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Seasonality with Trend and Cycle Interactions in Unobserved Components Models

Author

Listed:
  • Siem Jan Koopman

    (VU University Amsterdam)

  • Kai Ming Lee

    (VU University Amsterdam)

Abstract
This discussion paper resulted in a publication in the Journal of the Royal Statistical Society Series C (2009). Vol. 58, pages 427-448. Unobserved components time series models decompose a time series into a trend, a season, a cycle, an irregular disturbance, and possibly other components. These models have been successfully applied to many economic time series. The standard assumption of a linear model, often appropriate after a logarithmic transformation of the data, facilitates estimation, testing, forecasting and interpretation. However, in some settings the linear-additive framework may be too restrictive. In this paper, we formulate a non-linear unobserved components time series model which allows interactions between the trend-cycle component and the seasonal component. The resulting model is cast into a non-linear state space form and estimated by the extended Kalman filter, adapted for models with diffuse initial conditions. We apply our model to UK travel data and US unemployment and production series, and show that it can capture increasing seasonal variation and cycle dependent seasonal fluctuations.

Suggested Citation

  • Siem Jan Koopman & Kai Ming Lee, 0000. "Seasonality with Trend and Cycle Interactions in Unobserved Components Models," Tinbergen Institute Discussion Papers 08-028/4, Tinbergen Institute.
    • Handle: RePEc:tin:wpaper:20080028
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    File URL: https://papers.tinbergen.nl/08028.pdf
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    References listed on IDEAS

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    1. J. Durbin & S. J. Koopman, 2000. "Time series analysis of non‐Gaussian observations based on state space models from both classical and Bayesian perspectives," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 62(1), pages 3-56.
    2. [Reference to Proietti], Tommaso, 2000. "Comparing seasonal components for structural time series models," International Journal of Forecasting, Elsevier, vol. 16(2), pages 247-260.
    3. Matas-Mir, Antonio & Osborn, Denise R., 2004. "Does seasonality change over the business cycle? An investigation using monthly industrial production series," European Economic Review, Elsevier, vol. 48(6), pages 1309-1332, December.
    4. Gersch, Will & Kitagawa, Genshiro, 1983. "The Prediction of Time Series with Trends and Seasonalities," Journal of Business & Economic Statistics, American Statistical Association, vol. 1(3), pages 253-264, July.
    5. Dick van Dijk 1 & Birgit Strikholm & Timo Teräsvirta, 2003. "The effects of institutional and technological change and business cycle fluctuations on seasonal patterns in quarterly industrial production series," Econometrics Journal, Royal Economic Society, vol. 6(1), pages 79-98, June.
    6. Commandeur, Jacques J.F. & Koopman, Siem Jan, 2007. "An Introduction to State Space Time Series Analysis," OUP Catalogue, Oxford University Press, number 9780199228874.
    7. Findley, David F, et al, 1998. "New Capabilities and Methods of the X-12-ARIMA Seasonal-Adjustment Program: Reply," Journal of Business & Economic Statistics, American Statistical Association, vol. 16(2), pages 169-177, April.
    8. Proietti, Tommaso & Riani, Marco, 2007. "Transformations and Seasonal Adjustment: Analytic Solutions and Case Studies," MPRA Paper 7862, University Library of Munich, Germany.
    9. Siem Jan Koopman & Neil Shephard & Jurgen A. Doornik, 1999. "Statistical algorithms for models in state space using SsfPack 2.2," Econometrics Journal, Royal Economic Society, vol. 2(1), pages 107-160.
    10. Franses, Philip Hans, 1995. "Quarterly US Unemployment: Cycles, Seasons and Asymmetries," Empirical Economics, Springer, vol. 20(4), pages 717-725.
    11. Marianne Baxter & Robert G. King, 1999. "Measuring Business Cycles: Approximate Band-Pass Filters For Economic Time Series," The Review of Economics and Statistics, MIT Press, vol. 81(4), pages 575-593, November.
    12. Tommaso Proietti & Marco Riani, 2009. "Transformations and seasonal adjustment," Journal of Time Series Analysis, Wiley Blackwell, vol. 30(1), pages 47-69, January.
    13. Franses Philip Hans & de Bruin Paul, 2000. "Seasonal Adjustment and the Business Cycle in Unemployment," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 4(2), pages 1-14, July.
    14. Durbin, James & Koopman, Siem Jan, 2012. "Time Series Analysis by State Space Methods," OUP Catalogue, Oxford University Press, edition 2, number 9780199641178.
    15. Cecchetti, Stephen G & Kashyap, Anil K & Wilcox, David W, 1997. "Interactions between the Seasonal and Business Cycles in Production and Inventories," American Economic Review, American Economic Association, vol. 87(5), pages 884-892, December.
    16. Findley, David F, et al, 1998. "New Capabilities and Methods of the X-12-ARIMA Seasonal-Adjustment Program," Journal of Business & Economic Statistics, American Statistical Association, vol. 16(2), pages 127-152, April.
    17. S. J. Koopman & J. Durbin, 2003. "Filtering and smoothing of state vector for diffuse state‐space models," Journal of Time Series Analysis, Wiley Blackwell, vol. 24(1), pages 85-98, January.
    18. Harvey, Andrew & Scott, Andrew, 1994. "Seasonality in Dynamic Regression Models," Economic Journal, Royal Economic Society, vol. 104(427), pages 1324-1345, November.
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    Citations

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    Cited by:

    1. Altug, Sumru & Çakmaklı, Cem, 2016. "Forecasting inflation using survey expectations and target inflation: Evidence for Brazil and Turkey," International Journal of Forecasting, Elsevier, vol. 32(1), pages 138-153.
    2. Sumru Altug & Cem Cakmakli, 2014. "Inflation Targeting and Inflation Expectations: Evidence for Brazil and Turkey," Koç University-TUSIAD Economic Research Forum Working Papers 1413, Koc University-TUSIAD Economic Research Forum.
    3. Daniel Kinn, 2018. "Synthetic Control Methods and Big Data," Papers 1803.00096, arXiv.org.
    4. Steven Clark & T. Coggin, 2009. "Trends, Cycles and Convergence in U.S. Regional House Prices," The Journal of Real Estate Finance and Economics, Springer, vol. 39(3), pages 264-283, October.
    5. Irma Hindrayanto & Jan Jacobs & Denise Osborn, 2014. "On trend-cycle-seasonal interactions," DNB Working Papers 417, Netherlands Central Bank, Research Department.
    6. Paul Alagidede, 2012. "Trends And Cycles In The Net Barter Terms Of Trade For Sub-Saharan Africa's Primary Commodity Exporters," Journal of Developing Areas, Tennessee State University, College of Business, vol. 46(2), pages 213-229, July-Dece.

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    More about this item

    Keywords

    Seasonal interaction; Unobserved components; Non-linear state space models;
    All these keywords.

    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes

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