Hostname: page-component-77c89778f8-m8s7h Total loading time: 0 Render date: 2024-07-18T19:04:35.993Z Has data issue: false hasContentIssue false
  • English
  • Français

MIXING PROPERTIES OF A GENERAL CLASS OF GARCH(1,1) MODELS WITHOUT MOMENT ASSUMPTIONS ON THE OBSERVED PROCESS

Published online by Cambridge University Press:  30 August 2006

Christian Francq  and
Jean-Michel Zakoïan
Christian Francq
Affiliation:
Université Lille 3, GREMARS
Jean-Michel Zakoïan
Affiliation:
Université Lille 3, GREMARS and CREST
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider general, and possibly nonparametric, GARCH(1,1) processes. First we give conditions for the existence and the uniqueness of stationary ergodic solutions. Then we identify additional conditions for geometric ergodicity. These conditions consist of mild restrictions on the distribution of the latent independent process. No moment assumption is made on the generalized autoregressive conditionally heteroskedastic (GARCH) process. Applications to the asymptotic behavior of sample autocorrelations and to unit-root tests are proposed.This work was supported by INTAS (research project 03-51-3714). The authors gratefully acknowledge the quick and careful reading of the manuscript by Bruce Hansen and three referees. Their detailed comments led to a greatly improved presentation.

Type
Research Article
Information
Econometric Theory , Volume 22 , Issue 5 , October 2006 , pp. 815 - 834
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

