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I understand that differencing is a way to make a non stationary time series, stationary. From https://otexts.com/fpp2/stationarity.html they talk about first order differencing and second order differencing, but I'm trying to wrap my head around what this means for your forecast.

  1. When you forecast a first order differenced time series are you just forecasting the rate of change in the original data? I see that Hyndman states a second ordered time series would be the rate of change of change, but why is that useful to forecast?

  2. If point 1 is true, would it be possible to turn that result into a result on the scale of the non stationarized data (i.e. if you were to forecast the non differenced data, would it be possible to turn the first differenced forecast into a result of the non differenced data)

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If you have $z_t = \Delta y_t = y_t - y_{t-1}$ then you know that $y_t = y_{t-1}+z_t$.

If you forecast $z$ at time $t+1$ using all the information up to time $t$, $\hat{z}_{t+1}$ then the obvious forecast of $y_{t+1}$ would be $\hat{y}_{t+1} = y_t+\hat{z}_{t+1}$.

Similar (and natural) relationships can be derived for higher orders of differencing or longer horizons.

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