Stability and the Lyapounov exponent of threshold AR-ARCH models

Annals of Applied Probability

Volumn 14, Issue 4, 2004, Pages 1920-1949

  Cline, Daren B H ^a   Pu, Huay Min H ^a  

^a TEXAS A AND M UNIVERSITY   (United States)

Author keywords

ARCH; Ergodicity; Lyapounov exponent; Markov chain; Nonlinear time series

Indexed keywords

EID: 26844571127     PISSN: 10505164     EISSN: 10505164     Source Type: Journal    
DOI: 10.1214/105051604000000431     Document Type: Article

References (34)

1. BASRAK, B., DAVIS, R. and MIKOSCH, T. (2002). Regular variation of GARCH processes. Stochastic Process. Appl. 99 95-115.
2. BORKOVEC, M. (2000). Extremal behavior of the autoregressive process with ARCH(1) errors. Stochastic Process. Appl. 85 189-207.
3. BORKOVEC, M. and KLÜPPELBERG, C. (2001). The tail of the stationary distribution of an autoregressive process with ARCH(1) errors. Ann. Appl. Probab. 11 1220-1241.
4. BOUGEROL, P. (1987). Tightness of products of random matrices and stability of linear systems. Ann. Probab. 15 40-74.
5. BOUGEROL, P. and PICARD, N. (1992a). Strict stationarity of generalized autoregressive processes. Ann. Probab. 20 1714-1730.
6. BOUGEROL, P. and PICARD, N. (1992b). Stationarity of GARCH processes and some nonnegative time series. J. Econometrics 52 115-127.
7. n with stationary coefficients. Adv. in Appl. Probab. 18 211-220.
8. CHAN, K.-S. and TONG, H. (1985). On the use of the deterministic Lyapunov function for the ergodicity of stochastic difference equations. Adv. in Appl. Probab. 17 666-678.
9. CHAN, K.-S. and TONG, H. (1986). On estimating thresholds in autoregressive models. J. Time Ser. Anal. 7 179-190.
10. CLINE, D. B. H. and PU, H. H. (1998). Verifying irreducibility and continuity of a nonlinear time series. Statist. Probab. Lett. 40 139-148.
11. CLINE, D. B. H. and PU, H. H. (1999). Geometric ergodicity of nonlinear time series. Statist. Sinica 9 1103-1118.
12. CLINE, D. B. H. and PU, H. H. (2001). Geometric transience of nonlinear time series. Statist. Sinica 11 273-287.
13. CLINE, D. B. H. and PU, H. H. (2002). Stability of threshold-like ARMA time series. Dept. Statistics, Texas A&M Univ.
14. DIACONIS, P. and FREEDMAN, D. (1999). Iterated random functions. SIAM Rev. 41 45-76.
15. ENGLE, R. F. and BOLLERSLEV, T. (1986). Modelling the persistence of conditional variances. Econometric Rev. 5 1-50.
16. FERRANTE, M., FONSECA, G. and VIDONI, P. (2003). Geometric ergodicity, regularity of the invariant distribution and inference for a threshold bilinear Markov process. Statist. Sinica 13 367-384.
17. FURSTENBERG, H. and KESTEN, H. (1960). Products of random matrices. Ann. Math. Statist. 31 457-469.
18. GLOSTEN, L. R., JAGANNATHAN, R. and RUNKLE, D. (1993). On the relation between the expected value and the volatility of the nominal excess return on stocks. J. Finance 48 1779-1801.
19. GOLDIE, C. M. (1991). Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab. 1 126-166.
20. GOLDIE, C. M. and MALLER, R. A. (2000). Stability of perpetuities. Ann Probab. 28 1195-1218.
21. GOURIEROUX, C. and MONFORT, A. (1992). Qualitative threshold ARCH models. J. Econometrics 52 159-199.
22. HWANG, S. Y. and WOO, M.-J. (2001). Threshold ARCH(1) processes: Asymptotic inference. Statist. Probab. Lett. 53 11-20.
23. KESTEN, H. (1973). Random difference equations and renewal theory for products of random matrices. Acta Math. 131 207-248.
24. KINGMAN, J. F. C. (1973). Subadditive ergodic theory. Ann. Probab. 1 883-899.
25. LANNE, M. and SAIKKONEN, P. (2004). Nonlinear GARCH models for highly persistent volatility. Preprint.
26. LETAC, G. (1986). A contraction principle for certain Markov chains and its applications. In Random Matrices and Their Applications Conference 1984 (J. E. Cohen, H. Kesten and C. M. Newman, eds.) 263-273. Amer. Math. Soc., Providence, RI.
27. LI, C. W. and LI, W. K. (1996). On a double-threshold autoregressive heteroscedastic time series model. J. Appl. Econometrics 11 253-274.
28. LIU, J., LI, W. K. and LI, C. W. (1997). On a threshold autoregression with conditional heteroscedastic variances. J. Statist. Plann. Inference 62 279-300.
29. LU, Z. (1998). On the geometric ergodicity of a non-linear autoregressive model with an autoregressive conditional heteroscedastic term. Statist. Sinica 8 1205-1217.
30. 1 geometric ergodicity of a multivariate nonlinear AR model with an ARCH term. Statist. Probab. Lett. 51 121-130.
31. MEYN, S. P. and TWEEDIE, R. L. (1993). Markov Chains and Stochastic Stability. Springer, New York.
32. RABEMANAJARA, R. and ZAKOIAN, J. M. (1993). Threshold ARCH models and asymmetries in volatility. J. Appl. Econometrics 8 31-49.
33. TONG, H. (1990). Non-Linear Time Series Analysis: A Dynamical System Approach. Oxford Univ. Press.
34. ZAKOIAN, J.-M. (1994). Threshold heteroskedastic models. J. Econom. Dynam. and Control 18 931-955.