In this chapter, we discuss some of the mathematical properties of the linear innovations state space models described in Chap. 3. These results are based on Hyndman et al. (2008).
We provide conditions that ensure the model is of minimal dimension (Sect. 10.1) and conditions that guarantee the model is stable (Sect. 10.2). We will see that the non-seasonal models are already of minimal dimension, but that the seasonal models are slightly larger than necessary. The normalized seasonal models, introduced in Chap. 8, are of minimal dimension.
The stability conditions discussed in Sect. 10.2 can be used to derive the associated parameter space. We find that the usual parameter restrictions (requiring all smoothing parameters to lie between 0 and 1) do not always lead to stable models. Exact parameter restrictions are derived for all the linear models.
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© 2008 Springer-Verlag Berlin Heidelberg
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(2008). Some Properties of Linear Models. In: Forecasting with Exponential Smoothing. Springer Series in Statistics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-71918-2_10
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DOI: https://doi.org/10.1007/978-3-540-71918-2_10
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-71916-8
Online ISBN: 978-3-540-71918-2
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