Advertisement

An Introduction to Regime Switching Time Series Models

  • Theis Lange
  • Anders Rahbek
Chapter

Abstract

A survey is given on regime switching in econometric time series modelling. Numerous references to applied as well as methodological literature are presented. A distinction between observation switching (OS) and Markov switching (MS) models is suggested, where in OS models, the switching probabilities depend on functions of lagged observations. In contrast, in MS models the switching is a latent unobserved exogenous process. With an emphasis on OS and MS ARCH and cointegrated models, stationarity and ergodicity properties are discussed as well as likelihood-based estimation, asymptotic theory and hypothesis testing.

Keywords

GARCH Model Markov Switching Nonlinear Time Series Arch Model Regime Switching Model 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Akram, Q. and Nymoen, R. (2006): Econometric modelling of slack and tight labour markets. Economic Modelling 23, 579–596.CrossRefGoogle Scholar
  2. Alexander, C. and Lazar, E. (2006): Normal mixture GARCH(1,1): Applications to exchange rate modelling. Journal of Applied Econometrics 21, 307–336.CrossRefMathSciNetGoogle Scholar
  3. Altissimo, F. and Corradi, V. (2002): Bounds for inference with nuisance parameters present only under the alternative. The Econometrics Journal 5, 494–519.zbMATHCrossRefMathSciNetGoogle Scholar
  4. Amendola, A. and Niglio, M. (2004): Predictor distribution and forecast accuracy of threshold models. Statistical Methods & Applications 13, 3–14.zbMATHCrossRefMathSciNetGoogle Scholar
  5. Andrews, D. (1993): Tests for parameter instability and structural change with unknown change point. Econometrica 61, 821–856.zbMATHCrossRefMathSciNetGoogle Scholar
  6. Aslanidis, N. and Kouretas, G. (2005): Testing for two-regime threshold cointegration in the parallel and official markets for foreign currency in Greece. Economic Modelling 22, 665–682.CrossRefGoogle Scholar
  7. Balke, N. and Fomby, T. (1997): Threshold cointegration. International Economic Review 38, 627–645.zbMATHCrossRefMathSciNetGoogle Scholar
  8. Baum, C. and Karasulu, M. (1998): Modelling Federal Reserve discount policy. Computational Economics 11, 53–70.zbMATHCrossRefGoogle Scholar
  9. Bec, F. and Rahbek, A. (2004): Vector equilibrium correction models with non-linear discontinuous adjustments. The Econometrics Journal 7, 628–651.zbMATHCrossRefMathSciNetGoogle Scholar
  10. Bec, F., Rahbek, A. and Shephard, N. (2005): The ACR Model: A Multivariate Dynamic Mixture Autoregression. Oxford Bulletin of Economics and Statistics forthcoming.Google Scholar
  11. Bec, F., Salem, M.B. and Carrasco, M. (2004): Tests for unit-root versus threshold specification with an application to the purchasing power parity relationship. Journal of Business and Economic Statistics 22, 382–395.CrossRefMathSciNetGoogle Scholar
  12. Bollerslev, T. (1986): Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31, 307–327.zbMATHCrossRefMathSciNetGoogle Scholar
  13. Cai, J. (1994): A Markov model of switching-regime ARCH. Journal of Business and Economic Statistics 12, 309–316.CrossRefGoogle Scholar
  14. Caner, M. and Hansen, B. (2001): Threshold autoregression with a unit root. Econometrica 69, 1555–1596.zbMATHCrossRefMathSciNetGoogle Scholar
  15. Carrasco, M. (2002): Misspecified structural change, threshold, and Markov-switching models. Journal of Econometrics 109, 239–273.zbMATHCrossRefMathSciNetGoogle Scholar
  16. Carrasco, M. and Chen, X. (2002): Mixing and moment properties of various GARCH and stochastic volatility models. Econometric Theory 18, 17–39.zbMATHCrossRefMathSciNetGoogle Scholar
