Journal of Econometrics 97 (2000) 293}343
Structural analysis of vector error correction
models with exogenous I(1) variables
M. Hashem Pesaran *, Yongcheol Shin, Richard J. Smith
Faculty of Economics and Politics, Austin Robinson Building, University of Cambridge,
Sidgwick Avenue, Cambridge CB3 9DD, UK
Department of Economics, University of Edinburgh, UK
Department of Economics, University of Bristol, UK
Received 1 April 1997; received in revised form 1 October 1999; accepted 1 October 1999
Abstract
This paper generalizes the existing cointegration analysis literature in two respects.
Firstly, the problem of e$cient estimation of vector error correction models containing
exogenous I(1) variables is examined. The asymptotic distributions of the (log-)likelihood
ratio statistics for testing cointegrating rank are derived under di!erent intercept and
trend speci"cations and their respective critical values are tabulated. Tests for the
presence of an intercept or linear trend in the cointegrating relations are also developed
together with model misspeci"cation tests. Secondly, e$cient estimation of vector error
correction models when the short-run dynamics may di!er within and between equations
is considered. A re-examination of the purchasing power parity and the uncovered
interest rate parity hypotheses is conducted using U.K. data under the maintained
assumption of exogenously given foreign and oil prices. 2000 Elsevier Science S.A. All
rights reserved.
JEL classixcation: C12; C13; C32
Keywords: Structural vector error correction model; Cointegration; Unit roots; Likelihood ratio statistics; Critical values; Seemingly unrelated regression; Monte Carlo
simulations; Purchasing power parity; Uncovered interest rate parity
* Corresponding author. Tel.: #44 1223 335216; fax: #44 1223 335471.
E-mail address: [email protected] (M.H. Pesaran).
0304-4076/00/$ - see front matter 2000 Elsevier Science S.A. All rights reserved.
PII: S 0 3 0 4 - 4 0 7 6 ( 9 9 ) 0 0 0 7 3 - 1
294
M.H. Pesaran et al. / Journal of Econometrics 97 (2000) 293}343
1. Introduction
This paper generalizes the analysis of cointegrated systems advanced by
Johansen (1991,1995) in two important respects. Firstly, we consider a subsystem approach in which we regard a subset of random variables which are
integrated of order one (I(1)) as structurally exogenous; that is, any cointegrating
vectors present do not appear in the sub-system vector error correction model
(VECM) for these exogenous variables and the error terms in this sub-system are
uncorrelated with those in the rest of the system. This generalization is
particularly relevant in the macroeconometric analysis of &small open' economies where it is plausible to assume that some of the I(1) forcing variables, for
example, foreign income and prices, are exogenous. Similar considerations arise
in the empirical analysis of sectoral and regional models where some of the
economy-wide I(1) forcing variables may also be viewed as exogenous. This
extension paves the way for a more e$cient multivariate analysis of economic
time series for which data are typically only available over relatively short
periods. Secondly, we allow constraints on the short-run dynamics in the
VECM. This extension is also important in applied contexts where, due to data
limitations, researchers may wish to use a priori restrictions or model selection
criteria to choose the lag orders of the stationary variables in the model. As
Abadir et al. (1999) demonstrate, the inclusion of irrelevant stationary terms in
a VECM may result in substantial small sample estimator bias.
The plan of the paper is as follows. The basic vector autoregressive (VAR)
model and other notation are set out in Section 2. The importance of an
appropriate speci"cation of deterministic terms in the VAR model is also
highlighted here. In particular, unless the coe$cients associated with the intercept or the linear deterministic trend are restricted to lie in the column space of
the long-run multiplier matrix, the VAR model has the unsatisfactory feature
that quite di!erent deterministic behavior should be observed in the levels of the
variables for di!ering values of the cointegrating rank. The problem of e$cient
conditional estimation of a VECM containing I(1) exogenous variables is
addressed in Section 3. This section distinguishes between "ve di!erent cases
which are classi"ed by the deterministic behavior in the levels of the underlying
variables; that is, Case I: zero intercepts and linear trend coe$cients, Case II:
restricted intercepts and zero linear trend coe$cients, Case III: unrestricted
intercepts and zero linear trend coe$cients, Case IV: unrestricted intercepts
and restricted linear trend coe$cients, Case V: unrestricted intercepts and
unrestricted linear trend coe$cients. These cases have been considered by
A similar approach is taken in Harbo et al. (1998), an earlier version of which we became aware
of after the "rst version of this paper (Pesaran, Shin and Smith, 1997) was completed. This revision
identi"es the areas of overlap between the two papers.
M.H. Pesaran et al. / Journal of Econometrics 97 (2000) 293}343
295
Johansen (1995) when all the I(1) variables in the VAR are treated as endogenous. Section 4 weakens the distributional assumptions of earlier sections and
develops tests of cointegration rank for all "ve cases in Sections 4.1 and 4.2;
relevant asymptotic critical values are provided in Tables 6(a)}6(e). Modi"cations to the tests of Sections 4.1 and 4.2 necessitated in the presence of
exogenous variables which are integrated of order zero are brie#y addressed in
Section 4.3. Tests for the absence of an intercept or a trend in the cointegrating
relations are discussed in Section 4.4, and particular misspeci"cation tests
concerning the intercept and linear trend coe$cients are developed in Section
4.5 together with misspeci"cation tests for the weak exogeneity assumption. The
problem of e$cient conditional estimation of a VECM subject to restrictions on
the short-run dynamics is considered in Section 5. The subsequent two sections
consider the empirical relevance of the proposed tests. Section 6 addresses the
issue of the small sample performance of the proposed trace and maximum
eigenvalue tests using a limited set of Monte Carlo experiments. It compares the
size and power performance of the standard Johansen procedure with the new
tests that take account of exogenous I(1) variables as well as possible restrictions
on the short-run coe$cients. Section 7 presents an empirical re-examination of
the validity of the Purchasing Power Parity (PPP) and the Uncovered Interest
Parity (UIP) hypotheses using U.K. quarterly data over the period
1972(1)}1987(2) which was previously analyzed by Johansen and Juselius (1992)
and Pesaran and Shin (1996). In contrast to this earlier work, foreign prices are
assumed to be exogenously determined and a more satisfactory treatment of oil
price changes is provided in the analysis. Section 8 concludes the paper. Proofs
of results are collected in Appendix A and Appendix B describes the simulation
method for the computation of the asymptotic critical values provided in Tables
6(a)}(e).
2. The treatment of trends in VAR models
Let +z , denote an m-vector random process. The data generating process
R R
(DGP) for +z , is the vector autoregressive model of order p (VAR(p)) deR R
scribed by
U(¸)(z !l!ct)"e ,
R
R
t"1, 2,2,
(2.1)
where ¸ is the lag operator, l and c are m -vectors of unknown coe$cients, and
the (m, m) matrix lag polynomial of order p, U(¸),I !N U ¸G, comprises
K
G G
the unknown (m, m) coe$cient matrices +U ,N . For the purposes of exposition
G G
in this and the following section, the error process +e ,
is assumed to be
R R\
IN(0, X), X positive de"nite. The analysis that follows is conducted given the
initial values Z ,(z
, , z ).
\N> 2 296
M.H. Pesaran et al. / Journal of Econometrics 97 (2000) 293}343
It is convenient to re-express the lag polynomial U(¸) in a form which arises in
the vector error correction model discussed in Section 3; viz.
U(¸),!P¸#C(¸)(1!¸).
(2.2)
In (2.2), we have de"ned the long-run multiplier matrix
N
P,! I ! U
(2.3)
K
G
G
and the short-run response matrix lag polynomial C(¸),I !
K
N\C ¸G, C "!N
U , i"1,2, p!1. Hence, the VAR(p) model (2.1)
G G
G
HG> H
may be rewritten in the following form:
U(¸)z "a #a t#e , t"1, 2,2,
R
R
where
(2.4)
a ,!Pl#(C#P)c, a ,!Pc,
and the sum of the short-run coe$cient matrices C is given by
(2.5)
N\
N
C,I ! C "!P# iU .
K
G
G
G
G
The cointegration rank hypothesis is de"ned by
(2.6)
H : Rank[P]"r, r"0,2, m,
(2.7)
P
where Rank[.] denotes the rank of [.]. Under H of (2.7), we may express
P
P"ab,
(2.8)
where a and b are (m, r) matrices of full column rank. Correspondingly we may
de"ne (m, m!r) matrices of full column rank a and b whose columns form
,
,
bases for the null spaces (kernels) of a and b respectively; in particular, aa "0
,
and bb "0.
,
We now adopt the following assumptions.
Assumption 2.1. The (m, m) matrix polynomial U(z)"I !N U zG is such that
K
G G
the roots of the determinantal equation "U(z)""0 satisfy "z"'1 or z"1.
Assumption 2.1 rules out the possibility that the random process
+(z !l!ct), admits explosive roots or seasonal unit roots except at the
R
R
zero frequency.
Assumption 2.2. The (m!r, m!r) matrix a Cb is full rank.
, ,
M.H. Pesaran et al. / Journal of Econometrics 97 (2000) 293}343
297
Under Assumption 2.1, Assumption 2.2 is a necessary and su$cient condition
for the processes +b (z !l!ct), and +b(z !l!ct), to be integrated
, R
R
R
R
of orders one and zero respectively. Moreover, Assumption 2.2 speci"cally
excludes the process +(z !l!ct), being integrated of order two. Together
R
R
these assumptions permit the in"nite order moving average representations
described below. See Johansen (1991, Theorem 4.1, p. 1559) and Johansen (1995,
Theorem 4.2, p. 49).
The di!erenced process +*z , may be expressed under Assumptions 2.1
R R
and 2.2 from (2.4) as the in"nite vector moving average process
*z "C(¸)(a #a t#e )"b #b t#C(¸)e , t"1, 2,2,
(2.9)
R
R
R
where b ,Ca #CHa , b ,Ca . The matrix lag polynomial C(¸) is given
by
C(¸),I # C ¸H"C#(1!¸)CH(¸), CH(¸), CH¸H,
K
H
H
H
H
C, C ,
CH, CH.
(2.10)
H
H
H
H
Now, as C(¸)U(¸)"U(¸)C(¸)"(1!¸)I , PC"0 and CP"0; in particular,
K
C"b (a Cb )\a . Re-expressing (2.9) in levels,
, , ,
,
t(t#1)
#Cs #CH(¸)(e !e ),
(2.11)
z "z #b t#b
R
R
R
2
where the partial sum s ,R e , t"1, 2,2 .
R
Q Q
Adopting the VAR(p) formulation (2.1) rather than the more usual (2.4),
in which a and a are unrestricted, reveals immediately from (2.11) that
the restrictions (2.5) on a induce b "0 and ensure that the nature of the
deterministic trending behavior of the level process +z ,
remains
R R
invariant to the rank r of the long-run multiplier matrix P; that is, it is
linear. Hence, the in"nite moving average representation for the level process
+z , is
R R
z "l#ct#Cs #CH(¸)e ,
(2.12)
R
R
R
See Johansen (1995, De"nitions 3.2 and 3.3, p. 35). That is, de"ning the di!erence operator
D,(1!¸), the processes +b [D(z !l!ct)], , and +b(z !l!ct), admit stationary and
,
R
R
R
R
invertible ARMA representations; see also Engle and Granger (1987, De"nition, p. 252).
The
matrices
+C ,
can
be
obtained
from
the
recursions
G
C "N C U , i'1, C "I , C "!(I !U ), de"ning C "0 for i(0. Similarly, for the
G
H G\H H
K K
G
matrices +CH,, CH"C #CH , j'0, CH"I !C.
H
H
H
H\
K
From (2.2), as C(¸)U(¸)"(1!¸)I and, in particular, CP"0, CC!CHP"I .
K
K
298
M.H. Pesaran et al. / Journal of Econometrics 97 (2000) 293}343
where we have used the initialization z ,l#CH(¸)e . See also Johansen
(1994) and Johansen (1995, Section 5.7, pp. 80}84). If, however, a were not
subject to the restrictions (2.5), the quadratic trend term would be present in the
level equation (2.11) apart from in the full rank stationary case
H : Rank[P]"m or C"0. However, b would be unconstrained under the
K
null hypothesis of no cointegration; that is, H : Rank[P]"0, and C full rank.
In the general case H : Rank[P]"r of (2.7), this would imply the unsatisfactory
P
conclusion that quite di!erent deterministic trending behavior should be observed in the levels process +z , for di!ering values of the cointegrating rank
R R
r, the number of independent quadratic deterministic trends, m!r, decreasing
as r increases.
The above analysis further reveals that because cointegration is only concerned with the elimination of stochastic trends it does not therefore rule out the
possibility of deterministic trends in the cointegrating relations. Pre-multiplying
both sides of (2.12) by the cointegrating matrix b, we obtain the cointegrating
relations
bz "bl#(bc)t#bCH(¸)e , t"1, 2,2,
R
R
(2.13)
which are trend stationary. In general, the co-trending restriction (Park, 1992)
bc"0 if and only if a "0. In this case, the representation for the VAR(p)
model, (2.4), and the cointegrating regression, (2.13), will contain no deterministic trends. However, the restriction bc"0 may not prove to be satisfactory in
practice. Therefore, it is important that the composite bc in (2.13) or, equivalently, a "!Pc in (2.4) is estimated along with the other parameters of the
model and the co-trending restriction tested; see Section 4.4.
3. E7cient estimation of a structural error correction model
We now partition the m-vector of random variables z into the n-vector y and
R
R
the k-vector x , where k,m!n; that is, z "( y , x ), t"1, 2,2 . The primary
R
R
R R
concern of this paper is the structural modelling of the vector y conditional on
R
its past, y , y ,2, and current and past values of the vector of random
R\ R\
variables x , x , x ,2, t"1, 2,2 . The assumption of Section 2 concernR R\ R\
ing the error process +e ,
, e &IN(0, X), t"0,$1,2, where X is positive
R R\ R
Of course, the levels equation (2.12) could also have been obtained directly from (2.1) by noting
D(z !l!ct)"C(¸)e , t"1, 2,2 .
R
R
As the cointegration rank hypothesis (2.7) may be alternatively and equivalently expressed as
H : Rank[C]"m!r, r"0,2, m, it is interesting to note that, from (2.4) and (2.5), there are
P
r linearly independent deterministic trends and, from (2.11), m!r independent stochastic trends Cs ,
R
the combined total of which is m.
M.H. Pesaran et al. / Journal of Econometrics 97 (2000) 293}343
299
de"nite, permits a likelihood analysis and the conditional model interpretation
given below; see Harbo et al. (1998) for a similar development.
It is convenient to re-express the VAR(p) of (2.4) as the vector error correction
model (VECM):
N\
*z "a #a t# C *z #Pz #e , t"1, 2,2,
(3.1)
R
G R\G
R\
R
G
where the short-run response matrices +C ,N\ and the long-run multiplier
G G
matrix P are de"ned below (2.2).
By partitioning the error term e conformably with z "(y , x ) as
R
R
R R
e "(e , e ) and its variance matrix as
R
WR VR
X
X
WV
X" WW
X
X
VW
VV
we are able to express e conditionally in terms of e as
WR
VR
e "X X\e #u ,
(3.2)
WR
WV VV VR
R
where u &IN(0, X ), X ,X !X X\X and u is independent of e .
R
SS
SS
WW
WV VV VW
R
VR
Substitution of (3.2) into (3.1) together with a similar partitioning of the parameter vectors and matrices a "(a , a ), a "(a , a ), P"(P , P ),
W V
W V
W V
C"(C , C ), C "(C , C ), i"1,2, p!1, provides a conditional model for
W V
G
WG VG
*y in terms of z , *x , *z , *z , 2; viz.
