Model selection is a popular strategy in structural equation modeling (SEM). To select an “optimal” model, many selection criteria have been proposed. In this study, we derive the asymptotics of several popular selection procedures in SEM, including AIC, BIC, the RMSEA, and a two-stage rule for the RMSEA (RMSEA-2S). All of the results are derived under weak distributional assumptions and can be applied to a wide class of discrepancy functions. The results show that both AIC and BIC asymptotically select a model with the smallest population minimum discrepancy function (MDF) value regardless of nested or non-nested selection, but only BIC could consistently choose the most parsimonious one under nested model selection. When there are many non-nested models attaining the smallest MDF value, the consistency of BIC for the most parsimonious one fails. On the other hand, the RMSEA asymptotically selects a model that attains the smallest population RMSEA value, and the RESEA-2S chooses the most parsimonious model from all models with the population RMSEA smaller than the pre-specified cutoff. The empirical behavior of the considered criteria is also illustrated via four numerical examples.
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