Structural equation modeling with near singular covariance matrices

Conventional structural equation modeling involves fitting a structural model to the sample covariance matrix S. Due to collinearity or small samples with practical data, nonconvergences often occur in the estimation process. For a small constant aa, this paper proposes to fit the structural model to the covariance matrix Sa=S+aI. When treating Sa as the sample covariance matrix in the maximum likelihood (ML) procedure, consistent parameter estimates are still obtained. The asymptotic distributions of the parameter estimates and the corresponding likelihood ratio statistic are studied and compared to those by the conventional ML. Two rescaled statistics for the overall model evaluation with modeling Sa are constructed. Empirical results imply that the estimates from modeling Sa are more efficient than those of fitting the structural model to S even when data are normally distributed. Simulations and real data examples indicate that modeling Sa allows us to evaluate the overall model structure even when S is literally singular. Implications of modeling Sa in a broader context are discussed.