On the nature of SEM estimates of ARMA parameters

The aim of this article is to (a) reexamine the nature of structural equation modeling (SEM) estimates of autoregressive moving average (ARMA) parameters; (b) replicate S. Van Buuren's simulation experiment in light of P. C. M. Molenaar's comment; and (c) examine the behavior of the log-likelihood ratio test. It is concluded that estimates of ARMA parameters obtained with SEM software are identical to those obtained by univariate stochastic model preliminary estimation, and are not true maximum likelihood (ML) estimates. Still, these estimates, which may be viewed as moment estimates, have the same asymptotic properties as ML estimates for pure autoregressive (AR) processes. For pure moving average (MA) processes, they are biased and less efficient. The estimates from SEM software for mixed processes seem to have the same asymptotic properties as ML estimates. Furthermore, the log-likelihood ratio is reliable for pure AR processes, but this is not the case for pure MA processes. For mixed processes, the behavior of the log-likelihood ratio varies, and in this case these statistics should be handled with caution.