Identification in Parametric Models

Abstract

A theory of identification is developed for general stochastic model whose probability law is determined by finite number of parameters. It is shown under weak regularity conditions that local identifiability of the unknown parameter vector is equivalent to nonsingularity of the information matrix. The use of reduced-form parameters to establish identifiability is also analyzed. The general results are applied to the familiar problem of determining whether the coefficients of system of linear simultaneous equations are identifiable. THE IDENTIFICATION PROBLEM concerns the possibility of drawing inferences from observed samples to an underlying theoretical An important part of econometric theory involves the derivation of conditions under which given structure will be identifiable. The basic results for linear simultaneous equation systems under linear parameter constraints were given by Koopmans and Rubin [10] in 1950. Extensions to nonlinear systems and nonlinear constraints were made by Wald [15], Fisher [4, 5, 6], and others. A summary of these results can be found in Fisher's comprehensive study [7]. The identification problem has also been thoroughly analyzed in the context of the classical single-equation errors-in-variables model. The basic papers here are by Neyman [12] and Reiers0l [13]. Most of this previous work on the identification problem has emphasized the special features of the particular model being examined. This has tended to obscure the fact that the problem of structural identification is very general one. It is not restricted to simultaneous-equation or errors-in-variables models. As Koopmans and Reiers0l [9] emphasize, the identification problem is a general and fundamental problem arising, in many fields of inquiry, as concomitant of the scientific procedure that postulates the existence of structure. In their important paper Koopmans and Reiers0l define the basic characteristics of the general identification problem. In the present paper we shall, in the case of general parametric model, derive some identifiability criteria. These criteria include the standard rank conditions for linear models as special cases. Our approach is based in part on the information matrix of classical mathematical statistics. Since this matrix is measure of the amount of information about the unknown parameters available in the sample, it is not surprising that it should be related to identification. For lack of identification is simply the lack of sufficient information to distinguish between alternative structures. The following results make this relationship more precise.2