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8.1 Mixed models with non-multiplicative trend

In this class of models, there are two subclasses: one with non-multiplicative seasonal component (MAN, MAdN, MNA, MAA, MAdA) and another one with the multiplicative one (MAM; MAdM; ANM; AAM; AAdM). The conditional mean for the former models coincides with the point forecasts, while the conditional variance can be calculated using the following recursive formula (Hyndman et al. 2008): \[\begin{equation} \begin{aligned} & \text{V}(y_{t+h}|t) = (1+\sigma^2) \xi_h - \mu_{t+h|t}^2 \\ & \text{where } \xi_{1} = \mu_{t+1|t}^2 \text{ and } \xi_h = \mu_{t+h|t}^2 + \sigma^2 \sum_{j=1}^{h-1} c_{j}^2 \xi_{h-j} \end{aligned} , \tag{8.1} \end{equation}\]

where \(\sigma^2\) is the variance of the error term. As for the second subgroup, the conditional mean corresponds to the point forecasts, when the forecast horizon is less than or equal to the seasonal frequency of the component (i.e. \(h\leq m\)), and there is a cumbersome formula for calculating the conditional mean to some of models in this subgroup for the \(h>m\). When it comes to the conditional variance, there exists the formula for some of models in the second subgroup, but they are cumbersome as well. The interested reader is directed to Hyndman et al. (2008), page 85.

When it comes to the parameters bounds for the models in this group, the first subgroup of models should have similar bounds to the ones for the respective additive error models (because both underly the same exponential smoothing methods), but with additional restrictions, comming from the multiplicative error.

  1. The traditional bounds (aka "usual") should work fine for these models for the same reasons they work for the pure additive models, although they might be too restrictive in some cases;
  2. The admissible bounds for smoothing parameters for the models in this group might be too wide and violate the condition of \((1+ \alpha \epsilon_t)>0, (1+ \beta \epsilon_t)>0, (1+ \gamma \epsilon_t)>0\), which is important in order not to break the models.

The second subgroup is more challenging in terms of parameters bounds because of the multiplication of states by the seasonal components.

References

Hyndman, Rob J., Anne B. Koehler, J. Keith Ord, and Ralph D. Snyder. 2008. Forecasting with Exponential Smoothing. Springer Berlin Heidelberg.