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After reading several notes on this topic, there's one point that I seem to miss. Most texts assume these hypothesis in a linear regression model:

  • $Y=\beta_0+\beta_1X+\varepsilon$
  • $\varepsilon \sim N(0,\sigma^2)$
  • $Var(\varepsilon_{|X=x})=\sigma^2$ for every $x$
  • The errors are indepedent of each other

From these assumptions, some texts conclude that the distribution of $Y$ given $X=x$ is normal with mean $\beta_0+\beta_1x$ and variance $\sigma^2$. However, if I'm not missing something, shouldn't we impose an extra assumption of independence between $X$ and $\varepsilon$ in order to claim this?

kjetil b halvorsen
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DGRasines
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  • Isn't this implied by "The errors are indepedent of each other"? – spdrnl Dec 07 '15 at 11:28
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    Do they perhaps treat the X's as fixed constants rather than random variables? – Glen_b Dec 07 '15 at 12:11
  • Is it not the case that if errors were not independent of the X's, then the distribution of the errors would depend on the distribution of the X's leading to breach of the second assumption? – user20637 Dec 07 '15 at 15:53
  • Not necessarily. The errors can be identically distributed yet highly dependent. An extreme example would be the case where $\epsilon_1=\epsilon_2=\cdots=\epsilon_n$ and $\epsilon_1$ could be correlated with say, $X_1$. – whuber Dec 07 '15 at 17:16
  • @Glen_b They start by defining a model with "random design", which is the one above, and then they define a model with "fixed design", that is, setting $X=x$. What I don't know is whether they fix $x$ as a constant and then assume the same list of hypothesis changing $X$ by $x$, or if they are conditioning the model with "random design" to $X=x$, which seems to be the natural thing to do. – DGRasines Dec 07 '15 at 18:16
  • Thanks. Could you perhaps link to an example of (or give a reference to) what you're discussing so we can all be on the same page? – Glen_b Dec 07 '15 at 20:52
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    Possible duplicate of [What is a complete list of the usual assumptions for linear regression?](https://stats.stackexchange.com/questions/16381/what-is-a-complete-list-of-the-usual-assumptions-for-linear-regression) – kjetil b halvorsen Jun 21 '17 at 12:37

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