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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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Article provided by Cambridge University Press in its journal Econometric Theory.

Volume (Year): 26 (2010)
Issue (Month): 05 (October)
Pages: 1398-1436

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Handle: RePEc:cup:etheor:v:26:y:2010:i:05:p:1398-1436_99
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  1. Prasad V. Bidarkota & J. Huston McCulloch, . "Optimal Univariate Inflation Forecasting with Symmetric Stable Shocks," Computing in Economics and Finance 1997 116, Society for Computational Economics.
  2. B. N. Cheng & S. T. Rachev, 1995. "Multivariate Stable Futures Prices," Mathematical Finance, Wiley Blackwell, vol. 5(2), pages 133-153.
  3. 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.
  4. 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.
  5. 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.
  6. 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.
  7. Davidson, James, 2004. "Moment and Memory Properties of Linear Conditional Heteroscedasticity Models, and a New Model," Journal of Business & Economic Statistics, American Statistical Association, vol. 22(1), pages 16-29, January.
  8. Caner, Mehmet, 1998. "Tests for cointegration with infinite variance errors," Journal of Econometrics, Elsevier, vol. 86(1), pages 155-175, June.
  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. 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.
  11. 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.
  12. repec:cup:etheor:v:13:y:1997:i:3:p:353-67 is not listed on IDEAS
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