# Make 21-days ahead forecast with daily log return data?

I want to make daily 1-days and 21-days ahead forecasts of a stock price. I have used daily log return data for both 1-day and 21-days forecasts.

Now I'm not sure if that is correct for the 21-day case since I would try to forecast the daily change from the 20th to the 21th days rather than a 21 day return. Do I have to use a completely new time series and create daily 21-day returns to make the 21-day ahead forecasts? Or is the procedure I have tried correct?

• Could you write down your model? Dec 25, 2021 at 14:14
• @RichardHardy What do you mean exactly? I use multiple forecasting models. A LSTM neural network and a AR(1) model as benchmark.
– Ben
Dec 25, 2021 at 14:20
• You could start from writing down the AR(1) model and how exactly you obtain forecasts from it. Dec 25, 2021 at 14:26
• It is more of a conceptual questiom. The forecasting model is irrelevant. My question is: Do I have to use daily return data or 21 day return data to make 21-day ahead forecasts?
– Ben
Dec 25, 2021 at 14:35
• It depends on the model. Dec 25, 2021 at 15:49

If you have daily log returns like you do, the 21 day ahead forecast comes about by doing 21 separate 1 day ahead forecasts. Each return forecast $$x_{T+j+1}^*=F(x^*_{T+j})$$ for $$j=0,...,20$$ has input the previous forecast, except for the first forecast ofcourse. But this is just in the case of an AR(1).
If you have another model like $$return_t=0$$ you do not need to do anything with your data, you know your forecast is zero. Long story short, your model is absolutely relevant in producing "correct" forecasts. Without a forecasting model how else would you produce a forecast?