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2.7. FILTERING 73of s(ω). That is 32E(q 2 t ) = E[= E[==Z π−πZ π Z π−πZ π−πZ π−πZ πe iωt dz(ω)−π−πe iλt dz(λ)]e iωt e iλt dz(ω)dz(λ)]E[dz(ω)dz(λ)]s(ω)dω. (2.110)The spectral density and autocovariance generating functions. Theautocovariancegenerating function for a time series q t is deÞned to beg(z) =∞Xj=−∞γ j z j ,where γ j =E(q t q t−j ) is the j-th autocovariance of q t .Ifweletz = e −iω ,thenZ1 πg(e −iω )e iωk dω = 1 ∞XZ πγ j e iω(k−j) dω.2π −π2πj=−∞−πLet a = k−j. Thene iωa =cos(ωa)+i sin(ωa) and the integral becomes,R π−π cos(ωa)dω +i R π−π sin(ωa)dω =(1/a)sin(aω)|π −π −(i/a)cos(aω)| π −π.The second term is 0 because cos(−aπ) = cos(aπ). The Þrst termis 0 too because the sine of any nonzero integer multiple of π is 0and a is an integer. Therefore, the only value of a that matters isa = k − j = 0, which implies that γ k = 1 R π2π −π g(e−iω )e iωk dω. Settingk =0,youhaveγ 0 =Var(q t )= 1 R π2π −π g(e−iω )dω, but you know that ⇐(52)Var(q t )= R π−π s(ω)dω, so the spectral density function is proportionalto the autocovariance generating function with z = e −iω . Notice also,that when you set ω =0,thens(0) = P ∞j=−∞ γ j . The spectral densityfunction of q t at frequency 0 is the same thing as the long-run varianceof q t . It follows thatZ πVar(q t )= s(ω)dω = 1 Z πg(e −iω )dω, (2.111)−π 2π −πwhere g(z) = P ∞j=−∞ γ j z j .⇐(53)32 We obtain the last equality because dz(ω) is a process with independent incrementsso unless λ = ω, Edz(ω)dz(λ) =0.