Probabilistic properties and estimation methods for periodic threshold autoregressive stochastic volatility

Ahmed Ghezal; Omar Alzeley; Ahmed Ghezal; Omar Alzeley

doi:10.3934/math.2024578

AIMS Mathematics

2024, Volume 9, Issue 5: 11805-11832. doi: 10.3934/math.2024578

Previous Article Next Article

Research article Special Issues

Probabilistic properties and estimation methods for periodic threshold autoregressive stochastic volatility

Ahmed Ghezal ¹,
Omar Alzeley ^{2
,
,}

1.
Department of Mathematics, Abdelhafid Boussouf University Center of Mila, Algeria; Email: a.ghezal@centre-univ-mila.dz
2.
Department of Mathematics, Umm Al-Qura University, Al-Qunfudah University College, Saudi Arabia; Email: oazeley@uqu.edu.sa

Received: 19 January 2024 Revised: 15 March 2024 Accepted: 18 March 2024 Published: 26 March 2024
MSC : 62G05, 62M10

In an endeavor to encapsulate the dual aspects of volatility progression and periodicity inherent in autocorrelation frameworks demonstrated by various nonlinear time series, a novel conceptualization emerges—the periodic threshold autoregressive stochastic volatility (PTAR-SV) model. This model served as a viable alternative to the conventional periodic threshold generalized autoregressive conditional heteroskedasticity (TGARCH) process. The inherent probabilistic framework of the PTAR-SV model incorporated certain essential features, including strict periodic stationarity, enhancing its analytical robustness. Additionally, this study established the conditions for higher-order moments to exist within the PTAR-SV model. The autocovariance structure pertaining to the powers of the PTAR-SV process has been studied. The process of parameter estimation was scrutinized via the quasi-maximum likelihood technique. This estimation approach involved assessing likelihood using prediction error decomposition and Kalman filtering. Moreover, we extended our analysis to include a Bayesian Markov chain Monte Carlo (MCMC) method based on Griddy-Gibbs sampling, particularly suitable when the distribution of model innovations follows a standard Gaussian. Through a simulation study, we evaluated the performances of both the quasi-maximum likelihood (QML) and Bayesian Griddy Gibbs estimates, providing valuable insights into their respective strengths and weaknesses. Finally, we applied our newly developed methodology to model the spot rates of the euro against the Algerian dinar, demonstrating its applicability and efficacy in real-world financial modeling scenarios.
- periodic stochastic volatility,
- periodic stationarity,
- threshold autoregressive model,
- Kalman filter,
- QMLE,
- Bayesian MCMC estimation
Citation: Ahmed Ghezal, Omar Alzeley. Probabilistic properties and estimation methods for periodic threshold autoregressive stochastic volatility[J]. AIMS Mathematics, 2024, 9(5): 11805-11832. doi: 10.3934/math.2024578

Related Papers:

Abstract

In an endeavor to encapsulate the dual aspects of volatility progression and periodicity inherent in autocorrelation frameworks demonstrated by various nonlinear time series, a novel conceptualization emerges—the periodic threshold autoregressive stochastic volatility (PTAR-SV) model. This model served as a viable alternative to the conventional periodic threshold generalized autoregressive conditional heteroskedasticity (TGARCH) process. The inherent probabilistic framework of the PTAR-SV model incorporated certain essential features, including strict periodic stationarity, enhancing its analytical robustness. Additionally, this study established the conditions for higher-order moments to exist within the PTAR-SV model. The autocovariance structure pertaining to the powers of the PTAR-SV process has been studied. The process of parameter estimation was scrutinized via the quasi-maximum likelihood technique. This estimation approach involved assessing likelihood using prediction error decomposition and Kalman filtering. Moreover, we extended our analysis to include a Bayesian Markov chain Monte Carlo (MCMC) method based on Griddy-Gibbs sampling, particularly suitable when the distribution of model innovations follows a standard Gaussian. Through a simulation study, we evaluated the performances of both the quasi-maximum likelihood (QML) and Bayesian Griddy Gibbs estimates, providing valuable insights into their respective strengths and weaknesses. Finally, we applied our newly developed methodology to model the spot rates of the euro against the Algerian dinar, demonstrating its applicability and efficacy in real-world financial modeling scenarios.

