Estimating ARMAX systems for multivariate time series using the state approach to subspace algorithms

Abstract

This paper discusses the asymptotic properties of estimators of ARMAX systems under weak low-level assumptions on the joint input/output process. The prime representative of this class of algorithms is CVA [W.E. Larimore, System identification, reduced order filters and modeling via canonical variate analysis, in: H.S. Rao, P. Dorato (Eds.), Proc. 1983 Amer. Control Conference 2, Piscataway, NJ, 1983, pp. 445–451]. Sufficient assumptions for strong consistency of the transfer function estimators under the assumption of correct specification are derived and explicit bounds on the orders of convergence are given. The assumptions used on the exogenous inputs are considerably weaker than the ones used in the results available in the literature typically requiring the inputs to be ARMA processes themselves, such as is assumed e.g. in [K. Peternell, W. Scherrer, M. Deistler, Statistical analysis of novel subspace identification methods, Signal Processing 52 (1996) 161–177]. Further sufficient conditions for the asymptotic normality of the estimated parameters are given, again under the assumption of correct specification. Finally two order estimation methods are analyzed and conditions for their consistency are derived.