MAXIMUM-LIKELIHOOD PARAMETER AND RANK ESTIMATION IN REDUCED-RANK MULTIVARIATE LINEAR REGRESSIONS

This paper considers the problem of maximum likelihood (ML) estimation for reduced-rank linear regression equations with noise of arbitrary covariance, The rank-reduced matrix of regression coefficients is para meterized as the product of two full-rank factor matrices. This parame terization is essentially constraint free, but it is not unique, which renders the associated ML estimation problem rather nonstandard. Neve rtheless, the problem turns out to be tractable, and the following res ults are obtained: An explicit expression is derived for the ML estima te of the regression matrix in terms of the data covariances and their eigenelements. Furthermore, a detailed analysis of the statistical pr operties of the ML parameter estimate is performed. Additionally, a ge neralized likelihood ratio test (GLRT) is proposed for estimating the rank of the regression matrix. The paper also presents the results of some simulation exercises, which lend empirical support to the theoret ical findings.