Multivariate Linear Relationships: Maximum Likelihood Estimation and Regression Bounds

Abstract

Summary The well known result that, when the ratio of error variances is specified in normal theory bivariate linear relationships, the maximum likelihood estimator of the slope parameter takes the same form for both structural and functional relationships is extended to the multivariate case. Furthermore, it is well known that in the bivariate case the estimator of the slope is bounded by the two regression estimators. Analogous results are derived for the multivariate case. The asymptotic variance–covariance matrix of the maximum likelihood estimators of the coefficients of a multivariate linear relationship is determined, and an estimator of this matrix is proposed. Large sample confidence regions for the parameters are formulated and the relationship between these and normal theory regression confidence regions is discussed.