STATIONARITY AND β-MIXING PROPERTY OF A MIXTURE AR-ARCH MODELS

Abstract

We consider a MAR model with ARCH type conditional heteroscedasticity. MAR-ARCH model can be derived as a smoothed version of the double threshold AR-ARCH model by adding a random error to the threshold parameters. Easy to check sufficient conditions for strict stationarity, ${\beta}-mixing$ property and existence of moments of the model are given via Markovian representation technique.

Keywords

• mixture AR-ARCH model;
• Markov chain;
• stationarity;
• geometric ergodicity;
• ${\beta}-mixing$

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References

1. D. B. H. Cline and H. H. Pu, Stability and the Lyapounov exponent of threshold AR-ARCH models, Ann. Appl. Probab. 14 (2004), no. 4, 1920-1949 https://doi.org/10.1214/105051604000000431
2. D. Dijk, T. TerÄasvirta, and P. Franses, Smooth transition autoregressive models: a survey of recent developments, Econometric Rev. 21 (2002), no. 1, 1-47 https://doi.org/10.1081/ETC-120008723
3. R. F. Engle, Autoregressive conditional heteroscedasticity with estimates of the variance of the United Kingdom inflation, Econometrica 50 (1982), no. 4, 987-1007 https://doi.org/10.2307/1912773
4. M. Lanne and P. Saikkonen, Modeling the U.S. Short-term interest rate by mixture autoregressive processes, Journal of Financial Econometrics 1 (2003), no. 1, 96-125 https://doi.org/10.1093/jjfinec/nbg004
5. M. Lanne and P. Saikkonen, On mixture autoregressive models, 2004, preprint
6. N. D. Le, R. D. Martin, and A. E. Raftery, Modeling Flat Stretches, bursts and outliers in time series using mixture transition distribution models, J. Amer. Statist. Assoc. 91 (1996), no. 436, 1504-1515 https://doi.org/10.2307/2291576
7. O. Lee, Geometric ergodicity and existence of higher-order moments for DTARCH (p, q) process, J. Korean Statist. Society 32 (2003), no. 2, 193-202
8. S. P. Meyn and R. L. Tweedie, Markov chains and stochastic stability, Springer-Verlag, 1993
9. H. Tong, Nonlinear Time Series: A Dynamical System Approach, Oxford: Oxford University Press, 1990
10. C. S. Wong and W. K. Li, On a mixture autoregressive model, J. Royal Stat. Soc. Ser. B Stat. Methodl. 62 (2000), no. 1, 95-115
11. C. S. Wong and W. K. Li, On a mixture autogressive conditional heteroscedastic model, J. Amer. Statist. Assoc. 96 (2001), no. 455, 982-995 https://doi.org/10.1198/016214501753208645
12. C. S. Wong and W. K. Li, On a logistic mixture autoregressive model, Biometrika 88 (2001), no. 3, 833-846 https://doi.org/10.1093/biomet/88.3.833
13. A. J. Zeevi, R. Meir, and R. J. Adler, Nonlinear models for time series using mixtures of autoregressive models, 2000

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