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2.4. UNIT ROOTS 47on q t last forever, the random walk {ξ t } is called the permanent component.The stationary AR(1) part of the process, {z t },iscalledthetransitory component because the effect of v t shocks on z t and thereforeon q t eventually die out. This random walk—AR(1) model has anARIMA(1,1,1) representation. 18 To deduce the ARIMA formulation,take Þrst differences of (2.57) to get∆q t = ² t + ∆z t= ² t +(ρ∆z t−1 + ∆v t )+(ρ² t−1 − ρ² t−1 )= ρ[∆z t−1 + ² t−1 ]+(² t − ρ² t−1 + v t − v t−1 )= ρ∆q t−1 +(² t − ρ² t−1 + v t − v t−1 ), (2.58)| {z }(a)where ρ∆q t−1 is the autoregressive part. The term labeled (a) in thelast line of (2.58) is the moving-average part. To see the connection,write this term out as,² t + v t − (ρ² t−1 + v t−1 )=u t + θu t−1 , (2.59)where u t is an iid process with E(u t ) = 0 and E(u 2 t ) = σ 2 u. Nowyou want to choose θ and σ 2 u such that u t + θu t−1 is observationallyequivalent to ² t + v t − (ρ² t−1 + v t−1 ),whichyoucandobymatchingcorresponding moments. Let ζ t = ² t + v t − (ρ² t−1 + v t−1 )andη t = u t + θu t−1 . Then you have,⇐(28)E(ζt 2 ) = σ² 2 (1 + ρ 2 )+2σv,2E(ηt 2 ) = σ2 u (1 + θ2 ),E(ζ t ζ t−1 ) = −(σv 2 + ρσ2 ² ),E(η t η t−1 ) = θσ