Interview Question: Tale of Two Concepts in Regression

Sample Question #272 (statistics / econometrics)
My junior quant just told me that when I run a regression like y=bx+e, where y is the dependent variable and x the independent variable, the coefficient estimate on b is just the correlation between x and y. I’m not too sure about what he said. What do you think? Is the coefficient estimate the same as the correlation?
I also heard that the R2 from the regression can be interpreted as some kind of correlation. Can you explain?
(Comment: A lot of quants mix up these concepts)
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3 Responses to Interview Question: Tale of Two Concepts in Regression

  1. says:

    so R squared  is just corr(x,y)^2 as corr(x,y)=corr(b^hat*x, y), am I right?

  2. Brett says:

    Lionapoleon: you’re correct. I corrected and updated my answer to reflect your contribution. Thanks.

  3. Brett says:

    ANSWER (updated)
    I’ll restrict my answer to the univariate case.
    You should know the formula for obtaining the coefficient estimate: b_hat = cov(x,y)/var(x). But what’s the correlation coeffcient between x and y? It’s rho=cov(x,y)/sqrt(var(x)var(y)). Now can you see the difference?
    R^2 is actually the square of the correlation between y and the fitted value of y, i.e., it equals [corr(y, b_hat * x)]^2=[corr(y, x)]^2 in the univariate case.

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