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?


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.

  • $\begingroup$ I read something about the residuals being able to be thought of as the sum of independent errors, does that make sense? $\endgroup$
    – user321627
    Jun 1 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
    Jun 1 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. $\endgroup$ Jun 1 at 13:07
  • $\begingroup$ @BigBendRegion better provide a source! $\endgroup$
    – AdamO
    Jun 1 at 13:33
  • $\begingroup$ See here, bottom of p. 64. pzs.dstu.dp.ua/DataMining/mls/bibl/Gauss2Kalman.pdf $\endgroup$ Jun 1 at 14:11

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