R
R\
R
R\
R\
N\
*y "c #c t#K *x # W *z #P z #u ,
(3.3)
R
R
G R\G
WWV R\
R
G
t"1, 2,2,
where c , a !X X\a , c , a !X X\a , K"X X\, W ,
W
WV VV V W
WV VV V
WV VV
G
C !X X\C , i"1,2, p!1, and P ,P !X X\P .
WG
WV VV VG
WWV
W
WV VV V
Following Johansen (1992) and Boswijk (1992, Chapter 3), we assume that the
process +x , is weakly exogenous with respect to the matrix of long-run
R R
multiplier parameters P; viz.
Assumption 3.1. P "0.
V
Therefore,
P "P .
WWV
W
Speci"cation tests for Assumption 3.1 are presented in Section 4.5 below.
(3.4)
300
M.H. Pesaran et al. / Journal of Econometrics 97 (2000) 293}343
Consequently, under Assumption 3.1, from (3.1) and (3.3), the system of equations are rendered as
N\
*y "c #c t#K *x # W *z #P z #u ,
(3.5)
R
R
G R\G
W R\
R
G
N\
*x "a # C *z #e , t"1, 2,2,
(3.6)
R
V
VG R\G
VR
G
where now, because a "0, c ,a and the relations (2.5) are modi"ed to
V
W
c "!P l#(C !X X\C #P )c,
W
W
WV VV V
W
c "!P c.
W
(3.7)
The restriction P "0 of Assumption 3.1 implies that the elements of the vector
V
process +x , are not cointegrated among themselves as is evident from (3.6).
R R
Moreover, the information available from the di!erenced VAR(p!1) model
(3.6) for +x , is redundant for e$cient conditional estimation and inference
R R
concerning the long-run parameters P as well as the deterministic and shortW
run parameters c , c , K and W , i"1,2, p!1, of (3.5). Furthermore, under
G
Assumption 3.1, we may regard +x , as long run forcing for +y , ; see
R R
R R
Granger and Lin (1995).
The cointegration rank hypothesis (2.7) is therefore restated in the context of
(3.5) as
H : Rank[P ]"r,
P
W
r"0,2, n.
(3.8)
We di!erentiate between and delineate "ve cases of interest; viz.
Case I: (No intercepts; no trends.) c0 "0 and c "0. That is, l"0 and c"0.
Hence, the structural VECM (3.5) becomes
N\
*y "K *x # W *z #P z #u .
(3.9)
R
R
G R\G
W R\
R
G
Case II: (Restricted intercepts; no trends.) c "!P l and c "0. Here,
W
c"0. The structural VECM (3.5) is
N\
*y "(!P l)#K *x # W *z #P z #u .
(3.10)
R
W
R
G R\G
W R\
R
G
Case III: (Unrestricted intercepts; no trends.) c0 O0 and c "0. Again, c"0.
In this case, the intercept restriction c "!P l is ignored and the structural
W
Note that this restriction does not preclude +y , being Granger-causal for +x , in the short
R R
R R
run.
M.H. Pesaran et al. / Journal of Econometrics 97 (2000) 293}343
301
VECM estimated is
N\
*y "c #K *x # W *z #P z #u .
(3.11)
R
R
G R\G
W R\
R
G
Case IV: (Unrestricted intercepts; restricted trends.) c0 O0 and c "!P c.
W
Thus
N\
*y "c #(!P c)t#K *x # W *z #P z #u .
(3.12)
R
W
R
G R\G
W R\
R
G
Case V: (Unrestricted intercepts; unrestricted trends.) c0 O0 and c O0. Here,
the deterministic trend restriction c "!P c is ignored and the structural
W
VECM estimated is
N\
*y "c #c t#K *x # W *z #P z #u .
(3.13)
R
R
G R\G
W R\
R
G
It should be emphasized that the DGPs for Cases II and III are identical as
are those for Cases IV and V. However, as in the test for a unit root proposed by
Dickey and Fuller (1979) compared with that of Dickey and Fuller (1981) for
univariate models, estimation and hypothesis testing in Cases III and V proceed
ignoring the constraints linking, respectively, the intercept and trend coe$cient
vectors, c and c , to the parameter matrix P whereas Cases II and IV fully
W
incorporate the restrictions in (3.7).
We concentrate on Case IV, that is, (3.12), which may be simply revised to
yield the remainder. Firstly, note that under (3.8) we may express
P "a b,
(3.14)
W
W
where the (n, r) loading matrix a and the (m, r) matrix of cointegrating vectors
W
b are each full column rank and identi"ed up to an arbitrary (r, r) non-singular
matrix.
The estimation of the cointegrating matrix b has been the subject of much
intensive research for the case in which n"m or k"0, that is, no exogenous
variables. See, for example, Engle and Granger (1987), Johansen
(1988,1991,1995), Phillips (1991), Ahn and Reinsel (1990), Phillips and Hansen
(1990), Park (1992) and Pesaran and Shin (1999). More recently, Harbo et al.
(1998) have also considered the cointegration rank hypothesis (3.8) in the
context of a conditional model; that is, when k'0.
The procedure elucidated below is an adaptation for the case in which k'0
which gives the results for n"m as a special case. Rewrite (3.12) as
N\
*y "c #K *x # W *z #P zH #u ,
R
R
G R\G
WH R\
R
G
That is, (a K\)(Kb)"a b for any (r, r) non-singular matrix K.
W
W
t"1, 2,2,
(3.15)
302
M.H. Pesaran et al. / Journal of Econometrics 97 (2000) 293}343
where zH "(t, z ), and P "P (!c, I ). Note that Rank[P ]
R\
R\
WH
W
K
WH
"Rank[P ] and, thus, from (3.14)
W
P "a b ,
(3.16)
WH
W H
where
b "
H
!c
b.
I
K
Consequently, we may therefore restate the cointegration rank hypothesis (3.14)
as
H : Rank[P ]"r, r"0,2, n.
(3.17)
P
WH
Hence, we may adapt the reduced rank techniques of Johansen (1995) to
estimate the revised system (3.15); see also Boswijk (1995) and Harbo et al.
(1998).
If ¹ observations are available, stacking the structural VECM (3.15) results in
*Y"c ι #W *Z #P ZH #U,
(3.18)
2
\
WH \
where *Y,(*y ,2, *y ), ι is a ¹-vector of ones, *X,(*x ,2, *x ),
2 2
2
*Z ,(*z ,2, *z ), i"1,2, p!1, W,(K, W ,2,W
), *Z ,
\G
\G
2\G
N\
\
(*X, *Z ,2, *Z ), ZH ,(s , Z ), s ,(1,2, ¹), Z ,(z ,2, z
)
\
\N
\
2 \ 2
\
2\
and U,(u ,2, u ).
2
The log-likelihood function of the structural VECM model (3.18) is given by
n¹
¹
1
l (w; r)"!
ln 2p! ln"X "! ¹race(X\UU),
2
SS
SS
2
2
2
(3.19)
where the parameter vector w collects together the unknown parameters in X ,
SS
c , W and P . Successively concentrating out X , c and W, and a in (3.19)
WH
SS W
results in the concentrated log-likelihood function
n¹
¹
l (b ; r)"! (1#ln 2p)! ln"¹\*YK (I !ZK H b
2 H
2
\ H
2
2
;(b ZK H ZK H b )\b ZK H )*YK ",
(3.20)
H \ \ H
H \
where *YK and ZK H are respectively the OLS residuals from regressions of *Y
\
and ZH on (ι , *Z ). De"ning the sample moment matrices
\
2
\
S ,¹\*YK *YK , S ,¹\*YK ZK H , S ,¹\ZK H ZK H ,
(3.21)
77
78
\
88
\ \
The trend parameter vector c is no longer identi"ed in system (3.15) and (3.6). Only bc and
a "C c may be identi"ed.
V
V
The corresponding estimators are given by XK "¹\UK UK , where UK is de"ned via (3.18) and is
SS
a function of the unknown parameter matrices c , W and P , c( (P )"
WH
WH
(*Y!WK (P )*Z !P ZH )ι (ι ι )\,
WK (PWH)"(*Y!P ZH )PM ι *Z (*Z PM ι *Z )\,
WH
\
WH \ 2 2 2
WH \
\
\
\
where PM ι ,I !ι (ι ι )\ι , and a( (b )"*YK ZK H b (b ZK H ZK H b )\.
2
2 2 2
2
W H
\ H H \ \ H
M.H. Pesaran et al. / Journal of Econometrics 97 (2000) 293}343
303
the maximization of the concentrated log-likelihood function l (b ; r) of (3.20)
2 H
reduces to the minimization of
"S ""b (S !S S\S )b "
87 77 78 H
"S !S b (b S b )\b S "" 77 H 88
77
78 H H 88 H
H 87
"b S b "
H 88 H
with respect to b . The solution bK to this minimization problem, that is, the
H
H
maximum likelihood (ML) estimator for b , is given by the eigenvectors correH
sponding to the r largest eigenvalues jK '2'jK '0 of
P
"jK S !S S\S ""0;
(3.22)
88
87 77 78
cf. Johansen (1991, pp. 1553}1554). The ML estimator bK is identi"ed up to
H
post-multiplication by an (r, r) non-singular matrix; that is, r just-identifying
restrictions on b are required for exact identi"cation. The resultant maxiH
mized concentrated log-likelihood function l (b ; r) at bK of (3.20) is
2 H
H
n¹
¹
¹ P
l (r)"! (1#ln 2p)! ln"S "! ln(1!jK ).
(3.23)
2
77
G
2
2
2
G
Note that the maximized value of the log-likelihood l (r) is only a function of
2
the cointegration rank r (and n and k) through the eigenvalues +jK ,P de"ned
G G
by (3.22). See also Harbo et al. (1998, Section 2).
For the other four cases of interest, we need to modify our de"nitions of *YK
and ZK H and, consequently, the sample moment matrices S , S and S given
\
77 78
88
by (3.21). We state the required de"nitions below.
Case I: c "0 and c "0. *YK and ZK H are the OLS residuals from the
\
regression of *Y and Z
on *Z .
\
\
Case II: c "!P l and c "0. *YK and ZK H are the OLS residuals from
W
\
the regression of *Y and ZH on *Z , where ZH "(ι , Z ).
\
\
\
2 \
Case III: c O0 and c "0. *YK and ZK H are the OLS residuals from the
\
regression of *Y and Z
on (ι , *Z ).
\
2
\
Case IV: c O0 and c "!P c. *YK and ZK H are the OLS residuals from
W
\
the regression of *Y and ZH on (ι , *Z ), where ZH "(s , Z ).
\
2
\
\
2 \
Case V: c O0 and c O0. *YK and ZK
are the OLS residuals from the
\
regression of *Y and ZH on (ι , s , *Z ).
\
2 2
\
4. Structural tests for cointegration and tests of speci5cation
Our interest in this section is "ve-fold. Firstly, Section 4.1 addresses
testing the null hypothesis of cointegration rank r, H of (3.8), against the
P
Pesaran and Shin (1999) provide a comprehensive treatment of the imposition of exactly- and
over-identifying (non-linear) restrictions on b when n"m and, thus, k"0. In principle, their
approach may be adapted for our problem; see Section 7.
304
M.H. Pesaran et al. / Journal of Econometrics 97 (2000) 293}343
alternative hypothesis
H : Rank[P ]"r#1, r"0,2, n!1,
P>
W
in the structural VECM (3.5). Secondly, Section 4.2 presents a test of the null
hypothesis of cointegration rank r, H above, r"0,2, n!1, against the
P
alternative hypothesis of stationarity; that is
H : Rank[P ]"n.
L
W
Tables 6(a)}(e) of Appendix B provide the relevant asymptotic critical values.
Thirdly, Section 4.3 discusses testing H of (3.8) in the presence of weakly
P
exogenous explanatory variables which are integrated of order zero. Fourthly,
testing whether an intercept should be present in Case II, that is, c "0 or
bl"0, or whether a trend should be present in Case IV, that is, the co-trending
restriction c "0 or bc"0, is considered in Section 4.4. Finally, Section 4.5 is
concerned with providing speci"cation tests for the various assumptions embodied in our approach. Proofs of the results in this section may be found in
Appendix A.
We now weaken the independent normal distributional assumption of Sections 2 and 3 on the error process +e ,
in (2.1) and, hence, on the structural
R R\
error process +u ,
in (3.2).
R R\
Assumption 4.1. The error process +e ,
is such that
R R\
(a) (i) E+e "+z ,R\ , Z ,"0, (ii) var+e "+z ,R\ , Z ,"X, X positive de"R R\G G R R\G G nite;
(b) (i) E+u "x , +z ,R\ , Z ,"0, (ii) var+u "x , +z ,R\ , Z ,"X , where
R R R\G G R R R\G G SS
u ,e !X X\e and X ,X !X X\X ;
R
WR
WV VV VR
SS
WW
WV VV VW
(c) sup E+""e ""Q,(R for some s'2.
R
R
Assumption 4.1(a) states that the error process +e ,
is a martingale
R R\
di!erence sequence with constant conditional variance; hence, +e ,
is an
R R\
uncorrelated process. Therefore, the VECM (3.1) represents a conditional model
for *z given +*z ,N\ and z , t"1, 2,2 . Assumption 4.1(b) is a linear
R
R\G G
R\
conditional mean condition; that is, under Assumption 4.1(b)(i),
E+e "x ,+z ,R\ , Z ,"X X\e which, together with Assumption 4.1(b)(ii),
WR R R\G G WV VV VR
also ensures that var+e "x ,+z ,R\ , Z ,"X . Therefore, under this assumpWR R R\G G SS
tion, (3.15) can still be interpreted as a conditional model for *y given
R
*x ,+*z ,N\ and z , t"1, 2,2. Hence, (3.15) remains appropriate for
R
R\G G
R\
conditional inference. Moreover, the error process +u ,
is also a martingale
R R\
See Harbo et al. (1998, Theorem 1, Appendix) for a statement and proof of Theorem 4.2 below
for Cases I, II and IV under the distributional assumption of Sections 2 and 3. This analysis may be
straightforwardly adapted for Theorem 4.1 below under the same assumption. Case III which
corresponds to their c"0 is stated in Harbo et al. (1998, Theorem 2).
M.H. Pesaran et al. / Journal of Econometrics 97 (2000) 293}343
305
di!erence process with constant conditional variance and is uncorrelated with
the +e ,
process. Thus, Assumptions 4.1(a) (ii) and 4.1(b) (ii) rule out any
VR R\
conditional heteroskedasticity. Assumption 4.1(c) is quite standard and, together with Assumption 4.1(a), is required for the multivariate invariance principle stated in (4.1) below; see Phillips and Solo (1992, Theorem 3.15(a), p. 983).
Assumption 4.1(b) together with Assumption 4.1(c) implies the multivariate
invariance principle (4.2) below. Assumption 4.1(c) embodies a slight strengthening of that in Phillips and Durlauf (1986, Theorem 2.1(d), p. 475) which together
with Assumption 4.1(a) "rstly allows an invariance principle to be stated in
terms of the +z , process itself as the VAR(p) form (2.1) together with
R R
Assumptions 2.1 and 2.2 yields j""C ""(R, where ""A"",[tr(AA)]; see
H H
Phillips and Solo (1992, Theorem 3.15(b), p. 983). Secondly, terms involving
stationary components are asymptotically negligible relative to those involving
components integrated of order one. See Phillips and Solo (1992, Theorem 3.16,
Remark 3.17(iii), p. 983). Note also that ""CH""(R, and ""C""(R, PhilH H
lips and Solo (1992, Lemma 2.1, p. 972), which excludes the cointegrating
relations (2.13) being fractionally integrated of positive order.
