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

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

-brett

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.