Identification of a linear system from inexact data: A three-variable example☆

Abstract In 1901 Pearson formulated the general problem of how to fit a hyperplane in the most efficient way to a system of points in a data space. This problem is still not exactly solved in all its generality. As Kalman and Los have shown, all statistical attemps to solve the problem have failed, because each of them can provide only prejudicial and statistical, but not objective and mathematical solutions. However, exact mathematical solutions do exist for special cases. This paper's main principle of linear identification from inexact data provides the mathematical framework in which the problem and the deficiencies of the statistical solutions are conveniently discussed, in particular those of the least squares regression and statistical common factors schemes. It will be argued that the exact common factors, or Frisch scheme, offers most promise to direct us to complete and exact solutions, even though it imposes severe restrictions on the orders of the systems because of Wilson's inequality. Throughout this paper the problem and its various solution schemes are illustrated by an empirical example consisting of three data variables describing the profitability performance of some large U.S. bank holding companies. For this empirical example the Frisch scheme provides a unique solution, contrary to some earlier pessimistic conclusions.