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Stationarity And Memory Of Arch(∞) Models

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  • Zaffaroni, Paolo

Abstract

We establish the necessary and sufficient conditions for covariance stationarity of ARCH(∞), for both the levels and the squares. The result applies to any form of the conditional variance coefficients. This includes GARCH(p,q) and also specifications with hyperbolically decaying coefficients, such as the autoregressive coefficients of the autoregressive fractionally integrated moving average model. The covariance stationarity condition for the levels rules out long memory in the squares.I thank Peter M. Robinson for useful comments on previous versions of the paper. Also, I am grateful to the co-editor (Bruce E. Hansen) and an anonymous referee whose suggestions greatly improved the paper.

Suggested Citation

  • Zaffaroni, Paolo, 2004. "Stationarity And Memory Of Arch(∞) Models," Econometric Theory, Cambridge University Press, vol. 20(1), pages 147-160, February.
    • Handle: RePEc:cup:etheor:v:20:y:2004:i:01:p:147-160_20
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    Cited by:

    1. Royer, Julien, 2023. "Conditional asymmetry in Power ARCH(∞) models," Journal of Econometrics, Elsevier, vol. 234(1), pages 178-204.
    2. Josu Arteche, 2012. "Standard and seasonal long memory in volatility: an application to Spanish inflation," Empirical Economics, Springer, vol. 42(3), pages 693-712, June.
    3. Li, Muyi & Li, Wai Keung & Li, Guodong, 2015. "A new hyperbolic GARCH model," Journal of Econometrics, Elsevier, vol. 189(2), pages 428-436.
    4. Ruiz, Esther & Veiga, Helena, 2008. "Modelling long-memory volatilities with leverage effect: A-LMSV versus FIEGARCH," Computational Statistics & Data Analysis, Elsevier, vol. 52(6), pages 2846-2862, February.
    5. Bordignon, Silvano & Caporin, Massimiliano & Lisi, Francesco, 2007. "Generalised long-memory GARCH models for intra-daily volatility," Computational Statistics & Data Analysis, Elsevier, vol. 51(12), pages 5900-5912, August.
    6. Dominique Guegan, 2007. "La persistance dans les marchés financiers," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00179269, HAL.
    7. Conrad, Christian & Karanasos, Menelaos, 2006. "The impulse response function of the long memory GARCH process," Economics Letters, Elsevier, vol. 90(1), pages 34-41, January.
    8. Dominique Guegan, 2005. "How can we Define the Concept of Long Memory? An Econometric Survey," Econometric Reviews, Taylor & Francis Journals, vol. 24(2), pages 113-149.
    9. Conrad, Christian, 2010. "Non-negativity conditions for the hyperbolic GARCH model," Journal of Econometrics, Elsevier, vol. 157(2), pages 441-457, August.
    10. Robinson, Peter M. & Zafaroni, Paolo, 2005. "Pseudo-maximum likelihood estimation of ARCH models," LSE Research Online Documents on Economics 4544, London School of Economics and Political Science, LSE Library.
    11. Jondeau, Eric, 2015. "The dynamics of squared returns under contemporaneous aggregation of GARCH models," Journal of Empirical Finance, Elsevier, vol. 32(C), pages 80-93.
    12. Robinson, Peter M. & Zaffaroni, Paolo, 2005. "Pseudo-maximum likelihood estimation of ARCH(∞) models," LSE Research Online Documents on Economics 58182, London School of Economics and Political Science, LSE Library.
    13. Harry Vander Elst, 2015. "FloGARCH : Realizing long memory and asymmetries in returns volatility," Working Paper Research 280, National Bank of Belgium.
    14. repec:wyi:journl:002190 is not listed on IDEAS
    15. Peter M Robinson & Paolo Zaffaroni, 2005. "Pseudo-Maximum Likelihood Estimation of ARCH(8) Models," STICERD - Econometrics Paper Series 495, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
    16. Kwan, Wilson & Li, Wai Keung & Li, Guodong, 2012. "On the estimation and diagnostic checking of the ARFIMA–HYGARCH model," Computational Statistics & Data Analysis, Elsevier, vol. 56(11), pages 3632-3644.
    17. Souhir Ben Amor & Heni Boubaker & Lotfi Belkacem, 2022. "A Dual Generalized Long Memory Modelling for Forecasting Electricity Spot Price: Neural Network and Wavelet Estimate," Papers 2204.08289, arXiv.org.
    18. Muyi Li & Wai Keung Li & Guodong Li, 2013. "On Mixture Memory Garch Models," Journal of Time Series Analysis, Wiley Blackwell, vol. 34(6), pages 606-624, November.
    19. Darolles, Serge & Fol, Gaëlle Le & Lu, Yang & Sun, Ran, 2019. "Bivariate integer-autoregressive process with an application to mutual fund flows," Journal of Multivariate Analysis, Elsevier, vol. 173(C), pages 181-203.
    20. Dominique Guegan, 2005. "How can we Define the Concept of Long Memory? An Econometric Survey," Econometric Reviews, Taylor & Francis Journals, vol. 24(2), pages 113-149.
    21. Axel Groß‐KlußMann & Nikolaus Hautsch, 2013. "Predicting Bid–Ask Spreads Using Long‐Memory Autoregressive Conditional Poisson Models," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 32(8), pages 724-742, December.
    22. Carl Lönnbark, 2016. "Asymmetry with respect to the memory in stock market volatilities," Empirical Economics, Springer, vol. 50(4), pages 1409-1419, June.
    23. Royer, Julien, 2021. "Conditional asymmetry in Power ARCH($\infty$) models," MPRA Paper 109118, University Library of Munich, Germany.
    24. Bildirici, Melike & Ersin, Özgür, 2012. "Nonlinear volatility models in economics: smooth transition and neural network augmented GARCH, APGARCH, FIGARCH and FIAPGARCH models," MPRA Paper 40330, University Library of Munich, Germany, revised May 2012.
    25. Antypas, Antonios & Koundouri, Phoebe & Kourogenis, Nikolaos, 2013. "Aggregational Gaussianity and barely infinite variance in financial returns," Journal of Empirical Finance, Elsevier, vol. 20(C), pages 102-108.

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