SEM Modeling with Singular Moment Matrices Part I: ML-Estimation of Time Series

Abstract

A structural equation model (SEM) with deterministic intercepts is introduced. The Gaussian likelihood function does not contain determinants of sample moment matrices and is thus well-defined for only one statistical unit. The SEM is applied to the dynamic state space model and compared with the Kalman filter (KF) approach. The likelihood of both methods are shown to be equivalent, but for long time series numerical problems occur in the SEM approach, which are traced to the inversion of the latent state covariance matrix. Both approaches are compared on several aspects. The SEM approach is now open for idiographic (N = 1) analysis and estimation of panel data with correlated units.

Keywords:

• estimation
• Kalman filtering (KF)
• maximum likelihood (ML)
• state space models
• structural equation models (SEM)
• time series

Notes

¹Otherwise there are exact identities in the components of y and the singular normal distribution can be used (Mardia, Kent, and Bibby, 1979, p. 41). See also Footnote 4.

²Moreover, the system matrices may depend on lagged measurements Z ⁱ = {z _i, …, z ₀}, and the measurement matrices H_i,d _i R _i may depend on Z ⁱ⁻¹ in order to specify ARCH effects (AutoRegressive Conditional Heteroscedasticity [Bollerslev, Engle, and Nelson, 1994]). ARCH models are used in financial econometrics to model the volatility fluctuations of return series of stocks, exchange rates, etc. The so called conditional Gaussian model can be treated by the Kalman filter (Liptser and Shiryayev, 2001, Vol. 2), but not by SEM. This is because the SEM approach assumes a joint Gaussian distribution of the manifest and latent states, whereas in the recursive approach only conditional Gaussianity is required.

³SEM with N = 1; SEM states y and x (without index i) must be distinguished from the dynamical states y _i , x _i in Eqs. (9–10).

⁴Σ _y is positive definite (p.d.), if either (i) Σ_ϵ is p.d. (cf. Magnus and Neudecker, 1999, Theorem 22, p. 21) or (ii) if Σ_ζ is p.d. and ΛB ₁ has rank K = k(T + 1) (cf. Golub and Van Loan, 1996, Theorem 4.2.1, p. 141).

⁵The prediction errors ν _i+1 = z _i+1 − E[z _i+1|Z ⁱ ] are martingale differences, that is, E[ν_i+1|Z ⁱ ] = 0 and thus uncorrelated, since for i > j we have .

For system matrices independent of the measurements they are Gaussian and thus even independent (cf. Liptser and Shiryayev, 2001, Vol. 2, Ch. 13).

⁶Highest posterior density

⁷ http://www.fernuni-hagen.de/imperia/md/content/ls_statistik/sde.zip, http://www.fernuni-hagen.de/imperia/md/content/ls_statistik/publikationen/semarchive.exe