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TAIL AND NONTAIL MEMORY WITH APPLICATIONS TO EXTREME VALUE AND ROBUST STATISTICS

Published online by Cambridge University Press:  08 March 2011

Jonathan B. Hill
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Abstract

New notions of tail and nontail dependence are used to characterize separately extremal and nonextremal information, including tail log-exceedances and events, and tail-trimmed levels. We prove that near epoch dependence (McLeish, 1975; Gallant and White, 1988) and L0-approximability (Pötscher and Prucha, 1991) are equivalent for tail events and tail-trimmed levels, ensuring a Gaussian central limit theory for important extreme value and robust statistics under general conditions. We apply the theory to characterize the extremal and nonextremal memory properties of possibly very heavy-tailed GARCH processes and distributed lags. This in turn is used to verify Gaussian limits for tail index, tail dependence, and tail-trimmed sums of these data, allowing for Gaussian asymptotics for a new tail-trimmed least squares estimator for heavy-tailed processes.

Type
Research Article
Information
Econometric Theory , Volume 27 , Issue 4 , August 2011 , pp. 844 - 884
Copyright
Copyright © Cambridge University Press 2011

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Footnotes

The authors kindly thanks two anonymous referees and co-editor Yuichi Kitamura for helpful suggestions.

References

REFERENCES

An, H.Z. & Huang, F.C. (1996) The geometrical ergodicity of nonlinear autoregressive models. Statistica Sinica 6, 943–956.Google Scholar
Baillie, R.T., Bollerslev, T., & Mikkelsen, H.O. (1996) Fractionally integrated generalized autoregressive conditional heteroscedasticity, Journal of Econometrics 74, 3–30.Google Scholar
Basrak, , Davis, R.A., & Mikosch, T. (2002a) A characterization of multivariate regular variation, Annals of Applied Probability 12, 908–920.CrossRefGoogle Scholar
Basrak, B., Davis, R.A., & Mikosch, T. (2002b) Regular variation of GARCH processes. Stochastic Processes and Their Applications 99, 95–115.CrossRefGoogle Scholar
Beirlant, J., Vynckier, P., & Teugels, J.L. (1996) Practical Analysis of Extreme Values. Leuven University Press.Google Scholar
Bierens, H.J. (1987) ARMAX model specification testing, with an application to unemployment in the Netherlands. Journal of Econometrics 35, 161–190.CrossRefGoogle Scholar
Bingham, N.H., Goldie, C.M., & Teugels, J.L. (1987) Regular Variation. Cambridge University Press.10.1017/CBO9780511721434Google Scholar
Borkovec, M. & Klüppelberg, C. (2001) The tail of the stationary distribution of an autoregressive process with ARCH(1) errors, Annals of Applied Probability 11, 1220–1241.Google Scholar
Bougerol, P. & Picard, N. (1992) Stationarity of GARCH processes and of some nonnegative time series. Journal of Econometrics 52, 115–127.10.1016/0304-4076(92)90067-2CrossRefGoogle Scholar
Boussama, F. (1998) Ergodicité, Mélange et Estimation dans le Modelès GARCH. Ph.D. thesis, Université 7 Paris.Google Scholar
Carrasco, M. & Chen, X. (2002) Mixing and moment properties of various GARCH and stochastic volatility models, Econometric Theory 18, 17–39.CrossRefGoogle Scholar
Chernick, M.R. (1981) A limit theorem for the maximum of autoregressive processes with uniform marginal distribution. Annals of Probability 9, 145–149.Google Scholar
Chernick, M.R., Hsing, T., & McCormick, W.P. (1991) Calculating the extremal index for a class of stationary sequences. Advances in Applied Probability 23, 835–850.CrossRefGoogle Scholar
