is defined on a probabilityspace (X, F, P) and is assumedto be strictly
stationary and ergodic with zero mean, finite variance, and spectral density
matrix fg( X). We also requirethe partialsum processconstructedfrom {ft} to
satisfy a multivariateinvarianceprinciple.More specifically,for r E [0,11and as
T -s o we require
[Tr]
(Cl)
XT(r) = T-1/2 ,2i,
B(r)
(R-mixing)
1
where B(r) is m-vectorBrownianmotion with covariancematrix
(2)
(2
T
T
=f
)
)(0)
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168
P. C. B. PHILLIPS AND S. OULIARIS
Writing
00
k=1
we have
o+ ol + o1The convergencecondition (Cl) is Reyni-mixing(R-mixing).This requiresthe
random element XT(r) to be asymptoticallyindependentof each event E E F,
0
i.e.,
nE) -*P(BE
P({(XTE
T-*ox.
*)P(E),
In this sense, the randomelement XT may be thoughtof as escapingfrom its
own probabilityspacewhen R-mixingapplies.The readeris referredto Hall and
Heyde (1980, p. 57) for further discussion. Functional limit theorems under
R-mixing such as (Cl) are known to apply in very general situations. For
example,the theoremsof McLeish(1975)that wereused in the paperby Phillips
(1987) are all R-mixinglimit theorems.Extensionsto multipletime seriesfollow
as in Phillipsand Durlauf(1986).
It will he convenientfor muchof this paperto take (t to be the linearprocess
generatedby
(3)
00
00
00
E
E
C1y,
iiC1ii< o,
C(1)= E C,
j=-00
j=-00
j=-00
where the sequence of random vectors {et} is iid (0, 2) with 2 > 0 and JCj =
j} where Cj = (Cjkl). This includesall stationaryARNIAprocesses
maxk {-11IC1jk1l
and is thereforeof wide applicability.The process (t has a continuousspectral
density matrixgiven by
f (X)
(1/2)(
=
Cjeiix)E(
Cjeiix)*.
In addition to the absolutesummabilityof {Cj} in (3) we will use the following
condition (based on (5.37) of Hall and Heyde (1980)):
k=1[
j=k
j=k
which is again satisfiedby all stationaryARMA models.Note that (C2) holds for
all sequences{Cj} that are 1-summablein the sense of Brillinger(1981, equation
2.7.14). These assumptionsare not the weakestpossible but are generalenough
for our purposeshere.
Let {aj } be a scalarsequencethat is absolutelysummableand definethe new
process
00
00
(4)
{
E
j= -00
ajyt-j,
>l,-00
Eijislaji
<
oX,
s
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169
RESIDUAL BASED TESTS
and the associated random element
[Tr]
(5)
*
T-
XT(r)
1
Let a (1) = E 00ai. We shall make use of the following important lemma describing the asymptotic behavior of XT*(r).
LEMMA
(6)
2.1: If (C2) holds, then as T-* x
p )O
sup IXXT(r)-a(1)XT(r)I
O< r <1
and
XT*(r) - >B*(r)--a(l)B(r)
(7)
or vector Brownian motion with covariance matrix Q* =
a(1)2Q.
We now partition
= (y, x')' into the scalar variate Y' and the n-vector
z,
=
conformable partitions of 2 and B(r):
with
the
n
+
1)
following
x,(m
i n
Q222
[ '21
We shall assume
222 >
B2(r ) n
n
0 and use the block triangular decomposition of 2:
01
[ll~
(8)
~
L- [ 121 L22 J
=L'L,
with
(9)
1 =
11l
(4Q1
-
2
=
1Q1/5
)/122121 21
=
L
-1/2(42,
22
21
22
Let W(r) be m-vector standard Brownian motion and define:
rrau
1
21 22
lo
=(1
-a'
21]
at1
2
[~f2l
F2 2 ]
A-1),
K
Q(r) = WJ(r) -
(f1
(1
,fAF221)
)(f
<
(r)
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-
1/2
22
P. C. B. PHILLIPS AND S. OULIARIS
170
LEMMA2.2:
(a)
B(r)-L'W(r);
(b)
(c)
Lrq-111K,
(d)
71Q
Q W-11
q'B(r)
111Q(r);
71
q lB dB',
(e)
=
2K K;
11-2 QdQ;
1f
all 2
Q2;
0
where
@11 2 =
22 '21
@11 -21
=
'11
and
all.2
=
a1
-
1A11a21.
REMARKS:
(a) This lemma shows how to reformulate some simple linear and
quadratic functionals of the Brownian motion B(r) into distributionally equivalent functionals of standard Brownian motion. These representations turn out to
be very helpful in identifying key parameter dependencies in the original expressions. As is clear from (b)-(e) the conditional variance wll.2 is the sole carrier of
these dependencies in (b)-(e).
(b) Note that det Q = W11*2 det 222 and is zero iff 11*2 = 0 (given Q12 > 0)Note also that we may write
11.2 = w11(1-p2),
P =
@21022
@21/22
11'
where p2 is a squared correlation coefficient. When W112 = 0 (p2 = 1) then 2 is
singular and yt and x, are cointegrated, as pointed out in Phillips (1986). At the
other extreme when there is no correlation between the innovations of y, and x,
we have p2 = 0, 2 nonsingular, and a regression of y, on x, is spurious in the
sense of Granger and Newbold (1974).
(c) Consider the Hilbert space L2[0, 1] of square integrable functions on the
interval [0,1] with inner product Jofg for f, g E L2[0, 1]. In this space, Q is the
projection of W1 on the orthogonal complement of the space spanned by
the elements of W2.
3. RESIDUAL BASED TESTS OF COINTEGRATION
We consider the linear cointegrating regressions
(10)
x + "U
Residual based tests seek to test a null hypothesis of no cointegration using scalar
unit root tests applied to the residuals of (10).
This null may be formulated in terms of the conditional variance parameter
11.2 as the composite hypothesis
HO: (A)1. *0
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171
RESIDUAL BASED TESTS
The alternative is simply
H1: o11 2 =
leading to p2= 1 and cointegration, as pointed out above. Engle and Granger
(1987) discuss the procedure and suggest various tests. Their main recommendation is to use the augmented Dickey-Fuller (ADF) test and they provide some
critical values obtained by Monte Carlo methods for the case m = 2.
We shall consider the asymptotic properties of the following residual based
tests:
(i) Augmented Dickey Fuller: ADF= ta* in the regression Aiu=a* u*t_1 +
+Vp
I=PlqDi jut-i
(ii) Phillips' (1987)
Za test: Regress u, =
Za=T(a
a-1)
-(1/2)
&u-1 + k, and compute
2
( ST/-Sk2)T(
Eut2_
where
T
1
T
(1 1)
STI=
l
T
T-1 ,kt + 2T-1 Y. ws Y. k^tkt-s
s=1
1
t=s+l
+ 1).
for some choice of lag window such as ws= 1-s/(l
(iii) Phillips' (1987) Zt test: Regress ut= aut-1 + k, and compute
-1)/STI-
U 1)
Zt
(1/2)(sTI-
k)[sTI(T
mUt-1)]
with sk1 and sT1 as in (ii).
