Parameter regions for monthly data

 

 

 

Fig. 10.4. Light-shaded region: the forecastable region of α and β for model
ETS(A,A_d,A) with m = 12. Dark-shaded region: the usual region where 0 < α < 1 − γ
and 0 < β < α. The right column shows the region for model ETS(A,A,A) (when
φ = 1).

 

where

 

 

and

Note that this is equivalent to (10.5) if we reparameterize the model,
replacing α in (10.5) by α − γ/m. Therefore the forecastability conditions for

 

 

10.4 Exercises 161

the standard ETS(A,A_d,A) model are the same as the stability conditions for the normalized ETS(A,A_d,A) model, apart from this minor reparam-
eterization. In particular, the normalized models are stable (provided the parameters are within the stability regions).

 

10.3 Conclusions

With the non-seasonal exponential smoothing models, our results are clear: the models are of minimal dimension and are stable using the usual con-
straints. In fact, it is possible to allow parameters to take values in a larger space, and still retain a stable model.

With the seasonal exponential smoothing methods, the situation is more
complicated.Thereisaredundancyinthestatevectorbecausetheseasonalstates
are not constrained, making the models of larger dimension than necessary. The
same redundancy leads to a unit root in the discount matrix, causing all of the
linear seasonal models to be unstable for any values of the model parameters.
However, we have shown that the model can be made forecastable, and we
have provided conditions for the parameters to ensure forecastability.

The normalized model circumvents this problem by requiring the seasonal
states to sum to zero, thus removing the inherent redundancy in the seasonal
terms. This leads to both a minimal dimension model and a stable model.

 

10.4 Exercises

 

Exercise 10.1.

a. Show that the non-seasonal models ETS(A,N,N) and ETS(A,A_d,N) are of
minimal dimension.

b. Show that the seasonal models ETS(A,A,A) and ETS(A,A_d,A) are not of
minimal dimension.

c. Show that the normalized seasonal models ETS(A,N,A), ETS(A,A,A) and ETS(A,A_d,A) are of minimal dimension.

d. Show that the (unnormalized) seasonal models ETS(A,A,A) and

ETS(A,A_d,A) are of minimal dimension if the level component is omitted from the models. (This is an alternative to normalization).

Exercise 10.2. Complete Example 10.3 by showing that u is proportional to [−1, 0, 1,..., 1] for the ETS(A,A,A) model.

Exercise 10.3. The expression x_t = Dx_t−1 + gy_t also applies to some of the nonlinear models discussed in Chap. 4. Use this observation to write down the stability conditions for the relevant nonlinear models.