Ango Nze, P. (1992) Critères d'ergodicité de quelques modèles à représentation markovienne. Comptes Rendus de l'Academie des Sciences de Paris 315, 1301–1304.Google Scholar
Ango Nze, P. (1998) Critères d'ergodicité géométrique ou arithmétique de modèles linéaires perturbés à représentation markovienne. Comptes Rendus de l'Academie des Sciences de Paris 326, 371–376.Google Scholar
Basrak, B., R.A. Davis, & T. Mikosch (2002) Regular variation of GARCH processes. Stochastic Processes and Their Applications 99, 95–115.Google Scholar
Berkes, I., L. Horváth, & P.S. Kokoszka (2003) GARCH processes: Structure and estimation. Bernoulli 9, 201–227.Google Scholar
Bollerslev, T. (1986) Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31, 307–327.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
Boussama, F. (1998) Ergodicité, mélange et estimation dans les modèles GARCH. Ph.D. dissertation, Paris-7 University.
Bradley, R.C. (1986) Basic properties of strong mixing conditions. In E. Eberlein & M.S. Taqqu (eds.), Dependence in Probability and Statistics, A Survey of Recent Results, pp. 165–192. Birkhäuser.
Brockwell, P.J. & R.A. Davis (1991) Time Series: Theory and Methods. Springer-Verlag.
Bühlmann, P. & A.J. McNeil (2000) Nonparametric GARCH Models. Manuscript, ETH, Zürich.
Carrasco, M. & X. Chen (2002) Mixing and moment properties of various GARCH and stochastic volatility models. Econometric Theory 18, 17–39.Google Scholar
Chow, Y.S. & H. Teicher (1997) Probability Theory, 3rd ed. Springer-Verlag.
Davydov, Y. (1973) Mixing conditions for Markov chains. Theory of Probability and Its Applications 18, 313–328.Google Scholar
Ding, Z., C. Granger, & R.F. Engle (1993) A long memory property of stock market returns and a new model. Journal of Empirical Finance 1, 83–106.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 the United Kingdom inflation. Econometrica 50, 987–1007.Google Scholar
Feigin, P.D. & R.L. Tweedie (1985) Random coefficient autoregressive processes: A Markov chain analysis of stationarity and finiteness of moments. Journal of Time Series Analysis 6, 1–14.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
Francq, C. & J.-M. Zakoïan (1998) Estimating linear representations of nonlinear processes. Journal of Statistical Planning and Inference 68, 145–165.Google Scholar
Francq, C. & J.-M. Zakoïan (2004) Maximum likelihood estimation of pure GARCH and ARMA-GARCH processes. Bernoulli 10, 605–637.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. Teräsvirta (1999) Properties of moments of a family of GARCH processes. Journal of Econometrics 92, 173–192.Google Scholar
Herrndorf, N. (1984) A functional central limit theorem for weakly dependent sequences of random variables. Annals of Probability 12, 141–153.Google Scholar
Hwang, S.Y. & T.Y. Kim (2004) Power transformation and threshold modeling for ARCH innovations with applications to tests for ARCH structure. Stochastic Processes and Their Applications 110, 295–314.Google Scholar
Ibragimov, I.A. (1962) Some limit theorems for stationary processes. Theory of Probability and Its Applications 7, 349–382.Google Scholar
Jensen, S.T. & A. Rahbek (2004) Asymptotic normality of the QMLE estimator of ARCH in the nonstationary case. Econometrica 72, 641–646.Google Scholar
Klüppelberg, C., A. Lindner, & R. Maller (2004) A continuous time GARCH process driven by a Lévy process: Stationarity and second order behaviour. Journal of Applied Probability 41, 601–622.Google Scholar
Kristensen, D. (2006) Geometric ergodicity of a class of Markov chains with applications to time series models. Paper presented at the Econometric Society 2005 World Congress.
Lee, O. & D.W. Shin (2004) Strict stationarity and mixing properties of asymmetric power GARCH models allowing a signed volatility. Economics Letters 84, 167–173.Google Scholar
Lee, S.-W. & B.E. Hansen (1994) Asymptotic theory for the GARCH(1,1) quasi-maximum likelihood estimator. Econometric Theory 10, 29–52.Google Scholar
Ling, S., W.K. Li, & M. McAleer (2003) Estimation and testing for unit root processes with GARCH(1,1) errors: Theory and Monte Carlo evidence. Econometric Reviews 22, 179–202.Google Scholar
Ling, S. & M. McAleer (2002) Stationarity and the existence of moments of a family of GARCH processes. Journal of Econometrics 106, 109–117.Google Scholar
Lu, Z. (1996) A note on geometric ergodicity of autoregressive conditional heteroscedasticity (ARCH) model. Statistics and Probability Letters 30, 305–311.Google Scholar
Lumsdaine, R.L. (1996) Consistency and asymptotic normality of the quasi-maximum likelihood estimator in IGARCH(1,1) and covariance stationary GARCH(1,1) models. Econometrica 64, 575–596.Google Scholar
Meitz, M. & P. Saikkonen (2004) Ergodicity, Mixing, and Existence of Moments of a Class of Markov Models with Applications to GARCH and ACD Models. Paper presented at the Econometric Society 2005 World Congress.
Meyn, S.P. & R.L. Tweedie (1996) Markov Chains and Stochastic Stability, 3rd ed., Springer-Verlag.
Mokkadem, A. (1990) Propriétés de mélange des processus autorégressifs polynomiaux. Annales de l'Institut Henri Poincaré 26, 219–260.Google Scholar
Nelson, D.B. (1990) Stationarity and persistence in the GARCH(1,1) model. Econometric Theory 6, 318–334.Google Scholar
Pham, D.T. (1986) The mixing property of bilinear and generalised random coefficients autoregressive models. Stochastic Processes and Their Applications 23, 291–300.Google Scholar
Phillips, P.C.B. (1987) Time series regression with a unit root. Econometrica 55, 277–301.Google Scholar
Phillips, P.C.B. & P. Perron (1988) Testing for a unit root in time series regression. Biometrica 75, 335–346.Google Scholar
Rahbek, A., E. Hansen, & J. Dennis (2002) ARCH Innovations and Their Impact on Cointegration Rank Testing. Working paper, University of Copenhagen.
Romano, J.L. & L.A. Thombs (1996) Inference for autocorrelations under weak assumptions. Journal of the American Statistical Association 91, 590–600.Google Scholar
Schwert, G.W. (1989) Why does stock market volatility change over time? Journal of Finance 45, 1129–1155.Google Scholar
Taylor, S. (1986) Modelling Financial Time Series. Wiley.
Tjøstheim, D. (1990) Non-linear time series and Markov chains. Advances in Applied Probability 22, 587–611.Google Scholar
Yang, L. & R. Tschernig (2006) Nonparametric Estimation of Generalized Impulse Response Functions. Unpublished document, Michigan State University.
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