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The functional central limit theorem for a family of GARCH observations with applications

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  • Berkes, István
  • Hörmann, Siegfried
  • Horváth, Lajos

Abstract

We consider polynomial variables which define an important subclass of Duan's augmented processes. We prove functional central limit theorems for the observations as well as for the volatility process under the assumption of finite second moments. The results imply the convergence of CUSUM, MOSUM and Dickey-Fuller statistics under optimal conditions.

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  • Berkes, István & Hörmann, Siegfried & Horváth, Lajos, 2008. "The functional central limit theorem for a family of GARCH observations with applications," Statistics & Probability Letters, Elsevier, vol. 78(16), pages 2725-2730, November.
    • Handle: RePEc:eee:stapro:v:78:y:2008:i:16:p:2725-2730
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    Cited by:

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    2. Moritz Jirak, 2017. "On Weak Invariance Principles for Partial Sums," Journal of Theoretical Probability, Springer, vol. 30(3), pages 703-728, September.
    3. Jean-Yves Pitarakis, 2017. "A Simple Approach for Diagnosing Instabilities in Predictive Regressions," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 79(5), pages 851-874, October.
    4. Eunju Hwang, 2021. "Limit Theory for Stationary Autoregression with Heavy-Tailed Augmented GARCH Innovations," Mathematics, MDPI, vol. 9(8), pages 1-10, April.
    5. Lee, Oesook & Lee, Jungwha, 2014. "The functional central limit theorem for the multivariate MS–ARMA–GARCH model," Economics Letters, Elsevier, vol. 125(3), pages 331-335.
    6. Lee, Oesook, 2018. "Stationarity and functional central limit theorem for ARCH(∞) models," Economics Letters, Elsevier, vol. 162(C), pages 107-111.
    7. Jean-Yves Pitarakis, 2020. "A Novel Approach to Predictive Accuracy Testing in Nested Environments," Papers 2008.08387, arXiv.org, revised Oct 2023.
    8. Moritz Jirak, 2021. "Edgeworth expansions for volatility models," Papers 2111.00529, arXiv.org, revised Sep 2022.
    9. Moritz Jirak, 2016. "Optimal Rate of Convergence for Empirical Quantiles and Distribution Functions for Time Series," Journal of Time Series Analysis, Wiley Blackwell, vol. 37(6), pages 825-836, November.
    10. Marcel, Bräutigam & Marie, Kratz, 2019. "Bivariate FCLT for the Sample Quantile and Measures of Dispersion for Augmented GARCH(p, q) processes," ESSEC Working Papers WP1909, ESSEC Research Center, ESSEC Business School.
    11. Lee, O., 2013. "The functional central limit theorem for ARMA–GARCH processes," Economics Letters, Elsevier, vol. 121(3), pages 432-435.

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