Assessing the effect of model missspecifications on parameter estimates in structural equation models

Model misspecifications may have a systematic effect on parameters, causing biases in their estimates. In the application of structural equation models, every interesting model is fallible. When simultaneously evaluating a model, it is of interest to study whether all parameters are affected by a misspecification. This paper provides three procedures for evaluating such an effect: (1) analyzing the path, (2) using a functional relationship, and (3) using a significance test. Analyzing the path is illustrated through a confirmatory factor model. This method is ad hoc but intuitive. A more rigorous approach is built upon the concept of orthogonality of two sets of parameters. When parameter a is orthogonal to parameter b, omitting parameter b. will not affect the estimation of parameter a. The functional relationship of two sets of parameters is used to cheek their orthogonality. The distribution of the difference between estimates based on different models is obtained, which provides a Hausman-like way to check significant parameter differences that are due to biases. Examples illustrate that these procedures can provide valuable information on identifying parameter estimates that are systematically affected by a model misspecification.