What happens if you reject normality of residuals when estimating with least squares?

Is it too important to have normality on the residuals?

  • $\begingroup$ in what way are the residuals non-normal? $\endgroup$ – Macro Jun 3 '12 at 0:55
  • $\begingroup$ and how did you decide to reject normality? $\endgroup$ – John Jun 3 '12 at 2:07
  • $\begingroup$ And how many observations do you have? $\endgroup$ – jbowman Jun 3 '12 at 18:31

The $t$-statistic is assumed to be distributed asymptotically normally for hypothesis tests. If your residuals are severely non-normal, your $t$-statistics, $p$-values, and hypothesis tests will be meaningless.

Your $\hat{\beta}$ estimates are still okay, but you cannot express confidence in the $\beta$s.

You can try and use some form of robust standard error that controls for non-normality. Alternatively, if your dataset is very large and your $\hat{\beta}$ estimates are very far from zero, you might be able to get away with it.

  • $\begingroup$ gmacfarlane, the t-test (the unequal variance version) is actually equivalent to use of robust standard errors. In samples of at least modest size and with independent observations, the property that actually matters is constant variance; if the observations in your two groups have it, the equal variance t-test will be fine. If they don't, the unequal variance t-test will still be fine. Normality (or non-Normality) of the residuals is just not very important; the Central Limit Theorem takes care of the distribution of the t-statistic. $\endgroup$ – guest Jun 3 '12 at 3:50
  • $\begingroup$ What if the error distribution is not normal by any normality test, but is symmetric and majority of the residuals are near to zero? Something like a t distribution? Assumption satisfied? $\endgroup$ – user41490 Mar 7 '14 at 10:55
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    $\begingroup$ I believe that statistic inference is destroyed more than implied above. And non-normality of residuals is sometimes indicative of not having the best transformation of $Y$. $\endgroup$ – Frank Harrell Mar 7 '14 at 13:23

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