A Note on Algebraic Solutions to Identification

Abstract

Algebraic methods to establish the identification of structural equation models remain a viable option. However, sometimes it is unclear whether the algebraic solution establishes identification. One example is when there is more than one way to solve for the parameter, but one way leads to a single value and a second way leads to a function with more than one value. This article proves that one explicit and unique solution is sufficient for model identification even when other explicit solutions permit more than one solution. The results are illustrated with an example. The results are useful to attempts to use algebraic means to address model identification.

Keywords:

• identification
• nonlinear simultaneous equations
• structural equation models

ACKNOWLEDGMENTS

The authors thank the anonymous reviewers and the editor for helpful comments on this manuscript. Conversations with Greg Forest and Richard Smith in the early stages of this research were very helpful. Financial support from NSF SES 0617276 and NIDA DA013148-05A2 are gratefully acknowledged.

Notes

Long (1983) only mentions the variances and covariances of the observed variables. In some models, the means also can play a role.

Algebraic solutions can also be useful in formulating new rules of identification (e.g., O'Brien, 1994).

Higher-order moments in some situations provide additional information that would aid model identification. However, these higher-order moments are rarely used and we confine ourselves to the typical situation where a researcher only employs the variances and covariances, and sometimes the means of the observed variables to aid model identification.

A reviewer points out that if θ is discrete, these derivatives would not exist, but that there are cases in which a local identification of θ is well-defined (e.g., when θ is unidimensional and its states admit a total order).

We do not report the entire vectors or due to considerations of space.