Where does the tail begin? An approach based on scoring rules

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.

Keywords:

• Hill estimator
• optimal sample fraction
• risk management
• tail index
• value-at-risk

JEL Classification:

• C13 (Estimation)
• C14 (Semiparametric and Nonparametric Methods)
• G32 (Financial Risk and Risk Management)

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 $\left|X\right|$ has tail index α, then $\mathrm{E}|X{|}^{\alpha }<\infty$ for $\underset{¯}{\alpha }<\alpha$ and $\mathrm{E}|X{|}^{\overline{\alpha }}=\infty$ for $\overline{\alpha }>\alpha$ (de Haan and Ferreira, 2006, example 1.16).

2 Since $L(·)$ is slowly varying, (1(1) $$1-F(x)=x^{-1/\gamma }L(x),\hspace{1em}\text{where} L(·) \text{is} \text{slowly} \text{varying}.$$(1) ) is equivalent to $\underset{t\to \infty }{\lim }\frac{1-F(tx)}{1-F(t)}=x^{-1/\gamma }$ for all x > 0. Also, for non-decreasing functions $f_n(·)$ and $g(·)$ the relation $\underset{n\to \infty }{\lim }{f}_n(x)=g(x)$ implies $\underset{n\to \infty }{\lim }{f}_n^{\leftarrow }(x)=g^{\leftarrow }(x)$ under some regularity conditions. Using these two facts, the equivalence of (1(1) $$1-F(x)=x^{-1/\gamma }L(x),\hspace{1em}\text{where} L(·) \text{is} \text{slowly} \text{varying}.$$(1) ) and (2(2) $$\underset{t\to \infty }{\lim }\frac{U(ty)}{U(t)}=y^{\gamma }\hspace{1em}\text{for} \text{all} y>0,$$(2) ) may be shown; see the proof of theorem 1.2.1.1 in de Haan and Ferreira (2006) for details.

3 Recall that $p\gg 1-k/n$ is required for ${\widehat{x}}_p(k)$ in (3(3) $$F^{\leftarrow }(p)=U(1/[1-p])\approx U(n/k){\left(\frac{k}{n[1-p]}\right)}^{\gamma }\approx X_{(k+1)}{\left(\frac{k}{n[1-p]}\right)}^{\widehat{\gamma }(k)}=:{\widehat{x}}_p(k).$$(3) ) to make sense.

4 Mikosch and Stărică (2000, Thm. 2.1) show that the true tail indices of the (G)ARCH models can be computed as the unique positive solution $\alpha >0$ of $\mathrm{E}{[{\alpha }^{°}{\epsilon }_1^2+{\beta }^{°}]}^{\alpha /2}=1.$ 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 ${\widehat{\epsilon }}_1,\dots ,{\widehat{\epsilon }}_{10},$ because they are unreliable due to initialization effects in the variance Equation (12(11) $$\begin{array}{c}{X}_i={\sigma }_i{\epsilon }_i,\hspace{1em}{\epsilon }_i\stackrel{\mathrm{i}.\mathrm{i}.\mathrm{d}.}{\sim }(0,1),\\ {\sigma }_i^2={\omega }^{°}+{\alpha }^{°}{X}_{i-1}^2+{\beta }^{°}{\sigma }_{i-1}^2,\hspace{1em}{\omega }^{°}>0,{\alpha }^{°}\ge 0,{\beta }^{°}\ge 0.\end{array}$$(11) ).

Additional information

Funding

This work was supported by the German Research Foundation (Deutsche Forschungsgemeinschaft [DFG]) under Grant HO 6305/1-1.