We de"ne the partial sum process
2?
SC (a),¹\ e ,
2
Q
Q
where [¹a] denotes the integer part of ¹a, a3[0,1]. Under Assumption 4.1,
SC (a) satis"es the multivariate invariance principle (Phillips and Durlauf, 1986,
2
Theorem 2.1, p. 475)
S (a)NB (a), a3[0, 1],
(4.1)
2
K
where B (.) denotes an m-dimensional Brownian motion with variance matrix
K
X. We partition SC (a)"(SW (a), SV (a)) conformably with z "(y , x ) and
2
2
2
R
R R
the Brownian motion B (a)"(B (a), B (a)) likewise, a3[0, 1]. De"ne
K
L
I
SS (a),¹\2?
u , a3[0,1]. Hence, as u ,e !X X\e ,
2
Q Q
R
WR
WV VV VR
SS (a)NBH(a),
(4.2)
2
L
where BH(a),B (a)!X X\B (a) is a Brownian motion with variance matrix
L
L
WV VV I
X which is independent of B (a), a3[0, 1]. Consequently, the results described
SS
I
in Harbo et al. (1998) remain valid under Assumption 4.1.
Under Assumption 3.1, the (m, m!r) matrix a ,diag(a,, a,), where a, is
,
W V
V
a (k, k) non-singular matrix, is a basis for the orthogonal complement of the (m, r)
loadings matrix a"(a , 0). Hence, we de"ne the (m!r)-dimensional standard
W
We are grateful to Peter Boswijk for helpful discussions on this result; a proof of this assertion is
available from the authors on request.
306
M.H. Pesaran et al. / Journal of Econometrics 97 (2000) 293}343
Brownian motion W
(a),(W (a), W (a)) partitioned into the (n!r)- and
K\P
L\P
I
k-dimensional sub-vector independent standard Brownian motions
W (a),(a,X a,)\a,BH(a) and W (a),(a,X a,)\a,B (a), a3[0, 1].
L\P
W SS W
W L
I
V VV V
V I
See Pesaran et al. (1997, Appendix A) for further details. We will also require the
corresponding de-meaned (m!r)-vector standard Brownian motion
W
(a) da,
(4.3)
K\P
and de-meaned and de-trended (m!r)-vector standard Brownian motion
WI
(a),W
(a)!
K\P
K\P
1
WK
(a),WI
(a)!12 a!
K\P
K\P
2
1
a! WI
(a) da,
K\P
2
(4.4)
and their respective partitioned counterparts WI
(a)"(WI
(a), WI (a)), and
K\P
L\P
I
WK
(a)"(WK
(a), WK (a)), a3[0, 1].
K\P
L\P
I
4.1. Testing H against H
P
P>
The (log-) likelihood ratio statistic for testing H : Rank[P ]"r against
P
W
H : Rank[P ]"r#1 is given by
P>
W
LR(H "H )"!¹ ln(1!jK
),
(4.5)
P P>
P>
where jK is the rth largest eigenvalue from the determinantal equation (3.22),
P
r"0,2, n!1, with the appropriate de"nitions of *YK and ZK H and, thus, the
\
sample moment matrices S , S and S given by (3.21) to cover Cases I}V.
77 78
88
Theorem 4.1 (Limit distribution of LR(H "H )). Under H dexned by (3.8)
P P>
P
and Assumptions 2.1, 2.2, 3.1 and 4.1, the limit distribution of LR(H "H ) of
P P>
(4.5) for testing H against H
is given by the distribution of the maximum
P
P>
eigenvalue of
dW (a) F
(a)
L\P
K\P
F
(a)F
(a) da
K\P
K\P
\ F
(a) dW (a),
K\P
L\P
(4.6)
where
W
(a)
Case I
K\P
(W
(a), 1)
Case II
K\P
F
(a)" WI
(a)
Case III, a3[0, 1],
K\P
K\P
(WI
(a), a!) Case I<
K\P
WK
(a)
Case <
K\P
r"0,2, n!1, where Cases I}< are dexned following (3.8).
M.H. Pesaran et al. / Journal of Econometrics 97 (2000) 293}343
307
4.2. Testing H against H
P
L
The (log-) likelihood ratio statistic for testing H : Rank[P ]"r against
P
W
H : Rank[P ]"n is given by
L
W
L
LR(H "H )"!¹ ln(1!jK ),
(4.7)
P L
G
GP>
where jK is the ith largest eigenvalue from the determinantal equation (3.22).
G
Theorem 4.2 (Limit distribution of LR(H "H )). Under H dexned by (3.8) and
P L
P
Assumptions 2.1, 2.2, 3.1 and 4.1, the limit distribution of LR(H "H ) of (4.7) for
P L
testing H against H is given by the distribution of
P
L
\
¹race
dW (a) F
(a)
F
(a)F
(a) da
L\P
K\P
K\P
K\P
(a) dW (r) ,
; F
K\P
L\P
where F
(a), a3[0, 1], is dexned in Theorem 4.1 for Cases I}<,
K\P
r"0,2, n!1.
4.3. Testing H in the presence of I(0) weakly exogenous regressors
P
The (log-) likelihood ratio tests described above require that the process
+x , is integrated of order one as noted below (3.7) and implied by the weak
R R
exogeneity Assumption 3.1; see (3.6). However, many applications will include
current and lagged values of weakly exogenous regressors which are integrated
of order zero as explanatory variables in (3.5). In such circumstances, Theorems
4.1 and 4.2 no longer apply with the limiting distributions of the (log-) likelihood
ratio tests for cointegration now being dependent on nuisance parameters.
However, the above analysis may easily be adapted to deal with this di$culty.
Let +w , denote a k -vector process of weakly exogenous explanatory
R R
U
variables which is integrated of order zero. Therefore, the partial sum vector
process +R w , is integrated of order one. De"ning R w as a subQ Q R
Q Q
vector of x with the corresponding subvector of *x as w , t"1, 2,2, allows
R
R
R
the above analysis to proceed unaltered. With these re-de"nitions of x and *x
R
R
to include R w and w , t"1, 2,2, respectively, the partial sum R w will
Q Q
R
Q Q
now appear in the cointegrating relations (2.13) and the lagged level term z
in
R\
(3.5) although economic theory may indicate its absence; that is, the corresponding (k , r) block of the cointegrating matrix b is null. This constraint on the
U
cointegrating matrix b is straightforwardly tested using a likelihood ratio
statistic which will possess a limiting chi-squared distribution with rk degrees
U
of freedom under H . See Rahbek and Mosconi (1999) for further discussion.
P
308
M.H. Pesaran et al. / Journal of Econometrics 97 (2000) 293}343
4.4. Testing the co-trending hypothesis
The cointegration rank hypothesis does not preclude the presence of deterministic trends in the cointegrating relations; see (2.13). In particular, Section
2 demonstrates that in order to preserve similar deterministic trending behavior
in the level process +z , for di!ering values of the cointegration rank r it is
R R
necessary to restrict the trend (intercept) coe$cients. Accordingly, in the context
of the VECM formulation (3.5), the trend parameter is rendered as c "!P c
W
(see (3.7)). In the absence of a deterministic trend in (2.1), c"0, then the intercept
parameter in the VECM formulation (3.5) is similarly restricted as c "!P l,
W
and the cointegrating relationship (2.13) has an intercept given by bl. Therefore,
when the cointegrating rank is r, out of the n trend (intercept) terms, r of them
can take any values and the remaining n!r terms must satisfy prior restrictions.
This sub-section is concerned with the (log-) likelihood ratio test for the
co-trending restriction, that is, the absence of a trend in the cointegrating
relationship (2.13); viz. bc"0 or equivalently c "0 in the VECM formulation
(3.5) for Case IV which yields Case III. In the absence of a trend, c"0 (Case II),
we also consider the likelihood ratio test for the absence of an intercept in the
cointegrating relationship (2.13); viz. bl"0 or equivalently c "0 in the
VECM formulation (3.5) which yields Case I. In both cases, the requisite ML
estimators are obtained under the cointegrating hypothesis H of (3.8). See
P
Johansen (1995, Theorem 11.3, p. 162) for similar results when n"m and Harbo
et al. (1998, Theorem 3) which states Theorem 4.3 below under the assumptions
of Sections 2 and 3.
The following theorems detail the limiting behavior of both likelihood ratio
tests.
Theorem 4.3 (Limit distribution of the likelihood ratio statistic for c "0 in
Case IV). In Case I<, under H dexned by (3.8), Assumptions 2.1, 2.2, 3.1 and
P
4.1, and c "0, the likelihood ratio statistic for c "0 has a limiting chi-squared
distribution with r degrees of freedom, r"1,2, n.
Similarly,
Theorem 4.4 (Limit distribution of the likelihood ratio statistic for c "0 in
Case II). In Case II, under H dexned by (3.8), Assumptions 2.1, 2.2, 3.1 and 4.1,
P
and c "0, the likelihood ratio statistic for c "0 has a limiting chi-squared
distribution with r degrees of freedom, r"1,2, n.
We also note that under H of (3.8) the tests described in this sub-section are
P
asymptotically independent of the corresponding cointegration rank tests for
H of Sections 4.1 and 4.2. Consequently, the overall asymptotic size of the
P
M.H. Pesaran et al. / Journal of Econometrics 97 (2000) 293}343
309
resultant induced test for H and the co-trending or intercept hypothesis of this
P
sub-section based on the above likelihood ratio statistics is simply calculated
from the asymptotic sizes of the individual tests.
4.5. Specixcation tests
The framework adopted in this paper has imposed certain assumptions which
may be overly restrictive. For completeness, this sub-section describes speci"cation tests for these assumptions.
Firstly, as discussed in Section 2, we require that the trend parameters
obey certain parametric restrictions in order that the deterministic trending
behavior of the level process +z , is invariant to the cointegrating rank r.
R R
In the absence of a deterministic trend, the intercept parameter vector is
similarly restricted. In order to test these restrictions, we may use likelihood
ratio tests calculated under the cointegration rank hypothesis H de"ned by
P
(3.8): Case IV against Case V in the "rst instance and Case III against Case II in
the second. See Johansen (1995, Corollary 11.2, p. 163) for the full system case
n"m.
Theorem 4.5 (Limit distribution of the likelihood ratio statistic for
c "!P c). In Case < (dexned by (3.13)), under H dexned by (3.8), Assump
W
P
tions 2.1, 2.2, 3.1 and 4.1, and c "!P c, the likelihood ratio statistic for
W
c "!P c has a limiting chi-squared distribution with n!r degrees of freedom,
W
r"0,2, n!1.
Similarly,
Theorem 4.6 (Limit distribution of the likelihood ratio statistic for
c "!P l). In Case III (dexned by (3.11)), under H dexned by (3.8), Assump
W
P
tions 2.1, 2.2, 3.1 and 4.1, and c "!P l, the likelihood ratio statistic for
W
c "!P l has a limiting chi-squared distribution with n!r degrees of freedom,
W
r"0,2, n!1.
Secondly, in Assumption 3.1, we imposed the weak exogeneity restriction
P "0; that is, the level process +x , is integrated of order one and longV
R R
run forcing for +y , . This restriction takes two forms. Firstly, as noted in
R R
Section 3, the level process +x , is not mutually cointegrated. Secondly, the
R R
lagged cointegrating relationship does not enter the evolution of the di!erenced
process +*x , in (3.6). Our discussion below is concerned with Case IV
R R
de"ned analogously to that of earlier sections, the remaining cases are de"ned
likewise and their associated results may be obtained by a similar analysis to
that for Case IV.
310
M.H. Pesaran et al. / Journal of Econometrics 97 (2000) 293}343
Firstly, consider the following sub-system regression model:
N\
*x "a # C *z #P xH #e ,
(4.8)
R
V
VG R\G
VVH R\
VR
G
where P H ,P (c , I ), P ,(P , P ), c,(c , c ) and xH ,(t, x ),
VV
VV V I
V
VW VV
W V
R\
R\
t"1, 2,2; cf. (3.6). Eq. (4.8) embodies the possibility that the level process
+x , is mutually cointegrated. We denote the cointegration rank hypothesis
R R
in (4.8) by HV : Rank[P ]"r, r"0,2, k. Therefore, similarly to Theorems 4.1
P
V
and 4.2 (cf. Johansen, 1991).
Theorem 4.7 (Limit distribution of LR(HV "HV )). Under H dexned in (3.8) and
P
Assumptions 2.1, 2.2, 3.1 and 4.1, the limit distribution of the likelihood ratio
statistic LR(HV "HV ) for testing HV against HV in (4.8) is the distribution of the
maximum eigenvalue of
where
dW (a) F (a)
I
I
F (a)F (a) da
I
I
\ F (a) dW (a),
I
I
W (a)
Case I
I
(W (a), 1)
Case II
I
F (a)" WI (a)
Case III, a3[0, 1],
I
I
(WI (a), a!) Case I<
I
WK (a)
Case <
I
and Cases I}< are dexned following (3.8).
Theorem 4.8 (Limit distribution of LR(HV "HV )). Under H dexned in (3.8) and
P
I
Assumptions 2.1, 2.2, 3.1 and 4.1, the limit distribution of the likelihood ratio
statistic LR(HV "HV) for testing HV against HV in (4.8) is the distribution of
I
I
\ dW (a) F (a)
F (a)F (a) da
F (a) dW (r) ,
¹race
I
I
I
I
I
I
where F (a), a3[0, 1], is dexned in Theorem 4.3 for Cases I}<.
I
Secondly, consider the sub-system regression model
N\
*x "a # C *z #a bK zH #e( ,
(4.9)
R
V
VG R\G
VW H R\
VR
G
t"1,2, ¹, where bK denotes the ML estimator for the cointegrating vector
H
b under H in Cases I}V of the earlier sections. Eq. (4.9) allows for the
H
P
M.H. Pesaran et al. / Journal of Econometrics 97 (2000) 293}343
311
alternative possibility that the lagged cointegrating relationships +b zH,
H R R
might enter (3.6).
Theorem 4.9 (Limit distribution of the likelihood ratio statistic for a "0). UnVW
der H dexned in (3.8) and Assumptions 2.1, 2.2, 3.1 and 4.1, the likelihood ratio
P
statistic for a "0 in (4.9) has a limiting chi-squared distribution with kr degrees
VW
of freedom for Cases I}<, r"1,2, n.
5. E7cient estimation of a structural error correction model subject to constraints
on the short-run dynamics
The structural VECM model described in Section 3 does not permit any
restrictions to be imposed on the short-run dynamics of the model. In many
applications, due to data limitations or a priori restrictions, researchers would
potentially wish to impose such restrictions on the VECM form (3.5). Furthermore, Abadir et al. (1999) suggest that the inclusion of irrelevant terms in
systems such as the VECM (3.5) may result in substantial estimator bias. As the
restrictions only concern parameter vectors and matrices associated with stationary variables, the results of the previous section continue to hold under
H of (3.8) as will become evident below.
P
As in earlier sections, our exposition is in terms of Case IV. Consider the
following linear constraints on the parameter matrix W associated with the
short-run dynamics in the VECM (3.18):
vec(W)"Su,
(5.1)
where S is a (nmp!n, s) matrix of full column rank s with known elements and
u an s-vector of unknown parameters, s)nmp!n. For example, consider the
jth equation of (3.18), revised to incorporate exclusion restrictions on the
short-run dynamics,
*y "c ι #uH SH *Z #p ZH #u
(5.2)
H
H 2
\
WHH \
H
where *y ,(*y ,2,*y ) and u ,(u ,2, u ), j"1,2, n, and
H
H
2H
H
H
2H
P "(p ,2, p ). The formulation (5.2) allows certain elements of the
WH
WH
WHL
stationary component *Z to be excluded from the jth equation and, thus, also
\
the lag lengths to di!er across equations. These constraints are summarized by
wH"SHuH, with SH an appropriately de"ned selection matrix, j"1,2, n, and
W"(w,2, wL). Stacking these n equations yields
*Y"c ι #W*Z #P ZH #U,
2
\
WH \
(5.3)
Boswijk (1995) provides a Lagrange multiplier test for a "0 in (4.9). See also Johansen (1992)
VW
and Harbo et al. (1998, Section 4.1).