References

[1]	S. J. Taylor, Financial returns modelled by the product of two stochastic processes-a study of the daily sugar prices, 1961–1975, Time Ser. Anal. Theory Pract., 1 (1982), 203–226.
[2]	M. A. Carnero, D. Peña, E. Ruiz, Persistence and kurtosis in GARCH and stochastic volatility models, J. Financ. Econom., 2 (2004), 319–342. https://doi.org/10.1093/jjfinec/nbh012 doi: 10.1093/jjfinec/nbh012
[3]	H. Malmsten, T. Teräsvirta, Stylized facts of financial time series and three popular models of volatility, SSE/EFI Work. Pap. Ser. Econ. Financ., 563 (2004), 1–44.
[4]	F. Black, Studies of stock market volatility changes, In: 1976 Proceedings of the American Statistical Association Business and Economic Statistics section, 1976.
[5]	A. Ghezal, $QMLE$ for periodic time-varying asymmetric $\log GARCH$ models, Commun. Math. Stat., 9 (2021), 273–297. https://doi.org/10.1007/s40304-019-00193-4 doi: 10.1007/s40304-019-00193-4
[6]	A. Ghezal, I. Zemmouri, M-estimation in periodic threshold GARCH models: Consistency and asymptotic normality, Miskolc Math. Notes, 2023, In press.
[7]	E. Jacquier, N. G. Polson, P. E. Rossi, Bayesian analysis of stochastic volatility models with fat-tails and correlated errors, J. Econometrics, 122 (2004), 185–212. https://doi.org/10.1016/j.jeconom.2003.09.001 doi: 10.1016/j.jeconom.2003.09.001
[8]	F. J. Breidt, A threshold autoregressive stochastic volatility model, Ⅵ Latin American congress of probability and mathematical statistics (CLAPEM), Valparaiso, Citeseer, Chile, 1996.
[9]	H. Tong, On a threshold model, In: Pattern Recognition and Signal Processing, ed. by C. H. Chen. Amsterdam: Sijthoff and Noordhoff, 1978,575–586.
[10]	M. K. P. So, W. K. Li, K. Lam, A threshold stochastic volatility model, J. Forecast., 21 (2002), 473–500. https://doi.org/10.1002/for.840 doi: 10.1002/for.840
[11]	C. W. S. Chen, F. C. Liu, M. K. P. So, Heavy-tailed-distributed threshold stochastic volatility models in financial time series, Aust. N. Z. J. Stat., 50 (2008), 29–51. https://doi.org/10.1111/j.1467-842X.2007.00498.x doi: 10.1111/j.1467-842X.2007.00498.x
[12]	X. Mao, E. Ruiz, H. Veiga, Threshold stochastic volatility: Properties and forecasting, Int. J. Forecast., 33 (2017), 1105–1123. https://doi.org/10.1016/j.ijforecast.2017.07.001 doi: 10.1016/j.ijforecast.2017.07.001
[13]	A. Harvey, E. Ruiz, N. Shephard, Multivariate stochastic variance models, Rev. Econ. Stud., 61 (1994), 247–264. https://doi.org/10.2307/2297980 doi: 10.2307/2297980
[14]	E. Ghysels, A. C. Harvey, E. Renault, Stochastic volatility, Handbook Stat., 14 (1996), 119–191.
[15]	F. J. Breidt, N. Crato, P. De Lima, The detection and estimation of long memory in stochastic volatility, J. Econometrics, 83 (1998), 325–348. https://doi.org/10.1016/S0304-4076(97)00072-9 doi: 10.1016/S0304-4076(97)00072-9
[16]	J. Kim, D. S. Stoffer, Fitting stochastic volatility models in the presence of irregular sampling via particle methods and the EM algorithm, J. Time Ser. Anal., 29 (2008), 811–833. https://doi.org/10.1111/j.1467-9892.2008.00584.x doi: 10.1111/j.1467-9892.2008.00584.x
[17]	H. Ghosh, B. Gurung, Prajneshu, Kalman filter-based modelling and forecasting of stochastic volatility with threshold, J. Appl. Stat., 42 (2015), 492–507. https://doi.org/10.1080/02664763.2014.963524 doi: 10.1080/02664763.2014.963524
[18]	A. Aknouche, Periodic autoregressive stochastic volatility, Stat. Infer. Stoch. Pro., 20 (2017), 139–177. https://doi.org/10.1007/s11203-016-9139-z doi: 10.1007/s11203-016-9139-z
[19]	N. Boussaha, F. Hamdi, On periodic autoregressive stochastic volatility models: Structure and estimation, J. Stat. Comput. Simul., 88 (2018), 1637–1668. https://doi.org/10.1080/00949655.2017.1401626 doi: 10.1080/00949655.2017.1401626
[20]	I. Tsiakas, Periodic stochastic volatility and fat tails, J. Financ. Econom., 4 (2006), 90–135. https://doi.org/10.1093/jjfinec/nbi023 doi: 10.1093/jjfinec/nbi023
[21]	R. A. Davis, T. Mikosch, Probabilistic properties of stochastic volatility models, Handbook of financial time series, Springer, Berlin, 2009,255–267.
[22]	E. G. Gladyshev, Periodically correlated random sequences, Sov. Math. Dokl., 2 (1961), 385–388.
