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On the probabilistic structure of power threshold generalized arch stochastic processes

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  • Gonçalves, E.
  • Leite, J.
  • Mendes-Lopes, N.

Abstract

The aim of this paper is to develop a probabilistic study on a large and general class of conditionally heteroscedastic models, namely the δ-TGARCH processes. For this class of processes we establish necessary and sufficient conditions of strict stationarity, ergodicity and existence of moments. A discussion on the weak stationarity of an associated vectorial process, moments and weak stationarity up to the order δ of those processes is also presented. Finally, the minimal representation of a δ-TGARCH process is obtained developing, in a unique way, the corresponding conditional moment of order δ in terms of present and past observations.

Suggested Citation

  • Gonçalves, E. & Leite, J. & Mendes-Lopes, N., 2012. "On the probabilistic structure of power threshold generalized arch stochastic processes," Statistics & Probability Letters, Elsevier, vol. 82(8), pages 1597-1609.
    • Handle: RePEc:eee:stapro:v:82:y:2012:i:8:p:1597-1609
    DOI: 10.1016/j.spl.2012.04.014
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    1. Mittnik, Stefan & Paolella, Marc S. & Rachev, Svetlozar T., 2002. "Stationarity of stable power-GARCH processes," Journal of Econometrics, Elsevier, vol. 106(1), pages 97-107, January.
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    3. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
    4. Gonçalves, E. & Mendes-Lopes, N., 2010. "On the structure of generalized threshold arch processes," Statistics & Probability Letters, Elsevier, vol. 80(7-8), pages 573-580, April.
    5. Engle, Robert F, 1982. "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation," Econometrica, Econometric Society, vol. 50(4), pages 987-1007, July.
    6. Ding, Zhuanxin & Granger, Clive W. J. & Engle, Robert F., 1993. "A long memory property of stock market returns and a new model," Journal of Empirical Finance, Elsevier, vol. 1(1), pages 83-106, June.
    7. Bougerol, Philippe & Picard, Nico, 1992. "Stationarity of Garch processes and of some nonnegative time series," Journal of Econometrics, Elsevier, vol. 52(1-2), pages 115-127.
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    Cited by:

    1. Jeffrey Chu & Stephen Chan & Saralees Nadarajah & Joerg Osterrieder, 2017. "GARCH Modelling of Cryptocurrencies," JRFM, MDPI, vol. 10(4), pages 1-15, October.

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