7.5 Normalisation of seasonal indices in ETS models
One of the ideas arising from time series decomposition (Section 3.2), inherited by the conventional ETS, is the renormalisation of seasonal indices. It comes to one of the two principles, depending on the type of seasonality:
- If the model has additive seasonality, then the seasonal indices should add up to zero in a specific period of time, e.g. monthly indices should add up to zero over the yearly period;
- If the model has multiplicative seasonality, then the geometric mean of seasonal indices over a specific period should be equal to one.
Condition (2) in the conventional ETS is substituted by “the arithmetic mean of multiplicative indices should be equal to one”, which does not have reasonable grounds behind it. If we deal with the multiplicative effect, the geometric mean should be used, not the arithmetic one. Otherwise, we introduce bias in the model by multiplying components by indices.
While the normalisation is a natural element of the time series decomposition and works fine for the initial seasonal indices, renormalising the seasonal indices over time might not be natural for the ETS.
Hyndman et al. (2008) discuss different mechanisms for the renormalisation of seasonal indices, which, as the authors claim, are needed to make the principles (1) and (2) hold from period to period in the data. However, I argue that this is an unnatural idea for the ETS models. The indices should only be normalised during the initialisation of the model (at the moment \(t=0\)), and that they should vary independently for the rest of the sample. The rationale for this comes from the model itself. To illustrate it, I will use ETS(A,N,A), but the idea can be easily applied to any other seasonal ETS model with any types of components and any number of seasonal frequencies. Just a reminder, ETS(A,N,A) is formulated as: \[\begin{equation} \begin{aligned} y_t = &l_{t-1} + s_{t-m} + \epsilon_t \\ {l}_{t} = &l_{t-1} + \alpha\epsilon_t \\ s_t = &s_{t-m} + \gamma\epsilon_t \end{aligned}. \tag{7.4} \end{equation}\] Let’s assume that this is the “true model” (as discussed in Section 1.4) for whatever data we have for whatever reason. In this case, the set of equations (7.4) tells us that the seasonal indices change over time, depending on the value of the smoothing parameter \(\gamma\) and each specific value of \(\epsilon_t\), which is assumed to be i.i.d. All seasonal indices \(s_{t+i}\) in a particular period (e.g. monthly indices in a year) can be written down explicitly based on (7.4): \[\begin{equation} \begin{aligned} s_{t+1} = &s_{t+1-m} + \gamma\epsilon_{t+1} \\ s_{t+2} = &s_{t+2-m} + \gamma\epsilon_{t+2} \\ \vdots \\ s_{t+m} = &s_{t} + \gamma\epsilon_{t+m} \end{aligned}. \tag{7.5} \end{equation}\] If this is how the data is “generated” and the seasonality evolves over time, then there is only one possibility, for the indices \(s_{t+1}, s_{t+2}, \dots, s_{t+m}\) to add up to zero: \[\begin{equation} s_{t+1}+ s_{t+2}+ \dots+ s_{t+m} = 0 \tag{7.6} \end{equation}\] or after inserting (7.5) in (7.6): \[\begin{equation} s_{t+1-m}+ s_{t+2-m}+ \dots+ s_{t} + \gamma \left(\epsilon_{t+1}+ \epsilon_{t+2}+ \dots+ \epsilon_{t+m}\right) = 0 , \tag{7.7} \end{equation}\] meaning that:
- The previous indices \(s_{t+1-m}, s_{t+2-m}, \dots, s_{t}\) add up to zero and
- Either:
- \(\gamma=0\),
- the sum of error terms \(\epsilon_{t+1}, \epsilon_{t+2}, \dots, \epsilon_{t+m}\) is zero.
Note that we do not consider the situation \(s_{t+1-m}+ \dots+ s_{t} = -\gamma \left(\epsilon_{t+1}+ \dots+ \epsilon_{t+m}\right)\) as it does not make sense. The condition (1) can be considered reasonable if the previous indices are normalised. (2.a) means that the seasonal indices do not evolve over time. However, (2.b) implies that the error term is not independent, because \(\epsilon_{t+m} = -\epsilon_{t+1}- \epsilon_{t+2}- \dots- \epsilon_{t+m-1}\), which violates one of the basic assumptions of the model from Subsection 1.4.1, meaning that (7.5) cannot be considered as the “true” model anymore, as it omits some important elements. Thus renormalisation is unnatural for the ETS from the “true” model point of view.
Alternatively each seasonal index could be updated on each observation \(t\) (to make sure that the indices are renormalised). In this situation we have: \[\begin{equation*} \begin{aligned} &s_t = s_{t-m} + \gamma\epsilon_t \\ &s_{t-m+1}+ s_{t-m+2}+ \dots+ s_{t-1} + s_{t} = 0 \end{aligned}, \end{equation*}\] which can be rewritten as \(s_{t-m} + \gamma\epsilon_t = -s_{t+1-m}- s_{t+2-m}- \dots- s_{t-1}\), meaning that: \[\begin{equation*} \begin{aligned} s_{t-m}+ s_{t+1-m}+ s_{t+2-m}+ \dots+ s_{t-1} = -\gamma\epsilon_t \end{aligned}. \end{equation*}\] But due to the renormalisation, the sum on the left-hand side should be equal to zero, implying that either \(\gamma=0\) or \(\epsilon_t=0\). While the former might hold in some cases (deterministic seasonality), the latter cannot hold for all \(t=1,\dots,T\) and again violates the model’s assumptions. The renormalisation is thus impossible without changing the structure of the model. Hyndman et al. (2008) acknowledge that and propose in Chapter 8 some modifications for the seasonal ETS model (i.e. introducing new models), which we do not aim to discuss in this chapter.
The discussion in this section demonstrates that the renormalisation of seasonal indices is unnatural for the conventional ETS model and should not be used. This does not mean that the initial seasonal indices (corresponding to the observation \(t=0\)) cannot be normalised. On the contrary, this is the desired approach as it reduces the number of estimated parameters in the model. But this means that there is no need to implement an additional mechanism of renormalisation of indices in-sample for \(t>0\).