Page - 57 - in The Austrian Business Cycle in the European Context

57 5.2. 1 Coherence81 The coherence measures the linear relatedness of two stationary processes. It can be regarded as the frequency-domain-equiva- lent to the cross-correlation in the time domain. The output is de- fined over the interval [O, 1t] and shows the correlation of the cycli- cal (or at least stationary) component of the series at each fre- quency. It is defined by the squared cross-spectrum, divided by the product of the spectral density functions of both series (24) co(m)= lrb,s (m)l2 Yb.b (m) Ys,s (m) Applying this quadratic transformation ensures that values are real and symmetric. It can be interpreted as the frequency-domain- counterpart to R2, the well known coefficient of determination, as it shows the proportion of variance of one series explained by the other for a given frequency m. But this transformation has a substantial disadvantage. Croux - Forni - Reich/in ( 1999) stressed that this statistic "does not measure correlation at different frequencies, because it disregards the phase difference between variables"82• Thus, only the synchronised comovement of two time series, over some specified spectrum or at a certain frequency, can be observed. Whether these frequen- cies are phase-aligned or not is of no influence to this measure. Related to our problem, the coherence assumes high values if both series show similar frequency gains, irrespective of whether one time series is leading or lagging. 81 In the literature, this measure is sometimes called squared coherence. 82 See Croux - Forni - Reich/in ( 1999), p. 4.