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

  • Hongchang HuEmail author
  • Weifu Hu
  • Xinxin Yu
Regular Article


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)
$$\begin{aligned} y_t=x_t^T\beta +\varepsilon _t,t=1,2,\ldots ,n, \end{aligned}$$
$$\begin{aligned} \varepsilon _t=\Delta ^{-\delta }g(L;\varphi )\eta _t \end{aligned}$$
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.


Linear regression model Maximum likelihood estimator Fractional time series Asymptotic property 

Mathematics Subject Classification

62J05 62M10 



We would like to thank two anonymous referees for their valuable and constructive comments which significantly improved the presentation of the paper. The work was supported by Natural Science Foundation of China(No.11471105, 11471123).


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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsHubei Normal UniversityHuangshiChina

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