Hostname: page-component-857557d7f7-fn92c Total loading time: 0 Render date: 2025-11-20T06:02:16.763Z Has data issue: false hasContentIssue false
  • English
  • Français

CHANGE POINT TESTS FOR THE TAIL INDEX OF β-MIXING RANDOM VARIABLES

Published online by Cambridge University Press:  06 June 2016

Yannick Hoga
Yannick Hoga*
Affiliation:
University of Duisburg-Essen
*
*Address correspondence to Yannick Hoga, Faculty of Economics and Business Administration, University of Duisburg-Essen, Universitätsstraße 12, D-45117 Essen, Germany, tel. +49 201 1834365, e-mail: yannick.hoga@vwl.uni-due.de.
Get access Rights & Permissions [Opens in a new window]

Abstract

The tail index as a measure of tail thickness provides information that is not captured by standard volatility measures. It may however change over time. Currently available procedures for detecting those changes for dependent data (e.g., Quintos et al., 2001) are all based on comparing Hill (1975) estimates from different subsamples. We derive tests for a wide class of other tail index estimators. The limiting distribution of the test statistics is shown not to depend on the particular choice of the estimator, while the assumptions on the dependence structure allow for sufficient generality in applications. A simulation study investigates empirical sizes and powers of the tests in finite samples.

Information

Type
ARTICLES
Information
Econometric Theory , Volume 33 , Issue 4 , August 2017 , pp. 915 - 954
Copyright
Copyright © Cambridge University Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

The author would like to thank seminar participants at TU Dortmund for helpful discussion. Furthermore, the author is indebted to two anonymous referees and Christoph Hanck for their detailed comments, that greatly improved the quality of the paper. Full responsibility is taken for all remaining errors. Support of DFG (HA 6766/2-1) is gratefully acknowledged.

