A matrix equality useful in goodness-of-fit testing of structural equation models

The use of augmented moment matrices (replacing covariances) allows to carry out mean- and covariance-structure analysis using conventional software for covariance structure analysis. The present paper establishes the algebraic equality of two alternative goodness-of-fit test statistics in normal-theory GLS analysis of augmented moment matrices. In order to obtain a direct proof for the equality of the two statistics, a fundamental matrix equality involving generalized inverse matrices is elaborated. The results are developed in a general context of multiple group data with non-normal data. The approach adopted allows us to obtain asymptotic robustness results for normal-theory GLS goodness-of-fit test statistics, applicable to a general class of models and multiple group data. Possibly misspecified models and a possibly non-finite variance matrix of sample moments are allowed. Simulated data in a simple context of regression with errors in variables are used to illustrate the practical relevance of the theoretical results obtained.