this is an exam question of mine, but I am really struggling with it.

I have seen proofs online but they are too vague and do not connect the dots explicitly. Would someone be able to post a proof which shows the steps exactly?

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#### Best Answer

An Arima(0,1,1) can be written as $$ X_t = X_{t-1} + epsilon_t + thetaepsilon_{t-1}. $$ I think if you use the innovations algorithm to come up with minimum-MSE, recursive, one-step-ahead forecasts, you will get $$ hat{X}_t = X_{t-1} + theta (X_{t-1} – hat{X}_{t-1}) $$ for $t > 1$. If you just define $ alpha – 1 = theta$ then this equation above can be rewritten as $$ hat{X}_t = alpha X_{t-1} + (1-alpha)hat{X}_{t-1}. $$

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