Closed-form results for vector moving average models with a univariate estimation approach

The estimation of a vector moving average (VMA) process represents a challenging task since the likelihood estimator is extremely slow to converge, even for small-dimensional systems. An alternative estimation method is provided, based on computing several aggregations of the variables of the system and applying likelihood estimators to the resulting univariate processes; the VMA parameters are then recovered using linear algebra tools. This avoids the complexity of maximizing the multivariate likelihood directly. Closed-form results are presented and used to compute the parameters of the process as a function of its autocovariances, using linear algebra tools. Then, an autocovariance estimation method based on the estimation of univariate models only is introduced. It is proved that the resulting estimator is consistent and asymptotically normal. A Monte Carlo simulation shows the good performance of this estimator in small samples.