The Likelihood Ratio Test for the Rank of a Cointegration Submatrix*
提出一个似然比检验，用于检测协整矩阵中子矩阵的秩是否不足，适用于验证标准化有效性、永久-暂时分解及长期格兰杰非因果性等假设，检验统计量服从卡方极限分布。
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.