Decomposition Methods and Seasonal Indexes

Here is the demonstration of Classical Decomposition Multiplicative Model:

Y = TxCxSxe

(In order to browse the Excel file online, please use the Internet Explorer, or download to see them if Netscape is used)

Step 1.  Input some quarterly sales data into the Excel files: Sales Data

Step 2. Calculate a moving average equal to the length of the seasons in a year. In this case SMA(4) is used since there are four seasons in a year. The result is as following: SMA(4)

Step 3. Center the moving averages since the season length is an even number. The result is as following: SMA2C

Step 4. Calculate the actual as a proportion of the centered moving average to obtain the unadjusted seasonal index for each period. The result is as following: Percentage MA

Step 5. Adjust the total of the seasonal indexed to equal the number of seasons in a year. In this case, the 4-quarterly seasonal indexes should be summed to 4. The result is as following: SeasonaL indexes.

Step 6. Deseasonalize the time series by dividing it by the adjusted seasonal indexes----"salesa01".  The Excel results are presented as following: Deseasonalized Data (using manipulated seasonal indexes)

Step 7. Estimate the trend-cyclical regression equation using Deseasonalized Data. Before running regression, we need to generate a trend variable by the commands: choose "GENR", type "trend=@trend(1992.1)" in the dialogue box. 

When the deseasonalized series is used, the regression result of the fitted trend values from the manipulated seasonal indexes is as follow:

Click for regression estimation and fitted Trend= C + bt. From the regression result, you can get the values for coefficients "a" and "b", and the residuals--"Resid", then the " fitted trend" can be calculated by "FITTED_TREND = TREND - Resid

Then obtain the values of the fitted trend as: Trend01

Step 8. Multiply the fitted trend values by their appropriate seasonal factors to compute the fitted value. The result is as Fitted Values if the manipulate seasonal indexes is used. 

Step 9. Calculate the residual errors and then use the Rooted Sum Squared Errors (RSE) to measure the accuracy of the prediction. The RSE is 3.56709 in this method. (see the Excel file). 

Similar results for adjusted seasonal indexes can also be obtained by using EVIEWS:

Step 1:

The "Scaling Factors", that is Eviews' seasonal indexes, are also summed to be 4 which are very close to the manipulated seasonal indexes found in our previous Excel example file. Since these two methods get the similar results of seasonal indexes, either one method can be chosen because both method is very similarly to obtain the RSE. (Show as below).

Step 2. From Eviews, use the seasonal indexes to derive the deseasonalized series, the result is as following: Deseasonalizes Data 2 (using Eviews' seasonal indexes)

Step 3. Estimate the trend-cyclical regression equation using the deseasonalized data. Before running regression, we need to generate a trend variable by the commands: choose "GENR", type "trend=@trend(1992.1)" in the dialogue box. 

Then click for regression estimation and fitted Trend= C + bt.  

From the regression result, we can get the values for coefficients "a" and "b", and the residuals--"Resid", 

After run the trend regression and click the "Forecast" to generate the "fitted_trend02" or 

Generate the values of the fitted trend by calculated from "FITTED_TREND02 = SALESA - Resid" In this case, the estimated trend value is: Trend02

Step 4. Multiply the fitted trend values by their appropriate seasonal factors to compute the fitted value. The result is: Fitted Values 2 (or "forecast values for sale" = "fitted_trend02 x seasonal index") if the EVIEWS' seasonal indexes is used. The steps to generate the seasonal indexes are as followings: Firstly, generate a dummy seasonal factor for each season,

then combine the four seasonal dummies into a series of seasonal indexes.

Step 5. Calculate the residual errors and then RSE to measure the accuracy of the fit. The RSE is 3.567528 when the EVIEWS' seasonal indexes is used. As you can see, the RSE is almost same as in the previous Excel file, therefore, either the manipulated seasonal indexes or the EVIEWS' seasonal index can be used to carry out the decomposition and predictions.  As you can see, the fitted Values (or the forecasted values) in both methods are similar to each other for the same period.

Argument for centered moving average:

Some may argue that why you need to use the centered moving average method in the seasonal decomposition. The reason is simple: this step minimizes the RSE which is pursued by econometricians so as to make an accurate prediction. If you don't take this step, the RSE would be larger than the RSE from the centered moving average method. Click to see the example presented in the Excel file. The RSE = 4.481 which is greater than the previous one.

The End