The Asymptotic Properties of Estimates of the Parameters of a Single Equation in a Complete System of Stochastic Equations

Abstract

In a previous paper [2] the authors have given a method for estimating the coefficients of a single equation in a complete system of linear stochastic equations. In the present paper the consistency of the estimates and the asymptotic distributions of the estimates and the test criteria are studied under conditions more general than those used in the derivation of these estimates and criteria. The point estimates, which can be obtained as maximum likelihood estimates under certain assumptions including that of normality of disturbances, are consistent even if the disturbances are not normally distributed and (a) some predetermined variables are neglected (Theorem 1) or (b) the single equation is in a non-linear system with certain properties (Theorem 2). Under certain general conditions (normality of the disturbances not being required) the estimates are asymptotically normally distributed (Theorems 3 and 4). The asymptotic covariance matrix is given for several cases. The criteria derived in [2] for testing the hypothesis of over-identification have, asymptotically, $\chi^2$-distributions (Theorem 5). The exact confidence regions developed in [2] for the case that all predetermined variables are exogenous (that is, that the difference equations are of zero order) are shown to be consistent and to hold asymptotically even when this assumption is not true (Theorem 6).