The likelihood ratio test for the rank of a cointegration submatrix

This paper proposes a likelihood ratio test for rank-dficiency of a sub- matrix of the cointegrating matrix. Special cases of the test include the one of invalid normalization in systems of cointegrating equations, the feasibility of permanent-transitory decompositions and of subhypotheses related to neutrality and long run Granger noncausality. The proposed test has x2 limit distribution and indicates the validity of the normalization with probability one in the limit, for valid normalizations. The asymptotic properties of several derived estimators of the rank are also discussed. It is found that a testing procedure that starts from the hypothesis of minimal rank is preferable.