Hostname: page-component-89b8bd64d-x2lbr Total loading time: 0 Render date: 2026-05-08T01:50:06.999Z Has data issue: false hasContentIssue false
  • English
  • Français

MULTIVARIATE ECOGARCHPROCESSES

Published online by Cambridge University Press:  13 September 2010

Stephan Haug  and
Robert Stelzer
Get access Rights & Permissions [Opens in a new window]

Abstract

A multivariate extension of the exponential continuoustime GARCH (p, q)model (ECOGARCH) is introduced and studied.Stationarity and mixing properties of the newstochastic volatility model are investigated, andways to model a component-wise leverage effect arepresented.

Information

Type
Research Article
Information
Econometric Theory , Volume 27 , Issue 2 , April 2011 , pp. 344 - 371
Copyright
Copyright © Cambridge University Press 2010

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.)

Article purchase

Temporarily unavailable

Footnotes

The authors are grateful to the editor and twoanonymous referees for their very good commentsand also to Claudia Klüppelberg and Jean Jacod fortheir helpful remarks on a draft of this paper.The second author acknowledges financial supportfrom the Deutsche Forschungsgemeinschaft throughthe graduate program Angewandte AlgorithmischeMathematik at the Technische Universität Münchenduring the initial work on the contents of thispaper.

References

REFERENCES

Applebaum, D. (2004) Lévy Processes and Stochastic Calculus. Cambridge University Press.CrossRefGoogle Scholar
Barndorff-Nielsen, O. & Shepard, N. (2001) Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial econometrics (with discussion). Journal of the Royal Statistical Society, Series B 63, 167–241.CrossRefGoogle Scholar
Bhatia, R. (1997) Matrix Analysis, Graduate Texts in Mathematics 69. Springer-Verlag.Google Scholar
Bollerslev, T. (1986) Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31, 307–327.CrossRefGoogle Scholar
Brockwell, P.J. & Davis, R.A. (1991) Time Series: Theory and Methods, 2nd ed. Springer-Verlag.CrossRefGoogle Scholar
Brockwell, P.J. & Lindner, A. (2009) Existence and uniqueness of stationary Lévy-driven CARMA processes. Stochastic Processes and Their Applications 119, 2660–2681.CrossRefGoogle Scholar
Chiu, T.Y.M., Leonard, T., & Tsui, K.-W. (1996) The matrix-logarithmic covariance model. Journal of the American Statistical Association 91, 198–210.CrossRefGoogle Scholar
Czado, C. & Haug, S. (2010) An ACD-ECOGARCH(1, 1) model. Journal of Financial Econometrics 8, 335–344.CrossRefGoogle Scholar
Davydov, Y.A. (1973) Mixing conditions for Markov chains. Theory of Probability and Its Applications 18, 312–328.CrossRefGoogle Scholar
Doukhan, P. (1994) Mixing, Lecture Notes in Statistics 85. Springer-Verlag.Google Scholar
Engle, R.F. (1982) Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50, 987–1008.CrossRefGoogle Scholar
Gourieroux, C. (2006) Continuous time Wishart processes for stochastic risk. Econometric Reviews 25, 177–217.CrossRefGoogle Scholar
Haug, S. & Czado, C. (2007) An exponential continuous time GARCH process. Journal of Applied Probability 44, 960–976.CrossRefGoogle Scholar
Horn, R.A. & Johnson, C.R. (1991) Topics in Matrix Analysis. Cambridge University Press.CrossRefGoogle Scholar
Jacod, J. & Shiryaev, A.N. (2003) Limit Theorems for Stochastic Processes, 2nd ed. Springer-Verlag.CrossRefGoogle Scholar
Kawakatsu, H. (2006) Matrix exponential GARCH. Journal of Econometrics 134, 95–128.CrossRefGoogle Scholar
Klüppelberg, C., Lindner, A., & Maller, R.A. (2004) A continuous-time GARCH process driven by a Lévy process: Stationarity and second-order behaviour. Journal of Applied Probability 41, 601–622.CrossRefGoogle Scholar
Kurtz, T. & Protter, P. (1996) Weak convergence of stochastic integrals and differential equations. In Talay, D. & Tularo, L. (eds.), Probabilistic Models for Nonlinear Partial Differential Equations, Lecture Notes in Mathematics 1627, 1–41.Google Scholar
Maller, R.A., Müller, G., & Szimayer, A. (2008) GARCH modelling in continuous time for irregularly spaced time series data. Bernoulli 14, 519–542.CrossRefGoogle Scholar
Marquardt, T. & Stelzer, R. (2007) Multivariate CARMA processes. Stochastic Processes and Their Applications 117, 96–120.CrossRefGoogle Scholar
Nelson, D.B. (1990) ARCH models as diffusion approximations. Journal of Econometrics 45, 7–38.CrossRefGoogle Scholar
Nelson, D.B. (1991) Conditional heteroskedasticity in asset returns: A new approach. Econometrica 59, 347–370.CrossRefGoogle Scholar
Östermark, R. (2001) Genetic modelling of multivariate EGARCHX-processes: Evidence on the international asset return signal response mechanism. Computational Statistics & Data Analysis 38, 71–93.CrossRefGoogle Scholar
Pigorsch, C. & Stelzer, R. (2009) A multivariate Ornstein-Uhlenbeck type stochastic volatility model. Manuscript Technische Universität München; available at http://www.m4.ma.tum.de.Google Scholar
Protter, P. (2004) Stochastic Integration and Differential Equations, 2nd ed. Springer-Verlag.Google Scholar
Rowland, T. & Weisstein, E.W. (2009) Matrix Exponential. From MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/MatrixExponential.html.Google Scholar
Sato, K. (1999) Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.Google Scholar
Semeraro, P. (2008) A multivariate variance gamma model for financial applications. International Journal of Theoretical and Applied Finance 11, 1–18.CrossRefGoogle Scholar
Stelzer, R. (2009a) First jump approximation of a Lévy driven SDE process and an application to multivariate ECOGARCH processes. Stochastic Processes and Their Applications 119, 1932–1951.CrossRefGoogle Scholar
Stelzer, R. (2009b) Multivariate COGARCH(1,1) processes. Bernoulli 16, 80–115.Google Scholar
Tse, Y. & Hackard, J.C. (2004) Can Island provide liquidity and price discovery in the dark? Review of Quantitative Finance and Accounting 23, 149–166.CrossRefGoogle Scholar
Yang, S.-Y. & Doong, S.-C. (2004) Price and volatility spillovers between stock prices and exchange rates: Empirical evidence from the G-7 countries. International Journal of Business and Economics 3, 139–153.Google Scholar