Čižek, P. (2008) General trimmed estimation: Robust approach to nonlinear and limited dependent variable models, Econometric Theory 24, 1500–1529.CrossRefGoogle Scholar
Cline, D.B.H. (1983) Estimation and Linear Prediction for Regression, Autoregression and ARMA with Infinite Variance Data. Ph.D. Dissertation, Colorado State University.Google Scholar
Cline, D.B.H. (2007) Regular variation of order 1 nonlinear AR-ARCH models, Stochastic Processes and Their Applications 117, 840–861.CrossRefGoogle Scholar
Cline, D.B.H. & Pu, H-m. H. (2004) Stability and the Lyapunov exponent of threshold AR-ARCH models. Annals of Applied Probability 14, 1920–1949.CrossRefGoogle Scholar
Csörgő, S., Horváhth, L., & Mason, D.M. (1986) What portion of the sample makes a partial sum asymptotically stable or normal. Probability Theory and Related Fields 72, 1–16.Google Scholar
Davidson, J. (1992) A central limit theorem for globally nonstationary near-epoch dependent functions of mixing processes. Econometric Theory 8, 313–329.CrossRefGoogle Scholar
Davidson, J. (1994) Stochastic Limit Theory. Oxford University Press.Google Scholar
Davidson, J. (2004) Moment and memory properties of linear conditional heteroscedasticity models, and a new model. Journal of Business and Economics Statistics 22, 16–29.CrossRefGoogle Scholar
Davis, R.A. & Mikosch, T. (1998) The sample autocorrelations of heavy-tailed processes with applications to ARCH. Annals of Statistics 26, 2049–2080.Google Scholar
Davis, R.A. & Mikosch, T. (2009a) Extreme value theory for GARCH processes. In Andersen, T.G., Davis, R.A., Kreiss, J.-P., & Mikosch, T. (eds.), Handbook of Financial Time Series, pp. 187–200. Springer.Google Scholar
Davis, R.A. & Mikosch, T. (2009b) Extremes of stochastic volatility models. In Andersen, T.G., Davis, R.A., Kreiss, J.-P., & Mikosch, T. (eds.), Handbook of Financial Time Series, pp. 355–364. Springer.Google Scholar
Davis, R.A. & Mikosch, T. (2009c) The extremogram: A correlogram for extreme events. Bernoulli 15, 977–1009.Google Scholar
Davis, R.A. & Resnick, S. (1996) Limit theory for bilinear processes with heavy-tailed noise. Annals of Applied Probability 6, 1191–1210.CrossRefGoogle Scholar
Davison, A.C. & Smith, R.L. (1990) Models for exceedances over high thresholds. Journal of the Royal Statistical Society, Series B 52, 393–442.Google Scholar
de Haan, L., Resnick, S.I., Rootzén, H., & de Vries, C.G. (1989) Extremal behaviour of solutions to a stochastic difference equation with applications to ARCH processes. Stochastic Processes and Their Applications 32, 213–224.Google Scholar
de Jong, R.M. (1997) Central limit theorems for dependent heterogeneous random variables. Econometric Theory 13, 353–367.CrossRefGoogle Scholar
de la Peña, V.H., Ibragimov, R., & Sharakhmetov, S. (2003) On extremal distributions and sharp Lp-bounds for sums of multilinear forms. Annals of Probability 31, 630–675.Google Scholar
Doukhan, P. (1994) Mixing: Properties and Examples. Lecture Notes in Statistics 85. Springer.Google Scholar
Doukhan, P. & Louhichi, S. (1999) A new weak dependence condittion and applications to moment inequalities. Stochastic Processes and Their Applications 84, 313–342.CrossRefGoogle Scholar
Drees, H., Ferreira, A., & de Haan, L. (2004) On maximum likelihood estimation of the extreme value index. Annals of Applied Probability 14, 1179–1201.CrossRefGoogle Scholar
Embrechts, P., Klüppleberg, C., & Mikosch, T. (1997) Modelling Extremal Events for Insurance and Finance. Springer-Verlag.Google Scholar
