Directory UMM :Data Elmu:jurnal:J-a:Journal of Econometrics:Vol97.Issue2.Aug2000:
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: hashem.pesaran@econ.cam.ac.uk (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.1 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
1 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$cients. Section 7 presents an empirical re-examination of
the validity of the Purchasing Power Parity (PPP) and the Uncovered Interest
Parity (UIP) hypotheses using U.K. quarterly data over the period
1972(1)}1987(2) which was previously analyzed by Johansen and Juselius (1992)
and Pesaran and Shin (1996). In contrast to this earlier work, foreign prices are
assumed to be exogenously determined and a more satisfactory treatment of oil
price changes is provided in the analysis. Section 8 concludes the paper. Proofs
of results are collected in Appendix A and Appendix B describes the simulation
method for the computation of the asymptotic critical values provided in Tables
6(a)}(e).
2. The treatment of trends in VAR models
Let Mz N= denote an m-vector random process. The data generating process
t t/1
(DGP) for Mz N= is the vector autoregressive model of order p (VAR(p)) det t/1
scribed by
U(¸)(z !l!ct)"e ,
t
t
t"1, 2,2,
(2.1)
where ¸ is the lag operator, l and c are m -vectors of unknown coe$cients, and
the (m, m) matrix lag polynomial of order p, U(¸),I !+p U ¸i, comprises
i/1 i
m
the unknown (m, m) coe$cient matrices MU Np . For the purposes of exposition
i i/1
is assumed to be
in this and the following section, the error process Me N=
t t/~=
IN(0, X), X positive de"nite. The analysis that follows is conducted given the
initial values Z ,(z
, , z ).
0
~p`1 2 0
296
M.H. Pesaran et al. / Journal of Econometrics 97 (2000) 293}343
It is convenient to re-express the lag polynomial U(¸) in a form which arises in
the vector error correction model discussed in Section 3; viz.
U(¸),!P¸#C(¸)(1!¸).
(2.2)
In (2.2), we have de"ned the long-run multiplier matrix
A
B
p
P,! I ! + U
(2.3)
m
i
i/1
and the short-run response matrix lag polynomial C(¸),I !
m
U , i"1,2, p!1. Hence, the VAR(p) model (2.1)
+p~1C ¸i, C "!+p
j/i`1 j
i/1 i
i
may be rewritten in the following form:
U(¸)z "a #a t#e , t"1, 2,2,
t
0
1
t
where
(2.4)
a ,!Pl#(C#P)c, a ,!Pc,
0
1
and the sum of the short-run coe$cient matrices C is given by
(2.5)
p~1
p
C,I ! + C "!P# + iU .
m
i
i
i/1
i/1
The cointegration rank hypothesis is de"ned by
(2.6)
H : Rank[P]"r, r"0,2, m,
(2.7)
r
where Rank[.] denotes the rank of [.]. Under H of (2.7), we may express
r
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
M
M
bases for the null spaces (kernels) of a and b respectively; in particular, a@a "0
M
and b@b "0.
M
We now adopt the following assumptions.
Assumption 2.1. The (m, m) matrix polynomial U(z)"I !+p U zi is such that
i/1 i
m
the roots of the determinantal equation DU(z)D"0 satisfy DzD'1 or z"1.
Assumption 2.1 rules out the possibility that the random process
M(z !l!ct)N= admits explosive roots or seasonal unit roots except at the
t/1
t
zero frequency.
Assumption 2.2. The (m!r, m!r) matrix a@ Cb is full rank.
M M
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 Mb@ (z !l!ct)N= and Mb@(z !l!ct)N= to be integrated
t/1
t/1
t
M t
of orders one and zero respectively.2 Moreover, Assumption 2.2 speci"cally
excludes the process M(z !l!ct)N= being integrated of order two. Together
t/1
t
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 M*z N= may be expressed under Assumptions 2.1
t t/1
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)
t
0
1
t
0
1
t
where b ,Ca #CHa , b ,Ca . The matrix lag polynomial C(¸) is given
0
0
1 1
1
by3
=
=
C(¸),I # + C ¸j"C#(1!¸)CH(¸), CH(¸), + CH¸j,
j
m
j
j/1
j/0
=
=
(2.10)
C, + C ,
CH, + CH.
j
j
j/0
j/0
Now, as C(¸)U(¸)"U(¸)C(¸)"(1!¸)I , PC"0 and CP"0; in particular,
m
C"b (a@ Cb )~1a@ . Re-expressing (2.9) in levels,
M
M M M
t(t#1)
z "z #b t#b
#Cs #CH(¸)(e !e ),
(2.11)
t
0
0
1 2
t
t
0
where the partial sum s ,+t e , t"1, 2,2 .
s/1 s
t
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
0
1
the restrictions (2.5) on a induce b "0 and ensure that the nature of the
1
1
remains
deterministic trending behavior of the level process Mz N=
t t/1
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
Mz N= is4
t t/1
z "l#ct#Cs #CH(¸)e ,
(2.12)
t
t
t
2 See Johansen (1995, De"nitions 3.2 and 3.3, p. 35). That is, de"ning the di!erence operator
D,(1!¸), the processes Mb@ [D(z !l!ct)]N= , and Mb@(z !l!ct)N= admit stationary and
M
t
t/1
t
t/1
invertible ARMA representations; see also Engle and Granger (1987, De"nition, p. 252).
3 The
matrices
MC N
can
be
obtained
from
the
recursions
i
C "+p C U , i'1, C "I , C "!(I !U ), de"ning C "0 for i(0. Similarly, for the
i
j/1 i~j j
0
m 1
m
1
i
matrices MCHN, CH"C #CH , j'0, CH"I !C.
j
j
j
j~1
0
m
4 From (2.2), as C(¸)U(¸)"(1!¸)I and, in particular, CP"0, CC!CHP"I .
m
m
298
M.H. Pesaran et al. / Journal of Econometrics 97 (2000) 293}343
where we have used the initialization z ,l#CH(¸)e .5 See also Johansen
0
0
(1994) and Johansen (1995, Section 5.7, pp. 80}84).6 If, however, a were not
1
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
m
1
null hypothesis of no cointegration; that is, H : Rank[P]"0, and C full rank.
