## Identification of echelon canonical forms for vector linear processes using least squares.(English)Zbl 0756.62033

To identify stationary and invertible $$p$$-dimensional autoregressive- moving-average (ARMA) processes, the author considers models that are presumed to be represented in (reversed) echelon canonical form. The least squares method is used to estimate the model parameters. When the transfer function of the process is assumed to be rational and the model structure is given, the consistency of the least squares estimator is shown (theorem 3.1 and corollary 3.1). In section 4, a consistency procedure is presented to determine a (reversed) echelon canonical form structure (theorem 4.1). In section 5, when the transfer function of the process is nonrational, under little modifications of the assumptions on the model assumed above, the consistency of the least squares estimator still holds (but with a slower speed of convergence), and the result to determine the model structure remains true.

### MSC:

 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 62F12 Asymptotic properties of parametric estimators 93E12 Identification in stochastic control theory
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