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:

\[Y – T = S + R\]

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 will not be so. The seasonal component, therefore, needs adjustment. To do this the seasonal totals are averaged, for example in a quarterly time series the four quarterly totals are added and divided by twelve. This average, which may also be called the “adjustment factor,” is subtracted from each quarterly or monthly total. The adjusted totals are then averaged by dividing by the number of quarterly or monthly observations used to arrive at these totals. The averages which are finally obtained represent the seasonal component. The four seasonal components in the case of quarterly data, or the twelve seasonal components in case of monthly data, repeat 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 represents a series which jointly determines the trend and the residual, for

\[Y – S = T + R\]

The residual component may now be separated by 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 the differences $$\left( {Y – T} \right)$$
3. Determine the seasonal component $$S$$ as explained above
4. Write the de-seasonalised series by taking the 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 of the calculations, select any quarter or month and add the values for $$T$$, $$S$$ and $$R$$, which would be equal to $$Y$$