0
In the general case H : Rank[P]"r of (2.7), this would imply the unsatisfactory
r
conclusion that quite di!erent deterministic trending behavior should be observed in the levels process Mz N= for di!ering values of the cointegrating rank
t t/1
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
b@z "b@l#(b@c)t#b@CH(¸)e , t"1, 2,2,
t
t
(2.13)
which are trend stationary. In general, the co-trending restriction (Park, 1992)
b@c"0 if and only if a "0. In this case, the representation for the VAR(p)
1
model, (2.4), and the cointegrating regression, (2.13), will contain no deterministic trends. However, the restriction b@c"0 may not prove to be satisfactory in
practice. Therefore, it is important that the composite b@c in (2.13) or, equivalently, a "!Pc in (2.4) is estimated along with the other parameters of the
1
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
t
t
the k-vector x , where k,m!n; that is, z "( y@ , x@ )@, t"1, 2,2 . The primary
t t
t
t
concern of this paper is the structural modelling of the vector y conditional on
t
its past, y , y ,2, and current and past values of the vector of random
t~1 t~2
variables x , x , x ,2, t"1, 2,2 . The assumption of Section 2 concernt t~1 t~2
, e &IN(0, X), t"0,$1,2, where X is positive
ing the error process Me N=
t t/~= t
5 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 .
t
t
6 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
r
r linearly independent deterministic trends and, from (2.11), m!r independent stochastic trends Cs ,
t
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):
p~1
*z "a #a t# + C *z #Pz #e , t"1, 2,2,
(3.1)
t
0
1
i t~i
t~1
t
i/1
where the short-run response matrices MC Np~1 and the long-run multiplier
i i/1
matrix P are de"ned below (2.2).
By partitioning the error term e conformably with z "(y@ , x@ )@ as
t t
t
t
e "(e@ , e@ )@ and its variance matrix as
yt xt
t
X
X
yx
X" yy
X
X
xy
xx
we are able to express e conditionally in terms of e as
yt
xt
(3.2)
e "X X~1e #u ,
t
yt
yx xx xt
where u &IN(0, X ), X ,X !X X~1X and u is independent of e .
t
xt
t
uu
uu
yy
yx xx xy
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@ )@,
y x
y1 x1
1
y0 x0
0
C"(C@ , C@ )@, C "(C@ , C@ )@, i"1,2, p!1, provides a conditional model for
yi xi
i
y x
*y in terms of z , *x , *z , *z , 2; viz.
t
t~1
t
t~1
t~2
p~1
*y "c #c t#K *x # + W *z #P z #u ,
(3.3)
t
0
1
t
i t~i
yy.x t~1
t
i/1
t"1, 2,2,
A
B
where c , a !X X~1a , c , a !X X~1a , K"X X~1, W ,
i
yx xx
y1
yx xx x1
0
y0
yx xx x0 1
C !X X~1C , i"1,2, p!1, and P ,P !X X~1P .
yy.x
y
yx xx x
yi
yx xx xi
Following Johansen (1992) and Boswijk (1992, Chapter 3), we assume that the
process Mx N= is weakly exogenous with respect to the matrix of long-run
t t/1
multiplier parameters P; viz.7
Assumption 3.1. P "0.
x
Therefore,
P "P .
yy.x
y
7 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
p~1
*y "c #c t#K *x # + W *z #P z #u ,
(3.5)
t
0
1
t
i t~i
y t~1
t
i/1
p~1
*x "a # + C *z #e , t"1, 2,2,
(3.6)
t
x0
xi t~i
xt
i/1
where now, because a "0, c ,a and the relations (2.5) are modi"ed to
x1
1
y1
c "!P l#(C !X X~1C #P )c,
y
0
y
y
yx xx x
c "!P c.
1
y
(3.7)
The restriction P "0 of Assumption 3.1 implies that the elements of the vector
x
process Mx N= are not cointegrated among themselves as is evident from (3.6).
t t/1
Moreover, the information available from the di!erenced VAR(p!1) model
(3.6) for Mx N= is redundant for e$cient conditional estimation and inference
t t/1
concerning the long-run parameters P as well as the deterministic and shorty
run parameters c , c , K and W , i"1,2, p!1, of (3.5). Furthermore, under
0 1
i
Assumption 3.1, we may regard Mx N= as long run forcing for My N= ; see
t t/1
t t/1
Granger and Lin (1995).8
The cointegration rank hypothesis (2.7) is therefore restated in the context of
(3.5) as
H : Rank[P ]"r,
r
y
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.
1
Hence, the structural VECM (3.5) becomes
p~1
*y "K *x # + W *z #P z #u .
(3.9)
t
t
i t~i
y t~1
t
i/1
Case II: (Restricted intercepts; no trends.) c "!P l and c "0. Here,
0
y
1
c"0. The structural VECM (3.5) is
p~1
*y "(!P l)#K *x # + W *z #P z #u .
(3.10)
t
y
t
i t~i
y t~1
t
i/1
Case III: (Unrestricted intercepts; no trends.) c0 O0 and c "0. Again, c"0.
1
In this case, the intercept restriction c "!P l is ignored and the structural
0
y
8 Note that this restriction does not preclude My N= being Granger-causal for Mx N= in the short
t t/1
t t/1
run.
M.H. Pesaran et al. / Journal of Econometrics 97 (2000) 293}343
301
VECM estimated is
p~1
*y "c #K *x # + W *z #P z #u .
(3.11)
t
0
t
i t~i
y t~1
t
i/1
Case IV: (Unrestricted intercepts; restricted trends.) c0 O0 and c "!P c.
1
y
Thus
p~1
*y "c #(!P c)t#K *x # + W *z #P z #u .
(3.12)
t
0
y
t
i t~i
y t~1
t
i/1
Case V: (Unrestricted intercepts; unrestricted trends.) c0 O0 and c O0. Here,
1
the deterministic trend restriction c "!P c is ignored and the structural
1
y
VECM estimated is
p~1
*y "c #c t#K *x # + W *z #P z #u .
(3.13)
t
0
1
t
i t~i
y t~1
t
i/1
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
0
1
y
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)
y
y
where the (n, r) loading matrix a and the (m, r) matrix of cointegrating vectors
y
b are each full column rank and identi"ed up to an arbitrary (r, r) non-singular
matrix.9
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
p~1
*y "c #K *x # + W *z #P zH #u ,
t
t
0
t
i t~i
yH t~1
i/1
9 That is, (a K~1)(Kb@)"a b@ for any (r, r) non-singular matrix K.
y
y
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 ]
t~1
yH
y
m
yH
t~1
"Rank[P ] and, thus, from (3.14)10
y
(3.16)
P "a b@ ,
yH
y H
where
A B
b "
H
!c@
b.
