## 1. INTRODUCTION

Structural equation modeling (SEM) with latent variables is a powerful tool for social scientists, allowing researchers to simultaneously estimate relationships between latent variables and observed indicators, and structural relationships between latent variables. The SEM approach thus combines much of the analytic strengths of the psychometric tradition, with its emphasis on measurement, with those of the econometric tradition, with its emphasis on modeling multiequation relations between observed variables. This powerful combination has made SEM an increasingly popular methodological approach in a variety of disciplines.

By far the most popular estimator for SEM is the maximum likelihood (ML) estimator. This is a “full-information” estimator in that it arrives at all parameter estimates simultaneously by using information from the entire system. If the model under investigation is correctly specified and the observed variables do not have excessive kurtosis, the ML estimator is consistent, asymptotically unbiased, and asymptotically efficient (Browne 1984; Jöreskog and Sörbom 1996).^{1} An added advantage of the ML estimator is that it provides a variety of statistics that help analysts evaluate how well the model under investigation “fits” the available data. While these are highly desirable properties, the ML estimator and other full-information estimators do have drawbacks. A major one is that when any part of a model is misspecified, as is almost always the case, bias can spread to parts of the model that are not misspecified. For this reason, research on alternative, limited information estimators continues.

One of the most recent limited information estimators is Bollen's (1996) two-stage least-squares estimator for latent variable SEM (2SLS). This estimator is consistent and has the same asymptotic properties as the ML estimator, but it is efficient only among limited information estimators, not full-information estimators. A recent simulation study suggests that the 2SLS estimator performed somewhat better than the ML estimator under modest misspecification and that the efficiency advantage of the ML estimator was modest at best, even in perfectly specified models (Bollen et al. 2007). A drawback of Bollen's 2SLS estimator for latent variable models is that, unlike the ML estimator, little has been written on how to assess model fit. Bollen (1996: 117–18) briefly discusses possible overidentification tests for equations but does not provide a complete discussion on alternative tests and their relative performance.^{2} In this paper, we describe how to make use of several well-known econometric tests of the appropriateness of instrumental variables (IVs) so as to enable tests of model fit for each overidentified equation in a latent variable model. We explain how to use these tests not only to identify a misspecified model but also to help diagnose the source of misspecification within a model. This is something that goodness-of-fit tests from the ML and other full-information estimators do not provide, although Lagrange multiplier tests (“modification index”) suggest parameter restrictions in a model that might be lifted. Finally, we conduct a large Monte Carlo experiment to evaluate the finite sample properties of these IV tests. The results apply not only to Bollen's 2SLS estimator for SEMs but also to more traditional 2SLS applications found in sociology, economics, and other disciplines where overidentification tests of IVs are applicable.