(iv) Variance ratio test:
(12)
P=Tl
(T'
where t t112-c
(13)
Q
EU2
21222w21
T
=
and
l
T
+ T- s
T-Lttt
1
S=1
ttt-+ts
t=s+l
for some choice of lag window such as w= 1 - s/(l + 1) (see Phillips and
Durlauf (1986), Newey and West (1987)) and where { t} are the residuals from
the least squares regression:
z Fllz,-1 +{
(14)
(v) A multivariate trace statistic:
T
PZ= T tr(Mz-z 1),
Mz= T-1 Y.z z
1
where Q is as in (13).
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172
P. C. B. PHILLIPS AND S. OULIARIS
REMARKS: (a) Note that Za and Z, are constructed using an estimate T21that
is based on the residuals k, from the autoregression of ut on ut-1. When the
estimate STI is based on first differences Au,i in place of kt (as suggested by
the null of no cointegration) we shall denote the resulting tests by Za and Zt. The
distinction is important since these tests have very different properties under the
alternative hypothesis of cointegration, as we see below.
(b) In a similar way, Pu and P, are constructed using the covariance matrix
estimate 2 that is based on the residuals ( from the first order vector autoregression (14). When the estimate 2 is based on first differences {, =A z we denote
the resulting tests by Pu and Pz. Again the distinction is important since Pu and
Pz have different properties under the alternative from those of Pu and Pz.
(c) The variance ratio test Pu is new. Its construction is intuitively appealing.
P, measures the size of the residual variance from the cointegrating regression of
y, on x,, viz. T- EuT 2, against that of a direct estimate of the population
conditional variance of yt given xt, viz. th11 .2. If the model (1) is correct and has
no degeneracies (i.e. 2 nonsingular), then the variance ratio should stabilize
asymptotically. If there is a degeneracy in the model, then this will be picked up
by the cointegrating regression and the variance ratio should diverge.
(d) The multivariate trace statistic Pz is also new. Its appeal is similar to that of
P,. Thus, tSQis a direct estimate of the covariance matrix of zt, while Mzz is
simply the observed sample moment matrix. Any degeneracies in the model such
as cointegration ultimately manifest themselves in the behavior of Mzz and,
hence, that of the statistic Pz. This behavior will be examined in detail below.
Note that Pz is constructed in the form of Hotelling's T02 statistic, which is a
common statistic (see, e.g., Muirhead (1982, Chapter 10)) in multivariate analysis
for tests of multivariate disperision.
(e) Note that none of the tests (i)-(iv) are invariant to the formulation of the
regression equation (10). Thus, for these tests, different outcomes will occur
depending on the normalization of the equation. One way around this problem is
to employ regression methods in fitting (10) which are invariant to normalization.
The obvious candidate is orthogonal regression, leading to
(15)
where
'zt
=Ut5
b= argmin b'Mzzb
b'b = 1.
Here b is the direction of smallest variation in the observed moment matrix Mzz
and corresponds to the smallest principal component with
T
T-1 Eu2
=
b'Mzzb= Xmin(Mzz)
1
where Xmin(Mzz)is the smallest latent root of Mzz Smallest latent root tests
based on Xmin(Mzz)may be constructed. For example, an orthogonal regression
version of the variance ratio statistic Pu would be:
Px = TXnmiin(Q ) /Xmin (M77z)-
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173
RESIDUAL BASED TESTS
Unfortunately, Px has a limiting distribution which depends on the nuisance
parameter 2. The multivariate trace statistic P, offers a very convenient alternative. P, is a normalization invariance analogue of Pu and has the same general
appeal as statistics such as P.; yet, as we see below, its asymptotic distribution is
free of nuisance parameters.
(f) Each of the test statistics (i)-(iv) has been constructed using the residuals ut
of the least squares regression (10). These statistics may also be constructed using
the residuals Ut of the least squares regression
(16)
y = a + 8'xt +Ut
with a fitted intercept. In a similar way, for test (v) the statistic P, may be
constructed using M = T- ET(zt-Z-)(z-ti)'
and residuals {t from a VAR
such as (14) with a fitted intercept. These modifications do not affect the
interpretation of the tests but the alternate construction does have implications
for the asymptotic critical values. These will be considered below.
4. ASYMPTOTICTHEORY
Our first concern is to develop a limiting distribution theory for the tests
(i)-(v) under the null of no cointegration. In this case, the covariance matrix 2 is
positive definite. The statistic that presents the main difficulty in this analysis is
the ADF. We shall give the asymptotic theory for this test separately in the
second result below.
T
THEOREM
4.1: If {Z
z}t
-* oo:
(a)
Z
(b)
Z
(c)
PU-
(d)
(Z
is generatedby (1), if Q > 0, and if (Cl) holds, then as
4
f>RdR;
|1R dS;
(Q2
tr
;
lo
W)I
where notations are the same as in Lemma 2.2 and
R(r) =Q(r)
S(r)
Q(r)=(K'K
(jQ
2)
)1/2
We require the lag truncationparameter 1-x* o as T xo
and I
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o(T).
P. C. B. PHILLIPS AND S. OULIARIS
174
REMARKS:
(a) Recall from Lemma2.2 that
Q(r) = W1(r)
W1W2'( W22'
W2)
whose distributiondependson a singleparametern, the dimensionof W2.Recall
too that
K =(1,|
'
WlW2
W2W2)
whose distribution is also independent of nuisance parameters.
(b) We deduce from the preceding remark that the limiting distributions of Za,
Z, P", and Pz are free of nuisance parameters and are dependent only on the
known dimension number n (or m = n + 1 in the case of P,). These statistics
therefore lead to (asymptotically) similar tests. Critical values for these statistics
have been computed by simulation and are reported in Appendix B. For the case
of the Z, statisticdemeaned(i.e. computedfrom the regression(16) with a fitted
intercept)the values in Table Ilb in AppendixB correspondclosely with those
reported by Engle and Yoo (1987, Table 2, p. 157) for the Dickey Fuller t
statistic. Differencesoccuronly at the second decimalplace and are likely to be
the result of: (i) differencesin the actual sample sizes used in the simulations
(T = 200 in Engle and Yoo (1987);and T = 500 in ours);and (ii) samplingerror.
(c) The Z. and Z, tests have the same limitingdistributionin the generalcase
as the Dickey Fullerresidualbased a and t tests do in the highlyrestrictivecase
of iid (0, Q) errors.This point is discussedfurtherin the originalversionof the
paper which is availableas a technicalreport on request(Phillipsand Ouliaris
(1987)). Thus, the Z. and Z, tests have the same propertyin this context of
cointegratingregressionsfor which they were originally designed in Phillips
(1987) as scalar unit root tests, viz. that they eliminatenuisanceparametersand
lead to limit distributionswhich are the same as those possessedby the DickeyFuller tests in the iid errorenvironment.However,the limitingdistributionshere,
JoRdR and JoR dS, are different from the simple unit root case, and they are
dependenton the dimensionnumbern.
(d) We remarkthat the limiting distributionsof Z., Z., P",P, (the statistics
mentioned earlier which are based on first differencesk,= AU, and =,Azt
ratherthan regressionresidualsk, and (,) are the same as those of Z., Zt, Pu,Pz
given in Theorem4.1. This followsin a straightforward
way from the proof given
in the Appendix.
