# 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? Commented 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
Commented Dec 25, 2021 at 14:20
• You could start from writing down the AR(1) model and how exactly you obtain forecasts from it. Commented 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
Commented Dec 25, 2021 at 14:35
• It depends on the model. Commented Dec 25, 2021 at 15:49

As I read it your procedure is correct, at least for an AR(1) process. I want to point out though that a 1step ahead AR(1) forecast with 21 day return data it would amount to a 21 day ahead forecast. In this sense, the model and the frequency of your data are relevant with respect to the interpretation.

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?