The Theory of Parametric Identification

Abstract

This paper sets out a general criterion for the identifiability of a statistical system, based on Kullback's integral. It is shown that the general identification problem is equivalent to a maximisation problem, or where parameter restrictions are present, a problem in nonlinear programming. The relationship of this criterion to that based on the of the underlying distribution is also exhibited. THE COLLECTION OF results on the identification problem in econometrics is by now assuming the proportions of an imposing edifice. It is, however, a little surprising to note that this structure has been growing upwards and, more recently, outwards, without a corresponding strengthening in the foundations. It is true that in the case of work on linear simultaneous equation systems (and this, with the work of Koopmans and Reiersol [3] and Chapters 1-4 of Fisher [1] in particular, has almost assumed the status of a classical line of development), these results are founded on a pretty secure rock; to wit, the identifiability of the reduced form in the absence of any singularities in the data matrices. Nevertheless, with the development of other wings on our edifice, it seems desirable to look to more basic things. The recent paper by Thomas Rothenberg [5] provided a welcome attack on this subject. The identifiability of a parametric system is approached via the nonsingularity of R. A. Fisher's information matrix evaluated at the true value of the parameter. The present note generalizes this approach by providing a simple criterion for identifiability, which not only affords an approach to global identification, but also makes no assumptions about the regularity of the underlying distribution. Rothenberg's basic theorem emerges as a simple corollary to this result. The approach has a natural relationship with estimation theory and also provides a straightforward method for delineating the subspace of observationally equivalent parameters in the case of underidentification.