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2.4. UNIT ROOTS 43The extension to higher-ordered processes is straightforward.The nonstationary case.If ρ =1,q t has the driftless random walkprocess 14 q t = q t−1 + ² t .Setting ρ = 1 in (2.44) gives the analogous moving-average representationXt−1q t = q 0 + ² t−j .The effect on q t from an ² t−j shock is 1 regardless of how far in the pastit occurred. The ² t shocks therefore exert a permanent effect on q t .The statistical theory developed for estimating ρ for stationary timeseriesdoesn’t work for unit root processes because we have terms like1 − ρ in denominators and the variance of q t won’t exist. To seewhy that is the case, initialize the process by setting q 0 =0. Thenq t =(² t + ² t−1 + ··· + ² 1 ) ∼ N(0,tσ² 2 ). You can see that the varianceof q t grows linearly with t. Now a typical term in the numeratorof (2.45) is {q t ² t+1 } which is an independent sequence with meanE(q t ² t+1 ) = E(q t )E(² t+1 ) = 0 but the variance isVar(q t ² t+1 ) = E(qt 2 )E(² 2 t+1) = tσ²4 which goes to inÞnity over time.Since an inÞnite variance violates the regularity conditions of the usualcentral limit theorem, a different asymptotic distribution theory is requiredto deal with non-stationary data. Likewise, the denominator in(2.45) does not have a Þxed mean. In fact, E( 1 PT q2t )=σ 2 P t = T 2doesn’t converge to a Þnite number either.The essential point is that the asymptotic distribution of the OLSestimator of ρ is different when {q t } has a unit root than when theobservations are stationary and the source of this difference is that thevariance of the observations grows ‘too fast.’ It turns out that a differentscaling factor is needed on the left side of (2.45). In the stationary case,we scaled by √ T , but in the unit root case, we scale by T .T (ˆρ − ρ) =j=01 P T −1T1 P T −1T 2 t=1 qt2t=1 q t ² t+1, (2.49)14 When ρ = 1, we need to set α = 0 to prevent q t from trending. This willbecome clear when we see the Bhargava [12] formulation below.