Classical Decomposition Model

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

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