Journal of Statistical Theory and Practice

, Volume 2, Issue 4, pp 597–632 | Cite as

Improved Nonlinear Multivariate Financial Time Series Prediction with Mixed-State Latent Factor Models

  • Mohamed SaidaneEmail author
  • Christian Lavergne


The deficiencies of stationary models applied to financial time series are well documented. A special form of non-stationarity, where the underlying generator switches between (approximately) stationary regimes, seems particularly appropriate for financial markets. We use a dynamic switching (modelled by a hidden Markov model) combined with a linear conditionally heteroskedastic latent factor model in a hybrid mixed-state latent factor model (MSFM) and discuss the practical details of training such models with a new approximated version of the Viterbi algorithm in conjunction with the expectation-maximization (EM) algorithm to iteratively estimate the model parameters in a maximum-likelihood sense. The performance of the MSFM is evaluated on both simulated and financial data sets. On the basis of out-of-sample forecast encompassing tests as well as other measures for forecasting accuracy, our results indicate that the use of this new method yields overall better forecasts than those generated by competing models.


Latent factor models EM algorithm Conditional heteroskedasticity HMM Time series segmentation Forecasting 

AMS Subject Classification

62H25 62M05 62M10 62P20 


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  1. Amemiya, T., 1985. Advanced Econometrics, Cambridge Mass: Harvard University Press.Google Scholar
  2. Bar-Shalom, Y., Li, X.R., 1993. Estimation and Tracking: Principles, Techniques and Software, Artech House Inc., Boston, London.zbMATHGoogle Scholar
  3. Bauwens, L., Bos, C., van Dijk, H., 1999. Adaptive polar sampling with an application to a bayes measure of value-at-risk. Tinbergen Institute, Discussion Paper TI 99-082/4.Google Scholar
  4. Bera, A.K., Jarque, C.M., 1982. Model specification tests: a simultaneous approach. Journal of Econometrics, 20, 59–82.MathSciNetCrossRefGoogle Scholar
  5. Bollen, S., Gray, N., Whaley, R., 2000. Regime-switching in foreign exchange rates: evidence from currency option prices. Journal of Econometrics, 94, 239–276.CrossRefGoogle Scholar
  6. Bollerslev, T., 1986. Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 31, 307–327.MathSciNetCrossRefGoogle Scholar
  7. Bollerslev, T., 1987. A conditional heteroskedastic time series model for speculation prices and rates of return. Review of Economics and Statistics, 69, 542–547.CrossRefGoogle Scholar
  8. Bollerslev, T., Chou, R.Y., Kroner, K.F., 1992. ARCH modeling in finance: a review of the theory and empirical evidence. Journal of Econometrics, 52, 5–59.CrossRefGoogle Scholar
  9. Bollerslev, T., Wooldridge, J., 1992. Quasi-maximum likelihood estimation and inference in dynamic models with time-varying covariances. Econometric Reviews, 11, 143–172.MathSciNetCrossRefGoogle Scholar
  10. Boyen, X., Firedman, N., Koller, D., 1999. Discovering the hidden structure of complex dynamic systems. In Proceedings of the 15th Conference on Uncertainty in Artificial Intelligence, pp. 91–100. San Francisco: Morgan Kaufmann.Google Scholar
  11. Cai, J., 1994. A Markov model of switching-regime ARCH. Journal of Business and Economic Statistics, 12, 309–316.Google Scholar
  12. Carnero, M.A., Peña, D., Ruiz, E., 2004. Persistence and Kurtosis in GARCH and stochastic volatility models. Journal of Financial Econometrics, 2, 319–342.CrossRefGoogle Scholar
  13. Chaudhuri, K., Klaassen, F., 2002. Have east asian stock markets calmed down? Evidence from a regime-switching model. Discussion Paper, University of Amesterdam.Google Scholar
  14. Chong, Y.Y., Hendry, D.F., 1986. Econometric evaluation of linear macroeconomics models. Review of Economics Studies, 53, 671–690.CrossRefGoogle Scholar
