By contrast, for a unit root process, the effect of Et on Yi+s is seen from( 8) d (4)to be 0△Yt+s,O△Yt+s-1 a△Yt vs+s-1+…+t1+1( since 2△Yt+= vs from(4) An innovation Et has a permanent effect on the level of y that is captured by aY Example The following ARIMA(4, 1, 0)model was estimated for Yt △Yt=0.555+0.312△Y-1+0.122△Yt-2-0.116△Yt-3-0.081△yt-4+Et For this specification, the permanent effect of a one-unit change in Et on the level of Yt is estimated to be v(1) (1)(1-0.312-0.122+0.116+0.081) =1.31 2.2.4 Transformations to Achieve Stationarity A final difference between trend-stationary and unit root process that deserves comment is the transformation of the data needed to generate a stationary time series. If the process is really trend stationary as in(6), the appropriate treatment is to subtract at from Xt to produce a stationary representation. By contrast if the data were really generated by the unit root process(5), subtracting at from Yt, would succeed in removing the time-dependence of the mean but not the variance as seen in(5) There have been several papers that have studied the consequence of overdif fer encing and underdif ferencing 1. If the process is really TSP as in(6), difference it would b AXt=u+at-u-a(t-1)+v(L(1- LEt=a+u(L)Et (9)By contrast, for a unit root process, the effect of εt on Yt+s is seen from (8) and (4) to be ∂Yt+s ∂εt = ∂4Yt+s ∂εt + ∂4Yt+s−1 ∂εt + ... + ∂4Yt+1 ∂εt + ∂Yt ∂εt = ψs + ψs−1 + ... + ψ1 + 1 (since ∂4Yt+s ∂εt = ψs from (4)) An innovation εt has a permanent effect on the level of Y that is captured by lims→∞ ∂Yt+s ∂εt = 1 + ψ1 + ψ2 + ... = ψ(1). Example: The following ARIMA(4, 1, 0) model was estimated for Yt : 4Yt = 0.555 + 0.3124Yt−1 + 0.1224Yt−2 − 0.1164Yt−3 − 0.0814yt−4 + εˆt . For this specification, the permanent effect of a one-unit change in εt on the level of Yt is estimated to be ψ(1) = 1 φ(1) = 1 (1 − 0.312 − 0.122 + 0.116 + 0.081) = 1.31. 2.2.4 Transformations to Achieve Stationarity A final difference between trend-stationary and unit root process that deserves comment is the transformation of the data needed to generate a stationary time series. If the process is really trend stationary as in (6), the appropriate treatment is to subtract αt from Xt to produce a stationary representation. By contrast, if the data were really generated by the unit root process (5), subtracting αt from Yt , would succeed in removing the time-dependence of the mean but not the variance as seen in (5). There have been several papers that have studied the consequence of overdiffer − encing and underdifferencing: 1. If the process is really TSP as in (6), difference it would be 4Xt = µ + αt − µ − α(t − 1) + ψ(L)(1 − L)εt = α + ψ ∗ (L)εt . (9) 9