  17. Chan, K. (1990): Percentage points of likelihood ratio tests for threshold autoregression. Journal of the Royal Statistical Society. Series B 53, 691–696.Google Scholar
  18. Chan, K. (1993): Consistency and limiting distribution of the least squares estimator of a threshold autoregressive model. Annals of Statistics 21, 520–533.zbMATHCrossRefMathSciNetGoogle Scholar
  19. Chan, K. and Tong, H. (1990): On likelihood ratio tests for threshold autoregression. Journal of the Royal Statistical Society. Series B 53, 469–476.MathSciNetGoogle Scholar
  20. Chow, Y. (1998): Regime switching and cointegration tests of the efficiency of futures markets. Journal of Futures Markets 18, 871–901.CrossRefGoogle Scholar
  21. Clements, M. and Galvão, A. (2004): Testing the expectations theory of the term structure of interest rates in threshold models. Macroeconomic Dynamics 7, 567–585.Google Scholar
  22. Clements, M. and Krolzig, H. (1998): A comparison of the forecast performance of markov-switching and threshold autoregressive models of US GNP. The Econometrics Journal 1, C47–C75.CrossRefGoogle Scholar
  23. Cline, D. and Pu, H. (2004): Stability and the Lyapounov exponent of threshold AR-ARCH models. Annals of Applied Probability 14, 1920–1949.zbMATHCrossRefMathSciNetGoogle Scholar
  24. Cox, D. (1981): Statistical analysis of time series: Some recent developments. Scandinavian Journal of Statistics 8, 93–115.zbMATHGoogle Scholar
  25. Davidson, J. (2004): Forecasting Markov-switching dynamic, conditionally heteroscedastic processes. Statistics and Probability Letters 68, 137–147.zbMATHCrossRefMathSciNetGoogle Scholar
  26. Davies, R. (1977): Hypothesis testing when a nuisance parameter is present only under the alternative. Biometrika 64, 247–254.zbMATHCrossRefMathSciNetGoogle Scholar
  27. Davies, R. (1987): Hypothesis testing when a nuisance parameter is present only under the alternative. Biometrika 74, 33–43.zbMATHMathSciNetGoogle Scholar
  28. Diebold, F., Lee, J. and Weinbach, G. (1994): Nonstationary Time Series Analysis and Cointegration. Advanced Texts in Econometrics, 283-302. Oxford University Press.Google Scholar
  29. Douc, R., Moulines, E. and Rydén, T. (2004): Asymptotic properties of the maximum likelihood estimator in autoregressive models with Markov regime. Annals of Statistics 32, 2254–2304.zbMATHCrossRefMathSciNetGoogle Scholar
  30. Dueker, M. (1997): Markov switching in GARCH processes and mean-reverting stock-market volatility. Journal of Business and Economic Statistics 15, 26–34.CrossRefGoogle Scholar
  31. Dufrenot, G. and Mignon, V. (2002): Recent Developments in Nonlinear Cointegration with Applications to Macroeconomics and Finance. Kluwer Academic PublishersGoogle Scholar
  32. Dumas, B. (1992): Dynamic equilibrium and the real exchange rate in a spatially seperated world. Review of Financial Studies 5, 153–180.CrossRefGoogle Scholar
  33. Enders, W. and Granger, C. (1998): Unit-root tests and asymmetric adjustment with an example using the term structure of interest rates. Journal of Business and Economic Statistics 16, 304–311.CrossRefGoogle Scholar
  34. Filardo, A. (1994): Business-cycle phases and their transitional dynamics. Journal of Business and Economic Statistics 12, 299–308.CrossRefGoogle Scholar
  35. Filardo, A. and Gordon, S. (1998): Business cycle durations. Journal of Econometrics 85, 99–123.zbMATHCrossRefGoogle Scholar
  36. Fornari, F. and Mele, A. (1997): Sign- and volatility-switching ARCH models: Theory and applications to international stock markets. Journal of Applied Econometrics 12, 49–65.CrossRefGoogle Scholar
  37. Francq, C. and Roussignol, M. (1998): Ergodicity of autoregressive processes with Markov switching and consistency of the maximum-likelihood estimator. Statistics 32, 151–173.zbMATHCrossRefMathSciNetGoogle Scholar
