1. Cycle average:
- (40 + 50 + 60 + 35) / 4 = 46.25
- (30 + 50 + 60 + 30) / 4 = 42.5
- (35 + 60 + 80 + 40) / 4 = 53.75
- (50 + 70 + 100 + 50) / 4 = 67.5
2. Trend:
- (46.25 + 42.5) / 2 = 44.375
- (42.5 + 53.75) / 2 = 48.125
- (53.75 + 67.5) / 2 = 60.625
3. First residuals:
- Quarter 1, 1997: 40 - 44.375 = -4.375
- Quarter 2, 1997: 60 - 44.375 = 15.625
- Quarter 3, 1997: 80 - 48.125 = 31.875
- Quarter 4, 1997: 35 - 48.125 = -13.125
And so on for the rest of the quarters.
4. Seasonal factors:
- Average of the first residuals for Quarter 1 = (-4.375 - 2.5) / 4 = -1.46875
- Average of the first residuals for Quarter 2 = (15.625 - 2.5) / 4 = 3.28125
- Average of the first residuals for Quarter 3 = (31.875 - 2.5) / 4 = 7.34375
- Average of the first residuals for Quarter 4 = (-13.125 - 2.5) / 4 = -3.40625
5. Fitted values:
- Trend + seasonal factor
- Quarter 1, 1997: 44.375 - 1.46875 = 42.90625
- Quarter 2, 1997: 44.375 + 3.28125 = 47.65625
- Quarter 3, 1997: 48.125 + 7.34375 = 55.46875
- Quarter 4, 1997: 48.125 - 3.40625 = 44.71875
And so on for the rest of the quarters.
6. Second residuals:
- Observed value - fitted value
- Quarter 1, 1997: 40 - 42.90625 = -2.90625
- Quarter 2, 1997: 60 - 47.65625 = 12.34375
- Quarter 3, 1997: 80 - 55.46875 = 24.53125
- Quarter 4, 1997: 35 - 44.71875 = -9.71875
And so on for the rest of the quarters.
Question
A small ice-cream manufacturer who supplies a supermarket chain reports sales figures
of K50, 000 for the last quarter.
Is this amount good or bad? Can this single figure be used to forecast future sales?
With only one figure we never have sufficient data to make either a judgment or a
forecast.
We also need to compare the figure with:
-that for the previous quarter;
-that for the corresponding quarter of the previous year;
-that which would be expected from the current trend, if that can be identified.
We need to identify any past pattern and use it for future sales predictions.
We also need to estimate how accurate those predictions are likely to be.
In order to understand the situation more fully, the following data was collected. It
represents the quarterly sales figures for the four years preceding the figure of K50 000
quoted above.
Sales (K'000)
Year Quarter 1 Quarter 2 Quarter 3 Quarter 4
1997 40 60 80 35
1998 30 50 60 30
1999 35 60 80 40
2000 50 70 100 50
Method
Use an additive model to identify how much has been 'added' to the sales because it is a particular quarter of the year.
Additive model Observed
Sales value
=
Trend
value
+
Seasonal
factor
+
Random
factor
A = T + S + R
The Trend value, T, is the value of the sales had the seasonal effect been 'averaged out'.
The Seasonal factor, S, is the average effect of it being a particular quarter, e.g.
summery.
The Random factor, R, is the random variation due to neither of the previous variables.
This cannot be eliminated but is a useful measure of the goodness of fit of the model and
therefore of the quality of any forecasts made from it.
Step by stepe and by showing your work calculate the and find the:
1.cycle average which is the moving average of four consecutive quarters, i.e. one
from each season. As this value is not centered on any quarter its deviation from the
observed sales cannot be found.
2.The trend which is the moving average of two consecutive cycle averages and is
centered on a quarter. This second calculation is only necessary for cycles with even
numbers of periods as an odd number would produce a cycle average which was
already centered.
3.The first residual is the difference between the observed values and the trend. This will
be largely due to the seasonality of the data.
4.From these first residuals, calculate the seasonal factor which is the deviation from the trend due to it being a particular
season.
The fitted values are those produced by the model. i. e. trend + seasonal factor.
The second residual is the difference between the observed and fitted model values.
1 answer