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2.4. UNIT ROOTS 49ρ = 1, the increments ∆q t are independent and the numerator of VR kis Var(q t − q t−k )=Var(∆q t + ∆q t−1 + ···∆q t−k+1 )=kVar(∆q t ), whereVar(∆q t )=σ² 2 + σv. 2 In the absence of transitory component dynamics,VR k = 1 for all k ≥ 1.If 0 < ρ < 1, {q t } is still a unit root process, but its dynamicsare driven in part by the transitory part, {z t }.ToevaluateVR k , Þrstnote that γ0 z = σv/(1 2 − ρ 2 ). The k-th autocovariance of the transitorycomponent is γk z =E(z t z t−k )=ρ k γ0, z γ0 ∆z =E[∆z t ][∆z t ]=2(1− ρ)γ0 z ⇐(33)and the k-th autocovariance of ∆z t isγk ∆z =E[∆z t ][∆z t−k ]=−(1 − ρ) 2 ρ k−1 γ0 z < 0. (2.64)By (2.64), ∆z t is negatively correlated with its past values and thereforeexhibits mean reverting behavior because a positive change today isexpected to be reversed in the future. You also see that γ ∆q0 = σ² 2 +γ0∆zand for k>1γ ∆qk= γ ∆zk < 0. (2.65)By (2.65), the serial correlation in {∆q t } is seen to be determined bythe dynamics in the transitory component {z t }. Interactions betweenchanges are referred to as the short run dynamics of the process. Thus,working on (2.63), the variance ratio statistic for the random walk—AR(1) model can be written asVR k = 1− 2(1 − ρ)2 γ z 0P k−1j=1γ ∆q0³ ´1 −j ρj−1 k→ 1 − 2(1 − ρ)γz 0γ ∆q as k →∞0= 1− γ∆z 0σ² 2 + . (2.66)γ∆z 0VR ∞ − 1 is the fraction of the short run variance of ∆q t generated bychanges in the the transitory component. VR ∞ is therefore increasingin the relative size of the random walk component σ² 2 /γ0 ∆z .Near Observational EquivalenceBlough [16],Faust [50], and Cochrane [30] point out that for a samplewith Þxed T any unit root process is observationally equivalent to a