I
m
Consequently, we may therefore restate the cointegration rank hypothesis (3.14)
as
H : Rank[P ]"r, r"0,2, n.
(3.17)
r
yH
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
(3.18)
*Y"c ι@ #W *Z #P ZH #U,
~
yH ~1
0 T
where *Y,(*y ,2, *y ), ι is a ¹-vector of ones, *X,(*x ,2, *x ),
1
T T
1
T
*Z ,(*z ,2, *z ), i"1,2, p!1, W,(K, W ,2,W
), *Z ,
~i
1~i
T~i
1
p~1
~
)
(*X@, *Z@ ,2, *Z@ )@, ZH ,(s , Z@ )@, s ,(1,2, ¹)@, Z ,(z ,2, z
~1
0
T~1
~1
T ~1 T
1~p
~1
and U,(u ,2, u ).
1
T
The log-likelihood function of the structural VECM model (3.18) is given by
n¹
¹
1
l (w; r)"!
ln 2p! lnDX D! ¹race(X~1UU@),
T
uu
uu
2
2
2
(3.19)
where the parameter vector w collects together the unknown parameters in X ,
uu
c , W and P . Successively concentrating out X , c and W, and a in (3.19)
0
yH
uu 0
y
results in the concentrated log-likelihood function
¹
n¹
l# (b ; r)"! (1#ln 2p)! lnD¹~1*YK (I !ZK H@ b
~1 H
T
T H
2
2
(3.20)
](b@ ZK H ZK H@ b )~1b@ ZK H )*YK @D,
H ~1
H ~1 ~1 H
where *YK and ZK H are respectively the OLS residuals from regressions of *Y
~1
and ZH on (ι , *Z@ ). 11 De"ning the sample moment matrices
~
~1
T
(3.21)
S ,¹~1*YK *YK @, S ,¹~1*YK ZK H@ , S ,¹~1ZK H ZK H@ ,
~1 ~1
~1
ZZ
YY
YZ
10 The trend parameter vector c is no longer identi"ed in system (3.15) and (3.6). Only b@c and
a "C c may be identi"ed.
x0
x
11 The corresponding estimators are given by XK "¹~1UK UK @, where UK is de"ned via (3.18) and is
uu
a function of the unknown parameter matrices c , W and P , c( (P )"
0
yH
0 yH
(*Y!WK (P )*Z !P ZH )ι (ι@ ι )~1,
WK (PyH)"(*Y!P ZH )PM ι *Z@ (*Z PM ι *Z@ )~1,
~
~
~
yH
~
yH ~1 T T T
yH ~1
where PM ι ,I !ι (ι@ ι )~1ι@ , and a( (b )"*YK ZK H @b (b@ ZK H ZK H @b )~1.
T
T T T
T
y H
~1 H H ~1 ~1 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)
T H
reduces to the minimization of
DS DDb@ (S !S S~1S )b D
ZY YY YZ H
DS !S b (b@ S b )~1b@ S D" YY H ZZ
H ZY
YY
YZ H H ZZ H
Db@ S b D
H ZZ 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
1
r
(3.22)
DjK S !S S~1S D"0;
ZZ
ZY YY YZ
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, r2 just-identifying
restrictions on b are required for exact identi"cation.12 The resultant maxiH
mized concentrated log-likelihood function l# (b ; r) at bK of (3.20) is
T H
H
¹
¹ r
n¹
(3.23)
l# (r)"! (1#ln 2p)! lnDS D! + ln(1!jK ).
YY
i
T
2
2
2
i/1
Note that the maximized value of the log-likelihood l# (r) is only a function of
T
the cointegration rank r (and n and k) through the eigenvalues MjK Nr de"ned
i i/1
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
~1
YY YZ
ZZ
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
~1
0
1
regression of *Y and Z
on *Z .
~1
~
Case II: c "!P l and c "0. *YK and ZK H are the OLS residuals from
~1
0
y
1
the regression of *Y and ZH on *Z , where ZH "(ι , Z@ )@.
~1
T ~1
~1
~
Case III: c O0 and c "0. *YK and ZK H are the OLS residuals from the
~1
0
1
regression of *Y and Z
on (ι , *Z@ )@.
~
~1
T
Case IV: c O0 and c "!P c. *YK and ZK H are the OLS residuals from
~1
0
1
y
the regression of *Y and ZH on (ι , *Z@ )@, where ZH "(s , Z@ )@.
~1
T ~1
~
~1
T
Case V: c O0 and c O0. *YK and ZK
are the OLS residuals from the
0
1
~1
regression of *Y and ZH on (ι , s , *Z@ )@.
~
~1
T T
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
r
12 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,
r`1
y
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
r
alternative hypothesis of stationarity; that is
H : Rank[P ]"n.
n
y
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
r
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
0
b@l"0, or whether a trend should be present in Case IV, that is, the co-trending
restriction c "0 or b@c"0, is considered in Section 4.4. Finally, Section 4.5 is
1
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.13
We now weaken the independent normal distributional assumption of Secin (2.1) and, hence, on the structural
tions 2 and 3 on the error process Me N=
t t/~=
in (3.2).
error process Mu N=
t t/~=
Assumption 4.1. The error process Me N=
is such that
t t/~=
(a) (i) EMe DMz Nt~1 , Z N"0, (ii) varMe DMz Nt~1 , Z N"X, X positive de"t t~i i/1 0
t t~i i/1 0
nite;
(b) (i) EMu Dx , Mz Nt~1 , Z N"0, (ii) varMu Dx , Mz Nt~1 , Z N"X , where
uu
t t t~i i/1 0
t t t~i i/1 0
u ,e !X X~1e and X ,X !X X~1X ;
uu
yy
yx xx xy
t
yt
yx xx xt
(c) sup EMDDe DDsN(R for some s'2.