(e) Note, finally, that if the statisticsare based on the regression(16) with a
fitted interceptthen the limitingdistributionsof Z., Z,, and Pu have the same
form as in (a)-(c) but now R, S, and Q are functionalsof the demeanedstandard
Brownian motion W(r) = W(r) - joJW.Observe that W is the projection in
L210, 1] of W onto the orthogonalcomplementof the constant function, thereby
justifying this terminology.In a similar way, if the cointegratingregression
involves fitted time trendsthe limitingdistributionsin (a)-(c) continueto retain
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175
RESIDUAL BASED TESTS
their stated form but involvefunctionalsof correspondinglydetrendedstandard
Brownianmotion.
4.2: Let t z, } be generated by (1) and suppose { I,} follows a
stationary vector ARMA process. If Q > 0 and (Cl) holds, then as T -*oo
THEOREM
ADF
1R dS,
provided the order of the autoregressionin the ADF is such that p
and p = o(T /3)
-x
o as T -xoo
REMARKS: (a) Theorem4.2 shows that ADF and Zt have the same limiting
distribution.This distributionis convenientlyrepresentedas a stochasticintegral
in termsof the continuousstochasticprocesses(R(r), S(r)). Theseprocessesare,
in turn, continuousfunctionalsof the m-vectorstandardBrownianmotion W(r).
In accordwith our earlierremarksconcerningZ,, the limitingdistributionof the
ADF dependsonly on the dimensionnumbern (the numberof regressorsin (10)
or, equivalently,the system dimension m(= n + 1)). Given m, the ADF is an
asymptoticallysimilartest.
(b) The proof of Theorem4.2 dependscriticallyon the fact that the orderof
the autoregression p
-x 00.
While this behavior is also required for a general unit
root test in the scalarcase (see Said and Dickey (1984)) it is not requiredwhen
the scalar process is drivenby a finite order AR model with a unit root. It is
important to emphasizethat this is not the case when the ADF is used as a
residualbased test for cointegration.Thus, we still need p -* x even when the
vectorprocess {, is drivenby a finiteorderVAR. This is becausethe residualson
which the ADF is based are (random)linear combinationsof {,. These linear
combinations no longer follow simple AR processes. In general, they satisfy
(conditional) ARMA models and we need p
-x 00
in order to mimic their
behavior.
(c) We mentionone specialcase wherethe requirementp -x o is not needed.
This occurswhen the elementsof (, are drivenby a diagonalAR processof finite
order,viz.
p
b(L)~t = ?,,
b(L)
=
,
bjL,
bi= 1;
Et,iid(O, 2).
i=O
In this case
Q = (1l/b (1))2T 9
IQO (fIb(eix)
d
-x2)d
,
and we observethat Q is a scalarmultipleof QO.The exampleschosenby Engle
and Granger(1987) for their simulationexperiments(Tables II and III in their
paper)both fall withinthis specialcase.
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176
P. C. B. PHILLIPS AND S. OULIARIS
(d) Note that the ADF test is basically a t test in a long autoregression
involving the residualsui,. In this sense, the ADF is a simple extension of the
Dickey-Fullert test. Note that no such extensionof the Dickey-Fullera test is
recommendedby Said and Dickey (1984) since even as p -xoo the coefficient
estimate Te* has a limitingdistributionthatis dependenton nuisanceparameters
(cf. Said and Dickey (1984, p. 605)) in the scalarunit root case. In contrast,the
Z. statistic is an asymptoticallysimilartest. Thus, the nonparametriccorrection
of the Z, test successfullyeliminatesnuisanceparametersasymptoticallyeven in
the case of cointegratingregressions.This point will be of some importancelater
when we considerthe powerof these varioustests.
5. TEST CONSISTENCY
Our next concern is to considerthe behaviorof the tests based on Za, Zt,
ADF, Pu and P, underthe alternativeof cointegration.To be specificwe define
z, to be cointegratedif there exists a vector h on the unit sphere(h'h = 1) for
which q, = h'z, is stationary with continuous spectral density fqq(X). This ensures
that the action of the cointegratingvectorh reducesthe integratedprocessz, to a
stationary time series with propertiesbroadly in agreementwith those of the
innovations{, in (1). The spectraldensityof h'(z, - zt- ) satisfies
h'f(jX)h =fqq(X)I1- e A12
from which we deducethat
h'f(jX)h
=fqq(O)X2 + o(2),
X- 0.
This implies that h'Q?h= 0 so that SQis singular.
THEOREM 5.1: If {z,t) is generated by (1) and is a cointegrated system with
cointegrating vector h and S22 > 0, if q, = h'zt and fqq(o)> 0, then
(a)
Za
(b)
Zt O(T1/2)
Op(T)g
ADF =Op(- 1/2)
so that each of these tests is consistent.
(c)
REMARKS: (a) We see from Theorem5.1 that Z. divergesfaster as T -0
underthe alternativeof cointegrationthando eitherof the statisticsZ, and ADF.
This suggeststhat Z. is likelyto havehigherpowerthan Z, and ADF in samples
of moderatesize. It also suggeststhat the null distributionof Z. in finitesamples
is likely to be more sensitivethan Z, and ADF to changesin parameterswhich
move the null closer to the alternative(i.e. as CWll2 ?* 0 or p2 1).
(b) As is clearfromthe proofof Theorem5.1 the requirementthat fqq(O) > 0 is
needed for results (b) and (c). It is not needed for result (a). Moreover,when
fqq(O) = 0 we show in the proof that Z, = Op(T). In this case the cointegrating
vector does more than reduce z, to the stationaryprocess q, = h'z,. It actually
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RESIDUAL BASED TESTS
177
annihilatesall spectralpowerat the origin.When this happens,there should be
more evidencein the data for cointegrationand, correspondingly,the Zt statistic
divergesat the fasterrate Op(T).
(c) Note furtherthat the Z, statistic,whichis based directlyon the coefficient
estimate a in the residualbased regression,does not involve an estimateof the
standarderrorof regressionlike Z,. Its rate of convergenceis Op(T) under the
alternativeirrespectiveof the value of fqq(O).
> 0 in Theorem5.1 is not essentialto the validity
(d) The requirementthat S222
of results (a)-(c) providedy, and x, are still cointegrated.However,when it is
relaxedand we allow 222 to be singular,then we need to allow for cointegrated
regressors xt in (10). In such cases we have b = b + O(T-/2) in place of
b = b + Op(T-1) (see Park and Phillips(1989, Section 5.2) for details) and the
proofs become more complicated.
In order to developour next theoremlet us continueto assumethat 222> 0.
Define an orthogonalmatrixH= [H1,h] and the process:
[H1'~] [wlt In
(17)
Wt[=
j'
1'
say.
This is zero mean, stationarywith spectrumfWW(X),
say, underthe alternativeof
cointegration.
THEOREM 5.2: If { zt;
is generated by (1) and is a cointegrated system with
cointegrating vector h and fQ22> 0 and if f,,(0) > 0, then as T - oo:
^
(a)
(b)
= Op(T),
=
PZ Op(T),
so that each of these tests is consistent.
We observed earlierthat the Z and P tests may be constructedusing first
differencesratherthan residuals.Thus, we denoted by Za and Zt the statistics
which utilize the firstdifferencesAut ratherthan kt in the estimatorsSk and sT.
Similarly,we denotedby Pu and Pz the statisticswhichutilizethe firstdifferences
Azt = {t in place of the residuals(t fromthe VAR (14). Thesemodifiedtests have
very differentpropertiesunderthe alternativeas the followingresultshows.
