Regression coefficients and properties

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

• b_yx (regression coefficient of Y on X) is the slope of the line Y = a + bX. It is the change in Y for a unit change in X: b_yx = r·σy/σx = Sxy/Sxx
• b_xy (regression coefficient of X on Y) is the slope of the line X = a′ + b′Y. It is the change in X for a unit change in Y: b_xy = r·σx/σy = Sxy/Syy

Properties

1. Geometric mean: r = ±√(b_yx · b_xy), i.e. b_yx · b_xy = r² ≤ 1. The sign of r is the common sign of both coefficients.
2. Same sign: b_yx, b_xy and r always have the same sign (all positive or all negative).
3. Both cannot exceed 1: if one coefficient is greater than 1 in magnitude, the other must be less than 1 in magnitude.
4. Arithmetic mean ≥ |r|: |b_yx + b_xy|/2 ≥ |r|, because AM ≥ GM and both coefficients have the same sign. (For positive r this is simply (b_yx + b_xy)/2 ≥ r.)
5. Change of origin and scale: regression coefficients are independent of change of origin but not of scale. If u = (x − a)/h and v = (y − c)/k, then b_yx = (k/h)·b_vu.
6. Intersection: the two regression lines intersect at (x̄, ȳ), so both lines pass through the point of means.
7. If r = 0, then b_yx = b_xy = 0 and the lines are perpendicular; if r = ±1 the lines coincide.