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46 CHAPTER 2. SOME USEFUL TIME-SERIES METHODS(26)⇒and ξ t−2 = q t−2 − γ 0 − γ 1 (t − 2). Substitute these expressions into(2.55) and then substitute this result into (2.50) to get q t = α 0 + α 1 t + ⇐(25)ρ 1 q t−1 + ρ 2 q t−2 + ² t ,whereα 0 = γ 0 [1 − ρ 1 − ρ 2 ]+γ 1 [ρ 1 +2ρ 2 ], andα 1 = γ 1 [1 − ρ 1 − ρ 2 ]. Now subtract q t−1 from both sides of this result,add and subtract ρ 2 q t−1 to the right hand side, and you end up with∆q t = α 0 + α 1 t + βq t−1 + δ 1 ∆q t−1 + ² t , (2.56)where β =(ρ 1 + ρ 2 − 1), and δ 1 = −ρ 2 . (2.56) is called the augmentedDickey—Fuller (ADF) regression. Under the null hypothesis that q t hasaunitroot,β =0.As before, a test of the unit root null hypothesis can be conductedby estimating the regression (2.56) by OLS and comparing the studentizedcoefficient, τ on β (the t-ratio reported by standard regressionroutines) to the appropriate table of critical values. The distributionof τ, while dependent on the speciÞcation of the deterministic factors,is fortunately invariant to the number of lagged dependent variables inthe augmented Dickey—Fuller regression. 16Permanent-and-Transitory-Components RepresentationIt is often useful to model a unit root process as the sum of differentsub-processes. In section chapter 2.2.7 we will model the time-series asbeing the sum of ‘trend’ and ‘cyclical’ components. Here, we will thinkofaunitrootprocess{q t } as the sum of a random walk {ξ t } and anorthogonal stationary process, {z t }q t = ξ t + z t . (2.57)(27)⇒iidTo Þx ideas,letξ t = ξ t−1 + ² t be a driftless random walk with ² t ∼N(0, σ² 2 )andletz t = ρz t−1 + v t be a stationary AR(1) process withiid0 ≤ ρ < 1andv t ∼ N(0, σv). 2 17 Because the effect of the ² t shocks16 An alternative strategy for dealing with higher-order serial correlation is thePhillips and Perron [120] method. They suggest a test that employs a nonparametriccorrection of the OLS studentized coefficient for ˆβ so that its asymptoticdistribution is the same as that when there is no higher ordered serial correlation.We will not cover their method.