Chapter 11 Times Series 473
is provided and shows a profi le very similar to the one you saw earlier with
the boxplot.
A chart is also included, showing both the production data and the
adjusted production values. There is a clear increase in liquor sales in the
data set after adjusting for seasonal variation. To further explore this trend,
you can smooth the sales data using three-parameter exponential smoothing.
Three-Parameter Exponential Smoothing
You perform exponential smoothing on seasonal data using three smoothing
constants. This process is known as three-parameter exponential smooth-
ing or Winters’ method. The smoothing constants in the Winters’ method
involve location, trend, and seasonality. Winters’ method can be used
for either multiplicative or additive seasonality, though in this text, we’ll
assume only multiplicative seasonality. The equation for a time series variable
yt with a multiplicative seasonality adjustment is
yt 51 b 0 1b 1 t^23 Ip1et
and for additive seasonality adjustment the equation is
yt 51 b 0 1b 1 t^21 Ip1et
In these equations b 0 , b 1 , and et once again represent the location, trend,
and error parameters of the model and Ip represents the seasonal index at point
p in the seasonal data. For example, if we used the multiplicative seasonal
indexes shown in Figure 11-29, I 5 would equal 1.015. Once again, these para-
meters are not considered to be constant but can vary with time. The liquor
sales data are an example of such a series. The sales are seasonal, but there
is also a time trend to the data such that sales increase from year to year after
adjusting for seasonality.
Let’s concentrate on smoothing with a multiplicative seasonality factor.
The smoothing equations used in three-parameter exponential smoothing
are similar to equations you’ve already seen. For the smoothed location
value Sn and the smoothed trend value Tn, from a time series where the
length of the period is q, the recursive equations are
Sn 5 w
yn
In 2 q
1112 w^21 Sn 211 Tn 212
Tn 5 t 1 Sn 2 Sn 2121112 t 2 Tn 21
Note that the recursive equation for Sn is identical to the equation used in
two-parameter exponential smoothing except that the current observation yn
must be seasonally adjusted. Here, In 2 q is the seasonal index taken from the
