Abstract
We propose a state space model with Markov switching, whose regimes are associated with the model parameters and regime transition probabilities are modeled using wavelets. The estimation is based on the maximum likelihood method using the EM algorithm and a bootstrap method is proposed in order to assess the distribution of the maximum likelihood estimators. To evaluate the state variables and regime probabilities, the Kalman filter and a probability filter procedure conditional on each possible regime, at each instant, are used. These procedures are evaluated with simulated data and illustrated with the US monthly industrial production index from July 1968 to February 2011.
References
Alencar, A. P., C. M. C. Toloi, and P. A. Morettin. 2006. “State Space Model with Markovian Switching and Transition Probabilities Modelled using wavelets”. PhD Thesis, Institute of Mathematics and Statistics of University of São Paulo.Search in Google Scholar
Anderson, B. D. O., and J. B. Moore. 1979. Optimal Filtering. Prentice Hall: Englewood Cliffs.Search in Google Scholar
Burns, A., and W. Mitchell. 1946. Measuring business cycles. New York: NBER.Search in Google Scholar
Chauvet, M., and J. D. Hamilton. 1994. “Competitive Markets and Endogenous Cycles: An Evaluation.” In: Business Cycles: Theory and Empirical Methods, edited by C. Milas, P. Rothman, and D. van Dijk. Boston: Kluwer, 53–71.Search in Google Scholar
Crowley, P. 2005. “An Intuitive Guide to Wavelets for Economists.” Discussion Paper – Bank of Finland, 2005–01.Search in Google Scholar
Dahlhaus, R., M. H. Neumann, and R. von Sachs. 1999. “Wavelet Estimation of Time-Varying Autoregressive Processes.” Bernoulli 5: 873–906.10.2307/3318448Search in Google Scholar
Daubechies, I. 1992. Ten Lectures on Wavelets. Philadelphia: SIAM.10.1137/1.9781611970104Search in Google Scholar
Dempster, A. P., N. M. Laird, and D. B. Rubin, 1977. “Maximum Likelihood from Incomplete Data Via the em Algorithm.” Journal of the Royal Statistical Society B39: 1–38.10.1111/j.2517-6161.1977.tb01600.xSearch in Google Scholar
Diebold, F. X., J. H. Lee, and G. C.Weinbach. 1994. “Regime Switching with Time Varying Transition Probabilities,” In: Nonstationary time series analysis and cointegration, edited by C. Hargreaves. Oxford Press: Oxford, 283–302,Search in Google Scholar
Diebold, F. X., and G. D. Rudenbusch. 1996. “Measuring Business Cycles: A Modern Perspective.” The Review of Economics and Statistics 78: 67–77.10.2307/2109848Search in Google Scholar
Donoho, D. L., and I. M. Johnstone. 1998. “Minimax Estimation via Wavelet Shrinkage.” Annals of Statistics 26: 879–921.10.1214/aos/1024691081Search in Google Scholar
Donoho, D. L., and I. M. Johnstone. 1999. “Asymptotic Minimaxity of Wavelet Estimators with Sampled Data.” Statistica Sinica 9: 1–32.Search in Google Scholar
Doornik, J. A. 1996. Object-oriented Matrix Programming Using Ox. Oxford: Oxford University Press.Search in Google Scholar
Durbin, J., and S. J. Koopman. 2001. Time Series Analysis by State Space Methods. Oxford: Oxford University Press.Search in Google Scholar
Efron, B. 1979. “Bootstrap Methods: Another Look at Jacknife.” Annals of Statistics 7: 1–26.10.1214/aos/1176344552Search in Google Scholar
Fahrmeir, L. 1992. “State Space Modelling and Conditional Mode Estimation for Categorical Time Series,” In: New Directions in Time Series Analysis – Part I, edited by D. Brillinger, P. Caines, J. Geweke, E. Parzen, M. Rosenblatt, and M. S. Taqqu. New York: Springer, 87–109.Search in Google Scholar
Filardo, A. J. 1994. “Business Cycle Phases and their Transitional Dynamics.” Journal of Business and Economic Statistics 78: 111–125.Search in Google Scholar
Fletcher, R. 1987. Pratical Methods of Optmization. New York: John Wiley & Sons.Search in Google Scholar
Gençay, R., N. Gradojevic, F. Selçuk, and B. Whitcher. 2010. “Asymmetry of Information Flow between Volatilities Across Time Scales.” Quantitative Finance 10: 895–915.10.1080/14697680903460143Search in Google Scholar
Gençay, R., F. Selçuk, and B. Whitcher. 2001. An Introduction to Wavelets and Other Filtering Methods in Finance and Economics. San Diego: Academic Press.10.1016/B978-012279670-8.50004-5Search in Google Scholar
Goldfeld, S. M., and R. E. Quandt. 1973. “A Markov Model for Switching Regression.” Journal of Econometrics 1: 3–16.10.1016/0304-4076(73)90002-XSearch in Google Scholar
