A likelihood ratio test for the rank of a cointegration submatrix

This paper proposes a likelihood ratio test for rank deficiency of a submatrix
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 a chi-squared 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.