Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-14T19:19:37.223Z Has data issue: false hasContentIssue false
  • English
  • Français

Long memory properties and covariance structureof the EGARCH model

Published online by Cambridge University Press:  15 November 2002

Donatas Surgailis  and
Marie-Claude Viano
Donatas Surgailis
Affiliation:
Vilnius Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania; [email protected].
Marie-Claude Viano
Affiliation:
Université des Sciences et Technologies de Lille, 59655 Villeneuve-d'Ascq Cedex, France; [email protected].
Get access

Abstract

The EGARCH model of Nelson [29] is one of the most successfulARCH models which may exhibit characteristic asymmetries offinancial time series, as well as long memory. The paper studiesthe covariance structure and dependence properties of the EGARCHand some related stochastic volatility models. We show that thelarge time behavior of the covariance of powers of the (observed)ARCH process is determined by the behavior of the covariance ofthe (linear) log-volatility process; in particular, a hyperbolicdecay of the later covariance implies a similar hyperbolic decayof the former covariances. We show, in this case, that normalizedpartial sums of powers of the observed process tend to fractionalBrownian motion. The paper also obtains a (functional) CLT for thecorresponding partial sums' processes of the EGARCH model withshort and moderate memory. These results are applied to studyasymptotic behavior of tests for long memory using the R/Sstatistic.

Keywords

EGARCH modelslong-memorypartial sumsrescaled range.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 6: New directions in Time Series Analysis (Guest Editor: Philippe Soulier)New directions in Time Series Analysis (Guest Editor: Philippe Soulier) , 2002 , pp. 311 - 329
Copyright
© EDP Sciences, SMAI, 2002

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.)

References

T.W. Anderson, The Statistical Analysis of Time Series. Wiley, New York (1971).
Baillie, R.T., Long memory processes and fractional integration in econometrics. J. Econometrics 73 (1996) 5-59. CrossRef
P. Billingsley, Convergence of Probability Measures. Wiley, New York (1968).
Bollerslev, T. and Mikkelsen, H.O., Modeling and pricing long memory in stock market volatility. J. Econometrics 73 (1996) 151-184. CrossRef
Breidt, F.J., Crato, N. and de Lima, P., On the detection and estimation of long memory in stochastic volatility. J. Econometrics 83 (1998) 325-348. CrossRef
D.R. Brillinger, Time Series. Data Analysis and Theory. Holt, Rinehart and Winston, New York (1975).
Yu. Davydov, The invariance principle for stationary processes. Theory Probab. Appl. 15 (1970) 487-489. CrossRef
A. Demos, Moments and dynamic structure of a time-varying-parameter stochastic volatility in mean model. Preprint (2001).
Ding, Z. and Granger, C.W.J., Modeling volatility persistence of speculative returns: A new approach. J. Econometrics 73 (1996) 185-215. CrossRef
Engle, R.F., Autoregressive conditional heteroskedasticity with estimates of the variance of U.K. inflation. Econometrica 50 (1982) 987-1008. CrossRef
E. Ghysels, A.C. Harvey and E. Renault, Stochastic volatility, edited by G.S. Maddala and C.R. Rao. North Holland, Amsterdam, Handb. Statist. 14 (1993) 119-191.
L. Giraitis, P. Kokoszka and R. Leipus, Rescaled variance and related tests for long memory in volatility and levels. Preprint (1999).
Giraitis, L., Koul, H.L. and Surgailis, D., Asymptotic normality of regression estimators with long memory errors. Statist. Probab. Lett. 29 (1996) 317-335. CrossRef
L. Giraitis, R. Leipus, P.M. Robinson and D. Surgailis, LARCH, leverage and long memory. Preprint (2000).
L. Giraitis, P.M. Robinson and D. Surgailis, A model for long memory conditional heteroskedasticity. Ann. Appl. Probab. (2000) (forthcoming).
A. Harvey, Long memory in stochastic volatility, edited by J. Knight and S. Satchell, Forecasting Volatility in the Financial Markets. Butterworth & Heineman, Oxford (1998).
C. He, T. Teräsvirta and H. Malmsten, Fourth moment structure of a family of first order exponential GARCH models, Preprint. Econometric Theory (to appear).
Ho, H.-C. and Hsing, T., Limit theorems for functionals of moving averages. Ann. Probab. 25 (1997) 1636-1669. CrossRef
Hosking, J.R.M., Fractional differencing. Biometrika 68 (1981) 165-176. CrossRef
Hurst, H., Long term storage capacity of reservoirs. Trans. Amer. Soc. Civil Engrg. 116 (1951) 770-799.
D. Kwiatkowski, P.C. Phillips, P. Schmidt and Y. Shin, Testing the null hypothesis of stationarity against the alternative of a unit root: How sure are we that economic time series have a unit root? J. Econometrics 54 (1992) 159-178.
Long, A.C. Lo memory in stock market prices. Econometrica 59 (1991) 1279-1313.
Lobato, I.N. and Savin, N.E., Real and spurious long-memory properties of stock market data (with comments). J. Business Econom. Statist. 16 (1998) 261-283.
V.A. Malyshev and R.A. Minlos, Gibbs Random Fields. Kluwer, Dordrecht (1991).
Mandelbrot, B.B., Statistical methodology for non-periodic cycles: From the covariance to R/S analysis. Ann. Econom. Social Measurement 1 (1972) 259-290.
Mandelbrot, B.B., Limit theorems of the self-normalized range for weakly and strongly dependent processes. Z. Wahrsch. Verw. Geb. 31 (1975) 271-285. CrossRef
Mandelbrot, B.B. and Taqqu, M.S., Robust R/S analysis of long run serial correlation. Bull. Int. Statist. Inst. 48 (1979) 59-104.
Mandelbrot, B.B. and Wallis, J.M., Robustness of the rescaled range R/S in the measurement of noncyclic long run statistical dependence. Water Resources Research 5 (1969) 967-988. CrossRef
Nelson, D.B., Conditional heteroskedasticity in asset returns: A new approach. Econometrica 59 (1991) 347-370. CrossRef
Robinson, P.M., Testing for strong serial correlation and dynamic conditional heteroskedasticity in multiple regression. J. Econometrics 47 (1991) 67-84. CrossRef
Robinson, P.M., The memory of stochastic volatility models. J. Econometrics 101 (2001) 195-218. CrossRef
Robinson, P.M. and Zaffaroni, P., Nonlinear time series with long memory: A model for stochastic volatility. J. Statist. Plan. Inf. 68 (1998) 359-371. CrossRef
S. Taylor, Modelling Financial Time Series. Wiley, Chichester (1986).