ahead forecast is Xt+=H+a(t+s)+tt+v+1-1+v+2 hich are associated with forecast error X-X Hu+a(t+s)+Et+s+V1Et+s-1+2/++5-2 +ys-1Et+1+sEt +us+1Et-1+ -u+a(t+s)+sEt +as+1Et-1+. Et+s+v1Et+s-1+2Et+5-2+.+1s-1Et+1 The Mse of this forecast is E[X+-Xt+2={1+n1+n2+…+v21 The MsE increases with the forecasting horizon s. though as s becomes large the added uncertainty from forecasting farther into the future becomes negligible imEX+-X+2={1+2+n+…}a2 Note that the limiting MSE is just the unconditional variance of the stationary component v (L)Et To forecast the unit root process(4), recall that the change AYt is a stationary rocess that can be forecast using the standard formula △Yt+s E[(Y计+s-Yt+8-1)Y,Y-1 a+vsEt +2s+1Et-1+0s+2Et-2+ The level of the variable at date t+ s is simply the sum of the change between t nd t+s (Yt+s-Yt+s-1)+(Yt+s-1-Yt+s-2)+…+(Y+1-Y)+Y(7) △Yt+s+△Yt+s-1+…+△Yt+1+Y Therefore the s period ahead forecast error for the unit root process is {△Yt+8+△Yt+s-1+ Yt+1+Yt {△Y++△Yt+-1t+…+△Y+t+Y {t+s-1+v1 us-2=+1}+…+{et+}ahead forecast is Xˆ t+s|t = µ + α(t + s) + ψsεt + ψs+1εt−1 + ψs+2εt−2 + .... which are associated with forecast error Xt+s − Xˆ t+s|t = {µ + α(t + s) + εt+s + ψ1εt+s−1 + ψ2εt+s−2 + .... +ψs−1εt+1 + ψsεt + ψs+1εt−1 + ....} −{µ + α(t + s) + ψsεt + ψs+1εt−1 + ....} = εt+s + ψ1εt+s−1 + ψ2εt+s−2 + ... + ψs−1εt+1. The MSE of this forecast is E[Xt+s − Xˆ t+s|t ] 2 = {1 + ψ 2 1 + ψ 2 2 + ... + ψ 2 s−1 }σ 2 . The MSE increases with the forecasting horizon s, though as s becomes large, the added uncertainty from forecasting farther into the future becomes negligible: lims→∞ E[Xt+s − Xˆ t+s|t ] 2 = {1 + ψ 2 1 + ψ 2 2 + ...}σ 2 . Note that the limiting MSE is just the unconditional variance of the stationary component ψ(L)εt . To forecast the unit root process (4), recall that the change 4Yt is a stationary process that can be forecast using the standard formula: 4Yˆ t+s|t = Eˆ[(Yt+s − Yt+s−1)|Yt , Yt−1, ...] = α + ψsεt + ψs+1εt−1 + ψs+2εt−2 + ... The level of the variable at date t + s is simply the sum of the change between t and t + s: Yt+s = (Yt+s − Yt+s−1) + (Yt+s−1 − Yt+s−2) + ... + (Yt+1 − Yt) + Yt (7) = 4Yt+s + 4Yt+s−1 + ... + 4Yt+1 + Yt . (8) Therefore the s period ahead forecast error for the unit root process is Yt+s − Yˆ t+s|t = {4Yt+s + 4Yt+s−1 + ... + 4Yt+1 + Yt} −{4Yˆ t+s|t + 4Yˆ t+s−1|t + ... + 4Yˆ t+1|t + Yt} = {εt+s + ψ1εt+s−1 + ... + ψs−1εt+1} +{εt+s−1 + ψ1εt+s−2 + ... + ψs−2εt+1} + ... + {εt+1} = εt+s + [1 + ψ1]εt+s−1 + [1 + ψ1 + ψ2]εt+s−2 + ... + [1 + ψ1 + ψ2 + ... + ψs−1]εt+1, 7