Aitken Estimators as a Tool in Allocating Predetermined Aggregates

The problem considered in this paper is one of allocating a predetermined aggregate among its components. An example from econometrics is the allocation among various competing consumption items of a consumer's budget, the total size of which is usually taken as determined outside the allocation framework. Under the specification adopted, one may treat any choice of (m — 1) of the m components of the predetermined aggregate as stochastic, fitting the allocation equations for these (m — 1) variables by Aitken's principle. The parameters of the equation excluded from the estimation procedure may be found by differencing; the estimates obtained in this way are shown to be invariant under any choice of which (m — 1) equations are fitted. Further, this invariance property is distribution free. These results hold good whether the aggregate is simple or weighted. As an alternative interpretation, moreover, one may regard this problem simply as one of fitting m equations containing a common regressor under an exact linear constraint relating this regressor to the m re-gressands of the system.