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48 CHAPTER 2. SOME USEFUL TIME-SERIES METHODSσu 2 (1 + θ2 ) = σ² 2 (1 + ρ2 )+2σv 2 , (2.60)θσu 2 = −(σv 2 + ρσ² 2 ). (2.61)(30)⇒(31)⇒These are two equations in the unknowns, σu 2 and θ which can be solved.The equations are nonlinear in σu 2 and getting the exact solution ispretty messy. To sketch out what to do, Þrst get θ 2 =[σv 2 + ρσ² 2 ] 2 /(σu) 2 2from (2.61). Substitute it into (2.60) to get x 2 + bx + c =0wherex = σu 2,b= −[σ2 ² (1 + ρ2 )+2σv 2], and c =[σ2 v + ρσ2 ² ]2 . The solution forσu 2 can then be obtained by the quadratic formula.Variance Ratios(32)⇒The variance ratio statistic at horizon k is the variance of the k-periodchange of a variable divided by k times the one-period changeVR k = Var(q t − q t−k )kVar(∆q t )= Var(∆q t + ···+ ∆q t−k+1 ). (2.62)kVar(∆q t )The use of these statistics were popularized by Cochrane [29] who usedthem to conduct nonparametric tests of the unit root hypothesis inGNP and to measure the relative size of the random walk componentin a time-series.Denote the k-th autocovariance of the stationary time-series {x t } byγk x =Cov(x t ,x t−k ). The denominator of (2.62) is kγ ∆q0 , the numeratoris Var(q t −q t−k+1 )=k h γ ∆q0 + P k−1j=1(1 − j k )(γ∆q j + γ ∆q−j ) i , so the varianceratio statistic can be written asVR k = γ∆q 0 + P k−1j=1(1 − j k )(γ∆q j + γ ∆q−j )γ ∆q0= 1+ 2 P k−1j=1(1 − j k )γ∆q jγ ∆q(2.63)0k−1 X= 1+2 (1 − j k )ρ∆qj=1j ,where ρ ∆qj= γ ∆qj /γ ∆q0 is the j-th autocorrelation coefficient of ∆q t .Measuring the size of the random walk. Suppose that q t evolves accordingto the permanent—transitory components model of (2.57). If