t
t
is a martingale
Assumption 4.1(a) states that the error process Me N=
t t/~=
is an
di!erence sequence with constant conditional variance; hence, Me N=
t t/~=
uncorrelated process. Therefore, the VECM (3.1) represents a conditional model
for *z given M*z Np~1 and z , t"1, 2,2 . Assumption 4.1(b) is a linear
t~1
t
t~i i/1
conditional mean condition; that is, under Assumption 4.1(b)(i),
EMe Dx ,Mz Nt~1 , Z N"X X~1e which, together with Assumption 4.1(b)(ii),
yx xx xt
yt t t~i i/1 0
also ensures that varMe Dx ,Mz Nt~1 , Z N"X . Therefore, under this assumpuu
yt t t~i i/1 0
tion, (3.15) can still be interpreted as a conditional model for *y given
t
*x ,M*z Np~1 and z , t"1, 2,2. Hence, (3.15) remains appropriate for
t~1
t
t~i i/1
is also a martingale
conditional inference. Moreover, the error process Mu N=
t t/~=
13 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
process. Thus, Assumptions 4.1(a) (ii) and 4.1(b) (ii) rule out any
the Me N=
xt t/~=
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 Mz N= process itself as the VAR(p) form (2.1) together with
t t/1
Assumptions 2.1 and 2.2 yields += jDDC DD(R, where DDADD,[tr(A@A)]1@2; see
j/0 j
Phillips and Solo (1992, Theorem 3.15(b), p. 983).14 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 += DDCHDD(R, and DDCDD(R, Philj/0 j
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
*Ta+
Se (a),¹~1@2 + e ,
s
T
s/1
where [¹a] denotes the integer part of ¹a, a3[0,1]. Under Assumption 4.1,
Se (a) satis"es the multivariate invariance principle (Phillips and Durlauf, 1986,
T
Theorem 2.1, p. 475)
(4.1)
S% (a)NB (a), a3[0, 1],
T
m
where B (.) denotes an m-dimensional Brownian motion with variance matrix
m
X. We partition Se (a)"(Sy (a)@, Sx (a)@)@ conformably with z "(y@ , x@ )@ and
T
t
t t
T
T
the Brownian motion B (a)"(B (a)@, B (a)@)@ likewise, a3[0, 1]. De"ne
m
n
k
Su (a),¹~1@2+*Ta+ u , a3[0,1]. Hence, as u ,e !X X~1e ,
s/1 s
t
yt
yx xx xt
T
(4.2)
Su (a)NBH(a),
n
T
where BH(a),B (a)!X X~1B (a) is a Brownian motion with variance matrix
n
n
yx xx k
X which is independent of B (a), a3[0, 1]. Consequently, the results described
uu
k
in Harbo et al. (1998) remain valid under Assumption 4.1.
Under Assumption 3.1, the (m, m!r) matrix a ,diag(aM, aM), where aM is
x
y x
M
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
y
14 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
m~r
n~r
k
k-dimensional sub-vector independent standard Brownian motions
W (a),(aM@X aM)~1@2aM@BH(a) and W (a),(aM@X aM)~1@2aM@B (a), a3[0, 1].
x k
x xx x
k
y n
y uu y
n~r
See Pesaran et al. (1997, Appendix A) for further details. We will also require the
corresponding de-meaned (m!r)-vector standard Brownian motion
P
1
W
(a) da,
(4.3)
m~r
0
and de-meaned and de-trended (m!r)-vector standard Brownian motion
WI
(a),W
(a)!
m~r
m~r
A BP A B
1
WK
(a),WI
(a)!12 a!
m~r
m~r
2
1
1
a! WI
(a) da,
m~r
2
(4.4)
0
and their respective partitioned counterparts WI
(a)"(WI
(a)@, WI (a)@)@, and
m~r
n~r
k
WK
(a)"(WK
(a)@, WK (a)@)@, a3[0, 1].
m~r
n~r
k
4.1. Testing H against H
r
r`1
The (log-) likelihood ratio statistic for testing H : Rank[P ]"r against
r
y
H : Rank[P ]"r#1 is given by
r`1
y
LR(H DH )"!¹ ln(1!jK
),
(4.5)
r r`1
r`1
where jK is the rth largest eigenvalue from the determinantal equation (3.22),
r
r"0,2, n!1, with the appropriate de"nitions of *YK and ZK H and, thus, the
~1
sample moment matrices S , S and S given by (3.21) to cover Cases I}V.
YY YZ
ZZ
Theorem 4.1 (Limit distribution of LR(H DH )). Under H dexned by (3.8)
r r`1
r
and Assumptions 2.1, 2.2, 3.1 and 4.1, the limit distribution of LR(H DH ) of
r r`1
(4.5) for testing H against H
is given by the distribution of the maximum
r
r`1
eigenvalue of
P
1
0
AP
dW (a) F
(a)@
n~r
m~r
1
0
B P
F
(a)F
(a)@ da
m~r
m~r
~1 1
F
(a) dW (a)@,
m~r
n~r
0
(4.6)
where
G
W
(a)
Case I
m~r
(W
(a)@, 1)@
Case II
m~r
(a)
Case III, a3[0, 1],
F
(a)" WI
m~r
m~r
(WI
(a)@, a!1)@ Case I<
2
m~r
WK
(a)
Case <
m~r
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
r
n
The (log-) likelihood ratio statistic for testing H : Rank[P ]"r against
r
y
H : Rank[P ]"n is given by
n
y
n
LR(H DH )"!¹ + ln(1!jK ),
(4.7)
r n
i
i/r`1
where jK is the ith largest eigenvalue from the determinantal equation (3.22).
i
Theorem 4.2 (Limit distribution of LR(H DH )). Under H dexned by (3.8) and
r n
r
Assumptions 2.1, 2.2, 3.1 and 4.1, the limit distribution of LR(H DH ) of (4.7) for
r n
testing H against H is given by the distribution of
r
n
~1
1
1
F
(a)F
(a)@ da
dW (a) F
(a)@
¹race
m~r
m~r
n~r
m~r
0
0
1
] F
(a) dW (r)@ ,
m~r
n~r
0
where F
(a), a3[0, 1], is dexned in Theorem 4.1 for Cases I}
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$cient estimation of vector error
correction models when the short-run dynamics may di!er within and between equations
is considered. A re-examination of the purchasing power parity and the uncovered
interest rate parity hypotheses is conducted using U.K. data under the maintained
assumption of exogenously given foreign and oil prices. ( 2000 Elsevier Science S.A. All
rights reserved.