312
M.H. Pesaran et al. / Journal of Econometrics 97 (2000) 293}343
where *Y,(*y ,2, *y ), subject to the constraints vec(W)"Su as in (5.1)
L
with S,diag(S,2, SL) and u"(u ,2, uL ); cf. (3.18). Note that the constraints (5.1) also permit the possibility of cross-equation restrictions.
Let u denote a ¹-consistent estimator for u in the seemingly unrelated
regressions (SUR) VECM (5.3) subject to the constraints vec(W)"Su of (5.1);
that is, ¹(u !u)"O (1) and, thus, ¹(WI !W)"O (1) where
.
.
vec(WI )"Su . For example, under Assumptions 2.1, 2.2, 3.1 and 4.1, a ¹consistent estimator u may be obtained using SUR estimation on (5.3) subject
to vec(W)"Su, with P unrestricted and X replaced by an initial ¹WH
SS
consistent estimator based on residuals computed from ordinary least squares
equation-by-equation estimation, once again treating P as unrestricted.
WH
The next step subtracts the estimated short-run dynamics from the left-hand side
of (5.3), and applies the reduced rank technique to the following synthetic
multivariate regression
(*Y!WI *Z )"c ι #P ZH #UI ,
(5.4)
\
2
WH \
where vec(WI )"Su and UI ,U!(WI !W)*Z .
\
For this purpose consider the minimization with respect to b under H of
H
P
(3.17) of the determinantal criterion function
"¹\*YI (I !ZI * b (b ZI * ZI * b )\b ZI H )*YI ",
2
\ H H \ \ H
H \
cf. (3.20), where *YI ,(*Y!WI *Z )PM ι and ZI H ,ZH PM ι are respectively the
\
\
\
de-meaned versions of *Y!WI *Z and ZH , PM ι ,I !ι (ι ι )\ι , which
\
\
2
2 2 2
2
results from (5.4) by concentrating out a after substitution of P "a b .
W
WH
W H
De"ning the moment matrices
SI ,¹\*YI *YI , SI ,¹\*YI ZI H , SI ,¹\ZI H ZI H ,
(5.5)
\
88
\ \
77
78
cf. (3.21), the solution bI to the above minimization problem is given by the
H
eigenvectors corresponding to the r largest eigenvalues jI '2'jI '0 of (cf.
P
(3.22))
"jI SI !SI SI \SI ""0.
(5.6)
88
87 77 78
The modi"cations of the de"nitions of *YI and ZI H and, thus, the sample
\
moment matrices SI , SI
and SI
of (5.5) necessary in the determinantal
77 78
88
equation (5.6) for the "ve cases of interest are given as follows.
A proof of ¹-consistency of the SUR estimator of u can be established along the following
lines. Using results in Park and Phillips (1989), in particular, Theorems 3.1 and 3.2, p. 102, and their
Sections 5.2 and 5.4, an estimator for X based on residuals from equation-by-equation ordinary
SS
least squares treating P as unrestricted is ¹-consistent; see Park and Phillips (1989, Theorem
WH
3.4, p. 107). Consequently, an adaptation of Park and Phillips (1989, Proof of Theorem 3.1, p. 126)
may be used to show that the resultant SUR estimator for u is ¹-consistent.
M.H. Pesaran et al. / Journal of Econometrics 97 (2000) 293}343
313
Case I: c "0 and c "0. *YI ,(*Y!WI *Z ) and ZI H ,Z
where
\
\
\
vec(WI )"Su and u is a ¹-consistent estimator for u of vec(W)"Su obtained
from *Y"W*Z #P Z #U.
\
W \
Case II: c "!P l and c "0. *YI ,(*Y!WI *Z ) and ZI H ,ZH where
W
\
\
\
vec(WI )"Su , u is a ¹-consistent estimator for u of vec(W)"Su obtained
from *Y"W*Z #P ZH #U and ZH "(ι , Z ).
\
WH \
\
2 \
Case III: c O0 and c "0. *YI ,(*Y!WI *Z )PM ι and ZI H ,Z PM ι are
\
\
\
respectively *Y!WI *Z and Z
de-meaned, PM ι ,I !ι (ι ι )\ι , where
2
2 2 2
2
\
\
vec(WI )"Su , u is a ¹-consistent estimator for u of vec(W)"Su obtained
from *Y"c ι #W*Z #P Z #U.
2
\
W \
Case IV: c O0 and c "!P c. *YI ,(*Y!WI *Z )PM ι and ZI H ,ZH PM ι
W
\
\
\
are respectively *Y!WI *Z and ZH de-meaned, where vec(WI )"Su , u is
\
\
a ¹-consistent estimator for u of vec(W)"Su obtained from
*Y"c ι #W*Z #P ZH #U and ZH "(s , Z ).
2
\
WH \
\
2 \
Case V: c O0 and c O0. *YI ,(*Y!WI *Z )PM ι and ZI H ,Z PM ι are,
\ O
\
\ O
respectively, *Y!WI *Z
and Z
de-meaned and de-trended,
\
\
PM ι s ,PM ι !PM ι s (s PM ι s )\s PM ι , where vec(WI )"Su , u is a ¹-consistent
2 2 2
2
estimator for u of vec(W)"Su obtained from *Y"c ι #c s
2
2
#W*Z #P Z #U.
\
W \
The estimated eigenvalues +jI ,L
from (5.6) may be then used to test
G G
H against H , cf. Theorem 4.1, and H against H , cf. Theorem 4.2.
P
P>
P
L
"
"
Theorem 5.1 (Limit distributions of LR (H "H ) and LR (H "H )). If u is
P P>
P L
a ¹-consistent estimator for u, then, under H dexned by (3.8) and Assumptions
P
2.1, 2.2, 3.1
"and 4.1, the limit distributions of the pseudo-(log-)likelihood ratio
LR
statistics
(H "H ),!¹ ln(1!jI
) for testing H against H
and
"
P P>
P>
P
P>
LR (H "H ),!¹L
ln (1!jI ) for testing H against H are identical to
P L
GP>
G
P
L
those of LR(H "H ) of (4.5) and LR(H "H ) of (4.7) stated in Theorems 4.1 and
P P>
P L
4.2 respectively for Cases I}<, r"0,2, n!1.
For Case IV, let bI denote the (m#1, r) matrix of just-identixed eigenvectors
H
under H of (3.8) corresponding to the eigenvalues jI , i"1,2, r, of (5.6).
P
G
Consider the multivariate regression
*Y"c ι #W*Z #a bI ZH #UI ,
2
\
W H \
(5.7)
subject to the constraints vec(W)"Su of (5.1); cf. (5.3). Asymptotically, the
inclusion of the lagged estimated cointegrating relationships bI zH is the same
H R\
as the inclusion of the actual cointegrating relationships b zH , t"1,2, ¹.
H R\
Therefore, as the regression contains only variables which are asymptotically
stationary, SUR estimation of (5.7) will be asymptotically e$cient for the
short-run parameters u and the (n, r) loadings matrix a . The modi"cations of
W
(5.7) required for Cases I}III and V are obvious and are therefore omitted.
314
M.H. Pesaran et al. / Journal of Econometrics 97 (2000) 293}343
6. Finite sample properties
This section provides some preliminary evidence on the "nite sample properties of the Case IV cointegration rank statistics developed in the previous
sections using Monte Carlo techniques. The experimental design is an extension
of that used by Yamada and Toda (1998), who consider a VAR(2) model with
2 endogenous I(1) variables, that is, y "(y , y ). Their VAR(2) model is
R
R R
augmented by a scalar exogenous I(1) variable x , which results in the following
R
DGP:
(I !A¸)(I !B¸)z "e ,
R
R
where z "(y , x ),
R
R R
a
0
a
A" a
a
a , B"
0
0
1
(6.1)
0
b
b
0
0
0
0
0 ,
(6.2)
0
and e &IN(0, X). An advantage of this formulation is that the integration and
R
cointegration properties of the system (6.1) are controlled via a and a if we
assume "b b "(1. To ensure that the cointegration rank is unity, that is,
Rank[P]"1, we must have either a "1 or a "1 (but not both) and, for
reasons which will become clear below, we additionally require
a "!a(1!a ) and a "aa . As all of the Case IV cointegration rank
statistics considered below are invariant to the intercept and trend parameter
vectors l and c, we have set l"0 and c"0 in (6.1).
Consequently, the structural VECM (3.5) is given by
*y "k *x #W *z #P z #u ,
(6.3)
R
R
R\
W R\
R
where u &IN(0, X ),
R
SS
j
0
a b
0
k" ,
W "
,
j
a b
a b
0
and, for a cointegration rank of unity Rank[P ]"1, either, if a "1 and
W
"a "(1:
0
P "
(a #b [1!a ], ![1!a ]!a b , aa ),
W
1 or, if a "1 and "a "(1:
1!a
(1, !b , a).
P "!
W
!a
M.H. Pesaran et al. / Journal of Econometrics 97 (2000) 293}343
315
Because the cointegration rank statistics of earlier sections are invariant to scale,
we may set
1
X "
SS
o
o
1
and var(e )"1 yielding j and j as the covariances of y and y with x ,
VR
R
R
R
respectively.
The above DGP, characterized via k, W , P and X , contains 9 free
W
SS
parameters. It is clearly beyond the scope of the present paper to examine all the
possible parameter con"gurations that this design entails. Hence, for illustrative
purposes, in what follows we "x a "0.1, b "0.2, b "!0.2, j "0.3 and
j "!0.2. The remaining parameter values are chosen to ensure that the
cointegrating rank is unity, Rank[P ]"1, and that the population R for each
W
of the structural VECM equations falls in the range 0.10}0.30, which re#ects the
sort of values generally encountered in practice. A summary of the parameter values considered and the implied population R is given below:
Experiments: Set I
Experiments: Set II
Parameters
E
?
E
@
E
A
E
B
E
?
E
@
E
A
E
B
a
a
a
o
1.0
0.8
0
0
1.0
0.8
0
0.5
1.0
0.8
1
0
1.0
0.8
1
0.5
0.8
1.0
0
0
0.8
1.0
0
0.5
0.8
1.0
1
0
0.8
1.0
1
0.5
0.121
0.163
0.121
0.152
0.123
0.198
0.122
0.188
0.198
0.087
0.172
0.083
0.290
0.138
0.270
0.134
Implied R
R
WR
R
WR
Population R values of the structural VECM equations are obtained as
Var(*y "*x , *z , z )
GR
R
R\ R\
R GR "1!
W
Var(*y )
GR
1
"1!
, i"1,2.
Var(*y )
GR
The unconditional variances Var(*y ), i"1, 2, are given by the "rst two diagonal elements of the
GR
following matrix:
Cov(*z )" C XC ;
R
H H
H
see below (2.9) and fn. 3.
Other experiments with slightly larger values for these population R were also conducted and
yielded similar results to those described below.
316
M.H. Pesaran et al. / Journal of Econometrics 97 (2000) 293}343
For each of these eight experiments we computed three sets of Case IV (log-)
likelihood ratio statistics for testing H against H
and H :
P
P>
L
(i)
¸R1: the statistics due to Johansen (1991) which treat all three variables in
z as endogenous;
R
(ii) ¸R2: the statistics developed in Sections 4.1 and 4.2 which correctly treat
x as an exogenous I(1) variable;
R
(iii) ¸R3: the statistics of Section 5 which also incorporate the zero restrictions
on the short-run dynamics in the matrix W which is estimated by
equation-by-equation ordinary least squares treating P as unrestricted.
W
Sample sizes ¹"30, 50, 75, 100, 150 and 300 are considered. The nominal size
of each of the tests is set at 0.05 and the number of replications at 20,000.
The rejection frequencies for testing H : Rank(P )"r against H
and H ,
P
W
P>
L
r"0, 1, are summarized in Tables 1(a)}1(d) for the experiments in Set I. The
results in Table 1(a) for Experiment E show that all tests tend to be under?
sized when ¹)150. Because the size distortion is fairly uniform across the tests,
it is possible to directly compare their power properties. The power of the ¸R3
tests of Section 5 is signi"cantly higher than their ¸R1 and ¸R2 counterparts
with the power of the ¸R2 tests of Sections 4.1 and 4.2 being slightly better than
the ¸R1 tests of Johansen (1991) in most cases. There seems little to choose
between the ¸R3 tests for H against H
and H although the former test is
P
P>
L
slightly more powerful for ¹'50. Similar conclusions may be drawn from
Experiment E summarized in Table 1(b).
@
In Experiment E , see Table 1(c), all tests are slightly over-sized for ¹*150.
A
The ¸R3 tests still perform best in terms of power with the ¸R2 test also slightly
more powerful than the ¸R1 tests. In relative terms, the results are similar to
those in Table 1(a) but the powers of the tests are now signi"cantly greater than
those obtained for Experiment E . Consequently, a non-zero value for a (which
?
results in a higher value of the population R GR , i"1, 2) may improve the "nite
W
sample power performance of the tests. As above, there is little di!erence
between the results for Experiments E and E in Tables 1(c) and 1(d), respecA
B
tively, indicating that the structural error correlation parameter o has little e!ect
on the "nite sample performance of these tests for the experiments in Set I.
Tables 2(a)}2(d) summarize the experiments in Set II. The results in Table 2(a)
for Experiment E are very similar to those in Tables 1(a) and 1(b). However,
?
Table 2(b) for Experiment E indicates that as ¹ increases all tests, from being
@
under-sized, become slightly over-sized and then approach nominal size. Unlike
the experiments in Set I, the non-zero value of o, although it does not result in
a higher value for the population R GR , i"1, 2, improves the power of all tests
W
signi"cantly.
Similar results for Cases I}V to those for Case IV given below were also found and are available
from the authors on request.
M.H. Pesaran et al. / Journal of Econometrics 97 (2000) 293}343
317
Table 1(a)
Rejection frequencies for experiment E
?
