We now turn to the concepts of phase shift and dynamic correlation. Iden-
tifying lead-lag relationship between the series Yf and Xf in time domain is
carried out using the cross correlations at lags τ = ±1, ±2,. . . ,. In contrast,
in the frequency domain, this can be achieved frequency by frequency using
the cross spectrum. The cross spectrum, which is the Fourier transform of the
covariance function of Yf and Xf, is given by
f2,x(ω) = cyx(ω) - iqyx(ω), (5)
where cyx(ω) is the cospectrum and ‰-(ω) is the quadrature spectrum. It can
be used to derive the phase spectrum defined as
φyx(ω) = - arctan(<⅛(ω)∕⅛(ω)). (6)
The phase spectrum at frequency ω measures the lead of the cyclical compo-
nent of Yf at this frequency over the corresponding component of Xf. It can
be interpreted as the negative of the angle which would transform the compo-
nent in Xf into the best linear approximation of Yf. To facilitate an intuitive
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