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Level forecast from changes

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A series follows $ARIMA (1, 1, 0)$ with $Δ X_{t} = 1 + 0.5Δ X_{t - 1} + ϵ_{t}$ , where $ϵ_{t}$ has mean $0$ and is independent of the past. The last $2$ observed levels are $50$ and then $53$ . What is the forecast of the level $3$ steps after the last observation? Give an exact decimal.

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Hint

Forecast each change in turn, then add them all to the last level.

Solution

The last change is $53 - 50 = 3$ .
The next $3$ forecast changes are $1 + 0.5 \times 3 = 2.5$ , then $1 + 0.5 \times 2.5 = 2.25$ , then $1 + 0.5 \times 2.25 = 2.125$ .
Adding them to the last level gives $53 + 2.5 + 2.25 + 2.125 = 59.875$ .

Answer: $59.875$

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