Likelihood ratio tests on cointegrating vectors, disequilibrium adjustment vectors, and their orthogonal complements

"Cointegration theory provides a flexible class of statistical models that combine long-run relationships and short-run dynamics. This paper presents three likelihood ratio (LR) tests for simultaneously testing restrictions on cointegrating relationships and on how quickly the system reacts to the deviation from equilibrium implied by the cointegrating relationships. Both the orthogonal complements of the cointegrating vectors and of the vectors of adjustment speeds have been used to define the common stochastic trends of a nonstationary system. The restrictions implicitly placed on the orthogonal complements of the cointegrating vectors and of the adjustment speeds are identified for a class of LR tests, including those developed in this paper. It is shown how these tests can be interpreted as tests for restrictions on the orthogonal complements of the cointegrating relationships and adjustment vectors, which allow one to combine and test for economically meaningful restrictions on cointegrating relationships and on common stochastic trends"--Federal Reserve Board web site.