Application of the Periodic Self-Exciting Threshold Autoregressive Model
Keywords:
Periodic Self-Exciting Threshold Autoregressive models, LS estimation, LR test, LAN property, Algeria's temperature.
Abstract
In this paper, we analyze Algerian temperature data using the periodic self-exciting threshold autoregressive (PSETAR) model. Despite the significant advantages offered by the periodic SETAR model in capturing seasonal and threshold-based behaviors, it remains underutilized in practical applications. The goal of this work is to demonstrate the utility of this model by applying it to Algeria's temperature series. We examine the properties of the model and discuss its estimation using the least squares method. The linearity is tested using the likelihood ratio test, and we extend the local asymptotic normality to p regimes. This analysis provides a deeper understanding of the temperature dynamics in Algeria, highlighting the model's ability to capture seasonal variations and thresholds.References
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(2) Aknouche, A., Almohaimeed, B., and Dimitrakopoulos, S. (2022). Periodic autoregressive conditional duration. Journal of Time Series Analysis, 43, 5-29.
(3) Anderson, P. L., Sabzikar, F. and Meerschaert, M. M. (2020). Parsimonious time series modeling for high frequency climate data. Journal of Time Series Analysis, 42. DOI:10.1111/jtsa.12579.
(4) Bentarzi, M., and Djeddou, L. (2014). Adaptive Estimation of Periodic First-Order Threshold Autoregressive Model. Communications in Statistics - Simulation and Computation, 43(7), 1611-1630.
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(6) Beong-Soo (1991). On adaptive estimation in the non-linear threshold AR(1) model. Perdue university. Technical report 90-45.
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(8) Franses, P. and Paap, R. (2004). Periodic time series models. Oxford University Press.
(9) Ghezal, A., Balegh, M and Zemmouri, I. (2024). Markov-switching threshold stochastic volatility models with regime changes. AIMS Mathematics, 9(2), 3895-3910.
(10) Ghezal, A. and Alzeley, O. (2024).Probabilistic properties and estimation methods for periodic threshold autoregressive stochastic volatility. AIMS Mathematics, 9(5), 11805-11832.
(11) Hamdi, F. and Khalfi, A. (2019). Predictive density criterion for SETAR models. Communications in Statistics - Simulation and Computation, 51(2), 443--459. https://doi.org/10.1080/03610918.2019.1653915
(12) Lama, A., Singh, K.N., Singh, H. et al. (2021). Forecasting monthly rainfall of Sub-Himalayan region of India using parametric and non-parametric modelling approaches. Model. Earth Syst. Environ. https://doi.org/10.1007/s40808-021-01124-5
(13) Lewis, P. A. W., and Ray, B. K. (2002). Nonlinear modelling of periodic threshold autoregressions using TSMARS. Journal of Time Series Analysis, 23(4), 459-471.
(14) Swensen, A. R. (1985). The Asymptotic Distribution of the likelihood ratio for autoregressive time series with a regression trend. Journal of Multivariate Analysis, 16, 54-70.
(15) Tong, H. (1978). On threshold models. In C. H. Chen (Ed.), In Pattern Recognition and Signal Processing. Amsterdam: Sijhoff & Noordhoff.[vskip]
(2) Aknouche, A., Almohaimeed, B., and Dimitrakopoulos, S. (2022). Periodic autoregressive conditional duration. Journal of Time Series Analysis, 43, 5-29.
(3) Anderson, P. L., Sabzikar, F. and Meerschaert, M. M. (2020). Parsimonious time series modeling for high frequency climate data. Journal of Time Series Analysis, 42. DOI:10.1111/jtsa.12579.
(4) Bentarzi, M., and Djeddou, L. (2014). Adaptive Estimation of Periodic First-Order Threshold Autoregressive Model. Communications in Statistics - Simulation and Computation, 43(7), 1611-1630.
(5) Bentarzi, M. and Merzougui, M. (2009). Adaptive Test for Periodicity in Self-Exciting Threshold Autoregressive Models. Communications in Statistics -- Simulation and Computation. 38, 1592-1609.
(6) Beong-Soo (1991). On adaptive estimation in the non-linear threshold AR(1) model. Perdue university. Technical report 90-45.
(7) Chan, K. S., Petruccelli, J. D., Tong, H. and Woolford, S. W. (1985). A Multiple-Threshold AR(1) Model. Journal of Applied Probability, Vol. 22, No. 2. pp. 267-279.
(8) Franses, P. and Paap, R. (2004). Periodic time series models. Oxford University Press.
(9) Ghezal, A., Balegh, M and Zemmouri, I. (2024). Markov-switching threshold stochastic volatility models with regime changes. AIMS Mathematics, 9(2), 3895-3910.
(10) Ghezal, A. and Alzeley, O. (2024).Probabilistic properties and estimation methods for periodic threshold autoregressive stochastic volatility. AIMS Mathematics, 9(5), 11805-11832.
(11) Hamdi, F. and Khalfi, A. (2019). Predictive density criterion for SETAR models. Communications in Statistics - Simulation and Computation, 51(2), 443--459. https://doi.org/10.1080/03610918.2019.1653915
(12) Lama, A., Singh, K.N., Singh, H. et al. (2021). Forecasting monthly rainfall of Sub-Himalayan region of India using parametric and non-parametric modelling approaches. Model. Earth Syst. Environ. https://doi.org/10.1007/s40808-021-01124-5
(13) Lewis, P. A. W., and Ray, B. K. (2002). Nonlinear modelling of periodic threshold autoregressions using TSMARS. Journal of Time Series Analysis, 23(4), 459-471.
(14) Swensen, A. R. (1985). The Asymptotic Distribution of the likelihood ratio for autoregressive time series with a regression trend. Journal of Multivariate Analysis, 16, 54-70.
(15) Tong, H. (1978). On threshold models. In C. H. Chen (Ed.), In Pattern Recognition and Signal Processing. Amsterdam: Sijhoff & Noordhoff.[vskip]
Published
2025-01-29
How to Cite
Bezziche, N., & Merzougui, M. (2025). Application of the Periodic Self-Exciting Threshold Autoregressive Model. Statistics, Optimization & Information Computing. https://doi.org/10.19139/soic-2310-5070-2254
Issue
Section
Research Articles
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