  38. Francq, C., Roussignol, M. and Zakoïan, J. (2001): Conditional heteroskedasticity driven by hidden Markov chains. Journal of Time Series Analysis 22, 197–220.zbMATHCrossRefMathSciNetGoogle Scholar
  39. Francq, C. and Zakoïan, J. (2001): Stationarity of multivariate Markov-switching ARMA models. Journal of Econometrics 102, 339–364.zbMATHCrossRefMathSciNetGoogle Scholar
  40. Francq, C. and Zakoïan, J. (2005): The L 2-structures of standard and switching-regime GARCH models. Stochastic Processes and their Applications 115, 1557–1582.zbMATHCrossRefMathSciNetGoogle Scholar
  41. Franses, P. and van Dijk, D. (2000): Nonlinear Time Series Models in Empirical Finance. Cambridge University Press.Google Scholar
  42. Fuh, C. (2004): On Bahadur efficiency of the maximum likelihood estimator in hidden Markov models. Statistica Sinica 14, 127–155.zbMATHMathSciNetGoogle Scholar
  43. Garcia, R. (1998): Asymptotic null distribution of the likelihood ratio test in Markov switching models. International Economic Review 39, 763–788.CrossRefMathSciNetGoogle Scholar
  44. Giese, J. (2006): Characterising the yield curve’s derivatives in a regime-changing cointegrated VAR model. Working paper, Nuffield College University of Oxford.Google Scholar
  45. Glosten, L., Jaganathan, R. and Runkle, D. (1993): On the relation between the expected value and the volatility of the nominal excess return on stocks. Journal of Finance 48, 1779–1801.CrossRefGoogle Scholar
  46. Gonzalo, J. and Pitarakis, J. (2002): Estimation and model selection based inference in single and multiple threshold models. Journal of Econometrics 110, 319–352.zbMATHCrossRefMathSciNetGoogle Scholar
  47. Gourieroux, C. and Monfort, A. (1992): Qualitative threshold ARCH models. Journal of Econometrics 52, 159–199.zbMATHCrossRefMathSciNetGoogle Scholar
  48. Gouveia, P. and Rodrigues, P. (2004): Threshold cointegration and the PPP hypothesis. Journal of Applied Statistics 31, 115–127.zbMATHCrossRefMathSciNetGoogle Scholar
  49. Gray, S. (1996): Modeling the conditional distribution of interest rates as a regime-switching process. Journal of Financial Economics 42, 27–62.CrossRefGoogle Scholar
  50. Guidolin, M. and Timmermann, A. (2005): Economic implications of bull and bear regimes in UK stock and bond returns. The Economic Journal 115, 111–143.CrossRefGoogle Scholar
  51. Haas, M., Mittnik, S. and Paolella, M. (2004): Mixed normal conditional heteroskedasticity. Journal of Financial Econometrics 2, 211–250.CrossRefGoogle Scholar
  52. Haas, M., Mittnik, S. and Paolella, M. (2004): A new approach to Markov-switching GARCH models. Journal of Financial Econometrics 2, 493–530.CrossRefGoogle Scholar
  53. Hamilton, J. (1989): A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica 57, 357–384.zbMATHCrossRefMathSciNetGoogle Scholar
  54. Hamilton, J. (1990): Analysis of time series subject to changes in regime. Journal of Econometrics 45, 39–70.zbMATHCrossRefMathSciNetGoogle Scholar
  55. Hamilton, J. (1994): Time Series Analysis. Princeton University Press.Google Scholar
  56. Hamilton, J. and Lin, G. (1996): Stock market volatility and the business cycle. Journal of Applied Econometrics 11, 573–593.CrossRefGoogle Scholar
  57. Hamilton, J. and Raj, B. (2002): Advances in Markov-switching Models: Applications in Business Cycle Research and Finance. Physica-Verlag.Google Scholar
  58. Hamilton, J. and Susmel, R. (1994): Autoregressive conditional heteroskedasticity and changes in regime. Journal of Econometrics 64, 307–333.zbMATHCrossRefGoogle Scholar
  59. Hansen, B. (1992): The likelihood ratio test under nonstandard conditions: Testing the Markov switching model of GNP. Journal of Applied Econometrics 7, S61–S82.CrossRefGoogle Scholar
