Asymptotic properties of the maximum likelihood estimator in autoregressive models with Markov regime

Asymptotic properties of the maximum likelihood estimator in autoregressive models with Markov regime

Randal Douc, Éric Moulines, and Tobias Rydén

Source: Ann. Statist. Volume 32, Number 5 (2004), 2254-2304.

Abstract

An autoregressive process with Markov regime is an autoregressive process for which the regression function at each time point is given by a nonobservable Markov chain. In this paper we consider the asymptotic properties of the maximum likelihood estimator in a possibly nonstationary process of this kind for which the hidden state space is compact but not necessarily finite. Consistency and asymptotic normality are shown to follow from uniform exponential forgetting of the initial distribution for the hidden Markov chain conditional on the observations.

Primary Subjects: 62M09

Secondary Subjects: 62F12

Keywords: Asymptotic normality; autoregressive process; consistency; geometric ergodicity; hidden Markov model; identifiability; maximum likelihood; switching autoregression

Full-text: Open access

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Permanent link to this document: http://projecteuclid.org/euclid.aos/1098883789
Digital Object Identifier: doi:10.1214/009053604000000021
Mathematical Reviews number (MathSciNet): MR2102510
Zentralblatt MATH identifier: 1056.62028

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References

Bakry, D., Milhaud, X. and Vandekerkhove, P. (1997). Statistics of hidden Markov chains with finite state space. The nonstationary case. C. R. Acad. Sci. Paris Sér. I Math. 325 203--206. (In French.)

Mathematical Reviews (MathSciNet): MR1467078

Digital Object Identifier: doi:10.1016/S0764-4442(97)84600-9

Bar-Shalom, Y. and Li, X.-R. (1993). Estimation and Tracking. Principles, Techniques, and Software. Artech House, Boston.

Mathematical Reviews (MathSciNet): MR1262124

Zentralblatt MATH: 0853.62076

Baum, L. and Petrie, T. (1966). Statistical inference for probabilistic functions of finite state Markov chains. Ann. Math. Statist. 37 1554--1563.

Mathematical Reviews (MathSciNet): MR202264

Digital Object Identifier: doi:10.1214/aoms/1177699147

Project Euclid: euclid.aoms/1177699147

Bickel, P. and Ritov, Y. (1996). Inference in hidden Markov models. I. Local asymptotic normality in the stationary case. Bernoulli 2 199--228.

Mathematical Reviews (MathSciNet): MR1416863

Digital Object Identifier: doi:10.2307/3318520

Project Euclid: euclid.bj/1178291719

Bickel, P., Ritov, Y. and Rydén, T. (1998). Asymptotic normality of the maximum likelihood estimator for general hidden Markov models. Ann. Statist. 26 1614--1635.

Mathematical Reviews (MathSciNet): MR1647705

Digital Object Identifier: doi:10.1214/aos/1024691255

Project Euclid: euclid.aos/1024691255

Booth, J. G. and Hobert, J. P. (1999). Maximizing generalized linear mixed model likelihoods with an automated Monte Carlo EM algorithm. J. R. Stat. Soc. Ser. B Stat. Methodol. 61 265--285.

Booth, J. G., Hobert, J. P. and Jank, W. (2001). A survey of Monte Carlo algorithms for maximizing the likelihood of a two-stage hierarchical model. Stat. Modelling 1 333--349.

Cappé, O. (2001). Recursive computation of smoothed functionals of hidden Markovian processes using a particle approximation. Monte Carlo Methods Appl. 7 81--92.

Mathematical Reviews (MathSciNet): MR1828199

Chan, K. and Ledolter, J. (1995). Monte Carlo EM estimation for time series models involving counts. J. Amer. Statist. Assoc. 90 242--252.

Mathematical Reviews (MathSciNet): MR1325132

Digital Object Identifier: doi:10.2307/2291149

JSTOR: links.jstor.org

Chib, S., Nardari, F. and Shephard, N. (2002). Markov chain Monte Carlo methods for stochastic volatility models. J. Econometrics 108 281--316.

