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A STUDY ON GARCH(p, q) PROCESS

Lee, Oe-Sook

  • Published : 2003.07.01

Abstract

We consider the generalized autoregressive model with conditional heteroscedasticity process(GARCH). It is proved that if (equation omitted) β/sub i/ < 1, then there exists a unique invariant initial distribution for the Markov process emdedding the given GARCH process. Geometric ergodicity, functional central limit theorems, and a law of large numbers are also studied.

Keywords

ARCH/GARCH model;Markov chain;irreducibility;geometric ergodicity;functional central limit theorem

References

  1. Econometric Theory v.6 Stationarity and persistence in the GARCH(1,1) model D.B.Nelson https://doi.org/10.1017/S0266466600005296
  2. Adv. Appl. Prob. v.22 Nonlinear time series and Markov chains D.Tjøstheim https://doi.org/10.2307/1427459
  3. Bull. Kor Math. Soc. v.38 no.3 Strict stationarity and functional central limit theorem for ARCH/GARCH model O.Lee;J.Kim https://doi.org/10.1090/S0273-0979-01-00918-1
  4. J. Econometrics v.31 Generalized autoregressive conditional heteroskedasticity T.Bollerslev https://doi.org/10.1016/0304-4076(86)90063-1
  5. Stat. and Prob. Letters v.31 The geometric ergodicity and existence of moments for a class of nonlinear time series model H.An;M.Chen;F.Huang https://doi.org/10.1016/S0167-7152(96)00033-8
  6. J. Time Series Anal. v.13 A test for conditional heteroscedisticity in time series models A.K.Bera;M.L.Higgins https://doi.org/10.1111/j.1467-9892.1992.tb00123.x
  7. Economitrica v.50 Autoregressive conditional heteroscedasticity with estimates of the variance of the United Kingdom inflation R.F.Engle https://doi.org/10.2307/1912773
  8. Statistca Sinica v.4 Probabilistic properties of the β-ARCH model D.Guegan;J.Diebolt
  9. Non-linear Time Series: A Dynamical System Approach H.Tong
  10. Adv. Appl. Prob. v.30 A central limit theorem for random coefficient autoregressive models and ARCH/GARCH models A.Rudolph https://doi.org/10.1239/aap/1035227994
  11. Markov chains and Stochastic Stability S.P.Meyn;R.L.Tweedie
  12. Stat. and Prob. Letters v.46 On probabilistic properties of nonlinear ARMA(p.q) models O.Lee https://doi.org/10.1016/S0167-7152(99)00096-6
  13. J. Econometrics v.52 Stationarity of GARCH processes and of some nonnegative time series P.Bougerol;N.Picard https://doi.org/10.1016/0304-4076(92)90067-2
  14. J. Time Series Anal. v.5 ARMA models with ARCH errors A.A.Weiss https://doi.org/10.1111/j.1467-9892.1984.tb00382.x
  15. J. Econometrics v.11 On a double threshold autoregressive conditional heterokedastic time series model C.W.Li;W.K.Li https://doi.org/10.1002/(SICI)1099-1255(199605)11:3<253::AID-JAE393>3.0.CO;2-8
  16. J. Statist. Plann. Inference v.2 On a threshold autoregression with conditional heteroskedastic variances J.Liu;W.K.Li;C.W.Li
  17. Annals Prob. v.16 no.3 Asymptotics of a class of Markov processes which are not in general irreducible R.N.Bhattacharya;O.Lee https://doi.org/10.1214/aop/1176991694
  18. Stochastic Processes and their Applications v.85 Extremal behavior of the autoregressive process with ARCH(1) errors M.Borkovec https://doi.org/10.1016/S0304-4149(99)00073-3
  19. J. Appl. Prob. v.35 A central limit theorem for contractive stochastic dynamical systems M.Benda https://doi.org/10.1239/jap/1032192562
  20. Z. Wahrsch. Verw. Gebiete. v.67 Convergence in distribution of products of random matrices H.Kesten;F.Spitzer https://doi.org/10.1007/BF00532045
  21. Papers in Probabilit, Statistics and Analysis Criteria for rates of convergence of Markov chains with application to queueing theory, in J.F.C. Kingman and G.E.H. Reuter, eds. R.L.Tweedie
  22. Stat. and Prob. Lettrs v.30 A note on geometric ergodicity of autoregressive conditional heteroscedasticity(ARCH) model Z.Lu https://doi.org/10.1016/S0167-7152(95)00233-2
  23. Stochastic Processes and their Applications v.23 The mixing property of bilinear and generalised random coefficient autoregressive models D.T.Pham https://doi.org/10.1016/0304-4149(86)90042-6
  24. Stochastic Processes and their Applications v.34 A multicative ergodic theorem for Lipschitz maps J.Elton https://doi.org/10.1016/0304-4149(90)90055-W
  25. Ann. Prob. v.24 no.2 A Liapounov bound for solutions of the Poisson equation P.W.Glynn;s.P.Meyn https://doi.org/10.1214/aop/1039639370
  26. J. Appl. Prob. v.36 On the probabilistic properties of a double threshold ARMA conditional heteroskedastic model S.Ling https://doi.org/10.1239/jap/1032374627
  27. J. Kor. Math. Soc. v.33 Limit theorems for Markov processes generated by iterations of random maps O.Lee

Cited by

  1. Geometric ergodicity and β-mixing property for a multivariate CARR model vol.100, pp.1, 2008, https://doi.org/10.1016/j.econlet.2007.12.002
  2. DETECTING FOR SMOOTH STRUCTURAL CHANGES IN GARCH MODELS vol.32, pp.03, 2016, https://doi.org/10.1017/S0266466614000942