¸R1
¸R2
¸R3
H
P
H
L
H
P>
H
L
H
P>
H
L
H
P>
30
r"0
r"1
0.204
0.024
0.166
0.015
0.196
0.020
0.172
0.020
0.265
0.024
0.263
0.024
50
r"0
r"1
0.151
0.017
0.123
0.009
0.164
0.015
0.140
0.015
0.254
0.019
0.254
0.019
75
r"0
r"1
0.199
0.020
0.159
0.013
0.226
0.022
0.206
0.022
0.345
0.025
0.352
0.025
100
r"0
r"1
0.280
0.029
0.248
0.019
0.342
0.031
0.326
0.031
0.475
0.034
0.492
0.034
150
r"0
r"1
0.418
0.033
0.430
0.031
0.524
0.040
0.549
0.040
0.723
0.043
0.764
0.043
300
r"0
r"1
0.957
0.050
0.988
0.051
0.988
0.053
0.999
0.053
0.997
0.053
0.999
0.053
¹
Table 1(b)
Rejection frequencies for experiment E
@
¸R1
¸R2
¸R3
H
P
H
L
H
P>
H
L
H
P>
H
L
H
P>
30
r"0
r"1
0.193
0.021
0.164
0.014
0.184
0.018
0.165
0.018
0.304
0.025
0.309
0.025
50
r"0
r"1
0.135
0.013
0.112
0.008
0.144
0.012
0.125
0.012
0.280
0.020
0.284
0.020
75
r"0
r"1
0.162
0.017
0.133
0.010
0.187
0.017
0.169
0.017
0.350
0.025
0.356
0.025
100
r"0
r"1
0.227
0.022
0.196
0.016
0.278
0.025
0.257
0.025
0.461
0.032
0.478
0.032
150
r"0
r"1
0.419
0.036
0.429
0.035
0.519
0.036
0.547
0.036
0.724
0.042
0.768
0.042
300
r"0
r"1
0.958
0.048
0.991
0.047
0.989
0.049
0.998
0.049
0.997
0.050
1.00
0.050
¹
318
M.H. Pesaran et al. / Journal of Econometrics 97 (2000) 293}343
Table 1(c)
Rejection frequencies for experiment E
A
¸R1
¸R2
¸R3
H
P
H
L
H
P>
H
L
H
P>
H
L
H
P>
30
r"0
r"1
0.231
0.028
0.182
0.019
0.221
0.024
0.192
0.024
0.291
0.029
0.279
0.029
50
r"0
r"1
0.203
0.024
0.159
0.015
0.221
0.022
0.188
0.022
0.323
0.030
0.308
0.030
75
r"0
r"1
0.299
0.033
0.257
0.024
0.351
0.034
0.329
0.034
0.486
0.044
0.490
0.044
100
r"0
r"1
0.443
0.044
0.431
0.036
0.535
0.047
0.541
0.047
0.680
0.055
0.707
0.055
150
r"0
r"1
0.763
0.054
0.824
0.053
0.858
0.058
0.896
0.058
0.934
0.060
0.959
0.060
300
r"0
r"1
1.00
0.056
1.00
0.055
1.00
0.057
1.00
0.057
1.00
0.060
1.00
0.060
¹
Table 1(d)
Rejection frequencies for experiment E
B
¸R1
¸R2
¸R3
H
P
H
L
H
P>
H
L
H
P>
H
L
H
P>
30
r"0
r"1
0.226
0.027
0.182
0.017
0.219
0.022
0.188
0.022
0.304
0.025
0.309
0.025
50
r"0
r"1
0.198
0.023
0.157
0.015
0.220
0.022
0.188
0.022
0.351
0.033
0.339
0.033
75
r"0
r"1
0.289
0.033
0.252
0.023
0.345
0.036
0.322
0.036
0.515
0.047
0.521
0.047
100
r"0
r"1
0.433
0.041
0.422
0.033
0.526
0.046
0.529
0.046
0.707
0.060
0.734
0.060
150
r"0
r"1
0.751
0.054
0.811
0.052
0.848
0.057
0.888
0.057
0.946
0.063
0.968
0.063
300
r"0
r"1
1.00
0.054
1.00
0.056
1.00
0.057
1.00
0.057
1.00
0.060
1.00
0.060
¹
M.H. Pesaran et al. / Journal of Econometrics 97 (2000) 293}343
319
Table 2(a)
Rejection frequencies for experiment E
?
¸R1
¸R2
¸R3
H
P
H
L
H
P>
H
L
H
P>
H
L
H
P>
30
r"0
r"1
0.202
0.022
0.163
0.015
0.191
0.018
0.169
0.018
0.227
0.021
0.216
0.021
50
r"0
r"1
0.156
0.015
0.128
0.009
0.166
0.014
0.146
0.014
0.217
0.019
0.201
0.019
75
r"0
r"1
0.197
0.021
0.164
0.012
0.233
0.022
0.210
0.022
0.295
0.028
0.278
0.029
100
r"0
r"1
0.276
0.027
0.251
0.022
0.340
0.029
0.332
0.029
0.424
0.038
0.425
0.038
150
r"0
r"1
0.521
0.041
0.554
0.036
0.642
0.043
0.681
0.043
0.747
0.050
0.796
0.050
300
r"0
r"1
0.986
0.054
0.998
0.054
0.998
0.053
1.00
0.053
1.00
0.056
1.00
0.056
¹
Table 2(b)
Rejection frequencies for experiment E
@
¸R1
¸R2
¸R3
H
P
H
L
H
P>
H
L
H
P>
H
L
H
P>
30
r"0
r"1
0.258
0.035
0.197
0.023
0.247
0.028
0.211
0.028
0.276
0.030
0.251
0.030
50
r"0
r"1
0.262
0.032
0.210
0.021
0.291
0.030
0.254
0.030
0.349
0.038
0.323
0.038
75
r"0
r"1
0.408
0.047
0.377
0.034
0.474
0.046
0.469
0.046
0.559
0.054
0.559
0.054
100
r"0
r"1
0.597
0.055
0.616
0.049
0.691
0.055
0.718
0.055
0.773
0.066
0.804
0.066
150
r"0
r"1
0.897
0.060
0.942
0.057
0.951
0.057
0.972
0.057
0.977
0.060
0.989
0.060
300
r"0
r"1
1.00
0.060
1.00
0.060
1.00
0.057
1.00
0.057
1.00
0.059
1.00
0.059
¹
320
M.H. Pesaran et al. / Journal of Econometrics 97 (2000) 293}343
Table 2(c)
Rejection frequencies for experiment E
A
¸R1
¸R2
¸R3
H
P
H
L
H
P>
H
L
H
P>
H
L
H
P>
30
r"0
r"1
0.285
0.044
0.209
0.026
0.274
0.033
0.229
0.033
0.348
0.057
0.298
0.057
50
r"0
r"1
0.324
0.044
0.261
0.030
0.364
0.042
0.325
0.042
0.502
0.077
0.465
0.077
75
r"0
r"1
0.524
0.058
0.501
0.047
0.600
0.061
0.600
0.061
0.768
0.096
0.781
0.096
100
r"0
r"1
0.735
0.066
0.769
0.062
0.814
0.065
0.842
0.065
0.932
0.094
0.948
0.094
150
r"0
r"1
0.965
0.064
0.986
0.060
0.984
0.060
0.993
0.060
0.998
0.079
0.999
0.079
300
r"0
r"1
1.00
0.061
1.00
0.060
1.00
0.058
1.00
0.058
1.00
0.066
1.00
0.066
¹
Table 2(d)
Rejection frequencies for experiment E
B
¸R1
¸R2
¸R3
H
P
H
L
H
P>
H
L
H
P>
H
L
H
P>
30
r"0
r"1
0.392
0.071
0.288
0.044
0.388
0.056
0.328
0.056
0.496
0.093
0.435
0.093
50
r"0
r"1
0.539
0.074
0.490
0.057
0.595
0.072
0.579
0.072
0.770
0.119
0.763
0.119
75
r"0
r"1
0.814
0.085
0.837
0.073
0.875
0.080
0.893
0.080
0.964
0.113
0.972
0.113
100
r"0
r"1
0.958
0.082
0.978
0.078
0.978
0.078
0.987
0.078
0.997
0.103
0.998
0.103
150
r"0
r"1
1.00
0.068
1.00
0.066
1.00
0.066
1.00
0.066
1.00
0.079
1.00
0.079
300
r"0
r"1
1.00
0.063
1.00
0.061
1.00
0.061
1.00
0.061
1.00
0.068
1.00
0.068
¹
M.H. Pesaran et al. / Journal of Econometrics 97 (2000) 293}343
321
Table 2(c) for Experiment E demonstrates that the under-size problem of
A
previous experiments is not a general feature of these tests in small samples.
Both ¸R3 tests are now over-sized with the ¸R1 and ¸R2 tests oversized for
¹)100, for example, the ¸R3 tests have empirical size 0.096 for ¹"75.
Consequently, power comparisons of the di!erent tests are problematic and the
signi"cantly large power of the ¸R3 tests over the others' should be discounted.
Subject to this caveat, the comparisons between the tests are quite similar to
those reported earlier. Table 2(d) for Experiment E reveals similar conclusions.
B
However, the results for the experiments in Set II tend to be more sensitive to
non-zero-values of a and o than those for those in Set I.
In summary, the above results suggest that all tests perform reasonably
satisfactorily in most of the cases considered. Albeit quite slowly, the empirical
sizes of all tests tend to the nominal level as the sample size increases. Furthermore, the ¸R3 tests of Section 5 are more powerful than the other tests although
there are situations where these tests tend to over-reject in small samples.
7. An empirical application: A re-examination of the long-run validity of the PPP
and UIP hypotheses
In this section we re-examine the empirical evidence on the long-run validity
of the Purchasing Power Parity (PPP) and Uncovered Interest Parity (UIP)
hypotheses presented in Johansen and Juselius (1992) and Pesaran and Shin
(1996). In these applications domestic and foreign prices and interest rates are
assumed to be endogenously determined. Such a symmetric treatment of domestic and foreign variables does not, however, seem to be necessary in the case of
small open economies where it is unlikely that changes in domestic variables
have a signi"cant impact on the long-run evolution of foreign (world) prices or
interest rates. This earlier analysis also included changes in the logarithm of oil
prices and its lagged values as exogenous I(0) variables in the underlying
cointegrating VAR model. However, as discussed in Section 4.3, the appropriate
method of allowing for such e!ects is to include an integrated version of the I(0)
variables in the model which, in the context of the present application, implies
adding the logarithm of oil prices to the list of I(1) variables, and then testing the
validity of excluding the level of oil prices from the cointegrating relations.
For comparability purposes, we use the same quarterly observations employed by Johansen and Juselius. The data set relates to the U.K. economy and
covers the period 1972(1)}1987(2). The included I(1) variables are: the logarithm
of the e!ective exchange rate, e , the logarithm of the U.K. wholesale price index,
R
p , the logarithm of the trade-weighted foreign wholesale price index, pH, the
R
R
logarithm of oil prices, po , and the domestic and foreign interest rate variables,
R
r "ln (1#R /100) and rH"ln (1#RH/100), where R is the U.K. three-month
R
R
R
R
R
Treasury Bill rate, and RH is the three-month Eurodollar rate. Here we carry out
R
322
M.H. Pesaran et al. / Journal of Econometrics 97 (2000) 293}343
a cointegration analysis on the six I(1) variables, e , p , r , rH, pH, and po , treating
R R R R R
R
the last two variables as weakly exogenous and, thus, long-run forcing, in the
sense described in Section 3. We also considered treating the foreign interest rate
variable rH as exogenous but, given the importance of the U.K. in world
R
"nancial markets, it was felt that this was unlikely to be justi"ed.
Consequently, the appropriate framework for this application is the model
de"ned by (3.5) and (3.6). Following Johansen and Juselius (1992) we set p"2
but did not include seasonal dummies in the model. The seasonal e!ects were
only marginally signi"cant and given the limited sample size available it was
thought best to leave them out. With the foreign price and oil price variables,
pH and po , treated as I(1) and exogenous, we estimated the following system of
R
R
equations over the period 1972(3)}1987(2):
*y "c #(!P c)t#K *x #W *z #P z #u ,
R
W
R
R\
W R\
R
*x "a #C *z #e ,
(7.1)
R
V
V R\
VR
where y "(p , e , r , rH), x "(pH, po ), z "(y , x ), P and W are each (4, 6)
R
R R R R
R
R
R
R
R R
W
coe$cient matrices, and K a (4,2) coe$cient matrix. Note that in the above
speci"cation the trend coe$cients (!P c) are restricted as in Case IV de"ned
W
by (3.12) and, hence, the level of y will exhibit linear deterministic trends for all
R
values of the cointegration rank r"Rank(P ). These restrictions on the trend
W
coe$cients can be tested using standard chi-squared tests set out in Theorem
4.5. The (log-) likelihood ratio statistics for testing the trend restrictions (Case
IV) against the unrestricted trend speci"cation (Case V) for r"0, 1, 2, 3 are
7.61[9.49], 7.26[7.81], 4.73[5.99], and 2.24[3.84], respectively. The "gures in [.]
are the 0.05 critical values of the chi-squared distribution with 4!r degrees of
freedom. Therefore, irrespective of the value of r the trend restrictions cannot be
rejected.
Table 3 reports the cointegration rank test statistics de"ned by LR(H "H )
P L
and LR(H "H ) of (4.7) and (4.5) respectively, together with the correspondP P>
ing asymptotic critical values at the 0.05 and 0.10 signi"cance levels reproduced
from Table 6(d) with k"2 and n!r"4, 3, 2, 1.
Neither statistic rejects the hypothesis of no cointegration at the 0.05 signi"cance level but LR(H " H ) rejects the null of no cointegration at the 0.10
P L
level. If we con"ned our analysis solely to these statistical tests, we could only
conclude that there is at most one cointegrating relation between the six I(1)
variables under investigation.
These results are based on the assumption that pH ard po are I(1) and non-cointegrated.
However, it is possible to test this assumption (see Theorems 4.7 and 4.8). Using an unrestricted
trend VAR(3) model in these exogenous variables, augmented with two lagged changes in the
endogenous variables we could not reject the hypothesis that pH and po are non-cointegrated. The
choice of the order of the VAR in this application was based on the Schwarz Bayesian Criterion and
the choice of trend speci"cation was based on the speci"cation test of Theorem 4.5.
M.H. Pesaran et al. / Journal of Econometrics 97 (2000) 293}343
323
Table 3
Cointegration rank statistics
LR(H "H )
P L
LR(H "H )
P P>
H
P
Statistic
0.05 CV
0.10 CV
Statistic
0.05 CV
0.10 CV
r"0
r"1
r"2
r"3
80.71
46.62
21.03
6.93
81.20
56.43
35.37
18.08
76.68
52.71
32.51
15.82
34.09
25.59
14.10
6.93
37.85
31.68
24.88
18.08
35.04
29.00
22.53
15.82
However, economic theory suggests the existence of two long-run (cointegrating) relations in the above system, namely the PPP and UIP relations de"ned as
p !e !pH and r !rH, respectively. Since in this application the sample size is
R
R
R
R
R
relatively small (¹"60) and the dimension of the system is relatively large
(m"6), the evidence against the hypothesis of two cointegrating relations does
not seem to be particularly strong. Moreover, the Monte Carlo simulation
results of Section 6 indicate that these cointegrating rank test statistics generally
tend to under-reject in small samples. We therefore examine the case in which
restrictions on short-run dynamics are available and are imposed in the construction of the (log-) likelihood ratio tests of the cointegrating rank restrictions;
see Section 5. A preliminary analysis of the regressions for *p , *e , *r and *rH
R
R
R
R
suggest the following 24 zero restrictions on the short-run coe$cients:
K"
0 *
0 0
0 0
0 0
,
* 0 0 0 0 0
0 * * 0 0 *
W "
,
0 0 * 0 0 0
0 0 0 * * 0
(7.2)
where unrestricted coe$cients are denoted by *. The maximized log-likelihood
values of the restricted and the unrestricted model together with the associated
Akaike Information Criterion (AIC) and the Schwarz Bayesian Criterion (SBC)
are given by
LL
AIC
SBC
Unrestricted
Restricted
725.01
661.01
593.99
716.49
676.49
634.61
It is clear that the 24 restrictions are not rejected and that both AIC and SBC
also select the restricted model.
324
M.H. Pesaran et al. / Journal of Econometrics 97 (2000) 293}343
Table 4
Cointegration rank statistics for short-run restricted model
"
LR (H "H )
P L
"
LR (H "H )
P P>
H
Statistic
0.05 CV
0.10 CV
Statistic
0.05 CV
0.10 CV
r"0
r"1
r"2
r"3
139.62
63.89
26.80
7.21
81.20
56.43
35.37
18.08
76.68
52.71
32.51
15.82
75.73
37.08
19.60
7.21
37.85
31.68
24.88
18.08
35.04
29.00
22.53
15.82
"
Table 4 reports the cointegration rank statistics LR (H "H ) and
"
P L
LR (H "H ) of Section 5 based on the above restricted model.
