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6.3 Smoothing parameters bounds

Similar to the pure additive ADAM ETS, it is possible to have different types of bounds, including the classical, the usual and the admissible ones. However, in case of pure multiplicative models, the classical and the usual restrictions become more reasonable from the point of view of the model itself, while the derivation of admissible bounds becomes a challenging task. In order to see the former, consider the ETS(M,N,N) model, for which the level is updated using the following relation: \[\begin{equation} l_t = l_{t-1} (1 + \alpha\epsilon_t) = l_{t-1} (1-\alpha + \alpha(1+\epsilon_t)). \tag{6.15} \end{equation}\] As discussed previously, the main benefit of pure multiplicative models is in modelling positive data. So, it is reasonable to assume that \((1 + \epsilon_t)>0\), which then implies that the actual values will always be positive, and that each component of the model should also be positive. This means that \(\alpha(1 + \epsilon_t)>0\), which implies that \((1-\alpha + \alpha(1+\epsilon_t))>1-\alpha\) or equivalently based on (6.10) \((1 + \alpha\epsilon_t)>1-\alpha\) should always hold. Now in order for the model to make sense, the condition \((1 + \alpha\epsilon_t)>0\) should hold as well, ensuring that the level is always positive. This leads to the following set of inequalities: \[\begin{equation} \begin{aligned} (1 + \alpha\epsilon_t)> &0 \\ (1 + \alpha\epsilon_t)> &1-\alpha \end{aligned} . \tag{6.16} \end{equation}\] This can only be satisfied in the case of \(1-\alpha\geq0\) or \(\alpha\leq1\). Another bounds can be obtained by analysing the equation (6.15) and using the restriction for positivity of its elements: \((1-\alpha + \alpha(1+\epsilon_t))>0\) which can only be achieved, when \((1+\epsilon_t)>\frac{1-\alpha}{\alpha}\), leading to another two inequalities: \[\begin{equation} \begin{aligned} (1 + \epsilon_t)> &0 \\ (1 + \epsilon_t)> &\frac{1-\alpha}{\alpha} \end{aligned} , \tag{6.17} \end{equation}\] which can be satisfied only when \(\alpha\geq0\), because, as we have already shown, the condition \(1-\alpha\geq0\) should hold. So, in general the bounds \([0, 1]\) guarantee that the model ETS(M,N,N) will produce positive values only. The two special cases \(\alpha=0\) and \(\alpha=1\) make sense, because the level in (6.15) will be positive in this case, implying that for the former the model becomes equivalent to the global level, while for the latter the model is equivalent to Random Walk. Using similar logic, it can be shown that the classical restriction \(\alpha, \beta, \gamma \in [0, 1]\) guarantees that the model will always produce positive values.

The more restrictive condition of the usual bounds, discussed in Parameters Bounds section makes sense as well, although it might be more restrictive than needed, but it has a different idea: guaranteeing that the model exhibits averaging properties.

Finally, the admissible bounds might still make sense for the pure multiplicative models, but the condition for parameters bounds becomes more complicated and implies that the distribution of the error term becomes trimmed from below in order to satisfy (6.16) and (6.17). Very crudely, the conventional restriction from pure additive models can be used to get an approximation to the proper admissible bounds, given the limit (6.5), but this should be used with care, given the discussion above.