The setting is a simple OLS regression where the true model has regressor $x$ and error term $u$, but we can only measure $\bar{x}=x+v$ where $v$ is iid with mean 0.

According to the textbook:

$\mathbb{E}(u|x)=0$ is satisfied, but $\mathbb{E}(\bar{u}|\bar{x})=\mathbb{E}(u-\beta x|x+v)=-\beta v$

I can see that $x$ and $u$ are correlated, but why am I allowed to just take $\beta x$ out of the expectation when I "only" condition on the sum $x+v$, and not explicitly on $v$?


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