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I have a few questions where I'm trying to show if things are true or false. I'll say upfront that these are homework so I'd rather not get the entire answer just someone to point me in the right direction. So here we go,

If $y = x \beta + e\text{ and } E(e|x) = 0, \text{ then } E(x^2e) = 0$.

If $y = x \beta + e,\text{ and }E(xe) = 0,\text{ then } E(x^2e) = 0$

Thanks for any help.

Edit: Solved then removed one of them.

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    If you solve one of them, you could leave the question up and post your answer as a solution. Someone else might be interested in how you did it.2011-09-15

1 Answers 1

2

Hints:

1) The equations involving $y$ are irrelevant, since $y$ doesn't appear in the conclusion.

2) A fundamental principle is the "Theorem of Total Expectation": $E[ E[X | Y]] = E[X]$.

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    Hi Robert, thanks for the reply. I'm aware of the Theorem of Total Expectation, we call it Interated Expectation, but I'm still not sure how that'll help.2011-09-15
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    What's $E[x^2 e | x]$?2011-09-15
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    +1 Yes, your hints show (1.), but how do we prove (2.)? "If $E(xe)=0$, then $E(x^2e)=0$." In fact, that doesn't appear true to me.2011-09-15
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    So, $E[E[x^2e|x]] = E[xE[xe|x]] = xE[E[xe|x]] = 0$, when $E[xe] = 0$2011-09-15
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    @tsha For (2.), you are not given that $E(xe|x) = 0$ (which doesn't make sense anyway, since that is the same as saying "either $x$ is zero, or $E(e|x)=0$"); only that $E(ex)=0$.2011-09-15
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    For (2.), what happens if $e$ is constant? Can you find an example where $E(x) = 0$ but $E(x^2) \ne 0$?2011-09-15