Modeling and Multivariate Methods - SAS

380 Performing Time Series Analysis Chapter 14
Smoothing Models
The moving average form of the model is
∞
y t
= a t
+ ( 2α + ( j – 1)α )a t–
j
j = 1
Linear (Holt) Exponential Smoothing
The model for linear exponential smoothing is y t
= μ t
+ β t
t + a t .
The smoothing equations defined in terms of smoothing weights α and γ are
L t
= αy t
+ ( 1 – α) ( L t – 1
+ T t – 1
) andT t
= γ( L t
– L t – 1
) + ( 1 – γ)T t – 1
This model is equivalent to an ARIMA(0, 2, 2) model where
( 1 – B) 2 y t
= ( 1 – θB – θ 2 B 2 )a t with θ = 2 – α – αγ and θ 2
= α – 1 .
The moving average form of the model is
∞
y t
= a t
+ ( α + jαγ)a t – j
j = 1
Damped-Trend Linear Exponential Smoothing
The model for damped-trend linear exponential smoothing is y t
= μ t
+ β t
t+
a t .
The smoothing equations in terms of smoothing weights α, γ, and ϕ are
L t
= αy t
+ ( 1 – α) ( L t – 1
+ ϕT t – 1
) andT t
= γ( L t
– L t – 1
) + ( 1 – γ)ϕT t – 1
This model is equivalent to an ARIMA(1, 1, 2) model where
( 1 – ϕB) ( 1 – B)y t
= ( 1 – θ 1 B – θ 2 B 2 )a t with θ 1
= 1 + ϕ– α–
αγϕ and θ 2
= ( α – 1)ϕ.
The moving average form of the model is
∞
α + αγϕ( ϕ j – 1)
y t
= α t
+ ---------------------------------------
αt
ϕ – 1 – j
j = 1
Seasonal Exponential Smoothing
The model for seasonal exponential smoothing is y t
= μ t
+ st () + a t .
The smoothing equations in terms of smoothing weights α and δ are
L t
= α( y t
– S t – s
) + ( 1 – α)L t – 1 andS t
= δ( y t
– L t–
s
) + ( 1 – δ)ϕS t–
s