Engle, R. & Ng, V. (1993). Measuring and testing the impact of news on volablity. Journal of Finance 48, 1749–1778.Google Scholar
Feller, W. (1946) A limit theorem for random variables with infinite moments. American Journal of Mathematics 68, 257–262.10.2307/2371837Google Scholar
Feller, W. (1971) An Introduction to Probability Theory and Its Applications, 2nd ed., vol. 2. Wiley.Google Scholar
Gabaix, X. (2008) Power laws. In Durlauf, S.N. & Blume, L.E. (eds.), The New Palgrave Dictionary of Economics, 2nd ed. Palgrave Macmillan.Google Scholar
Galambos, J. (1987) The Asymptotic Theory of Extreme Order Statistics. Kreiger.Google Scholar
Gallant, A.R. & White, H. (1988) A Unified Theory of Estimation and Inference for Nonlinear Dynamic Models. Basil Blackwell.Google Scholar
Giraitis, L., Kokoszka, P., & Leipus, R. (2000) Stationary ARCH models: Dependence structure and central limit theorem. Econometric Theory 16, 3–22.Google Scholar
González-Rivera, G. (1998) Smooth-Transition GARCH models, Studies in Non-Linear Dynamics and Econometrics 3, 61–78.Google Scholar
Guegan, D. & Ladoucette, S. (2001) Non-mixing properties of long memory processes. Comptes Rendus de l’Academie des Sciences, Series I Mathematics 333, 373–376.Google Scholar
Haeusler, E. & Teugels, J.L. (1985) On asymptotic normality of Hill’s estimator for the exponent of regular variation. Annals of Statistics 13, 743–756.Google Scholar
Hahn, M.G., Kuelbs, J., & Samur, J.D. (1987) Asymptotic normality of trimmed sums of mixing random variables. Annals of Probability 15, 1395–1418.CrossRefGoogle Scholar
Hahn, M.G., Kuelbs, J., & Weiner, D.C. (1990) The asymptotic joint distribution of self-normalized censored sums and sums of squares. Annals of Probability 18, 1284–1341.CrossRefGoogle Scholar
Hahn, M.G. & Weiner, D.C. (1992) Asymptotic behavior of self-normalized trimmed sums: Nonnormal limits. Annals of Probability 20, 455–482.Google Scholar
Hall, P. (1982) On some estimates of an exponent of regular variation. Journal of the Royal Statistical Society, Series B 44, 37–42.Google Scholar
Hall, P. & Yao, Q. (2003) Inference in ARCH and GARCH models with heavy-tailed errors. Econometrica 71, 285–317.Google Scholar
He, X., Jurecková, J., Koenker, R., & Portnoy, S. (1990) Tail behavior of regression estimators and their breakdown points. Econometrica 58, 1195–1214.Google Scholar
Hill, B.M. (1975) A simple general approach to inference about the tail of a distribution. Annals of Mathematical Statistics 3, 1163–1174.Google Scholar
Hill, J.B. (2008) Robust Estimation and Inference for Extremal Dependence in Time Series. University of North Carolina at Chapel Hill.Google Scholar
Hill, J.B. (2009a). Central Limit Theory for Kernel-Self Normalized Tail-Trimmed Sums of Dependent Data with Applications. Working paper, University of North Carolina at Chapel Hill.Google Scholar
Hill, J.B. (2009b) 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.Google Scholar
Hill, J.B. (2010) On tail index estimation for dependent, heterogeneous data. Econometric Theory 26, 1398–1436.CrossRefGoogle Scholar
Hill, J.B. (2011) Extremal memory of stochastic volatility with an application to tail shape inference. Journal of Statistical Planning and Inference 141, 663–676.Google Scholar
Hill, J.B. & Renault, E. (2010) Generalized Method of Moments with Tail Trimming. Working paper, University of North Carolina at Chapel Hill.Google Scholar
Hill, J.B. & Shneyerov, A. (2010) Are There Common Values in First-Price Auctions? A Tail-Index Nonparametric Test. Working paper, University of North Carolina at Chapel Hill.Google Scholar
