Modeling and Multivariate Methods - SAS

Chapter 14 Performing Time Series Analysis 379
Smoothing Models
The example shown here has the Level weight (α) fixed at a value of 0.3 and the Trend weight (γ) bounded
by 0.1 and 0.8. In this case, the value of the Trend weight is allowed to move within the range 0.1 to 0.8
while the Level weight is held at 0.3. Note that you can specify all the smoothing weights in advance by
using these custom constraints. In that case, none of the weights would be estimated from the data although
forecasts and residuals would still be computed. When you click Estimate, the results of the fit appear in
place of the dialog.
Simple Exponential Smoothing
The model for simple exponential smoothing is y t
= μ t
+ α t .
The smoothing equation, L t = αy t +(1– α)L t-1 , is defined in terms of a single smoothing weight α. This
model is equivalent to an ARIMA(0, 1, 1) model where
( 1 – B)y t
= ( 1 – θB)α t with θ = 1 – α .
The moving average form of the model is
∞
y t
= a t
+ αa t–
j
j – 1
Double (Brown) Exponential Smoothing
The model for double exponential smoothing is y t
= μ t
+ β 1
t+
a t .
The smoothing equations, defined in terms of a single smoothing weight α are
L t
= αy t
+ ( 1 – α)L t – 1 and T t
= α( L t
– L t – 1
) + ( 1 – α)T t – 1 .
This model is equivalent to an ARIMA(0, 1, 1)(0, 1, 1) 1 model
( 1 – B) 2 y t
= ( 1 – θB) 2 a t where θ 11 ,
= θ 21 , with θ = 1 – α .