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Journal of Financial Econometrics Advance Access published online on September 6, 2006
Journal of Financial Econometrics, doi:10.1093/jjfinec/nbl004
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© The Author 2006. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org
Received July 10, 2006
Revised July 31, 2006
Accepted August 9, 2006

Article

Stationarity of a Markov-Switching GARCH Model

Ji-Chun Liu 1 *
     1 School of Mathematical Science, Xiamen University, Xiamen, 361005, PR China

* To whom correspondence should be addressed.
Ji-Chun Liu, E-mail: liujichun65{at}126.com


  Abstract

This article investigates some structural properties of the Markov-switching GARCH process introduced by Haas, Mittnik, and Paolella. First, a sufficient and necessary condition for the existence of the weakly stationary solution of the process is presented. The solution is weakly stationary, and the causal expansion of the Markov-switching GARCH process is also established. Second, the general conditions for the existence of any integer-order moment of the square of the process are derived. The technique used in this article for the weak stationarity and the high-order moments of the process is different from that used by Haas, Mittnik, and Paolella and avoids the assumption that the process started in the infinite past with finite variance. Third, a sufficient and necessary condition for the strict stationarity of the Markov-switching GARCH process with possibly infinite variance is given. Finally, the strict stationarity of the so-called integrated Markov-switching GARCH process is also discussed.

Keywords: ergodicity, existence of moments, integrated Markov-switching GARCH process, Markov-switching GARCH process, strict stationarity, weak stationarity.
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