References

REFERENCES

Aguilar, M. & Hill, J. (2015) Robust score and portmanteau tests of volatility spillover. Journal of Econometrics 184, 37–61.CrossRefGoogle Scholar
Berkes, I., Horváth, L., & Kokoszka, P. (2003) Estimation of the maximal moment exponent of a GARCH(1,1) sequence. Econometric Theory 19, 565–586.CrossRefGoogle Scholar
Berkes, I., Horváth, L., & Kokoszka, P. (2004) Testing for parameter constancy in GARCH(p,q) models. Statistics & Probability Letters 70, 263–273.CrossRefGoogle Scholar
Bickel, P. & Wichura, M. (1971) Convergence criteria for multiparameter stochastic processes and some applications. The Annals of Statistics 42, 1656–1670.CrossRefGoogle Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley.Google Scholar
Bingham, N., Goldie, C., & Teugels, J. (1987) Regular Variation. Cambridge Universtiy Press.CrossRefGoogle Scholar
Bücher, A. (2015) A note on weak convergence of the sequential multivariate empirical process under strong mixing. Journal of Theoretical Probability 28, 1028–1037.CrossRefGoogle Scholar
Chan, N., Li, D., Peng, L., & Zhang, R. (2013) Tail index of an AR(1) model with ARCH(1) errors. Econometric Theory 29, 920–940.CrossRefGoogle Scholar
Cramér, H. & Leadbetter, M. (1967) Stationary and Related Stochastic Processes. Wiley.Google Scholar
Csörgő, S. & Viharos, L. (1998) Estimating the tail index. In Szyszkowicz, B. (ed.), Asymptotic Methods in Probability and Statistics, pp. 833–881. Elsevier.CrossRefGoogle Scholar
Daníelsson, J., Jansen, D., & de Vries, C. (1996) The method of moments ratio estimator for the tail shape parameter. Communications in Statistics - Theory and Methods 25, 711–720.CrossRefGoogle Scholar
de Haan, L. & Ferreira, A. (2006) Extreme Value Theory. Springer.CrossRefGoogle Scholar
de Haan, L. & Peng, L. (1998) Comparison of tail index estimators. Statistica Neerlandica 52, 60–70.CrossRefGoogle Scholar
Drees, H. (1998a) A general class of estimators of the extreme value index. Journal of Statistical Planning and Inference 66, 95–112.CrossRefGoogle Scholar
Drees, H. (1998b) On smooth statistical tail functionals. Scandinavian Journal of Statistics 24, 187–210.CrossRefGoogle Scholar
Drees, H. (2000) Weighted approximations of tail processes for β-mixing random variables. The Annals of Applied Probability 10, 1274–1301.CrossRefGoogle Scholar
Drees, H. (2003) Extreme quantile estimation for dependent data, with applications to finance. Bernoulli 9, 617–657.CrossRefGoogle Scholar
Drees, H. (2008) Some aspects of extreme value statistics under serial dependence. Extremes 11, 35–53.CrossRefGoogle Scholar
Eberlein, E. (1984) Weak convergence of partial sums of absolutely regular sequences. Statistics & Probability Letters 2, 291–293.CrossRefGoogle Scholar
Einmahl, J., de Haan, L., & Zhou, C. (2016) Statistics of heteroscedastic extremes. Journal of the Royal Statistical Society: Series B 78, 31–51.CrossRefGoogle Scholar
Einmahl, J., Gantner, M., & Sawitzki, G. (2010) Asymptotics of the shorth plot. Journal of Statistical Planning and Inference 140, 3003–3012.CrossRefGoogle Scholar
Galbraith, J. & Zernov, S. (2004) Circuit breakers and the tail index of equity returns. Journal of Financial Econometrics 2, 109–129.CrossRefGoogle Scholar
Hill, B. (1975) A simple general approach to inference about the tail of a distribution. The Annals of Statistics 3, 1163–1174.CrossRefGoogle Scholar
Hill, J. (2009) On functional central limit theorems for dependent, heterogeneous arrays with applications to tail index and tail dependence estimation. Journal of Statistical Planning and Inference 139, 2091–2110.CrossRefGoogle Scholar
Kim, M. & Lee, S. (2009) Test for tail index change in stationary time series with Pareto-type marginal distribution. Bernoulli 15, 325–356.CrossRefGoogle Scholar
Kim, M. & Lee, S. (2011) Change point test for tail index for dependent data. Metrika 74, 297–311.CrossRefGoogle Scholar
Kim, M. & Lee, S. (2012) Change point test of tail index for autoregressive processes. Journal of the Korean Statistical Society 41, 305–312.CrossRefGoogle Scholar
Koedijk, K., Schafgans, M., & de Vries, C. (1990) The tail index of exchange rate returns. Journal of International Economics 29, 93–108.CrossRefGoogle Scholar
Lin, Z. & Choi, YK. (1999) Some limit theorems for fractional Lévy Brownian fields. Stochastic Processes and their Applications 82, 229–244.CrossRefGoogle Scholar
Móricz, F. (1982) A general moment inequality for the maximum of partial sums of single series. Acta Scientiarum Mathematicarum 44, 67–75.Google Scholar
Quintos, C., Fan, Z., & Phillips, P. (2001) Structural change tests in tail behaviour and the Asian crisis. Review of Economic Studies 68, 633–663.CrossRefGoogle Scholar
Resnick, S. (2007) Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. Springer.Google Scholar
Rootzén, H. (2009) Weak convergence of the tail empirical process for dependent sequences. Stochastic Processes and their Applications 119, 468–490.CrossRefGoogle Scholar
Shorack, G. & Wellner, J. (1996) Weak Convergence and Empirical Processes. Springer.Google Scholar
Vogelsang, T. (1997) Wald-type tests for detecting breaks in the trend function of a dynamic time series. Econometric Theory 13, 818–849.CrossRefGoogle Scholar
Vogelsang, T. (1999) Sources of nonmonotonic power when testing for a shift in mean of a dynamic time series. Journal of Econometrics 88, 283–299.CrossRefGoogle Scholar
Wagner, N. & Marsh, T. (2004) Tail index estimation in small samples: Simulation results for independent and ARCH-type financial return models. Statistical Papers 45, 545–562.CrossRefGoogle Scholar
Werner, T. & Upper, C. (2004) Time variation in the tail behavior of bund future returns. Journal of Future Markets 24, 387–398.CrossRefGoogle Scholar
Wichura, M. (1970) On the construction of almost uniformly convergent random variables with given weakly convergent image laws. The Annals of Mathematical Statistics 41, 284–291.CrossRefGoogle Scholar