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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?

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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

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  • $\begingroup$ I always though it was the slope that was fixed and X was a random variable. My much used tome"Understanding Regression Assumptions" By William Berry list one of the regression assumptions as,....i.e., each independent variable is uncorrelated with the error term. That has always been my understanding. You can see this in number 3 in this link as well. statisticsbyjim.com/regression/… $\endgroup$ – user54285 Apr 8 at 20:22
  • $\begingroup$ Yeah, like I said it is often listed in terms of diagnostics as well. I was simply answering based on the theorem as described / derived by Wilks, where it is not among the assumptions involved in the proof of the theorem $\endgroup$ – Rick Hass Apr 8 at 21:18
  • $\begingroup$ Also, X is a random variable, but the values are taken to be fixed as a part of the derivation of the equations. Slopes are fixed in that they are parameters, and parameters are fixed constants (with sampling error) in frequentist statistics $\endgroup$ – Rick Hass Apr 8 at 21:20
  • $\begingroup$ sorry Rick Hass. I thought we were talking, the op was, about what was and was not in the Gauss Markov assumptions. My mistake. Thanks for the explanation. $\endgroup$ – user54285 Apr 8 at 21:46
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    $\begingroup$ thanks. I think the strict exogenity is common in the areas I do analysis, social science generally. That if a variable is excluded it can not be correlated with both a predictor and Y. Of course I have never been able to understand how that could be true in the real world. There are huge number of excluded variables, some will likely violate this rule. it is the type of issue that raises doubt about statistics to me a practitioner only. $\endgroup$ – user54285 2 days ago

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