# Do we assume that the regressors are uncorrelated with the unobserved error $\epsilon$ for least squares?

I recall seeing sources in the past state that the Gauss-Markov assumptions assume that the regressors are uncorrelated with $$\epsilon$$ in order to make $$E[\hat{\beta}] = \beta$$. But is this necessarily an assumption? Because if $$X$$ is fixed, then by definition, it's going to be uncorrelated with anything.

It seems to only be an assumption if $$X$$ was taken as a random variable, but I think in most contexts of linear regression, we take $$X$$ to be fixed?

• – markowitz Apr 8 at 19:11

From Wilks's (1962) derivation on p. 283-289 in Mathematical Statistics the assumptions state nothing about correlation between $$X$$ and $$\epsilon$$ only that the set of $$X_i$$ vectors are linearly independent. And yes, the $$X$$'s are taken to be fixed.
It should be noted that the theorem itself is in terms of expectations, in which case the $$\epsilon$$ vanish.
Edit: from the Wiki article if and thanks to the commenter, if $$X$$ is not considered fixed, but the assumptions are instead taken as conditional on $$X$$ then we have the Strict exogeneity assumption in which $$X$$ and $$\epsilon$$ are uncorrelated, which is common in econometrics