Solved – What does the assumption: “The independent variable is not random.” in OLS mean

Let's start with what the assumption means.

OLS usually motivates the outcome as random. We usually write

$$ y vert x sim mathcal{N}(x^Tbeta, sigma^2) $$

The $y vert x$ is a bit of an abuse of notation. It means that, assuming I already know $x$, then I can consider $y$ as a random draw from a normal distribution with specified mean and variance. So the assumption here is not really that $x$ isn't random, its just that whatever distribution that $x$ has will not affect our inferences about $y$ because we are to know what $x$ is. We are talking about the conditional distribution of $y$. Conditioned on what? $x$.

Verification of this assumption is not required. In fact, it is patently false! But that doesn't matter for OLS, because OLS takes $x$ as given. We know it at the time of doing inference on $y$.