Can someone clarify the difference between regression slope (i.e. b0 = COV(Y,X)/Var(X)) and correlation coefficient (i.e. r = COV(Y,X)/var(x)var(y)) Mathematically, I see the differences, but intuitively, I thought the correlation coefficient was the slope. Can someone please clarify why they’re different? Many thanks
correlation coefficient = cov(x,y) / (sigma x * sigma y) not var(x) * var(y) as you have got it above. and regression slope is the beta. (from SML days) beta = cov(x,y)/var(x) correlation coeff is not the slope, beta is the slope. Hope I am right!
Ok, got ya. I actually made a typo and indeed the denominator in the correlation calculation should be sigma x*sigma y. Put differently then, how would you express these definitions in layman’s terms: “slope” = as you move through y, you get a certain change in x “correlation” = for a change in one value, you get a certain % change in another value Sounds the same… I’m trying to make clear these differences, as they are mathematically. Thanks
both correlation and the regression slope are two different things correlation is not the change of one value for a % change in another value. It is the “strength” of the linear relation between the two variables in a single variable regression. (and which is definitely not the slope).
Great, thanks.
in a univariate regression, the two measures are related as follows start with beta = cov(x,y)/var(x) = cov(x,y)/(sigma x * sigma x) then substitute = correlation(x,y)* (sigma x * sigma y) into the above equation cancel out the sigma x in the numerator and denominator, and you get beta = correlation (x,y) * (sigma y)/(sigma x) so, the beta is just the correlation adjusted for the relative dispersions of x and y. correlation coefficient = cov(x,y) / (sigma x * sigma y) not var(x) * var(y) as you have got it above. and regression slope is the beta. (from SML days) beta = cov(x,y)/var(x)
oops - forget the mess at the end of my previous post - sloppy cut and pasting.