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ASYMPTOTIC THEORY FOR A FACTOR GARCH MODEL

Published online by Cambridge University Press:  01 April 2009

Christian M. Hafner*
Affiliation:
Université catholique de Louvain
Arie Preminger
Affiliation:
University of Haifa
*
*Address correspondence to Christian M. Hafner, Institut de statistique and CORE, Université catholique de Louvain, Voie du Roman Pays 20, B-1348, Louvain-la-Neuve, Belgium; e-mail: christian.hafner@uclouvain.be.

Abstract

This paper investigates the asymptotic theory for a factor GARCH (generalized autoregressive conditional heteroskedasticity) model. Sufficient conditions for asymptotic stability and existence of moments are established. These conditions allow for volatility spillover and integrated GARCH. We then show the strong consistency and asymptotic normality of the quasi–maximum likelihood estimator (QMLE) of the model parameters. The results are obtained under the finiteness of the fourth-order moment of the innovations.

Type
ARTICLES
Copyright
Copyright © Cambridge University Press 2009

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References

REFERENCES

Alexander, C.O. (2001) Orthogonal GARCH. In Alexander, C.O. (ed.), Mastering Risk, vol. II, pp. 21–38. Prentice-Hall.Google Scholar
Andrews, D.W.K. (1999) Estimation when a parameter is on a boundary. Econometrica 67, 1341–1383.Google Scholar
Baillie, R.T. & Chung, H. (2001) Estimation of GARCH models from the autocorrelations of the squares of a process. Journal of Time Series Analysis 22, 631–650.Google Scholar
Bauwens, L., Laurent, S., & Rombouts, J.V.K. (2006) Multivariate GARCH models: A survey. Journal of Applied Econometrics 21, 79–109.Google Scholar
Berkes, I., Horváth, L., & Kokoszka, P.S. (2003) GARCH processes: Structure and estimation. Bernoulli 9, 201–227.CrossRefGoogle Scholar
Bollerslev, T. (1990) Modelling the coherence on short-run nominal exchange rates: A multivariate generalized ARCH model. Review of Economics and Statistics 72, 498–505.Google Scholar
Bollerslev, T. & Engle, R.F. (1993) Common persistence in conditional variances. Econometrica 61, 167–186.Google Scholar
Bollerslev, T., Engle, R.F., & Wooldridge, J.M. (1988) A capital asset pricing model with time varying covariances. Journal of Political Economy 96, 116–131.Google Scholar
Bollerslev, T. & Wooldridge, J. (1992) Quasi maximum likelihood estimation and inference in dynamic models with time varying covariances. Econometric Reviews 11, 143–172.Google Scholar
Bougerol, P. & Picard, N. (1992a) Strict stationarity of generalized autoregressive processes. Annals of Probability 20, 1714–1730.Google Scholar
Bougerol, P. & Picard, N. (1992b) Stationarity of GARCH processes and of some nonnegative time series. Journal of Econometrics 52, 115–127.Google Scholar
Boussama, F. (1998) Ergodicity, mixing and estimation in GARCH models. Ph.D. Dissertation, University Paris 7.Google Scholar
Boussama, F. (2000) Asymptotic normality for the quasi-maximum likelihood estimator of a GARCH model. Comptes Rendus de l'Académie des Sciences de Paris, series I 331, 81–84.Google Scholar
Brooks, C., Burke, S.P., & Persand, G. (2003) Multivariate GARCH models: Software choice and estimation issues. Journal of Applied Econometrics 18, 725–734.CrossRefGoogle Scholar
Carrasco, M. & Chen, X. (2002) Mixing and moments properties of various GARCH and stochastic models. Econometric Theory 18, 17–39.Google Scholar
Comte, F. & Lieberman, O. (2000) Second order noncausality in multivariate GARCH processes. Journal of Time Series Analysis 21, 535–557.CrossRefGoogle Scholar
Comte, F. & Lieberman, O. (2003) Asymptotic theory for multivariate GARCH processes. Journal of Multivariate Analysis 84, 61–84.Google Scholar
Davydov, Y. (1973) Mixing conditions for Markov chains. Theory of Probability and Its Applications 18, 313–328.Google Scholar
Drost, F.C., Klaassen, C.A.J. & Werker, B.J.M. (1997) Adaptive estimation in time series models. Annals of Statistics 25, 786–817.CrossRefGoogle Scholar
Engle, R.F. (2002) Dynamic conditional correlation — A simple class of multivariate GARCH models. Journal of Business & Economic Statistics 20, 339–350.Google Scholar
