On the Nature of SEM Estimates of ARMA Parameters

Abstract

Several papers have been devoted to the use of structural equation modeling (SEM) software in fitting autoregressive moving average (ARMA) models to a univariate series observed in a single subject. Van Buuren (1997) went beyond specification and examined the nature of the estimates obtained with SEM software. Although the results were mixed, he concluded that these parameter estimates resemble true maximum likelihood (ML) 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 article is to (a) reexamine the nature of SEM estimates of ARMA parameters; (b) replicate Van Buuren's simulation experiment in light of Molenaar's comment; and (c) examine the behavior of the log-likelihood ratio test. We conclude that estimates of ARMA parameters obtained with SEM software are identical to those obtained by univariate stochastic model preliminary estimation, and are not true 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.