P P>
These statistics do not reject the null hypothesis of r"2 which is in line with
the prediction of economic theory. Consequently, we proceed as if there are two
co-integrating relations. We return subsequently to re-consider the case of
cointegrating rank r"1 to see the extent to which our conclusions based on
r"2 regarding the long-run validity of the PPP hypothesis are a!ected by this
choice of r.
We now examine the validity of the PPP and UIP hypotheses using the
long-run structural modelling techniques advanced in Pesaran and Shin (1999).
This approach enables us to test the validity of these hypotheses and to identify
factors that might be responsible for their possible breakdown. Denote the two
cointegrating vectors comprising bH"(!c, I )b associated with
K
zH"(t, p , e , r , rH, pH, po ) by bH"(bH , b , b , b , b , b , b ) and
R
R R R R R
R
bH"(bH , b , b , b , b , b , b ) respectively viewing bH as explaining
domestic prices and bH as explaining the domestic interest rate. Exact identi"ca
tion of these vectors requires the imposition of two restrictions per vector. We
chose the following exactly identifying constraints.
H :
#
1 * * 0 * *
,
b " *
H
0
*
* 1 * * *
which yielded the estimate
bK "
H#
0.0195 1
(0.0229)
!0.0062 0
(0.0038)
¸¸ "714.50,
#
!0.701
!6.005 0
(0.345)
(2.088)
!0.091 1
!0.791
(0.095)
(0.326)
!2.564
(1.943)
0.493
(0.334)
0.206
(0.274)
,
!0.054
(0.046)
M.H. Pesaran et al. / Journal of Econometrics 97 (2000) 293}343
325
where ¸¸ is the maximized value of the log-likelihood function for the just#
identi"ed case. Asymptotic standard errors are given in parentheses.
A number of hypotheses of interest may now be tested using the above exactly
identi"ed model. Consider "rstly the co-trending hypothesis H , namely that
the trend coe$cients are zero in the two cointegrating relations:
0 1 * * 0 * *
b "
,
H
0 0 * 1 * * *
H :
Under H , we obtained the estimate
0 1
bK " 0 0
H
!0.881
(0.333)
!0.037
(0.101)
!5.511
(1.748)
1
0
!0.854
!(0.280)
!0.929
!0.067
(0.414)
(0.053)
!0.025
(0.048)
0.023 ,
(0.015)
¸¸ "713.30.
Therefore, the log-likelihood ratio statistic for testing the co-trending hypothesis
is equal to 2(714.50}713.30)"2.40, which is below the 0.05 critical value of the
chi-squared distribution with 2 degrees of freedom. Hence, the co-trending
hypothesis is not rejected.
We now consider the hypothesis that level of oil prices do not enter the
cointegrating relations which we denote by H . To save space, we only report
the test of this hypothesis jointly with the co-trending hypothesis, H 5H ,
which yields
0 1
bK "
H
0 0
!0.834
(0.327)
!0.062
(0.129)
!5.646
(1.850)
1
0
!0.803
(0.432)
!0.925 0
(0.227)
,
!0.005 0
(0.063)
¸¸
"711.49.
All the computations reported in this section are carried out using Microxt 4.0. See Pesaran and
Pesaran (1997).
Recall that the primary motivation behind the inclusion of oil prices in this model is to capture
the short-run e!ects of changes in oil prices on the PPP and the UIP relations.
326
M.H. Pesaran et al. / Journal of Econometrics 97 (2000) 293}343
The (log-) likelihood ratio statistic associated with these 4 over-identifying
restrictions is 6.01 which, thus, does not reject H 5H ; similar results obtain
for the individual test of H undertaken separately from H . Accordingly, in
what follows we carry out tests of the PPP and the UIP hypotheses given
H 5H .
Under the UIP hypothesis and H 5H :
0 1 * *
0 * 0
H : b "
,
3'.
H
0 0 0 1 !1 0 0
the estimate of which is
bK "
H3'.
0 1
0 0
!0.965
(0.281)
0
!5.132
0
(1.257)
1
!1
!0.943 0
(0.167)
,
0
0
¸¸ "709.61.
3'.
Therefore, the (log-) likelihood ratio statistic for testing the UIP hypotheses is
9.78 which is well below the 0.05 critical value of the chi-squared distribution
with 7 degrees of freedom. Thus, the hypothesis of UIP (jointly with H 5H )
is not rejected even at the 0.01 level.
Next, testing the PPP hypothesis given H 5H :
0 1 !1 0 0 !1 0
,
H : b "
...
H
0 0
* 1 *
* 0
we obtained the value of 33.37 for the (log-) likelihood ratio statistic which is
well above the 0.05 critical value of the chi-squared distribution with 7 degrees
of freedom. Therefore, the PPP hypothesis considered jointly with H 5H is
strongly rejected.
However, as can be easily deduced from the above exactly-identi"ed estimation results, the rejection of the PPP hypothesis is mainly due to the statistically
signi"cant e!ect of domestic interest rates on the real exchange rate (de"ned by
e #pH!p ). To see this, consider the following modi"ed version of the PPP
R
R
R
hypothesis (denoted by H H ) that allows the e!ect of domestic interest rate on
...
Notice that the number of over-identifying restrictions in this case is equal to 7("11}4).
M.H. Pesaran et al. / Journal of Econometrics 97 (2000) 293}343
327
the real exchange rate to be unrestricted:
0 1 !1 * 0 !1 0
H H : b "
.
...
H
0 0
* 1 *
* 0
Under these restrictions, we obtained the following estimate:
0 1
bK H " 0 0
H...
!1
!0.022
(0.097)
!5.363
(0.876)
1
0
!1
0
!0.778 !0.002 0 ,
(0.325)
(0.045)
¸¸ H "709.81.
...
The (log-) likelihood ratio statistics for testing H H is 9.37 which is below
...
the 0.05 critical value of the chi-squared distribution with 6 degrees of
freedom. Thus, the main cause of the breakdown of the PPP hypothesis in
the present application seems to be the existence of a statistically signi"cant
negative relation between the real exchange rate and the domestic interest
rate.
Finally, we estimated the cointegrating relations under the modi"ed PPP
and the UIP restrictions (jointly with H 5H ) yielding the following
estimate:
bK H
H... 3'.
0 1 !1
0
" 0 0
¸¸ H "709.03.
... 3'.
!4.775
0 !1 0
(0.627) !1
0 0 ,
1
(7.3)
The log-likelihood ratio statistic is now 10.93 which is well below the 0.05
critical value of the chi-squared distribution with 9 degrees of freedom. Hence,
we are unable to reject the modi"ed PPP and the UIP hypotheses.
Similar results also follow if instead of the domestic interest rate the coe$cient of the foreign
interest rate variable is left unrestricted.
328
M.H. Pesaran et al. / Journal of Econometrics 97 (2000) 293}343
Table 5
Estimates of the error correction coe$cients and diagnostic statistics
Equation
a
W
a
W
RM *p
R
!0.062
!0.063
0.784
(0.016)
(0.066)
0.124
0.049
(0.070)
(0.285)
*e
R
*r
R
*rH
R
0.044
!0.036
(0.025)
(0.103)
0.075
0.421
(0.028)
(0.114)
0.154
0.070
0.172
s (4)
1!
s (1)
$$
s (2)
,
s (1)
&#
3.35
3.31
3.15
0.02
[0.50]
[0.07]
[0.21]
[0.88]
0.76
0.72
0.13
0.06
[0.94]
[0.40]
[0.94]
[0.81]
1.87
1.35
5.02
0.13
[0.76]
[0.25]
[0.08]
[0.72]
9.75
1.35
29.32
0.023
[0.045]
[0.25]
[0.00]
[0.88]
Conditional on the above long-run estimates we have the following expressions for the error correction terms:
bK z "p !e !pH !4.775r ,
R\
R\
R\
R\
R\
bK z "r !rH .
R\
R\
R\
To check the resultant model's statistical adequacy, we also estimated the
following VECM:
*y "c !a bct#K *x #W *z #a bz #u
R
W
R
R\
W R\
R
"c !a bct#K *x #W *z #a b z #a b z #u ,
W
R
R\
W R\
W R\
R
where a and a are 4-dimensional vectors of adjustment (error correction)
W
W
coe$cients. The estimates of these adjustment coe$cients together with a number of diagnostic test statistics are presented in Table 5 below.
The "gures in ( . ) are estimated asymptotic standard errors whereas those in
[ ) ] are the corresponding p-values. The bold-faced estimates denote statistical
signi"cance at the 0.05 level. The error correction equations pass almost all of
the diagnostic tests with the exception of the equation for *rH, which conclusiveR
ly fails the normality test and marginally the residual serial correlation test. The
domestic price equation performs best, explaining 0.78 of the price variation
Full estimation results are available on request. s (4) is the Lagrange multiplier statistic for
1!
testing the null of no serial correlation, s (1) is Ramsey's RESET test statistic, s (2) is the
$$
,
Jarque}Bera statistic for testing the null of Gaussian errors, and s (1) is the statistic for testing the
&
null of no heteroskedasticity. The number in ( ) ) indicates the degrees of freedom.
M.H. Pesaran et al. / Journal of Econometrics 97 (2000) 293}343
329
over the sample period. The error correction term associated with the "rst
cointegrating relation explaining long-run price movements also has a signi"cant but small negative impact on current price changes, suggesting an equilibrating but slow adjustment process for U.K. prices in response to changes in the
domestic interest rate, the exchange rate and foreign prices.
We now return to consider the estimation results under cointegration rank
r"1 as suggested by the test results in Table 3. For this case, we write
b "(bH, b ,2, b , b ). Under the just-identifying normalization restriction
H
b "1, we obtained the estimates
0.004 1 !0.923 !3.567 !1.929 !1.362 0.074
bK "
,
H
(0.018)
(0.323)
(1.891)
(1.434)
(1.553) (0.207)
¸¸"701.70.
Only the coe$cients of e and r are statistically signi"cant. Imposing the
R
R
restrictions implied by H 5H and setting the coe$cient of rH to zero yields
R
0 1 !0.782 !5.198 0 !0.998 0
bK "
, ¸¸"700.43.
H
(0.273)
(1.519)
(0.182)
The log-likelihood ratio statistic for testing these three restrictions is 2.52.
Further, imposing the restrictions that the coe$cients of e and pH are both
R
R
equal to !1, as indicated by the PPP hypothesis, we "nally obtain
bK "
H...
0 1 !1 !5.401
(0.861)
0 !1 0
, ¸¸"699.26.
This result is remarkably close to that obtained assuming r"2 for the modi"ed
PPP relation given above by (7.3). Therefore, as far as the test of the PPP
hypothesis is concerned our main conclusion would seem to be una!ected by
assuming the cointegration rank to be r"1 or r"2. The above result continues
to hold if instead of setting b "1, we normalize on the coe$cients of e , r or pH.
R R
R
Clearly, we cannot normalize on the coe$cient of rH if we wish to test its possible
R
signi"cance in the PPP relationship.
However, normalizing on the coe$cient of rH is not inappropriate if we
R
believe that the single cointegrating relationship in fact represents the UIP
hypothesis. Using this normalization and estimating the cointegrating relation
subject to the UIP restrictions (jointly with H 5H ), namely, b "1 and
bH"b "b "b "b "0, we obtain
bK "(0 0 0 !1 1 0 0), ¸¸"692.93.
H3'.
The log-likelihood ratio statistic for testing these 6 restrictions is equal to
2(701.70!692.93)"17.54 and strongly rejects the UIP hypothesis if r"1. We
also tried a number of modi"cations of the UIP hypothesis but they were all
330
M.H. Pesaran et al. / Journal of Econometrics 97 (2000) 293}343
rejected. Thus, we conclude that with r"1 only a modi"ed version of the PPP
hypothesis would appear to be compatible with the data.
8. Concluding remarks
The results derived in this paper are intended to complement the pioneering
contributions of Johansen reviewed recently in Johansen (1995). We extend his
work in two directions: we relax the assumption that all the I(1) variables in the
system are endogenously determined and allow for parametric restrictions on
the coe$cients of the stationary variables in the vector error correction form of
the cointegrating VAR model. We also highlight the importance of appropriate
speci"cations of intercepts and/or trend coe$cients in these models. Our reexamination of the validity of the PPP and UIP relations provides an illustration of the utility of these extensions in practice where the available time series
observations are often limited relative to the size of the VAR models being
estimated. As with many other contributions in this area our results are asymptotic. In this paper we also provide some limited Monte Carlo evidence on the
small sample performance of alternative tests of the cointegrating rank restrictions. These results are encouraging and suggest that tests that allow for the
presence of exogenous I(1) variables and/or restrictions on the short-run coe$cients are likely to perform better in small samples. However, more work in this
area is clearly necessary before more de"nite conclusions can be reached.
Acknowledgements
Partial "nancial support from the ESRC (Grant Nos. R000233608 and
R000237334) and the Isaac Newton Trust of Trinity College, Cambridge, for this
research is gratefully acknowledged. We are also grateful to an Editor, two
anonymous referees, Peter Boswijk, Cheng Hsiao, Bent Nielsen, and Michael
Binder for helpful comments on the previous version of this paper (Pesaran,
Shin and Smith, 1997).
Appendix A. Proofs of results
In order to prove Theorems 4.1}4.9 and 5.1, we need a number of additional
subsidiary results. We consider Case IV and brie#y detail the alterations
necessary for the other cases below Lemma A.1.
De"ne the (m#1, m!r) matrix d as
d,
!c
I
K
b ,
,
M.H. Pesaran et al. / Journal of Econometrics 97 (2000) 293}343
331
where b is a (m, m!r) matrix whose columns are a basis for the orthogonal
,
complement of b. Hence, (b, b ) is a basis for RK. Let n be the (m#1)-unit
,
vector (1, 0). Then, (b , n, d) is a basis for RK>. It therefore follows from (2.11)
H
that for (t!1)/¹)a(t/¹, a3[0, 1]
¹\dzH"¹\b l#b CSC#b ¹\CH(¸)e Nb CB (a),
,
R
,
R
, K
R
,
see Phillips and Solo (1992, Theorem 3.15, p. 983). Also
(A.1)
¹\nzH"¹\tNa.
R
Similarly, we note that
(A.2)
b zH"bl#bCH(¸)e "O (1).
(A.3)
H R
R
.
Hence, from (A.3) and Phillips and Solo (1992, Theorem 3.16, p. 983), it follows
that
R ,
b S b P
@@
H 88 H
where R
is a (r, r) positive-de"nite
@@
b S "b S P #¹\b ZK H U,
H 87
H 88 WH
H \
matrix.
Similarly,
as
R "R a ,
b S P
H 87
@W
@@ W
and,
as
S "¹\(P ZK H #U ) (PM ι !PM ι *Z (*Z PM ι *Z )\ *Z PM ι )
\
\
\
\
77
WH \
(P ZK H #U ),
WH \
R "X #a R a .
S P
77
WW
SS
W @@ W
Cf. Johansen (1991, Lemma A.3, p. 1569) and Johansen (1995, Lemma 10.3, p.
146).
