Page - 72 - in The Austrian Business Cycle in the European Context

72 The calculation of means of leads and lags can also be done in the frequency domain, looking at the cross-spectrum r( w) of both series b and s98• As this in general takes on complex values as shown in (23), it has to be transformed into polar coordinates which yield {36) Yb,s (a>)= lh., (w)jje-iPh(w) with the argument Ph(w) indicating the shift in the number of peri- ods as a multiple of the frequency for which it is observed {the so called phase angle shift). Averaging over a certain frequency band that lies between the boundaries w" and w1 by (37) "'11 Ph(w) f-w-dw "'I allows looking at leads and lags over this band. Positive values in- dicate that the observed series z, is leading the reference series zh, with negative values indicating lags. 7 .2 Contingency tables for turning points Artis -Krolzig -Toro (2004) proposed a further method of analysing business cycle comovements. With a binary variable indicating whether the economy is expanding or contracting {which can be derived from the turning points) they constructed 2 x 2 contin- gency tables for each pair of series of the form of Table 1. 98 See Joint Research Centre of the European Commission ( 1993).