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I'm trying to determine whether the above statement is true or false, and provide a short proof. My instinct is that this is false, and I just need a simple counterexample.

cashman
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    Please make $X$ and $x$ consistent. – Zhanxiong Mar 30 '24 at 12:17
  • ... and clarify your notation. Is "$X$" a random variable or not? Does "$Xe$" refer to a scalar or perhaps a dot product? Are these expectations conditional on $X$ or not? What additional assumptions are you making about $e$? – whuber Mar 30 '24 at 13:30
  • Counterexample: https://stats.stackexchange.com/questions/190703/non-linear-endogeneity/190800#190800 – Christoph Hanck Mar 30 '24 at 14:56
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    Fixed lowercase $x$, sorry was late at night. I believe it can be assumed that $X$, $e$, $Y$ and $\beta$ are all scalars. $X$ and $e$ are both random variables. – cashman Mar 30 '24 at 20:57
  • Perhaps it is as simple as a counterexample like letting $e = \frac{1}{x}$... – cashman Mar 30 '24 at 20:59
  • Thanks Christoph. – cashman Mar 30 '24 at 21:08
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    Your question has nothing to do with $Y,$ so why mention it in the title? This suggests you haven't formulated the question you really want to ask. – whuber Mar 31 '24 at 02:32

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