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Overidentified Model

An overidentified model is a model for which there is more than enough information in the data to estimate the model parameters. By contrast, an underidentified model has insufficient information from the data to estimate the free parameters, and a just-identified model has just enough information to solve for the free parameters. Identification is an important part of many statistical models but is most frequently and extensively discussed in the context of structural equation modeling. In structural equation modeling, the information available from the data is contained in the observed variance–covariance matrix and the parameters consist of the freely estimated parameters of the model. A model must be just-identified or overidentified in order to estimate parameters. Overidentified models are particularly important in structural equation modeling because such models allow the analyst to examine indices of model fit or measures of how well the tested model describes the observed data.

In this entry, overidentified models are described in the broader context of issues of model identification. First, this entry examines model identification using examples based on solving systems of equations. Next, the focus shifts to model identification issues in structural equation modeling, addressing both what it means for a model to be overidentified and practical approaches to determining whether a model is identified. The entry concludes by describing the importance of overidentified models in structural equation modeling for the purpose of assessing model fit and comments on the parallels between identification issues and model degrees of freedom.

Identification in Systems of Equations

A common way to think about the identification problem is by analogy to the process of solving systems of equations. The important players are the number of unknowns in the system of equations (typically represented by letters such as x, y, z, etc.) and the number of equations in the system. For the purposes of this entry, one can think of the unknowns as model parameters to be estimated and of the equations as sources of information. (In the study of systems of equations, systems are said to be determined, underdetermined, or overdetermined. These terms are parallel to the terms just-identified, underidentified, and overidentified.)

When the number of unknowns is greater than the number of equations, the system is underidentified. There is not sufficient information from the equations to solve for the unknowns and there may be an infinite number of values for the unknowns that would satisfy the system. This is the case for the following two equations with three unknowns. There are many possible solutions for the unknowns. For example, the unknowns could be x = 14, y = 4, z = –1, or x = 20, y = 10, z = –7, but there’s not enough information in the system to identify a unique solution.

x – y =10.

z + y = 3.

When the number of unknowns equals the number of equations, the system is just-identified. It will be possible to use the information in the system to solve for the unknowns. In these equations, one can solve for x and y to find their respective values. The solution can only be x = 7 and y = −2.

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