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For convenience and context, I'm looking at the formula for standard error of the slope coefficient from here: https://www.statology.org/standard-error-of-regression-slope/

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If X is not normally distributed but the residuals are, is the estimated slope coefficient's distribution characterized by the T-distribution?

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  • $\begingroup$ Residuals are never independently Normally distributed, so you likely are trying to ask about the errors in the model. $\endgroup$
    – whuber ♦
    May 31 at 21:00
  • $\begingroup$ @whuber, yes your comment makes sense. $\endgroup$
    – Jitty
    Jun 1 at 1:55
  • $\begingroup$ In Kutner's Applied Linear Statistical Models Ch2 Page 42, it explains that the sampling distribution of $b_1$ is normal because $b_1$ is a linear combination of $Y_i$ and if we assume $Y_i$ is normally distributed, then a linear combination of $Y_i$ is also normally distributed. I think this is the answer I was looking for, if I'm not misunderstanding the passage. Would love feedback on this.. $\endgroup$
    – Jitty
    Jun 19 at 0:58
  • $\begingroup$ It contradicts what you write. Because the estimated slope is Normal, it is not characterized by a Student t distribution. $\endgroup$
    – whuber ♦
    Jun 19 at 17:01

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Yes! The usual linear regression model does not assume that X is normal. The assumptions are independence, homoscedasticity, linearity and normality of residuals.

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    $\begingroup$ Your comment about residuals is not quite correct, as my linked answer discusses. $\endgroup$
    – Dave
    May 31 at 19:47
  • $\begingroup$ The model makes no assumptions whatsoever about residuals: you seem to use "residual" (observed difference between response and fitted value) instead of "error" (a random variable in the model). $\endgroup$
    – whuber ♦
    May 31 at 21:01

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