×

Limit theory for the sample autocorrelations and extremes of a GARCH \((1,1)\) process. (English) Zbl 1105.62374

Summary: The asymptotic theory for the sample autocorrelations and extremes of a GARCH (1, 1) process is provided. Special attention is given to the case when the sum of the ARCH and GARCH parameters is close to 1, that is, when one is close to an infinite variance marginal distribution. This situation has been observed for various financial log-return series and led to the introduction of the IGARCH model. In such a situation, the sample autocorrelations are unreliable estimators of their deterministic counterparts for the time series and its absolute values, and the sample autocorrelations of the squared time series have nondegenerate limit distributions. We discuss the consequences for a foreign exchange rate series.

MSC:

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62P05 Applications of statistics to actuarial sciences and financial mathematics
62E20 Asymptotic distribution theory in statistics
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Babillot, M., Bougerol, P. and Elie, L. (1997). The random difference equation Xn = AnXn-1 + BN in the critical case. Ann. Probab. 25 478-493. · Zbl 0873.60045
[2] Baillie, R. T. and Bollerslev, T. (1989). The message in daily exchange rates: a conditional-variance tale. J. Bus. Econom. Statist. 7 297-304.
[3] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation. Cambridge Univ. Press. · Zbl 0617.26001
[4] Bollerslev, T., Chou, R. Y. and Kroner, K. F. (1992). ARCH models in finance: a review of the theory and evidence. J. Econometrics 52 5-59. · Zbl 0825.90057
[5] Bougerol, P. and Picard, N. (1992). Strict stationarity of generalized autoregressive processes. Ann. Probab. 20 1714-1730. · Zbl 0763.60015
[6] Bougerol, P. and Picard, N. (1992). Stationarity of GARCH processes and of some nonnegative time series. J. Econometrics 52 115-127. · Zbl 0746.62087
[7] Brandt, A. (1986). The stochastic equation Yn+1 = AnYn + Bn with stationary coefficients. Adv. Appl. Probab. 18 211-220. JSTOR: · Zbl 0588.60056
[8] Breiman, L. (1965). On some limit theorems similar to the arc sin law. Theory Probab. Appl. 10 323-331. · Zbl 0147.37004
[9] Dacorogna, U. A., M üller, U. A., Nagler, R. J., Olsen, R. B. and Pictet, O. V. (1993). A geographical model for the daily and weekly seasonal volatility in the foreign exchange market. J. Internat. Money and Finance 12 413-438.
[10] Davis, R. A. and Hsing, T. (1995). Point process and partial sum convergence for weakly dependent random variables with infinite variance. Ann. Probab. 23 879-917. · Zbl 0837.60017
[11] Davis, R. A. and Mikosch, T. (1998). The sample autocorrelations of heavy-tailed processes with applications to ARCH. Ann. Statist. 26 2049-2080. · Zbl 0929.62092
[12] Davis, R. A., Mikosch, T. and Basrak, B. (1999). The sample ACF of solutions to multivariate stochastic recurrence equations. Preprint. Available at www.math.rug.nl/ mikosch. URL: · Zbl 0997.60012
[13] de Haan, L., Resnick, S. I., Rootzén, H. and de Vries, C. G. (1989). Extremal behaviour of solutions to a stochastic difference equation with applications to ARCH processes. Stochastic Process Appl. 32 213-224. · Zbl 0679.60029
[14] Embrechts, P., Kl üppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance. Springer, Berlin. · Zbl 0873.62116
[15] Embrechts, P. and Mikosch, T. (1991). A bootstrap procedure for estimating the adjustment coefficient. Insurance Math. Econom. 10 181-190. · Zbl 0747.62104
[16] Engle, R. F. (ed.) (1995). ARCH Selected Readings. Oxford Univ. Press.
[17] Engle, R. F. and Bollerslev, T. (1986). Modelling the persistence of conditional variances. (With comments and reply by authors). Econemetric Rev. 5 1-87. · Zbl 0619.62105
[18] Falk, M., H üsler, J. and Reiss, R.-D. (1994). Laws of Small Numbers: Extremes and Rare Events. Birkhäuser, Basel. · Zbl 0817.60057
