Linear regression with normalized variables

Question

Suppose I have two variables X and Y such that mean(X) = 0 = mean(Y) and sd(X) = 1 = sd(Y). The slope of the linear regression line for Y vs X is cov(X,Y)/var(X) = corr(X,Y) since X and Y are normalized. Suppose corr(X,Y) < 1. By the same token, the slope of the linear regression line for X vs Y is corr(Y,X) = corr(X,Y) < 1. However, the slope of the best fit line for X vs Y should be the reciprocal of the slope of the best fit line for Y vs X (by reflecting and rotating the co-ordindate axes, the best fit line should not change). So the slope of the best fit line of X vs Y should be greater than 1. What am I missing here?

Answer 1

The slope of the "inverse regression" is not the reciprocal. That is, if

$$Y = b_0 + b_1 X + e$$

and

$$X = a_0 + a_1 Y + \epsilon$$

then $a_1 \ne 1/b_1$ (except in the trivial case). You have given a nice reason why not!