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Asymptotic Inference For Ar Models With Heavy-Tailed G-Garch Noises

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  • Zhang, Rongmao
  • Ling, Shiqing

Abstract

It is well known that the least squares estimator (LSE) of an AR(p) model with i.i.d. (independent and identically distributed) noises is n1/αL(n)-consistent when the tail index α of the noise is within (0,2) and is n1/2-consistent when α ≥ 2, where L(n) is a slowly varying function. When the noises are not i.i.d., however, the case is far from clear. This paper studies the LSE of AR(p) models with heavy-tailed G-GARCH(1,1) noises. When the tail index α of G-GARCH is within (0,2), it is shown that the LSE is not a consistent estimator of the parameters, but converges to a ratio of stable vectors. When α ε [2,4], it is shown that the LSE is n1–2/α-consistent if α ε (2,4), logn-consistent if α = 2, and n1/2 / logn-consistent if α = 4, and its limiting distribution is a functional of stable processes. Our results are significantly different from those with i.i.d. noises and should warn practitioners in economics and finance of the implications, including inconsistency, of heavy-tailed errors in the presence of conditional heterogeneity.

Suggested Citation

  • Zhang, Rongmao & Ling, Shiqing, 2015. "Asymptotic Inference For Ar Models With Heavy-Tailed G-Garch Noises," Econometric Theory, Cambridge University Press, vol. 31(4), pages 880-890, August.
    • Handle: RePEc:cup:etheor:v:31:y:2015:i:04:p:880-890_00
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    Cited by:

    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).
    2. Ke Zhu, 2018. "Statistical inference for autoregressive models under heteroscedasticity of unknown form," Papers 1804.02348, arXiv.org, revised Aug 2018.
    3. Jiang, Feiyu & Li, Dong & Zhu, Ke, 2020. "Non-standard inference for augmented double autoregressive models with null volatility coefficients," Journal of Econometrics, Elsevier, vol. 215(1), pages 165-183.
    4. Guili Liao & Qimeng Liu & Rongmao Zhang & Shifang Zhang, 2022. "Rank test of unit‐root hypothesis with AR‐GARCH errors," Journal of Time Series Analysis, Wiley Blackwell, vol. 43(5), pages 695-719, September.
    5. Francq, Christian & Zakoian, Jean-Michel, 2021. "Testing the existence of moments and estimating the tail index of augmented garch processes," MPRA Paper 110511, University Library of Munich, Germany.
    6. Hwang, Eunju, 2019. "A note on limit theory for mildly stationary autoregression with a heavy-tailed GARCH error process," Statistics & Probability Letters, Elsevier, vol. 152(C), pages 59-68.
    7. 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.
    8. Eunju Hwang, 2021. "Limit Theory for Stationary Autoregression with Heavy-Tailed Augmented GARCH Innovations," Mathematics, MDPI, vol. 9(8), pages 1-10, April.
    9. Pedersen, Rasmus Søndergaard, 2017. "Robust inference in conditionally heteroskedastic autoregressions," MPRA Paper 81979, University Library of Munich, Germany.
    10. She, Rui & Ling, Shiqing, 2020. "Inference in heavy-tailed vector error correction models," Journal of Econometrics, Elsevier, vol. 214(2), pages 433-450.
    11. Feiyu Jiang & Dong Li & Ke Zhu, 2019. "Non-standard inference for augmented double autoregressive models with null volatility coefficients," Papers 1905.01798, arXiv.org.

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