Seasonal Models. The characteristic equation for matrix D in the (un-normalized) ETS(A,N,A) model is f (λ) = (1 − λ)P(λ) = 0

The characteristic equation for matrix D in the (un-normalized) ETS(A,N,A) model is f (λ) = (1 − λ)P(λ) = 0, where

(10.4)

Thus, D has a unit eigenvalue regardless of the values of the model parameters, and so the model is always unstable.

 

 

156 10 Some Properties of Linear Models

Similarly, the characteristic equation of D for model ETS(A,A_d,A) is f(λ) = (1 − λ)P(λ) = 0, where

P(λ) = λ^m+1 + (α + β − φ)λ^m + (α + β − αφ)λ^m−1 + · · · + (α + β − αφ)λ²

+ (α + β − αφ + γ − 1)λ + φ(1 − α − γ). (10.5)

 

 

The same argument applies to all of the seasonal models. Thus, the ETS(A,N,A), ETS(A,A,A) and ETS(A,A_d,A) models are forecastable if and only if the roots of P(λ) lie inside the unit circle. Hyndman et al. (2008) use this result to derive the specific conditions for forecastability; these conditions are summarized in Table 10.2.

The inequalities involving only α and γ provide necessary conditions
for forecastability that are easily implemented. The final condition (giving
a range for β) is more complicated to use than finding the numerical roots
of (10.5). Therefore, we suggest that, in practice, these conditions on α and
γ be imposed when estimating the model; the roots of (10.5) can then be
calculated and tested.

To visualize these regions, we have plotted them in Figs. 10.2-10.4. The light-shaded regions represent the forecastability regions; the dark-shaded regions are the usual regions given by

0 < α < 1, 0<β<α, 0 < γ < 1− α, and 0 < φ < 1.

The forecastable region for α and γ is illustrated in Fig. 10.2. For large
values of φ, the upper limit of γ is obtained when the upper limit of α equals
the lower limit of α. For φ = 1, this simplifies to γ < 2m/(m − 1), as given by
Archibald (1991), but for smaller values of φ the upper limit of γ is smaller
than this.

The right hand column of Fig. 10.2 shows that the usual parameter
region of an ETS(A,N,A) model is entirely within the forecastability region.

 

10.2 Stability and the Parameter Space 157

Table 10.2. Forecastability conditions for models ETS(A,N,A) and ETS(A,A_d,A).

 

 

 

Therefore ETS(A,N,A) models obtained using the usual constraints are always forecastable.

The forecastable region for α and β is depicted in Figs. 10.3 and 10.4 for
m = 4 and m = 12 respectively. For m = 4, the usual parameter region is
entirely contained within the forecastability region for all values of φ and
γ, except when both φ and γ are relatively small. However, for m = 12
(Fig. 10.4), it can be seen that the usual parameter region and the forecastabil-
ity region intersect for model ETS(A,A_d,A), but neither is contained within
the other, even when φ = 1. Therefore, models obtained using the usual
constraints may often not be forecastable.

Consequently, we recommend that the usual parameter regions not be
used. Instead, when parameters are estimated, the optimization routine
should be constrained to return values within the forecastability region. If
we constrain the parameters to lie in the intersection of the usual region
and the forecastability region, we can retain the interpretation of the model
equations as weighted averages. However, such constraints may produce
inferior forecasts when the best-fitting model lies outside the more restricted
region.

 

158 10 Some Properties of Linear Models

 

 

Fig. 10.2. Light-shaded region: the forecastable region of α and γ for model ETS(A,A_d,A). Dark-shaded region: the usual region where 0 < α < 1 and 0 < γ < 1− α. The right column shows the regions for model ETS(A,A,A) (when φ = 1). These are also the regions for model ETS(A,N,A) as they are independent of β.