Kybernetika 55 no. 6, 915-942, 2019

Probabilistic properties of a Markov-switching periodic GARCH process

Billel Aliat and Fayçal HamdiDOI: 10.14736/kyb-2019-6-0915

Abstract:

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.

Keywords:

Markov-switching models, periodic $GARCH$ models, periodic stationarity, higher-order moments, Markov-switching $PGARCH$ models, $GMM$ method

Classification:

62M10, 60G10

paper.pdf

References:

  1. A. Aknouche and M. Bentarzi: On the existence of higher-order moments for periodic $GARCH$ models. Statist. Probab. Lett. 78 (2008), 3262-3268.   DOI:10.1016/j.spl.2008.06.010
  2. A. Aknouche and A. Bibi: Quasi-maximum likelihood estimation of periodic $GARCH$ and periodic $ARMA$-$GARCH$ processes. J. Time Series Anal. 28 (2009), 19-46.   DOI:10.1016/j.spl.2008.06.010
  3. A. Aknouche and H. Guerbyenne: Periodic stationarity of random coefficient periodic autoregressions. Statist. Probab. Lett. 79 (2009), 990-996.   DOI:10.1016/j.spl.2008.12.012
  4. B. Aliat and F. Hamdi: On markov-switching periodic $ARMA$ models. Comm. Statist. Theory Methods 47 (2018), 344-364.   DOI:10.1080/03610926.2017.1303734
  5. M. Augustyniak: Maximum likelihood estimation of the Markov-switching $GARCH$ model. Comput. Statist. Data Anal. 76 (2014), 61-75.   DOI:10.1016/j.csda.2013.01.026
  6. M. Augustyniak, M. Boudreault and 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:10.1007/s11009-016-9541-4
  7. L. Bauwens, A. Preminger and J. V. K. Rombouts: Theory and inference for Markov switching $GARCH$ model. Econometr. J. 13 (2010), 218-244.   DOI:10.1111/j.1368-423x.2009.00307.x
  8. M. Bentarzi and F. Hamdi: Mixture periodic autoregressive conditional heteroskedastic models. Comput. Statist. Data Anal. 53 (2008), 1-16.   DOI:10.1016/j.csda.2008.06.019
  9. M. Bentarzi and F. Hamdi: Mixture periodic autoregression with aeriodic $ARCH$ errors. Adv. Appl. Statist. 8 (2008), 219-46.   CrossRef
  10. A. Bibi and A. Aknouche: On periodic $GARCH$ processes: Stationarity Statist. 17 (2008), 305-316.    DOI:10.3103/s1066530708040029
  11. A. Bibi and 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:10.1016/j.spl.2010.06.007
  12. A. Bibi and I. Lescheb: A conditional least squares approach to $PGARCH$ and $PARMA$-$PGARCH$ time series estimation. Comptes Rendus Math. 348 (2010), 1211-1216.   DOI:10.1016/j.crma.2010.10.019
  13. A. Bibi and I. Lescheb: Estimation and asymptotic properties in periodic $GARCH(1,1)$ models. Comm. Statist. Theory Methods 42 (2013), 3497-3513.   DOI:10.1080/03610926.2011.633201
  14. M. Billio, R. Casarin and A. Osuntuyi: Efficient Gibbs sampling for Markov switching $GARCH$ models. Comput. Statist. Data Anal. 100 (2016), 37-57.   DOI:10.1016/j.csda.2014.04.011
  15. T. Bollerslev: Generalized autoregressive conditional heteroskedasticity. J. Econometr. 31 (1986), 307-327.   DOI:10.1016/0304-4076(86)90063-1
  16. T. Bollerslev and E. Ghysels: Periodic autoregressive conditional heteroskedasticity. J. Business Econom. Statist. 14 (1996), 139-152.   DOI:10.2307/1392425
  17. P. Bougerol and N. Picard: Strict stationarity of generalized autoregressive processes. Ann. Probab. 20 (1992), 1714-1730.   DOI:10.1214/aop/1176989526
