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Invertibility Conditions for Exponential Smoothing Models


  • Rob J. Hyndman


  • Muhammad Akram
  • Blyth Archibald


In this article we discuss invertibility conditions for some state space models, including the models that underly simple exponential smoothing, Holt's linear method, Holt-Winters' additive method and damped trend versions of Holt's and Holt-Winters' methods. The parameter space for which the model is invertible is compared to the usual parameter regions. We find that the usual parameter restrictions (requiring all smoothing parameters to lie between 0 and 1) do not always lead to invertible models. Conversely, some invertible models have parameters which lie outside the usual region. We also find that all seasonal exponential smoothing methods are non-invertible when the usual equations are used. However, this does not affect the forecast mean. Alternative models are presented which solve the problem while retaining the basic exponential smoothing ideas.

Suggested Citation

  • Rob J. Hyndman & Muhammad Akram & Blyth Archibald, 2003. "Invertibility Conditions for Exponential Smoothing Models," Monash Econometrics and Business Statistics Working Papers 3/03, Monash University, Department of Econometrics and Business Statistics.
  • Handle: RePEc:msh:ebswps:2003-3

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    Cited by:

    1. Ralph D Snyder, 2005. "A Pedant's Approach to Exponential Smoothing," Monash Econometrics and Business Statistics Working Papers 5/05, Monash University, Department of Econometrics and Business Statistics.
    2. Ralph D. Snyder, 2004. "Exponential Smoothing: A Prediction Error Decomposition Principle," Monash Econometrics and Business Statistics Working Papers 15/04, Monash University, Department of Econometrics and Business Statistics.

    More about this item


    exponential smoothing; invertibility; state space models.;

    JEL classification:

    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes
    • C53 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Forecasting and Prediction Models; Simulation Methods

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