THE OBSERVATION that large disparity in the variances of the disturbances from different structural equations in a simultaneous system can aid identification is as old as the discovery of the identification problem itself. Thus, in the classic example of E. J. Working [10], it is observed that whereas in the Marshallian cross neither the supply nor the demand curve is identified, this is not the case if one of the curves shifts about a great deal relative to the other. If such shifts do occur, then the relatively stable relationship is approximately traced out by the equilibrium points of intersection. This can, of course, be taken as an early statement of the fact that if one of the equations contains a shifting variable not in the other, the latter equation will be identified by the usual rank condition criterion;2 however, it is clear that even if there is no such explicit variable and all shifts come from the disturbance terms, something is still gained towards the identification of the relatively stable relationship. In other words, the example can be read as implying that information on the variances of the disturbance terms of a multiple equation system can be used for identification of the equation with the smallest disturbance variance.

In estimating a function, certain assumptions are made about the random term. This paper deals with the influence of such assumptions on estimators for the function y = x0. The case of a normally distributed random term, both in its multiplicative and additive forms, is considered here. The major part of the article is devoted to a hypothesis concerning a lognormally distributed random term, for which consistent estimators are derived. THE EXPONENTIAL function is very familiar in economic research. For instance, the relationship between consumption and income is often expressed as y=x', where y denotes consumption and x is income. Usually it is assumed that the actual observations on y are determined by the function x' and by an additional random term s. In estimating this function, the statistical properties of the estimators of a and E(y) will depend on the assumptions made about the random term.2 In this paper some alternative specifications of this random term will be given and their influence on the estimates considered.