For this paper I have gathered quarterly data
on the sales of Calloway Golf

Company from 1995 to the third quarter of
1999,and will attempt to fit a time

series model using the Classical
Decomposition Method, which uses a multifactor

model shown below: Yt =
f(T,C,S,e) where Yt = actual value of the time series at

time t f =
mathematical function of T = trend C = cyclical influences S =

seasonal
influences e = error The trend component (T) in a time series is the

long-run
general movement caused by long-term economic, demographic, weather
and

technological movements. The cyclical component (C) is an influence of
about

three to nine years caused by economic, demographic, weather, and
technological

changes in an industry or economy. The seasonal variations (S)
are the result of

weather and man-made conventions such as holidays. These
can occur every year,

month week, or 24 hours. The error term (e) is simply
the residual component of

a time series that is not explained by T, C, and S.
There are two general types

of decomposition models that can be used. They
are the additive and

multiplicative decomposition models. Additive: Y = T + C
+ S + e Multiplicative:

Y = T * C * S * e As you can see above the type
of seasonality can be determined

by looking at the plot of the data. The
determination of whether seasonal

influences are additive or multiplicative
is usually evident from the plot of

the data, but this is not the case with
the data for Calloway as you can see

from the first graph of the quarterly
sales. While it is my pretension that the

seasonal influences for Calloway
are multiplicative, I will use both methods and

compare the two models to
determine which is a better fit for the quarterly data

for Calloway Golf.
Multiplicative Model In the multiplicative decomposition

model, which is the
most frequently used model, Y is a product of the four

components, T, C, S,
and e. C and S are indexes that are proportions centered on

1. Only the
trend, T, is measured in the same units as the items being

forecasted. The
first step in the decomposition method is to find the seasonal

indexes, as
shown in table 1, in this case by performing a four-period moving

average and
using a method called the ratio to moving average method. It is

necessary to
measure the seasonality first because it is difficult to measure

the trend of
a highly seasonal series. By looking at the final seasonal indexes

we can see
that there is seasonality in the series, because the indexes are

smaller in
the first and fourth quarters. One would expect this, because the

sales of
golf equipment are more likely to occur in the spring and summer,

rather than
the fall and winter. Once the final seasonal indexes are calculated

and
adjusted we can move on to the next step of the decomposition method. Once

we
have identified the seasonal component of demand, the trend-cycle of
the

series can be estimated. Decomposing the trend-cycle is done by
deseasonalizing

the actual sales. This is shown in table 2 and was calculated
using the

following equation: Y/S = TCSe/S = Tce Where S = the seasonal index
for period

t. Once the deseasonalized sales have been calculated, one must
use a simple

linear regression to determine the trend in sales. This is shown
in graph 2,

where the deseasonalized sales have been plotted and a regression
(trend) line

has been added with the equation above the chart. We simply plug
the t values

into the equation to find the trend (Tt) values as shown in
table 2. The next

step in the multiplicative decomposition model is to
calculate the fitted values

(TS) by multiplying the trend (T) by its
appropriate seasonal factor. This is

shown in table 3, the fitted
decomposition time-series model. Once this is done

I calculated the
errors of the model, as shown in table 3, and measured the

accuracy of the
fit using the known actuals. As you can see, the adjusted

R-squared
equals .698, which means that nearly 70% of the original variance
of

Y(45.594^2) has been removed by decomposing it into its seasonal and
trend

components. Although the RSE is fairly high, the R-squared is also
quite high,

so I would conclude that the model is a fairly good fit. Additive
Model The

steps of the additive decomposition method are very similar to
those of the

multiplicative model, which I have described above. The first
difference,

though, is that with the additive method Y is the sum of its four
components, T,

S, C, and e. Because the steps are so similar between the
two methods I am not

going to go into a detailed explanation of the steps,
but I will describe the

major differences in the two models. Like the
multiplicative method, we must

first calculate a four-period moving average
and center it to estimate the trend

cycle. Next we must subtract the centered
moving average from the actual sales

to obtain the seasonal error factor for
each period. Next, we use these error

terms to calculate the unadjusted
seasonal indexes. This is where the methods in

the two models differ. The
mean of the unadjusted seasonal indexes must be

determined and then
subtracted from each of the unadjusted terms to calculate

the final seasonal
indexes. In the additive model, the sum of the final seasonal

indexes must be
equal to 0. All of this is shown at the bottom of table 5. Now

that we have
the final seasonal indexes, we can calculate the deseasonalized

sales by
subtracting the seasonal index from the actual sales for each
period.

These values are simply estimates of trend-cyclical error. The
deseasonalized

sales are shown in the eighth column of table 5. Once we have
determined the

deseasonalized sales, we can plot the data and find a trend
line, which will

help us to determine the equation for the trend of the
deseasonalized data. The

plot of the deseasonalized data is shown in graph 3,
with the trend line and

equation added in. With that trend equation we can
estimate the fitted trend

values, which are shown in column ten of table 5.
Lastly, to find the fitted Yt

values, we add the fitted trend to its
appropriate seasonal index. Now that we

have estimated all of the fitted Yt
values, we must now estimate the errors of

the model, and measure the
accuracy of the fit using the known actuals just like

in the multiplicative
model. As you can see from table 5, the adjusted R-squared

equals .2795,
which means that nearly 28% of the original variance of

Y(45.594^2) has
been removed by decomposing it into its seasonal and trend

components. In
this model the RSE is very high and the R-squared is quite low,

so I would
conclude that the model is not a very good fit. Conclusion While it

was
difficult to say from looking at the plot of the data whether the
seasonal

influences where additive or multiplicative, the analysis of the RSE
and the

R-squared reinforced my hypothesis that the model with the best
fit for the

quarterly data of Calloway Golf is the multiplicative
decomposition method.