COINTEGRATION, ERROR CORRECTION, AND AGGREGATION IN DYNAMIC MODELS: A COMMENT*
| Author | Mananao Aoki |
| DOI | http://doi.org/10.1111/j.1468-0084.1988.mp50001007.x |
| Published date | 01 February 1988 |
| Date | 01 February 1988 |
OXFORD BULLETIN OF ECONOMICS AND STATISTICS, 50,1(1988)
0305-9049 $3.00
COINTEGRATION, ERROR CORRECTION, AND
AGGREGATION IN DYNAMIC MODELS:
A COMMENT*
Mananao Aoki
One of the ways the notion of cointegration is motivated in Granger (1983,
1986) and Engle and Granger (1987) is to point to a better integrated treat-
ment of short-run dynamics and longer-run equilibrium dynamics by an error
correcting model of Sargan (1964) in which the time difference of some
variable Ax1 is related, among other things, to the level variable z_1. The
interpretation is that z1 refers to a combination of two (or more) level
variables, x1 ay1, say, where x = ay is an equilibrium relation, and the error
correcting models allow deviations from the equilibrium relation to affect the
shorter-run behaviour of x1. Basic in this scheme is the separation of dynamic
modes into fast-acting (shorter-run) and slower (longer-run) responses.
Cointegration recognizes the fact that when (two or more) variables are
cointegrated, then slower dynamic modes (i.e., longer-run responses) can be
aggregated out from their dynamic behaviour by a suitable linear combina-
tion of the original variables.
This note points out the relationship between the notion of cointegration
that has recently become central in modelling time series with unit roots, as
for example in Stock and Watson (1986) and King, Plosser and Watson
(1987), and the notion of dynamic aggregation introduced by Aoki (1968,
1971). Aoki (1968) introduced the notion of dynamic aggregation to the
control literature as a dynamic generalization of the concept of static aggrega-
tion in the economics literature. This notion was originally introduced as a
way of approximating complex (high-dimensional) dynamics by simpler
(lower-dimensional) dynamics. Later, this notion was shown to produce exact
lower-dimensional dynamics when dynamic matrices are of special structure,
Aoki(1971).
The key idea of the dynamic aggregation in Aoki (1968) is to separate the
eigenvalues of the dynamics into two mutually exclusive classes c1 and c2, and
retain only eigenvalues of one class to build lower-dimensional dynamic
models. Although the notion was introduced originally to continuous deter-
ministic dynamics, there is a natural counterpart in discrete-time and
stochastic dynamics systems. (See Aoki and Huddle (1967) for an indication
of this.) In typical applications, eigenvalues are classified into those near the
unit circle and those near the origin. Since the eigenvalues near the origin
* Research supported in part by a grant from the National Science Foundation.
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