Engle, R.F. & Kroner, K.F. (1995) Multivariate simultaneous generalized ARCH. Econometric Theory 11, 122–150.Google Scholar
Engle, R.F., Ng, V., & Rothschild, M. (1990) Asset pricing with a factor-ARCH structure: Empirical estimates for Treasury bills. Journal of Econometrics 45, 213–237.CrossRefGoogle Scholar
Francq, C. & Zakoïan, J.-M. (2004) Maximum likelihood estimation of pure GARCH and ARMA-GARCH processes. Bernoulli 10, 605–637.Google Scholar
Francq, C. & Zakoïan, J.-M. (2006) Mixing properties of a general class of GARCH(1,1) models without moment assumptions. Econometric Theory 22, 815–834.Google Scholar
Francq, C. & Zakoïan, J.-M. (2007) Quasi-maximum likelihood inference in GARCH processes when some coefficients are equal to zero. Stochastic Processes and Their Applications 117, 1265–1284.CrossRefGoogle Scholar
Goldsheid, I.Y. (1991) Lyapounov exponents and asymptotic behavior of the product of random matrices. In Arnold, L., Crauel, H., & Eckmann, J.-P.(eds.), Lyapunov Exponents, pp. 23–37. Lecture Notes in Mathematics 1486. Springer-Verlag.Google Scholar
Gouriéroux, C. (1997) ARCH Models and Financial Applications. Springer-Verlag.CrossRefGoogle Scholar
Hafner, C.M. (2003) Fourth moment structure of multivariate GARCH models. Journal of Financial Econometrics 1, 26–54.Google Scholar
Hafner, C.M. & Rombouts, J. (2007) Semiparametric multivariate volatility models. Econometric Theory 23, 251–280.Google Scholar
Jeantheau, T. (1998) Strong consistency of estimators for multivariate ARCH models. Econometric Theory 14, 70–86.Google Scholar
Kingman, J.F.C. (1973) Subadditive ergodic theory. Annals of Probability 1, 883–909.Google Scholar
Kristensen, D. (2007) Geometric Ergodicity of a Class of Markov Chains with Applications to Time Series Models. Working paper, University of Wisconsin.Google Scholar
Kristensen, D. & Linton, O. (2006) A closed form estimator for the GARCH(1,1) model. Econometric Theory 22, 323–337.Google Scholar
Kristensen, D. & Rahbek, A. (2005) Asymptotics of the QMLE for a class of ARCH(q) models. Econometric Theory 21, 946–961.Google Scholar
Lanne, M. & Saikkonen, P., P. (2007) A multivariate generalized orthogonal factor GARCH model. Journal of Business & Economic Statistics 25, 61–75.Google Scholar
Lee, S.-W. & Hansen, B.E. (1994) Asymptotic theory for the GARCH(1,1) quasi- maximum likelihood estimator. Econometric Theory 10, 29–52.Google Scholar
Ling, S. & McAleer, M. (2003) Asymptotic theory for a vector ARMA-GARCH model. Econometric Theory 19, 280–310.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
Lütkepohl, H. (1996) Handbook of Matrices. Wiley.Google Scholar
Meitz, M. & Saikkonen, P. (2004) Ergodicity, Mixing and Existence of Moments of a Class of Markov Models with Applications to GARCH and ACD Models. Working Paper Series in Economics and Finance, Stockholm School of Economics.Google Scholar
Meyn, S.P. & Tweedie, R.L. (1993) Markov Chains and Stochastic Stability. Springer-Verlag.Google Scholar
Nelson, D.B. (1990) Stationarity and persistence in the GARCH(1,1) model. Econometric Theory 6, 318–334.Google Scholar
Pfanzagl, J. (1969) On the measurability and consistency of minimum contrast estimates. Metrika 14, 249–272.Google Scholar
Preminger, A. & Storti, G. (2006)A GARCH(1,1) Estimate with (almost) No Moment Conditions on the Error Term. CORE Discussion paper 68, Université catholique de Louvain.Google Scholar
Scott, D.J. (1973) Central limit theorems for martingales and for processes with stationary increments using a Skorokhod representation approach. Advances in Applied Probability 5, 119–137.CrossRefGoogle Scholar
Storti, G. (2006) Minimum distance estimation of GARCH(1,1) models. Computational Statistics and Data Analysis 51, 1803–1821.Google Scholar
Tse, Y.K. & Tsui, A.K.C. (2002) A multivariate GARCH model with time-varying correlations. Journal of Business & Economic Statistics 20, 351–362.Google Scholar
van der Weide, R. (2002) GO-GARCH: A multivariate generalized orthogonal GARCH model. Journal of Applied Econometrics 17, 549–564.Google Scholar
Vrontos, I.D., Dellaportas, P., & Politis, D.N. (2003) A full-factor multivariate GARCH model. Econometrics Journal 6, 311–333.Google Scholar
White, H. (2001) Asymptotic Theory for Econometricians. Academic Press.Google Scholar