Example 10.2: Local level model with drift

The local level model with drift is equivalent to the ETS(A,A,N) model with β =0. Thus, the discount matrix for this model is

D =

1 − α 1 − α

, so that U =

(1 − α)/ q

and V =

−

(1 −α)/ α

,

α / q

q / α

where q = ^√ 1 − 2 α + 2 α ². The corresponding roots of D are 1 and 1 − α, and corresponding to the unit root we have:

w^_u = [1, 1]	(1 − α)/ q	= 1/ q	and v^_g = [0, q / α ]	α	= 0.
	α / q

Thus, the model is not stable as D has a unit root. However, it can be forecastable as the unit root satisfies the third condition of Theorem 10.2.

The other root is 1 − α with

w^_u = [1, 1]		= 1 and	v^_g = [1, − (1 − α)/ α ]	α	= α.

So the model will be forecastable if 0 < α < 2 (condition 1 of Theorem 10.2), or if α = 0 (condition 3 of Theorem 10.2). If α = 0, the model is equivalent to the linear regression model y_t = _ ₀ + bt + ε _t.

The stability conditions for models without seasonality (i.e., ETS(A,N,N), ETS(A,A,N) and ETS(A,A_d,N)) are summarized in Table 10.1. These are given

10.2 Stability and the Parameter Space

Table 10.1. Stability conditions for models without seasonality.

ETS(A,N,N):		< α < 2
ETS(A,A,N):		< α < 2
		< β < 4 − 2 α
ETS(A,Ad,N):	1 − 1/ φ < α < 1 + 1/ φ
	α (φ − 1) < β < (1+ φ)(2 − α)
		< φ ≤ 1

φ = 0.5

φ = 0.9

φ = 1

β

β

β

−1

−1

−1

−1

−1

−1

α

α

α

Fig. 10.1. Parameter spaces for model ETS(A,A_d,N). The right hand graph shows theregion for model ETS(A,A,N) (when φ = 1). In each case, the light-shaded regions rep-resent the stability regions; the dark-shaded regions are the usual regions constructed by restricting each parameter in the conventional parameterization to lie between 0 and 1.

in McClain and Thomas (1973) for the ETS(A,A,N) model; results for the ETS(A,A_d,N) and ETS(A,N,N) models are obtained in a similar way. To visualize these regions, we have plotted them in Fig. 10.1. The light-shaded regions represent the stability regions; the dark-shaded regions are the usual regions defined by 0 < β < α < 1. Note that the usual parameter region is entirely within the stability region in each case. Therefore non-seasonal models obtained using the usual constraints are always stable (and always forecastable).

10.2.1 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, where

P (λ) = λ^m + αλ^{m− 1}+ αλ^{m− 2}+ · · · + αλ ²+ αλ + (α + γ − 1). (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) λ ²

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

(10.5)

Example 10.3: Additive Holt-Winters’ model ETS(A,A,A)

In this case,

α 1

α 0 ^_

α

−β 1

m 1

^−β 0 m_^− 1

−

β

D =

−γ

^−γ 0 _

−

1 ^−

γ

.

−

−

m− 1

−

0 m

1 0 m

1 ^Im

0 m

−

−

−

−

Solving the equations corresponding to the unit root case (Exercise 10.2) shows that u is proportion to [ − 1, 0, 1,..., 1]. It follows that w^_u = 0. Con-sequently, the model is forecastable if the remaining roots are inside the unit circle.

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 becalculated 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 < 2 m /(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

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

ETS(A,N,A):

max(

−

mα, 0) < γ < 2

−

α and

− 2

< α < 2

−

γ

m

−

ETS(A,Ad,A):

0 < φ ≤ 1

max(1 − 1/ φ − α, 0) < γ < 1 + 1/ φ − α

1 − 1/ φ − γ (1 − m + φ + φm)/(2 φm) < α < (B + C)/(4 φ)

− (1 − φ)(γ / m + α) < β < D + (φ − 1) α

where

B = φ (4 − 3 γ) + γ (1 − φ)/ m

C =^! B ^2 − 8^_ φ ^2(1 − γ)^2+2(φ − 1)(1 − γ) − 1^_+8 γ ^2(1 − φ)/ m

D =min θ ^"(φ − φα +1)(1 − cos θ)

− γ ^_

(

+ φ)(

1 − cos

θ

mθ) +cos(m

−

1) θ + φ

(m +

) θ

_ #

^− ^cos2(1 − cos mθ)

cos

and θ is a solution to

φα − φ +1+

(φ − 1)(1 + cos θ − cos mθ) + cos(m − 1) θ − φ cos(m + 1) θ

= 0

γ

2(1 + cos θ)(1 − cos mθ)

Conditions for ETS(A,A,A) can be obtained from ETS(A,A_d,A) by setting φ = 1.

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 isentirely 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.

