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This question is regarding applying power transformations on input time series.

In literature the idea behind applying transformations to input time series is to stabilize variance. But I have seen that power transformations can make time series normally distributed.

The question is can we argue that for the special case of linear models (like ARIMA, prophet) making time series normally distirbuted means there is higher probability of getting residuals normally distributed. Therefore stablizing variance is not the only reason for applying transformations.

Or making input time series normal and stablizing variance are related to each other is some way.

rookie_mathematician
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  • It is almost never the case that a power transformation will make anything Normally distributed. All you can hope is to make the distribution approximately symmetric. Many sources confuse the aim of symmetrization with the vain hope of achieving Normality. The relationship between achieving symmetry and stabilizing variance is not universal, but it happens to hold in a variety of practical situations. https://stats.stackexchange.com/questions/4831 addresses these issues. Also see https://stats.stackexchange.com/questions/298 and https://stats.stackexchange.com/questions/24227. – whuber Feb 08 '23 at 17:57
  • Thanks @whuber for those detalied links. I also got to know about John Tukey's book you mentioned in that link. Going to read that too. – rookie_mathematician Feb 09 '23 at 11:25
  • @whuber I see you have consistently used the word "symmetry", I checked your other links. So by symmetry do you mean skewness or something else too. – rookie_mathematician Feb 09 '23 at 16:57
  • I use it in the sense of "approximate symmetry" employed in related threads at https://stats.stackexchange.com/a/3530/919, https://stats.stackexchange.com/a/4833/919. The practical application of this concept is illustrated in my post at https://stats.stackexchange.com/a/96684/919. Recently I offered a formal definition at https://stats.stackexchange.com/a/603002/919. A formal definition of (exact) symmetry of a distribution -- the ideal we are aiming for -- is given at https://stats.stackexchange.com/a/29010/919. – whuber Feb 09 '23 at 17:06

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