Cov(e,X) in the population regression Say the population regression function is:
$$ Y_i = \beta_0 + \beta_1 X_i + \varepsilon_i $$
(In the econometrics context) While I can't just assume that $E[\varepsilon_i | X_i] = 0$, can I not say that $Cov(\varepsilon_i, X_i) = 0$ just as an algebraic consequence of OLS?
Of course, OLS only that says $\widehat{Cov}(\varepsilon_i,X_i) = 0$, but assuming I'm running OLS on the entire population dataset, then this "sample" covariance is just the population covariance, right?
What am I missing here? Is it that we assume that there's a data generating process and thus I can't just say that $\varepsilon_i$ is the residual from OLS on the population dataset? Or is the error term in the population regression function not a residual from OLS but some structural error term of this model?
But if I run OLS on the entire population dataset, it is the residual.

The assumption that $Cov(X,\epsilon)=0$ is not even needed. If you replace it with the more intuitive and practically relevant assumption that $E(Y|X=x) = \beta_0+\beta_1 x$, then the covariance condition is automatically true. So instead of worrying about the covariance assumption, you can instead worry about whether the conditional mean function is truly linear.
