Manufacturing Earnings and Cycles: New Evidence

We now turn to the concepts of phase shift and dynamic correlation. Iden-
tifying lead-lag relationship between the series Y_f and X_f 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 Y_f and X_f, is given by

f₂,_x(ω) = c_yx(ω) - iq_yx(ω),                        (5)

where c_yx(ω) is the cospectrum and ‰-(ω) is the quadrature spectrum. It can
be used to derive the phase spectrum defined as

φ_yx(ω) = - arctan(<⅛(ω)∕⅛(ω)).                (6)

The phase spectrum at frequency ω measures the lead of the cyclical compo-
nent of Y_f at this frequency over the corresponding component of X_f. It can
be interpreted as the negative of the angle which would transform the compo-
nent in X_f into the best linear approximation of Y_f. To facilitate an intuitive

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