Asymptotic properties of autoregressive regime-switching models

Madalina Olteanu; Joseph Rynkiewicz

ESAIM: Probability and Statistics (2012)

  • Volume: 16, page 25-47
  • ISSN: 1292-8100

Abstract

top
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.

How to cite

top

Olteanu, Madalina, and Rynkiewicz, Joseph. "Asymptotic properties of autoregressive regime-switching models." ESAIM: Probability and Statistics 16 (2012): 25-47. <http://eudml.org/doc/273607>.

@article{Olteanu2012,
abstract = {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.},
author = {Olteanu, Madalina, Rynkiewicz, Joseph},
journal = {ESAIM: Probability and Statistics},
keywords = {likelihood ratio test; switching times series; hidden Markov model},
language = {eng},
pages = {25-47},
publisher = {EDP-Sciences},
title = {Asymptotic properties of autoregressive regime-switching models},
url = {http://eudml.org/doc/273607},
volume = {16},
year = {2012},
}

TY - JOUR
AU - Olteanu, Madalina
AU - Rynkiewicz, Joseph
TI - Asymptotic properties of autoregressive regime-switching models
JO - ESAIM: Probability and Statistics
PY - 2012
PB - EDP-Sciences
VL - 16
SP - 25
EP - 47
AB - 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.
LA - eng
KW - likelihood ratio test; switching times series; hidden Markov model
UR - http://eudml.org/doc/273607
ER -

References

top
  1. [1] R.C. Bradley, Basic properties of strong mixing conditions. A survey and some open questions. Probab. Surveys 2 (2005) 107–144. Zbl1189.60077MR2178042
  2. [2] D. Dacunha-Castelle and E. Gassiat, The estimation of the order of a mixture model. Bernoulli3 (1997) 279–299. Zbl0889.62012MR1468306
  3. [3] D. Dacunha-Castelle and E. Gassiat, Testing in locally conic models and application to mixture models. ESAIM : PS 1 (1997) 285–317. Zbl1007.62507MR1468112
  4. [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. Zbl0957.62073MR1740115
  5. [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. Zbl1056.62028MR2102510
  6. [6] P. Doukhan, Mixing : properties and examples. Springer-Verlag, New York. Lect. Notes in Stat. 85 (1994). Zbl0801.60027MR1312160
  7. [7] P. Doukhan, P. Massart and E. Rio, Invariance principles for absolutely regular empirical processes. Ann. Inst. Henri Poincaré31 (1995) 393–427. Zbl0817.60028MR1324814
  8. [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. [9] C. Francq and M. Roussignol, Ergodicity of autoregressive processes with Markov-switching and consistency of the maximum likelihood estimator. Statistics32 (1998) 151–173. Zbl0954.62104MR1708120
  10. [10] K. Fukumizu, Likelihood ratio of unidentifiable models and multilayer neural networks. Ann. Stat.31 (2003) 833–851. Zbl1032.62020MR1994732
  11. [11] R. Garcia, Asymptotic null distribution of the likelihood ratio test in Markov switching models. Internat. Econ. Rev.39 (1998) 763–788. MR1638204
  12. [12] E. Gassiat, Likelihood ratio inequalities with applications to various mixtures. Ann. Inst. Henri Poincaré38 (2002) 897–906. Zbl1011.62025MR1955343
  13. [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. Zbl0982.62016MR1780964
  14. [14] J.D. Hamilton, A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica57 (1989) 357–384. Zbl0685.62092MR996941
  15. [15] J.D. Hamilton, Analysis of time series subject to changes in regime. J. Econom.64 (1990) 307–333. Zbl0825.62950MR1067230
  16. [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. [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. [18] B.E. Hansen, Inference when a nuisance parameter is not identified under the null hypothesis. Econometrica64 (1996) 413–430. Zbl0862.62090MR1375740
  19. [19] J. Henna, On estimating the number of constituents of a finite mixture of continuous distributions. Ann. Inst. Statist. Math.37 (1985) 235–240. Zbl0577.62031MR799237
  20. [20] A.J. Izenman and C. Sommer, Philatelic mixtures and multivariate densities. J. Am. Stat. Assoc.83 (1988) 941–953. 
  21. [21] C. Keribin, Consistent estimation of the order of mixture models. Sankhya : The Indian Journal of Statistics 62 (2000) 49–66. Zbl1081.62516MR1769735
  22. [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. Zbl0906.62088MR1628184
  23. [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. [24] B.G. Leroux, Maximum penalized likelihood estimation for independent and Markov-dependent mixture models. Biometrics48 (1992) 545–558. 
  25. [25] B.G. Leroux, Consistent estimation of a mixing distribution. Ann. Stat.20 (1992) 1350–1360. Zbl0763.62015MR1186253
  26. [26] B.G. Lindsay, Moment matrices : application in mixtures. Ann. Stat.17 (1983) 722–740. Zbl0672.62063MR994263
  27. [27] X. Liu and Y. Shao, Asymptotics for likelihood ratio tests under loss of identifiability. Ann. Stat.31 (2003) 807–832. Zbl1032.62014MR1994731
  28. [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. Zbl1320.62194MR2460859
  29. [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. Zbl0798.62004MR1294074
  30. [30] T. Ryden, Estimating the order of hidden Markov models. Statistics26 (1995) 345–354. Zbl0836.62057MR1365683
  31. [31] G.W. Schwert, Business cycles, financial crises and stock volatility. Carnegie-Rochester Conf. Ser. Public Policy31 (1989) 83–125. 
  32. [32] A.W. Van der Vaart, Asymptotic Statistics. Cambridge University Press (2000). Zbl0910.62001MR1652247
  33. [33] C.S. Wong and W.K. Li, On a mixture autoregressive model. J. R Stat. Soc. Ser. B62 (2000) 95–115. Zbl0941.62095MR1747398
  34. [34] J.F. Yao and J.G. Attali, On stability of nonlinear AR processes with Markov switching. Adv. Appl. Probab.32 (2000) 394–407. Zbl0961.60076MR1778571

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.