Mathematical Reviews (MathSciNet): MR1894758

Digital Object Identifier: doi:10.1016/S0304-4076(01)00137-3

Churchill, G. A. (1989). Stochastic models for heterogeneous DNA sequences. Bull. Math. Biol. 51 79--94.

Mathematical Reviews (MathSciNet): MR978904

Digital Object Identifier: doi:10.1016/S0092-8240(89)80049-7

de Jong, P. and Shephard, N. (1995). The simulation smoother for time series models. Biometrika 82 339--350.

Mathematical Reviews (MathSciNet): MR1354233

Zentralblatt MATH: 0823.62072

Digital Object Identifier: doi:10.1093/biomet/82.2.339

JSTOR: links.jstor.org

Del Moral, P. and Guionnet, A. (2001). On the stability of interacting processes with applications to filtering and genetic algorithms. Ann. Inst. H. Poincaré Probab. Statist. 37 155--194.

Mathematical Reviews (MathSciNet): MR1819122

Digital Object Identifier: doi:10.1016/S0246-0203(00)01064-5

Del Moral, P. and Miclo, L. (2000). Branching and interacting particle systems approximations of Feynman--Kac formulae with applications to non-linear filtering. Séminaire de Probabilités XXXIV. Lecture Notes in Math. 1729 1--145. Springer, Berlin.

Mathematical Reviews (MathSciNet): MR1768060

Del Moral, P. and Miclo, L. (2001). Particle approximations of Lyapunov exponents connected to Schrödinger operators and Feynman--Kac semigroups. Technical report, Univ. Toulouse III.

Douc, R. and Matias, C. (2001). Asymptotics of the maximum likelihood estimator for general hidden Markov models. Bernoulli 7 381--420.

Mathematical Reviews (MathSciNet): MR1836737

Project Euclid: euclid.bj/1080004757

Digital Object Identifier: doi:10.2307/3318493

Doucet, A., de Freitas, N. and Gordon, N. (2001). An introduction to sequential Monte Carlo methods. In Sequential Monte Carlo Methods in Practice (A. Doucet, N. de Freitas and N. Gordon, eds.) 3--14. Springer, New York.

Mathematical Reviews (MathSciNet): MR1847784

Zentralblatt MATH: 1056.93576

Doucet, A., Logothetis, A. and Krishnamurthy, V. (2000). Stochastic sampling algorithms for state estimation of jump Markov linear systems. IEEE Trans. Automat. Control 45 188--202.

Mathematical Reviews (MathSciNet): MR1756516

Digital Object Identifier: doi:10.1109/9.839943

Durrett, R. (1996). Probability: Theory and Examples, 2nd ed. Duxbury, Belmont, CA.

Mathematical Reviews (MathSciNet): MR1609153

Fort, G. and Moulines, E. (2003). Convergence of the Monte Carlo expectation maximization for curved exponential families. Ann. Statist. 31 1220--1259.

Mathematical Reviews (MathSciNet): MR2001649

Digital Object Identifier: doi:10.1214/aos/1059655912

Project Euclid: euclid.aos/1059655912

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

Mathematical Reviews (MathSciNet): MR1708120

Digital Object Identifier: doi:10.1080/02331889808802659

Francq, C. and Zakoian, J. M. (2001). Stationarity of multivariate Markov-switching ARMA models. J. Econometrics 102 339--364.

Mathematical Reviews (MathSciNet): MR1842246

Digital Object Identifier: doi:10.1016/S0304-4076(01)00057-4

Fredkin, D. R. and Rice, J. A. (1987). Correlation functions of a function of a finite-state Markov process with application to channel kinetics. Math. Biosci. 87 161--172.

Mathematical Reviews (MathSciNet): MR929996

Digital Object Identifier: doi:10.1016/0025-5564(87)90072-1

Geyer, G. J. (1994). On the convergence of Monte Carlo maximum likelihood computations. J. Roy. Statist. Soc. Ser. B 56 261--274.

