Regression question - Weak exogeneity

Question

The typical assumption of linear regression, weak exogeneity, states, $$E(\bf{\epsilon_{i}})=0$$ when the regressors are fixed and $$E(\bf{\epsilon_{i}}|\bf{x_{i}})=0$$ when the regressors are random. I can't figure out for the life of me why you don't still need to condition upon your regressors when they are fixed. If we are going to use our model to extrapolate y values for x values other than the fixed ones we selected, won't we need to assume that the expected value of the epsilons is zero at those points as well?

Answer 1

You don't generally condition on things that are fixed. I can write down $E(\epsilon_i \mid 1+1)$, but the conditioning is meaningless.

If you are interested in predicting for an unknown future $x_i$, then you need to condition on that when you write down your prediction distribution. But fitting the linear regression model is not the same as predicting.