Maximum Likelihood Estimation of the Parameters of a System of Simultaneous Regression Equations

Procedures for computing the full information maximum likelihood (FIML) estimates of the parameters of a system of simultaneous regression equations have been described by Koopmans, Rubin, and Leipnik, Chernoff and Divinsky, Brown, and Eisenpress. However, all of these methods are rather complicated since they are based on estimating equations that are expressed in an inconvenient form. In this paper, a transformation of the maximum likelihood (ML) equations is developed which not only leads to simpler computations but which also simplifies the study of the properties of the estimates. The equations are obtained in a form which is capable of solution by a modified Newton-Raphson iterative procedure. The form obtained also shows up very clearly the relation between the maximum likelihood estimates and those obtained by the three-stage least squares method of Zellner and Theil.