  60. Hansen, B. (1996): Inference when a nuisance parameter is not identified under the null hypothesis. Econometrica 64, 413–430.zbMATHCrossRefMathSciNetGoogle Scholar
  61. Hansen, B. (1997): Inference in TAR models. Studies in Nonlinear Dynamics and Econometrics 2, 1–14.CrossRefMathSciNetGoogle Scholar
  62. Hansen, B. (2000): Sample splitting and threshold estimation. Econometrica 68, 575–603.zbMATHCrossRefMathSciNetGoogle Scholar
  63. Hansen, B. and Seo, B. (2002): Testing for two-regime threshold cointegration in vector error-correction models. Journal of Econometrics 110, 293–318.zbMATHCrossRefMathSciNetGoogle Scholar
  64. Holst, U., Lindgren, G., Holst, J. and Thuvesholmen, M. (1994): Recursive estimation in switching autoregressions with a Markov regime. Journal of Time Series Analysis 15, 489–506.zbMATHCrossRefMathSciNetGoogle Scholar
  65. Johansen, S. (2008): Cointegration: Overview and development. In: Andersen, T.G., Davis, R.A., Kreiss, J.-P. and Mikosch, T. (Eds.): Handbook of Financial Time Series, 671–693. Springer, New York.Google Scholar
  66. Jong, R. de (2002): Nonlinear minimization estimators in the presence of cointegrating relations. Journal of Econometrics 110, 241–259.zbMATHCrossRefMathSciNetGoogle Scholar
  67. Kim, C. (2004): Markov-switching models with endogenous explanatory variables. Journal of Econometrics 122, 127–136.CrossRefMathSciNetGoogle Scholar
  68. Klaassen, F. (2001): Improving GARCH volatility forecasts with regime-switching GARCH. Empirical Economics 27, 363–394.CrossRefGoogle Scholar
  69. Krishnamurthy, V. and Rydén, T. (1998): Consistent estimation of linear and non-linear autoregressive models with Markov regime. Journal of Time Series Analysis 19, 291–307.zbMATHCrossRefMathSciNetGoogle Scholar
  70. Kristensen, D. and Rahbek, A. (2005): Asymptotics of the QMLE for a class of ARCH(q) models. Econometric Theory 21, 946–961.zbMATHCrossRefMathSciNetGoogle Scholar
  71. Kristensen, D. and Rahbek, A. (2007): Likelihood-based inference in Non-linear Error-Correction Models. Working paper, University of Copenhagen and CREATES.Google Scholar
  72. Krolzig, H. (1997): Markov-Switching Vector Autoregressions. Springer, Berlin.zbMATHGoogle Scholar
  73. Krolzig, H., Marcellino, M. and Mizon, G. (2002): A Markov-switching vector equilibrium correction model of the UK labour market. Empirical Economics 27, 233–254.CrossRefGoogle Scholar
  74. Lamoureux, C. and Lastrapes, W. (1990): Persistence in variance, structural change, and the GARCH model. Journal of Business and Economic Statistics 8, 225–234.CrossRefGoogle Scholar
  75. Lanne, M. and Saikkonen, P. (2003): Modeling the U.S. short-term interest rate by mixture autoregressive processes. Journal of Financial Econometrics 1, 96–125.CrossRefGoogle Scholar
  76. Lee, O. (2005): Probabilistic properties of a nonlinear ARMA process with Markov switching. Communications in Statistics: Theory and Methods 34, 193–204.zbMATHCrossRefMathSciNetGoogle Scholar
  77. Li, M. and Lin, H. (2004): Estimating Value-at-Risk via Markov switching ARCH models - an empirical study on stock index returns. Applied Economics Letters 11, 679–691.CrossRefGoogle Scholar
  78. Li, W. and Lam, K. (1995): Modelling asymmetry in stock returns by a threshold autoregressive conditional heteroscedastic model. Statistician 44, 333–341.CrossRefGoogle Scholar
  79. Liu, J., Li, W. and Li, C. (1997): On a threshold autoregression with conditional heteroscedastic variances. Journal of Statistical Planning and Inference 62, 279–300.zbMATHCrossRefMathSciNetGoogle Scholar
  80. Liu, X. and Shao, Y. (2004): Asymptotics for the likelihood ratio test in a two-component normal mixture model. Journal of Statistical Planning and Inference 123, 61–81.zbMATHCrossRefMathSciNetGoogle Scholar
