Journal of the Korean Statistical Society

The Korean Statistical Society (한국통계학회)
권호 표지

STATIONARY $\beta-MIXING$ FOR SUBDIAGONAL BILINEAR TIME SERIES

  • Lee Oe-Sook (Department of Statistics, Ewha Womans University)
  • Published : 2006.03.01
Download PDF
⟨ Previous Next ⟩

Abstract

We consider the subdiagonal bilinear model and ARMA model with subdiagonal bilinear errors. Sufficient conditions for geometric ergodicity of associated Markov chains are derived by using results on generalized random coefficient autoregressive models and then strict stationarity and ,a-mixing property with exponential decay rates for given processes are obtained.

Keywords

References

  1. BIBI, A. AND OYET, A. (2002). 'A note on the properties of some time varying bilinear models', Statistics and Probability Letters, 58, 399-411 https://doi.org/10.1016/S0167-7152(02)00153-0
  2. CARRASCO, M. AND CHEN, X. (2002). 'Mixing and moment properties of various GARCH and stochastic volatility models', Econometric Theory, 18, 17-39 https://doi.org/10.1017/S0266466602181023
  3. CHANDA, K. C. (1992). 'Stationarity and central limit theorem associated with bilinear time series models', Journal of Time Series Analysis, 12, 301-313 https://doi.org/10.1111/j.1467-9892.1991.tb00085.x
  4. 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
  5. DOUKHAN, P. (1994). Mixing. Properties and examples, Lecture Notes in Statistics, 85, Springer-Verlag, New York
  6. ENGLE, R. F. (1982). 'Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation.', Econometrica, 50, 987-1008 https://doi.org/10.2307/1912773
  7. FRANCQ, C. (1999). 'ARMA models with bilinear innovations', Communications in Statistics-Stochastic models, 15, 29-52 https://doi.org/10.1080/15326349908807524
  8. GRANGER, C. W. J. AND ANDERSEN, A. P. (1978). An Introduction to Bilinear Time Series Models, Vandenhoeck and Ruprecht, Gottingen
  9. HAMILTON, J. D. (1989). 'A new approach to the economic analysis of nonstationary time series and the business cycle', Econometrica, 57, 357-384 https://doi.org/10.2307/1912559
  10. LIU, J. (1992). 'On stationarity and asymptotic inference of bilinear time series models', Statistica Sinica, 2, 479-494
  11. LIU, J. AND BROCKWELL, P. J. (1988). 'On the general bilinear time series model', Journal of Applied Probability, 25, 553-564 https://doi.org/10.2307/3213984
  12. MEYN, S. P. AND TWEEDIE, R. L. (1993). Markov Chains and Stochastic Stability, Communications and Control Engineering Series, Springer-Verlag, London
  13. PHAM, D. T. (1985). 'Bilinear Markovian representation and bilinear models', Stochastic Processes and Their Applications, 20, 295-306 https://doi.org/10.1016/0304-4149(85)90216-9
  14. PHAM, D. T. (1986). 'The mixing property of bilinear and generalised random coefficient autoregressive models.', Stochastic Processes and Their Applications, 23, 291-300 https://doi.org/10.1016/0304-4149(86)90042-6
  15. RAO, M. BHASKARA, SUBBA RAO, T. AND WALKER, A. M. (1983). 'On the existence of some bilinear time series models', Journal of Time Series Analysis, 4, 95-110 https://doi.org/10.1111/j.1467-9892.1983.tb00362.x
  16. SUBBA RAO, T. (1981). 'On the theory of bilinear time series models', Journal of Royal Statistical Society, Ser. B, 43, 244-255
  17. SUBBA RAO, T. AND GABR, M. M. (1984). An Introduction to Bispectral Analysis and Bilinear Time Series Models, Lecture Notes in Statistics, 24, Springer-Verlag, New York
  18. TERDIK, G. (1999). Bilinear Stochastic Models and Related Problems of Nonlinear Time Series Analysis. A Frequency Domain Approach, Lecture Notes in Statistics, 142, Springer-Verlag, New York
  19. TONG, H. (1978). 'On a threshold model', In Pattern Recognition and Signal Processing (C. H. Chen, ed.), Sijthoff and Noordhoff, Amsterdam
  20. TONG, H. (1981). 'A note on a Markov bilinear stochastic process in discrete time', Journal of Time Series Analysis, 2, 279-284 https://doi.org/10.1111/j.1467-9892.1981.tb00326.x
  21. WEISS, A. A. (1986). 'ARCH and bilinear time series models: Comparison and combination', Journal of Business and Economic Statistics, 4, 59-70 https://doi.org/10.2307/1391387