The Rank of a Sub-Matrix of Cointegration

This paper proposes a test of the rank of the sub-matrix of b, where b is a cointegrating matrix. In addition, the sub-matrix of d, an orthogonal complement to b, is investigated. We show that information on the rank of the sub-matrix of b and/or d is useful in several situations. We construct the test statistic by using the eigenvalues of the quadratic form of the sub-matrix. We show that the test statistic has a limiting chi-squared distribution when the data is non-trending, and we propose a conservative test when the data is trending. Finite sample simulations show that, although the simulation settings are limited, the proposed test works well.