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In Regression coefficients that flip sign after including other predictors, ars's answer states that "Basically, if your variables are positively correlated, then the coefficients will be negatively correlated, which can lead to a wrong sign on one of the coefficients."

This is not intuitive to me, and I was wondering if someone can provide (1) an intuitive explanation (preferably without equations) and (2) a mathematical proof of this effect.

24n8
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  • This is discussed extensively in [related threads](https://stats.stackexchange.com/search?tab=votes&q=regression%20change%20sign%20multicoll*%20score%3a2)--please check them out. – whuber Jul 10 '20 at 12:33
  • @whuber I did see several posts on this subject, but I still don't understand high (multi)collinearity leads to a negation in sign of the coefficient. I don't think multicollinearity is guaranteed to call this effect, but it is one possible effect? In other words, there is no guarantee multicollinearity will negate the coefficient? – 24n8 Jul 10 '20 at 15:30
  • Absolutely not! But contemplating the extreme case where one variable is a positive multiple of the other can be helpful: when the estimated coefficient of the variable increases, the *only* thing the model can do to compensate is to decrease the estimated coefficient of the other variable. – whuber Jul 10 '20 at 15:32
  • @whuber This is more so for me than a response to you, but I found this https://stats.stackexchange.com/questions/229052/when-a-and-b-are-positively-related-variables-can-they-have-opposite-effect-on. The top answer shows this geometrically. – 24n8 Jul 11 '20 at 18:15

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