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68 CHAPTER 2. SOME USEFUL TIME-SERIES METHODSThe Spectral Representation of a Time SeriesIn section 2.4, a unit-root time series was decomposed into the sum ofa random walk and a stationary AR(1) component. Here, we want tothink of the time-series observations as being built up of underlyingcyclical (cosine) functions each with different amplitudes and exhibitingcycles of different frequencies. A key question in spectral analysisis, which of these frequency components are relatively important indetermining the behavior of the observed time-series?To Þx ideas, begin with the deterministic time-series, q t = a cos(ωt),where time is measured in years. This function exhibits a cycle everyt = 2π years. By choosing values of ω between 0 and π, youcangetωthe process to exhibit cycles at any length that you desire. This isillustrated in Figure 2.1 where q 1t = a cos(t) exhibits a cycle every2π =6.28 years and q 2t = a cos(πt) displays a cycle every 2 years.1.510.5-0.500 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2 3.6 4 4.4 4.8 5.2 5.6 6-1-1.5Figure 2.1: Deterministic Cycles—q 1t = cos(t) (dashed)cyclesevery2π =6.28 years and q 2t =cos(πt) (solid) cycles every 2 years.Something is clearly missing at this point and it is randomness.We introduce uncertainty with a random phase shift. If you compareq 1t = a cos(t) toq 3t = a cos(t + π/2), q 3t is just q 1t with a phase shift(horizontal movement) of π . This phase shift is illustrated in Figure 2.22