Seasonal Component Multiplicative Model

Using the multiplicative model, i.e.$$Y = T \times S \times R$$, the ratio detrended series may be obtained by dividing the actual observations by the corresponding trend values:

\[\frac{Y}{T} = S \times R\]

The remainder now consists of the seasonal and the residual components. The seasonal component may be isolated from the ratio-detrended series by averaging the detrended ratios for each month or quarter. The adjustment seasonal totals are, however, obtained by multiplying the seasonal totals by the following adjustment factor.

\[{\text{Adjustment}}\,{\text{Factor}}\,{\text{ = }}\,\frac{{{\text{Total}}\,{\text{Number}}\,{\text{of}}\,{\text{Observations}}}}{{{\text{Sum}}\,{\text{of}}\,{\text{Detrended}}\,{\text{Ratios}}}}\]

These adjustment seasonal totals are then averaged over the number of detrended ratios in each quarter or month. The obtained averages represent the seasonal component. After having determined the seasonal component S, the de-seasonalised series may be obtained by dividing the actual observations Y by the corresponding seasonal component. The de-seasonalised series so obtained determines the trend and the residual, for

\[\frac{Y}{T} = S \times R\]

The residual component may now be separated by a further division of the de-seasonalised series by the trend, for

\[\frac{Y}{{S \times T}} = R\]


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

1. Write $$Y$$ and $$T$$
2. Obtained the detrended ratios, i.e. $$\frac{Y}{T}$$
3. Determine he seasonal component S as explained above
4. Obtain the de-seasonalised series by finding $$\frac{Y}{S}$$
5. Further divide the ratios obtained in step $$4$$ by $$T$$ to separate out the residual $$R$$
6. As a final check of the calculations, select any quarter or month and multiply out $$T$$, $$S$$ and $$R$$. This product would be equal to $$Y$$.