Seite - 56 - in The Austrian Business Cycle in the European Context

56 correlation is significantly higher than for other leads and lags in close neighbourhood. 5.2 Frequency domain statistics for analysing comovements Sometimes it is easier to interpret comovements not in the time domain, with discrete time data points, but in the frequency do- main. This is especially the case in business cycle analysis where special frequencies or frequency bands are the focus of interest. These statistics can be derived by the Fourier transformation of time domain statistics. So is the cross-spectra the frequency- do- main-equivalent of the cross correlation Pb.i (.), like the spectral density function is the equivalent of the auto-correlation function. The cross-spectra of two series for a certain frequency tu is defined as (23) I <X) Yb.s (w) = -2 LPb,s (r) e-iwr ,r r=-oo with tu being a frequency within [-n;.n) and Pb., being the cross- correlation as defined in (22). As can be seen in (23), the cross- spectrum contains a complex part which does not cancel out be- cause the cross-correlation function is not symmetric, i.e. Pb.s (,)*Pb.,(-,). Therefore, this statistic cannot be interpreted in order to determine leads and lags directly, but has to be trans- formed into another statistic like the coherence, the phase spec- tra or the mean delay.