Question 1:
The All American Toy Company has a very seasonal sales pattern. Sales are high during the
fall quarter, drop off substantially in the winter quarter and are more typical during spring and
summer quarters. The following historical data exist for the past 16 quarters.
Based on these data, develop a seasonally adjusted forecast for the four quarters of 2002 using
a linear trend regression model.
ANSWER:
The time-series plot illustrates that there is a pronounced seasonal component as well
as a trend component.
The first step in developing a seasonally adjusted forecast for the four quarters of 2002 is to
calculate the seasonal indexes for the four quarters. We can do this using the multiplicative
model:
yt  Tt x St x Ct x I t .
To compute the seasonal indexes, we go through several steps. First, for quarterly data, we
compute 4 period moving averages shown as follows:
Now, we need to compute the centered moving averages by averaging each successive pair of
moving averages as shown:
The centered moving averages provide an estimate for the Tt x Ct portion of the
multiplicative model. To isolate the seasonal and irregular portion, we compute the ratio-tomoving-average by dividing each y value by the centered moving average value as shown:
Note, we now have three ratio to moving average values for each quarter. We can arrange
these in a table format as follows:
The next step is to average these ratio-to-moving-averages for each quarter as follows:
The averages represent the seasonal indexes for each quarter. For example, the index for Fall
quarter is 1.34, indicating that fall sales are 134 percent of the “normal” quarter sales.
These indexes can be used to adjust the forecast made for each quarter.
Next, we need to deseasonalize the data by dividing each sales value by its appropriate
seasonal index. The results are shown as follows:
.
Linear regression can be used to develop a trend-based forecasting model. The model will
take the following form:
yˆ  b0  b1t .
The intercept and slope are calculated using the least squares equations:
 ty   n

( t )
t  
t
b1
y
2
2
n
and
b0  y  b1t .
Note, the y values that are used are the deseasonalized values.
The results of the calculations are:
yˆ  4779.3  153.3t .
To generate the forecast for periods 17-20, we substitute t = 17, t=18, t=19, and t=20 into the
forecast equation as follows:
For Winter Quarter 2002 (t=17)
yˆ  4779.3  153.3(17)  7385.4
For Spring Quarter 2002 (t=18)
yˆ  4779.3  153.3(18)  7538.70
For Summer Quarter 2002 (t-19)
yˆ  4779.3  153.3(19)  7692.0
For Fall Quarter 2002 (t=20)
yˆ  4779.3  153.3(20)  7845.3
These forecasts are referred to as seasonally unadjusted forecasts. The final step is to adjust
these for the seasonal component by multiplying each forecast by the associated seasonal
index.
Winter Quarter 2002:
7385.4 x 0.60 = 4431.24
Spring Quarter 2002:
7538.7 x 1.04 = 7840.25
Summer Quarter 2002:
7692.0 x 1.03 = 7922.76
Fall Quarter 2002:
7845.3 x 1.34 = 10512.70
These are the seasonally adjusted forecasts for the 2002 year.
Question 2:
The Norton Industrial Company has 12 quarters of historical sales data and is interested in
forecasting sales for quarter 13. The following data are available:
Based on this data, use simple exponential smoothing with a smoothing constant value equal
to 0.10 and a starting forecast value equal to the sales in quarter 1 to forecast sales for period
13.
ANSWER:
The simple exponential smoothing equation is:
Ft 1  Ft   ( yt  Ft )
Given a starting forecast value for period 1 equal to sales in period 1 we get the following:
Thus, the forecast for period 13 is 436.3.