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On Tail Index Estimation For Dependent, Heterogeneous Data


  • Hill, Jonathan B.


In this paper we analyze the asymptotic properties of the popular distribution tail index estimator by Hill (1975) for dependent, heterogeneous processes. We develop new extremal dependence measures that characterize a massive array of linear, nonlinear, and conditional volatility processes with long or short memory. We prove that the Hill estimator is weakly and uniformly weakly consistent for processes with extremes that form mixingale sequences and asymptotically normal for processes with extremes that are near epoch dependent (NED) on some arbitrary mixing functional. The extremal persistence assumptions in this paper are known to hold for mixing, L -NED, and some non- L -NED processes, including ARFIMA, FIGARCH, explosive GARCH, nonlinear ARMA-GARCH, and bilinear processes, and nonlinear distributed lags like random coefficient and regime-switching autoregressions.

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  • Hill, Jonathan B., 2010. "On Tail Index Estimation For Dependent, Heterogeneous Data," Econometric Theory, Cambridge University Press, vol. 26(05), pages 1398-1436, October.
  • Handle: RePEc:cup:etheor:v:26:y:2010:i:05:p:1398-1436_99

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    References listed on IDEAS

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    2. repec:cup:etheor:v:13:y:1997:i:3:p:353-67 is not listed on IDEAS
    3. Prasad Bidarkota & J Huston Mcculloch, 2004. "Testing for persistence in stock returns with GARCH-stable shocks," Quantitative Finance, Taylor & Francis Journals, vol. 4(3), pages 256-265.
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    6. Akgiray, Vedat & Booth, G Geoffrey, 1988. "The Stable-Law Model of Stock Returns," Journal of Business & Economic Statistics, American Statistical Association, vol. 6(1), pages 51-57, January.
    7. Davidson, James, 1993. "An L1-convergence theorem for heterogeneous mixingale arrays with trending moments," Statistics & Probability Letters, Elsevier, vol. 16(4), pages 301-304, March.
    8. Chan, Ngai Hang & Tran, Lanh Tat, 1989. "On the First-Order Autoregressive Process with Infinite Variance," Econometric Theory, Cambridge University Press, vol. 5(03), pages 354-362, December.
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    10. 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.
    11. Davidson, James, 1992. "A Central Limit Theorem for Globally Nonstationary Near-Epoch Dependent Functions of Mixing Processes," Econometric Theory, Cambridge University Press, vol. 8(03), pages 313-329, September.
    12. Davidson, James, 1993. "The Central Limit Theorem for Globally Nonstationary Near-Epoch Dependent Functions of Mixing Processes: The Asymptotically Degenerate Case," Econometric Theory, Cambridge University Press, vol. 9(03), pages 402-412, June.
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    Cited by:

    1. Sio Chong U & Jacky So & Deng Ding & Lihong Liu, 2016. "An efficient Fourier expansion method for the calculation of value-at-risk: Contributions of extra-ordinary risks," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 3(01), pages 1-27, March.
    2. Catani, P.S. & Ahlgren, N.J.C., 2017. "Combined Lagrange multiplier test for ARCH in vector autoregressive models," Econometrics and Statistics, Elsevier, vol. 1(C), pages 62-84.
    3. Rossi, Eduardo & Santucci de Magistris, Paolo, 2013. "Long memory and tail dependence in trading volume and volatility," Journal of Empirical Finance, Elsevier, vol. 22(C), pages 94-112.
    4. Aguilar, Mike & Hill, Jonathan B., 2015. "Robust score and portmanteau tests of volatility spillover," Journal of Econometrics, Elsevier, vol. 184(1), pages 37-61.
    5. Iglesias, Emma M., 2015. "Value at Risk of the main stock market indexes in the European Union (2000–2012)," Journal of Policy Modeling, Elsevier, vol. 37(1), pages 1-13.
    6. Bryan Kelly & Hao Jiang, 2013. "Tail Risk and Asset Prices," NBER Working Papers 19375, National Bureau of Economic Research, Inc.
    7. Hill, Jonathan B. & Aguilar, Mike, 2013. "Moment condition tests for heavy tailed time series," Journal of Econometrics, Elsevier, vol. 172(2), pages 255-274.
    8. Moosup Kim & Sangyeol Lee, 2016. "On the tail index inference for heavy-tailed GARCH-type innovations," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 68(2), pages 237-267, April.
    9. João Nicolau & Paulo M.M. Rodrigues, 2015. "A New Regression-Based Tail Index Estimator: An Application to Exchange Rates," Working Papers w201514, Banco de Portugal, Economics and Research Department.
    10. Iglesias, Emma M. & Linton, Oliver, 2009. "Estimation of tail thickness parameters from GJR-GARCH models," UC3M Working papers. Economics we094726, Universidad Carlos III de Madrid. Departamento de Economía.
    11. 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.
    12. Jondeau, Eric, 2016. "Asymmetry in tail dependence in equity portfolios," Computational Statistics & Data Analysis, Elsevier, vol. 100(C), pages 351-368.
    13. Nieto, Maria Rosa & Ruiz, Esther, 2016. "Frontiers in VaR forecasting and backtesting," International Journal of Forecasting, Elsevier, vol. 32(2), pages 475-501.
    14. Giovanni Caggiano & Efrem Castelnuovo, 2008. "Long Memory and Non-Linearities in International Inflation," "Marco Fanno" Working Papers 0076, Dipartimento di Scienze Economiche "Marco Fanno".
    15. Trapani, Lorenzo, 2016. "Testing for (in)finite moments," Journal of Econometrics, Elsevier, vol. 191(1), pages 57-68.
    16. 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.
    17. Prono, Todd, 2016. "Closed-Form Estimation of Finite-Order ARCH Models: Asymptotic Theory and Finite-Sample Performance," Finance and Economics Discussion Series 2016-083, Board of Governors of the Federal Reserve System (U.S.), revised Jul 2017.
    18. Iglesias, Emma M., 2015. "Value at Risk and expected shortfall of firms in the main European Union stock market indexes: A detailed analysis by economic sectors and geographical situation," Economic Modelling, Elsevier, vol. 50(C), pages 1-8.

    More about this item

    JEL classification:

    • C1 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General
    • C2 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables
    • C3 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables
    • C4 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics
    • C5 - Mathematical and Quantitative Methods - - Econometric Modeling
    • C8 - Mathematical and Quantitative Methods - - Data Collection and Data Estimation Methodology; Computer Programs


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