Abstract
Learning about the tail shape of time series is important in, e.g., economics, finance, and risk management. However, it is well known that estimates of the tail index can be very sensitive to the choice of the number k of tail observations used for estimation. We propose a procedure that determines where the tail begins by choosing k in a data-driven fashion using scoring rules. So far, scoring rules have mainly been used to compare density forecasts. We also demonstrate how our proposal can be used in multivariate applications in the system risk literature. The advantages of our choice of k are illustrated in simulations and an empirical application to Value-at-Risk forecasts for five U.S. blue-chip stocks.
Acknowledgments
The author is indebted to two anonymous referees for their detailed comments, that greatly improved the quality of the paper. Furthermore, the author would like to thank Christoph Hanck and Till Massing for carefully reading an early draft of this manuscript.
Notes
1 If a random variable has tail index α, then
for
and
for
(de Haan and Ferreira, Citation2006, example 1.16).
2 Since is slowly varying, (Equation1
(1)
(1) ) is equivalent to
for all x > 0. Also, for non-decreasing functions
and
the relation
implies
under some regularity conditions. Using these two facts, the equivalence of (Equation1
(1)
(1) ) and (Equation2
(2)
(2) ) may be shown; see the proof of theorem 1.2.1.1 in de Haan and Ferreira (Citation2006) for details.
3 Recall that is required for
in (Equation3
(3)
(3) ) to make sense.
4 Mikosch and Stărică (Citation2000, Thm. 2.1) show that the true tail indices of the (G)ARCH models can be computed as the unique positive solution of
We have obtained the true tail indices of models (M6) and (M7) by solving this equation for α.
5 All data have been taken from finance.yahoo.com.
6 We discard the first 10 standardized residuals because they are unreliable due to initialization effects in the variance Equation (Equation12
(11)
(11) ).