March 2023 A two-step estimation procedure for locally stationary ARMA processes with tempered stable innovations
Shu Wei Chou-Chen, Pedro A. Morettin
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Braz. J. Probab. Stat. 37(1): 155-176 (March 2023). DOI: 10.1214/23-BJPS565


The class of locally stationary processes assumes a time-varying (tv) spectral representation and finite second moment. Different areas have observed phenomena with heavy tail distributions or infinite variance. Using stable distribution as a heavy-tailed innovation is an attractive option. However, its estimation is difficult due to the absence of a closed expression for the density function and the non-existence of second moment. In this paper, we propose the tvARMA model with tempered stable innovations, which have lighter tails than the stable distribution and have finite moments. A two-step method is proposed to estimate this parametric model. In the first step, we use the blocked Whittle estimation to estimate the time-varying structure of the process. In the second step, we recover residuals from the first step and use the maximum likelihood method to estimate the rest of the parameters related to the standardized classical tempered stable (stdCTS) innovations. We perform simulation studies to evaluate the consistency of the maximum likelihood estimation of independent stdCTS samples. Then, we execute simulations to study the two-step estimation of our model. Finally, an empirical application is illustrated.

Funding Statement

The authors are grateful to the support of a CNPq grant (141607/2017-3) and the University of Costa Rica (SWC), and a FAPESP grant 2018/04654-9 (PAM).


The authors are grateful for the comment of an Associate Editor and three anonymous referees for their valuable comments and suggestions that improved the quality of this paper.


Download Citation

Shu Wei Chou-Chen. Pedro A. Morettin. "A two-step estimation procedure for locally stationary ARMA processes with tempered stable innovations." Braz. J. Probab. Stat. 37 (1) 155 - 176, March 2023.


Received: 1 July 2022; Accepted: 1 February 2023; Published: March 2023
First available in Project Euclid: 27 April 2023

MathSciNet: MR4580889
zbMATH: 07692854
Digital Object Identifier: 10.1214/23-BJPS565

Keywords: blocked Whittle estimation , locally stationary process , tempered stable distribution , two-step estimation

Rights: Copyright © 2023 Brazilian Statistical Association


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Vol.37 • No. 1 • March 2023
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