The Specific Analysis of Structural Equation Models

Some new algebraic results overcome the limitations of DAG theory and give a specific form of structural equation analysis that checks identifiability, tests the implied constraints, equation by equation, and gives consistent estimators of the parameters in closed form from the equations.

Abstract

Conventional structural equation modeling fits a covariance structure implied by the equations of the model. This treatment of the model often gives misleading results because overall goodness of fit tests do not focus on the specific constraints implied by the model. An alternative treatment arising from Pearl's directed acyclic graph theory checks identifiability and lists and tests the implied constraints. This approach is complete for Markov models, but has remained incomplete for models with correlated disturbances. Some new algebraic results overcome the limitations of DAG theory and give a specific form of structural equation analysis that checks identifiability, tests the implied constraints, equation by equation, and gives consistent estimators of the parameters in closed form from the equations. At present the method is limited to recursive models subject to exclusion conditions. With further work, specific structural equation modeling may yield a complete alternative to the present, rather unsatisfactory, global covariance structure analysis.