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. If each element of a vector of time series x first achieves
stationarity after differencing, but a linear combination a'x is already stationary,
the time series x are said to be co-integrated with co-integrating vector a. There
may be several such co-integrating vectors so that a becomes a matrix. Interpreting
a'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. 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 proposed.
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. 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 GNP is co-integrated with M2, but not M1, M3,
or aggregate liquid assets.