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74 CHAPTER 2. SOME USEFUL TIME-SERIES METHODSLinear FiltersYou can see how a Þlter changes the character of a time series by comparingthe spectral density function of the original observations withthat of the Þltered data.Let the original data q t have the Wold moving-average representation,q t = b(L)² t where b(L) = P ∞j=0 b j L j and ² t ∼ iid with E(² t )=0and Var(² t )=σ 2 ² . The k-th autocovariance isγ k = E(q t q t−k )=E[b(L)² t b(L)² t−k ]⎛⎞∞X X ∞= E⎝b j ² t−j b s ² t−s−k⎠ = σ²2j=0s=0⎛∞X⎝j=0and the autocovariance generating function for q t isg(z) === σ 2 ²⎛∞X∞X ∞Xγ k z k = σ²2 ⎝k=−∞ k=−∞ j=0⎛∞X ∞X⎝σ ²2k=−∞ j=0⎛∞X⎝ b j z j X ∞j=0 k=jb j b j−k⎞⎠ z kb j b j−k⎞⎠ z k z j z −j = σ 2 ²∞X∞Xk=−∞ j=0b j−k z −(j−k) ⎞⎠ = σ 2 ² b(z)b(z −1 ).b j b j−k⎞⎠ ,b j z j b j−k z −(j−k)(54)⇒But from (2.111), you know that s(ω) = g(eiω ). To summarize, these2πresults, the spectral density of q t can be represented ass(ω) = 12π g(e−iω )= 12π σ2 ² b(e −iω )b(e iω ). (2.112)(55)⇒Let the transformed (Þltered) data be given by ˜q t = a(L)q t wherea(L) = P ∞j=−∞ a j L j . Then ˜q t = a(L)q t = a(L)b(L)² t = ˜b(L)² t ,where˜b(L) =a(L)b(L). Clearly, the autocovariance generating function ofthe Þltered data is ˜g(z) =σ²˜b(z)˜b(z 2 −1 )=σ² 2 a(z)b(z)b(z −1 )a(z −1 )=a(z)a(z −1 )g(z), and letting z = e −iω , the spectral density function ofthe Þltered data is˜s(ω) =a(e −iω )a(e iω )s(ω). (2.113)