[19] Furstenberg, H. and Kesten, H. (1960). Products of random matrices. Ann. Math. Statist. 31 457-469. · Zbl 0137.35501
[20] Goldie, C. M. (1991). Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab. 1 126-166. · Zbl 0724.60076
[21] Gourieroux, C. (1997). ARCH Models and Financial Applications. Springer, New York. · Zbl 0861.62077
[22] Gourieroux, C., Monfort, A. and Trognon, A. (1984). Pseudo maximum likelihood methods: theory. Econometrica 52 681-700. JSTOR: · Zbl 0575.62031
[23] Guillaume, D. M. Pictet O. V. and Dacorogna, M. M. (1995). On the intra-daily performance of GARCH processes. Olsen & Associates (Z ürich). Internal Document DMG. 1994-07-31.
[24] Hsing, T. (1991). Estimating the parameters of rare events. Stochastic Process. Appl. 37 117-139. · Zbl 0722.62021
[25] Hsing, T. (1993). On some estimates based on sample behavior near high level excursions. Probab. Theory Related Fields 95 331-356. · Zbl 0791.60024
[26] Hsing, T., H üsler, J. and Leadbetter, M. R. (1988). On the exceedance point process for a stationary sequence. Probab. Theory Related Fields 78 97-112. · Zbl 0619.60054
[27] Ibragimov, I. A. and Linnik, Yu. V. (1971). Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff, Groningen. · Zbl 0219.60027
[28] Kallenberg, O. (1983). Random Measures, 3rd ed. Akademie, Berlin. · Zbl 0544.60053
[29] Kesten, H. (1973). Random difference equations and renewal theory for products of random matrices. Acta Math. 131 207-248. · Zbl 0291.60029
[30] Leadbetter, M. R. (1983). Extremes and local dependence of stationary sequences. Z. Wahrsch. Verw. Gebiete 65 291-306. · Zbl 0506.60030
[31] Leadbetter, M. R., Lindgren, G. and Rootzén H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer, Berlin. · Zbl 0518.60021
[32] Li, W. K. and Mak, T. K. (1994). On the squared residual autocorrelations in non-linear time series with conditional heteroskedasticity. J. Time Ser. Anal. 15 627-636. · Zbl 0807.62070
[33] Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, London. · Zbl 0925.60001
[34] Mikosch, T. and St aric a, C. (1998). Limit theory for the sample autocorrelations and extremes of a GARCH 1 1 process. Preprint. Available at www.math.rug.nl/ mikosch. URL:
[35] Mikosch, T. and St aric a, C. (1999). Change of structure in financial time series, long range dependence and the GARCH model. Preprint. Available via www.math.rug.nl/ mikosch. URL:
[36] Mori, T. (1977). Limit distributions of two-dimensional point processes generated by strongmixing sequences. Yokohama Math. J. 25 155-168. · Zbl 0374.60010
[37] Nelson, D. B. (1990). Stationarity and persistence in the GARCH 1 1 model. Econometric Theory 6 318-334. JSTOR:
[38] Pitts, S. M., Gr übel, R. and Embrechts, P. (1996). Confidence bounds for the adjustment coefficient. Adv. in Appl. Probab. 28 802-827. JSTOR: · Zbl 0856.62094
[39] Resnick, S. I. (1986). Point processes, regular variation and weak convergence. Adv. Appl. Probab. 18 66-138. JSTOR: · Zbl 0597.60048
[40] Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York. · Zbl 0633.60001
[41] Resnick, S. I. (1997). Heavy tail modeling and teletraffic data. (With discussion and rejoinder by author). Ann Statist. 25 1805-1869. · Zbl 0942.62097
[42] Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random processes. Stochastic Models with Infinite Variance. Chapman and Hall, London. · Zbl 0925.60027
[43] Smith, R. L. and Weissman, I. (1994). Estimating the extremal index. J. Roy. Statist. Soc. Ser. B 56 515-528. JSTOR: · Zbl 0796.62084
[44] St aric a, C and Pictet, O. (1997). The tales the tails of ARCH processes tell. Technical report. Chalmers Univ., Gothenberg.
[45] Vervaat, W. (1979). On a stochastic difference equation and a representation of nonnegative infinitely divisible random variables. Adv. Appl. Probab. 11 750-783. JSTOR: · Zbl 0417.60073
[46] Wald, A. (1947). Sequential Analysis. Wiley, New York. · Zbl 0029.15805
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.