[23]	A. Ghezal, A. Bibi, QMLE of periodic bilinear models and of PARMA models with periodic bilinear innovations, Kybernetika, 54 (2018), 375–399. http://doi.org/10.14736/kyb-2018-2-0375 doi: 10.14736/kyb-2018-2-0375
[24]	A. Brandt, The stochastic equation $Y_{n+1} = A_{n}Y_{n}+B_{n}$ with stationary coefficients, Adv. Appl. Probab., 18 (1986), 211–220. http://doi.org/10.2307/1427243 doi: 10.2307/1427243
[25]	P. Bougerol, N. Picard, Strict stationarity of generalized autoregressive processes, Ann. Probab., 20 (1992), 1714–1730. http://doi.org/10.1214/aop/1176989526 doi: 10.1214/aop/1176989526
[26]	C. Francq, J. M. Zakoïan, Linear-representation based estimation of stochastic volatility models, Scand. J. Stat., 33 (2006), 785–806. https://doi.org/10.1111/j.1467-9469.2006.00495.x doi: 10.1111/j.1467-9469.2006.00495.x
[27]	S. Meyn, R. Tweedie, Markov chains and stochastic stability, 2 Eds., Springer Verlag, New York, 2009.
[28]	L. Ljung, P. E. Caines, Asymptotic normality of prediction error estimators for approximate system models, Stochastics, 3 (1979), 29–46. https://doi.org/10.1109/CDC.1978.268066 doi: 10.1109/CDC.1978.268066
[29]	M. S. Arulampalam, S. Maskell, N. Gordon, T. Clapp, A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking, IEEE T. Signal Proces., 50 (2002), 174–188. https://doi.org/10.1109/78.978374 doi: 10.1109/78.978374
[30]	A. Doucet, A. M. Johansen, A tutorial on particle filtering and smoothing: Fifteen years later, Handbook Nonlinear Filt., 12 (2011), 656–704.
[31]	A. P. Dempster, N. M. Laird, D. B. Rubin, Maximum likelihood from incomplete data via the EM algorithm (with discussion), J. R. Stat. Soc. Ser. B, 39 (1977), 1–38. https://doi.org/10.1111/j.2517-6161.1977.tb01600.x doi: 10.1111/j.2517-6161.1977.tb01600.x
[32]	W. Dunsmuir, A central limit theorem for parameter estimation in stationary vector time series and its applications to models for a signal observed with noise, Ann. Stat., 7 (1979), 490–506. https://doi.org/10.1214/aos/1176344671 doi: 10.1214/aos/1176344671
[33]	E. Ruiz, Quasi-maximum likelihood estimation of stochastic variance models, J. Econometrics, 63 (1994), 284–306. https://doi.org/10.1016/0304-4076(93)01569-8 doi: 10.1016/0304-4076(93)01569-8
[34]	E. Jacquier, N. G. Polson, P. E. Rossi, Bayesian analysis of stochastic volatility models, J. Bus. Econ. Stat., 12 (1994), 371–389. Available from: https://EconPapers.repec.org/RePEc: bes: jnlbes: v: 12: y: 1994: i: 4: p: 371-89.
[35]	R. S. Tsay, Analysis of financial time series: Financial econometrics, 3 Eds., Wiley, 2010.
[36]	H. Sakai, S. Ohno, On backward periodic autoregressive processes, J. Time Ser. Anal., 18 (1997), 415–427. https://doi.org/10.1111/1467-9892.00059 doi: 10.1111/1467-9892.00059
[37]	A. Ghezal, M. Balegh, I. Zemmouri, Markov-switching threshold stochastic volatility models with regime changes, AIMS Math., 9 (2024), 3895–3910. https://doi.org/ 10.3934/math.2024192 doi: 10.3934/math.2024192
[38]	M. Bentarzi, L. Djeddou, Adaptive estimation of periodic first-order threshold autoregressive model, Commun. Stat.-Simul. Comput., 43 (2014), 1611–1630. https://doi.org/10.1080/03610918.2012.740123 doi: 10.1080/03610918.2012.740123
[39]	A. Aknouche, N. Demmouche, S. Dimitrakopoulos, N. Touche, Bayesian MCMC analysis of periodic asymmetric power GARCH models, Stud. Nonlinear Dyn. E., 24 (2020), 1–24. https://doi.org/10.1515/snde-2018-0112 doi: 10.1515/snde-2018-0112
[40]	A. Aknouche, E. Al-Eid, N. Demouche, Generalized quasi-maximum likelihood inference for periodic conditionally heteroskedastic models, Stat. Infer. Stoch. Pro., 21 (2018), 485–511. https://doi.org/10.1007/s11203-017-9160-x doi: 10.1007/s11203-017-9160-x
[41]	A. Aknouche, B. S. Almohaimeed, S. Dimitrakopoulos, Periodic autoregressive conditional duration, J. Time Ser. Anal., 43 (2022), 5–29. https://doi.org/10.1111/jtsa.12588 doi: 10.1111/jtsa.12588
[42]	S. Kim, N. Shephard, S. Chib, Stochastic volatility: Likelihood inference and comparison with ARCH models, Rev. Econ. Stud., 65 (1998), 361–393. https://doi.org/10.1111/1467-937X.00050 doi: 10.1111/1467-937X.00050
[43]	S. Chib, F. Nardari, N. Shephard, Markov chain Monte Carlo methods for stochastic volatility models, J. Econometrics, 108 (2002), 281–316. https://doi.org/10.1016/S0304-4076(01)00137-3 doi: 10.1016/S0304-4076(01)00137-3

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)