Long memory properties and covariance structure of the EGARCH model

Donatas Surgailis; Marie-Claude Viano

ESAIM: Probability and Statistics (2010)

  • Volume: 6, page 311-329
  • ISSN: 1292-8100

Abstract

top
The EGARCH model of Nelson [29] is one of the most successful ARCH models which may exhibit characteristic asymmetries of financial time series, as well as long memory. The paper studies the covariance structure and dependence properties of the EGARCH and some related stochastic volatility models. We show that the large time behavior of the covariance of powers of the (observed) ARCH process is determined by the behavior of the covariance of the (linear) log-volatility process; in particular, a hyperbolic decay of the later covariance implies a similar hyperbolic decay of the former covariances. We show, in this case, that normalized partial sums of powers of the observed process tend to fractional Brownian motion. The paper also obtains a (functional) CLT for the corresponding partial sums' processes of the EGARCH model with short and moderate memory. These results are applied to study asymptotic behavior of tests for long memory using the R/S statistic.

How to cite

top

Surgailis, Donatas, and Viano, Marie-Claude. "Long memory properties and covariance structure of the EGARCH model." ESAIM: Probability and Statistics 6 (2010): 311-329. <http://eudml.org/doc/104295>.

@article{Surgailis2010,
abstract = { The EGARCH model of Nelson [29] is one of the most successful ARCH models which may exhibit characteristic asymmetries of financial time series, as well as long memory. The paper studies the covariance structure and dependence properties of the EGARCH and some related stochastic volatility models. We show that the large time behavior of the covariance of powers of the (observed) ARCH process is determined by the behavior of the covariance of the (linear) log-volatility process; in particular, a hyperbolic decay of the later covariance implies a similar hyperbolic decay of the former covariances. We show, in this case, that normalized partial sums of powers of the observed process tend to fractional Brownian motion. The paper also obtains a (functional) CLT for the corresponding partial sums' processes of the EGARCH model with short and moderate memory. These results are applied to study asymptotic behavior of tests for long memory using the R/S statistic. },
author = {Surgailis, Donatas, Viano, Marie-Claude},
journal = {ESAIM: Probability and Statistics},
keywords = {EGARCH models; long-memory; partial sums; rescaled range.},
language = {eng},
month = {3},
pages = {311-329},
publisher = {EDP Sciences},
title = {Long memory properties and covariance structure of the EGARCH model},
url = {http://eudml.org/doc/104295},
volume = {6},
year = {2010},
}

TY - JOUR
AU - Surgailis, Donatas
AU - Viano, Marie-Claude
TI - Long memory properties and covariance structure of the EGARCH model
JO - ESAIM: Probability and Statistics
DA - 2010/3//
PB - EDP Sciences
VL - 6
SP - 311
EP - 329
AB - The EGARCH model of Nelson [29] is one of the most successful ARCH models which may exhibit characteristic asymmetries of financial time series, as well as long memory. The paper studies the covariance structure and dependence properties of the EGARCH and some related stochastic volatility models. We show that the large time behavior of the covariance of powers of the (observed) ARCH process is determined by the behavior of the covariance of the (linear) log-volatility process; in particular, a hyperbolic decay of the later covariance implies a similar hyperbolic decay of the former covariances. We show, in this case, that normalized partial sums of powers of the observed process tend to fractional Brownian motion. The paper also obtains a (functional) CLT for the corresponding partial sums' processes of the EGARCH model with short and moderate memory. These results are applied to study asymptotic behavior of tests for long memory using the R/S statistic.
LA - eng
KW - EGARCH models; long-memory; partial sums; rescaled range.
UR - http://eudml.org/doc/104295
ER -

