# Setting intercept to zero: Will this change both standard deviations and the error term?

After running a single regression with a forced zero intercept, I understand that $\beta$ (slope(s)) will change as $\alpha$ (intercept) will be set to zero. Easy.

$\rho = \beta(\sigma_x / \sigma_y)$ is left in question...

by way of.... $\beta = \frac{\mathrm{Cov}(X,Y)}{\mathrm{Var}(X)}$.....or......${\mathrm{Cov}(X,Y)}= \beta(\sigma_x^2)$

Given the forced change, will the $\sigma_x$ and $\sigma_y$ (std.devs) both remain as their non-forced original values, with $\rho$ (correlation) changing to match the new forced slope? What will happen to the error term given the forced zero regression?

Note: Calculations confirmed in Excel via using StdevP & VarP functions in small sample group.

• I don't understand your notation $\rho_y$ and $\rho_y$. The formula you give should probably be meant $\beta_x = b_x(\sigma_x/\sigma_y)$. This formula of converting b in beta is valid also when there is no intercept; but then $\sigma_x$ and $\sigma_y$ must be standard deviations not from the means but from 0, - they are root mean squares then. – ttnphns Oct 18 '13 at 1:27
• @ttnphns Please see edits for better clarification – Bob Hopez Oct 18 '13 at 2:27
• It'd be nice to follow more standard notation: $b$ = regr. coef; $\beta$= standardized regr. coef; $r$ = empirical correlation etc. – ttnphns Oct 18 '13 at 2:36