By C. W. J. Granger, Karl Shell, Paul Newbold
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Additional info for Forecasting Economic Time Series
An e:x:ample 1s when X,_ = -a y; since then dz xC w) = -adz_vC w), w~ch may be wntten dzx(w) = ae''IT dzy(w) and so
Dw dA.. With these processes, frequency components are no longer independent of each other, so that the amplitude of the seasonal frequency, say, can be a function of the business cycle. i I ' I I ! ) is everywhere differentiable, then one can write I •' ! ) I I possible. Some time-domain models that correspond to the slowly varymg frequency-domain models are discussed in Chapter 10. A wider class of nonstationary processes, with an attractive interpretation, are known as harmonizable processes and have been discussed by Loeve , Blane-Lapierre and Fortet , and Joyeux  among others.
Further, tl_le covanance between s( w ) and s( w2), that is, the covariance between estrmates at two 1 different frequencies, does tend to 'zero as n -7 oo, _s~ _that for l~ge n, s_( w) has a tendency to become very jagged and the possibility of finding_ spunous "cycles" in one's data is enhanced. This is the reason the penodogram proved to be an unsatisfactory tool of analysis and has been largely superA seded by the spectrum. . 4) and is proportional to the periodogram. Thus, since kn(A) takes its main maximum at A= 0, sk(w) for given w is seen to be a weighted average of periodogram-type estimates at frequencies centered on w.