ARMA-based SEM when the number of time points T exceeds the number of cases N: Raw data maximum likelihood

Longitudinal structural equation modeling has generally addressed the time-dependent covariance structure of a relatively small number of repeated measures, T, observed in a relatively large representative sample, N. In contrast, the literature on autoregressive moving average modeling is usually directed at a single realization comprising many observations, i.e. N=1, and T>50. This paper deals with autoregressive moving average based structural equation modeling of time series data, in the situation that N is small, T is intermediate, and T>N. The aims of this paper are: (a) to give a brief overview of the development of alternative formulations of the likelihood function to obtain estimates of autoregressive moving average parameters, in particular the formulation that lies at the basis of Melard's algorithm; (b) show the equivalence between the likelihood function to obtain estimates for these parameters, and the raw data likelihood method that can be used in structural equation modeling programs like Mx, and demonstrate this equivalence through simulation experiments; and (c) provide illustrations of this use of Mx with real data.