Pseudo-maximum likelihood estimators in linear regression models with fractional time series

Abstract

Fractal time series and linear regression models are known to play an important role in many scientific disciplines and applied fields. Although there have been enormous development after their appearance, nobody investigates them together. The paper studies a linear regression model (or trending fractional time series model) $$\begin{aligned} y_t=x_t^T\beta +\varepsilon _t,t=1,2,\ldots ,n, \end{aligned}$$ where $$\begin{aligned} \varepsilon _t=\Delta ^{-\delta }g(L;\varphi )\eta _t \end{aligned}$$ with parameters \(0\le \delta \le 1,\varphi ,\beta ,\sigma ^2\) and \(\eta _t\) i.i.d. with zero mean and variance \(\sigma ^2\). Firstly, the pseudo-maximum likelihood (ML) estimators of \(\varphi ,\beta ,\sigma ^2\) are given. Secondly, under general conditions, the asymptotic properties of the ML estimators are investigated. Lastly, the validity of method is illuminated by a real example.