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Quasi‐maximum likelihood estimation of periodic GARCH and periodic ARMA‐GARCH processes

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  • Abdelhakim Aknouche
  • Abdelouahab Bibi

Abstract

. This article establishes the strong consistency and asymptotic normality (CAN) of the quasi‐maximum likelihood estimator (QMLE) for generalized autoregressive conditionally heteroscedastic (GARCH) and autoregressive moving‐average (ARMA)‐GARCH processes with periodically time‐varying parameters. We first give a necessary and sufficient condition for the existence of a strictly periodically stationary solution of the periodic GARCH (PGARCH) equation. As a result, it is shown that the moment of some positive order of the PGARCH solution is finite, under which we prove the strong consistency and asymptotic normality of the QMLE for a PGARCH process without any condition on its moments and for a periodic ARMA‐GARCH (PARMA‐PGARCH) under mild conditions.

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  • Abdelhakim Aknouche & Abdelouahab Bibi, 2009. "Quasi‐maximum likelihood estimation of periodic GARCH and periodic ARMA‐GARCH processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 30(1), pages 19-46, January.
    • Handle: RePEc:bla:jtsera:v:30:y:2009:i:1:p:19-46
    DOI: 10.1111/j.1467-9892.2008.00598.x
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    Cited by:

    1. Daniel Dzikowski & Carsten Jentsch, 2024. "Structural Periodic Vector Autoregressions," Papers 2401.14545, arXiv.org.
    2. Ikhlaas Gurrib & Elgilani Elsharief & Firuz Kamalov, 2020. "The Effect of Energy Cryptos on Efficient Portfolios of Key Energy Listed Companies in the S&P Composite 1500 Energy Index," International Journal of Energy Economics and Policy, Econjournals, vol. 10(2), pages 179-193.
    3. Regnard, Nazim & Zakoïan, Jean-Michel, 2011. "A conditionally heteroskedastic model with time-varying coefficients for daily gas spot prices," Energy Economics, Elsevier, vol. 33(6), pages 1240-1251.
    4. Aknouche, Abdelhakim & Demmouche, Nacer & Touche, Nassim, 2018. "Bayesian MCMC analysis of periodic asymmetric power GARCH models," MPRA Paper 91136, University Library of Munich, Germany.
    5. Christian Francq & Roch Roy & Abdessamad Saidi, 2011. "Asymptotic Properties of Weighted Least Squares Estimation in Weak PARMA Models," Journal of Time Series Analysis, Wiley Blackwell, vol. 32(6), pages 699-723, November.
    6. Bibi, Abdelouahab & Lescheb, Ines, 2014. "A note on integrated periodic GARCH processes," Statistics & Probability Letters, Elsevier, vol. 87(C), pages 121-124.
    7. Aknouche, Abdelhakim, 2013. "Periodic autoregressive stochastic volatility," MPRA Paper 69571, University Library of Munich, Germany, revised 2015.
    8. Ausloos, Marcel & Nedic, Olgica & Dekanski, Aleksandar & Mrowinski, Maciej J. & Fronczak, Piotr & Fronczak, Agata, 2017. "Day of the week effect in paper submission/acceptance/rejection to/in/by peer review journals. II. An ARCH econometric-like modeling," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 468(C), pages 462-474.
    9. repec:dau:papers:123456789/5529 is not listed on IDEAS
    10. Aknouche, Abdelhakim & Al-Eid, Eid & Demouche, Nacer, 2016. "Generalized quasi-maximum likelihood inference for periodic conditionally heteroskedastic models," MPRA Paper 75770, University Library of Munich, Germany, revised 19 Dec 2016.
    11. Nadia Boussaha & Faycal Hamdi & Saïd Souam, 2018. "Multivariate Periodic Stochastic Volatility Models: Applications to Algerian dinar exchange rates and oil prices modeling," EconomiX Working Papers 2018-14, University of Paris Nanterre, EconomiX.

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