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I'm a bit confused by the following statement found in "Introductory Econometrics, A Modern Approach" (Wooldridge, 2019):

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The phrase seems to imply the assumption is about independence and Normality. But as far as I can tell, $u \sim N(0,\sigma^2)$ already implies complete independence between $u$ and the explanatory variables. Maybe he is just being explicit about the implications of Normality without going onto the details of what implies what, details that might confuse a novice reader (this might be supported by the title of the assumption, which is "Normality" rather than "Independence and Normality").

Still, my point is: $u \sim N(0,\sigma^2)$ implies independence not only for the first two moments of $u$ but for the whole distribution. Is this the case? Am I misunderstanding Wooldridge's point?

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Stating the distribution of $u$ does not imply independence.

Consider $v = 2u$. Thus, $v \sim \mathcal N (0, 4\sigma^2)$, yet $v$ is not independent of $u$.

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$u\sim\mathcal N(0,\sigma^2)$ does not imply independence, as @Firebug stated. But I would add that $u|x_1,x_2,\cdots,x_k\sim\mathcal N(0,\sigma^2)$ does imply independence, since it means that the conditional distribution of $u$ is always the same, regardless of the values of the covariates.

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