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## 7.3 Mixed models with non-multiplicative trend

There are two subclasses in this class of models:

The conditional mean for the former models coincides with the point forecasts, while the conditional variance can be calculated using the following recursive formula : \begin{aligned} & \mathrm{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{7.3} where $$\sigma^2$$ is the variance of the error term. Still, the predictive distribution from these models does not have a closed form, and as a result, in general the simulations need to be used to get the correct quantiles.

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 a formula for some of models in the second subgroup, but they are cumbersome as well. For all of these, 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 has bounds similar to the ones for the respective additive error models (Section 5.4) because they both underlie the same Exponential Smoothing methods, but with additional restrictions, coming from the multiplicative error (Section 6.4).

1. The traditional bounds (aka “usual”) should work fine for these models for the same reasons they work for the pure additive ones, 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 $$(1+ \alpha \epsilon_t)>0$$, 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. In general, to be on the safe side, the bounds should not be wider than [0, 1] for the smoothing parameters $$\alpha$$ and $$\gamma$$ in these models.

Finally, some models in this group are difficult to motivate from the application point of view. For example, ETS(M,A,A) assumes that the trend and seasonal components change in units (e.g. sales increase for several units in January), while the error term reflects the percentage changes. Similarly, ETS(A,A,M) has a difficult explanation because it assumes unit changes due to the error term and percentage changes due to the seasonality.

### References

• Hyndman, R.J., Koehler, A.B., Ord, J.K., Snyder, R.D., 2008. Forecasting with Exponential Smoothing: The State Space Approach. Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-540-71918-2