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I was studying linear regression lately and checking the assumptions for Ordinary Least Squares method for the regression problem. I was not sure about the intuition behind the difference of squares being used as the error measure (instead of something like x^4 or x^6) and I found an answer somewhere on stackoverflow that said:

The main idea of OLS is to not minimising the error but making it equal to zero. The errors (residuals) follow normal distribution with a variance in the order of the square. So to minimise the errors we need to use least squares.

And then I got to the square difference of values being used in variance. So why is square function employed in the definition of variance. Is it just convenience or is there any reason for not using functions in the order of x^4 or x^6.

kaushalpranav
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  • Related: https://stats.stackexchange.com/questions/118/why-square-the-difference-instead-of-taking-the-absolute-value-in-standard-devia , https://stats.stackexchange.com/questions/225734/why-isnt-variance-defined-as-the-difference-between-every-value-following-each – kjetil b halvorsen Feb 01 '19 at 12:49
  • I googled the first sentence of your "quotation" but only your post shows up. It's unlikely anybody actually stated this because it's not remotely close to correct, but if you think they did then could you please (a) give an accurate quotation and (b) reference it so we can read it in context? – whuber Feb 01 '19 at 20:06
  • [This](https://math.stackexchange.com/a/2849651) is a very good explanation to your question. I was startled first by your question. Haven't really thought about it before. But the answer I linked to explained it pretty well, although he goes into LR only at the end. Have a look. – Humpelstielzchen Feb 01 '19 at 18:57

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