The Impact of Specification Error on the Estimation, Testing, and Improvement of Structural Equation Models

The purpose of this paper is to assess the impact of misspecification on the estimation, testing, and improvement of structural equation models. A population study is conducted whereby a prototypical latent variable model is misspecified in various ways. Measurement model and structural model misspecifications are considered separately and together. The maximum likelihood estimator (ML) is compared to a limited information two-stage least squares (2SLS) estimator implemented in LISREL. The ratio of chi-square to its degrees of freedom and power of the likelihood ratio test is assessed for each misspecification. The modification index provided by LISREL is also studied. Results indicate that ML and 2SLS estimates of measurement and structural parameters are both affected by measurement model misspecification. For misspecification of the structural part, ML is shown to propagate errors throughout the structural parameters whereas 2SLS isolates errors only in the parameters of the misspecified equation. Results also show that relying on the ratio of chi-square to degrees of freedom as an index of fit may lead to accepting models with severe parameter bias. Finally, the modification index is shown to be an unreliable indicator of the location of a specification error.