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Limit theory of quadratic forms of long-memory linear processes with heavy-tailed GARCH innovations

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  • Chan, Ngai Hang
  • Zhang, Rong-Mao

Abstract

Let Xt=∑j=0∞cjεt−j be a moving average process with GARCH (1, 1) innovations {εt}. In this paper, the asymptotic behavior of the quadratic form Qn=∑j=1n∑s=1nb(t−s)XtXs is derived when the innovation {εt} is a long-memory and heavy-tailed process with tail index α, where {b(i)} is a sequence of constants. In particular, it is shown that when 1<α<4 and under certain regularity conditions, the limit distribution of Qn converges to a stable random variable with index α/2. However, when α≥4, Qn has an asymptotic normal distribution. These results not only shed light on the singular behavior of the quadratic forms when both long-memory and heavy-tailed properties are present, but also have applications in the inference for general linear processes driven by heavy-tailed GARCH innovations.

Suggested Citation

  • Chan, Ngai Hang & Zhang, Rong-Mao, 2013. "Limit theory of quadratic forms of long-memory linear processes with heavy-tailed GARCH innovations," Journal of Multivariate Analysis, Elsevier, vol. 120(C), pages 18-33.
    • Handle: RePEc:eee:jmvana:v:120:y:2013:i:c:p:18-33
    DOI: 10.1016/j.jmva.2013.04.007
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    References listed on IDEAS

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    1. Basrak, Bojan & Davis, Richard A. & Mikosch, Thomas, 2002. "Regular variation of GARCH processes," Stochastic Processes and their Applications, Elsevier, vol. 99(1), pages 95-115, May.
    2. Ling S., 2003. "Adaptive Estimators and Tests of Stationary and Nonstationary Short- and Long-Memory ARFIMA-GARCH Models," Journal of the American Statistical Association, American Statistical Association, vol. 98, pages 955-967, January.
    3. Kokoszka, Piotr S. & Taqqu, Murad S., 1997. "The asymptotic behavior of quadratic forms in heavy-tailed strongly dependent random variables," Stochastic Processes and their Applications, Elsevier, vol. 66(1), pages 21-40, February.
    4. Baillie, Richard T & Chung, Ching-Fan & Tieslau, Margie A, 1996. "Analysing Inflation by the Fractionally Integrated ARFIMA-GARCH Model," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 11(1), pages 23-40, Jan.-Feb..
    5. Rong‐Mao Zhang & Zheng‐Yan Lin, 2012. "Limit theory for a general class of GARCH models with just barely infinite variance," Journal of Time Series Analysis, Wiley Blackwell, vol. 33(1), pages 161-174, January.
    6. Wu, Wei Biao & Shao, Xiaofeng, 2007. "A Limit Theorem For Quadratic Forms And Its Applications," Econometric Theory, Cambridge University Press, vol. 23(5), pages 930-951, October.
    7. Koul, Hira L. & Surgailis, Donatas, 2001. "Asymptotics of empirical processes of long memory moving averages with infinite variance," Stochastic Processes and their Applications, Elsevier, vol. 91(2), pages 309-336, February.
    8. Carrasco, Marine & Chen, Xiaohong, 2002. "Mixing And Moment Properties Of Various Garch And Stochastic Volatility Models," Econometric Theory, Cambridge University Press, vol. 18(1), pages 17-39, February.
    9. Surgailis, Donatas, 0. "Stable limits of empirical processes of moving averages with infinite variance," Stochastic Processes and their Applications, Elsevier, vol. 100(1-2), pages 255-274, July.
    10. Francq, Christian & Zakoïan, Jean-Michel, 2006. "Mixing Properties Of A General Class Of Garch(1,1) Models Without Moment Assumptions On The Observed Process," Econometric Theory, Cambridge University Press, vol. 22(5), pages 815-834, October.
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    1. Hwang, Eunju & Hong, Won-Tak, 2021. "A multivariate HAR-RV model with heteroscedastic errors and its WLS estimation," Economics Letters, Elsevier, vol. 203(C).

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