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Time Series Forecasting: The Case for the Single Source of Error State Space

Listed author(s):
  • J Keith Ord
  • Ralph D Snyder


  • Anne B Koehler
  • Rob J Hyndman


  • Mark Leeds

The state space approach to modelling univariate time series is now widely used both in theory and in applications. However, the very richness of the framework means that quite different model formulations are possible, even when they purport to describe the same phenomena. In this paper, we examine the single source of error [SSOE] scheme, which has perfectly correlated error components. We then proceed to compare SSOE to the more common version of the state space models, for which all the error terms are independent; we refer to this as the multiple source of error [MSOE] scheme. As expected, there are many similarities between the MSOE and SSOE schemes, but also some important differences. Both have ARIMA models as their reduced forms, although the mapping is more transparent for SSOE. Further, SSOE does not require a canonical form to complete its specification. An appealing feature of SSOE is that the estimates of the state variables converge in probability to their true values, thereby leading to a formal inferential structure for the ad-hoc exponential smoothing methods for forecasting. The parameter space for SSOE models may be specified to match that of the corresponding ARIMA scheme, or it may be restricted to meaningful sub-spaces, as for MSOE but with somewhat different outcomes. The SSOE formulation enables straightforward extensions to certain classes of non-linear models, including a linear trend with multiplicative seasonals version that underlies the Holt-Winters forecasting method. Conditionally heteroscedastic models may be developed in a similar manner. Finally we note that smoothing and decomposition, two crucial practical issues, may be performed within the SSOE framework.

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Paper provided by Monash University, Department of Econometrics and Business Statistics in its series Monash Econometrics and Business Statistics Working Papers with number 7/05.

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Length: 33 pages
Date of creation: Apr 2005
Handle: RePEc:msh:ebswps:2005-7
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  1. James C. Morley & Charles Nelson & Eric Zivot, 2000. "Why Are Beveridge-Nelson and Unobserved-Component Decompositions of GDP So Different?," Working Papers 0013, University of Washington, Department of Economics.
  2. Tim Bollerslev, 1986. "Generalized autoregressive conditional heteroskedasticity," EERI Research Paper Series EERI RP 1986/01, Economics and Econometrics Research Institute (EERI), Brussels.
  3. Peter R. Winters, 1960. "Forecasting Sales by Exponentially Weighted Moving Averages," Management Science, INFORMS, vol. 6(3), pages 324-342, April.
  4. Koehler, Anne B. & Snyder, Ralph D. & Ord, J. Keith, 2001. "Forecasting models and prediction intervals for the multiplicative Holt-Winters method," International Journal of Forecasting, Elsevier, vol. 17(2), pages 269-286.
  5. Proietti, Tommaso & Harvey, Andrew, 2000. "A Beveridge-Nelson smoother," Economics Letters, Elsevier, vol. 67(2), pages 139-146, May.
  6. Durbin, James & Koopman, Siem Jan, 2001. "Time Series Analysis by State Space Methods," OUP Catalogue, Oxford University Press, number 9780198523543.
  7. Snyder, R.D. & Ord, J.K. & Koehler, A.B., 1997. "Prediction Intervals for Arima Models," Monash Econometrics and Business Statistics Working Papers 8/97, Monash University, Department of Econometrics and Business Statistics.
  8. Forbes, C.S. & Snyder, R.D. & Shami, R.S., 2000. "Bayesian Exponential Smoothing," Monash Econometrics and Business Statistics Working Papers 7/00, Monash University, Department of Econometrics and Business Statistics.
  9. Hyndman, R.J. & Koehler, A.B. & Ord, J.K. & Snyder, R.D., 2001. "Prediction Intervals for Exponential Smoothing State Space Models," Monash Econometrics and Business Statistics Working Papers 11/01, Monash University, Department of Econometrics and Business Statistics.
  10. H. Theil & S. Wage, 1964. "Some Observations on Adaptive Forecasting," Management Science, INFORMS, vol. 10(2), pages 198-206, January.
  11. Harvey, A.C. & Koopman, S.J.M., 1999. "Signal Extraction and the Formulation of Unobserved Components Models," Discussion Paper 1999-44, Tilburg University, Center for Economic Research.
  12. Rob Hyndman & Muhammad Akram & Blyth Archibald, 2008. "The admissible parameter space for exponential smoothing models," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 60(2), pages 407-426, June.
  13. Engle, Robert F, 1982. "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation," Econometrica, Econometric Society, vol. 50(4), pages 987-1007, July.
  14. Lawton, Richard, 1998. "How should additive Holt-Winters estimates be corrected?," International Journal of Forecasting, Elsevier, vol. 14(3), pages 393-403, September.
  15. Baki Billah & Maxwell L King & Ralph D Snyder & Anne B Koehler, 2005. "Exponential Smoothing Model Selection for Forecasting," Monash Econometrics and Business Statistics Working Papers 6/05, Monash University, Department of Econometrics and Business Statistics.
  16. I. Gijbels & A. Pope & M. P. Wand, 1999. "Understanding exponential smoothing via kernel regression," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 61(1), pages 39-50.
  17. Harvey, Andrew & Ruiz, Esther & Sentana, Enrique, 1992. "Unobserved component time series models with Arch disturbances," Journal of Econometrics, Elsevier, vol. 52(1-2), pages 129-157.
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