Identifying Multiple Cointegrating Relations
The Johansen test for cointegration addresses many of the limitations of the Engle-Granger method. It avoids two-step estimators and provides comprehensive testing in the presence of multiple cointegrating relations. Its maximum likelihood approach incorporates the testing procedure into the process of model estimation, avoiding conditional estimates. Moreover, the test provides a framework for testing restrictions on the cointegrating relations B and the adjustment speeds A in the VEC model.
At the core of the Johansen method is the relationship between the rank of the impact matrix C = AB′ and the size of its eigenvalues. The eigenvalues depend on the form of the VEC model, and in particular on the composition of its deterministic terms (see The Role of Deterministic Terms). The method infers the cointegration rank by testing the number of eigenvalues that are statistically different from 0, then conducts model estimation under the rank constraints. Although the method appears to be very different from the Engle-Granger method, it is essentially a multivariate generalization of the augmented Dickey-Fuller test for unit roots. See, e.g., .
The Johansen test is implemented in Econometrics Toolbox™ by the function
jcitest. For an example, see Test for Cointegration Using the Johansen Test.