# Contradiction in ARIMA: How does ARIMA predict a stationary series?

I know that for ARIMA to run, the series first needs to be made stationary using differencing. But stationary series are not predictable. So, how is this happening actually?

Quote from Stationarity and differencing:

In general, a stationary time series will have no predictable patterns in the long-term. Time plots will show the series to be roughly horizontal (although some cyclic behaviour is possible), with constant variance.

• yes, sitting at time $t$, the long term expectations are the mean and variance of the stationary series. But, short term, that's not necessarily going to be true empirically because there are dependencies short term depending on the particular model. Dec 6, 2022 at 15:32
• Also, differencing addresses only a single very specific kind of non-stationarity. I very much recommend doi.org/10.1007/s10618-022-00894-5 Dec 6, 2022 at 16:14

This means long-term forecasts will tend to the mean value. However, short-term predictions may well differ from the mean. Consider an MA(1) process as an example: $$x_t=c+\theta_1 \varepsilon_{t-1}+\varepsilon_t.$$ A one-step-ahead point forecast targeting the conditional mean is $$\hat x_{t+1|t}=\hat c+\hat\theta_1 \hat\varepsilon_{t}.$$ This is generally different from the estimated mean $$\hat c$$ that is also the long-term point forecast (where "long" in the context of MA(1) is $$h>1$$ steps ahead): $$\hat x_{t+h|t}=\hat c.$$