For the ARIMA(1, 1, 0) model with drift, (1 B)(1 B)xt = + wt, let yt = (1 B)xt = xt.

For the ARIMA(1, 1, 0) model with drift, (1 − φB)(1 − B)xt = δ + wt, let yt = (1 − B)xt = ∇xt.

(a) Noting that yt is AR(1), show that, for j ≥ 1, yn n+j = δ [1 + φ + · · · + φj−1] + φj yn.

(b) Use part

(a) to show that, for m = 1, 2, . . . , xn n+m = xn +

δ

1 − φ

h m − φ(1 − φm)

(1 − φ)

i

+ (xn − xn−1)

φ(1 − φm)

(1 − φ) .

Hint: From (a), xn n+j − xn n+j−1 = δ 1−φj 1−φ + φj (xn − xn−1). Now sum both sides over j from 1 to m.

(c) Use (3.144) to find P n n+m by first showing that ψ∗

0 = 1, ψ∗

1 = (1 + φ), and

ψ∗

j − (1 + φ)ψ∗

j−1 + φψ∗

j−2 = 0 for j ≥ 2, in which case ψ∗

j = 1−φj+1 1−φ , for j ≥ 1. Note that, as in Example 3.36, equation (3.144) is exact here.