Example 6.4: Forecast variance for the ETS(A,A,A) model 1 страница

To obtain the forecast variance for the ETS(A,A,A) model, we could take the limit as φ → 1 in (6.24) and apply L’Hospital’s rule. However, in many ways it is simpler to go back to (6.23) and replace φ_j with j. This yields (Exercise 6.2)

vn + h|n = σ ^2^_	1 + (h − 1)^_ α ^2 + αβh + 6^1	β ^2 h (2 h − 1)^_	(6.25)
	+ γhm { 2 α + γ + βm (hm + 1) } ^_.

The forecast variance expressions for all other models can be obtained as special cases of either (6.24) or (6.25):

• For (A,A_d,N), we use the results of (A,A_d,A) with γ = 0 and s_t = 0 for all t.

• For (A,A,N), we use the results of (A,A,A) with γ = 0 and s_t = 0 for all t.

• The results for (A,N,N) are obtained from (A,A,N) by further setting β = 0 and b_t = 0 for all t.

• The results for (A,N,A) are obtained as a special case of (A,A,A) with β = 0 and b_t = 0 for all t.

Derivation of Results for Class 2

The models in Class 2 can all be written using the following state space model:

yt = w^_xt− 1(1+ ε t),	(6.26)
xt = (F + gw^_ε t) xt− 1,	(6.27)

where w, g, F, x_t and ε _t are the same as for the corresponding Class 1 model. The lower tail of the error distribution is truncated so that 1 + ε _t is positive.

Appendix: Derivations

The truncation is usually negligible as σ is usually relatively small for these models.

Let ^mn + h|n = E(^xn + h | ^xn) and ^Vn + h|n = V(^xn + h | ^xn) as in Sect. 6.2. The forecast means for Class 2 have the same form as for Class 1, namely

µ_{n + h|n} = w^_m_{n + h− 1 |n} = w^_F ^{h− 1} x_n.

From (6.26), it can be seen that the forecast variance is given by

v_{n + h|n} = w^_V_{n + h− 1 |n}w (1+ σ ²) + σ ² w^_m_{n + h− 1 |n}m^__{n + h− 1 |n}w = w^_V_{n + h− 1 |n}w (1 + σ ²) + σ ² µ ² _{n + h|n}.

To obtain V_{n + h− 1 |n}, first note that x_t = F x_{t− 1} + ge_t, where e_t = y_t − w^_x_{t− 1}= w^_x_{t− 1} ε _t. Then it is readily seen that V_{n + h|n} = F V_{n + h− 1 |n}F ^_ +

gg^_ V(e_{n + h} | x_n). Now let θ_h be defined such that V(e_{n + h} | x_n) = θ_h σ ². Then,by repeated substitution,

h− 1
Vn + h|n = σ ^2∑ F ^j gg^_ (F ^j) ^_θh−j.
j =0
Therefore,
h− 1
w^_Vn + h− 1 |nw = σ ^2∑ c ^2 j θh−j,	(6.28)

j =1

where c_j = w^_F ^{j− 1} g. Now

_ _

^en + h ^{= w_ ( x}n + h− 1 ^{− m}n + h− 1 |n ^{) + w_m}n + h− 1 |n ^ε n + h ^,

which we square and take expectations to give θ_h = w^_V_{n + h− 1 |n}w + µ ² _{n + h|n}. Substituting (6.28) into this expression for θ_h gives

		h− 1	^θh−j ^+ ^µ ^2 n + h
		θh = σ ^2∑ c ^2 j	n,	(6.29)
		j =1	|
where θ 1	= µ ^2	. The forecast variance is then given by
	n +1 |n
		vn + h|n = (1+ σ ^2) θh − µ ^2 n + h|n.	(6.30)

Derivation of Results for Class 3

Note that we can write (see p. 85)

y_t = w ₁ ^_x_{t− 1} z_t^__{− 1} w ₂(1+ ε _t).

100 6 Prediction Distributions and Intervals

−→

So let ^Qh ^{= x}n + h^z_n^_ _{+ h} , ^Mh ⁼ E^{( Q}h | ^xn, ^zn ⁾ and ^Vn + h|n ⁼ V^{( Q} h | ^xn, ^zn ⁾

−→

where Q _h = vec(Q_h). Note that

^Qh = (^F 1 ^xn + h− 1+ ^G 1 ^xn + h− 1 ^ε n + h)(^z_n^_ + _h− ₁ ^F 2 ^_ + ^z_n^_ + _h− ₁ ^G 2 ^_ε n + h)

= F ₁ Q_{h− 1} F ₂ ^_ + (F ₁ Q_{h− 1} G ₂ ^_ + G ₁ Q_{h− 1} F ₂ ^_) ε _{n + h} + G ₁ Q_{h− 1} G ₂ ^_ε ² _{n + h}.

It follows that M ₀ = x_nz_n^_ and

Mh = F 1 Mh− 1 F 2 ^_ + G 1 Mh− 1 G 2 ^_σ ^2.

(6.31)

For the variance of Qh, we find V 0 = 0, and

V

=

^_vec

(F Q

F

) + vec(F Q

G

+ G

Q

F

) ε

n + h|n

V

h− 1

2 _

+ h

h− 1

2 _

h− 1

2 _

n + h

+ vec(G 1 Qh

1 G 2 ^_) ε

= (F

F) V

−

(F

n

F _

) ^_

2 ^⊗

n + h− 1 |n

⊗

Q

G

⊗

F

+

F

⊗

G

1) ^_

+ (G 2 ⊗ F 1 + F 2 ⊗ G 1)V(^−→h− 1 ε n + h )(

Q

G

2 ^⊗

G

1) ^_

+ (G 2 ⊗ G 1)V(^−→h− 1 ε n + h )(

Q

Q

G

G

1) ^_

^ε n + h ^)(

2 ^⊗

+ (F 2 ⊗ F 1)Cov(^−→h− 1, ^−→h− 1

Q

Q

F

F

1) ^_.

+ (G 2 ⊗ G 1)Cov(^−→h− 1 ε n + h , ^−→h− 1)(

2 ⊗

Next we find that

Q

Q

Q

V(^−→h− 1

ε n + h ) =E[ ^−→h− 1(^−→h− 1) ^_ε n + h ]

=

σ

V

n + h

M

M

1) ^_

,

_ Q

1 n + ^−→h

1(^−→h

Q

−Q|

4 ^−

− Q ^_

Q

V(^−→h− 1

ε n + h ) =E

^−→h− 1(^−→h− 1) ^_ε n + h − E(^−→h− 1)E(^−→h− 1) ^_σ

=

σ ^4 V

+ M (M

)

_

M

(M

)

_σ

_

^_ Vn + h− 1 |n

−→h

_

−→

h

^_ −

−→h

−→

h

=

σ

M

− 1

M

− 1

− 1

− 1

and

_3

n + h− 1 |n + 2 ^−→h− 1(^−→h− 1) ^_ ^_,

Q

Q

Q

Q

Q

Q

Cov(^−→h− 1,

^−→h− 1 ε n + h ) =

E ^−→h− 1

(^−→h− 1) ^_ε n + h

− E(^−→

h− 1)E(^−→h− 1) ^_σ

=

M

M

2 M

M

σ ^_(Vn + h− 1 |n + ^−→h− 1

(^_ ^−→h− 1) _ ) − σ

^−→h− 1(^−→h− 1) ^_

= σ ² V_{n + h− 1 |n}.

Дата добавления: 2015-10-24