Quasi-least-squares estimation in semimartingale regression models. (English) Zbl 0586.62137

Christopeit, Norbert

Stochastics 16, 255-278 (1986).

The class of linear regression models \[ y_ t=y_ o+B\int^{t}_{0}x_ sd\theta_ s+M_ t,\quad t\geq 0, \] is considered. The m-dimensional process \(\{y_ t\}\) is a semimartingale which is generated as the sum of the k-dimensional input process \(\{x_ t\}\), integrated with respect to some increasing process \(\{\theta_ t\}\) and weighted with some \(m\times k\) parameter matrix B, plus an unobservable disturbance \(\{M_ t\}\) which is assumed to be an m- dimensional martingale.
Denoting \(\lambda_{\max}(t)\) resp. \(\lambda_{\min}(t)\) the maximal resp. minimal eigenvalue of \(Z_ t=\int^{t}_{o}x_ sx'\!_ xd\theta_ s\), it is supposed that the assumption \[ \lambda_{\min}(T)\to \infty \quad a.e.\quad and\quad (\log \lambda_{\max}(T))^{1-\delta}=O(\lambda_{\min}(T))\quad a.e. \] for some \(\delta >0\) or, in its weaker form: \[ \lambda_{\min}(T)\to \infty \quad a.e.\quad and\quad \log \lambda_{\max}(T)=o(\lambda_{\min}(T))\quad a.e. \] holds. Under this assumption and some other technical condition it is shown that the estimator \[ \hat B'\!_ T=(\int^{T}_{0}x_ sx'\!_ sd\theta_ s)^{-1}\int^{T}_{0}x_ sdy'\!_ s \] converges a.e. to the true parameter matrix B. Finally some examples are investigated using the main theorem.

Reviewer: D.Jaruškova

MSC:

62M09	Non-Markovian processes: estimation
62M10	Time series, auto-correlation, regression, etc. in statistics (GARCH)
60G44	Martingales with continuous parameter
62M05	Markov processes: estimation; hidden Markov models

Keywords:

quasi-least-squares estimation; stochastic regressors; local martingale; maximal eigenvalue of the design matrix; minimal eigenvalue; transformation formula; stability properties; strong consistency; linear regression models; semimartingale

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