SEM modeling with singular moment matrices Part III: GLS estimation

We discuss generalized least squares (GLS) and maximum likelihood (ML) estimation for structural equations models (SEM), when the sample moment matrices are possibly singular. This occurs in several instances, for example, for panel data when there are more panel waves than independent replications or for time series data where the number of time points is large, but only one unit is observed. In previous articles, it was shown that ML estimation of the SEM is possible by using a correct Gaussian likelihood function. In this article, the usual GLS fit function is modified so that it is also defined for singular sample moment matrices S. In large samples, GLS and ML estimation perform similarly, and the modified GLS approach is a good alternative when S becomes nearly singular. Both GLS approaches do not work for N = 1, since here S = 0 and the modified GLS approach yields biased estimates. In conclusion, ML estimation (and pseudo ML under misspecification) is recommended for all sample sizes including N = 1.