Page - 89 - in The Austrian Business Cycle in the European Context

89 Less clear-cut is, from visual inspection, the leading property of one of the series compared with the other. Concerning the length of the phases - i.e. the time span between peaks and following troughs - again visual inspection can not reveal any differences. In order to evaluate the leading or lagging properties of all other filtered time series with respect to the Austrian business cycle, some descriptive statistics have been calculated. Tables A 1 a to c in the annex show the cross-correlations, the coherences, the av- erage spectra and the mean delay for all observed time series in relation to the reference series according to all three methods of cycle extraction. Cross-correlations are calculated as given by (22) in chapter 5. For four leads and lags, correlations have been calculated for the filtered time series with rmax showing the maxi- mum cross-correlation. 8.2.1.1 Robustness of cross-correlation statistics r0 shows the cross-correlation between a specific series and the reference series at zero lag, i.e. the coincident cross-correlation. The values can range from 1 (perfect cross-correlation at a spe- cific lead or lag) to 0. By construction, they are symmetric and only defined for positive values. The column labelled tmax shows the lead (positive figures) or lag (negative figures) at which the highest cross-correlation has been observed. In most cases ,max is observed at lag zero, which indicates that this series is coincident with the reference series1os. Not surprisingly, for all three filtering methods cross-correlations are higher the closer the definition of the series comes to the refer- ence series. Gross value added (this time including agriculture and 105 The reader is reminded of the fact that Croux - Forni - Reich/in ( 1999) had proven that simple static cross-correlation of band-pass filtered data is the same as dynamic correlation over the same frequency band. For this reason, dynamic cor- relation is not presented in the following.