4.7 Parameters bounds
While many practitioners and academics accept that the smoothing parameters of Exponential Smoothing methods should lie between zero and one, this is not entirely true for the ETS models. There are, in fact, several possible restrictions on smoothing parameters, and it is worth discussing them separately:
Classical or conventional bounds are \(\alpha, \beta, \gamma \in (0,1)\). The idea behind them originates from the simple Exponential Smoothing method (Section 3.4), where it is logical to restrict the bounds with this region because then the smoothing parameters regulate what weight the actual value \(y_t\) will have and what weight will be assigned to the predicted one \(\hat{y}_t\). Hyndman et al. (2008) showed that this condition is sometimes too loose and, in other cases, is too restrictive for some ETS models. Brenner et al. (1968) were some of the first to show that the bounds are wider than this region for many Exponential Smoothing methods. Still, the conventional restriction is the most often used in practice, just because it is easy to interpret.
Usual or traditional bounds are those that satisfy the set of the following equations: \[\begin{equation} \begin{aligned} &\alpha \in [0, 1)\\ &\beta \in [0, \alpha) \\ &\gamma \in [0, 1-\alpha) \end{aligned}. \tag{4.27} \end{equation}\] This set of restrictions guarantees that the weights decline over time exponentially (see Section 3.4.2), and the ETS models have the property of “averaging” the values over time. In the lower boundary condition, the model’s components become deterministic, and we can say that they are calculated as the global averages of the values over time.
Admissible bounds, satisfying the stability condition. The idea here is that the most recent observation should have a higher weight than the older ones, which is regulated via the smoothing parameters. However, in this case, we do not impose the restriction of exponential decay of weights on the models, so they can oscillate or decay harmonically as long as their absolute values decrease over time. The condition is more complicated mathematically than the previous two. It will be discussed later in the monograph for the pure additive models (see Section 5.1), but here are several examples for bounds, satisfying this condition (from Chapter 10 of Hyndman et al., 2008):
- ETS(A,N,N): \(\alpha \in (0, 2)\);
- ETS(A,A,N): \(\alpha \in (0, 2); \beta \in (0, 4-2\alpha)\);
- ETS(A,N,A): \(\alpha \in \left(\frac{-2}{m-1}, 2-\gamma\right); \gamma \in (\max(-m\alpha, 0), 2-\alpha)\);
As you see, the admissible bounds are much wider than the conventional and usual ones. In fact, in this case, smoothing parameters can become either negative or greater than one in some cases for some models, which is hard to interpret but might indicate that the data is difficult to predict. Furthermore, the admissible bounds correspond to the restrictions of the parameters for ARIMA models, underlying some of pure additive ETS models (see discussion in Section 8.4). In a way, they are more natural for the ETS models than the other two because they follow the formulation and arise naturally. However, their usage in practice has been met with mixed success, with only a handful of papers using them instead of (1) or (2) Snyder et al. (2017).