Likelihood Ratio Tests in Reduced Rank Regression with Applications to Econometric Structural Equations

Likelihood ratio criteria are developed for hypotheses concerning multivariate regression matrices when reduced rank is assumed or equivalently hypotheses concerning linear restrictions on regression matrices. In an econometric simultaneous equation model a single restriction may be called a ”structural equation.” A model of reduced rank in statistics is often termed a ”linear functional relationship.” In this paper the likelihood ratio test that the regression matrix satisfies some specified restriction is developed under the assumption of normality. The test for the corresponding hypothesis that the regression matrix of given rank is a specified matrix is also developed. The asymptotic distribution of the test criterion is found under several alternative assumptions on the sequence of models. The ”cointegration model” is included in this study. The test for one structural equation is an advancement on the test statistic proposed by Anderson and Rubin (1949 and 1950).