Consider two variables, \(y\) and \(x\). Do a regression of \(y\) on \(x\) to get a slope coefficient that we call \(b_{1, x, y}\). Do another regression of \(x\) on \(y\) to get a slope coefficient that we call \(b_{1, y, x}\). Show that the correlation coefficient between \(x\) and \(y\) is the geometric mean of the two slope coefficients up to sign; that is, show that \(|r|=\sqrt{b_{1, x, y} b_{1, y, x}}\).