JEL classixcation: C12; C13; C32
Keywords: Structural vector error correction model; Cointegration; Unit roots; Likelihood ratio statistics; Critical values; Seemingly unrelated regression; Monte Carlo
simulations; Purchasing power parity; Uncovered interest rate parity
* Corresponding author. Tel.: #44 1223 335216; fax: #44 1223 335471.
E-mail address: hashem.pesaran@econ.cam.ac.uk (M.H. Pesaran).
0304-4076/00/$ - 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.1 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
1 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$cients. Section 7 presents an empirical re-examination of
the validity of the Purchasing Power Parity (PPP) and the Uncovered Interest
Parity (UIP) hypotheses using U.K. quarterly data over the period
1972(1)}1987(2) which was previously analyzed by Johansen and Juselius (1992)
and Pesaran and Shin (1996). In contrast to this earlier work, foreign prices are
assumed to be exogenously determined and a more satisfactory treatment of oil
price changes is provided in the analysis. Section 8 concludes the paper. Proofs
of results are collected in Appendix A and Appendix B describes the simulation
method for the computation of the asymptotic critical values provided in Tables
6(a)}(e).
2. The treatment of trends in VAR models
Let Mz N= denote an m-vector random process. The data generating process
t t/1
(DGP) for Mz N= is the vector autoregressive model of order p (VAR(p)) det t/1
scribed by
U(¸)(z !l!ct)"e ,
t
t
t"1, 2,2,
(2.1)
where ¸ is the lag operator, l and c are m -vectors of unknown coe$cients, and
the (m, m) matrix lag polynomial of order p, U(¸),I !+p U ¸i, comprises
i/1 i
m
the unknown (m, m) coe$cient matrices MU Np . For the purposes of exposition
i i/1
is assumed to be
in this and the following section, the error process Me N=
t t/~=
IN(0, X), X positive de"nite. The analysis that follows is conducted given the
initial values Z ,(z
, , z ).
0
~p`1 2 0
296
M.H. Pesaran et al. / Journal of Econometrics 97 (2000) 293}343
It is convenient to re-express the lag polynomial U(¸) in a form which arises in
the vector error correction model discussed in Section 3; viz.
U(¸),!P¸#C(¸)(1!¸).
(2.2)
In (2.2), we have de"ned the long-run multiplier matrix
A
B
p
P,! I ! + U
(2.3)
m
i
i/1
and the short-run response matrix lag polynomial C(¸),I !
m
U , i"1,2, p!1. Hence, the VAR(p) model (2.1)
+p~1C ¸i, C "!+p
j/i`1 j
i/1 i
i
may be rewritten in the following form:
U(¸)z "a #a t#e , t"1, 2,2,
t
0
1
t
where
(2.4)
a ,!Pl#(C#P)c, a ,!Pc,
0
1
and the sum of the short-run coe$cient matrices C is given by
(2.5)
p~1
p
C,I ! + C "!P# + iU .
m
i
i
i/1
i/1
The cointegration rank hypothesis is de"ned by
(2.6)
H : Rank[P]"r, r"0,2, m,
(2.7)
r
where Rank[.] denotes the rank of [.]. Under H of (2.7), we may express
r
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
M
M
bases for the null spaces (kernels) of a and b respectively; in particular, a@a "0
M
and b@b "0.
M
We now adopt the following assumptions.
Assumption 2.1. The (m, m) matrix polynomial U(z)"I !+p U zi is such that
i/1 i
m
the roots of the determinantal equation DU(z)D"0 satisfy DzD'1 or z"1.
Assumption 2.1 rules out the possibility that the random process
M(z !l!ct)N= admits explosive roots or seasonal unit roots except at the
t/1
t
zero frequency.
Assumption 2.2. The (m!r, m!r) matrix a@ Cb is full rank.
M M
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 Mb@ (z !l!ct)N= and Mb@(z !l!ct)N= to be integrated
t/1
t/1
t
M t
of orders one and zero respectively.2 Moreover, Assumption 2.2 speci"cally
excludes the process M(z !l!ct)N= being integrated of order two. Together
t/1
t
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 M*z N= may be expressed under Assumptions 2.1
t t/1
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)
t
0
1
t
0
1
t
where b ,Ca #CHa , b ,Ca . The matrix lag polynomial C(¸) is given
0
0
1 1
1
by3
=
=
C(¸),I # + C ¸j"C#(1!¸)CH(¸), CH(¸), + CH¸j,
j
m
j
j/1
j/0
=
=
(2.10)
C, + C ,
CH, + CH.
j
j
j/0
j/0
Now, as C(¸)U(¸)"U(¸)C(¸)"(1!¸)I , PC"0 and CP"0; in particular,
m
C"b (a@ Cb )~1a@ . Re-expressing (2.9) in levels,
M
M M M
t(t#1)
z "z #b t#b
#Cs #CH(¸)(e !e ),
(2.11)
t
0
0
1 2
t
t
0
where the partial sum s ,+t e , t"1, 2,2 .
s/1 s
t
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
0
1
the restrictions (2.5) on a induce b "0 and ensure that the nature of the
1
1
remains
deterministic trending behavior of the level process Mz N=
t t/1
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
Mz N= is4
t t/1
z "l#ct#Cs #CH(¸)e ,
(2.12)
t
t
t
2 See Johansen (1995, De"nitions 3.2 and 3.3, p. 35). That is, de"ning the di!erence operator
D,(1!¸), the processes Mb@ [D(z !l!ct)]N= , and Mb@(z !l!ct)N= admit stationary and
M
t
t/1
t
t/1
invertible ARMA representations; see also Engle and Granger (1987, De"nition, p. 252).
3 The
matrices
MC N
can
be
obtained
from
the
recursions
i
C "+p C U , i'1, C "I , C "!(I !U ), de"ning C "0 for i(0. Similarly, for the
i
j/1 i~j j
0
m 1
m
1
i
matrices MCHN, CH"C #CH , j'0, CH"I !C.
j
j
j
j~1
0
m
4 From (2.2), as C(¸)U(¸)"(1!¸)I and, in particular, CP"0, CC!CHP"I .
m
m
298
M.H. Pesaran et al. / Journal of Econometrics 97 (2000) 293}343
where we have used the initialization z ,l#CH(¸)e .5 See also Johansen
0
0
(1994) and Johansen (1995, Section 5.7, pp. 80}84).6 If, however, a were not
1
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
m
1
null hypothesis of no cointegration; that is, H : Rank[P]"0, and C full rank.