  18. J. Cai: A Markov model of Switching-regime $ARCH$. J. Business Econom. Statist. 12 (1994), 309-316.   DOI:10.2307/1392087
  19. C. Francq, M. Roussignol and J. M. Zako\"ıan: Conditional heteroskedasticity driven by hidden Markov chains. J. Time Ser. Anal. 22 (2001), 197-220.   DOI:10.1111/1467-9892.00219
  20. C. Francq and J. M. Zako\"ıan: The $L^{2}$-structures of standard and switching-regime $GARCH$ models. Stoch. Process. Appl. 115 (2005), 1557-1582.   CrossRef
  21. C. Francq and J. M. Zako\"ıan: Deriving the autocovariances of powers of Markov-switching $GARCH$ models with applications to statistical inference. Computat. Statist. Data Anal. 52 (2008), 3027-3046.   DOI:10.1016/j.csda.2007.08.003
  22. P. H. Franses and R. Paap: Modeling changing day-of-the-week seasonality in stock returns and volatility. Appl. Financ. Econom. 52 (2000), 3027-3046.   CrossRef
  23. E. G. Gladyshev: Periodically correlated random sequences. Doklady Akademii Nauk SSSR 137 (1961), 1026-1029.   CrossRef
  24. S. F. Gray: Modeling the conditional distribution of interest rates as a regime-switching process. J. Financ. Econom. 42 (1996), 27-62.   DOI:10.1016/0304-405x(96)00875-6
  25. M. Haas and M. S. Paolella: Mixture and regime-switching $GARCH$ models. In: Handbook of Volatility Models and their Applications 2012, pp. 71-102.   DOI:10.1002/9781118272039.ch3
  26. M. Haas, S. Mittnik and M. S. Paolella: A new approach to Markov-switching $GARCH$ models. J. Financ. Econometr. 2 (2004), 493-530.   DOI:10.1093/jjfinec/nbh020
  27. F. Hamdi and 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:10.1109/icmsao.2013.6552570
  28. F. Hamdi and S. Souam: Mixture periodic $GARCH$ models: theory and applications. Empir. Econom. 55 (2018), 1925-1956.   DOI:10.1007/s00181-017-1348-9
  29. J. D. Hamilton: A new approach to the economie analysis of nonstationary time series and the business cycle. Econometrica 57 (1989), 357-384.   DOI:10.2307/1912559
  30. J. D. Hamilton and R. Susmel: Autoregressive condiational heteroskedasticity and changes in regime. J. Econometr. 64 (1994), 307-333.   DOI:10.1016/0304-4076(94)90067-1
  31. J. S. Henneke, S. T. Rachev, F. J. Fabozzi and M.Nikolov: MCMC-based estimation of Markov switching $ARMA$-$GARCH$ models. Appl. Econom. 43 (2011), 259-271.   DOI:10.1080/00036840802552379
  32. R. A. Horn and C. R. Johnson: Matrix Analysis. Second edition. Cambridge University Press, New York 2013.   CrossRef
  33. F. Klaassen: Improving $GARCH$ volatility forecasts. Empir. Econometr. 27 (2002), 363-394.   DOI:10.1007/s001810100100
  34. P. Lancaster and M. Tismenetsky: The Theory of Matrices. Academic Press, New York 1985.   CrossRef
  35. O. Lee and D. W. Shin: Geometric ergodicity and moment conditions for a seasonal $GARCH$ model with periodic coefficients. Comm. Statist. Theory Methods 39 (2009), 38-51.   DOI:10.1080/03610920802715032
  36. W. K. Newey and K. D. West: Hypothesis testing with efficient method of moments estimation. Int. Econom. Rev. 28 (1987), 3, 777-787.   DOI:10.2307/2526578
  37. N. Regnard and J. M. Zako\"ıan: Structure and estimation of a class of nonstationary yet nonexplosive $GARCH$ models. J. Time Series Anal. 31 (2010), 348-364.   DOI:10.1111/j.1467-9892.2010.00669.x