The paper explores the relationship between co-integration and error correction models and develops estimation procedures and tests for co- integration. The components of the vector are co-integrated of order b,d, if all components of are integrated of order d and there exists a vector ( such that is integrated of order d-b, . The vector is called the co-integrating vector. Special emphasis is given to the case in which . This situation has the immediate interpretation of a long-run equilibrium so that co-integration implies that deviations from equilibrium are stationary with finite variance despite of the fact that the series themselves are nonstationary and have infinite variances.
By use of the second author’s representation theorem [Co-integrated variables and error-correcting models. UCSD discussion paper 83-13 (1983)] the authors show how co-integrated systems are connected by moving average, autoregressive, and error correction representations. Furthermore, the authors present an asymptotically efficient two-step estimator. Testing for co-integration combines the problems of testing for unit roots and tests with parameters which are not identified under the null. Several test statistics are suggested and the critical values for these statistics obtained by a Monte Carlo simulation are given. The paper closes with some applications.