5.3: If (1) is a cointegratedsystem with cointegrating vector h and
then as T- oo
THEOREM
Q22 > 0,
Z0,
Zt5
Pu5
Pz =
op(i)
and each of these tests is inconsistent.
The use of residualsratherthan first differencesin the constructionof these
tests has a big impact on their asymptoticbehaviorunder the alternativeof
cointegration. Clearly, the formulation in terms of residuals leading to
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178
P. C. B. PHILLIPS AND S. OULIARIS
Z4,, Z,, P, Pz is preferable.Phillipsand Durlauf(1986)reacheda similarconclu-
sion in a relatedcontext,dealingwith multivariateunit root tests.
6. COMMON CONCEPrUAL
PITFALLS
Since it is the hypothesisof cointegrationthat is of primaryinterestratherthan
the hypothesisof no cointegrationit is often arguedthat cointegrationwould be
the better choice of the null hypothesis.For example,in a recent surveyEngle
(1987) concludes that a "null hypothesisof cointegrationwould be far more
In spite
useful in empiricalresearchthan the naturalnull of non-cointegration."
of such commonlyexpressedviews, no residualbased stastisticaltest of cointegrationproceedsalong these lines.
A major source of difficultylies in the estimationof Q under (the null of)
cointegration.In order to assess whethera multipletime series is cointegrated,
residual based tests seek, in effect, to determinewhether there exists a linear
combinationof the serieswhosevarianceis an orderof magnitude(in T) smaller
than that of the individualseries. Equivalently,one can work directlywith the
covariancematrixSQand seekto determinewhetherits smallestlatentroot is zero
and SQis singular.Let us assumethat the supposedcointegratinglinearcombination h were known. In such a case, we would seek to test
Ho': h'Qh = O.
In order to test Ho'it would seem appropriateto estimateSQby SQusing the first
differences (, =,Az, of the data and base some test statistic on h'Qh. Since
0 under the null Ho', but not under the alternative (h'Qh > 0), we might
h'Qh P O
expect a suitablyrescaledversionof h'S2hto providegood discriminatorypower.
Of course, this approachrelieson criticalvaluesfor the statisticthat is based on
h'Q2hbeing workedout. Likewise,if h werenot known,we would seek to test
Hot":QIsingular.
If Q22 > 0 then the obviousapproachwould be to base some test statisticon the
estimatedconditionalvariance
11=2
11 -21
22
21
Since c'11.2-* 0 underthe null Ho', similarconsiderationsapply.
The followinglemmaindicatesthe pitfallsinherentin this approach.It will be
convenientfor the proof to employthe smoothedperiodogramestimateof S2:
(18)
S=
21+ 1
I
_ (2,
sT
denotes the periodogram and wx(X)=
IXx(X) =
w,(X)wx(X)*
the finite Fouriertransformof a (multiple)time series { xt}.
(2gT)-l/212Tx,eiAt
In (18) the bandwidth parameter/ plays a similar role to that of the lag
truncationparameterin the weightedcovarianceestimator(14). We shall assume
that / = o(T1/2). We have the followinglemma.
where
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RESIDUAL BASED TESTS
LEMMA
179
6.1:
(a)
h Qh = T1(qT-
(b)
@l 2 = h'
q0)o + op
+ op(T-
are Op(l).
REMARKS: (a) We see fromLemma6.1 that both Th'Qhand TCZ11.2
Indeed, both of these statisticsare weaklyconvergentin a trivialway, viz.
Th'Qh, TC11.2= (q -qo)
(19)
where q. is a random variable signifyingthe (weak) limit of the stationary
sequence {qT} as T -* oo. Tests that are based on these statisticsthereforeresult
in inconsistent tests. Note also that the limiting distributiongiven by (19) is
dependenton that of the (stationary)sequenceq,, whichin turndependson that
of the data zt. Thus,no centrallimit theoryis applicablein this context.And any
statistical tests that are based on h'Qh or 611.2 under the null of cointegration
would need criticalvaluestailoredto the distributionof the data. Suchspecificity
is highly undesirable.
(b) The above resultssuggestthat classicalproceduresdesignedto test a null of
cointegrationcan have seriousdefects.Statisticsthat are based on Q or 611.2 are
not to be recommended.An alternativeapproachthat is inspiredby principal
componentstheoryis not to test Ho' directlybut to examinewhetherany of the
latent roots of Q are small enough to be deemed negligible. This approach
proceeds under the hypothesisthat Q > 0 (no cointegration)and is well established in multivariateanalysis(e.g., Anderson(1984)).It has been exploredin the
present context by Phillipsand Ouliaris(1988).
7. ADDITIONAL
ISSUES
The results of this paperare all asymptotic.They are broadlyconsistentwith
simulation findingsreportedin Engle and Granger(1987) for the ADF and in
Phillipsand Ouliaris(1988)for the Za, Z,, and ADF tests. However,it is certain
that there are parametersensitivitiesthat are likely to affect the finite sample
propertiesof these tests in importantways. This is becauseas we approachthe
alternative hypothesis of cointegration,the model undergoes a fundamental
degeneracy.This seemsdestinedto manifestitself in the finitesamplebehaviorof
the tests in differingdegrees,dependingon theirconstruction.
Some guidanceon this issue is given by the performanceof the Za,, Z, and
ADF tests in simpletests for the presenceof a unit root in raw time series(rather
than regressionresiduals).Simulationfindingsin this contexthave been reported
by Schwert (1986) and Phillips and Perron(1988). These studies indicate the
power advantagesof the Za test that we have establishedby asymptoticarguments in this paper. But they also show that size distortionscan be substantial
for all of the tests in modelswith parametersapproachingthe stabilityregion.It
seems likely that similarconclusionswill hold for residualbased unit root tests.
However, the issues deserveto be exploredsystematicallyin simulationexperiments.
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180
P. C. B. PHILLIPS AND S. OULIARIS
Cowles Foundationfor Research in Economics, Yale University,New Haven, CT
06520, U.S.A.
ManuscriptreceivedJuly, 1987; final revisionreceived February,1989.
APPENDIX A
2.1: The first part of the argument relies on the construction of a sequence { Y,
PROOFOFLEMMA
of stationary and ergodic martingale differences which are representative of the sequence { t,}. Under
(C2) this construction may be performed as in the proof of Theorem 5.5 of Hall and Heyde (1980, pp.
141-142). We then have
(Al)
(t = Yt+ Z'+ - Z>+I
where { Y,= C(1)et } is the required martingale difference sequence and Zt+ is strictly stationary.
= Q and Zt+ is square integrable. Now
Note that with this construction E(YOYO')
00
(r)E
00
+ E
a-jY,-j
-00
aj(Zt+j j-(Ztj+()
-00
and
[Tr]
XT*(r) = T- 1/2
E
oc
oc
Yt j + T- 1/2 E, aj Z1+-j -
Eaj
Z[TI
-j +1) -
As in Hall and Heyde (1980, p. 143) it follows that
[Tr]
(A2)
1
sup IX*(r)-Tl/2O<rl1
where Y,t =
,*It
P
1
. a.,'Y-j1.
The new sequence { Y,* } is strictly stationary, ergodic, and square integrable with spectral density
matrix
fyy(AX)
E
00aei00
E a1e
ajezj )
=(1/2Xr)(
= (1/2i
) E aje2j |
.
