Estimation and testing stationarity for double-autoregressive models

Journal of the Royal Statistical Society. Series B: Statistical Methodology

Volumn 66, Issue 1, 2004, Pages 63-78

  Ling, Shiqing ^a  

^a HONG KONG UNIVERSITY OF SCIENCE AND TECHNOLOGY   (Hong Kong)

Author keywords

Asymptotic normality; Brownian motion; Consistency; Double autoregressive model; Lagrange multiplier test; Maximum likelihood estimator; Stationarity

Indexed keywords

EID: 1042302548     PISSN: 13697412     EISSN: None     Source Type: Journal    
DOI: 10.1111/j.1467-9868.2004.00432.x     Document Type: Article

References (41)

1. Amemiya, T. (1985) Advanced Econometrics. Cambridge: Harvard University Press.
2. Billingsley, P. (1968) Convergence of Probability Measures. New York: Wiley.
3. Borkovec, M. (2001) Asymptotic behavior of the sample autocovariance and autocorrelation function of the AR(1) process with ARCH(1). Bernoulli, 6, 847-872.
4. Borkovec, M. and Kluppelberg, C. (1998) The tail of the stationary distribution of an autoregressive process with ARCH(1) errors. Ann. Appl. Probab., 11, 1220-1241.
5. Bougerol, P. and Picard, N. (1992) Strict stationarity of generalized autoregressive processes. Ann. Probab., 20, 1714-1730.
6. Chan, N. H. and Tran, L. T. (1989) On the first-order autoregressive process with infinite variance. Econometr. Theory, 5, 354-362.
7. Davis, R. A., Knight, K. and Liu, J. (1992) M-estimation for autoregressions with infinite variance. Stoch. Processes Appl. 40, 145-180.
8. Dickey, D. A. and Fuller, W. A. (1979) Distribution of the estimators for autoregressive time series with a unit root. J. Am. Statist. Ass., 74, 427-431.
9. Engle, R. F. (1982) Autoregressive conditional heteroskedasticity with estimates of variance of U. K. inflation. Econometrica, 50, 987-1008.
10. Feigin, P. D. and Tweedie, R. L. (1985) Random coefficient autoregressive processes: a Markov chain analysis of stationarity and finiteness of moments. J. Time Ser. Anal., 6, 1-14.
11. Francq, C. and Zakol̈an, J. M. (1998) Estimating linear representations of nonlinear processes. J. Statist. Planng Inf., 68, 145-165.
12. Francq, C. and Zakol̈an, J. M. (2000) Covariance matrix estimation for estimators of mixing weak ARMA models. J. Statist. Planng Inf., 83, 369-394.
13. Guégan, D. and Diebolt, J. (1994) Probabilistic properties of the β-ARCH-model. Statist. Sin., 4, 71-87.
14. Hall, P. and Heyde, C. C. (1980) Martingale Limit Theory and Its Applications. San Diego: Academic Press.
15. Hamilton, J. D. (1994) Time Series Analysis. Princeton: Princeton University Press.
16. Jeganathan, P. (1988) On the strong approximation of the distributions of estimators in linear stochastic models: I, II, Stationary and explosive AR models. Ann. Statist., 16, 1283-1314.
17. Kluppelberg, C., Mailer, R. A., Van De Vyver, M. and Wee, D. (2002) Testing for reduction to random walk in autoregressive conditional heteroscedasticity models. Econometr. J., 5, 387-416.
18. Koul, H. L. and Pflug, G. (1990) Weakly adaptive estimators in explosive autoregression. Ann. Statist., 18, 939-960.
19. Kurtz, T. G. and Protter, P. (1991) Weak limit theorems to stochastic integrals and stochastic differential equations. Ann. Probab., 19, 1035-1070.
20. Lee, S.-W and Hansen, B. E. (1994) Asymptotic theory for GARCH(1,1) quasi-maximum likelihood estimator. Econometr. Theory, 10, 29-52.
21. Li, W. K. (1992) On the asymptotic standard errors of residual autocorrelations in nonlinear time series modeling. Biometrika, 79, 435-437.
22. Li, W. K., Ling, S. and Wong, H. (2001) Estimation for partially nonstationary multivariate autoregressive models with conditional heteroscedasticity. Biometrika, 88, 1135-1152.
23. Li, W. K. and Mak, T. K. (1994) On the squared residual autocorrelations in non-linear time series with conditional heteroskedasticity. J. Time Ser. Anal., 15, 627-636.
24. Ling, S. (1999) On probability properties of a double threshold ARMA conditional heteroskedasticity model. J. Appl. Probab., 36, 688-705.
25. Ling, S. (2003) A double AR(p) model: structure and estimation. Working Paper. Department of Mathematics, Hong Kong University of Science and Technology, Hong Kong.
26. Ling, S. and Li, W. K. (1997) On fractionally integrated autoregressive moving-average time series models with conditional heteroskedasticity. J. Am. Statist. Ass., 92, 1184-1194.
27. Ling, S. and Li, W. K. (1998) Limiting distributions of maximum likelihood estimators for unstable ARMA models with GARCH errors. Ann. Statist., 26, 84-125.
28. Ling, S. and Li, W. K. (2003) Asymptotic inference for unit root processes with GARCH (1, 1) errors. Econometr. Theory, 19, 541-564.
29. Ling, S. and McAleer, M. (2003) Asymptotic theory for a new vector ARMA-GARCH model. Econometr. Theory, 19, 280-310.
30. Lumsdaine, R. (1996) Consistency and asymptotic normality of the quasi-maximum likelihood estimator in IGARCH(1,1) and covariance stationary GARCH(1,1) models. Econometrica, 3, 575-596.
31. Mikosch, T., Gadrich, T., Kluppelberg, C. and Adler, R. J. (1995) Parameter estimation for ARMA models with infinite variance innovations. Ann. Statist., 23, 305-326.
32. Nicholls, D. F. and Quinn, B. G. (1982) Random coefficient autoregressive models: an introduction. Lect. Notes Statist., 11.
33. Phillips, P. C. B. (1987) Time series regression with a unit root. Econometrica, 55, 277-301.
34. Phillips, P. C. B. (1989) Partially identified econometric models. Econometr. Theory, 5, 181-240.
35. Robinson, P. M. (1994) Efficient tests of nonstationary hypotheses. J. Am. Statist. Ass., 89, 1420-1437.
36. Tjøstheim, D. (1986) Estimation in nonlinear time series models. Stoch. Processes Appl., 21, 251-273.
37. Tjøstheim, D. (1990) Nonlinear time series and Markov chains. Adv. Appl. Probab., 22, 587-611.
38. Tong, H. (1990) Nonlinear Time Series: a Dynamical System Approach. Oxford: Clarendon.
39. Tsay, R. S. (1987) Conditional heteroscedastic time series models. J. Am. Statist. Ass., 82, 590-604.
40. Weiss, A. A. (1986) Asymptotic theory for ARCH models: estimation and testing. Econometr. Theory, 2, 107-131.
41. Wong, H. and Li, W. K. (1997) On a multivariate conditional heteroscedastic model. Biometrika, 84, 111-123.