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Gauss-Newton and M-estimation for ARMA processes with infinite variance

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  • Davis, Richard A.
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    Abstract

    We consider two estimation procedures, Gauss-Newton and M-estimation, for the parameters of an ARMA (p,q) process when the innovations belong to the domain of attraction of a nonnormal stable distribution. The Gauss-Newton or iterative least squares estimate is shown to have the same limiting distribution as the maximum likelihood and Whittle estimates. The latter was derived recently by Mikosch et al. (1995). We also establish the weak convergence for a class of M-estimates, including the case of least absolute deviation, and show that, asymptotically, the M-estimate dominates both the Gauss-Newton and Whittle estimates. A brief simulation is carried out comparing the performance of M-estimation with iterative and ordinary least squares. As suggested by the asymptotic theory, M-estimation, using least absolute deviation for the loss function, outperforms the other two procedures.

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    Bibliographic Info

    Article provided by Elsevier in its journal Stochastic Processes and their Applications.

    Volume (Year): 63 (1996)
    Issue (Month): 1 (October)
    Pages: 75-95

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    Handle: RePEc:eee:spapps:v:63:y:1996:i:1:p:75-95

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    Keywords: Gauss-Newton estimate Heavy-tails Stable distributions M-estimation ARMA processes;

    References

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    1. Kl├╝ppelberg, Claudia & Mikosch, Thomas, 1993. "Spectral estimates and stable processes," Stochastic Processes and their Applications, Elsevier, vol. 47(2), pages 323-344, September.
    2. Davis, Richard A. & Knight, Keith & Liu, Jian, 1992. "M-estimation for autoregressions with infinite variance," Stochastic Processes and their Applications, Elsevier, vol. 40(1), pages 145-180, February.
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    Cited by:
    1. Li, Jinyu & Liang, Wei & He, Shuyuan, 2011. "Empirical likelihood for LAD estimators in infinite variance ARMA models," Statistics & Probability Letters, Elsevier, vol. 81(2), pages 212-219, February.
    2. SBRANA, Giacomo & SILVESTRINI, Andrea, 2010. "Aggregation of exponential smoothing processes with an application to portfolio risk evaluation," CORE Discussion Papers 2010039, Universit├ę catholique de Louvain, Center for Operations Research and Econometrics (CORE).

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