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Shrinkage estimator for exponential smoothing models

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  • Pritularga, Kandrika F.
  • Svetunkov, Ivan
  • Kourentzes, Nikolaos

Abstract

Exponential smoothing is widely used in practice and has shown its efficacy and reliability in many business applications. Yet there are cases, for example when the estimation sample is limited, where the estimated smoothing parameters can be erroneous, often unnecessarily large. This can lead to over-reactive forecasts and high forecast errors. Motivated by these challenges, we investigate the use of shrinkage estimators for exponential smoothing. This can help with parameter estimation and mitigating parameter uncertainty. Building on the shrinkage literature, we explore ℓ1 and ℓ2 shrinkage for different time series and exponential smoothing model specifications. From a simulation and an empirical study, we find that using shrinkage in exponential smoothing results in forecast accuracy improvements and better prediction intervals. In addition, using bias–variance decomposition, we show the interdependence between smoothing parameters and initial values, and the importance of the initial value estimation on point forecasts and prediction intervals.

Suggested Citation

  • Pritularga, Kandrika F. & Svetunkov, Ivan & Kourentzes, Nikolaos, 2023. "Shrinkage estimator for exponential smoothing models," International Journal of Forecasting, Elsevier, vol. 39(3), pages 1351-1365.
    • Handle: RePEc:eee:intfor:v:39:y:2023:i:3:p:1351-1365
    DOI: 10.1016/j.ijforecast.2022.07.005
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    1. Bunn, Derek W. & Vassilopoulos, Angelos I., 1999. "Comparison of seasonal estimation methods in multi-item short-term forecasting," International Journal of Forecasting, Elsevier, vol. 15(4), pages 431-443, October.
    2. Fildes, Robert & Hibon, Michele & Makridakis, Spyros & Meade, Nigel, 1998. "Generalising about univariate forecasting methods: further empirical evidence," International Journal of Forecasting, Elsevier, vol. 14(3), pages 339-358, September.
    3. Hyndman, Rob J. & Koehler, Anne B. & Snyder, Ralph D. & Grose, Simone, 2002. "A state space framework for automatic forecasting using exponential smoothing methods," International Journal of Forecasting, Elsevier, vol. 18(3), pages 439-454.
    4. Kourentzes, Nikolaos & Petropoulos, Fotios & Trapero, Juan R., 2014. "Improving forecasting by estimating time series structural components across multiple frequencies," International Journal of Forecasting, Elsevier, vol. 30(2), pages 291-302.
    5. Shanika L. Wickramasuriya & George Athanasopoulos & Rob J. Hyndman, 2019. "Optimal Forecast Reconciliation for Hierarchical and Grouped Time Series Through Trace Minimization," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 114(526), pages 804-819, April.
    6. Kourentzes, Nikolaos & Barrow, Devon & Petropoulos, Fotios, 2019. "Another look at forecast selection and combination: Evidence from forecast pooling," International Journal of Production Economics, Elsevier, vol. 209(C), pages 226-235.
    7. Schäfer Juliane & Strimmer Korbinian, 2005. "A Shrinkage Approach to Large-Scale Covariance Matrix Estimation and Implications for Functional Genomics," Statistical Applications in Genetics and Molecular Biology, De Gruyter, vol. 4(1), pages 1-32, November.
    8. Makridakis, Spyros & Spiliotis, Evangelos & Assimakopoulos, Vassilios, 2018. "The M4 Competition: Results, findings, conclusion and way forward," International Journal of Forecasting, Elsevier, vol. 34(4), pages 802-808.
    9. Juan R Trapero & Nikolaos Kourentzes & Robert Fildes, 2015. "On the identification of sales forecasting models in the presence of promotions," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 66(2), pages 299-307, February.
    10. Miller, Don M. & Williams, Dan, 2003. "Shrinkage estimators of time series seasonal factors and their effect on forecasting accuracy," International Journal of Forecasting, Elsevier, vol. 19(4), pages 669-684.
    11. Hyndman, Rob J. & Billah, Baki, 2003. "Unmasking the Theta method," International Journal of Forecasting, Elsevier, vol. 19(2), pages 287-290.
    12. Trapero, Juan R. & Cardós, Manuel & Kourentzes, Nikolaos, 2019. "Empirical safety stock estimation based on kernel and GARCH models," Omega, Elsevier, vol. 84(C), pages 199-211.
    13. Zou, Hui, 2006. "The Adaptive Lasso and Its Oracle Properties," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 1418-1429, December.
    14. Meira, Erick & Cyrino Oliveira, Fernando Luiz & Jeon, Jooyoung, 2021. "Treating and Pruning: New approaches to forecasting model selection and combination using prediction intervals," International Journal of Forecasting, Elsevier, vol. 37(2), pages 547-568.
    15. Assimakopoulos, V. & Nikolopoulos, K., 2000. "The theta model: a decomposition approach to forecasting," International Journal of Forecasting, Elsevier, vol. 16(4), pages 521-530.
    16. Ming Chen & Zhi-Long Chen, 2018. "Robust Dynamic Pricing with Two Substitutable Products," Manufacturing & Service Operations Management, INFORMS, vol. 20(2), pages 249-268, May.
    17. Lee, Yun Shin & Scholtes, Stefan, 2014. "Empirical prediction intervals revisited," International Journal of Forecasting, Elsevier, vol. 30(2), pages 217-234.
    18. Greis, Noel P. & Gilstein, C. Zachary, 1991. "Empirical Bayes methods for telecommunications forecasting," International Journal of Forecasting, Elsevier, vol. 7(2), pages 183-197, August.
    19. Makridakis, Spyros & Hibon, Michele, 2000. "The M3-Competition: results, conclusions and implications," International Journal of Forecasting, Elsevier, vol. 16(4), pages 451-476.
    20. Koenker, Roger W & Bassett, Gilbert, Jr, 1978. "Regression Quantiles," Econometrica, Econometric Society, vol. 46(1), pages 33-50, January.
    21. Sagaert, Yves R. & Kourentzes, Nikolaos & De Vuyst, Stijn & Aghezzaf, El-Houssaine & Desmet, Bram, 2019. "Incorporating macroeconomic leading indicators in tactical capacity planning," International Journal of Production Economics, Elsevier, vol. 209(C), pages 12-19.
    22. Michael J. Daniels & Robert E. Kass, 2001. "Shrinkage Estimators for Covariance Matrices," Biometrics, The International Biometric Society, vol. 57(4), pages 1173-1184, December.
    23. 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.
    24. Gneiting, Tilmann & Raftery, Adrian E., 2007. "Strictly Proper Scoring Rules, Prediction, and Estimation," Journal of the American Statistical Association, American Statistical Association, vol. 102, pages 359-378, March.
    25. Sanders, Nada R. & Graman, Gregory A., 2009. "Quantifying costs of forecast errors: A case study of the warehouse environment," Omega, Elsevier, vol. 37(1), pages 116-125, February.
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