Estimation and Simultaneous Correlation in Complete Equation Systems

Abstract

In this paper the problem of estimating the parameters of complete systems of simultaneous equations is considered. A class of estimates is proposed, based on Aitken’s generalized minimurn-variance approach. It is proved, under certain conditions, that a subclass yields consistent estimates, moreover that a subclass of this yields estimates with the same asymptotic covariance matrix as that of the limitedinformation maximum-likelihood estimates. The joint covariance matrix of estimates of different equations is also given. Both this matrix and the covariance matrix of the estimates of one equation remain valid if the system is nonlinear, under suitable conditions. It is proved that the asymptotic generalized variance of the estimates of one equation is never smaller, and in general larger, than the value obtained for this variance by a mechanical application of Fisher’s formula. Finally, a coefficient of simultaneous correlation for a complete linear system is proposed, which is the natural generalization of the coefficient of multiple correlation; it is closely related to Hotelling’s vector alienation coefficient Section 1 contains a brief introduction to the subject. In section 2 the estimation procedure is described and justified in an elementary way; only those members of the above-mentioned class are considered which appear to be especially promising. In section 2.6 the computational requirements are exposed; it appears that the procedure proposed is much simpler than limited-information maximumlikelihood. Both section 1 and section 2 are written in an expository style. Assumptions and theorems are to be found in section 3.

This article first appeared as a mimeographed memorandum of the Central Planning Bureau, The Hague, June 23, 1953.