It follows that
)
(I
E(2(
-
2fy
YY( )
a (1)2Q
(see Ibragimov and Linnik (1971, Theorem 18.2.1)). Under (4) we may now obtain a martingale
representation of Y,* analogous to (Al) for (,, viz.
(A3)
y* = Qt + Z* - Z,*+I
where Q, = a (1) Y, is a stationary ergodic sequence of martingale differences with covariance matrix
a(1)2g. We deduce from (A3) that
[Tr]
[Tr]
T-1/2 EY*=T
+T
1/2E
1
1/2 ( Z1*-_Z[*Tr
1
and as before
[Tr]
(A4)
sup
O< r
T-1/2
I
E (Y,*-Q,)
1
-,0.
P
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181
RESIDUAL BASED TESTS
Similarly,
[Tr]
(A5)
T-
sup
o <r
Y,)
-t
/
1
0.
1P
Noting that XT(r) = T-l/2y1Trr%, we now obtain from (A2), (A4), and (A5),
sup IXT (r')-a (1) XT (r)I -0
O<r
1
P
as required for (6); (7) follows directly.
PROOF OF LEMMA2.2: Part (a) is immediate from (8). To prove (b) note that
J02
121-L22A21a2iJ
[-A(j:w2w?)
2
giving the results required. Next
(A6)
=111
q'B(r) -=,'L'W(r)
'W(r)
=
111Q(r)
and (c)-(e) follow directly.
PROOF OF THEOREM4.1: We first observe that
(A7)
where
T=7,
'=( 1,-4')
(Phillips (1986)) and then
TT
(A8)
b=(T2-LIbT 2AZ
T-2y
=
all2.
To prove (a) we write
Za=T(a- 1 )-(1/2)
=(T'E~ U,_ aU
( S2
-
_
-2
U,_2
S2) /(T
S2
(1/2) ( STI
) /(T2
E
,_2
Now
I
T
- Sir,
so that
t=A,-(&-1)u,1=b'{5,-(&-1)z,1}
(
(A9)
k,k,_
t=s+l
s=l
and kt =u
Z
= T-1 E w/
(1/2)(s2_-sS2)
Z='T-1
I
T
E
Ws/E
s=l
t=s+l
T
EZ,_1,-
T-1
1
[(,_5 + (1-a0Zt-5-1]
X[~t+ (1&Zt_])(IU1
Since (, is strictly stationary with continuous spectral density matrix fiC(X) we have
I
T-1
,
.s=1
T
WSZ
E _t-st' -Qi
t=s+l
P
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t-I
182
P. C. B. PHILLIPS AND S. OULIARIS
provided I -l oo as T - oo with I = o(T). Moreover, as in the proof of Theorem 2.6 of Phillips (1988),
we find
T
T
I
T- 1 Ezt-
T-ly
lt-
E
s=1
1
'
wS/,
t=s+l
{t-st
0
BdB'.
Finally, since 1 - a= Op(T- 1) we deduce from (A8) and (A9) that
R dR
IB
'j'BdB'q/aII.2-
with the final equivalence following from parts (d) and (e) of Lemma 2.2.
The proof of part (b) follows in the same manner. To prove (c) and (d) we observe that
t
= i, + Op(T- 1) from (14) and hence
p
as T - oo provided
I -l oo and I = o(T). We deduce that
'W11 2 '
p
@11 2
and using (A8) and Lemma 2.2 (e) we obtain
Q2
1
as required for (c). Part (d) follows by noting that
P. =* tr(Q(j
tr{SL
BB')
=tr(f
(j1WW')
L'1
WWI)
as required.
4.2: The ADF test statistic is the usual t ratio for a' in the regression
PROOF OF THEOREM
p
(A10)
Au,
a *u,
+
+ v,p
E(,/u_
i=1
In conventional regression notation this statistic takes the form
ADF=
u' lQx u_ I)
a*/sv
where Xp is the matrix of observations on the p regressors (Au,_1,...A \ut_P), u.. is the vector of
P
observations of
and
T IE[42,,. Now
Qx =1- Xp(Xp'Xp)-'Xp
sPv-
(All)
(UlQXu)"-)
/
-
(T-2u
IQ_X- )
/2(T-lu' Qx Au)
and
(A12)
T-2uLQxpu-l
q' jBB'rl =
=
T-2uLu1
12j
I
+ o(1)
Q2
Moreover under (Cl) we have
b
*-
( R-mixing
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RESIDUAL BASED TESTS
183
so that the limit variate q is asymptotically independent of each event E e F = U91
where
f?
u;= a(t,, j < i). This independence makes it possible to condition on n without affecting the
probability of events E e F. Note that
(A13)
= gt,
A ut =Ybt'=>'t
say.
Since (, is a stationary (vector) ARMA process by assumption, it is clear that the new scalar process
,= ',
given 71,is also a stationary ARMA process (see, for instance, Lutkepohl (1984)). We write
its AR representation as
00
(A14)
v,=
d,g,_,=d(L)g,
E
1=o
where L is the backshift operator. Note that the sequence {dj } is majorized by geometrically
declining weights and is therefore absolutely summable. Moreover, given q, vt is an orthogonal
(0, q2(q)) sequence with
(Al15)
a 2 1)= d ()71Q7
We now note that the ADF procedure requires the lag order p in the autoregression (AIO) to be large
enough to capture the correlation structure of the errors. Even if {t is itself driven by a finite order
vector AR model, the scalar process g, will follow an ARMA model with a nonzero MA component.
It is therefore always necessary to let p -. oo in (A10) in order to capture the time series behavior of
g,. The only exception occurs when {, is itself an orthogonal sequence. Formally, in the context of
unit root tests, Said and Dickey (1984) require p to increase with T in such a way that p = o(T1/3).
= Op(T- 1), we see that (A10) converges to (A14), conditional on
When this happens, noting that
7q.In particular, we have
(A16)
T-u'-
Qx au'= T- lu' Iv+op()
T
=
b'T-1
+
oZ
z,1v,
(1)
T
= q'T-
?
Zt v to+(l).
Now write
v,
d (L
)tt
d (L)(t'-
and note that by Lemma 2.1
(Tr I
(TrI
(Al17)
T- 1/2 EVt = d(l)
1
7- 1/2
t
q
+ opl
1
uniformly in r (from (6)). Also
[Trl
(Al18)
T- 1/2
tB(r)
and we obtain from (A16)-(A18)
T
(A19)
q'T- 1 E Zt- Ivt
d (1),q'
B dB',q.
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184
P. C. B. PHILLIPS AND S. OULIARIS
We deduce from (All), (A12), (A15), and (A19) that
d (l)'
A1DF
BB'7}
q
7}
(q
r'f
q BdB'tq
BdB',q
(7'qJ BB7)'
(y0})
ov,0
1/2
flQdQ
(
l
)1/2
R dS
=
as required.
PROOFOFTHEOREM
5.1: First observe that since the system is cointegrated, we have:
b _A]=
=
b=ch,
c=(b'b)1"2
_say
4]
b,[
|
where
so that h and b are collinear. We have
b=b+
Op(T-)
from Phillips and Durlauf (1986, Theorem 4.1) and thus
ut = b'z,= b'z, + Op(T-2)-cq,
+ O (T-12)
In the residual based regression
u, = &u,_ I + k,,
we now obtain
a
=
yq(r) = E(q,q,-r)
yq(1)/yq(0),
y
P
with lal <1, and
k, = c(q, -aq,-j)
+ Op(T-l/2
= kt + Op(T-1/)
say,
where k, is stationary with continuous spectral density
fkk
(A)
=
c2 1-ae'
12fqq(X).
