Cross-Spectrum and Coherence

Two series can correlate strongly overall while actually only moving together at specific frequencies — say, both series share a clear quarterly cycle but move independently at higher frequencies. An ordinary correlation coefficient blends all of this into one number. The cross-spectrum and coherence break the relationship apart frequency by frequency instead.

What each one measures

The cross-spectrum is the frequency-domain analog of covariance: at each frequency, it's a complex number whose magnitude tells you how much joint power the two series share at that frequency, and whose phase tells you whether one series leads or lags the other there. Coherence normalizes this by each series' own power at that frequency, producing a value between 0 and 1 at every frequency — much like a squared correlation coefficient, but computed separately at each frequency rather than as one overall number.

Worked example

Two macro series, industrial production and a leading sentiment index, show coherence near 0.85 at the business-cycle frequency (roughly a 4-year cycle) but coherence near 0.1 at high frequencies (week-to-week noise). This tells you the two series are tightly linked at business-cycle length but essentially unrelated day to day — information a single correlation coefficient computed on raw levels would have blended into one misleading middling number.