The Likelihood Ratio Test for the Rank of a Cointegration Submatrix*

Abstract

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.

Regression-based analysis of cointegration systems

“…2 in Boswijk, 1996, or the discussion in Luukkonen et al, 1999, which both mention the identification scheme implicit in a triangular form), noting that our particular identification scheme is valid if we estimate correctly the true r and our chosen common trends are true ones (which is the case wpa1 if r = r). The validity of our identification restrictions can be checked by the procedures of Boswijk (1996) or Luukkonen et al (1999) (which provide tests for the validity of identifying restrictions of a more general form), or by the tests for the rank of a cointegration submatrix of Kurozumi (2005) and Paruolo (2006). Note, however, that the validity of all these procedures relies on the knowledge of the cointegrating rank.…”