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Maximum likelihood estimators in linear regression models with Ornstein-Uhlenbeck process. (English) Zbl 1373.62354

Summary: The paper studies the linear regression model \[ y_t=x^T_t\beta+\varepsilon_t,\quad t=1,2,\dots,n, \] where \[ d\varepsilon_t=\lambda(\mu-\varepsilon_t)dt+\sigma dB_t, \] with parameters \(\lambda\), \(\sigma\in \mathbb{R}^+\), \(\mu\in \mathbb{R}\) and \(\{B_t,t\geq 0\}\) the standard Brownian motion. Firstly, the maximum likelihood (ML) estimators of \(\beta\), \(\lambda\) and \(\sigma^2\) are given. Secondly, under general conditions, the asymptotic properties of the ML estimators are investigated. And then, limiting distributions for likelihood ratio test statistics of the hypothesis are also given. Lastly, the validity of the method are illuminated by two real examples.

MSC:

62J05 Linear regression; mixed models
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
60J60 Diffusion processes
62F12 Asymptotic properties of parametric estimators
62F05 Asymptotic properties of parametric tests
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