Regression coefficients when equal; properties

Regression coefficients

The two regression coefficients are the slopes of the regression lines:

• b_yx = r·σy/σx: regression coefficient of Y on X. It gives the average change in Y for a unit change in X.
• b_xy = r·σx/σy: regression coefficient of X on Y. It gives the average change in X for a unit change in Y.

When the two coefficients are equal

If b_yx = b_xy, then r·σy/σx = r·σx/σy. For r ≠ 0 this gives σx = σy. Then

b_yx = b_xy = r

So the two coefficients are equal when both variables have the same standard deviation, and each then equals the correlation coefficient. (If r = 0, both coefficients are 0 and are trivially equal.) Since b_yx·b_xy = r², equal coefficients also give r² = b², i.e. r = b, because b has the same sign as r.

Three properties

1. r = ±√(b_yx · b_xy): the correlation coefficient is the geometric mean of the two regression coefficients, with the same sign as both.
2. Both coefficients have the same sign as r, and they cannot both exceed 1 in magnitude (b_yx·b_xy = r² ≤ 1).
3. The arithmetic mean of the coefficients is at least |r| in magnitude: |b_yx + b_xy|/2 ≥ |r| (for positive r, (b_yx + b_xy)/2 ≥ r). Also, regression coefficients are independent of a change of origin but depend on a change of scale.