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A Tour in the Asymptotic Theory of GARCH Estimation

  • Christian FrancqEmail author
  • Jean-Michel Zakoïan
Chapter

Abstract

The main estimation methods of the univariate GARCH models are reviewed. A special attention is given to the asymptotic results and the quasi-maximum likelihood method.

Keywords

Asymptotic Theory Asymptotic Normality GARCH Model ARMA Model Arch Model 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Andrews, D.W.K. (1999): Estimation when a parameter is on a boundary. Econometrica 67, 1341-1384.zbMATHCrossRefMathSciNetGoogle Scholar
  2. Berkes, I. and Horváth, L. (2003): The rate of consistency of the quasi-maximum likelihood estimator. Statistics and Probability Letters 61, 133-143.zbMATHCrossRefMathSciNetGoogle Scholar
  3. Berkes, I. and Horváth, L. (2004): The efficiency of the estimators of the parameters in GARCH processes Annals of Statistics 32, 633-655.zbMATHCrossRefMathSciNetGoogle Scholar
  4. Berkes, I., Horváth, L. and Kokoszka, P.S. (2003): GARCH processes: structure and estimation. Bernoulli 9, 201-227.zbMATHCrossRefMathSciNetGoogle Scholar
  5. Billingsley, P. (1961): The Lindeberg-Levy theorem for martingales. Proceedings of the American Mathematical Society 12, 788-792.zbMATHCrossRefMathSciNetGoogle Scholar
  6. Bose, A. and Mukherjee, K. (2003): Estimating the ARCH parameters by solving linear equations. Journal of Time Series Analysis 24, 127-136.zbMATHCrossRefMathSciNetGoogle Scholar
  7. Bougerol, P. and Picard, N. (1992): Stationarity of GARCH processes and of some non-negative time series. Journal of Econometrics 52, 115-127.zbMATHCrossRefMathSciNetGoogle Scholar
  8. Boussama, F. (1998): Ergodicité, mélange et estimation dans les modèles GARCH. PhD Thesis, Université Paris-7, Paris.Google Scholar
  9. Brockwell, P.J. and Davis, R.A. (1991): Time series: theory and methods. Springer, New-York.Google Scholar
  10. Chen, M. and An, H.Z. (1998): A note on the stationarity and the existence of moments of the GARCH model. Statistica Sinica 8, 505-510.zbMATHMathSciNetGoogle Scholar
  11. Demos, A. and Sentana, E. (1998): Testing for GARCH effects: a one-sided approach. Journal of Econometrics 86, 97-127.zbMATHCrossRefGoogle Scholar
  12. Drost, F.C. and Klaassen, C.A.J. (1997): Efficient estimation in semiparametric GARCH models. Journal of Econometrics 81, 193-221.zbMATHCrossRefMathSciNetGoogle Scholar
  13. Engle, R.F. (1982): Autoregressive conditional heteroskedasticity with estimates of the variance of the United Kingdom inflation. Econometrica 50, 987-1007.zbMATHCrossRefMathSciNetGoogle Scholar
  14. Engle, R.F. and González-Rivera, G. (1991): Semiparametric ARCH models. Journal of Business and Econometric Statistics 9, 345-359.CrossRefGoogle Scholar
  15. Escanciano, J.C. (2007): Quasi-maximum likelihood estimation of semi-strong GARCH models. Working documents Google Scholar
  16. Francq, C. and Zakoïan, J.M. (2000): Estimating weak GARCH representations. Econometric Theory 16, 692-728.zbMATHCrossRefMathSciNetGoogle Scholar
  17. Francq, C. and Zakoïan, J.M. (2004): Maximum likelihood estimation of pure GARCH and ARMA-GARCH processes. Bernoulli 10, 605-637.zbMATHCrossRefMathSciNetGoogle Scholar
  18. Francq, C. and Zakoïan, J.M. (2006): On efficient inference in GARCH processes. In:Bertail, P., Doukhan, P., Soulier, P. (Eds): Statistics for dependent data, 305-327. Springer, New-York.Google Scholar
  19. Francq, C. and Zakoïan, J.M. (2007): Quasi-maximum likelihood estimation in GARCH processes when some coefficients are equal to zero. Stochastic Processes and their Applications 117, 1265-1284.zbMATHCrossRefMathSciNetGoogle Scholar
  20. Giraitis, L., Leipus, R. and Surgailis, D. (2008): ARCH(∞) models and long-memory properties. In: Andersen, T.G., Davis, R.A., Kreiss, J.-P. and Mikosch, T. (Eds.): Handbook of Financial Time Series, 70-84 Springer, New York.Google Scholar
  21. González-Rivera, G. and Drost, F.C. (1999): Efficiency comparisons of maximum-likelihood based estimators in GARCH models. Journal of Econometrics 93, 93-111.zbMATHCrossRefMathSciNetGoogle Scholar
  22. Gouriéroux, C. (1997): ARCH models and financial applications. Springer, New York.zbMATHGoogle Scholar
  23. Geweke, J. (1989): Exact predictive densities for linear models with ARCH disturbances. Journal of Econometrics 40, 63-86.zbMATHCrossRefGoogle Scholar
  24. Giraitis, L. and Robinson, P.M. (2001): Whittle estimation of ARCH models. Econometric Theory 17, 608-631.zbMATHCrossRefMathSciNetGoogle Scholar
