FUTURE DEVELOPMENTS IN THE STUDY OF COINTEGRATED VARIABLES*
| Published date | 01 August 1996 |
| Date | 01 August 1996 |
| Author | Norman Swanson,C. W. J. Granger |
| DOI | http://doi.org/10.1111/j.1468-0084.1996.mp58003007.x |
OXFORD BULLETIN OF ECONOMICS AND STATISTICS, 58,3(1996)
0305-9049
FUTURE DEVELOPMENTS IN THE STUDY OF
COINTE GRATED VARIABLES *
C. W J. Granger and Norman Swanson
I. INTRODUCTION
Since the publication of a paper with a similar title, Granger (1986), there
has been considerable interest and activity concerning cointegration. This
is illustrated by the books by Engle and Granger (1991), Banerjee,
Dolado, Gaibraith and Hendry (1993), Johansen (1995) and Hatanaka
(1995) plus many papers, both theoretical and applied. Much of the work
has been highly technical, and impressive and very useful but has not
necessarily helped economists interpret their data. This work has often
accepted the constraints imposed by the early papers and has not ques-
tioned these constraints. It is the objective of this paper to suggest and
examine generalizations whilst maintaining the main idea of cointegration
and consequently to, hopefully, provide ways of making interpretations of
the results of cointegration analysis both more realistic and more useful.
This paper raises more questions than it solves and so can be thought of
as a research agenda rather than a completed project. The same also was
clearly true for the 1986 paper.
The standard theory begins with a vector of n components x, all of
which are 1(1), so that each component of Ax1 is stationary in the simplest
form. Assume that there exists an n xr, r
zt='xt (1)
has components that are all 1(0), or zero mean stationary processes,
where z has r
to be cointegrated. It was clear from the beginning that cointegration
could only arise if. one had a 'common factor', later called a 'common
trend' representation
X, =Dw, +x,
where D is an n x m matrix, with m =n r, WI is an m x 1 1(1) vector and
x, is an n x 1 vector of 1(0) components. The z's have the 1(0) property
*This study was supported to NSF award SBR-93-08295 and by a Penn State University
Research and Graduate Studies Office Faculty Award.
537
© Blackwell Publishers 1996. Published by Blackwell Publishers, 108 Cowley Road, Oxford 0X4 11F,
UK & 238 Main Street, Cambridge, MA 02142, USA.
538 BULLETIN
because there are fewer common factors, the w's, than x's, so that there
must exist linear combinations of the x's that eliminate the w's. The other
crucial property of the common trend representation is that the 1(1)
property dominate the 1(0) property, so that an 1(1) variable plus an 1(0)
one is always 1(1). An important consequence of cointegration is that the
x's must at least appear to have been generated by an error-correction
system of equations
A(B) ¿x,=yz,_1+e, (2)
where y is an n x r matrix, e, is a stationary multivariate disturbance, and
A (B) is the usual lag polynomial with A(0) =1 and A (1) having all finite
elements.
If one assumes that w, can be written as a linear combination of x's, a
somewhat more constrained but more interesting representation can be
obtained. A popular way of doing this is to take
wt=y'xt (3)
where y 'y = O, y is an m x n matrix and O here is m x r as discussed by
Warne (1991) and Gonzalo and Granger (1995). This definition has the
advantage that there is no causality from z, to w, at zero frequency, as
discussed in Granger and Lin (1995), so that it is natural to view the w's
as contributors to the permanent components, and the z's as contributors
to the transitory components of the system. As there are now n equations
relating w's and z's to x's these can be inverted to give (from Warne
(1991))
x,=
= permanent component + transitory component. (4)
It is well known that the actual z terms are not identified, since any linear
combination of z's will still be 1(0), but replacing z, by pz,, where p is a
square r x r matrix such that p'p =1 does not affect the decomposition
(4).Multiplying the error-correction equation (2) first by x' and then, sepa-
rately, by yl and using (4) to replace lagged Ax, by lagged Az, and Aw,
one gets the transformed VAR model
z, = (I - 'y)z,_1 + lags of Aw, + lags of Az, + innovations
= lags of Aw, + lags of Az, + innovations
It should be noted that constants may have to be added to these equa-
tions to ensure that each component of z, has zero mean.
Some papers assume that the common trends should be random walks,
possibly following from the Beveridge and Nelson (1981) decomposition
of an 1(1) variable to permanent and transitory components. The above
decomposition which is used throughout this paper may not have this
© Blackwell Pubhshers 1996.
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