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I often hear that when the residuals depart from normality, the central limit theorem can be used to fix things. I do not quite understand how this works, since the central limit theorem is a statement about scaled sums of random variables. How exactly is the CLT used to make the data normal?

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The CLT does not make the data normal. For OLS the CLT is a result about the regression parameters. Indeed, they are expressed as a sum of random variables.

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  • $\begingroup$ I read something about the residuals being able to be thought of as the sum of independent errors, does that make sense? $\endgroup$
    – user321627
    Commented Jun 1, 2021 at 7:16
  • $\begingroup$ Not really, the residuals are $Y-\hat{Y}$, so you can think of them as a sum, but they don't tend to normal as $n \rightarrow \infty$. $\endgroup$
    – AdamO
    Commented Jun 1, 2021 at 12:59
  • $\begingroup$ I read somewhere that Gauss (the Gauss) had an interesting (but incorrect) argument as to why regression residuals should be normal. Even the giants make mistakes. $\endgroup$ Commented Jun 1, 2021 at 13:07
  • $\begingroup$ @BigBendRegion better provide a source! $\endgroup$
    – AdamO
    Commented Jun 1, 2021 at 13:33
  • $\begingroup$ See here, bottom of p. 64. pzs.dstu.dp.ua/DataMining/mls/bibl/Gauss2Kalman.pdf $\endgroup$ Commented Jun 1, 2021 at 14:11

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