Use predicted or actual values for 'unknown' independent variables in linear regression?

Question

When performing linear regression for the sake of prediction, I am left with the conundrum of whether I should use the actual values of the independent variables or the values I would have predicted for those independent variables. This obviously only concerns independent variables that have to be predicted, and not ones whose values are known before the observation of the dependent variable took place.

For example, let’s say you’re trying to predict the annual growth of a tree, using last year’s tree growth rate and current year rainfall (and assume you’re going to use linear regression for this prediction). When running the regression over a dataset of, say, 50 years - should you use what the rainfall actually was for each year or what you would have had predicted the rainfall would be for each year? Keeping in mind that, when it comes to predicting ‘n.ahead’, you are going to have to use a predicted value for that year’s rainfall. Note also that you won’t have this problem with the other independent variable (last year’s tree growth rate) as this is known at the point of prediction.

And if you should use predicted values but you can’t assemble the necessary data to make the predictions for the previous 50 years, is it a mathematically sound alternative to use actual values instead?

Answer 1

That depends on what the purpose of your prediction task is exactly:

• If you are really interested in predicting say in the next 50 years, given only the past data (at time 0), you should use the predicted values at each step as the actual values are not available during the prediction task.
• On the other hand, it's possible that you only want to predict the next year given the past year. In this case you can use the actual values as it's an iterative prediction.