# How to work with multi-step forecasting on differenced time series

I have a financial time series that I wish to make 5 step ahead (t+5) forecasts on.

As the series is non-stationary, I have differenced the series.

For every time step t, the response variable is equal to the value at t + 5.

Now, if I wish to predict the price at t + 5, while the series is differenced, the model will essentially predict the expected change from step t+4. However, the price at this step is not known at step t, hence I will not be able to retrieve the prediction for t+5, just the expected change from t+4.

Should I instead difference each observation from the observation at 5 steps before?

Let $$x_t$$ denote the value of the time series of interest at time $$t$$.
Let $$\Delta x_t:=x_t-x_{t-1}$$ denote the increment in $$x_t$$ from $$t-1$$ to $$t$$.
Let hats ($$\widehat{}$$) denote predictions.
If you have $$x_t$$ and $$\widehat{\Delta x}_{t+1},\dots,\widehat{\Delta x}_{t+5}$$ available, then it is straightforward to obtain $$\hat x_{t+5}:$$ $$\hat x_{t+5}=x_t+\widehat{\Delta x}_{t+1}+\dots+\widehat{\Delta x}_{t+5}.$$
• @Kaangy, If you are working with a differenced time series $\Delta x_t$, this is the way to do it. If your time series were stationary and you were working directly with $x_t$, then you could try predicting $x_{t+5}$ either directly or step by step. – Richard Hardy Nov 29 '19 at 14:18