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 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.