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

What does the assumption: "The independent variable is not random." in OLS mean? How can you verify that hypothesis?

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 $y$.

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