# Interpretation of residuals in linear regression

Assume I have the following regression model:

$Y=\beta_0 + \beta_1*X_1 + \beta_2*X_2 + \beta_3*X_3 + \epsilon$

If I run the same model without introducing the variable $X_3$, can I interpret the new residuals as the values of $Y$ minus the effects of covariates $X_1$ and $X_2$ (but the effect of $X_3$ on $Y$ remains)?

• That's an interesting question. The variance in the residuals of the new model indeed contains the variance introduced by $X_3$. However, the estimates of the effects for $X_1$ and $X_2$ will most likely have changed after excluding $X_3$, so it is not as simple as $\epsilon_{\text{old}} + \beta_3\cdot X_3$. – Frans Rodenburg Aug 27 '18 at 11:05
• – whuber Aug 27 '18 at 11:30