| Authors: |
Robert F. Engle
and C. W. J. Granger
|
| Tags: |
Autoregression,
Co-Integration,
Correction,
Dickey-Fuller
Error
Multivariate
Roots,
Series,
Tests.
Time
Unit
Vector
|
| Abstract: |
The relationship between co-integration and error correction
models, first suggested in Granger (1981), is here extended and
used to develop estimation procedures, tests, and empirical
examples. \par If each element of a vector of time series x'
first achieves stationarity after differencing, but a linear
combination Alpha' x' is already stationary, the time series x'
are said to be co-integrated with co-integrating vector alpha.
There may be several such co-integrating vectors so that alpha
becomes a matrix. Interpreting Alpha' x' = 0 as a long run
equilibrium, co-integration implies that deviations from
equilibrium are stationary, with finite variance, even though the
series themselves are nonstationary and have infinite variance.
\par The paper presents a representation theorem based on Granger
(1983), which connects the moving average, autoregressive, and
error correction representations for co-integrated systems. A
vector autoregression in differenced variables is incompatible
with these representations. Estimation of these models is
discussed and a simple but asymptotically efficient two-step
estimator is propsed. Testing for co-integration combines the
problems of unit root tests and tests with parameters
unidentified under the null. Seven statistics are formulated and
analyzed. The critical values of these statistics are calculated
based on a Monte Carlo simulation. Using these critical values,
the power properties of the tests are examined and one test
procedure is recommended for application. \par In a series of
examples, it is found that consumption and income are
co-integrated, wages and prices are not, short and long interest
rates are, and nominal GDP is co-integrated with M2, but not M1,
M3 or aggregate liquid assets. |
@article{Engl-Gran-1987,
title = {Co-Integration and Error Correction: Representation, Estimation,
and Testing},
author = {Robert F. Engle and C. W. J. Granger},
journal = {Econometrica},
month = {March},
number = {2},
pages = {251-276},
volume = {55},
year = {1987},
abstract = {The relationship between co-integration and error correction
models, first suggested in Granger (1981), is here extended and
used to develop estimation procedures, tests, and empirical
examples. \par If each element of a vector of time series x'
first achieves stationarity after differencing, but a linear
combination Alpha' x' is already stationary, the time series x'
are said to be co-integrated with co-integrating vector alpha.
There may be several such co-integrating vectors so that alpha
becomes a matrix. Interpreting Alpha' x' = 0 as a long run
equilibrium, co-integration implies that deviations from
equilibrium are stationary, with finite variance, even though the
series themselves are nonstationary and have infinite variance.
\par The paper presents a representation theorem based on Granger
(1983), which connects the moving average, autoregressive, and
error correction representations for co-integrated systems. A
vector autoregression in differenced variables is incompatible
with these representations. Estimation of these models is
discussed and a simple but asymptotically efficient two-step
estimator is propsed. Testing for co-integration combines the
problems of unit root tests and tests with parameters
unidentified under the null. Seven statistics are formulated and
analyzed. The critical values of these statistics are calculated
based on a Monte Carlo simulation. Using these critical values,
the power properties of the tests are examined and one test
procedure is recommended for application. \par In a series of
examples, it is found that consumption and income are
co-integrated, wages and prices are not, short and long interest
rates are, and nominal GDP is co-integrated with M2, but not M1,
M3 or aggregate liquid assets.},
keywords = {Autoregression, Co-Integration, Correction, Dickey-Fuller Error Multivariate Roots, Series, Tests. Time Unit Vector }
}
::