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On functional limits of short- and long-memory linear processes with GARCH(1,1) noises

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  • Zhang, Rong-Mao
  • Sin, Chor-yiu (CY)
  • Ling, Shiqing

Abstract

This paper considers the short- and long-memory linear processes with GARCH (1,1) noises. The functional limit distributions of the partial sum and the sample autocovariances are derived when the tail index α is in (0,2), equal to 2, and in (2,∞), respectively. The partial sum weakly converges to a functional of α-stable process when α<2 and converges to a functional of Brownian motion when α≥2. When the process is of short-memory and α<4, the autocovariances converge to functionals of α/2-stable processes; and if α≥4, they converge to functionals of Brownian motions. In contrast, when the process is of long-memory, depending on α and β (the parameter that characterizes the long-memory), the autocovariances converge to either (i) functionals of α/2-stable processes; (ii) Rosenblatt processes (indexed by β, 1/2<β<3/4); or (iii) functionals of Brownian motions. The rates of convergence in these limits depend on both the tail index α and whether or not the linear process is short- or long-memory. Our weak convergence is established on the space of càdlàg functions on [0,1] with either (i) the J1 or the M1 topology (Skorokhod, 1956); or (ii) the weaker form S topology (Jakubowski, 1997). Some statistical applications are also discussed.

Suggested Citation

  • Zhang, Rong-Mao & Sin, Chor-yiu (CY) & Ling, Shiqing, 2015. "On functional limits of short- and long-memory linear processes with GARCH(1,1) noises," Stochastic Processes and their Applications, Elsevier, vol. 125(2), pages 482-512.
  • Handle: RePEc:eee:spapps:v:125:y:2015:i:2:p:482-512
    DOI: 10.1016/j.spa.2014.09.016
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    1. Peligrad, Magda & Sang, Hailin, 2012. "Asymptotic Properties Of Self-Normalized Linear Processes With Long Memory," Econometric Theory, Cambridge University Press, vol. 28(3), pages 548-569, June.
    2. Agnieszka Jach & Tucker McElroy & Dimitris N. Politis, 2012. "Subsampling inference for the mean of heavy‐tailed long‐memory time series," Journal of Time Series Analysis, Wiley Blackwell, vol. 33(1), pages 96-111, January.
    3. Phillips, P.C.B., 1990. "Time Series Regression With a Unit Root and Infinite-Variance Errors," Econometric Theory, Cambridge University Press, vol. 6(1), pages 44-62, March.
    4. Tucker McElroy & Agnieszka Jach, 2012. "Subsampling inference for the autocovariances and autocorrelations of long-memory heavy- tailed linear time series," Journal of Time Series Analysis, Wiley Blackwell, vol. 33(6), pages 935-953, November.
    5. 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.
    6. 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.
    7. Sowell, Fallaw, 1992. "Maximum likelihood estimation of stationary univariate fractionally integrated time series models," Journal of Econometrics, Elsevier, vol. 53(1-3), pages 165-188.
    8. 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.
    9. 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.
    10. Giraitis, Liudas & Surgailis, Donatas, 0. "ARCH-type bilinear models with double long memory," Stochastic Processes and their Applications, Elsevier, vol. 100(1-2), pages 275-300, July.
    11. Biao Wu, Wei & Min, Wanli, 2005. "On linear processes with dependent innovations," Stochastic Processes and their Applications, Elsevier, vol. 115(6), pages 939-958, June.
    12. Basrak, Bojan & Segers, Johan, 2009. "Regularly varying multivariate time series," Stochastic Processes and their Applications, Elsevier, vol. 119(4), pages 1055-1080, April.
    13. Kokoszka, Piotr S. & Taqqu, Murad S., 1995. "Fractional ARIMA with stable innovations," Stochastic Processes and their Applications, Elsevier, vol. 60(1), pages 19-47, November.
    14. Jushan Bai, 1993. "On The Partial Sums Of Residuals In Autoregressive And Moving Average Models," Journal of Time Series Analysis, Wiley Blackwell, vol. 14(3), pages 247-260, May.
    15. 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..
    16. Hosking, Jonathan R. M., 1996. "Asymptotic distributions of the sample mean, autocovariances, and autocorrelations of long-memory time series," Journal of Econometrics, Elsevier, vol. 73(1), pages 261-284, July.
    17. 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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    Cited by:

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    2. Zhang, Xingfa & Zhang, Rongmao & Li, Yuan & Ling, Shiqing, 2022. "LADE-based inferences for autoregressive models with heavy-tailed G-GARCH(1, 1) noise," Journal of Econometrics, Elsevier, vol. 227(1), pages 228-240.
    3. Sin, C.Y. (Chor-yiu) & Lee, Cheng-Few, 2021. "Using heteroscedasticity-non-consistent or heteroscedasticity-consistent variances in linear regression," Econometrics and Statistics, Elsevier, vol. 18(C), pages 117-142.
    4. 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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