On the nature of SEM estimates of ARMA parameters

Several papers have been devoted to the use of structural equation modeling software in fitting autoregressive moving average models to an univariate series observed in a single subject. Van Buuren went beyond specification and examined the nature of the estimates obtained with structural equation modeling software. Though the results were mixed, he concluded these parameter estimates resemble true maximum likelihood estimates. Molenaar (1999) argued that the negative findings for pure moving average models might be due to the absence of invertibility constraints in Van Buuren's simulation experiment. The aim of this paper is to: (a) reexamine the nature of structural equation modeling estimates of autoregressive moving average parameters; (b) replicate Van Buuren's simulation experiment in the light of Molenaar's comment; and (c) examine the behavior of the log-likelihood ratio test. We conclude that estimates of autoregressive moving average parameters obtained with structural equation modeling software are identical to those obtained by univariate stochastic model preliminary estimation, and are not true maximum likelihood estimates. Still, these estimates, which may be viewed as moment estimates, have the same asymptotic properties as maximum likelihood estimates for pure autoregressive processes. For pure moving average processes they are biased and less efficient. The estimates from structural equation modeling software for mixed processes seem to have the same asymptotic properties as maximum likelihood estimates. Furthermore the log-likelihood ratio is reliable for pure autoregressive processes, but this is not the case for pure moving average processes. For mixed processes, the behavior of the log-likelihood ratio varies, and in this case these statistics should be handled with caution.