Asymptotic properties of autoregressive regime-switching models
ESAIM: Probability and Statistics, Tome 16 (2012), pp. 25-47.

The statistical properties of the likelihood ratio test statistic (LRTS) for autoregressive regime-switching models are addressed in this paper. This question is particularly important for estimating the number of regimes in the model. Our purpose is to extend the existing results for mixtures [X. Liu and Y. Shao, Ann. Stat. 31 (2003) 807-832] and hidden Markov chains [E. Gassiat, Ann. Inst. Henri Poincaré 38 (2002) 897-906]. First, we study the case of mixtures of autoregressive models (i.e. independent regime switches). In this framework, we give sufficient conditions to keep the LRTS tight and compute its the asymptotic distribution. Second, we consider the extension of the ideas in Gassiat [Ann. Inst. Henri Poincaré 38 (2002) 897-906] to autoregressive models with regimes switches according to a Markov chain. In this case, it is shown that the marginal likelihood is no longer a contrast function and cannot be used to select the number of regimes. Some numerical examples illustrate the results and their convergence properties.

DOI : 10.1051/ps/2011153
Classification : 62M10, 62F5, 62F12
Mots clés : likelihood ratio test, switching times series, hidden Markov model
@article{PS_2012__16__25_0,
     author = {Olteanu, Madalina and Rynkiewicz, Joseph},
     title = {Asymptotic properties of autoregressive regime-switching models},
     journal = {ESAIM: Probability and Statistics},
     pages = {25--47},
     publisher = {EDP-Sciences},
     volume = {16},
     year = {2012},
     doi = {10.1051/ps/2011153},
     mrnumber = {2911020},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ps/2011153/}
}
TY  - JOUR
AU  - Olteanu, Madalina
AU  - Rynkiewicz, Joseph
TI  - Asymptotic properties of autoregressive regime-switching models
JO  - ESAIM: Probability and Statistics
PY  - 2012
SP  - 25
EP  - 47
VL  - 16
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/ps/2011153/
DO  - 10.1051/ps/2011153
LA  - en
ID  - PS_2012__16__25_0
ER  - 
%0 Journal Article
%A Olteanu, Madalina
%A Rynkiewicz, Joseph
%T Asymptotic properties of autoregressive regime-switching models
%J ESAIM: Probability and Statistics
%D 2012
%P 25-47
%V 16
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/ps/2011153/
%R 10.1051/ps/2011153
%G en
%F PS_2012__16__25_0
Olteanu, Madalina; Rynkiewicz, Joseph. Asymptotic properties of autoregressive regime-switching models. ESAIM: Probability and Statistics, Tome 16 (2012), pp. 25-47. doi : 10.1051/ps/2011153. http://www.numdam.org/articles/10.1051/ps/2011153/

[1] R.C. Bradley, Basic properties of strong mixing conditions. A survey and some open questions. Probab. Surveys 2 (2005) 107-144. | MR | Zbl

[2] D. Dacunha-Castelle and E. Gassiat, The estimation of the order of a mixture model. Bernoulli 3 (1997) 279-299. | MR | Zbl

[3] D. Dacunha-Castelle and E. Gassiat, Testing in locally conic models and application to mixture models. ESAIM : PS 1 (1997) 285-317. | Numdam | MR | Zbl

[4] D. Dacunha-Castelle and E. Gassiat, Testing the order of a model using locally conic parametrization : population mixtures and stationary ARMA processes. Ann. Stat. 27 (1999) 1178-1209. | MR | Zbl

[5] R. Douc, E. Moulines and T. Rydén, Asymptotic properties of the maximum likelihood estimator in autoregressive models with Markov regime. Ann. Stat. 32 (2004) 2254-2304. | MR | Zbl

[6] P. Doukhan, Mixing : properties and examples. Springer-Verlag, New York. Lect. Notes in Stat. 85 (1994). | MR | Zbl

[7] P. Doukhan, P. Massart and E. Rio, Invariance principles for absolutely regular empirical processes. Ann. Inst. Henri Poincaré 31 (1995) 393-427. | Numdam | MR | Zbl

[8] Ch. Engel and J.D. Hamilton, Long swings in the dollar : are they in the data and do markets know it? Am. Econ. Rev. 80 (1990) 689-713.

