The long term cyclical movements in a time series can be studied more or less in the same way as the seasonal movements. The data for the study of cyclical movements consists of a large number of yearly observations, and seasonal variations are assumed to be absent. The multiplicative model, as indicated earlier, is used to analyse the cyclical movements. The model can be stated as
Using the above model, the ratio de-trended series may be obtained by the division of actual observations by the corresponding trend, i.e.
The series so obtained is composed of cyclical and residual movement. From this series the cyclical component may be separated using moving averages. As the length of the moving average is concerned, it is a matter of value judgment. However, the larger the lengths of the moving average the more cycles will be eliminated. Experience has shown that a three yearly moving average proves quite satisfactory for most economics data. The moving averages so obtained represent cyclical component in terms of ratios. The de-trended ratios are then divided by the cyclical ratios to provide the residual, i.e.
The entire analysis described above may be briefly summarized in the following steps:
1. Write and
2. Find the de-trended ratios, i.e.
3. Determine the moving averages of a suitable length, and these averages would represent
4. Divide the de-trended ratios by to obtain the residuals .
5. Check calculations by finding the product of , and for any period, and this would be equal to .