Using weather forecasts as exogenous data for timeseries forecasting I'm using Facebook Prophet as a forecasting model and I want to use weather data (temp for example) as additional regressor (exogenous data or external variable).
Additional regressors are integrated in a linear way into Prophet and need future values at forecast time.
My question is: If I know that I can have pretty good weather forecasts say at day-4:

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*Should I train the model using D-4 weather forecasts in the past and use the D-4 weather forecasts at prediction time ?

*Or, train the model using day's actual temperature (D-0) and then use the D-4 temperature forecast at prediction time (as if it was the same as D-0 or actual weather in 4 days) ?

EDIT
I assume in this case that the predictand has a relationship only with the actual temperature values and not with the lagged weather forecasts.

I'd definitely go with your option (1).  Here's a couple of reasons why:

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*It's possible that the weather forecasts will show a (hopefully small) bias by D-4. Then option (2) would result in biased forecasts from your Prophet model.


*Consider the situation where your predictand has a strong relationship to the observed temperature at the same point in time. Even if D-4 temperature forecasts have zero bias, they won't be perfect (some will be lower than the true temperature and some higher), so the relationship between your predictand and the D-4 forecasts will be weaker than that between your predictand and the observed temperature. Then option (2) would place too much weight on temperature over other regressors (whether endogenous or exogenous) in your fitted model.
In the jargon of weather forecasting, your option (2) is the perfect prognostic method ('perfect prog') - see American Meteorological Society's glossary. It was used in the past, partly because it's computationally cheaper, but is now rarely used. That page contrasts it with model output statistics (MOS), which corresponds to your approach (1). This is often used by weather forecast providers to 'post-process' output from physics-based computer models (numerical weather prediction models).
