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?

Thanks in advance!


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 $\hat x_{t+5}:$ $$ \hat x_{t+5}=x_t+\widehat{\Delta x}_{t+1}+\dots+\widehat{\Delta x}_{t+5}. $$

  • $\begingroup$ Okay, so I would have to predict each step, I couldn't make a one-shot prediction for t+5? $\endgroup$
    – Kaangy
    Nov 29 '19 at 13:56
  • $\begingroup$ @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. $\endgroup$ Nov 29 '19 at 14:18
  • $\begingroup$ Okay, thanks! Do you think it would be a problem if I only difference the predictors, and not the dependent variable? Alternatively, could I ignore differencing altogether, as my main goal is the best possible prediction and not inference? $\endgroup$
    – Kaangy
    Nov 29 '19 at 14:28
  • $\begingroup$ @Kaangy, There are some guidelines for dealing with nonstationary time series. Integrated series usually need to be differenced (even if the goal is prediction rather than inference), unless they are cointegrated, in which case something like a vector error correction model (VECM) can be used. $\endgroup$ Nov 29 '19 at 15:35
  • $\begingroup$ stats.stackexchange.com/questions/423406/… contains some pointers on when and how and why one uses predictor series. $\endgroup$
    – IrishStat
    Nov 29 '19 at 17:52

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