Hosking, J.R.M. (1981) Fractional differencing. Biometrika 68, 165–176.CrossRefGoogle Scholar
Hsing, T. (1991) On tail index estimation using dependent data. Annals of Statistics 19, 1547–1569.Google Scholar
Hsing, T. (1993) Extremal index estimation for a weakly dependent stationary sequence. Annals of Statistics 21, 2043–2071.CrossRefGoogle Scholar
Hsing, T., Hüsler, J., & Leadbetter, M.R. (1989) On the exceedance point process for a stationary Sequence. Probability Theory and Related Fields 78, 97–112.Google Scholar
Ibragimov, I.A. (1962) Some limit theorems for stationary processes. Theory of Probability and Its Applications 7, 349–382.CrossRefGoogle Scholar
Ibragimov, I.A. & Linnik, Y.V. (1971) Independent and Stationary Sequences of Random Variables. Wolters-Noordhof.Google Scholar
Ibragimov, R. (2009) Heavy-Tailed densities. In Durlauf, S.N. & Blume, L.E. (eds.), The New Palgrave Dictionary of Economics Online. Palgrave Macmillan.Google Scholar
Iglesias, E.M. & Linton, O. (2009) Estimation of Tail Thickness Parameters from GJR-GARCH Models. Mimeo, London School of Economics.Google Scholar
Joe, H. (1997) Multivariate Models and Dependence Concepts. Monograms in Statistics and Applied Probability 73, Chapman and Hall.Google Scholar
Jurečková, J. (1981) Tail behavior of location estimators. Annals of Statistics 9, 578–585.Google Scholar
Klüppelberg, C., Kuhn, G., & Peng, L. (2008) Semi-Parametric models for the multivariate tail dependence function—the asymptotically dependent case. Scandinavian Journal of Statistics 35, 701–718.CrossRefGoogle Scholar
Leadbetter, M.R. (1974) On extreme values in stationary sequences. Zeitschrift für Wahrscheinlichkeits Theorie und Verwandte Gebiete 28, 289–303.Google Scholar
Leadbetter, M.R. (1983) Extremes and local dependence in stationary sequences. Zeitschrift für Wahrscheinlichkeits Theorie und Verwandte Gebiete 65, 291–306.10.1007/BF00532484Google Scholar
Leadbetter, M.R., Lindgren, G., & Rootzén, H. (1983) Extremes and Related Properties of Random Sequences and Processes. Springer-Verlag.10.1007/978-1-4612-5449-2CrossRefGoogle Scholar
Leadbetter, M.R., Rootzén, H., & Choi, H. (2001) On central limit theory for random additive functions under weak dependence restrictions. Lecture Notes-Monograph Series 36, 464–476.CrossRefGoogle Scholar
Ledford, A.W. & Tawn, J.A. (1997) Modeling dependence within joint tail regions. Journal of the Royal Statistical Society, Series B 59, 475–499.Google Scholar
Ledford, A.W. & Tawn, J.A. (2003) Diagnostics for dependence within time series extremes. Journal of the Royal Statistical Society, Series B 65, 521–543.Google Scholar
Ling, S. (1999) On the probabilistic properties of a double threshold ARMA conditional heteroskedastic model. Journal of Applied Probability 36, 688–705.CrossRefGoogle Scholar
Linton, O., Pan, J., & Wang, H. (2010) Estimation for a nonstationary semi-strong GARCH(1,1) model with heavy-tailed errors. Econometric Theory 26, 1–28.Google Scholar
Longin, F. & Solnik, B. (2001) Extreme correlation of international equity markets. Journal of Finance 56, 649–676.Google Scholar
Loynes, R.M. (1965) Extreme values in uniformly mixing stationary stochastic processes. Annals of Mathematical Statistics 36, 993–999.CrossRefGoogle Scholar
McLeish, D.L. (1975) A maximal inequality and dependent strong law. Annals of Probability 3, 329–339.CrossRefGoogle Scholar
Meitz, M. & Saikkonen, P. (2008) Stability of nonlinear AR-GARCH models. Journal of Time Series Analysis 29, 453–475.CrossRefGoogle Scholar
