Estimating Linear Dynamical Systems Using Subspace Methods

This paper provides a survey on a class of so-called subspace methods whose main proponent is CCA proposed by Larimore (1983, Proceedings of the 1983 American Control Conference 2). Because they are based on regressions these methods for the estimation of ARMAX systems are attractive as a result of their conceptual simplicity and their numerical advantages as compared to traditional estimators based on criterion optimization. Under the assumption of correct specification the methods provide consistent and asymptotically normal estimates for stationary ARMAX processes where the innovations may be conditionally heteroskedastic and the exogenous inputs are strictly stationary of sufficiently short memory. For stationary autoregressive moving average (ARMA) processes with independent and identically distributed (i.i.d.) Gaussian innovations the estimates are even asymptotically efficient. For I(1) ARMA processes the estimates of both the long-run and the short-run dynamics are consistent without using the knowledge that the data are integrated in the algorithm. Additionally the algorithms provide easily accessible information on the appropriateness of the chosen model complexity. The algorithms include a number of design parameters that have to be set by the user. These include the order of the estimated system. This paper collects up-to-date knowledge on the effects of these design parameters, leading to a number of suggested automated choices to obtain a fully automated estimation procedure.