Estimating Linear Restrictions on Regression Coefficients for Multivariate Normal Distributions

In this paper linear restrictions on regression coefficients are studied. Let the p × q₂ matrix of coefficients of regression of the p dependent variates on q₂ of the independent variates be B₂. Maximum likelihood estimates of an m × p matrix Γ satisfying Γ' B₂ = 0 and certain other conditions are found under the assumption that the rank of B₂ is $p - m$ and the dependent variates are normally distributed (Section 2). Confidence regions for Γ under various conditions are obtained (Section 5). The likelihood ratio test of the hypothesis that the rank of B₂ is a given number is obtained (Section 3). A test of the hypothesis that Γ 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).