0
In the general case H : Rank[P]"r of (2.7), this would imply the unsatisfactory
r
conclusion that quite di!erent deterministic trending behavior should be observed in the levels process Mz N= for di!ering values of the cointegrating rank
t t/1
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
b@z "b@l#(b@c)t#b@CH(¸)e , t"1, 2,2,
t
t
(2.13)
which are trend stationary. In general, the co-trending restriction (Park, 1992)
b@c"0 if and only if a "0. In this case, the representation for the VAR(p)
1
model, (2.4), and the cointegrating regression, (2.13), will contain no deterministic trends. However, the restriction b@c"0 may not prove to be satisfactory in
practice. Therefore, it is important that the composite b@c in (2.13) or, equivalently, a "!Pc in (2.4) is estimated along with the other parameters of the
1
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
t
t
the k-vector x , where k,m!n; that is, z "( y@ , x@ )@, t"1, 2,2 . The primary
t t
t
t
concern of this paper is the structural modelling of the vector y conditional on
t
its past, y , y ,2, and current and past values of the vector of random
t~1 t~2
variables x , x , x ,2, t"1, 2,2 . The assumption of Section 2 concernt t~1 t~2
, e &IN(0, X), t"0,$1,2, where X is positive
ing the error process Me N=
t t/~= t
5 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 .
t
t
6 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
r
r linearly independent deterministic trends and, from (2.11), m!r independent stochastic trends Cs ,
t
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):
p~1
*z "a #a t# + C *z #Pz #e , t"1, 2,2,
(3.1)
t
0
1
i t~i
t~1
t
i/1
where the short-run response matrices MC Np~1 and the long-run multiplier
i i/1
matrix P are de"ned below (2.2).
By partitioning the error term e conformably with z "(y@ , x@ )@ as
t t
t
t
e "(e@ , e@ )@ and its variance matrix as
yt xt
t
X
X
yx
X" yy
X
X
xy
xx
we are able to express e conditionally in terms of e as
yt
xt
(3.2)
e "X X~1e #u ,
t
yt
yx xx xt
where u &IN(0, X ), X ,X !X X~1X and u is independent of e .
t
xt
t
uu
uu
yy
yx xx xy
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@ )@,
y x
y1 x1
1
y0 x0
0
C"(C@ , C@ )@, C "(C@ , C@ )@, i"1,2, p!1, provides a conditional model for
yi xi
i
y x
*y in terms of z , *x , *z , *z , 2; viz.
t
t~1
t
t~1
t~2
p~1
*y "c #c t#K *x # + W *z #P z #u ,
(3.3)
t
0
1
t
i t~i
yy.x t~1
t
i/1
t"1, 2,2,
A
B
where c , a !X X~1a , c , a !X X~1a , K"X X~1, W ,
i
yx xx
y1
yx xx x1
0
y0
yx xx x0 1
C !X X~1C , i"1,2, p!1, and P ,P !X X~1P .
yy.x
y
yx xx x
yi
yx xx xi
Following Johansen (1992) and Boswijk (1992, Chapter 3), we assume that the
process Mx N= is weakly exogenous with respect to the matrix of long-run
t t/1
multiplier parameters P; viz.7
Assumption 3.1. P "0.
x
Therefore,
P "P .
yy.x
y
7 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
p~1
*y "c #c t#K *x # + W *z #P z #u ,
(3.5)
t
0
1
t
i t~i
y t~1
t
i/1
p~1
*x "a # + C *z #e , t"1, 2,2,
(3.6)
t
x0
xi t~i
xt
i/1
where now, because a "0, c ,a and the relations (2.5) are modi"ed to
x1
1
y1
c "!P l#(C !X X~1C #P )c,
y
0
y
y
yx xx x
c "!P c.
1
y
(3.7)
The restriction P "0 of Assumption 3.1 implies that the elements of the vector
x
process Mx N= are not cointegrated among themselves as is evident from (3.6).
t t/1
Moreover, the information available from the di!erenced VAR(p!1) model
(3.6) for Mx N= is redundant for e$cient conditional estimation and inference
t t/1
concerning the long-run parameters P as well as the deterministic and shorty
run parameters c , c , K and W , i"1,2, p!1, of (3.5). Furthermore, under
0 1
i
Assumption 3.1, we may regard Mx N= as long run forcing for My N= ; see
t t/1
t t/1
Granger and Lin (1995).8
The cointegration rank hypothesis (2.7) is therefore restated in the context of
(3.5) as
H : Rank[P ]"r,
r
y
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.
1
Hence, the structural VECM (3.5) becomes
p~1
*y "K *x # + W *z #P z #u .
(3.9)
t
t
i t~i
y t~1
t
i/1
Case II: (Restricted intercepts; no trends.) c "!P l and c "0. Here,
0
y
1
c"0. The structural VECM (3.5) is
p~1
*y "(!P l)#K *x # + W *z #P z #u .
(3.10)
t
y
t
i t~i
y t~1
t
i/1
Case III: (Unrestricted intercepts; no trends.) c0 O0 and c "0. Again, c"0.
1
In this case, the intercept restriction c "!P l is ignored and the structural
0
y
8 Note that this restriction does not preclude My N= being Granger-causal for Mx N= in the short
t t/1
t t/1
run.
M.H. Pesaran et al. / Journal of Econometrics 97 (2000) 293}343
301
VECM estimated is
p~1
*y "c #K *x # + W *z #P z #u .
(3.11)
t
0
t
i t~i
y t~1
t
i/1
Case IV: (Unrestricted intercepts; restricted trends.) c0 O0 and c "!P c.
1
y
Thus
p~1
*y "c #(!P c)t#K *x # + W *z #P z #u .
(3.12)
t
0
y
t
i t~i
y t~1
t
i/1
Case V: (Unrestricted intercepts; unrestricted trends.) c0 O0 and c O0. Here,
1
the deterministic trend restriction c "!P c is ignored and the structural
1
y
VECM estimated is
p~1
*y "c #c t#K *x # + W *z #P z #u .
