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Given the following linear regression model as following, with two explanatory variables $x_1$ and $x_2$ and response $y$ $$y_i=a+bx_{i1}+cx_{i2}+\epsilon_{i}$$ We say that $\hat{a}, \hat{b}, \hat{c}$ are the least square estimators for $a,b,c$. I just wonder if we have the sample correlation of the two explanatory variables $x_1$ and $x_2$ is positive, then how can I see that the correlation between $\hat{b}$ and $\hat{c}$ will be negative? Is there a way to show that this is true? I'm trying to understand this and I cannot see why this should be negative since our correlation between the two explanatory variables $x_1$ and $x_2$ is positive?

kjetil b halvorsen
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Amir Hassan
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  • Variants of this have been asked many times on this site, but is difficult to search for. The one dup I found is https://stats.stackexchange.com/questions/569639/negative-correlation-between-hat-beta-1-and-hat-beta-2, which does not have answers ...This phenomenon often shows up in another way, as seen in posts such as https://stats.stackexchange.com/questions/242113/linear-regression-positively-correlated-predictors-and-negative-coefficients, https://stats.stackexchange.com/questions/99634/positive-correlations-to-dependent-variable-but-negative-coefficients – kjetil b halvorsen Apr 10 '22 at 12:13
  • This site search produces many insightful threads about closely related issues. @Kjetil It does not matter that a duplicate has no answers: we need only one version of any given question. Our [help] explains what to do to encourage answers to questions. – whuber Apr 10 '22 at 12:13

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