I want to do a rolling window forecast on a time series but it seems the series is white noise ARIMA (0,0,0) with non-zero mean. But when I difference the dataset and model it with an ARIMA(0,1,1) I get a really good forecast compared to the ARIMA (0,1,0).

Why, and is the differenced forecast not valid?


Red Line = ARIMA (0,1,1) and Blue line = ARIMA (0,0,0)

The Arima(0,0,0) and Arima(0,1,1) fitted on a rolling window and same original dataset with the below out-of sample forecast results. Should the forecasts not be equal in the case of white noise?

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  • $\begingroup$ 1. What is the MA coefficient in your ARIMA(0,1,1) model? 2. Are you sure these are forecasts rather than in-sample fitted values? 3. What evaluation metric do you use to support the claim that one forecast is better than the other? Have you compared out-of-sample MSEs, MAEs or something? $\endgroup$ – Richard Hardy Jan 21 at 9:39
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    $\begingroup$ Given your answers to 1. and 2., your forecasts from ARIMA(0,0,0) and ARIMA(0,1,1) should be identical: the MA coefficient of $-1$ completely cancels out the effect of differencing (this is to be expected if the original series is white noise, hence my question 1.). Are you depicting forecasts for the original variable in both cases, or by any chance for the original variable in case of ARIMA(0,0,0) and for the differenced variable in case of ARIMA(0,1,1)? $\endgroup$ – Richard Hardy Jan 21 at 10:52
  • $\begingroup$ Actually the blue line is a ARIMA (0,1,0) model. I will update the post....Nevertheless: 1. The MA-coefficient is -1 and significant 2. Yes, I am sure. They are out-of sample forecasts. I can send u the R code if you want to take a look. 3. Hmm...ARIMA (0,1,0) has lower RMSE and MAE but higher ME. So good point! It seemed like it created worse forecasting results by visual inspection but it seems the opposite is true! $\endgroup$ – endorphinus Jan 21 at 10:52
  • $\begingroup$ I am depicting forecasts on a dataset that is differenced manually one time and then run with the Arima(0,0,1) and Arima(0,0,0) model in it: 1) What is the reason for MA -1 to completely cancelling the effects of the differencing? $\endgroup$ – endorphinus Jan 21 at 10:55
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    $\begingroup$ Original: $x_t=\varepsilon_t$ (white noise), differenced: $x_t-x_{t-1}=\varepsilon_t-\varepsilon_{t-1}$ which is the same as $\Delta x_t=\varepsilon_t+\theta_1\varepsilon_{t-1}$ with $\theta_1=-1$. Hence, white noise becomes MA(1) with $\theta_1=-1$ when differenced. $\endgroup$ – Richard Hardy Jan 21 at 11:50

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