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Invariance principles for tempered fractionally integrated processes

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  • Sabzikar, Farzad
  • Surgailis, Donatas

Abstract

We discuss invariance principles for autoregressive tempered fractionally integrated moving averages in α-stable (1<α≤2) i.i.d. innovations and related tempered linear processes with vanishing tempering parameter limN→∞λ∕N=λ∗. We show that the limit of the partial sums process takes a different form in the weakly tempered (λ∗=0), strongly tempered (λ∗=∞), and moderately tempered (0<λ∗<∞) cases. These results are used to derive the limit distribution of the ordinary least squares estimate of AR(1) unit root with weakly, strongly, and moderately tempered moving average errors.

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  • Sabzikar, Farzad & Surgailis, Donatas, 2018. "Invariance principles for tempered fractionally integrated processes," Stochastic Processes and their Applications, Elsevier, vol. 128(10), pages 3419-3438.
    • Handle: RePEc:eee:spapps:v:128:y:2018:i:10:p:3419-3438
    DOI: 10.1016/j.spa.2017.11.004
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    1. Leipus, Remigijus & Paulauskas, Vygantas & Surgailis, Donatas, 2005. "Renewal regime switching and stable limit laws," Journal of Econometrics, Elsevier, vol. 129(1-2), pages 299-327.
    2. Sabzikar, Farzad & Surgailis, Donatas, 2018. "Tempered fractional Brownian and stable motions of second kind," Statistics & Probability Letters, Elsevier, vol. 132(C), pages 17-27.
    3. Giraitis, Liudas & Kokoszka, Piotr & Leipus, Remigijus & Teyssiere, Gilles, 2003. "Rescaled variance and related tests for long memory in volatility and levels," Journal of Econometrics, Elsevier, vol. 112(2), pages 265-294, February.
    4. Lo, Andrew W, 1991. "Long-Term Memory in Stock Market Prices," Econometrica, Econometric Society, vol. 59(5), pages 1279-1313, September.
    5. 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.
    6. Giraitis, Liudas & Kokoszka, Piotr & Leipus, Remigijus, 2000. "Stationary Arch Models: Dependence Structure And Central Limit Theorem," Econometric Theory, Cambridge University Press, vol. 16(1), pages 3-22, February.
    7. Phillips, P C B, 1987. "Time Series Regression with a Unit Root," Econometrica, Econometric Society, vol. 55(2), pages 277-301, March.
    8. Lavancier, Frédéric & Leipus, Remigijus & Philippe, Anne & Surgailis, Donatas, 2013. "Detection Of Nonconstant Long Memory Parameter," Econometric Theory, Cambridge University Press, vol. 29(5), pages 1009-1056, October.
    9. C. W. J. Granger & Roselyne Joyeux, 1980. "An Introduction To Long‐Memory Time Series Models And Fractional Differencing," Journal of Time Series Analysis, Wiley Blackwell, vol. 1(1), pages 15-29, January.
    10. Phillips, P C B, 1987. "Time Series Regression with a Unit Root," Econometrica, Econometric Society, vol. 55(2), pages 277-301, March.
    11. Sowell, Fallaw, 1990. "The Fractional Unit Root Distribution," Econometrica, Econometric Society, vol. 58(2), pages 495-505, March.
    12. GIRAITIS, Liudas & KOKOSZKA, Piotr & LEIPUS, Remigijus & TEYSSIÈRE, Gilles, 2003. "Rescaled variance and related tests for long memory in volatility and levels," LIDAM Reprints CORE 1594, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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    Cited by:

    1. Sabzikar, Farzad & Wang, Qiying & Phillips, Peter C.B., 2020. "Asymptotic theory for near integrated processes driven by tempered linear processes," Journal of Econometrics, Elsevier, vol. 216(1), pages 192-202.
    2. Beran, Jan & Sabzikar, Farzad & Surgailis, Donatas & Telkmann, Klaus, 2020. "On the empirical process of tempered moving averages," Statistics & Probability Letters, Elsevier, vol. 167(C).
    3. Farzad Sabzikar & Piotr Kokoszka, 2023. "Tempered functional time series," Journal of Time Series Analysis, Wiley Blackwell, vol. 44(3), pages 280-293, May.
    4. Kris Brabanter & Farzad Sabzikar, 2021. "Asymptotic theory for regression models with fractional local to unity root errors," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 84(7), pages 997-1024, October.
    5. dos Santos, Maike A.F., 2019. "Analytic approaches of the anomalous diffusion: A review," Chaos, Solitons & Fractals, Elsevier, vol. 124(C), pages 86-96.
    6. Sepideh Mosaferi & Mark S. Kaiser, 2021. "Nonparametric Cointegrating Regression Functions with Endogeneity and Semi-Long Memory," Papers 2111.00972, arXiv.org, revised Aug 2022.

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