Short‐Run Dynamics in Cointegrated Systems
Date | 01 August 1997 |
Author | Tommaso Proietti |
Published date | 01 August 1997 |
DOI | http://doi.org/10.1111/1468-0084.00073 |
OXFORD BULLETIN OF ECONOMICS AND STATISTICS, 59, 3 (1997)
0305-9049
SHORT-RUN DYNAMICS IN COINTEGRATED
SYSTEMS
Tommaso Proietti
I. INTRODUCTION
The problem of decomposing a time series into components bearing
meaningful interpretation is at the core of time series econometrics.
Although a wide range of univariate techniques has been proposed and
their properties ascertained, it is commonly acknowledged that multi-
variate time series modelling proves superior for this task, since it allows
the identification of innovation sources and dynamic effects by exploiting
the information on series subject to the same overall environment or
related by causal links.
An approach enjoying vast popularity is rooted in Sims’s (1980) influ-
ential critique concerning the identification restrictions in simultaneous
equation systems, and consists in estimating a reduced form model such
as a vector autoregression, possibly incorporating cointegration restric-
tions, and inverting it so as to gain information about trends and cycles.
It is assumed that the estimated model is a good statistical representation
of the data generating process (Johansen and Juselius, 1992).
Stock and Watson (1988) extended the univariate Beveridge–Nelson
decomposition showing that cointegrated systems can be represented in
terms of a reduced number of common stochastic trends, whose data
generating process is the random walk, plus transitory or stationary
components.
Later on, Vahid and Engle (1993), building on the notion of common
feature introduced by Engle and Kozicki (1993), determined the set of
conditions under which the cyclical component can be represented in
terms of a smaller number of common stochastic cycles. Applications
followed, the most successful of which deal with the particular case
arising when the number of common cycles and common trends adds up
to the dimension of the system; as a matter of fact that is the only case
for which an explicit solution is provided, with the cointegration relation-
ships acting as cycle generators. The common cycles model can be viewed
here as a refinement of the Stock and Watson model and thus a particu-
lar case of the multivariate Beveridge–Nelson decomposition.
405
© Blackwell Publishers Ltd, 1997. Published by Blackwell Publishers, 108 Cowley Road, Oxford
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A different permanent-transitory decomposition has been proposed by
Gonzalo and Granger (1995) who define the components (factors) as
linear combinations of the original series so that they are ‘observable’; of
course, the permanent component is in general no longer represented by
a multivariate random walk, although it is an I(1) process responsible for
the non-stationary behaviour of the series. The feature rendering this
decomposition attractive is the linearity in the observations, which often
eases the common factors’ interpretation.
The main purpose of this paper is to derive an explicit expression for
the unobserved components of the decompositions mentioned above in
terms of the matrices of the interim multiplier representation of a cointe-
grated system. Apart from having computational advantages in the extrac-
tion of the latent variables, as only quantities already available are
involved in the calculations, the results presented are relevant for the
interpretation of both long- and short-run dynamics of a cointegrated
system and give insight on the relation between the Beveridge–Nelson
and the Gonzalo–Granger decompositions. Moreover, it is possible to
extend the trend cycle decomposition also to systems in which the number
of common trends and cycles do not add up to the dimension of the
system; the issue of choosing a suitable basis for the common components
is addressed too.
Here is a brief outline of the paper: the next section deals with the
interim multiplier representation which is generally adopted for estima-
tion and testing of a cointegrated system. One of its advantages is that it
lends itself to be cast in state space form quite straightforwardly, which is
done in Section III. In the subsequent section the algorithm devised by
Beveridge and Nelson for the extraction of a trend from a difference
stationary process is applied and the results are discussed. Section V deals
with the case when a non-zero drift term is present and Section VI
illustrates the results with reference to a data sets quite popular in the
common cycle literature, namely the trivariate system investigated by
King et al. (1991). In Section VII we conclude.
II. INTERIM MULTIPLIER REPRESENTATION OF A COINTEGRATED
SYSTEM
Consider the p-th order autoregression for the NÅ1 time series vector Xt:
Xt\P1Xtµ1+...+PpXtµp+et,t\1, .. . , T, (1)
where et1NID(0,S) and Xµp+1,...,X0fixed. Denoting P(L)\
INµP1Lµ...µPpLp, we are concerned with the situation !P(L)!80
when !L!1s, i.e. there are roots on or outside the unit circle.
On writing P(L)\P(1)Lp+DG(L), where G(L)\INµG1Lµ...
µGpµ1Lpµ1and Gj\µIN+Sj
i\1Pj, j\1,...,pµ1, the model can be
reparameterized as an Error Correction Model:
© Blackwell Publishers 1997
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