Seite - 26 - in The Austrian Business Cycle in the European Context

26 (3) y,=Y,-1+c,+&, Simple arithmetics isolates the cycle as It can be seen, that the cyclical component includes the error term, which makes it very erratic and does not show up in the smoothness one would expect from a pure cyclical variation. Fur- thermore, forming first-order differences does not correspond to a symmetric filter as it generates only differences to past values. This leads to a phase shift of the cyclical component and therefore yields a dating output different from the case of using symmetric filters. In order to look at the properties of this first-order difference method at the frequency domain, Figure 2 depicts the gain func- tion when filtering data that are already seasonally adjusted. The grey shaded area marks the frequency band of the typical busi- ness cycle between ,r/3 (which corresponds to cycles of 6 quarters length) and 1r/16 (corresponding to a length of 32 quarters). The bold red line reveals that the first-order difference filter indeed cancels out frequencies that are located close to zero and there- fore can be regarded as variations of the trend component. But it also wipes out mistakenly some spectral mass of the business cycle frequencies in the shaded area. At ,r/2, which corresponds to cy- cles of 4 quarters) there is no spectral mass at all, as the data have been seasonally adjusted beforehand. A remarkable property of this filter is that it superimposes all higher frequencies, raising the gain for very high frequencies above 1 . This explains the very erratic output of the filter as one can see from looking at growth rate series. Applying first-order differences to processes which are either trend stationary or show a higher degree of integration biases the out- come for business cycle interpretation. In the first case, this would