On the Computation of Full-Information Maximum Likelihood Estimates for Nonlinear Equation Systems

Abstract

N this paper, I will generalize the modified Newton method previously applied in Chow (1968) to the computation of full-information maximum likelihood estimates of parameters of a system of linear structural equations to the case of a system of nonlinear structural equations. The success of that method for linear systems 1 has stimulated my present attempt to generalize it for nonlinear systems. The subject of maximum likelihood estimation of nonlinear simultaneous equation systems has been studied by Eisenpress and Greenstadt (1966). There are three main differences between their approach and ours. First, their basic formulation is more general, assuming that all parameters in the system may appear in every equation,2 whereas we assume as the basic setup that there is a distinct set of parameters belonging to each equation. Second, partly because of the first, we are able to obtain simpler and more explicit expressions for the derivatives of likelihood function required in the calculations. Third, and also partly because of the first, we can conveniently deal with the important problem of linear restrictions on the parameters in the same equation or in different equations. A fourth feature of this paper, and a feature which has partly motivated it, is the contrast of the linear with the nonlinear case. As it will be shown, there are many similarities in the computations of both. This demonstration can enhance our understanding of the nature of the estimation equations. Two additional features of this paper are the treatments of identities in the system and of residuals which may follow an autoregressive scheme. We will derive in section II the estimation equations for nonlinear systems, under the assumptions that each structural equation contains a distinct set of parameters, that the parameters are not subject to any linear restrictions, and that the (additive) residuals are serially uncorrelated. Section III treats the special case when some equations are linear, and contrasts this case with the nonlinear case. Section IV deals with identities and linear restrictions on the parameters. Section V is concerned with the problem of autoregressive residuals.