Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-09T20:15:43.205Z Has data issue: false hasContentIssue false
  • English
  • Français

ON THE TAIL BEHAVIORS OF A FAMILY OF GARCH PROCESSES

Published online by Cambridge University Press:  30 August 2006

Ji-Chun Liu
Ji-Chun Liu
Affiliation:
Xiamen University
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, some structural properties of a family of generalized autoregressive conditionally heteroskedastic (GARCH) processes are considered. First, a sufficient and necessary condition for the strict stationarity of this family of GARCH processes is given. Second, some simple conditions for the existence of the moments of the family of GARCH processes are also derived. Finally, we describe the tail of the marginal distribution of the family of GARCH processes.I am grateful to Bruce E. Hansen and an anonymous referee for helpful comments on the initial version.

Type
Research Article
Information
Econometric Theory , Volume 22 , Issue 5 , October 2006 , pp. 852 - 862
Copyright
© 2006 Cambridge University Press

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

REFERENCES

Adler, R.J., R. Feldaman, & M. Taqqu (eds.) (1997) A User's Guide to Heavy Tails: Statistical Techniques for Analyzing Heavy Tailed Distribution and Processes. Birkhauser.
Alsmeyer, G. (1991) Erneuerungstheorie. Teubner.
Basrak, B., R.A. Davis, & T. Mikosch (2002) Regular variation of GARCH processes. Stochastic Processes and Their Applications 99, 95–115.Google Scholar
Bollerslev, T. (1986) Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31, 307–327.Google Scholar
Borkovec, M. & C. Kluppelberg (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. & N. Picard (1992) Stationarity of GARCH processes and of some nonnegative time series. Journal of Econometrics 52, 115–127.Google Scholar
Breiman, L. (1965) On some limit theorems similar to the arc sin law. Theory of Probability and Its Applications 10, 323–331.Google Scholar
Davis, R. & S.I. Resnick (1996) Limit theory for bilinear processes with heavy tailed noise. Annals of Applied Probability 6, 1191–1210.Google Scholar
De Haan, L., S.I. Resnick, H. Rootzen, & G.G. De Vries (1989) Extremal behavior of solutions to a stochastic difference equation with applications to ARCH processes. Stochastic Processes and Their Applications 32, 213–224.Google Scholar
Embrechts, P., C. Kluppelberg, & T. Mikosch (1997) Modelling Extremal Events for Insurance and Finance. Springer-Verlag.
Engle, R.F. (1982) Autoregressive conditional heteroskedasticity with estimates of the variance of U.K. inflation. Econometrica 50, 987–1007.Google Scholar
Engle, R.F. (1990) Discussion: Stock market volatility and the crash of 87. Review of Financial Studies 3, 103–106.Google Scholar
Fornari, F. & A. Mele (1997) Sign- and volatility-switching ARCH models: Theory and applications to international stock markets. Journal of Applied Econometrics 12, 49–65.Google Scholar
Glosten, L.R., R. Jagannathan, & D. Runkle (1993) On the relation between the expected value and the volatility of the nominal excess return on stocks. Journal of Finance 48, 1779–1801.Google Scholar
He, C. & T. Terasvirta (1999) Properties of moments of a family of GARCH processes. Journal of Econometrics 92, 173–192.Google Scholar
Ling, S. & M. McAleer (2002) Stationarity and the existence of moments of a family GARCH processes. Journal of Econometrics 106, 109–117.Google Scholar
Mikosch, T. & C. Starica (2000) Limit theory for the sample autocorrelations and extremes of a GARCH(1,1) process. Annals of Statistics 28, 1427–1451.Google Scholar
Resnick, S.I. (1987) Extreme Values, Regular Variation, and Point Processes. Springer-Verlag.
Schwert, G.W. (1989) Why does stock market volatility change over time? Journal of Finance 45, 1129–1155.Google Scholar
Sentana, E. (1995) Quadratic ARCH models. Review of Economic Studies 62, 639–661.Google Scholar
Taylor, S.J. (1986) Modelling Financial Time Series. Wiley.
Yang, M. & R. Bewley (1995) Moving average conditional heteroskedastic processes. Economics Letters 49, 367–372.Google Scholar
Zakoïan, J.M. (1994) Threshold heteroskedastic models. Journal of Economic Dynamics and Control 18, 931–955.Google Scholar