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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1$\begingroup$ You appear to have a common misconception about the central limit theorem: stats.stackexchange.com/questions/473455/…. $\endgroup$– DaveJun 1, 2021 at 15:31
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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? $\endgroup$ Jun 1, 2021 at 7:16
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$\begingroup$ 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 $n \rightarrow \infty$. $\endgroup$– AdamOJun 1, 2021 at 12:59
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$\begingroup$ 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, 2021 at 13:07
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$\begingroup$ See here, bottom of p. 64. pzs.dstu.dp.ua/DataMining/mls/bibl/Gauss2Kalman.pdf $\endgroup$ Jun 1, 2021 at 14:11