It follows that
(A20)
ST
2S7f,
(0)
P
=
2XC2(1
a
-a)2fqq (0)
> 0
and
s2 k-var(k,)
= c2{ (1 +a2)yq(0)-2ayq(1)}
= c2( yq (0)2 - yq (1))
yq (0).
Now
T
T
(A21)
T- 1 UtI
1 P(1) Yq (0)
P
TIU--CY
1
P
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185
RESIDUAL BASED TESTS
and then
T
T
T-
1-T-
u
1
Sk2)
1
C2{ Yq
(A22)
- (1/2)(ST2
u2
E
(1/2)y(O)
-
a)
r
()
_ (1/2)yq(1)2/yq(O)}-
P
It follows that
Ztl =
-
T- E utt- I T
T(T
(1/2)(
STI
Sk
(T-
Ut
1)
=OP(T)
as required for part (a). Similarly, we find that
T=T{T-E
=O(Tl
,t
-T
-(1/2)(sTI
-Eu
si)}(T
/sTl(T
1Z1)}
/2)
in view of (A20)-(A22). Note, however, that if fqq(O)= 0 (so that q, has an MA unit root) we have, as
in Lemma 6.1,
TsI1= Op(1)
and in this case
Zt= Op(T)
as for Z,. This proves part (b). To prove (c) we observe that when fqq(O)> 0, qt has an AR
representation,
00
(A23)
Y
a,q,_l=e,
ao=1,
J=O
where { e, ) is an orthogonal (0, up). We take {a.) to be absolutely summable and then, following
Fuller (1976, p. 374) we write (A23) in alternate form as:
00
(A24)
Aqt =(1
-
+E Z
1)qt1
+ e,
k+lAqt-k
k=l
where O,= L=aj (i = 2, 3,...) and 1 = -EJ1a..
ADF regression (A10) (as p
oo) we find that
52
Since qt is stationary we know that 01 # 1. In the
+(2,
p
&* (61 - 1)# O,
p
T-1u'LQX u-,1
-
1 + (fai)2)
ue
and hence
ADF= Op(T112)
as required for (c).
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186
P. C. B. PHILLIPS AND S. OULIARIS
PROOFOF THEOREM5.2: Note that both P. and P, rely on the covariance matrix estimate Sagiven
by (13). This estimate relies on the residuals ', from the VAR (14), i.e.
+ ft-
Z, =nz,
As T -- oo we have (Park and Phillips (1988))
I-H[
gjH'=H
'-i
say,
where
g =E (ww2
1)/E(
w22)-
We may write
Ht+ (I[-
ft
=HH' + Ht
[
=H
Wt
2
[O
1?-[
g])
IVz,+
*
[w2
H [Wt j
H
+ Op(T-(/2)
)Zt-}
Op(T-
)
OW2t-(1 +OP(T-1
+ Op(T-1/2
-9W2t-1}
)
g]} w, + OP(T-1/2)
I-[O
=t+O (T-1/2),
say.
Now t, is zero mean, stationary with spectral density matrix
ftt(X) = H I- [0, g]e'x }fww(X){ I- [0, g]e'x
} *H'.
Observe that
(A25)
= H(I - [0, g])fww ()(J-
D2= f(0)
g,,=
I)/E(
E(w2tw2,-
[0, g])'H'
* 0 where
is positive definite, since fww(0)> 0 and 1 w22).
We now obtain
(A26)
Q
>
5?,
0
p
as T -- oo. It follows that
(A27)
l11-2
=
@
(2122221
'II-
1
l-2
=
l
Z-21-22221
p
where we partition Q conformably with Q?.Hence, using (A21) and (A27) we find that
PU= Op(T)
as required for part (a).
To prove part (b) we first note that by Proposition 5.4 of Park and Phillips (1989)
(A28)
= h (h'MA&.h)lh'+ Op(T-1)
M7-21
=hh'/mqq+
)
Op(T-
where
T
mqq= T- l
1
T
w22t= T- 1tq2.
1
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187
RESIDUAL BASED TESTS
It follows that
PZ= T h'f2h/mqq + Op (T-1)}
(A29)
= Op(T)
as required for (b).
PROOF OF THEOREM5.3: The Z. and Z, tests use
k, =u, - u,_1= b~t,= b~t,+ Op(T= C(q, - qt-i)
+ Op(T-1).
We therefore have
s2-
-4*
0
p
and, as in Lemma 6.1,
TS2 =Op().
Now
T-1F
+ (1/2)sk
-U,1)
tU,_(U,
T
=
c2T-1 E qt-(qt
-
qt--) + (1/2)c2T -ly
(qt
+
-qt_)2
p(T-1)
= Op(T-)
so that
1
Z. = Tt T- F,
(G-
_ 1))(1/2)(
5TI-Sk )
(T-
1
Ut 1)
= Op(l)
as T -. oo. The result for Zt follows in the same way.
In the case of P, it is easy to see from (A27) and (A28) that
P, = T{(h'fih/mqq)
+ Op(T-1)}
where Q is constructed using the first differences Azt = (t. However, since the system is cointegrated
the limit matrix Q2of Q is singular. Indeed h'Q2h= 0 and, further, since h's, = qt - qt- 1 we find
(A30)
Th's2h= Op(l).
We deduce that
PZ= Op(1)
and the test is inconsistent, as stated. In the case of Pu we observe that
(A31)
wl1.2 = det Q2/detf22
= det (H'QhH)/det
=
i
vp(T-h
{h'Qh -
Q22
h'bH1( Hl' H)
Hi'[2h} det ( H1'12H1
) /det Q22
)
in view of (A30) and the fact that
ht H, = On(T- 1/2)
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188
P. C. B. PHILLIPS AND S. OULIARIS
(see Lemma6.1). We deducethat
PU =OP(1)
as stated.
PROOFOFLEMMA
6.1: Q2is a consistent estimator of Q2based on (,. h'Q2his a consistent estimator
of h'Q2h= 0 based on r, = h't, = q, - q, 1. Consider the following smoothed periodogram estimates
h'Qh= 2/+1
Note that for s
=
-1, -
=-I
s
_Irr(2S/T)
+ 1.
we have
T
= (21rT)12
Wr(2qrs/T)
?1 (iqt
-
T
T
1/2qteei2,Tst/T-(21T) 1/2 qt lei2s{(t-
= (21rT)
1)+1}/T
e2Ts-qo) + O (
= (2rrT)l/2(q
qO)+ Op
= (21rT-l/2)(qT-
and, thus, for /= o(T- 1/2) we deduce that
(A32)
wr(21rsIT) = (21rT-112)(qT-
qo) + op(T-1/2)
uniformly in s. Hence,
h'Q2h= T- (qT-
qo)2 + op (T
It follows that
Th'Q2h= Op(l)
as requiredfor part(a). The resultcontinuesto hold for otherchoicesof spectralestimator.To prove
(b) note from (A31) that
(A)2
{h'Qh
2H(H-
H)
Hl'Qh}det(
H'2H)/(det
Q222)
Now S222 - ?Q22> 0, H1'92H1-- H1'92H1> 0, and
h'[ H, =21+ 1
Irwi(2,zs/T)
s= =-I
Now wr(2Ts/T)
= Op(T 1/2) uniformly in s and
(21+1)
1
s=-I
w,+,,(21rs1T)= Op(1)
so that
h'S2H1= Op(T-1/2)
We deduce that
arqi
h'dfpp(T-a)
2= h +
as required for part (b).
