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In OLS regression, when the assumption of normally distributed residuals is rejected, is bootstrap (and block bootstrap) the way to deal with it?

Is this the right way to go or non-normally distributed residuals should be handled differently?

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By rejected I assume you mean a statistical test is applied. Beware as these tests do not have a power of 1.0 as needed for your approach, and they can also, for very large N, reject the null hypothesis of normality when the non-normality is very small. But more to your point, OLS regression is only optimal if you have normal residuals and constant variance. If for example you should have analyzed log(Y) instead of Y, all the coefficients will be virtually meaningless. It is far better to think of a more general and robust solution. One path forward is semiparametric regression, e.g., proportional odds ordinal logistic regression or proportional hazards model. Examples in BBR and RMS course notes show how to do such robust modeling for continuous Y.

Don't use the bootstrap to get the right standard error of the wrong quantity.

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  • $\begingroup$ So do I understand correctly that bootstrap is not the appropriate approach to deal with non normally distributed residuals. Right? $\endgroup$
    – adrCoder
    Oct 21 '20 at 11:48
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    $\begingroup$ The bootstrap can get you the right standard error of the wrong parameter. So in general, no. Better to relax model assumptions. $\endgroup$ Oct 21 '20 at 12:05
  • $\begingroup$ What do you mean by relaxing model assumptions? To not care about not normal residuals? $\endgroup$
    – adrCoder
    Oct 21 '20 at 12:29
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    $\begingroup$ Yes, and linearity, etc. Semiparametric models (e.g. proportional odds model) for continuous Y make no assumption about the distribution of Y for a given combination of predictor settings. $\endgroup$ Oct 21 '20 at 12:39
  • $\begingroup$ very interesting. I will look at these models as I have never used them before $\endgroup$
    – adrCoder
    Oct 21 '20 at 12:58

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