Hostname: page-component-76fb5796d-x4r87 Total loading time: 0 Render date: 2024-04-25T11:24:28.686Z Has data issue: false hasContentIssue false
  • English
  • Français

STATIONARITY AND MEMORY OF ARCH(∞) MODELS

Published online by Cambridge University Press:  05 March 2004

Paolo Zaffaroni
Paolo Zaffaroni
Affiliation:
Banca d'Italia
Get access Rights & Permissions [Opens in a new window]

Abstract

We establish the necessary and sufficient conditions for covariance stationarity of ARCH(∞), for both the levels and the squares. The result applies to any form of the conditional variance coefficients. This includes GARCH(p,q) and also specifications with hyperbolically decaying coefficients, such as the autoregressive coefficients of the autoregressive fractionally integrated moving average model. The covariance stationarity condition for the levels rules out long memory in the squares.I thank Peter M. Robinson for useful comments on previous versions of the paper. Also, I am grateful to the co-editor (Bruce E. Hansen) and an anonymous referee whose suggestions greatly improved the paper.

Type
Research Article
Information
Econometric Theory , Volume 20 , Issue 1 , February 2004 , pp. 147 - 160
Copyright
© 2004 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

Bollerslev, T. (1986) Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31, 302–327.Google Scholar
Brockwell, P. & R. Davis (1987) Time Series: Theory and Methods. Springer-Verlag.
Engle, R.F. (1982) Autoregressive conditional heteroskedasticity with estimates of the variance of the United Kingdom. Econometrica 50, 987–1007.Google Scholar
Giraitis, L., P. Kokoska, & R. Leipes (2000) Stationary ARCH models: Dependence structure and central limit theorem. Econometric Theory 16, 3–22.Google Scholar
Giraitis, L. & P.M. Robinson (2001) Whittle estimation of ARCH models. Econometric Theory 17, 608–631.Google Scholar
He, C. & T. Teräsvirta (1999) Fourth moment structure of the GARCH(p,q) process. Econometric Theory 15, 824–846.Google Scholar
Hong, Y. (1997) One-sided testing for conditional heteroskedasticity in time series models. Journal of Time Series Analysis 18, 253–277.Google Scholar
Karanasos, M. (1999) The second moment and the autocovariance function of the squared errors of the GARCH model. Journal of Econometrics 90, 63–76.Google Scholar
Ling, S. (1999) On the probabilistic properties of a double threshold ARMA conditional heteroskedasticity model. Journal of Applied Probability 36, 688–705.Google Scholar
Ling, S. & M. McAleer (2002) Necessary and sufficient moment conditions for the GARCH(r, s) and asymmetric power GARCH(r, s) models. Econometric Theory 18, 722–729.Google Scholar
Loève, M. (1978) Probability Theory II, 4th ed. Springer-Verlag.
Milhøj, A. (1985) The moment structure of ARCH processes. Scandinavian Journal of Statistics 12, 281–292.Google Scholar
Nelson, D. (1990) Stationarity and persistence in the GARCH(1,1) model. Econometric Theory 6, 318–334.Google Scholar
Nelson, D. & C. Cao (1992) Inequality constraints in the univariate GARCH model. Journal of Business and Economic Statistics 10, 229–235.Google Scholar
Robinson, P.M. (1991) Testing for strong serial correlation and dynamic conditional heteroskedasticity in multiple regression. Journal of Econometrics 47, 67–84.Google Scholar