(3.13)
t
0
1
t
i t~i
y t~1
t
i/1
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
0
1
y
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)
y
y
where the (n, r) loading matrix a and the (m, r) matrix of cointegrating vectors
y
b are each full column rank and identi"ed up to an arbitrary (r, r) non-singular
matrix.9
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
p~1
*y "c #K *x # + W *z #P zH #u ,
t
t
0
t
i t~i
yH t~1
i/1
9 That is, (a K~1)(Kb@)"a b@ for any (r, r) non-singular matrix K.
y
y
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 ]
t~1
yH
y
m
yH
t~1
"Rank[P ] and, thus, from (3.14)10
y
(3.16)
P "a b@ ,
yH
y H
where
A B
b "
H
!c@
b.
I
m
Consequently, we may therefore restate the cointegration rank hypothesis (3.14)
as
H : Rank[P ]"r, r"0,2, n.
(3.17)
r
yH
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
(3.18)
*Y"c ι@ #W *Z #P ZH #U,
~
yH ~1
0 T
where *Y,(*y ,2, *y ), ι is a ¹-vector of ones, *X,(*x ,2, *x ),
1
T T
1
T
*Z ,(*z ,2, *z ), i"1,2, p!1, W,(K, W ,2,W
), *Z ,
~i
1~i
T~i
1
p~1
~
)
(*X@, *Z@ ,2, *Z@ )@, ZH ,(s , Z@ )@, s ,(1,2, ¹)@, Z ,(z ,2, z
~1
0
T~1
~1
T ~1 T
1~p
~1
and U,(u ,2, u ).
1
T
The log-likelihood function of the structural VECM model (3.18) is given by
n¹
¹
1
l (w; r)"!
ln 2p! lnDX D! ¹race(X~1UU@),
T
uu
uu
2
2
2
(3.19)
where the parameter vector w collects together the unknown parameters in X ,
uu
c , W and P . Successively concentrating out X , c and W, and a in (3.19)
0
yH
uu 0
y
results in the concentrated log-likelihood function
¹
n¹
l# (b ; r)"! (1#ln 2p)! lnD¹~1*YK (I !ZK H@ b
~1 H
T
T H
2
2
(3.20)
](b@ ZK H ZK H@ b )~1b@ ZK H )*YK @D,
H ~1
H ~1 ~1 H
where *YK and ZK H are respectively the OLS residuals from regressions of *Y
~1
and ZH on (ι , *Z@ ). 11 De"ning the sample moment matrices
~
~1
T
(3.21)
S ,¹~1*YK *YK @, S ,¹~1*YK ZK H@ , S ,¹~1ZK H ZK H@ ,
~1 ~1
~1
ZZ
YY
YZ
10 The trend parameter vector c is no longer identi"ed in system (3.15) and (3.6). Only b@c and
a "C c may be identi"ed.
x0
x
11 The corresponding estimators are given by XK "¹~1UK UK @, where UK is de"ned via (3.18) and is
uu
a function of the unknown parameter matrices c , W and P , c( (P )"
0
yH
0 yH
(*Y!WK (P )*Z !P ZH )ι (ι@ ι )~1,
WK (PyH)"(*Y!P ZH )PM ι *Z@ (*Z PM ι *Z@ )~1,
~
~
~
yH
~
yH ~1 T T T
yH ~1
where PM ι ,I !ι (ι@ ι )~1ι@ , and a( (b )"*YK ZK H @b (b@ ZK H ZK H @b )~1.
T
T T T
T
y H
~1 H H ~1 ~1 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)
T H
reduces to the minimization of
DS DDb@ (S !S S~1S )b D
ZY YY YZ H
DS !S b (b@ S b )~1b@ S D" YY H ZZ
H ZY
YY
YZ H H ZZ H
Db@ S b D
H ZZ 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
1
r
(3.22)
DjK S !S S~1S D"0;
ZZ
ZY YY YZ
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, r2 just-identifying
restrictions on b are required for exact identi"cation.12 The resultant maxiH
mized concentrated log-likelihood function l# (b ; r) at bK of (3.20) is
T H
H
¹
¹ r
n¹
(3.23)
l# (r)"! (1#ln 2p)! lnDS D! + ln(1!jK ).
YY
i
T
2
2
2
i/1
Note that the maximized value of the log-likelihood l# (r) is only a function of
T
the cointegration rank r (and n and k) through the eigenvalues MjK Nr de"ned
i i/1
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
~1
YY YZ
ZZ
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
~1
0
1
regression of *Y and Z
on *Z .
~1
~
Case II: c "!P l and c "0. *YK and ZK H are the OLS residuals from
~1
0
y
1
the regression of *Y and ZH on *Z , where ZH "(ι , Z@ )@.
~1
T ~1
~1
~
Case III: c O0 and c "0. *YK and ZK H are the OLS residuals from the
~1
0
1
regression of *Y and Z
on (ι , *Z@ )@.
~
~1
T
Case IV: c O0 and c "!P c. *YK and ZK H are the OLS residuals from
~1
0
1
y
the regression of *Y and ZH on (ι , *Z@ )@, where ZH "(s , Z@ )@.
~1
T ~1
~
~1
T
Case V: c O0 and c O0. *YK and ZK
are the OLS residuals from the
0
1
~1
regression of *Y and ZH on (ι , s , *Z@ )@.
~
~1
T T
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
r
12 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,
r`1
y
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
r
alternative hypothesis of stationarity; that is
H : Rank[P ]"n.
n
y
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
r
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
0
b@l"0, or whether a trend should be present in Case IV, that is, the co-trending
restriction c "0 or b@c"0, is considered in Section 4.4. Finally, Section 4.5 is
1
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.13
We now weaken the independent normal distributional assumption of Secin (2.1) and, hence, on the structural
tions 2 and 3 on the error process Me N=
t t/~=
in (3.2).
error process Mu N=
t t/~=
Assumption 4.1. The error process Me N=
is such that
t t/~=
(a) (i) EMe DMz Nt~1 , Z N"0, (ii) varMe DMz Nt~1 , Z N"X, X positive de"t t~i i/1 0
t t~i i/1 0
nite;
(b) (i) EMu Dx , Mz Nt~1 , Z N"0, (ii) varMu Dx , Mz Nt~1 , Z N"X , where
uu
t t t~i i/1 0
t t t~i i/1 0
u ,e !X X~1e and X ,X !X X~1X ;
uu
yy
yx xx xy
t
yt
yx xx xt
(c) sup EMDDe DDsN(R for some s'2.
