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On Tail Index Estimation Using Dependent,Heterogenous Data

  • Jonathan B. Hill

    ()

    (Department of Economics, Florida International University)

In this paper we analyze the asymptotic properties of the popularly used distribution tail estimator by B. Hill (1975), for heavy-tailed heterogenous, dependent processes. We prove the Hill estimator is weakly consistent for functionals of mixingales and L1-approximable processes with regularly varying tails, covering ARMA, GARCH, and many IGARCH and FIGARCH processes. Moreover, for functionals of processes near epoch-dependent on a mixing process, we prove a Gaussian distribution limit exists. In this case, as opposed to all existing prior results in the literature, we do not require the limiting variance of the Hill estimator to be bounded, and we develop a Newey-West kernel estimator of the variance. We expedite the theory by defining "extremal mixingale" and "extremal NED" properties to hold exclusively in the extreme distribution tails, disbanding with dependence restrictions in the non-extremal support, and prove a broad class of linear processes are extremal NED. We demonstrate that for greater degrees of serial dependence more tail information is required in order to ensure asymptotic normality, both in theory and practice.

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File URL: http://casgroup.fiu.edu/pages/docs/2247/1275232484_05-12.pdf
File Function: First version, 2005
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Paper provided by Florida International University, Department of Economics in its series Working Papers with number 0512.

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Length: 43 pages
Date of creation: Aug 2005
Date of revision:
Handle: RePEc:fiu:wpaper:0512
Contact details of provider: Postal: Miami, FL 33199
Phone: (305) 348-2316
Fax: (305) 348-1524
Web page: http://casgroup.fiu.edu/Economics/

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  1. Caner, Mehmet, 1998. "Tests for cointegration with infinite variance errors," Journal of Econometrics, Elsevier, vol. 86(1), pages 155-175, June.
  2. Loretan, Mico & Phillips, Peter C. B., 1994. "Testing the covariance stationarity of heavy-tailed time series: An overview of the theory with applications to several financial datasets," Journal of Empirical Finance, Elsevier, vol. 1(2), pages 211-248, January.
  3. 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.
  4. 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.
  5. 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.
  6. 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.
  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. Jonathan B. Hill, 2004. "Gaussian Tests of "Extremal White Noise" for Dependent, Heterogeneous, Heavy Tailed Time Series with an Application," Econometrics 0411014, EconWPA, revised 09 Dec 2004.
  9. Phillips, Peter C. B. & Loretan, Mico, 1991. "The Durbin-Watson ratio under infinite-variance errors," Journal of Econometrics, Elsevier, vol. 47(1), pages 85-114, January.
  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. 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.
  12. 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.
  13. B. N. Cheng & S. T. Rachev, 1995. "Multivariate Stable Futures Prices," Mathematical Finance, Wiley Blackwell, vol. 5(2), pages 133-153.
  14. repec:cup:etheor:v:13:y:1997:i:3:p:353-67 is not listed on IDEAS
  15. 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.
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