Page - 59 - in The Austrian Business Cycle in the European Context

59 but this time defined only for co within the interval [0, 1t). This meas- ure is related to coherence - as it shows its real part - but instead of rendering values real by a quadratic transformation, the com- plex term is cancelled out by summing negative and positive waves of the same frequency. This can be seen in a different rep- resentation of (28) - ( )_Yb,,(m)+Yb_,(-m) Pb,s (u - 2 This should not be a drawback as (co) and (-co) have the same pe- riodicity, which is the focus of interest in business cycle analysis. Furthermore, the authors stressed that dynamic correlation ob- served at a special frequency band is theoretically fully equivalent to static correlation applied to band-pass pre-filtered data ac- cordingly. 5.2.4 Cohesion In order to analyse the comovement of more than one time series, Cro ux - Forni - Reich/in ( 1999) proposed a multivariate version of the dynamic correlation index. It is constructed by weighing to- gether all dynamic correlation coefficients as defined in (27) and is called cohesion by them (29) ""· . W;W1•Ps-s·(m) h L..,,~J I J CO b (m) = "" L..,;~j W; Wj the p_, being again the dynamic correlation coefficients as de- fined in (27) over all combinations of time series except their di- agonal elements. The authors suggest choosing the weights ac- cording to their economic significance (e.g. by their proportion of contributing to GDP). This statistic has the advantage of allowing to capture common comovements of the underlying set of time series at all frequencies at one glance. This makes it more appro- priate for certain tasks than looking at cross-correlations because