  81. Lo, M. and Zivot, E. (2001): Threshold cointegration and nonlinear adjustment to the law of one price. Macroeconomic Dynamics 5, 506–532.Google Scholar
  82. Martens, M., Kofman, P. and Vorst, T. (1998): A threshold error-correction model for intraday futures and index returns. Journal of Applied Econometrics 13, 245–263.CrossRefGoogle Scholar
  83. McLachlan, G. and Peel, D. (2000): Finite Mixture Models. Wiley Series in Probability and Statistics. Wiley, New York.Google Scholar
  84. Mikosch, T. and Stărică, C. (2004): Non-stationarities in financial time series, the long range dependence and the IGARCH effects. Review of Economics and Statistics 86, 378–390.CrossRefGoogle Scholar
  85. Psadarakis, Z. and Spagnolo, N. (2003): On the determination of the number of regimes in Markov-switching autoregressive models. Journal of Time Series Analysis 24, 237–252.CrossRefMathSciNetGoogle Scholar
  86. Ruud, P. (1997): Extensions of estimation methods using the EM algorithm. Journal of Econometrics 49, 305–341.CrossRefMathSciNetGoogle Scholar
  87. Saikkonen, P. (2005): Stability results for nonlinear error correction models. Journal of Econometrics 127, 69–81.CrossRefMathSciNetGoogle Scholar
  88. Saikkonen, P. (2008): Stability of regime switching error correction models. Econometric Theory forthcoming.Google Scholar
  89. Seo, M. (2006): Estimation of threshold cointegration. London School of Economics, unpublished manuscript.Google Scholar
  90. Susmel, R. (2000): Switching volatility in private international equity markets. International Journal of Finance and Economics 5, 265–283.CrossRefGoogle Scholar
  91. Tjøstheim, D. (1990): Non-linear time series and Markov chains. Advances in Applied Probability 22, 587–611.CrossRefMathSciNetGoogle Scholar
  92. Tong, H. (1990): Non-Linear Time Series. Oxford Statistical Science Series. Oxford University Press, Oxford.Google Scholar
  93. Tsay, R. (1998): Testing and modeling multivariate threshold models. Journal of the American Statistical Association 93, 1188–1202.zbMATHCrossRefMathSciNetGoogle Scholar
  94. Ulloa, R. (2006): Essays in Nonlinear Time Series. Ph.D. thesis, University of Warwick, Department of Economics.Google Scholar
  95. Wong, C. and Li, W. (2000): On a mixture autoregressive model. Journal of the Royal Statistical Society, Series B 62, 95–115.zbMATHMathSciNetGoogle Scholar
  96. Wong, C. and Li, W. (2001): On a logistic mixture autoregressive model. Biometrika 88, 833–846.zbMATHCrossRefMathSciNetGoogle Scholar
  97. Wong, C. and Li, W. (2001): On a mixture autoregressive conditional heteroscedastic model. Journal of the American Statistical Association 96, 982–995.zbMATHCrossRefMathSciNetGoogle Scholar
  98. Yang, M. (2000): Some properties of vector autoregressive processes with Markov-switching coefficients. Econometric Theory 16, 23–43.zbMATHCrossRefMathSciNetGoogle Scholar
  99. Yang, M. (2001): Closed-form likelihood function of Markov-switching models. Economics Letters 70, 319–326.zbMATHCrossRefMathSciNetGoogle Scholar
  100. Yao, J. (2001): On square-integrability of an AR process with Markov switching. Statistics & Probability Letters 52, 265–270.zbMATHCrossRefMathSciNetGoogle Scholar
  101. Yao, J. and Attali, J. (2000): On stability of nonlinear AR processes with Markov switching. Advances in Applied Probability 32, 394–407.zbMATHCrossRefMathSciNetGoogle Scholar
  102. Zakoïan, J. (1994): Threshold heteroskedastic models. Journal of Economic Dynamics and Control 18, 931–955.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  1. 1.Department of EconomicsUniversity of CopenhagenCopenhagen KDenmark

Personalised recommendations