Mathematical Reviews (MathSciNet): MR1257812

JSTOR: links.jstor.org

Geyer, G. J. and Thompson, E. A. (1992). Constrained Monte Carlo maximum likelihood for dependent data (with discussion). J. Roy. Statist. Soc. Ser. B 54 657--699.

Mathematical Reviews (MathSciNet): MR1185217

JSTOR: links.jstor.org

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

Mathematical Reviews (MathSciNet): MR996941

Digital Object Identifier: doi:10.2307/1912559

JSTOR: links.jstor.org

Hamilton, J. D. (1990). Analysis of time series subject to changes in regime. J. Econometrics 45 39--70.

Mathematical Reviews (MathSciNet): MR1067230

Digital Object Identifier: doi:10.1016/0304-4076(90)90093-9

Holst, U., Lindgren, G., Holst, J. and Thuvesholmen, M. (1994). Recursive estimation in switching autoregressions with Markov regime. J. Time Series Anal. 15 489--506.

Mathematical Reviews (MathSciNet): MR1292163

Digital Object Identifier: doi:10.1111/j.1467-9892.1994.tb00206.x

Hürzeler, M. and Künsch, H. R. (2001). Approximating and maximising the likelihood for a general state space model. In Sequential Monte Carlo Methods in Practice (A. Doucet, N. de Freitas and N. Gordon, eds.) 159--175. Springer, New York.

Mathematical Reviews (MathSciNet): MR1847791

Jensen, J. L. and Petersen, N. V. (1999). Asymptotic normality of the maximum likelihood estimator in state space models. Ann. Statist. 27 514--535.

Mathematical Reviews (MathSciNet): MR1714719

Digital Object Identifier: doi:10.1214/aos/1018031205

Project Euclid: euclid.aos/1018031205

Juang, B.-H. and Rabiner, L. R. (1991). Hidden Markov models for speech recognition. Technometrics 33 251--272.

Mathematical Reviews (MathSciNet): MR1132665

Digital Object Identifier: doi:10.2307/1268779

JSTOR: links.jstor.org

Kim, C. and Nelson, C. (1999). State-Space Models with Regime Switching: Classical and Gibbs-Sampling Approaches with Applications. MIT Press.

Krishnamurthy, V. and Rydén, T. (1998). Consistent estimation of linear and non-linear autoregressive models with Markov regime. J. Time Series Anal. 19 291--307.

Mathematical Reviews (MathSciNet): MR1628184

Digital Object Identifier: doi:10.1111/1467-9892.00093

Krolzig, H.-M. (1997). Markov-Switching Vector Autoregressions. Modelling, Statistical Inference, and Application to Business Cycle Analysis. Lecture Notes in Econom. and Math. Systems 454. Springer, Berlin.

Mathematical Reviews (MathSciNet): MR1473720

Zentralblatt MATH: 0879.62107

Künsch, H. R. (2001). State space and hidden Markov models. In Complex Stochastic Systems (O. E. Barndorff-Nielsen, D. R. Cox and C. Klüppelberg, eds.) 109--173. Chapman and Hall/CRC Press, Boca Raton, FL.

Mathematical Reviews (MathSciNet): MR1893412

Le Gland, F. and Mevel, L. (2000). Exponential forgetting and geometric ergodicity in hidden Markov models. Math. Control Signals Systems 13 63--93.

Mathematical Reviews (MathSciNet): MR1742140

Digital Object Identifier: doi:10.1007/PL00009861

Leroux, B. G. (1992). Maximum likelihood estimation for hidden Markov models. Stochastic Process. Appl. 40 127--143.

Mathematical Reviews (MathSciNet): MR1145463

Digital Object Identifier: doi:10.1016/0304-4149(92)90141-C

Lindvall, T. (1992). Lectures on the Coupling Method. Wiley, New York.

