Modeling and Multivariate Methods - SAS

Chapter 14 Performing Time Series Analysis 381
Smoothing Models
This model is equivalent to a seasonal ARIMA(0, 1, 1)(0, 1, 0) S model where we define
so
with
θ 1
= θ 11 , , θ = 2
θ = 2, s
θ 2,
s , and θ = – θ 3 1 ,
θ 1 2,
s
( 1 – B) ( 1 – B s )y t
= ( 1 – θ 1 B – θ 2 B 2 – θ 3 B s + 1 )a t
θ 1
= 1 – α , θ 2
= δ( 1 – α)
, and θ 3
= ( 1 – α) ( δ–
1)
.
The moving average form of the model is
∞
y t
= a t
+ ψ j
a t–
j whereψ
j = 1
=
α for jmods ≠ 0
α+ δ( 1 – α) forjmods = 0
Winters Method (Additive)
The model for the additive version of the Winters method is y t
= μ t
+ β t
t + s() t + a t .
The smoothing equations in terms of weights α, γ, and δ are
L t
= α( y t
– S t – s
) + ( 1 – α) ( L t – 1
+ T t – 1
), T t
= γ( L t
– L t – 1
) + ( 1 – γ)T t – 1 , and
S t
= δ( y t
– L t
) + ( 1 – δ)S t – s .
This model is equivalent to a seasonal ARIMA(0, 1, s+1)(0,1,0)s model
( 1 – B) ( 1 – B 2 s + 1
)y t
1 θ i B i
= –
a t
i = 1
The moving average form of the model is
where
∞
y t
= a t
+ Ψ j
a t–
j
j = 1
α
+ jαγ , jmods ≠ 0
ψ =
α + jαγ + δ( 1 – α) , jmods = 0