The Rank of a Sub-matrix of Cointegration

This paper proposes a test of the rank of the sub-matrix of β, where β is a cointegrating matrix. In addition, the sub-matrix of β⊥, an orthogonal complement to β, is investigated. We show that information on the rank of the sub-matrix of β and/or β⊥ 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 chisquared 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. ∗Tel.: 81-42-580-8485; fax: 81-42-580-8195 E-mail address: [email protected] I owe special thanks to Pierre Perron and Taku Yamamoto. All errors are my responsibility. This research was supported by the Ministry of Education, Culture, Sports, Science and Technology under Grants-in-Aid No. 13730023 and No. 14203003.