# What does a differenced time series mean for forecast?

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)

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 $$\hat{y}_{t+1} = y_t+\hat{z}_{t+1}$$.