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Estimating FARIMA models with uncorrelated but non-independent error terms

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Abstract

In this paper we derive the asymptotic properties of the least squares estimator (LSE) of fractionally integrated autoregressive moving-average (FARIMA) models under the assumption that the errors are uncorrelated but not necessarily independent nor martingale differences. We relax the independence and even the martingale difference assumptions on the innovation process to extend considerably the range of application of the FARIMA models. We propose a consistent estimator of the asymptotic covariance matrix of the LSE which may be very different from that obtained in the standard framework. A self-normalized approach to confidence interval construction for weak FARIMA model parameters is also presented. All our results are done under a mixing assumption on the noise. Finally, some simulation studies and an application to the daily returns of stock market indices are presented to corroborate our theoretical work.

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Notes

  1. To cite few examples of nonlinear processes, let us mention the self-exciting threshold autoregressive (SETAR), the smooth transition autoregressive (STAR), the exponential autoregressive (EXPAR), the bilinear, the random coefficient autoregressive (RCA), the functional autoregressive (FAR) (see Tong (1990) and Fan and Yao (2008) for references on these nonlinear time series models).

  2. Recall that the fractional version of Cesàro’s Lemma states that for \((h_t)_t\) a sequence of positive reals, \(\kappa >0\) and \(c\ge 0\) we have

    $$\begin{aligned} \lim _{t\rightarrow \infty }h_tt^{1-\kappa }=\left| \kappa \right| c \Rightarrow \lim _{n\rightarrow \infty }\frac{1}{n^{\kappa }}\sum _{t=0}^n h_t=c. \end{aligned}$$

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Acknowledgements

We sincerely thank the anonymous referees and Editor for helpful remarks.

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Authors and Affiliations

  1. Université Bourgogne Franche-Comté, Laboratoire de mathématiques de Besançon, UMR CNRS 6623, 16 route de Gray, 25030, Besançon, France

    Yacouba Boubacar Maïnassara, Youssef Esstafa & Bruno Saussereau

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  1. Yacouba Boubacar Maïnassara
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Correspondence to Yacouba Boubacar Maïnassara.

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Boubacar Maïnassara, Y., Esstafa, Y. & Saussereau, B. Estimating FARIMA models with uncorrelated but non-independent error terms. Stat Inference Stoch Process 24, 549–608 (2021). https://doi.org/10.1007/s11203-021-09243-7

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