References

top
  1. T.W. Anderson, The Statistical Analysis of Time Series. Wiley, New York (1971).  
  2. R.T. Baillie, Long memory processes and fractional integration in econometrics. J. Econometrics73 (1996) 5-59.  
  3. P. Billingsley, Convergence of Probability Measures. Wiley, New York (1968).  
  4. T. Bollerslev and H.O. Mikkelsen, Modeling and pricing long memory in stock market volatility. J. Econometrics73 (1996) 151-184.  
  5. F.J. Breidt, N. Crato and P. de Lima, On the detection and estimation of long memory in stochastic volatility. J. Econometrics83 (1998) 325-348.  
  6. D.R. Brillinger, Time Series. Data Analysis and Theory. Holt, Rinehart and Winston, New York (1975).  
  7. Yu. Davydov, The invariance principle for stationary processes. Theory Probab. Appl.15 (1970) 487-489.  
  8. A. Demos, Moments and dynamic structure of a time-varying-parameter stochastic volatility in mean model. Preprint (2001).  
  9. Z. Ding and C.W.J. Granger, Modeling volatility persistence of speculative returns: A new approach. J. Econometrics73 (1996) 185-215.  
  10. R.F. Engle, Autoregressive conditional heteroskedasticity with estimates of the variance of U.K. inflation. Econometrica50 (1982) 987-1008.  
  11. 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.  
  12. L. Giraitis, P. Kokoszka and R. Leipus, Rescaled variance and related tests for long memory in volatility and levels. Preprint (1999).  
  13. L. Giraitis, H.L. Koul and D. Surgailis, Asymptotic normality of regression estimators with long memory errors. Statist. Probab. Lett.29 (1996) 317-335.  
  14. L. Giraitis, R. Leipus, P.M. Robinson and D. Surgailis, LARCH, leverage and long memory. Preprint (2000).  
  15. L. Giraitis, P.M. Robinson and D. Surgailis, A model for long memory conditional heteroskedasticity. Ann. Appl. Probab. (2000) (forthcoming).  
  16. A. Harvey, Long memory in stochastic volatility, edited by J. Knight and S. Satchell, Forecasting Volatility in the Financial Markets. Butterworth & Heineman, Oxford (1998).  
  17. C. He, T. Teräsvirta and H. Malmsten, Fourth moment structure of a family of first order exponential GARCH models, Preprint. Econometric Theory (to appear).  
  18. H.-C. Ho and T. Hsing, Limit theorems for functionals of moving averages. Ann. Probab.25 (1997) 1636-1669.  
  19. J.R.M. Hosking, Fractional differencing. Biometrika68 (1981) 165-176.  
  20. H. Hurst, Long term storage capacity of reservoirs. Trans. Amer. Soc. Civil Engrg.116 (1951) 770-799.  
  21. 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. Econometrics54 (1992) 159-178.  
  22. A.C. Lo, Long memory in stock market prices. Econometrica59 (1991) 1279-1313.  
  23. I.N. Lobato and N.E. Savin, Real and spurious long-memory properties of stock market data (with comments). J. Business Econom. Statist.16 (1998) 261-283.  
  24. V.A. Malyshev and R.A. Minlos, Gibbs Random Fields. Kluwer, Dordrecht (1991).  
  25. B.B. Mandelbrot, Statistical methodology for non-periodic cycles: From the covariance to R/S analysis. Ann. Econom. Social Measurement1 (1972) 259-290.  
  26. B.B. Mandelbrot, Limit theorems of the self-normalized range for weakly and strongly dependent processes. Z. Wahrsch. Verw. Geb.31 (1975) 271-285.  
  27. B.B. Mandelbrot and M.S. Taqqu, Robust R/S analysis of long run serial correlation. Bull. Int. Statist. Inst. 48 (1979) 59-104.  
  28. B.B. Mandelbrot and J.M. Wallis, Robustness of the rescaled range R/S in the measurement of noncyclic long run statistical dependence. Water Resources Research5 (1969) 967-988.  
  29. D.B. Nelson, Conditional heteroskedasticity in asset returns: A new approach. Econometrica59 (1991) 347-370.  
  30. P.M. Robinson, Testing for strong serial correlation and dynamic conditional heteroskedasticity in multiple regression. J. Econometrics47 (1991) 67-84.  
  31. P.M. Robinson, The memory of stochastic volatility models. J. Econometrics101 (2001) 195-218.  
  32. P.M. Robinson and P. Zaffaroni, Nonlinear time series with long memory: A model for stochastic volatility. J. Statist. Plan. Inf.68 (1998) 359-371.  
  33. S. Taylor, Modelling Financial Time Series. Wiley, Chichester (1986).  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.