Lemma A.1. Let B ,(d, ¹\n) and dexne G(a)"(G (a), G (a)), where
2
G (a),b CBI (a),
BI (a)["(BI (a), BI (a))]"B (a)! B (a) da,
and
, K
K
L
I
K
K
G (a),a!, a3[0, 1]. Then
¹\B S B N G(a)G(a) da,
(A.4)
2 88 2
(A.5)
B (S !S P )N G(a) dBI H(a),
2 87
88 WH
L
where BI H(a),[BI (a)!X X\BI (a)], a3[0, 1], and S , S and S are deL
L
WV VV I
88 87
77
xned in (3.21).
332
M.H. Pesaran et al. / Journal of Econometrics 97 (2000) 293}343
Proof. These statements follow from (A.1) and (A.2) by applying the results of
Phillips and Solo (1992, Theorem 3.15, p. 983) and using (4.1) and (4.2). Cf.
Johansen (1991, Lemma A.3, p. 1569) and Johansen (1995, Lemma 10.3, p. 146)
and Phillips and Durlauf (1986). 䊐
For the remaining cases, we need only make minor modi"cations to Case IV.
In Case I, d"b with (b, b ) a basis for RK and B "d. For Case II, where
,
,
2
ZH "(ι , Z ), we have
\
2 \
!l
b "
b
H
I
K
and, consequently, we de"ne n as in Case IV,
d"
!l
b
and B "(d, n).
,
2
I
K
Case III is similar to Case I as is Case V.
Proof of Theorems 4.1 and 4.2. See Pesaran et al. (1997, Appendix A). 䊐
Lemmas A.2}A.4 below follow the analysis of Johansen (1995, Chapter 13, pp.
177}200). See also Johansen (1994). We concentrate on the representation for
the limiting distribution of the ML estimator bK
in Case IV under
H
H : Rank[P ]"r of (3.8).
P
W
Firstly, we note that we may decompose the ML estimator bK via orthogonal
H
projection as
bK "b bM bK #ddM bK #nnM bK ,
H
H H H
H
H
where bM ,P,d n b (b P,d n b )\ with P,d n the orthogonal projector onto the
H
H H H
orthogonal complement of (d, n) and dM and nM de"ned similarly. We then de"ne
bI ,bK (bM bK )\
H
H H H
"b #ddM bI #nnM bI "b #B U ,
(A.6)
H
H
H
H
2 2
where B is de"ned in Lemma A.1 and
2
U ,(dM , ¹nM )bI ;
2
H
cf. Johansen (1995, (13.1), (13.2), p. 179). The normalization bM bI "I provides
H H
P
the r restrictions required to just-identify bI . The ML estimator for X is given
H
SS
from (3.20) by
XK "¹\*YK (I !ZK H bK (bK ZK H ZK H bK )\bK ZK H )*YK .
SS
2
\ H H \ \ H
H \
M.H. Pesaran et al. / Journal of Econometrics 97 (2000) 293}343
333
Lemma A.2. Under H dexned in (3.8) and Assumptions 2.1, 2.2, 3.1 and 4.1,
P
bI !b "o (¹\), bI S bI "R #o (1),
H
H
.
H 88 H
@@
.
bI S "R #o (1), XK "X #o (1).
H 87
@W
.
SS
SS
.
Proof. De"ning A ,(b , ¹\B ) and noting that A\"(bM , ¹dM , ¹nM ),
2
H
2
2
H
these results are obtained by the application of an argument similar to that used
in the proof of Lemma 13.1 of Johansen (1995, pp. 180}181). 䊐
Lemma A.3. Under H dexned in (3.8) and Assumptions 2.1, 2.2, 3.1 and 4.1,
P
¹U "(¹dM , ¹nM )(bI !b )
2
H
H
\ N
G(a)G(a) da
G(a) dBH(a)X\a (a X\a )\,
(A.7)
L
SS W W SS W
which is mixed normal with conditional variance given by
\
G(a)G(a) da
,
(A.8)
where BH(a),[B (a)!X X\B (a)] and G(a) is dexned in Lemma A.1,
L
L
WV VV I
a3[0, 1]. Hence, (bI !b )"O (¹\).
H
H
.
(a X\a )\
W SS W
Proof. It is straightforward to show that bI satis"es the likelihood equations for
H
bK derived from (3.20); viz.
H
a XK \[S !a bI S ]"0,
(A.9)
W SS 78
W H 88
where a ,a( (bK )bK bM and, from Lemma A.2, a !a "o (1). Eq. (A.9) after
W
W H H H
W
W
.
post-multiplication by B may be rearranged as
2
a XK \[(S !a b S )!a (bI !b )S !(a !a )b S ]B "0.
W SS
78
W H 88
W H
H 88
W
W H 88 2
Now, b S B "O (1); cf. Johansen (1995, Lemma 10.3, (10.18), p. 146). SubstiH 88 2
.
tuting for bI !b from (A.6) yields
H
H
¹U "(¹\B S B )\B (S !S b a )X\a (a X\a )\#o (1)
2
2 88 2
2 87
88 H W SS W W SS W
.
\ N
G(a)G(a) da
G(a) dBI H(a)X\a (a X\a )\,
L
SS W W SS W
by the consistency of XK , Lemma A.1 and a !a "o (1). The "rst equality in
SS
W
W
.
(A.7) results from the de"nition of U and the representation (A.7) follows by
2
noting that BI H(.) may be replaced by BH( . ) in (A.5). As G( . ) is independent of
L
L
(a X\a )\a X\BH( . ), ¹U is asymptotically mixed normal with conditional
W SS W
W SS L
2
variance (A.8). Cf. Johansen (1995, Lemma 13.2, pp. 181}183). 䊐
334
M.H. Pesaran et al. / Journal of Econometrics 97 (2000) 293}343
Lemma A.4. Under H dexned in (3.8) and Assumptions 2.1, 2.2, 3.1 and 4.1, the
P
likelihood ratio statistic
!2[l (b ; r)!l (bK ; r)]
2 H
2 H
N ¹race (a X\a )\a X\
W SS W
W SS
dBH(a)G(a)
L
G(a)G(a) da
\
G(a) dBH(a)X\a
(A.10)
L
SS W
which is distributed as a chi-squared random variable with r(m!r#1) degrees of
freedom.
Proof. The proof mirrors that of Lemma 13.8 of Johansen (1995, pp. 192}194).
Applications of Lemmas A.2 and A.3 allow us to show that
!2[l (b ; r)!l (bK ; r)] equals
2 H
2 H
¹;¹race+[(R !R R\R )\!R\](bI !b )S (bI !b ),
@@
@W WW W@
@@
H
H 88 H
H
# O (¹\)
.
and this is equal to
¹race+a X\a ¹U (¹\B S B )¹U ,#O (¹\);
W SS W
2
2 88 2
2
.
cf. Johansen (1995, Eq. (10.9), p. 142). The representation (A.10) follows from the
representations (A.5) of Lemma A.1 and (A.7) of Lemma A.3. Now, conditional
on G( . ), the random (m!r#1)r-vector vec(a X\ dBH(a)G(a)) is normally
W SS L
distributed with mean zero and conditional variance matrix
G(a)G(a) daa X\a .
W SS W
Hence, (A.10) is conditionally s[r(m!r#1)] which is independent of G(.)
which implies the result holds unconditionally. 䊐
Proof of Theorems 4.3 and 4.4. We discuss the proof of Theorem 4.4; Theorem
4.3 follows similarly. In Case III and under H , analogous to (A.10), the
P
likelihood ratio statistic !2[l (b; r)!l (bK ; r)], where bK denotes the ML
2
2
estimator for b in Case III, has the limiting representation
¹race (a X\a )\a X\
W SS W
W SS
dBH(a) G (a)
L
\
G (a)G (a) da
G (a) dBH(a) X\a ,
L
SS W
which is s[r(m!r)] distributed. The likelihood ratio statistic for c "0 is
given by !2[l (bK ; r)!l (bK ; r)] which by partitioned inversion of
2
2 H
M.H. Pesaran et al. / Journal of Econometrics 97 (2000) 293}343
335
( G(a)G(a) da)\ has the limiting representation under H and c "0 given by
P
\
¹race (a X\a )\a X\ dBH(a) G (a)
G (a) ds
W SS W
W SS
L
G (a) dBH(a) X\a ,
L
SS W
where G (a),G (a)! G (a)G (a) da( G (a)G (a) da)\G (a), a3[0, 1].
Now, conditional on G(.), the random r-vector G (a) dBH(a) is normally
L
distributed with mean zero and conditional variance matrix
a X\a ( G (a) da). Hence, in Case IV, under H and c "0, the likelihood
P
W SS W ratio statistic !2[l (bK ; r)!l (bK ; r)] has a limiting s(r) distribution. 䊐
2
2 H
Proofs of Theorems 4.5 and 4.6. We discuss the proof of Theorem 4.6; Theorem
4.5 follows by a similar argument. The likelihood ratio statistic is the di!erence
of likelihood ratio statistics for H against H in Cases IV and V as the
P
L
maximized likelihoods for H are identical in both cases. Hence, from Theorem
L
4.2, the limiting representation for the likelihood ratio statistic for H against
P
H in Case IV is given by partitioned inversion as
L
1
1
12
a! dW (a)
a! dW (a)
L\P
L\P
2
2
\
dW (a)WK
(a)
WK
(a)WK
(a) da
# ¹race
L\P
K\P
K\P
K\P
; WK
(a) dW (a) ,
K\P
L\P
the latter being the limiting representation for the likelihood ratio statistic for
H against H in Case V. As (a!) dW (a)&N(0, (1/12)I ), the result
P
L
L\P
L\P
follows. Cf. the proof of Theorem 2.3 of Johansen (1991, p. 1571). 䊐
Proof of Theorems 4.7 and 4.8. These results follow by similar arguments to
those in the Proofs of Theorems 4.1 and 4.2 given in Pesaran et al. (1997,
Appendix A). 䊐
Proof of Theorem 4.9. By Lemma A.3, bI !b "O (¹\). Hence, asympH
H
.
totically, the inclusion of the lagged estimated cointegrating relationships
bK zH is the same as the inclusion of the actual cointegrating relationships
H R\
b zH , t"1,2, ¹. Therefore, as the regression contains only variables which
H R\
are stationary, the result follows; see Phillips and Durlauf (1986) and Phillips
and Solo (1992). 䊐
336
M.H. Pesaran et al. / Journal of Econometrics 97 (2000) 293}343
Proof of Theorem 5.1. By an argument similar to that above Lemma A.1,
RI , where RI
b SI b P
is a (r, r) positive-de"nite matrix. As b SI "
H 88 H
@@
@@
H 87
b SI P #¹\b ZI H UI ,
H 88 WH
H \
¹\b ZI H UI "¹\b ZI H U!¹\b ZI H *Z (WI !W)
H \
H \ \
H \
and ¹(WI !W)"O (1), b SI P RI "RI a . Similarly, as SI "
.
H 87
@W
@@ W
77
¹\(P ZI H #UI )PM ι (P ZI H #UI ), SI P.RI "X #a RI a . It also
WH \
WH \
77
WW
SS
W @@ W
straightforwardly
seen
that
¹\B SI B "¹\B ZH PM ι ZHY B N
2 88 2
2 \
\ 2
G(a)G(a) da and B (SI !SI P )"¹\B ZH PM ι UI N G(a) dBI H(a),
2 87
88 WH
2 \
L
where B is de"ned in Lemma A.1; cf. (A.4) and (A.5). Therefore, the result is
2
obtained by employing similar arguments to those in the Proof of Theorems 4.1
and 4.2 given in Pesaran et al. (1997, Appendix A).
Appendix B. Computation of critical values
The calculations of asymptotic critical values for the (log-) likelihood ratio
cointegration rank statistics LR(H "H ) and LR(H "H ), de"ned by (4.5) and
P P>
P L
(4.7) respectively, are carried out as follows.
For a given cointegrating rank hypothesis H de"ned by (3.8), we generate
P
m!r independent random walk processes
z "z #e , t"1, 2,2, ¹,
R
R\
R
with z "0 and e &IN(0, I ). We then partition z "(y , x ) where y and x are
R
K
R
R R
R
R
respectively (n!r)- and k-vectors of random walk variables and e "(w , t ) is
R
R R
partitioned conformably with z "(y , x ), t"1, 2,2, ¹. For Cases I}V, we
R
R R
compute the following statistic:
\ 2
2
2
w f f f
f w ,
R R
R R
R R
R
R
R
where
(B.1)
z
Case I
R\
(z ,1)
Case II
R\
Case III,
f , z
R
R\
(z , t!) Case IV
R\
z(
Case V
R\
z
is de-meaned z
(or the OLS residual from the regression of z
on 1)
R\
R\
R\
and z(
is de-meaned and de-trended z
(or the OLS residual from the
R\
R\
304.4
258.7
215.4
175.9
141.2
110.1
83.18
59.33
39.81
24.05
12.36
4.16
295.7
251.2
208.0
169.8
135.6
105.4
78.47
55.42
36.69
21.46
10.25
3.04
0.05
12
11
10
9
8
7
6
5
4
3
2
1
0.10
12
11
10
9
8
7
6
5
4
3
2
1
320.6
273.0
228.8
189.1
152.3
120.1
91.90
66.14
45.95
29.08
15.83
6.49
329.4
280.5
235.8
195.9
158.2
125.9
96.79
69.87
49.56
32.16
17.79
8.13
1
344.8
295.4
249.3
207.7
168.9
135.5
104.9
77.11
55.07
36.20
20.93
9.52
353.5
303.1
256.7
214.9
175.3
141.4
110.2
81.45
59.19
39.40
23.37
11.41
2
368.9
317.2
269.6
226.2
185.6
150.4
117.5
87.89
64.13
43.24
25.81
12.21
378.0
326.0
277.4
233.5
192.1
156.7
123.3
92.92
68.70
46.79
28.64
14.19
3
392.1
338.9
289.9
244.7
202.3
164.7
130.5
99.28
72.93
49.87
30.68
14.91
402.0
347.9
298.7
252.1
209.3
171.4
136.0
104.2
77.60
53.72
33.68
17.24
4
415.8
360.6
310.1
262.7
218.5
179.0
143.0
109.9
81.39
56.55
35.26
17.40
426.2
370.3
318.6
271.0
225.7
185.6
148.7
115.3
86.38
60.47
38.10
19.76
5
67.95
62.62
56.60
50.60
45.00
39.39
33.48
27.57
21.58
15.57
9.28
3.04
71.25
65.86
59.62
53.91
47.94
42.30
36.27
29.95
23.92
17.68
11.03
4.16
0
j
71.51
65.76
59.83
54.30
48.23
42.72
36.93
30.70
24.84
18.96
12.83
6.49
74.99
68.98
63.21
57.54
51.30
45.45
39.76
33.14
27.27
21.16
17.79
8.13
1
74.74
68.99
63.13
57.41
51.43
45.88
40.02
33.93
28.16
22.18
16.01
9.52
78.30
72.21
66.35
60.62
54.53
48.81
42.86
36.59
30.60
24.50
18.08
11.41
2
77.88
72.03
66.21
60.60
54.44
48.86
43.16
37.15
31.26
25.23
18.95
12.21
81.38
75.45
69.65
63.82
57.81
51.99
46.26
39.92
33.98
27.93
21.25
14.19
3
80.98
74.99
69.29
63.55
57.57
52.11
46.03
40.07
34.37
28.20
21.72
14.91
84.68
78.45
72.58
67.16
60.80
55.38
49.13
42.92
37.24
30.65
24.04
17.24
4
83.91
78.07
72.04
66.56
60.60
55.05
48.96
43.14
37.25
31.05
24.48
17.40
87.66
81.60
75.43
70.20
63.78
58.28
52.07
46.19
40.14
33.42
27.03
19.76
5
The critical values of j
and j
statistics are computed as the maximum eigenvalue and the trace of the matrix de"ned by (B.1) for sample size ¹"500 and 10,000
replications. Case I is de"ned by (3.9). n and k refer to the number of endogenous and exogenous variables, respectively, and r is the number of cointegrating relations de"ned by
(3.8). The critical values at other quantiles are available from the authors upon request.
