On Structural Equation Model Equivalence

A necessary and sufficient condition for equivalence of structural equation models is presented. Compared to existing rules for equivalent model generation (Stelzl, 1986; Lee & Hershberger, 1990; Hershberger, 1994), it is applicable to a more general class including models with parameter restrictions and models that may or may not fulfil assumptions of the rules, to show that two models are nonequivalent, or to nonidentified models. The validity of the replacement rule by Lee and Hershberger, Stelzl's rules, and Hershberger's inverse indicator rule is implied from the present method. Its application for studying model equivalence or lack thereof is demonstrated on a series of empirical examples.