SEM modeling with singular moment matrices Part III: GLS estimation

Abstract

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.

Keywords:

• Generalized least squares (GLS) estimation
• maximum likelihood (ML) estimation
• panel data
• pseudo maximum likelihood (PML) estimation
• structural equation models (SEM)

Acknowledgments

I would like to thank the anonymous reviewers for valuable suggestions, which have improved the presentation of the article.

Notes

¹ AutoRegressive Integrated Moving Average with eXogenous variables (see, e.g., Box et al., 2015).

² This matrix is positive definite as long as there are no exact identities in the manifest variables.

³ The dependence of μ_n(ψ) and Σ(ψ) will be displayed only when necessary.

⁴ |A| denotes the determinant and tr(A) = ∑_i a_ii the trace of matrix A.

⁵ ML and LS estimates coincide only in simple cases such as the linear regression, when the error covariance matrix does not depend on the parameters (but cf., e.g. Hamerle et al., 1991).

⁶ Otherwise the singular normal distribution can be used (Mardia et al., 1979, p. 41). This case occurs in the presence of restrictions between the components of y_n.

⁷ For example, σ_ikσ_jl + σ_ilσ_jk = (2)(Σ ⊗ Σ).

⁸ tr[ABCD] = row’(A)(D’ ⊗ B)row(C’) (see Appendix I).

⁹ In the main text, we used , but here the dimension could be confounded with the commutation matrix .