= 1 − s _{1, t}

158 10 Some Properties of Linear Models

		φ = 0.5, m = 12					φ = 0.9, m = 12					φ = 1, m = 12
2.0					2.0					2.0
γ					γ					γ
1.0					1.0					1.0
0.0					0.0					0.0
−1					−1					−1
		α					α					α
		φ = 0.5, m = 4					φ = 0.9, m = 4					φ = 1, m = 4
2.0					2.0					2.0
γ					γ					γ
1.0					1.0					1.0
0.0					0.0					0.0
−1					−1					−1
		α					α					α

Fig. 10.2. Light-shaded region: the forecastable region of α and γ for modelETS(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.

10.2.2 Normalized Models

Archibald (1984, 1990) discussed the stable region for the normalized ver-sion of ETS(A,A,A), and Archibald (1991) provided some preliminary steps towards finding the stable region for the normalized version of ETS(A,A_d,A). Hyndman et al. (2008) extended this analysis and derived the results described below.

In Chap. 8, we showed that the normalized models can be written in state

space form with the state vector x_t = (__t, b_t, s _{1, t} ,..., s_{m− 1, t} ) ^_, where s_{i , t} is the estimate of the seasonal factor for the i th month ahead made at time t. Note

that s_{m , t} ≡ s _{0, t} − · · · − s_{m− 1, t} . Following Roberts (1982, Sect. 3), the

seasonal updating is defined as follows:

s 0, t = s 1, t− 1+ γ (1 − _m ¹) et, si, t = si +1, t− 1 − _m^γ et.

The level and trend equations are updated as with the standard model. Then w^_ = [1, 1, 1,0 ^__{m− 2}],

0 ^_

m 2

F =

φ

0 m_^− 2

,

g =

α

0 m

0 m

0 m

−

β

−

−

−

^Im

−

(γ / m)1

m

−

−

1 m^_

−

−

−

10.2 Stability and the Parameter Space

Parameter regions for quarterly data

γ = 0.1,

φ = 0.5

γ = 0.1,

φ = 0.9

γ = 0.1,

φ = 1

1.5

1.5

1.5

0.5

0.5

0.5

β

β

β

−0.5

−0.5

−0.5

−1.5

−1.5

−1.5

−1

−1

−1

α

α

α

γ = 0.5,

φ = 0.5

γ = 0.5,

φ = 0.9

γ = 0.5,

φ = 1

1.5

1.5

1.5

0.5

0.5

0.5

β

β

β

−0.5

−0.5

−0.5

−1.5

−1.5

−1.5

−1

−1

−1

α

α

α

γ = 0.9,

φ = 0.5

γ = 0.9,

φ = 0.9

γ = 0.9,

φ = 1

1.5

1.5

1.5

0.5

0.5

0.5

β

β

β

−0.5

−0.5

−0.5

−1.5

−1.5

−1.5

−1

−1

−1

α

α

α

Fig. 10.3. Light-shaded region: the forecastable region of α and β for modelETS(A,A_d,A) with m = 4. 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).

and

1 α

α

α

0 ^_

−β

φ

−

β

−

β

m 2

0 m^_

− 2

D =

−

−

−

− 2

,

(γ / m)1 m

−

2 (γ / m)1 m

−

(γ / m)1 m

−

Im

−

γ / m

γ / m

γ / m

1 m^_

−

−

−

where 1 _k denotes a k -vector of ones. The characteristic equation for D is given by

m +1
f (λ) =∑ θi λ^m ^+^1 ^−^i,	(10.6)

i =0

160 10 Some Properties of Linear Models

Parameter regions for monthly data

−1.5 −0.5 0.5 1.5 β

−1

−1.5 −0.5 0.5 1.5 β

−1

−1.5 −0.5 0.5 1.5 β

−1

γ = 0.1,		φ = 0.5
				1.5
				0.5
				β
				−0.5
				−1.5
				−1
	α
γ = 0.5,		φ = 0.5
				1.5
				0.5
				β
				−0.5
				−1.5
				−1
	α
γ = 0.9,		φ = 0.5
				1.5
				0.5
				β
				−0.5
				−1.5
				−1

α

γ = 0.1,		φ = 0.9
				1.5
				0.5
				β
				−0.5
				−1.5
				−1
	α
γ = 0.5,		φ = 0.9
				1.5
				0.5
				β
				−0.5
				−1.5
				−1
	α
γ = 0.9,		φ = 0.9
				1.5
				0.5
				β
				−0.5
				−1.5
				−1

α

γ = 0.1, φ = 1

	α
γ = 0.5,	φ = 1

	α
γ = 0.9,	φ = 1

0 1 2 3

α

Fig. 10.4. Light-shaded region: the forecastable region of α and β for modelETS(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

θ ₀=1

θ ₁= α + β − γ / m − φ

θ_i = α (1 − φ) + β − (1 − φ) γ / m, i =2,..., m − 1

θ_m = α (1 − φ) + β + γ [1 − (1 − φ)/ m ] − 1

and

θ_{m +1}= φ [1 − γ (1 − 1/ m) − α ].

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

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 arenotconstrained,makingthemodelsoflargerdimensionthannecessary. 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

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