Hostname: page-component-745bb68f8f-lrblm Total loading time: 0 Render date: 2025-01-11T02:51:59.170Z Has data issue: false hasContentIssue false
  • English
  • Français

THE DIFFUSION LIMIT OF A TVP-GQARCH-M(1,1) MODEL

Published online by Cambridge University Press:  05 March 2004

Stelios Arvanitis
Stelios Arvanitis
Affiliation:
Athens University of Economics and Business
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper presents results on the issue of weak convergence of a TVP-GQARCH-M(1,1) process. These suggest that the weak limit of this endogenous volatility model is an exogenous (stochastic) volatility continuous time process. Under the appropriate assumptions, we derive the invariant distributions at which the process converges in various cases. They reveal an approximate distributional relation between the GARCH or the asymmetric GQARCH variance processes and the continuous time Ornstein–Uhlenbeck models with respect to appropriate nonnegative Levy processes.I am grateful to Ritsa Panagiotou, Phyllis Alexander, Antonis Demos, Bruce Hansen, Nicholas Magginas, Nour Meddahi, Enrique Sentana, Paolo Zaffaroni, two anonymous referees, and the seminar participants at the Department of International and European Economic Studies for their valuable comments.

Type
Research Article
Information
Econometric Theory , Volume 20 , Issue 1 , February 2004 , pp. 161 - 175
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

Barndorff-Nielsen, O.E. (1997) Normal inverse Gaussian distributions and stochastic volatility modelling. Scandinavian Journal of Statistics 24, 1–13.Google Scholar
Barndorff-Nielsen, O.E. & N. Shephard (2001) Non-Gaussian based models and some of their uses in financial econometrics. Journal of the Royal Statistical Society, Series B 63, 167–241.Google Scholar
Barndorff-Nielsen, O.E. & N. Shephard (2001) Modelling by Levy processes for financial econometrics. In O.E. Barndorff-Nielsen, T. Mikosch, & S. Resnick (eds.), Levy Processes-Theory and Applications, pp. 283–318. Birkhauser.
Bollerslev, T. (1986) Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31, 307–327.Google Scholar
Corradi, V. (2000) Reconsidering the continuous time limit of the GARCH(1,1) process. Journal of Econometrics 96, 145–153.Google Scholar
Davidson, J. (1994) Stochastic Limit Theory: Advanced Texts in Econometrics. Oxford University Press.
Demos, A. (2000) The Autocorrelation Function of Conditionally Heteroskedastic in Mean Models. Discussion paper 2000-07, Department of International and European Economic Studies, Athens University of Economics and Business.
Drost, F.C. & T.E. Nijman (1993) Temporal aggregation of GARCH processes. Econometrica 61, 909–1027.Google Scholar
Drost, F.C. & B.J.M. Werker (1996) Closing the GARCH gap: Continuous time GARCH modelling. Journal of Econometrics 74, 31–57.Google Scholar
Duan, J.-C. (1997) Augmented GARCH(p,q) process and its diffusion limit. Journal of Econometrics 79, 97–127.Google Scholar
Engle, R.F. (1982) Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50, 987–1007.Google Scholar
Ethier, S.N. & T.G. Kurtz (1986) Markov Processes: Characterization and Convergence. Wiley.
Gourieroux, C., A. Monfort, & E. Renault (1993) Indirect inference. Journal of Applied Econometrics 8, 85–118.CrossRefGoogle Scholar
Gradshteyn, I.S. & I.M. Ryznik (1994) Table of Integrals, Series, and Products, 5th ed. Academic Press.
Hentschel, L. (1995) All in the family: Nesting Symmetric and Asymmetric GARCH models. Journal of Financial Economics 39, 71–104.Google Scholar
Karatzas, I. & E.S. Shreeve (1991) Brownian Motion and Stochastic Calculus, 2nd ed. Springer-Verlag.
Kushner, H.J. (1983) Approximation and Weak Convergence Methods for Random Processes with Applications to Stochastic Systems Theory. MIT Press.
Nelson, D. (1990) ARCH models as diffusion approximations. Journal of Econometrics 45, 97–127.Google Scholar
Skorohod, A.V. (1996) Lectures on the Theory of Stochastic Processes. Trans-Atlantic Publications.
Sentana, E. (1995) Quadratic ARCH models. Review of Economic Studies 62, 639–661.Google Scholar
Stroock, D.W. & S.R.S. Varadhan (1979) Multidimensional Diffusion Processes. (corrected second printing, 1997) Springer-Verlag.