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Having a simple model like: Y = a + bX + u where u represent the error term, if I know that the observation of X and Y are i.i.d, also the error term is i.i.d?

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marked as duplicate by gung regression Jan 3 at 12:42

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    $\begingroup$ stats.stackexchange.com/questions/94872 answers the question about independence, because $u = Y-a-bX$ is a function of the iid variables $(X,Y).$ As far as whether the $u_i$ have identical distributions, perhaps you can answer that part of the question yourself? $\endgroup$ – whuber Dec 29 '18 at 18:34
  • $\begingroup$ @whuber ok, thank you, therefore is independent. Although is trivial I still have some doubt about identically distribution. Supposing that both X and Y are N(0,1) then u is N(-a,2). Therefore, are not identically distributed. is it correct? $\endgroup$ – Albert Dec 29 '18 at 21:01
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    $\begingroup$ If $X$ and $Y$ are both standard Normal, then perforce $a=0$ and $b=1$ and you must still conclude the $u_i$ are identically distributed. $\endgroup$ – whuber Dec 29 '18 at 22:57
  • $\begingroup$ @whuber The variance of the sum of two independent standard normal, should not be the sum of the variance of each r.v.? As a consequence u should not have a different variance? $\endgroup$ – Albert Dec 30 '18 at 9:04
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    $\begingroup$ You haven't supplied assumptions that imply $u$ is the sum of independent standard Normal variables. $\endgroup$ – whuber Dec 30 '18 at 16:07