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

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Let's start with what the assumption means.

OLS usually motivates the outcome as random. We usually write

$$ y \vert x \sim \mathcal{N}(x^T\beta, \sigma^2) $$

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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Regression analysis is always done conditional on the explanatory ("independent") variables. Thus, these latter variables are always observed quantities. Mathematically, the theory of probability results in regression analysis are all conditional on these values, so they do not involve treatment of the $x_i$ valeus as random variables. Rather than saying that "the independent variable is not random" it is simpler just to say that "the independent variable is an observed value (so its value is known)".

Probability theory and statistics go to great pains to avoid getting into the weeds on what "randomness" actually is (which is a matter in the domain of philosophy). In probability theory we simply refer to a "random variable" as a particular type of function that gives rise to events to which probability statements are applied. In operational terms, in both probability and statistics, once you observe a random variable its value is then known, and so you now treat it as a fixed value rather than a random variable.

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