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Tail And Nontail Memory With Applications To Extreme Value And Robust Statistics


  • Hill, Jonathan B.


New notions of tail and nontail dependence are used to characterize separately extremal and nonextremal information, including tail log-exceedances and events, and tail-trimmed levels. We prove that near epoch dependence (McLeish, 1975; Gallant and White, 1988) and L 0 -approximability (Pötscher and Prucha, 1991) are equivalent for tail events and tail-trimmed levels, ensuring a Gaussian central limit theory for important extreme value and robust statistics under general conditions. We apply the theory to characterize the extremal and nonextremal memory properties of possibly very heavy-tailed GARCH processes and distributed lags. This in turn is used to verify Gaussian limits for tail index, tail dependence, and tail-trimmed sums of these data, allowing for Gaussian asymptotics for a new tail-trimmed least squares estimator for heavy-tailed processes.

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  • Hill, Jonathan B., 2011. "Tail And Nontail Memory With Applications To Extreme Value And Robust Statistics," Econometric Theory, Cambridge University Press, vol. 27(04), pages 844-884, August.
  • Handle: RePEc:cup:etheor:v:27:y:2011:i:04:p:844-884_00

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    References listed on IDEAS

    1. Matzkin, Rosa L, 1992. "Nonparametric and Distribution-Free Estimation of the Binary Threshold Crossing and the Binary Choice Models," Econometrica, Econometric Society, vol. 60(2), pages 239-270, March.
    2. Imbens, Guido W, 1992. "An Efficient Method of Moments Estimator for Discrete Choice Models with Choice-Based Sampling," Econometrica, Econometric Society, vol. 60(5), pages 1187-1214, September.
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    6. Newey, Whitney K. & McFadden, Daniel, 1986. "Large sample estimation and hypothesis testing," Handbook of Econometrics,in: R. F. Engle & D. McFadden (ed.), Handbook of Econometrics, edition 1, volume 4, chapter 36, pages 2111-2245 Elsevier.
    7. Donald W.K. Andrews, 1986. "Consistency in Nonlinear Econometric Models: A Generic Uniform Law of Large Numbers," Cowles Foundation Discussion Papers 790, Cowles Foundation for Research in Economics, Yale University.
    8. Horowitz, Joel L, 1992. "A Smoothed Maximum Score Estimator for the Binary Response Model," Econometrica, Econometric Society, vol. 60(3), pages 505-531, May.
    9. de Jong, Robert M., 1997. "Central Limit Theorems for Dependent Heterogeneous Random Variables," Econometric Theory, Cambridge University Press, vol. 13(03), pages 353-367, June.
    10. Andrews, Donald W.K., 1988. "Laws of Large Numbers for Dependent Non-Identically Distributed Random Variables," Econometric Theory, Cambridge University Press, vol. 4(03), pages 458-467, December.
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    12. Eichengreen, Barry & Watson, Mark W & Grossman, Richard S, 1985. "Bank Rate Policy under the Interwar Gold Standard: A Dynamic Probit Model," Economic Journal, Royal Economic Society, vol. 95(379), pages 725-745, September.
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    Cited by:

    1. Hill, Jonathan B. & Aguilar, Mike, 2013. "Moment condition tests for heavy tailed time series," Journal of Econometrics, Elsevier, vol. 172(2), pages 255-274.
    2. Hill, Jonathan B. & Prokhorov, Artem, 2016. "GEL estimation for heavy-tailed GARCH models with robust empirical likelihood inference," Journal of Econometrics, Elsevier, vol. 190(1), pages 18-45.
    3. Bollerslev, Tim & Todorov, Viktor & Li, Sophia Zhengzi, 2013. "Jump tails, extreme dependencies, and the distribution of stock returns," Journal of Econometrics, Elsevier, vol. 172(2), pages 307-324.
    4. Ilić, Ivana, 2012. "On tail index estimation using a sample with missing observations," Statistics & Probability Letters, Elsevier, vol. 82(5), pages 949-958.
    5. repec:bla:istatr:v:85:y:2017:i:1:p:108-142 is not listed on IDEAS
    6. Hill, Jonathan B. & Shneyerov, Artyom, 2013. "Are there common values in first-price auctions? A tail-index nonparametric test," Journal of Econometrics, Elsevier, vol. 174(2), pages 144-164.

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