Hostname: page-component-77c89778f8-gq7q9 Total loading time: 0 Render date: 2024-07-19T05:11:48.698Z Has data issue: false hasContentIssue false
  • English
  • Français

ON THE STATIONARITY OF MARKOV-SWITCHING GARCH PROCESSES

Published online by Cambridge University Press:  05 April 2007

Ari Abramson  and
Israel Cohen
Ari Abramson
Affiliation:
Technion—Israel Institute of Technology
Israel Cohen
Affiliation:
Technion—Israel Institute of Technology
Get access Rights & Permissions [Opens in a new window]

Abstract

Generalized autoregressive conditional heteroskedasticity (GARCH) models with Markov-switching regimes are often used for volatility analysis of financial time series. Such models imply less persistence in the conditional variance than the standard GARCH model and potentially provide a significant improvement in volatility forecast. Nevertheless, conditions for asymptotic wide-sense stationarity have been derived only for some degenerated models. In this paper, we introduce a comprehensive approach for stationarity analysis of Markov-switching GARCH models, which manipulates a backward recursion of the model's second-order moment. A recursive formulation of the state-dependent conditional variances is developed, and the corresponding conditions for stationarity are obtained. In particular, we derive necessary and sufficient conditions for the asymptotic wide-sense stationarity of two different variants of Markov-switching GARCH processes and obtain expressions for their asymptotic variances in the general case of m-state Markov chains and (p,q)-order GARCH processes.The authors thank Professor Rami Atar for helpful discussions. The authors thank the co-editor Bruce Hansen and the three anonymous referees for their helpful comments and suggestions and in particular the referee who proposed a generalization of the proof in Appendix B. This research was supported by the Israel Science Foundation (grant 1085/05).

Type
Research Article
Information
Econometric Theory , Volume 23 , Issue 3 , June 2007 , pp. 485 - 500
Copyright
© 2007 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Alexander, C. & E. Lazar (2004) Normal Mixture GARCH(1,1) Applications to Exchange Modelling. ISMA Centre Discussion Papers in Finance 2004–06, the Business School for Financial Markets at University of Reading.
Bollerslev, T. (1986) Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31, 307–327.Google Scholar
Cai, J. (1994) A Markov model of switching-regime ARCH. Journal of Business & Economic Statistics 12, 309–316.Google Scholar
Dueker, M.J. (1997) Markov switching in GARCH processes and mean reverting stock market volatility. Journal of Business & Economic Statistics 15, 26–34.Google Scholar
Francq, C., M. Roussignol, & J.-M. Zakoïan (2001) Conditional heteroskedasticity driven by hidden Markov chains. Journal of Time Series Analysis 22, 197–220.Google Scholar
Francq, C. & J.-M. Zakoïan (2001) Stationarity of multivariate Markov-switching ARMA models. Journal of Econometrics 102, 339–364.Google Scholar
Francq, C. & J.-M. Zakoïan (2002) Comments on the paper by Minxian Yang: “Some properties of vector autoregressive processes with Markov-switching coefficients.” Econometric Theory 18, 815–818.Google Scholar
Francq, C. & J.-M. Zakoïan (2005) The l2-structure of standard and switching-regime GARCH models. Stochastic Processes and Their Applications 115, 1557–1582.Google Scholar
Frömmel, M. (2004) Modelling Exchange Rate Volatility in the Run-up to EMU Using Markov Switching GARCH Model. Discussion paper, Hannover University.
Gray, S.F. (1996) Modeling the conditional distribution of interest rates as a regime-switching process. Journal of Financial Economics 42, 27–62.Google Scholar
Haas, M., S. Mittnik, & M.S. Paolella (2004a) Mixed normal conditional heteroskedasticity. Journal of Financial Econometrics 2, 211–250.Google Scholar
Haas, M., S. Mittnik, & M.S. Paolella (2004b) A new approach to Markov-switching GARCH models. Journal of Financial Econometrics 2, 493–530.Google Scholar
Hamilton, J.D. & R. Susmel (1994) Autoregressive conditional heteroskedasticity and changes in regime. Journal of Econometrics 64, 307–333.Google Scholar
Hawkins, D. & H. Simon (1949) Note: Some conditions of macroeconomics stability. Econometrica 17, 245–248.Google Scholar
Horn, R.A. & C.R. Johnson (1985) Matrix Analysis. Cambridge University Press.
Klaassen, F. (2002) Improving GARCH volatility forecasts with regime-switching GARCH. Empirical Economics 27, 363–394.Google Scholar
Lancaster, P. & M. Tismenetsky (1985) The Theory of Matrices, 2nd ed. Academic Press.
Marcucci, J. (2005) Forecasting stock market volatility with regime-switching GARCH models. Studies in Nonlinear Dynamics and Econometrics 9, 1145.Google Scholar
Metzler, L.A. (1950) A multiple-region theory of income and trade. Econometrica 18, 329–354.Google Scholar
Timmermann, A. (2000) Moments of Markov switching models. Journal of Econometrics 96, 75–111.Google Scholar
Wong, C.S. & W.K. Li (2001) On a mixture autoregressive conditional heteroscedastic model. Journal of the American Statistical Association 96, 982–985.Google Scholar
Yang, M. (2000) Some properties of vector autoregressive processes with Markov-switching coefficients. Econometric Theory 16, 23–43.Google Scholar
Yao, J. (2001) On square-integrability of an AR process with Markov switching. Statistics and Probability Letters 52, 265–270.Google Scholar