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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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    1. Drost, Feike C & Nijman, Theo E, 1993. "Temporal Aggregation of GARCH Processes," Econometrica, Econometric Society, vol. 61(4), pages 909-927, July.
    2. Ling, Shiqing & McAleer, Michael, 2003. "Asymptotic Theory For A Vector Arma-Garch Model," Econometric Theory, Cambridge University Press, vol. 19(2), pages 280-310, April.
    3. Bollerslev, Tim & Ghysels, Eric, 1996. "Periodic Autoregressive Conditional Heteroscedasticity," Journal of Business & Economic Statistics, American Statistical Association, vol. 14(2), pages 139-151, April.
    4. Jensen, Søren Tolver & Rahbek, Anders, 2004. "Asymptotic Inference For Nonstationary Garch," Econometric Theory, Cambridge University Press, vol. 20(6), pages 1203-1226, December.
    5. Nicholas Taylor, 2004. "Modeling discontinuous periodic conditional volatility: Evidence from the commodity futures market," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 24(9), pages 805-834, September.
    6. Mohamed Bentarzi & Marc Hallin, 1994. "On The Invertibility Of Periodic Moving‐Average Models," Journal of Time Series Analysis, Wiley Blackwell, vol. 15(3), pages 263-268, May.
    7. Martin Martens & Yuan‐Chen Chang & Stephen J. Taylor, 2002. "A Comparison of Seasonal Adjustment Methods When Forecasting Intraday Volatility," Journal of Financial Research, Southern Finance Association;Southwestern Finance Association, vol. 25(2), pages 283-299, June.
    8. Lee, Sang-Won & Hansen, Bruce E., 1994. "Asymptotic Theory for the Garch(1,1) Quasi-Maximum Likelihood Estimator," Econometric Theory, Cambridge University Press, vol. 10(1), pages 29-52, March.
    9. Søren Tolver Jensen & Anders Rahbek, 2004. "Asymptotic Normality of the QMLE Estimator of ARCH in the Nonstationary Case," Econometrica, Econometric Society, vol. 72(2), pages 641-646, March.
    10. Franses, Philip Hans & Paap, Richard, 2004. "Periodic Time Series Models," OUP Catalogue, Oxford University Press, number 9780199242030.
    11. Ghysels,Eric & Osborn,Denise R., 2001. "The Econometric Analysis of Seasonal Time Series," Cambridge Books, Cambridge University Press, number 9780521565882.
    12. Lumsdaine, Robin L, 1996. "Consistency and Asymptotic Normality of the Quasi-maximum Likelihood Estimator in IGARCH(1,1) and Covariance Stationary GARCH(1,1) Models," Econometrica, Econometric Society, vol. 64(3), pages 575-596, May.
    13. 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.
    14. Francq, Christian & Zakoian, Jean-Michel, 2007. "Quasi-maximum likelihood estimation in GARCH processes when some coefficients are equal to zero," Stochastic Processes and their Applications, Elsevier, vol. 117(9), pages 1265-1284, September.
    15. Nelson, Daniel B., 1990. "Stationarity and Persistence in the GARCH(1,1) Model," Econometric Theory, Cambridge University Press, vol. 6(3), pages 318-334, September.
    16. Taylor, James W., 2006. "Density forecasting for the efficient balancing of the generation and consumption of electricity," International Journal of Forecasting, Elsevier, vol. 22(4), pages 707-724.
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    Cited by:

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    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.
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    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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