Hamilton, J. D. 1989. “A New Approach to the Economic Analysis of Nonstationary Time Series and Business Cycle.” Econometrica 57: 357–384.10.2307/1912559Search in Google Scholar
Harrison, P. J., and C. F. Stevens. 1976. “Bayesian Forecasting.” Journal of the Royal Statistical Society B 38: 205–247.10.1111/j.2517-6161.1976.tb01586.xSearch in Google Scholar
Harvey, A. C. 1989. Forecasting, Structural Time Series Models and the Kalman Filter. Cambridge: Cambridge University Press.10.1017/CBO9781107049994Search in Google Scholar
Kalman, R. E. 1960. “A New Approach to Linear Filtering and Prediction Problems.” Journal of Basic Engineering. Transactions of the ASME D 82: 35–45.10.1115/1.3662552Search in Google Scholar
Kim, C., and C. R. Nelson. 1999. State space models with Markov switching – classical and Gibbs sampling approach with applications. Cambridge: MIT Press.Search in Google Scholar
Kitagawa, G., and W. Gersch. 1996. Smoothness Priors Analysis of Time Series. New York: Lecture Notes – Springer-Verlag.10.1007/978-1-4612-0761-0Search in Google Scholar
Koch, K. D. 1999. Parameter Estimation and Hypothesis Testing in Linear Models. Berlin: Springer-Verlag.10.1007/978-3-662-03976-2Search in Google Scholar
Meinhold, R. J., and N. D. Singpurwalla. 1989. “Robustification of Kalman Filter Models.” Journal of the American Statistical Association 84: 479–486.10.1080/01621459.1989.10478794Search in Google Scholar
Morettin, P. A. 1997. “Wavelets in Statistics.” São Paulo Journal of Mathematical Sciences 3: 211–272.Search in Google Scholar
Quandt, R. E. 1972. “A New Approach to Estimating Switching Regressions.” Journal of the American Statistical Association 67: 306–310.10.1080/01621459.1972.10482378Search in Google Scholar
Ramsey, J. B. 2010. “Wavelets in Economics and Finance: Past and Future”. New York: University Working Papers.Search in Google Scholar
Shumway, R. H., and D. S. Stoffer. 2006. Time Series Analysis and its Applications – with R examples. New York: Springer-Verlag.Search in Google Scholar
Stock, J.H., and M. W. Watson. 1989. “New Indexes of Coincident and Leading Economic Indicators.” In: NBER Macroeconomics Annual, edited by O. Blanchard, and S. Fischer. Cambridge: MIT Press, 351–394.Search in Google Scholar
Stock, J.H., and M. W. Watson. 1993. “A Procedure for Predicting Recessions with Leading Indicators: Econometric Issues and Recent Experience,” In: Business cycles, indicators and forecasting, edited by J.H. Stock, and M. W. Watson. Chicago: University of Chicago Press for NBER, 255–284.Search in Google Scholar
Stoffer, D. S., and K. Wall. 1991. “Bootstrapping State Space Models: Gaussian Maximum Likelihood Estimation and the Kalman Filter.” Journal of the American Statistical Association 86: 1024–1033.10.1080/01621459.1991.10475148Search in Google Scholar
Vidakovic, B. 1999. Statistical modelling by wavelets. New York: Wiley-Interscience.10.1002/9780470317020Search in Google Scholar
West, M., and J. Harrison. 1997. Bayesian Forecasting and Dynamic Models. New York: Springer-Verlag.Search in Google Scholar
Whitcher, B., P. Guttorp, and D. B. Percival. 2000. “Wavelet Analysis of Covariance with Applications to Atmospheric Time Series.” Journal of Geophysical Research 105, D11(D11):14941–14962.Search in Google Scholar
©2013 by Walter de Gruyter Berlin Boston
Articles in the same Issue
- Stochastically weighted average conditional moment tests of functional form
- Do Latin American Central Bankers Behave Non-Linearly? The Experiences of Brazil, Chile, Colombia and Mexico
- Empirical analysis of ARMA-GARCH models in market risk estimation on high-frequency US data
- Quasi-maximum likelihood estimation of multivariate diffusions
- Time-varying cointegration, identification, and cointegration spaces
- Noncausality and asset pricing
- State space Markov switching models using wavelets
Articles in the same Issue
- Stochastically weighted average conditional moment tests of functional form
- Do Latin American Central Bankers Behave Non-Linearly? The Experiences of Brazil, Chile, Colombia and Mexico
- Empirical analysis of ARMA-GARCH models in market risk estimation on high-frequency US data
- Quasi-maximum likelihood estimation of multivariate diffusions
- Time-varying cointegration, identification, and cointegration spaces
- Noncausality and asset pricing
- State space Markov switching models using wavelets