Modeling and Multivariate Methods - SAS

Chapter 14 Performing Time Series Analysis 377
Smoothing Models
Smoothing models represent the evolution of a time series by the model:
y t
= μ t
+ β t
t+ s() t + a t
where
μ t is the time-varying mean term,
β t
is the time-varying slope term,
st () is one of the s time-varying seasonal terms,
a t are the random shocks.
β t
0
Models without a trend have = and nonseasonal models have st () = 0 . The estimators for these
time-varying terms are
L t smoothed level that estimatesμ t
T t
is a smoothed trend that estimatesβ t
S t–
j for j = 01… , , , s – 1 are the estimates of the st ().
Each smoothing model defines a set of recursive smoothing equations that describes the evolution of these
estimators. The smoothing equations are written in terms of model parameters called smoothing weights.
They are
α, the level smoothing weight
γ, the trend smoothing weight
ϕ, the trend damping weight
δ, the seasonal smoothing weight.
While these parameters enter each model in a different way (or not at all), they have the common property
that larger weights give more influence to recent data while smaller weights give less influence to recent data.
Each smoothing model has an ARIMA model equivalent. You may not be able to specify the equivalent
ARIMA model using the ARIMA command because some smoothing models intrinsically constrain the
ARIMA model parameters in ways the ARIMA command will not allow.
Simple Moving Average
A simple moving average model (SMA) produces forecasted values that are equal to the average of
consecutive observations in a time window. The forecasts can be uncentered or centered in the time window.
To fit a simple moving average model, select Smoothing > Simple Moving Average. A window appears
with the following options:
Enter smoothing window width
Enter the width of the smoothing window.
Centered
Choose whether to center the forecasted values.