  15. Darrat, A.F., Zhong, M., 2000. On testing the random-walk hypothesis: a model comparison approach. The Financial Review, 35, 105–124.CrossRefGoogle Scholar
  16. Demos, A., Sentana, E., 1998. An EM algorithm for conditionally Heteroskedastic factor models. Journal of Business & Economic Statistics, 16, 357–361.Google Scholar
  17. Dempster, A., Laird, N., Rubin, D., 1977. Maximum Likelihood from incomplete data via the EM algorithm. Journal of Royal Statistical Society Series B, 39, 1–38.MathSciNetzbMATHGoogle Scholar
  18. Diebold, F., 1986. Comment on modeling the persistence of conditional variances. Econometric Reviews, 5, 51–56.CrossRefGoogle Scholar
  19. Donaldson, R.G., Kamstra, M., 1997. An artificial neural network-GARCH model for international stock return volatility, Journal of Empirical Finance, 4, 17–46.CrossRefGoogle Scholar
  20. Dueker, M., 1997. Markov switching in GARCH processes in mean reverting stock market volatility. Journal of Business and Economics Statistics, 15, 26–34.Google Scholar
  21. Dunis, C.L., Huang, X., 2002. Forecasting and trading currency volatility: an application of recurrent neural regression and model combination. Journal of Forecasting, 21, 317–354.CrossRefGoogle Scholar
  22. Ghahramani, Z., Hinton, G.E., 1998. Switching statespace models. Technical report, Department of Computer Science, University of Toronto. http:/ Scholar
  23. Gray, S.F., 1996. Modeling the conditional distribution of interest rates as a Regime-Switching process. Journal of Financial Economics, 42, 27–62.CrossRefGoogle Scholar
  24. Haas, M., Mittnik, S., Paolella, M., 2004a. Mixed normal conditional heteroskedasticity. Journal of Financial Econometrics, 2, 211–250.CrossRefGoogle Scholar
  25. Haas, M., Mittnik, S., Paolella, M., 2004b. A new approach to Markov-Switching GARCH models. Journal of Financial Econometrics, 2, 493–530.CrossRefGoogle Scholar
  26. Hamilton, J.D., and Susmel, R., 1994. Autoregressive Conditional Heteroskedasticity and Changes in Regime. Journal of Econometrics, 64, 307–333.CrossRefGoogle Scholar
  27. Harvey, A., Ruiz, E., Sentana, E., 1992. Unobserved component time series models with ARCH disturbances. Journal of Econometrics, 52, 129–157.CrossRefGoogle Scholar
  28. Juang, B.H., Rabiner, L.R., 1985a. A probabilistic distance measure for hidden Markov models. AT&T Technical Journal, 64, 391–408.MathSciNetCrossRefGoogle Scholar
  29. Juang, B.H., Rabiner, L.R., 1985b. Mixture autoregressive hidden Markov models for speech signals. IEEE Trans. Acoustics, Speech and Signal Processing ASSP, 33, 1404–1413.CrossRefGoogle Scholar
  30. Kadirkamanathan, V., Kadirkamanathan, M., 1996. Recursive estimation of dynamic modular RBF networks. In G. Tesauro, D.S. Touretsky and T.K. Leen (editors), Advances in Neural Information Processing Systems, Vol. 8, pp. 239–245. MIT Press.Google Scholar
  31. Kim, C-J., 1994. Dynamic linear models with markov switching. Journal of Econometrics, 60, 1–22.MathSciNetCrossRefGoogle Scholar
  32. Klaassen, F., 2002. Improving GARCH volatility forecasts with regime-switching GARCH. Empirical Economics, 27, 363–394.CrossRefGoogle Scholar
  33. Lamoureux, C., Lastrapes, W., 1990. Persistence in variance, structural change, and the GARCH model. Journal of Business & Economic Statistics, 8, 225–234.Google Scholar
  34. Lanne, M., Saikkonen, P., 2003. Modeling the U.S. short term interest rate by mixture of autoregressive processes. Journal of Financial Econometrics, 1, 96–125.CrossRefGoogle Scholar
  35. Lauritzen, S., 1996. Graphical models. Oxford Statistical Science Series, 17.Google Scholar
  36. Le, N.D., Martin, D., Raftery, A.E., 1996. Modeling flat stretches, bursts and outliers in time series using mixture transition distribution models. Journal of the American Statistical Association, 91, 1504–1515.MathSciNetzbMATHGoogle Scholar