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189
RESIDUAL BASED TESTS
APPENDIXB
Tables I-IV present estimates of the critical values for the Z,,
Zt, Pu,
and
PAstatistics. The tables
allow for cointegratingregressionswith up to five explanatoryvariables(n < 5). Criticalvaluesare
providedfor Models(10) and (16) and for cointegratingregressionswith a constanttermand trend.
The criticalvalues were generatedusing the Monte Carlomethodwith 10000iterationsand 500
observations.All the computationswereperformedon an IBM/AT using the GAUSSprogramming
language.The randominnovationsweredrawnfromthe standardnormalrandomnumbergenerator
in GAUSS (i.e., "RNDNS").Thus Q = I and p2 = 0 for the generateddata, therebysimplifyingthe
computationof the statistics.
Approximate95%confidenceintervalsfor the criticalvalues were computedusing the method
describedin Rohatgi (1984, pp. 496-500). In order to providesome indicationof the degree of
precisionin the estimates,we presentthe approximate95%confidenceintervalsfor n = 1 (referto the
rows labelled Al). Confidenceintervalsfor n > 2 are availablefrom the authorson request.
Usage
For Tables I and II (Z. and Z,): Rejectthe null hypothesisof no cointegrationif the computed
value of the statisticis smaller thanthe appropriatecriticalvalue.For example,for a regressionwith
a constant term and one explanatoryvariable(i.e. n = 1), we rejectat the 5%level if the computed
value of Za is less than - 20.4935or the computedvalueof Z, is less than - 3.3654.
For TablesIII and IV (P. and P,): Rejectthe null hypothesisof no cointegrationif the computed
value of the statisticis greater thanthe appropriatecriticalvalue.For example,for a regressionwith
two explanatoryvariables(i.e., n = 2) but no constantterm,we rejectat the 5%level if the computed
value of P, is greaterthan 32.9392or the computedvalueof P, is greaterthan71.2751.
TABLEIa
CRITICALVALUESFOR THE Za STATISTIC(STANDARD)
Size
0.1500
0.1250
0.1000
0.0750
0.0500
0.0250
0.0100
1 -10.7444
-11.5653
-12.5438
-13.8123
-15.6377
- 18.8833
- 22.8291
2
- 17.0148
- 18.1785
- 19.6142
- 21.4833
- 25.2101
- 29.2688
-22.6211
- 27.3952
- 32.2654
- 23.9225
- 28.8540
- 33.7984
- 25.5236
- 30.9288
- 35.5142
- 27.8526
- 33.4784
- 38.0934
- 31.5432
- 37.4769
-42.5473
- 36.1619
-42.8724
-48.5240
(-0.1866)
Al (-0.2009)
(+ 0.2283) (+ 0.2338)
(-0.2210)
(+0.2941)
(-0.2863)
(+0.3163)
(-0.5282)
(+0.3899)
(-0.5053)
(+0.6036)
(-0.8794)
(+0.6801)
- 16.0164
3 - 21.5353
4 - 26.1698
5 - 30.9022
TABLE Ib
CRITICALVALUESFOR THE Za STATISTIC(DEMEANED)
Size
.,
0.1500
0.1250
0.1000
1 - 14.9135 - 15.9292 -17.0390
2 -19.9461
3 - 25.0537
4 - 29.8765
5
- 34.1972
0.0750
0.0500
0.0250
0.0100
-18.4836
- 20.4935 - 23.8084 - 28.3218
- 21.0371
- 26.2262
- 31.1512
-22.1948
- 27.5846
- 32.7382
- 23.8739
- 29.5083
- 34.7110
-26.0943
- 32.0615
- 37.1508
- 29.7354
- 35.7116
-41.6431
- 34.1686
-41.1348
-47.5118
- 35.4801
- 37.0074
- 39.1100
- 41.9388
- 46.5344
- 52.1723
A1 (-0.2646) (-0.2664) (-0.3035) (-0.2660) (-0.4174) (-0.6163) (-0.9824)
(+0.1834) (+0.3011) (+0.3329) (+0.3348) (+0.4319) (+0.4834) (+ 1.1440)
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P. C. B. PHILLIPS AND S. OULIARIS
190
TABLE Ic
CRITICALVALUESFOR THE Z.f STATISTIC(DEMEANEDAND DETRENDED)
Size
0.1000
0.0750
0.0500
0.0250
0.0100
1 -20.7931
2 - 25.2884
3 - 30.2547
- 21.8068
- 26.4865
- 31.6712
-23.1915
- 27.7803
- 33.1637
- 24.7530
- 29.7331
- 34.9951
- 27.0866
- 32.2231
- 37.7304
-30.8451
-36.1121
-42.5998
-35.4185
-40.3427
-47.3590
4
5
- 36.0288
- 40.5939
- 37.7368
-42.3231
- 39.7286
- 44.5074
- 42.4593
- 47.3830
- 47.1068
- 52.4874
- 53.6142
- 58.1615
(-0.3946)
(+0.3020)
(-0.3466)
(+0.3044)
(-0.3908)
(+0.4081)
(-0.5445)
(+0.5049)
(-0.6850)
(+0.6158)
(-0.9235)
(+0.8219)
n
0.1250
0.1500
- 34.6336
- 38.9959
A1 (-0.2514)
(+0.2771)
TABLE Ila
CRITICALVALUESFOR THE Z, AND ADF STATISTICS(STANDARD)
0.1500
it
1
2
3
4
5
-
2.2584
2.7936
3.2639
3.6108
3.9438
0.1250
0.1000
Size
0.0750
0.0500
0.0250
0.0100
- 2.3533
- 2.8797
- 3.3529
- 3.7063
-4.0352
- 2.4505
- 2.9873
-3.4446
- 3.8068
-4.1416
- 2.5822
-3.1105
- 3.5716
- 3.9482
-4.2521
- 2.7619
- 3.2667
- 3.7371
-4.1261
-4.3999
- 3.0547
- 3.5484
- 3.9895
-4.3798
-4.6676
- 3.3865
- 3.8395
-4.3038
-4.6720
-4.9897
A1 (-0.0232) (-0.0247)
(-0.0328)
(-0.0600)
(-0.0269)
(-0.0439)
(-0.0382)
(+ 0.0211) (+ 0.0228) (+ 0.0218) (+ 0.0347) (+ 0.0318) (+ 0.0601) (+ 0.0755)
TABLE Ilb
CRITICALVALUESFOR THE Zt AND ADF STATISTICS(DEMEANED)
0.1500
it
1
2
3
4
5
-
2.8639
3.2646
3.6464
3.9593
4.2355
A1 (-0.0290)
(+0.0186)
0.1250
0.1000
Size
0.0750
0.0500
0.0250
0.0100
- 2.9571
