Uniqueness of Prediction Error Estimates of Multivariable Moving Average Models

Abstract

In prediction error (PE) identification the parameter estimates are given by the global minimum of a scalar-valued function of the innovation sample covariance matrix. It may happen that the loss function has multiple local minimum points so that a numerical search routine can fail to find the global minimum. Such a situation, usually referred to as lack of uniqueness of the estimates, has been experienced in practice and also theoretically examined for various model structure. A unique minimum of the criterion is also crucial for convergence of recursive PE algorithms.Here multivariable moving average (MA) models are considered. It is proved that for such models any reasonable PE criterion has asymptotically a unique stationary point. Furthermore it is shown that this stationary point is a (global) minimum which corresponds to the true parameter vector. This extends the result known for univariate MA models to the multivariate case.