Mikosch, T. & Stărică, C. (2000) Limit theory for the sample autocorrelations and extremes of a GARCH(1,1) process. Annals of Statistics 28, 1427–1451.CrossRefGoogle Scholar
Nagev, S.V. (1979) Large deviations of sums of independent random variables. Annals of Probability 7, 745–789.Google Scholar
Nagev, S.V. (1998) Some refinements of probabilistic and moment inequalities. Theory of Probability and Their Applications 42, 707–713.CrossRefGoogle Scholar
Naveau, P. (2003). Almost sure relative stability of the maximum of a stationary sequence. Advances in Applied Probability 35, 721–736.CrossRefGoogle Scholar
Nze, P.A., Buhlmann, P., & Doukhan, P. (2002) Weak dependence beyond mixing and asymptotics for nonparametric regression. Annals of Statistics 30, 397–430.Google Scholar
Nze, P.A. & Doukhan, P. (2004) Weak dependence: Models and applications to econometrics. Econometric Theory 20, 995–1045.Google Scholar
O’Brien, G.L. (1974) The maximum term of uniformly mixing stationary sequences. Zeitschrift für Wahrscheinlichkeit Theorie und Verwandte Gebeite 30, 57–63.Google Scholar
Pötscher, B.M. & Prucha, I.R. (1991) Basic structure of the asymptotic theory in dynamic nonlinear econometrics models, Part I: Consistency and approximation concepts. Econometric Reviews 10, 125–216.Google Scholar
Pruitt, W. (1985) Sums of independent random variables with the extreme terms excluded. In Srivastava, J.N. (ed.), Probability and Statistics: Essays in Honor of Franklin A. Graybill. Elsevier.Google Scholar
Ramos, A. & Ledford, A. (2009) A new class of models for bivariate joint tails. Journal of the Royal Statistical Society, Series B 71, 219–241.Google Scholar
Resnick, S. (1987) Extreme Values, Regular Variation and Point Processes. Springer-Verlag.Google Scholar
Rootzén, H. (1978) Extremes of moving averages of stable processes. Annals of Probability 6, 847–869.Google Scholar
Rootzén, H. (2008) Weak convergence to the tail empirical function for dependent sequences. Stochastic Processes and their Applications 119, 468–490.Google Scholar
Schmidt, R. & Stadtmüller, U. (2006) Non-Parametric estimation of tail dependence. Scandinavian Journal of Statistics 33, 67–335.CrossRefGoogle Scholar
Sklar, A. (1959) Fonctions de répartitions á n dimensions et leurs marges. Publications de L’Institut de Statistique de L’Université de Paris 8, 229–231.Google Scholar
Smith, R. (1984) Threshold methods for sample extremes. In Tiago de Oliveira, J. (ed.), Statistical Extremes and Applications, pp. 621–638. Reidel.Google Scholar
Smith, R.L. (1992) The extremal index for a markov chain. Journal of Applied Probability 29, 37–45.Google Scholar
Smith, R.L. & Weissman, I. (1994) Estimating the extremal index. Journal of the Royal Statistical Society, Series B 56, 515–528.Google Scholar
Stărică, C. (1999) Multivariate extremes for models with constant conditional correlations. Journal of Empirical Finance 6, 515–553.Google Scholar
Stigler, S.M. (1973) The asymptotic distribution of the trimmed mean. Annals of Statistics 1, 472–477.Google Scholar
Tong, H. & Lim, K. S. (1980) Threshold autoregression, limit cycles and cyclical data. Journal of the Royal Statistical Society, Series B 42, 245–292.Google Scholar
Wu, W.B. (2005) On the Badahur representation of sample quantiles for dependent sequences. Annals of Statistics 15, 20–36.Google Scholar
Wu, W.B. & Min, M. (2005) On linear processes with dependent innovations. Stochastic Processes and Their Applications 115, 939–958.Google Scholar