0
n!r!k
j
Case I: no intercepts and no trends
Table 6(a)
M.H. Pesaran et al. / Journal of Econometrics 97 (2000) 293}343
337
341.2
291.7
245.7
204.0
166.1
132.5
102.6
75.98
53.48
34.87
20.18
9.16
332.0
284.2
238.3
197.7
160.3
127.2
97.87
71.81
49.95
31.93
17.88
7.53
0.05
12
11
10
9
8
7
6
5
4
3
2
1
0.10
12
11
10
9
8
7
6
5
4
3
2
1
356.4
305.4
259.0
216.4
176.4
142.2
110.5
82.17
59.07
39.12
22.76
10.50
365.1
314.2
265.9
223.8
182.7
148.0
116.3
86.58
62.75
42.40
25.23
12.45
1
380.3
328.1
279.0
234.7
192.9
157.0
123.5
92.93
67.83
45.89
27.58
13.21
389.4
336.4
287.0
242.3
199.7
163.1
128.9
97.57
72.15
49.43
30.46
15.27
2
403.8
350.1
299.3
253.0
209.2
171.5
136.0
103.6
76.69
52.71
32.28
15.68
413.6
358.7
307.2
260.8
216.6
178.0
141.7
108.6
81.25
56.28
35.46
17.80
3
Case II is de"ned by (3.10). Also see the footnote to Table 6(a).
0
n!r!k
j
Table 6(b)
Case II: restricted intercepts and no trends
427.3
371.6
319.2
271.4
226.1
185.7
148.3
114.4
85.34
59.23
36.84
18.24
437.1
381.0
327.8
279.5
232.8
192.5
154.4
119.8
90.60
63.10
39.94
20.63
4
450.6
393.4
338.9
289.3
241.8
199.5
160.6
125.2
93.92
65.69
41.33
20.57
461.5
403.4
348.5
297.9
249.3
206.5
167.0
130.8
99.37
69.72
44.56
23.02
5
72.09
66.47
60.66
54.91
49.04
43.44
37.65
31.73
25.80
19.86
13.81
7.53
75.51
69.76
64.11
57.95
52.06
46.47
40.53
34.40
28.27
22.04
15.87
9.16
0
j
75.35
69.48
63.77
58.17
52.23
46.59
40.93
34.99
29.01
22.98
16.74
10.50
78.57
73.11
66.85
61.34
55.26
49.57
43.76
37.48
31.48
25.54
18.88
12.45
1
78.55
72.75
66.78
61.21
55.38
49.67
44.05
37.81
32.00
26.08
19.67
13.21
81.74
76.02
70.17
64.47
58.21
52.87
46.90
40.57
34.69
28.49
21.92
15.27
2
81.40
75.71
69.98
64.27
58.31
52.67
47.09
40.86
35.08
28.83
22.54
15.68
84.82
79.23
73.47
67.77
61.59
55.81
49.97
43.62
37.83
31.56
24.97
17.80
3
84.36
78.68
72.86
67.35
61.28
55.61
49.85
43.80
38.03
31.73
25.27
18.24
87.90
82.36
76.20
70.86
64.40
58.85
52.88
46.77
40.19
34.15
27.82
20.63
4
87.24
81.72
75.88
70.32
64.24
58.57
52.67
46.87
40.84
34.56
27.87
20.57
91.11
85.20
79.28
73.65
67.45
61.74
55.65
49.80
43.58
37.19
30.50
23.02
5
338
M.H. Pesaran et al. / Journal of Econometrics 97 (2000) 293}343
328.5
279.8
235.0
194.4
157.8
124.6
95.87
70.49
48.88
31.54
17.86
8.07
319.6
272.7
227.7
187.9
152.0
119.7
91.40
66.23
45.70
28.78
15.75
6.50
0.05
12
11
10
9
8
7
6
5
4
3
2
1
0.10
12
11
10
9
8
7
6
5
4
3
2
1
343.5
294.1
248.3
207.1
168.4
134.9
104.4
76.95
54.84
35.88
20.75
9.53
352.4
302.2
255.5
214.4
174.2
140.5
109.6
81.45
58.63
38.93
23.32
11.47
1
367.5
316.9
268.5
225.3
184.8
149.7
117.3
87.93
63.57
42.67
25.63
12.27
376.5
324.4
276.5
232.7
191.6
155.9
122.8
92.42
68.06
46.44
28.42
14.35
2
391.1
338.8
288.8
243.7
201.3
164.2
130.0
98.45
72.69
49.56
30.37
14.76
400.8
347.7
297.2
251.6
208.3
170.7
135.4
103.7
77.21
53.41
33.35
16.90
3
Case III is de"ned by (3.11). Also see the footnote to Table 6(a).
0
n!r!k
j
Table 6(c)
Case III: unrestricted intercepts and no trends
415.0
360.1
309.1
262.1
217.7
178.3
142.1
109.3
81.40
56.08
35.06
17.39
424.3
369.5
317.4
270.4
224.8
185.6
148.4
114.6
86.42
60.19
38.15
19.73
4
438.0
381.8
328.9
280.2
233.6
192.3
154.6
120.4
89.90
62.65
39.59
19.63
448.8
391.5
337.7
288.6
241.2
199.8
160.9
125.9
95.14
66.94
42.73
22.19
5
71.36
65.63
59.85
54.10
48.23
42.70
36.84
31.02
24.99
19.02
12.98
6.50
74.62
68.91
63.32
57.20
51.15
45.53
39.83
33.64
27.42
21.12
14.88
8.07
0
j
74.61
68.80
63.00
57.56
51.47
45.74
40.21
34.10
28.27
22.15
15.98
9.53
77.72
72.31
65.98
60.66
54.44
48.91
42.95
36.80
30.71
24.59
18.06
11.47
1
77.65
71.95
65.97
60.49
54.54
48.96
43.25
37.15
31.30
25.21
18.78
12.27
80.87
75.22
69.40
63.68
57.54
52.22
46.09
39.85
33.87
27.75
21.07
14.35
2
80.60
75.02
69.13
63.59
57.55
51.93
46.30
40.02
34.30
28.11
21.67
14.76
83.93
78.42
72.56
67.04
60.80
55.04
49.10
42.92
37.08
30.74
24.22
16.90
3
83.55
77.86
72.08
66.59
60.46
54.68
49.06
42.95
37.21
30.85
24.37
17.39
87.13
81.56
75.48
70.20
63.54
58.05
52.06
45.98
39.99
33.63
26.94
19.73
4
86.49
80.86
74.91
69.53
63.40
57.85
51.90
45.97
39.99
33.67
27.05
19.63
90.23
84.41
78.58
72.98
66.61
61.03
54.84
49.02
42.68
36.38
29.79
22.19
5
M.H. Pesaran et al. / Journal of Econometrics 97 (2000) 293}343
339
364.8
314.1
265.8
222.6
183.0
147.3
115.9
87.17
63.00
42.34
25.77
12.39
355.9
305.8
258.0
215.9
176.9
141.8
110.6
82.88
59.16
39.34
23.08
10.55
0.05
12
11
10
9
8
7
6
5
4
3
2
1
0.10
12
11
10
9
8
7
6
5
4
3
2
1
379.8
327.1
278.9
235.1
192.8
157.0
123.3
93.13
68.04
46.00
27.96
13.31
389.2
335.5
286.8
242.4
199.1
163.0
128.8
97.83
72.10
49.36
30.77
15.44
1
403.3
349.2
298.9
253.2
209.1
171.6
136.2
103.7
76.68
52.71
32.51
15.82
413.5
358.0
307.0
260.4
215.8
177.8
141.7
108.9
81.20
56.43
35.37
18.08
2
426.8
371.0
319.2
271.2
225.5
185.9
148.4
114.7
85.59
59.39
37.07
18.19
437.3
380.5
327.7
279.3
232.9
192.8
154.3
120.0
90.02
63.54
40.37
20.47
3
Case IV is de"ned by (3.12). Also see the footnote to Table 6(a).
0
n!r!k
j
Table 6(d)
Case IV: unrestricted intercepts and restricted trends
450.3
392.4
338.7
289.5
241.7
199.9
160.9
125.1
93.98
65.90
41.57
20.73
460.7
402.3
347.9
297.3
249.2
207.1
166.9
130.6
99.11
69.84
45.10
23.17
4
473.6
414.1
358.9
306.9
257.5
214.0
173.0
135.8
102.5
72.33
46.10
23.11
484.3
423.7
367.8
315.6
265.3
221.4
179.6
141.2
107.6
76.82
49.52
25.70
5
75.02
69.45
63.60
58.09
52.08
46.54
40.76
35.04
29.13
27.10
17.18
10.55
78.42
72.50
67.05
61.27
55.14
49.32
43.61
37.86
31.79
25.42
19.22
12.39
0
j
78.09
72.72
66.78
61.42
55.25
49.70
44.01
37.92
32.12
26.10
19.79
13.31
81.85
75.77
70.20
64.53
58.08
52.62
46.97
40.89
34.70
28.72
22.16
15.44
1
81.31
75.59
69.90
64.28
58.18
52.69
47.08
40.94
35.04
29.00
22.53
15.82
84.73
78.97
73.82
67.67
61.22
55.83
50.10
43.72
37.85
31.68
24.88
18.08
2
84.34
78.45
72.94
67.43
61.19
55.51
49.78
43.92
38.04
31.89
25.28
18.19
87.82
81.99
76.15
70.71
64.42
59.01
53.08
46.84
40.98
34.65
27.80
20.47
3
87.34
81.52
75.64
70.30
63.99
58.58
52.73
46.74
41.01
34.66
27.86
20.73
90.74
85.08
78.87
73.62
67.45
62.00
55.83
49.76
43.75
37.44
30.55
23.17
4
90.08
84.23
78.53
73.07
67.02
61.37
55.62
49.59
43.66
37.28
30.54
23.11
93.75
87.98
81.89
76.51
70.48
64.61
58.78
52.63
46.66
40.12
33.26
25.70
5
340
M.H. Pesaran et al. / Journal of Econometrics 97 (2000) 293}343
352.1
302.4
255.1
213.4
174.9
140.2
109.2
82.23
58.93
39.33
23.83
11.54
343.4
294.1
247.8
206.4
168.2
134.5
104.3
77.55
55.01
36.28
21.23
9.75
0.05
12
11
10
9
8
7
6
5
4
3
2
1
0.10
12
11
10
9
8
7
6
5
4
3
2
1
367.4
316.2
268.7
226.0
184.6
149.9
117.2
88.25
63.98
43.01
26.02
12.43
376.8
323.9
276.4
233.3
190.9
156.0
122.3
92.96
68.10
46.58
28.76
14.53
1
391.0
337.9
289.0
244.0
201.0
164.5
130.2
98.82
72.71
49.86
30.54
15.04
401.2
346.6
296.3
251.2
207.9
170.6
135.5
104.0
77.14
53.48
33.39
17.14
2
415.1
359.9
308.8
261.9
217.4
178.6
142.3
109.7
81.72
56.52
35.06
17.31
425.3
368.8
317.5
270.1
224.8
185.3
148.3
115.1
86.30
60.54
38.50
19.62
3
Case V is de"ned by (3.13). Also see the footnote to Table 6(a).
0
n!r!k
j
Case V: unrestricted intercepts and unrestricted trends
Table 6(e)
438.3
381.6
328.4
279.9
234.1
193.0
155.0
120.5
90.31
63.09
39.63
19.79
449.1
390.8
337.4
288.1
241.2
199.9
160.8
125.5
94.88
67.22
43.45
22.35
4
461.4
402.9
348.5
297.6
249.4
206.8
167.0
130.8
98.57
69.44
44.31
22.31
472.3
412.4
357.6
306.1
257.2
214.3
173.3
136.5
103.8
74.01
47.84
24.82
5
74.25
68.56
62.80
57.37
51.26
45.75
39.90
34.16
28.32
22.26
16.28
9.75
77.73
71.84
66.17
60.48
54.17
48.57
42.67
37.07
31.00
24.35
18.33
11.54
0
j
77.27
71.83
66.07
60.54
54.46
48.97
43.22
37.13
31.31
25.28
18.98
12.43
81.00
75.01
69.38
63.80
57.38
51.97
46.11
40.00
33.97
27.96
21.21
14.53
1
80.43
74.79
69.17
63.50
57.43
52.03
46.30
40.16
34.25
28.28
21.69
15.04
83.98
77.97
72.46
66.96
60.37
54.94
49.29
42.94
37.08
30.92
24.18
17.14
2
83.50
77.61
72.11
66.62
60.32
54.75
49.07
43.12
37.33
31.11
24.47
17.31
87.05
81.18
75.49
69.81
63.62
58.20
52.24
46.03
40.06
33.81
26.95
19.62
3
86.53
80.64
74.91
69.43
63.16
57.71
51.94
46.01
40.13
33.85
27.06
19.79
90.01
84.29
78.09
72.80
66.67
61.30
54.97
49.03
43.03
36.59
29.74
22.35
4
89.28
83.42
77.79
72.30
66.24
60.60
54.75
48.89
42.86
36.41
29.60
22.31
92.88
87.25
81.04
75.70
69.60
63.86
58.00
51.83
45.77
39.28
32.53
24.82
5
M.H. Pesaran et al. / Journal of Econometrics 97 (2000) 293}343
341
342
M.H. Pesaran et al. / Journal of Econometrics 97 (2000) 293}343
regression of z
on (1, t)), t"1, 2,2, ¹. We compute the maximum eigenR\
value and the trace of the matrix (B.1). Using ¹"500, the above steps are
repeated 10,000 times. Tables 6(a)}(e) provide the 0.05 and 0.10 upper critical
values from these empirical distributions, denoted j
and j
respectively, as
estimates of the corresponding critical values of the limiting distributions of the
cointegration rank statistics LR(H "H ) and LR(H "H ) for Cases I}V with
P P>
P L
n!r"1, 2,2, 12 and k"0,2, 5.
For Cases I, II and IV, the asymptotic critical values reported in Tables 6(a),
6(b) and 6(d) are similar to those in Tables 3}6 of Harbo et al. (1998). However,
for Cases III and V, Tables 6(c) and 6(e), the critical values when k"0 di!er
from those tabulated in Johansen (1995, Chapter 15) due to di!erences
in the speci"cation of the deterministic components which are summarized
below:
Intercept
Case
Case
Case
Case
Case
I
II
III
IV
V
Linear trend
Quadratic trend
PSS
JOH
PSS
JOH
PSS
JOH
N
Restricted
Unrestricted
Unrestricted
Unrestricted
N
Restricted
Unrestricted
Unrestricted
Unrestricted
N
N
N
Restricted
Unrestricted
N
N
Y
Restricted
Unrestricted
N
N
N
N
N
N
N
N
N
Y
In Case III, Johansen computes critical values including a linear trend in the
VAR, but we do not. Similarly, in Case V, the critical values reported by
Johansen (and Osterwald-Lenum (1992)) are based on a VAR model containing
both linear and quadratic trends, but in our computations we only include linear
trends with unrestricted coe$cients. See also Section 2 and the discussion in
Perron and Campbell (1993). We note in passing that the critical values reported
for Cases I, III and V may be regarded as multivariate generalizations of the
t-statistics q( , q( and q( in Dickey and Fuller (1979) while those reported for
I
O
Cases II and IV are multivariate equivalents of the F-statistics U and U in
Dickey and Fuller (1981).
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