Stability Analysis of Threshold Autoregressive Distributed Lag (TARDL(1,1,1)) Models Using Dynamical Approach: Applied Study of Limit Cycle Condition and Orbital Oscillations
DOI:
https://doi.org/10.62933/6d8mf890Keywords:
TARDL(1,1,1), Limit Cycle condition, Dynamical Approach, Orbital stability , OscillationsAbstract
This research aims to study the stability condition of the Threshold Autoregressive Distributed Lag [TARDL(1,1,1)] model using dynamic Approach. The limit cycle condition was applied to the model, where the stability of the model was analysed through differential equations that reflect oscillations and cycles, firstly will prove and present the limit cycle condition for this model. Two examples were presented: the first one satisfies the stability of limit cycle and stabilizes at a constant value, while the second one does at satisfy the condition and continuous to oscillate. MATLAB programing was used to plot the model trajectories and explain whether it is stable or not. The results showed that the model stabilizes when the condition is satisfied, while not satisfying it leads to continuous oscillations. Finally a practical application was conducted on real data to determine the stability or not of the models at the limit cycle.
References
[1] Tong, H. (1978). On a threshold model. In Pattern Recognition and Signal Processing (pp. 101-141). Springer.
[2] Hansen, B. E. (1999). Threshold effects in non-dynamic panels: Estimation, testing, and inference. Journal of Econometrics, 93(2), 345-368.
[3] Polemis, Michael L., and Markos Tselekounis. "Threshold effects in the regulation-innovation nexus: evidence from the telecommunications industry." Journal of Regulatory Economics 60.1 (2021): 74-93.
[4] Xu, Xin-Jian, Shuang He, and Li-Jie Zhang. "Dynamics of the threshold model on hypergraphs." Chaos: An Interdisciplinary Journal of Nonlinear Science 32.2 (2022).
[5] Qiang, Zhecheng, Eduardo L. Pasiliao, and Qipeng P. Zheng. "Target set selection in social networks with tiered influence and activation thresholds." Journal of combinatorial optimization 45.5 (2023): 117.
[6] Guerra, Ernesto, et al. "Endogenous thresholds in energy prices: Modeling and empirical estimation." Energy Economics 121 (2023): 106669.
[7] Pesaran, M. H., Shin, Y., & Smith, R. J. (2001). Bounds testing approaches to the analysis of level relationships. Journal of Applied Econometrics, 16(3), 289-326.
[8] Ozaki, T. (1985). Nonlinear time series models and dynamical systems. Handbook of Statistics, 5, 25-83.
[9] Tong, H. (1990). Non-linear time series: a dynamical system approach. Oxford University Press.
[10] Mohammad , A.A. and Salim, A.J. (2007). Stability of Logistic Autoregressive model. Qatar Univesity of scince journal, 27:17–28.
[11] Mohammad, A.A. and Gannam, A.K. (2010). Stability of Cauchy Autoregressive model. journal of pure and applied scince, Salahaddin University Hawler (special Issue):52-62.
[12] Mohammad, A.A. and Ghaffar, M.K. (2016). Astudy on stability of Conditional variance for GARCH models with application. Tikrit journal of pure scince, 21 (4):160-169.
[13] Mohammad, A.A. and Mudhir, A.A. (2018). Dynamical approach in studying stability condition of exponential (GARCH) models. Journal of King Saud University - Science, Vol.32 (1):272-278.
[14] Noori, Nooruldeen A., and Azher A. Mohammad. 2021. "Dynamical approach in studying GJR-GARCH (Q, P) models with application." Tikrit Journal of Pure Science 26(2): 145-156.
[15] Abdullah, Hiba H., and Azher A. Mohammad. "Stability Study of Exponential Double Autoregressive model with application." NeuroQuantology 20.6 (2022): 4812.
[16] Abdullah, Hiba H., and Azher A. Mohammad. "Stability Study of Logarithmic Double Autoregressive Model with Application." Journal of Algebraic Statistics 13.3 (2022): 3206-3218.
[17] Mohammed, Rouaa I., and Azher A. Mohammad. "Stability conditions for limit cycle of Smooth Transition Hyperbolic Tangent Autoregressive model." Journal of Algebraic Statistics 13.2 (2022): 2346-2357.
[18] Ali, Nezar E., and Azher A. Mohammad. "Stability conditions of limit cycle for Gompertz Autoregressive model." Tikrit Journal of Pure Science 28.2 (2023).
[19] Jordan, Soren, and Andrew Q. Philips. "Dynamic Simulation and Testing for Single-Equation Cointegrating and Stationary Autoregressive Distributed Lag Models." R Journal 10.2 (2018).
[20] Cho, Jin Seo, Matthew Greenwood‐Nimmo, and Yongcheol Shin. "Recent developments of the autoregressive distributed lag modelling framework." Journal of Economic Surveys 37.1 (2023): 7-32.
[21] Chang, Bisharat Hussain, et al. "Asymmetric effect of extreme changes in the exchange rate volatility on the US imports: Evidence from multiple threshold nonlinear autoregressive distributed lag model." Studies in economics and finance 37.2 (2020): 293-309.
[22] Chang, Bisharat Hussain, et al. "Asymmetric effect of extreme changes in the exchange rate volatility on the US imports: Evidence from multiple threshold nonlinear autoregressive distributed lag model." Studies in economics and finance 37.2 (2020): 293-309.
[23] Khalaf, Nihad S., Hiba H. Abdullah, and Nooruldeen A. Noori. "The Impact of Overall Intervention Model on Price of Wheat." Iraqi Journal of Science (2024): 853-862.
[24] Abdullah, Hiba, and Nooruldeen A. Noori. "Comparison of non-linear time series models (Beta-t-EGARCH and NARMAX models) with Radial Basis Function Neural Network using Real Data." Iraqi Journal For Computer Science and Mathematics 5.3 (2024): 26-44.
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