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For the following

$$Y = B_0 + B_1X_1 + B_2X_2,$$

when will $B_1$ be equal to the correlation between $Y$ and $X_1$?
When will $B_2$ be equal to the correlation between $Y$ and $X_2$?
Why?

Richard Hardy
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user141053
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  • If this is "homework" you should read http://stats.stackexchange.com/tags/self-study/info. – Carl Dec 04 '16 at 04:20
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    Related: https://stats.stackexchange.com/questions/32464/how-does-the-correlation-coefficient-differ-from-regression-slope/32466#32466 – Paul Dec 04 '16 at 05:07
  • @Paul Thank you for finding that thread. There may be a subtle difference, though: it concerns *estimated* coefficients whereas in the present case the question might be about the *model* coefficients. – whuber Dec 04 '16 at 19:13

1 Answers1

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The answer to both questions is that a slope coefficient will equal the correlation value when you have the equation in standardized form (i.e. when the Y and X variables are z-scores) and when the predictors are independent of each other.

Not a full proof of what I said - but you can see the two-predictor case here and you'll observe that the standardized coefficient will equal the correlation between Y and Xi when the correlation between X1 and X2 is r = 0

J Taylor
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