  25. Hall, P. and Yao, Q. (2003): Inference in ARCH and GARCH models with heavy-tailed errors. Econometrica 71, 285-317.zbMATHCrossRefMathSciNetGoogle Scholar
  26. Hamilton, J.D. (1994): Time series analysis. Princeton University Press, Princeton.zbMATHGoogle Scholar
  27. Horváth, L. and Liese, F. (2004): Lp-estimators in ARCH models. Journal of Statistical Planning and Inference 119, 277-309.zbMATHCrossRefMathSciNetGoogle Scholar
  28. Jensen, S.T. and Rahbek, A. (2004a): Asymptotic normality of the QMLE estimator of ARCH in the nonstationary case. Econometrica 72, 641-646.zbMATHCrossRefMathSciNetGoogle Scholar
  29. Jensen, S.T. and Rahbek, A. (2004b): Asymptotic inference for nonstationary GARCH. Econometric Theory 20, 1203-1226.zbMATHCrossRefMathSciNetGoogle Scholar
  30. Jordan, H. (2003): Asymptotic properties of ARCH(p) quasi maximum likelihood estimators under weak conditions. PhD Thesis, University of Vienna.Google Scholar
  31. Lee, S.W. and Hansen, B.E. (1994): Asymptotic theory for the GARCH(1,1) quasi-maximum likelihood estimator. Econometric Theory 10, 29-52.zbMATHCrossRefMathSciNetGoogle Scholar
  32. Lindner, A. (2008): Stationarity, mixing, distributional properties and moments of GARCH(p,q)-processes. In: Andersen, T.G., Davis, R.A., Kreiss, J.-P. and Mikosch, T. (Eds.): Handbook of Financial Time Series, 43-69. Springer, New York.Google Scholar
  33. Ling, S. (2003): Self-weighted LSE and MLE for ARMA-GARCH models. Unpublished working paper, HKUST.Google Scholar
  34. Ling, S. (2006): Self-weighted and local quasi-maximum likelihood estimators for ARMA-GARCH/IGARCH models. Journal of Econometrics. To appear.Google Scholar
  35. Ling, S. and Li, W.K. (1997): On fractionally integreted autoregressive moving-average time series models with conditional heteroscedasticity. Journal of the American Statistical Association 92, 1184-1194.zbMATHCrossRefMathSciNetGoogle Scholar
  36. Ling, S. and Li, W.K. (1998): Limiting distributions of maximum likelihood estimators for unstable ARMA models with GARCH errors. Annals of Statistics 26, 84-125.zbMATHCrossRefMathSciNetGoogle Scholar
  37. Ling, S. and McAleer, M. (2003): Asymptotic theory for a vector ARMA-GARCH model. Econometric Theory 19, 280-310.MathSciNetGoogle Scholar
  38. Ling, S. and McAleer, M. (2003): Adaptive estimation in nonstationry ARMA models with GARCH noises. Annals of Statistics 31, 642-674.zbMATHCrossRefMathSciNetGoogle Scholar
  39. Linton, O. (1993): Adaptive estimation in ARCH models. Econometric Theory 9, 539-564.CrossRefMathSciNetGoogle Scholar
  40. 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.zbMATHCrossRefMathSciNetGoogle Scholar
  41. Mikosch, T., Gadrich, T., Klüppelberg, C. and Adler, R.J. (1995): Parameter estimation for ARMA models with infinite variance innovations. Annals of Statistics 23, 305-326.zbMATHCrossRefMathSciNetGoogle Scholar
  42. Mikosch, T. and Straumann, D. (2002): Whittle estimation in a heavy-tailed GARCH(1,1) model. Stochastic Processes and their Application 100, 187-222.zbMATHCrossRefMathSciNetGoogle Scholar
  43. Nelson, D.B. (1990): Stationarity and persistence in the GARCH(1,1) model. Econometric Theory 6, 318-334.CrossRefMathSciNetGoogle Scholar
  44. Peng, L. and Yao, Q. (2003): Least absolute deviations estimation for ARCH and GARCH models. Biometrika 90, 967-975.CrossRefMathSciNetGoogle Scholar
  45. Rich, R.W., Raymond, J. and Butler, J.S. (1991): Generalized instrumental variables estimation of autoregressive conditional heteroskedastic models. Economics Letter 35, 179-185.CrossRefMathSciNetGoogle Scholar
  46. Straumann, D. (2005): Estimation in conditionally heteroscedastic time series models. Lecture Notes in Statistics. Springer, Berlin Heidelberg.Google Scholar
  47. Straumann, D. and Mikosch, T. (2006): Quasi-MLE in heteroscedastic time series: a stochastic recurrence equations approach. Annals of Statistics 34, 2449-2495.zbMATHCrossRefMathSciNetGoogle Scholar
  48. Wald, A. (1949): Note on the consistency of the maximum likelihood estimate. The Annals of Mathematical Statistics 20, 595-601.zbMATHCrossRefMathSciNetGoogle Scholar
  49. Weiss, A.A. (1986): Asymptotic theory for ARCH models: estimation and testing. Econometric Theory 2, 107-131.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  1. 1.University Lille IIIEQUIPPE-GREMARS, Domaine du Pont de boisVilleneuve d’Ascq CedexFrance
  2. 2.University Lille IIIEQUIPPE-GREMARS, and CRESTMalakoff CedexFrance

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