A fix-point theorem with econometric background: Part I. The theorem

The theorem provides a new approach to a problem that has originated in econometrics, namely the statistical estimation of parameters of so-called interdependent systems. We formulate the problem generally as one of orthogonal projection in Hilbert vector space, an interpretation based on the definition of interdependent systems in terms of conditional expectations; The subspace on which the vectors are projected is spanned by vectors some of which are given, whereas others--and this is a nonlinear feature of the problem--are unknown and to be determined by the projection. The problem is solved by an iterative least squares procedure that rests on a new application of the principle of contraction mapping.