# 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?

• Commented May 3, 2022 at 0:39

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