- 3.3513
- 3.7306
-4.0528
- 4.3288
-3.0657
- 3.4494
- 3.8329
-4.1565
-4.4309
- 3.1982
- 3.5846
- 3.9560
-4.2883
-4.5553
- 3.3654
- 3.7675
-4.1121
-4.4542
-4.7101
- 3.6420
-4.0217
-4.3747
-4.7075
-4.9809
- 3.9618
-4.3078
-4.7325
- 5.0728
- 5.2812
(-0.0261)
(-0.0232)
(-0.0296)
(-0.0424)
(-0.0389)
(-0.0582)
(+0.0263)
(+0.0317)
(+0.0380)
(+0.0304)
(+0.0415)
(+ 0.0501)
TABLE Ilc
CRITICALVALUESFOR THE Zt AND ADF STATISTICS(DEMEANEDAND DETRENDED)
0.1500
0.1250
0.1000
Size
0.0750
0.0500
0.0250
0.0100
- 3.3283
- 3.6613
- 3.9976
-4.2751
-4.5455
- 3.4207
- 3.7400
-4.0808
-4.3587
-4.6248
- 3.5184
- 3.8429
-4.1950
-4.4625
-4.7311
- 3.6467
- 3.9754
- 4.3198
-4.5837
-4.8695
- 3.8000
-4.1567
-4.4895
-4.7423
- 5.0282
-4.0722
-4.3854
-4.7699
- 5.0180
- 5.3056
-4.3628
-4.6451
- 5.0433
- 5.3576
- 5.5849
A1 (- 0.0259) (- 0.0246)
(+ 0.0246) (+0.0281)
(- 0.0244)
(- 0.0259)
(- 0.0350)
(- 0.0469)
(- 0.0629)
(+0.0205)
(+0.0301)
(+0.0288)
(+0.0507)
(+0.0722)
n
1
2
3
4
5
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191
RESIDUAL BASED TESTS
TABLE Illa
CRITICALVALUESFOR THE PU STATISTIC(STANDARD)
2
3
4
5
A1
0.1500
0.1250
0.1000
Size
0.0750
0.0500
0.0250
0.0100
17.2146
22.9102
28.9811
34.5226
39.7187
18.6785
24.6299
31.0664
36.4575
41.7669
20.3933
26.7022
33.5359
39.2826
44.3725
22.7588
29.4114
36.5407
41.8969
47.6970
25.9711
32.9392
40.1220
46.2691
51.8614
31.8337
39.2236
46.3395
53.3683
59.6040
38.3413
46.4097
55.7341
63.2149
69.4939
(-0.3833)
(-0.5797)
(-0.6274)
(-1.3218)
(-1.4320)
(-0.4356) (-0.3845)
(+ 0.3777) (+ 0.3842) (+ 0.4706) (+ 0.5793) (+ 0.6159) (+ 0.8630) (+ 1.4875)
TABLE IlIb
CRITICALVALUESFOR THE PU STATISTIC(DEMEANED)
it
0.1500
0.1250
0.1000
Size
0.0750
0.0500
0.0250
0.0100
1
2
3
4
5
24.1833
29.3836
35.1077
40.5469
45.3177
25.8456
31.4238
37.4543
42.5683
47.6684
27.8536
33.6955
39.6949
45.3308
50.3537
30.3123
36.4757
42.8111
48.6675
53.5654
33.7130
40.5252
46.7281
53.2502
57.7855
39.9288
46.6707
53.9710
61.2555
65.8230
48.0021
53.8731
63.4128
71.5214
76.7705
A1 (-0.3913)
(+0.4424)
(-0.4662)
(+0.5441)
(-0.5310)
(+0.4507)
(-0.5323)
(+0.7081)
(-0.5064)
(+0.8326)
(-1.0507)
(-1.6209)
(+ 1.3312) (+ 2.1805)
TABLE IlIc
CRITICALVALUESFORTHE PU STATISTIC(DEMEANEDAND DETRENDED)
it
0.1500
0.1250
0.1000
Size
0.0750
0.0500
0.0250
0.0100
1
2
3
4
5
36.9055
41.2115
46.9643
51.9689
56.0522
38.8150
43.4320
49.2906
54.3205
58.6310
41.2488
46.1061
52.0015
57.3667
61.6155
s44.2416
49.3671
55.4625
60.8175
65.3514
48.8439
53.8300
60.2384
65.8706
70.7416
56.0886
60.8745
68.4051
74.4712
79.0043
65.1714
69.2629
78.3470
84.5480
91.0392
A1 (-0.5294)
( +0.5171)
(-0.5724)
( +0.5187)
(-0.6764)
(+0.6762)
(-0.7143)
( +0.7989)
(-1.0116)
( +0.8773)
(-1.2024)
( +1.2936)
(-2.1849)
(+ 2.2679)
TABLE IVa
CRITICALVALUESFOR THE PZ STATISTIC(STANDARD)
n
0.1500
0.1250
0.1000
Size
0.0750
0.0500
0.0250
0.0100
1
2
3
4
5
30.0137
56.7679
92.7621
135.2724
186.4277
31.7517
59.1613
95.7974
138.9636
190.6337
33.9267
62.1436
99.2664
143.0775
195.6202
36.6646
65.6162
103.8454
148.4109
201.9621
40.8217
71.2751
109.7426
155.8019
210.2910
47.2452
79.5177
119.3793
166.3516
224.0976
55.1911
89.6679
131.5716
180.4845
237.7723
A1 (-0.4804)
(+0.4633)
(-0.4493)
(+0.5042)
(-0.5646)
(+0.6770)
(-0.7120)
(+0.9202)
(-0.8406)
(+0.7319)
(-1.1662)
(+1.5961)
(-1.2202)
(+1.7356)
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192
P. C. B. PHILLIPS AND S. OULIARIS
TABLE IVb
CRITICALVALUESFOR THE P, STATISTIC(DEMEANED)
n
0.1500
1
2
3
4
5
42.5452
74.1493
113.5617
160.8156
215.2089
A1 (-0.4629)
( +0.4873)
0.1000
Size
0.0750
0.0500
0.0250
0.0100
47.5877
80.2034
120.3035
168.8572
225.2303
50.7511
84.4027
125.4579
174.2575
232.4652
55.2202
89.7619
132.2207
182.0749
241.3316
61.4556
97.8734
142.5992
194.7555
255.5091
71.9273
109.4525
153.4504
209.8054
270.5018
(-0.7383)
(-0.6744)
(+ 0.7355) (+0.5972)
(-0.6903)
( +0.6662)
(-1.0214)
( +0.7440)
0.1250
44.8266
76.8850
116.6933
164.7394
219.5757
(-1.8177)
(-0.7998)
(+ 2.0530) (+ 2.4081)
TABLE IVC
CRITICALVALUESFORTHE P, STATISTIC(DEMEANEDAND DETRENDED)
n
0.1500
0.1250
0.1000
Size
0.0750
0.0500
0.0250
0.0100
1
2
3
4
5
66.2417
106.6198
154.8402
210.3150
273.3064
68.8271
109.9751
158.6619
214.3858
277.9294
71.9586
113.4929
163.1050
219.5098
284.0100
75.7349
118.3710
168.7736
225.6645
291.2705
81.3812
124.3933
175.9902
234.2865
301.0949
90.2944
133.6963
188.1265
247.3640
315.7299
102.0167
145.8644
201.0905
264.4988
335.9054
A (-0.5433)
(+0.6819)
(-0.7346)
(+0.5862)
(-0.8305)
(+0.7373)
(-2.3915)
(-0.6905)
(-0.8651)
(-1.6500)
(+ 1.1280) (+ 1.4149) (+ 1.8572) (+ 2.1024)
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