Parametrization of factor analysis models

Abstract

For general Factor Analysis (or equivalently, for error-in-variables) models the so-called ‘identifiability problem’ is apparently still unsolved although raised and discussed by K.Pearson already in 1901. In this paper we discuss the identifiability or, what we rather prefer to name model selection problem for factor analysis models representing two blocks of variables. We show that there is a continuum of (minimal) models which connect together two extreme representations of the pure regression type. This continuum of models can be parametrized in terms of a projection matrix describing the part of the modelled vector which is represented exactly (i.e., with no random modelling error) by the factor analysis model. Any procedure of model selection is just a procedure for choosing the ‘exact part’ of the modelled variables. Any choice results in certain modelling error for the remaining variables, whose variance can be computed explicitly. The analysis is done in the static case but carries over almost without changes to dynamic factor analysis models [see Picci and Pinzoni (1986)] by use of spectral representation.