Page - 58 - in The Austrian Business Cycle in the European Context

58 5.2.2 Phase spectra and mean delay In order to supplement the coherence with a statistic that informs about leading and lagging properties of time series, the phase spectrum can be calculated. This can be done by combining the real and the complex part of the cross-spectrum as defined in (23) and calculating the arcustangens of it in order to obtain the phase angle: where PH{ m) is the phase-delay-generating function multiplied by a scalar, and further (26) J!( ) 1 ~ Pb,s (r) .,, m = arctan - ~ 2,r r=-<X> PH (m) r ~(ffi) is the phase spectrum over all frequencies, indicating how large the lead (positive numbers) or lag (negative) is. Averaging over a specified frequency band yields the mean delay for this term. In this way one can look at the leading and lagging proper- ties only for frequencies within the business cycle boundaries or for other frequencies in the focus of interest. 5.2.3 Dynamic correlation Another way of overcoming the deficiency of the coherence of not accounting for phase shifts of frequencies has been proposed by Croux -Forni -Reich/in (1999). They recommended a measure that looks quite similar to its time-domain-equivalent, the cross- correlation, shown in (22): (27) - ( ) Yb.s (m) Pb Ci) = ,r===== ,s .Jrb,b (m)Y,,s (m)