t
t
is a martingale
Assumption 4.1(a) states that the error process Me N=
t t/~=
is an
di!erence sequence with constant conditional variance; hence, Me N=
t t/~=
uncorrelated process. Therefore, the VECM (3.1) represents a conditional model
for *z given M*z Np~1 and z , t"1, 2,2 . Assumption 4.1(b) is a linear
t~1
t
t~i i/1
conditional mean condition; that is, under Assumption 4.1(b)(i),
EMe Dx ,Mz Nt~1 , Z N"X X~1e which, together with Assumption 4.1(b)(ii),
yx xx xt
yt t t~i i/1 0
also ensures that varMe Dx ,Mz Nt~1 , Z N"X . Therefore, under this assumpuu
yt t t~i i/1 0
tion, (3.15) can still be interpreted as a conditional model for *y given
t
*x ,M*z Np~1 and z , t"1, 2,2. Hence, (3.15) remains appropriate for
t~1
t
t~i i/1
is also a martingale
conditional inference. Moreover, the error process Mu N=
t t/~=
13 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
process. Thus, Assumptions 4.1(a) (ii) and 4.1(b) (ii) rule out any
the Me N=
xt t/~=
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 Mz N= process itself as the VAR(p) form (2.1) together with
t t/1
Assumptions 2.1 and 2.2 yields += jDDC DD(R, where DDADD,[tr(A@A)]1@2; see
j/0 j
Phillips and Solo (1992, Theorem 3.15(b), p. 983).14 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 += DDCHDD(R, and DDCDD(R, Philj/0 j
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
*Ta+
Se (a),¹~1@2 + e ,
s
T
s/1
where [¹a] denotes the integer part of ¹a, a3[0,1]. Under Assumption 4.1,
Se (a) satis"es the multivariate invariance principle (Phillips and Durlauf, 1986,
T
Theorem 2.1, p. 475)
(4.1)
S% (a)NB (a), a3[0, 1],
T
m
where B (.) denotes an m-dimensional Brownian motion with variance matrix
m
X. We partition Se (a)"(Sy (a)@, Sx (a)@)@ conformably with z "(y@ , x@ )@ and
T
t
t t
T
T
the Brownian motion B (a)"(B (a)@, B (a)@)@ likewise, a3[0, 1]. De"ne
m
n
k
Su (a),¹~1@2+*Ta+ u , a3[0,1]. Hence, as u ,e !X X~1e ,
s/1 s
t
yt
yx xx xt
T
(4.2)
Su (a)NBH(a),
n
T
where BH(a),B (a)!X X~1B (a) is a Brownian motion with variance matrix
n
n
yx xx k
X which is independent of B (a), a3[0, 1]. Consequently, the results described
uu
k
in Harbo et al. (1998) remain valid under Assumption 4.1.
Under Assumption 3.1, the (m, m!r) matrix a ,diag(aM, aM), where aM is
x
y x
M
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
y
14 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
m~r
n~r
k
k-dimensional sub-vector independent standard Brownian motions
W (a),(aM@X aM)~1@2aM@BH(a) and W (a),(aM@X aM)~1@2aM@B (a), a3[0, 1].
x k
x xx x
k
y n
y uu y
n~r
See Pesaran et al. (1997, Appendix A) for further details. We will also require the
corresponding de-meaned (m!r)-vector standard Brownian motion
P
1
W
(a) da,
(4.3)
m~r
0
and de-meaned and de-trended (m!r)-vector standard Brownian motion
WI
(a),W
(a)!
m~r
m~r
A BP A B
1
WK
(a),WI
(a)!12 a!
m~r
m~r
2
1
1
a! WI
(a) da,
m~r
2
(4.4)
0
and their respective partitioned counterparts WI
(a)"(WI
(a)@, WI (a)@)@, and
m~r
n~r
k
WK
(a)"(WK
(a)@, WK (a)@)@, a3[0, 1].
m~r
n~r
k
4.1. Testing H against H
r
r`1
The (log-) likelihood ratio statistic for testing H : Rank[P ]"r against
r
y
H : Rank[P ]"r#1 is given by
r`1
y
LR(H DH )"!¹ ln(1!jK
),
(4.5)
r r`1
r`1
where jK is the rth largest eigenvalue from the determinantal equation (3.22),
r
r"0,2, n!1, with the appropriate de"nitions of *YK and ZK H and, thus, the
~1
sample moment matrices S , S and S given by (3.21) to cover Cases I}V.
YY YZ
ZZ
Theorem 4.1 (Limit distribution of LR(H DH )). Under H dexned by (3.8)
r r`1
r
and Assumptions 2.1, 2.2, 3.1 and 4.1, the limit distribution of LR(H DH ) of
r r`1
(4.5) for testing H against H
is given by the distribution of the maximum
r
r`1
eigenvalue of
P
1
0
AP
dW (a) F
(a)@
n~r
m~r
1
0
B P
F
(a)F
(a)@ da
m~r
m~r
~1 1
F
(a) dW (a)@,
m~r
n~r
0
(4.6)
where
G
W
(a)
Case I
m~r
(W
(a)@, 1)@
Case II
m~r
(a)
Case III, a3[0, 1],
F
(a)" WI
m~r
m~r
(WI
(a)@, a!1)@ Case I<
2
m~r
WK
(a)
Case <
m~r
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
r
n
The (log-) likelihood ratio statistic for testing H : Rank[P ]"r against
r
y
H : Rank[P ]"n is given by
n
y
n
LR(H DH )"!¹ + ln(1!jK ),
(4.7)
r n
i
i/r`1
where jK is the ith largest eigenvalue from the determinantal equation (3.22).
i
Theorem 4.2 (Limit distribution of LR(H DH )). Under H dexned by (3.8) and
r n
r
Assumptions 2.1, 2.2, 3.1 and 4.1, the limit distribution of LR(H DH ) of (4.7) for
r n
testing H against H is given by the distribution of
r
n
~1
1
1
F
(a)F
(a)@ da
dW (a) F
(a)@
¹race
m~r
m~r
n~r
m~r
0
0
1
] F
(a) dW (r)@ ,
m~r
n~r
0
where F
(a), a3[0, 1], is dexned in Theorem 4.1 for Cases I}