Statistics · Ch 3 — Linear Regression

Regression Coefficients: byx and bxy

The regression coefficient of $Y$ on $X$ , denoted $b_{yx}$ , measures the amount of change in $Y$ for a unit change in $X$ :

b_{yx} = r \cdot \frac{\sigma_y}{\sigma_x} = \frac{\text{Cov}(X,Y)}{\sigma_x^{2}} = \frac{n\Sigma XY - \Sigma X \Sigma Y}{n\Sigma X^{2} - (\Sigma X)^{2}}

The regression coefficient of $X$ on $Y$ , denoted $b_{xy}$ , measures the amount of change in $X$ for a unit change in $Y$ :

b_{xy} = r \cdot \frac{\sigma_x}{\sigma_y} = \frac{\text{Cov}(X,Y)}{\sigma_y^{2}} = \frac{n\Sigma XY - \Sigma X \Sigma Y}{n\Sigma Y^{2} - (\Sigma Y)^{2}}

Properties of regression coefficients (frequently tested in Gujarat Board Std-12 Statistics exams):

#	Property
1	$b_{yx}$ and $b_{xy}$ always carry the same sign, which is also the sign of $r$ .
2	The correlation coefficient is the geometric mean of the two regression coefficients: $r = \pm\sqrt{b_{yx} \cdot b_{xy}}$
3	Since $-1 \le r \le 1$ , it follows that $b_{yx} \cdot b_{xy} \le 1$ always — the product of the two regression coefficients can never exceed 1.
4	If one regression coefficient is greater than 1 in magnitude, the other must be less than 1 (their product still cannot exceed 1), unless $r = \pm 1$ , when both can equal $\pm 1$ .
5	Regression coefficients are independent of change of origin but not independent of change of scale.

Definition 1Cov(X, Y)

Covariance of X and Y — the average of the product of deviations of X and Y from their respective means, $\frac{\Sigma(X-\bar X)(Y-\bar Y)}{n}$ ; it is the numerator shared b …

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