[9] C. Francq and M. Roussignol, Ergodicity of autoregressive processes with Markov-switching and consistency of the maximum likelihood estimator. Statistics 32 (1998) 151-173. | MR | Zbl

[10] K. Fukumizu, Likelihood ratio of unidentifiable models and multilayer neural networks. Ann. Stat. 31 (2003) 833-851. | MR | Zbl

[11] R. Garcia, Asymptotic null distribution of the likelihood ratio test in Markov switching models. Internat. Econ. Rev. 39 (1998) 763-788. | MR

[12] E. Gassiat, Likelihood ratio inequalities with applications to various mixtures. Ann. Inst. Henri Poincaré 38 (2002) 897-906. | Numdam | MR | Zbl

[13] E. Gassiat and C. Keribin, The likelihood ratio test for the number of components in a mixture with Markov regime. ESAIM : PS 4 (2000) 25-52. | Numdam | MR | Zbl

[14] J.D. Hamilton, A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica 57 (1989) 357-384. | MR | Zbl

[15] J.D. Hamilton, Analysis of time series subject to changes in regime. J. Econom. 64 (1990) 307-333. | MR | Zbl

[16] B.E. Hansen, The likelihood ratio test under nonstandard conditions : testing the Markov switching model of GNP. J. Appl. Econom. 7 (1992) 61-82.

[17] B.E. Hansen, Erratum : The likelihood ratio test under nonstandard conditions : testing the Markov switching model of GNP. J. Appl. Econom. 11 (1996) 195-198.

[18] B.E. Hansen, Inference when a nuisance parameter is not identified under the null hypothesis. Econometrica 64 (1996) 413-430. | MR | Zbl

[19] J. Henna, On estimating the number of constituents of a finite mixture of continuous distributions. Ann. Inst. Statist. Math. 37 (1985) 235-240. | MR | Zbl

[20] A.J. Izenman and C. Sommer, Philatelic mixtures and multivariate densities. J. Am. Stat. Assoc. 83 (1988) 941-953.

[21] C. Keribin, Consistent estimation of the order of mixture models. Sankhya : The Indian Journal of Statistics 62 (2000) 49-66. | MR | Zbl

[22] V. Krishnamurthy and T. Rydén, Consistent estimation of linear and non-linear autoregressive models with Markov regime. J. Time Ser. Anal. 19 (1998) 291-307. | MR | Zbl

[23] P.-S. Lam, The Hamilton model with a general autoregressive component : estimation and comparison with other models of economic time series. J. Monet. Econ. 26 (1990) 409-432.

[24] B.G. Leroux, Maximum penalized likelihood estimation for independent and Markov-dependent mixture models. Biometrics 48 (1992) 545-558.

[25] B.G. Leroux, Consistent estimation of a mixing distribution. Ann. Stat. 20 (1992) 1350-1360. | MR | Zbl

[26] B.G. Lindsay, Moment matrices : application in mixtures. Ann. Stat. 17 (1983) 722-740. | MR | Zbl

[27] X. Liu and Y. Shao, Asymptotics for likelihood ratio tests under loss of identifiability. Ann. Stat. 31 (2003) 807-832. | MR | Zbl

[28] R. Rios and L.A. Rodriguez, Penalized estimate of the number of states in Gaussian linear AR with Markov regime. Electron. J. Stat. 2 (2008) 1111-1128. | MR

[29] K. Roeder, A graphical technique for determining the number of components in a mixture of normals. J. Am. Stat. Assoc. 89 (1994) 487-495. | MR | Zbl

[30] T. Ryden, Estimating the order of hidden Markov models. Statistics 26 (1995) 345-354. | MR | Zbl

[31] G.W. Schwert, Business cycles, financial crises and stock volatility. Carnegie-Rochester Conf. Ser. Public Policy 31 (1989) 83-125.

[32] A.W. Van Der Vaart, Asymptotic Statistics. Cambridge University Press (2000). | MR | Zbl

[33] C.S. Wong and W.K. Li, On a mixture autoregressive model. J. R Stat. Soc. Ser. B 62 (2000) 95-115. | MR | Zbl

[34] J.F. Yao and J.G. Attali, On stability of nonlinear AR processes with Markov switching. Adv. Appl. Probab. 32 (2000) 394-407. | MR | Zbl

Cité par Sources :