Probabilistic properties of a Markov-switching periodic $GARCH$ process

Aliat, Billel ; Hamdi, Fayçal

Kybernetika, Tome 55 (2019) no. 6, pp. 915-942

Abstract (VO)

In this paper, we propose an extension of a periodic $GARCH$ ($PGARCH$) model to a Markov-switching periodic $GARCH$ ($MS$-$PGA$ $RCH$), and provide some probabilistic properties of this class of models. In particular, we address the question of strictly periodically and of weakly periodically stationary solutions. We establish necessary and sufficient conditions ensuring the existence of higher order moments. We further provide closed-form expressions for calculating the even-order moments as well as the autocovariances of the powers of a $MS$-$PGARCH$ process. We thus show how these moments and autocovariances can be used for estimating model parameters using $GMM$ method.

Détail

DOI : 10.14736/kyb-2019-6-0915

Classification : 60G10, 62M10
Keywords: Markov-switching models; periodic $GARCH$ models; periodic stationarity; higher-order moments; Markov-switching $PGARCH$ models; $GMM$ method

Bibliographie

[1] A., Aknouche,, M., Bentarzi,: On the existence of higher-order moments for periodic $GARCH$ models. Statist. Probab. Lett. 78 (2008), 3262-3268. | DOI | MR

[2] A., Aknouche,, A., Bibi,: Quasi-maximum likelihood estimation of periodic $GARCH$ and periodic $ARMA$-$GARCH$ processes. J. Time Series Anal. 28 (2009), 19-46. | DOI | MR

[3] A., Aknouche,, H., Guerbyenne,: Periodic stationarity of random coefficient periodic autoregressions. Statist. Probab. Lett. 79 (2009), 990-996. | DOI | MR

[4] B., Aliat,, F., Hamdi,: On markov-switching periodic $ARMA$ models. Comm. Statist. Theory Methods 47 (2018), 344-364. | DOI | MR

[5] M., Augustyniak,: Maximum likelihood estimation of the Markov-switching $GARCH$ model. Comput. Statist. Data Anal. 76 (2014), 61-75. | DOI | MR

[6] M., Augustyniak,, M., Boudreault,, M., Morales,: Maximum likelihood estimation of the Markov-switching GARCH model based on a general collapsing procedure. Methodol. Comput. Appl. Probab. 20 (2018), 165-188. | DOI | MR

[7] L., Bauwens,, A., Preminger,, K., Rombouts, J. V.: Theory and inference for Markov switching $GARCH$ model. Econometr. J. 13 (2010), 218-244. | DOI | MR

[8] M., Bentarzi,, F., Hamdi,: Mixture periodic autoregressive conditional heteroskedastic models. Comput. Statist. Data Anal. 53 (2008), 1-16. | DOI | MR

[9] M., Bentarzi,, F., Hamdi,: Mixture periodic autoregression with aeriodic $ARCH$ errors. Adv. Appl. Statist. 8 (2008), 219-46. | MR

[10] A., Bibi,, A., Aknouche,: On periodic $GARCH$ processes: Stationarity Statist. 17 (2008), 305-316. | DOI | MR

[11] A., Bibi,, I., Lescheb,: Strong consistency and asymptotic normality of least squares estimators for $PGARCH$ and $PARMA$-$PGARCH$ models. Statist. Probab. Lett. 80 (2010), 1532-1542. | DOI | MR

[12] A., Bibi,, I., Lescheb,: A conditional least squares approach to $PGARCH$ and $PARMA$-$PGARCH$ time series estimation. Comptes Rendus Math. 348 (2010), 1211-1216. | DOI | MR

[13] A., Bibi,, I., Lescheb,: Estimation and asymptotic properties in periodic $GARCH(1,1)$ models. Comm. Statist. Theory Methods 42 (2013), 3497-3513. | DOI | MR

[14] M., Billio,, R., Casarin,, A., Osuntuyi,: Efficient Gibbs sampling for Markov switching $GARCH$ models. Comput. Statist. Data Anal. 100 (2016), 37-57. | DOI | MR

[15] T., Bollerslev,: Generalized autoregressive conditional heteroskedasticity. J. Econometr. 31 (1986), 307-327. | DOI | MR

