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72 CHAPTER 2. SOME USEFUL TIME-SERIES METHODSq t =Z πe −iωt du(ω)0| {z }(a)Z π+0e iωt dz(ω)| {z }(b)Z π−0e −iωt dz(ω) . (2.107)| {z }(c)Since u(−ω) =u(ω), the term labeled (a) in (2.107) can be written asR π0 e−iωt du(ω) = R 0−π eiωt du(ω). The third term labeled (c) in (2.107) isR π0 e−iωt dz(ω) = 1 R π2 0 e−iωt du(ω)+ 1 R π2 0 ie−iωt dv(ω) = 1 R 02 −π eiωt du(ω) −(51)⇒1 R 02 −π ieiωt dv(ω). Substituting these results back into (2.107) and cancelingterms you get, q t = 1 R 02 −π eiωt [du(ω) +idv(ω)] + R π0 eiωt dz(ω)= R π−π eiωt dz(ω). This is known as the Cramer Representation of q t ,which we restate asq t = limNXN→∞j=1a j cos(ω j t + λ j )=Z π−πe iωt dz(ω). (2.108)The point of all this is that any time-series can be thought of as beingbuilt up from a set of underlying subprocesses whose individualfrequency components exhibit cycles of varying frequency. The otherside of this argument is that you can, in principle, take any time-seriesq t and Þgure out what fraction of its variance is generated from thosesubprocesses that cycle within a given frequency range. The businesscycle frequency, which lies between 6 and 32 quarters is of key interestto, of all people, business cycle researchers.Notice that the process dz(ω) is built up from independent increments.Forcoincidentincrements,youcandeÞne the function s(ω)dωto be(s(ω)dω λ = ωE[dz(ω)dz(λ)] =0 otherwise , (2.109)where an overbar denotes the complex conjugate. 31 Sincee iωt e iωt = cos 2 (ωt) +sin 2 (ωt) = 1 at frequency ω, it follows thatE[e iωt e iωt dz(ω)dz(ω)] = s(ω)dω. That is, s(ω)dω is the variance ofthe ω−frequency component of q t , and is called the spectral densityfunction of q t . Since by (2.108), q t is built up from frequency componentsranging from [−π, π], the total variance of q t must be the integral31 If a and b are real numbers and z = a + bi is a complex number, the complexconjugate of z is ¯z = a − bi. The product z¯z = a 2 + b 2 is real.