These notes are based on the following article:
- Rothenberg, Thomas J. (1971) Identification in Parametric Models. Econometrica 39, 577–591.
Given an observed sample, we want to perform inference about an underlying structure. Identification refers to the question of whether such inference is even possible. Or, under what conditions is it possible?
Rothenberg is interested in conditions for conditions for identification in general parametric models, with rank conditions for linear models presented as a special case. The broader structural identification problem is more general than that of parametric identification. See Koopmans and Reiersøl (1950) for a general formulation.
The conditions laid out are related to the information matrix, which measures the information available from the sample about the parameters. Lack of identification is simply lack of sufficient information to distinguish between different structures.
The results are for unconditional models , but extensions to conditional models are straightforward.
Let and suppose that the distribution of is a member of some family of distribution functions . A structure is a set of hypotheses which imply a unique distribution . Let be the collection of all structures.
Two structures and in are observationally equivalent if they imply the same distribution of : . A structure is identifiable if there is no other structure in which is observationally equivalent.
These definitions apply to general sets and , but can be specialized to the case of parametric models. Suppose that every structure in can be described by a real vector and let be the set of all possible values of . Now, suppose that the distribution of is known to be a continuous density function of the form for some unknown value of . Let be the parametric family of all such functions. We have thus reduced the problem of identifying a general structure to that of identifying a single point in the parameter space .
We can now redefine the definitions above in terms of a generic parametric model. Two parameters (structures) and in are observationally equivalent if they imply the same distribution of : for all . Similarly, a parameter is identifiable if there is no other parameter in which is observationally equivalent.
Rothenberg considers two notions of identification—local and global. Global identification is simply a more precise way of labeling the definition above, that no other in the entire parameter space is observationally equivalent to . Local identification relaxes this requirement, and only requires there to be some open set containing over which no other is observationally equivalent. Clearly, global identification implies local identification.
So, to summarize, the (parametric) identification problem is to find conditions on and which guarantee that is identifiable, at least locally, but perhaps globally.
A parametric model is regular if the following conditions hold:
- is an open set,
- is a proper density for all ,
- the support of under is the same for all ,
- is smooth in ,
- the elements of the information matrix are continuous functions of .
The results that follow are for regular parametric models.
The information matrix is the usual matrix
A point is a regular point of a matrix if there exists an open set containing over which has constant rank.
Rothenberg’s main result on local identification in regular models is the following.
Theorem: If is a regular point of the information matrix , then is locally identifiable if and only if is nonsingular.
Following this is a theorem considering the case where is known to satisfy a set of constraints of the form .
Proving global identification is harder. Two cases are considered.
A function is a member of the exponential family of densities if it can be written where is differentiable in .
Theorem: If is a member of the exponential family and is nonsingular in a convex set containing , then every is globally identifiable.
Theorem: Suppose there exist known functions such that for all , for all . Then every is globally identifiable.
- Koopmans, T.C. and O. Reiersøl (1950). The Identification of Structural Characteristics. Annals of Mathematical Statistics 21, 165–181.