[16] T., Bollerslev,, E., Ghysels,: Periodic autoregressive conditional heteroskedasticity. J. Business Econom. Statist. 14 (1996), 139-152. | DOI

[17] P., Bougerol,, N., Picard,: Strict stationarity of generalized autoregressive processes. Ann. Probab. 20 (1992), 1714-1730. | DOI | MR

[18] J., Cai,: A Markov model of Switching-regime $ARCH$. J. Business Econom. Statist. 12 (1994), 309-316. | DOI

[19] C., Francq,, M., Roussignol,, M., Zako\"ıan, J.: Conditional heteroskedasticity driven by hidden Markov chains. J. Time Ser. Anal. 22 (2001), 197-220. | DOI | MR

[20] C., Francq,, M., Zako\"ıan, J.: The $L^{2}$-structures of standard and switching-regime $GARCH$ models. Stoch. Process. Appl. 115 (2005), 1557-1582. | MR

[21] C., Francq,, M., Zako\"ıan, J.: Deriving the autocovariances of powers of Markov-switching $GARCH$ models with applications to statistical inference. Computat. Statist. Data Anal. 52 (2008), 3027-3046. | DOI | MR

[22] H., Franses, P., R., Paap,: Modeling changing day-of-the-week seasonality in stock returns and volatility. Appl. Financ. Econom. 52 (2000), 3027-3046.

[23] G., Gladyshev, E.: Periodically correlated random sequences. Doklady Akademii Nauk SSSR 137 (1961), 1026-1029. | MR

[24] F., Gray, S.: Modeling the conditional distribution of interest rates as a regime-switching process. J. Financ. Econom. 42 (1996), 27-62. | DOI

[25] M., Haas,, S., Paolella, M.: Mixture and regime-switching $GARCH$ models. In: Handbook of Volatility Models and their Applications 2012, pp. 71-102. | DOI | MR

[26] M., Haas,, S., Mittnik,, S., Paolella, M.: A new approach to Markov-switching $GARCH$ models. J. Financ. Econometr. 2 (2004), 493-530. | DOI

[27] F., Hamdi,, S., Souam,: Mixture periodic $GARCH $ models: Applications to exchange rate modeling. In: Modeling, Simulation and Applied Optimization (ICMSAO), 5th International Conference on IEEE, 2013. | DOI

[28] F., Hamdi,, S., Souam,: Mixture periodic $GARCH$ models: theory and applications. Empir. Econom. 55 (2018), 1925-1956. | DOI

[29] D., Hamilton, J.: A new approach to the economie analysis of nonstationary time series and the business cycle. Econometrica 57 (1989), 357-384. | DOI | MR

[30] D., Hamilton, J., R., Susmel,: Autoregressive condiational heteroskedasticity and changes in regime. J. Econometr. 64 (1994), 307-333. | DOI

[31] S., Henneke, J., T., Rachev, S., J., Fabozzi, F., M.Nikolov: MCMC-based estimation of Markov switching $ARMA$-$GARCH$ models. Appl. Econom. 43 (2011), 259-271. | DOI

[32] A., Horn, R., R., Johnson, C.: Matrix Analysis. Second edition. Cambridge University Press, New York 2013. | MR

[33] F., Klaassen,: Improving $GARCH$ volatility forecasts. Empir. Econometr. 27 (2002), 363-394. | DOI

[34] P., Lancaster,, M., Tismenetsky,: The Theory of Matrices. Academic Press, New York 1985. | MR

[35] O., Lee,, W., Shin, D.: Geometric ergodicity and moment conditions for a seasonal $GARCH$ model with periodic coefficients. Comm. Statist. Theory Methods 39 (2009), 38-51. | DOI | MR

[36] K., Newey, W., D., West, K.: Hypothesis testing with efficient method of moments estimation. Int. Econom. Rev. 28 (1987), 3, 777-787. | DOI | MR

[37] N., Regnard,, M., Zako\"ıan, J.: Structure and estimation of a class of nonstationary yet nonexplosive $GARCH$ models. J. Time Series Anal. 31 (2010), 348-364. | DOI | MR