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 large number of yearly observations and seasonal variations are assumed to be absent. The multiplicative model, as indicated earlier, is used to analyses the cyclical movements. The model can be stated as

Using the above model the ratio de-trended series may be obtained by division of actual observations by the corresponding trend, i.e.

The series so obtained composed of cyclical and residual movement. From this series the cyclical component may be separated using moving averages. As to the length of moving average is concerned, it is a matter of value judgment. However, the larger the lengths of moving average the large will the number of cycles be eliminated. Experience has shown that a three yearly moving average prove 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 de-trended ratios, i.e.

**3.** Determine moving averages of suitable length, these averages would represents

**4.** Divide de-trended ratios by to obtained residuals .

**5.** Check calculations by finding product of , and for any period, this would be equal to .