Estimating Linear Restrictions on Regression Coefficients for Multivariate Normal Distributions

Abstract

In this paper linear restrictions on regression coefficients are studied. Let the $p \times q_2$ matrix of coefficients of regression of the $p$ dependent variates on $q_2$ of the independent variates be $\mathbf{\bar B}_2$. Maximum likelihood estimates of an $m \times p$ matrix $\Gamma$ satisfying $\Gamma'\mathbf{\bar B}_2 = 0$ and certain other conditions are found under the assumption that the rank of $\mathbf{\bar B}_2$ is $p - m$ and the dependent variates are normally distributed (Section 2). Confidence regions for $\Gamma$ under various conditions are obtained (Section 5). The likelihood ratio test of the hypothesis that the rank of $\mathbf{\bar B}_2$ is a given number is obtained (Section 3). A test of the hypothesis that $\Gamma$ is a certain matrix is given (Section 4). These results are applied to the "$q$-sample problem" (Section 7) and are extended for certain econometric models (Section 6).