Sum of residuals in multiple regression equals 0

Question

I understand that in multiple regression $$\sum_{}^{} X_{i,j}\hat{u}_{i} = 0 $$ but I do not understand how my textbook says that if we include the intercept in the regression ($X_{i,0} = 1$)then we get: $$\sum_{}^{} \hat{u}_{i} = 0$$

Answer 1

Completing the notation with the indices $$ \forall j: \sum_{i=1}^{n} X_{i,j}\hat{u}_{i} = 0. $$ As you say, if $X_0$ is the constant then $$ \forall i: X_{i,0} = 1. $$ Inputing $j = 0$ into the first equation $$ \begin{align*} \sum_{i=1}^{n} X_{i,0}\hat{u}_{i} & = 0 \\ \\ \sum_{i=1}^{n} 1\hat{u}_{i} & = 0 \\ \\ \sum_{i=1}^{n} \hat{u}_{i} & = 0. \end{align*} $$

Answer 2

When an intercept is included, sum of residuals in multiple regression equals 0.

In multiple regression, $$ \hat{y}_i = \beta_0 + \beta_1x_{i,1} + \beta_2x_{i,2} +…+ \beta_px_{i,p} $$ In Least squares regression, the sum of the squares of the errors is minimized. $$ SSE=\displaystyle\sum\limits_{i=1}^n \left(e_i \right)^2= \sum_{i=1}^n\left(y_i - \hat{y_i} \right)^2= \sum_{i=1}^n\left(y_i -\beta_0- \beta_1x_{i,1}-\beta_2x_{i,2}-…- \beta_px_{i,p} \right)^2 $$ Take the partial derivative of SSE with respect to $\beta_0$ and setting it to zero. $$ \frac{\partial{SSE}}{\partial{\beta_0}} = \sum_{i=1}^n 2\left(y_i -\beta_0- \beta_1x_{i,1}-\beta_2x_{i,2}-…- \beta_px_{i,p} \right)^1 (-1) = -2\displaystyle\sum\limits_{i=1}^n(y_i-\hat{y_i})=-2\displaystyle\sum\limits_{i=1}^ne_i=0 $$ Hence, when an intercept is included, sum of residuals in multiple regression equals 0.