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Zbl 0741.62083
Duflo, M.; Senoussi, R.; Touati, A.
Propriétés asymptotiques presque sûres de l'estimateur des moindres carrés d'un modèle autorégressif vectoriel. (Almost sure asymptotic properties of the least squares estimators in a vectorial autoregressive model). (French)
[J] Ann. Inst. Henri Poincaré, Probab. Stat. 27, No.1, 1-25 (1991). ISSN 0246-0203

The authors study a vectorial autoregressive model and give the almost sure asymptotic behaviour of the trajectories of stable, unstable and explosive models (theorem 3).\par To estimate the parameters the least squares method is used. There are essentially two hypotheses, controllability and the regularity which is needed when the model is explosive. If the model is regular, the speed of convergence of the estimation errors is always $O((\log(n))\sp{1+\gamma}/n)$ ($\gamma=0$ when the noise has a moment $\geq 2$) (theorem 1). Furthermore, when the model is explosive the speed is exponential $(O(\sqrt{n} \beta\sp n), \beta<1)$ (theorem 5). The predictor of this model and the empirical estimator of the noise covariance are also studied (consistent estimators are given in proposition $A$).\par The paper is very good, extending results for scalar autoregressive models, and contains useful results for estimation theory.
[M.Boutahar (Marseille)]

MSC 2000:: *62M10 Time series, etc. (statistics)
62F12 Asymptotic properties of parametric estimators

Keywords: vectorial autoregressive model; almost sure asymptotic behaviour; controllability; regularity; speed of convergence; predictor; empirical estimator of the noise covariance; consistent estimators; scalar autoregressive models; least squares estimators; strong consistency

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