Mathematical Reviews (MathSciNet): MR1180522

Zentralblatt MATH: 0850.60019

Louis, T. A. (1982). Finding the observed information matrix when using the EM algorithm. J. Roy. Statist. Soc. Ser. B 44 226--233.

Mathematical Reviews (MathSciNet): MR676213

JSTOR: links.jstor.org

MacDonald, I. L. and Zucchini, W. (1997). Hidden Markov and Other Models for Discrete-Valued Time Series. Chapman and Hall, London.

Mathematical Reviews (MathSciNet): MR1692202

Zentralblatt MATH: 0868.60036

Mann, H. B. and Wald, A. (1943). On the statistical treatment of linear stochastic difference equations. Econometrica 11 173--220.

Mathematical Reviews (MathSciNet): MR9291

Digital Object Identifier: doi:10.2307/1905674

JSTOR: links.jstor.org

Mevel, L. (1997). Statistique asymptotique pour les modèles de Markov cachés. Ph.D. thesis, Univ. Rennes 1.

Meyn, S. P. and Tweedie, R. L. (1993). Markov Chains and Stochastic Stability. Springer, Berlin.

Mathematical Reviews (MathSciNet): MR1287609

Zentralblatt MATH: 0925.60001

Ng, W., Larocque, J. and Reilly, J. (2001). On the implementation of particle filters for DOA tracking. In Proc. IEEE International Conference on Acoustics, Speech and Signal Processing 5 2821--2824. IEEE, Washington.

Nielsen, S. F. (2000). The stochastic EM algorithm: Estimation and asymptotic results. Bernoulli 6 457--489.

Mathematical Reviews (MathSciNet): MR1762556

Digital Object Identifier: doi:10.2307/3318671

Project Euclid: euclid.bj/1081616701

Orton, M. and Fitzgerald, W. (2002). A Bayesian approach to tracking multiple targets using sensor arrays and particle filters. IEEE Trans. Signal Process. 50 216--223.

Mathematical Reviews (MathSciNet): MR1946389

Digital Object Identifier: doi:10.1109/78.978377

Pitt, M. K. (2002). Smooth particle filters for likelihood evaluation and maximisation. Technical Report 651, Dept. Economics, Univ. Warwick.

Shiryaev, A. N. (1996). Probability, 2nd ed. Springer, New York.

Mathematical Reviews (MathSciNet): MR1368405

Susmel, R. (2000). Switching volatility in private international equity markets. Internat. J. Finance Economics 5 265--283.

Tanner, M. A. (1996). Tools for Statistical Inference. Methods for the Exploration of Posterior Distributions and Likelihood Functions, 3rd ed. Springer, New York.

Mathematical Reviews (MathSciNet): MR1396311

Zentralblatt MATH: 0846.62001

Thorisson, H. (2000). Coupling, Stationarity and Regeneration. Springer, New York.

Mathematical Reviews (MathSciNet): MR1741181

Zentralblatt MATH: 0949.60007

Tugnait, J. K. (1982). Adaptive estimation and identification for discrete systems with Markov jump parameters. IEEE Trans. Automat. Control 27 1054--1065. [Correction (1984) 29 286.]

Mathematical Reviews (MathSciNet): MR696299

Digital Object Identifier: doi:10.1109/TAC.1982.1103061

Wald, A. (1949). Note on the consistency of the maximum likelihood estimate. Ann. Math. Statist. 20 595--601.

Mathematical Reviews (MathSciNet): MR32169

Digital Object Identifier: doi:10.1214/aoms/1177729952

Project Euclid: euclid.aoms/1177729952

Yao, J.-F. and Attali, J.-G. (2000). On stability of nonlinear AR processes with Markov switching. Adv. in Appl. Probab. 32 394--407.

Mathematical Reviews (MathSciNet): MR1778571

Digital Object Identifier: doi:10.1239/aap/1013540170

Project Euclid: euclid.aap/1013540170

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