# Solved – How to prove that ARIMA(0,1,1) is equivalent to simple exponential smoothing

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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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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# Solved – How to prove that ARIMA(0,1,1) is equivalent to simple exponential smoothing

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?

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 $$hat{X}_t = alpha X_{t-1} + (1-alpha)hat{X}_{t-1}.$$