  37. Li, W.K., Wong, C.S., 2000. On a mixture autoregressive model. Journal of the Royal Statistical Society B, 62, 95–115.MathSciNetzbMATHGoogle Scholar
  38. Li, W.K., Wong, C.S., 2001. On a logistic mixture autoregressive model. Biometrika, 88, 833–846.MathSciNetCrossRefGoogle Scholar
  39. Ljung, G., Box, G., 1978. On a measure of lack of fit in time series models. Biometrika, 65, 297–303.CrossRefGoogle Scholar
  40. MacDonald, R., Taylor, P.M., 1992. Exchange rate economics: a survey. IMF Staff Papers, 39, 1–57.CrossRefGoogle Scholar
  41. Meese, R.A., Rogoff, K., 1983. Empirical exchange rates models of the seventies: do they fit out-of-sample. Journal of International Economics, 14, 2–24.CrossRefGoogle Scholar
  42. Mikosch, T., Starica, C., 2004. Nonstationarities in financial time series, the long-range dependence, and the IGARCH effects. Review of Economics and Statistics, 86, 378–390.CrossRefGoogle Scholar
  43. Murphy, K.P., 1998. Learning switching Kalman filter models. Technical Report 98–10, Compaq Cambridge Research Lab.Google Scholar
  44. Pagan, A.R., Schwert, G.W., 1990. Alternative models for conditional stock volatility. Journal of Econometrics, 45, 267–290.CrossRefGoogle Scholar
  45. Rauch, H.E., 1963. Solutions to the linear smoothing problem. IEEE Transactions on Automatic Control, 8, 371–372.CrossRefGoogle Scholar
  46. Rauch, H.E., Tung, F., Striebel, C.T., 1965. Maximum likelihood estimates of linear dynamic systems. AIAA Journal, 3, 1445–1450.MathSciNetCrossRefGoogle Scholar
  47. Rosti, A.V.I., Gales, M.J.F., 2001. Generalised Linear Gaussian Models. Tech. Rep. CUED/F-INFENG/TR.420, Cambridge University Engineering Department.Google Scholar
  48. Saidane, M., Lavergne, C., 2006. On factorial HMMs for time series in finance. The Kyoto Economic Review, 75, 63–90.zbMATHGoogle Scholar
  49. Saidane, M., Lavergne, C., 2007a. Conditionally heteroskedastic factorial HMMs for time series in finance. Applied Stochastic Models in Business and Industry, 23, 503–529.MathSciNetCrossRefGoogle Scholar
  50. Saidane, M., Lavergne, C., 2007b. A structured variational learning approach for switching latent factor models. Advances in Statistical Analysis — Journal of the German Statistical Society, 91, 245–268.MathSciNetCrossRefGoogle Scholar
  51. Schwarz, G., 1978. Estimating the dimension of a model. Annals of Statistics, 6, 461–464.MathSciNetCrossRefGoogle Scholar
  52. Schwert, G., 1989. Why does stock market volatility change over time? Journal of Finance, 44, 1115–1153.CrossRefGoogle Scholar
  53. Sentana, E., Fiorentini, G., 2001. Identification, estimation and testing of conditionally Heteroskedastic factor models. Journal of Econometrics, 102, 143–164.MathSciNetCrossRefGoogle Scholar
  54. Sentana, E., 1995. Quadratic ARCH models. Review of Economic Studies, 62, 639–661.CrossRefGoogle Scholar
  55. White, H., 1980. A heteroskedasticity consistent covariance matrix estimator and a direct test for heteroskedasticity. Econometrica, 48, 817–838.MathSciNetCrossRefGoogle Scholar
  56. Wilfiling, B., Männig, W., 2001. Exchange rate dynamics in anticipation of time contingent regime switching: modelling the effects of a possible delay. Journal of International Money and Finance, 20, 91–113.CrossRefGoogle Scholar
  57. Wong, C., Li, W., 2001. On a mixture autoregressive conditional heteroskedastic model. Journal of the American Statistical Association, 96, 982–995.MathSciNetCrossRefGoogle Scholar

Copyright information

© Grace Scientific Publishing 2008

Authors and Affiliations

  1. 1.The University of 7 November at Carthage, ISCC BizerteZarzounaTunisia
  2. 2.University of Montpellier 2, UMR-CNRS 5149France

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