Seasonal Component Additive Model

Using the additive model i.e. Y = T + S + R, the de-trended series may be obtained by subtracting the trend values from the actual observations i.e.,

Y - T  = S + R

The remainder now consists of the seasonal and the residual components. The trend values that are subtracted might have been obtained by any of the methods described earlier, however, the moving trend or the least square trend are preferable.

The residual component may be eliminated from the de-trended series by averaging the de-trended values for each month or quarter separately. For an efficient trend the sum of these averages must have been zero, however, generally it would not be so. The seasonal component, therefore, need adjustment. To do this the seasonal totals are averaged i.e. in a quarterly time series the four quarterly totals are added and divided by twelve. This average, which may also be called as “adjustment factor” is subtracted from each quarterly or monthly totals. Adjusted totals are then averaged by dividing by the number of quarterly or monthly observations used to arrive at these totals. These averages which are finally obtained represent the seasonal component. The four seasonal components, in case of quarterly data or the twelve seasonal components, in case of monthly data, repeat itself during the subsequent years.

After having determined the seasonal component S, the de-seasonalised series may be obtained by subtracting the seasonal component S from the actual observations Y. The de-seasonlised series so obtained also represent a series which jointly determine the trend and the residual, for                 

Y - S  = T + R

The residual component may now be separated by a further subtracting the trend from the seasonally adjusted series for,

Y - S  - T = R

The entire analysis described above may be briefly summarized in the following steps.

1. Write Y and T in adjacent columns
2. Write the de-trended series by taking differences \left( {Y - T} \right)
3. Determine seasonal component S as explained above
4. Write the de-seasonalised series by taking differences \left( {T - S} \right)
5. Subtract the trend T from the de-seasonlised series to separate out the residual component R.
6. As a final check on the calculations, select any quarter or month and add the values for T, S and R, which would be equal to Y.