# Why the sum of residuals equals 0 when we do a sample regression by OLS?

That’s my question, I have looking round online and people post a formula by they don’t explain the formula. Could anyone please give me a hand with that ? cheers

If the OLS regression contains a constant term, i.e. if in the regressor matrix there is a regressor of a series of ones, then the sum of residuals is exactly equal to zero, as a matter of algebra.

For the simple regression,
specify the regression model

Then the OLS estimator $(\hat a, \hat b)$ minimizes the sum of squared residuals, i.e.

For the OLS estimator to be the argmin of the objective function, it must be the case as a necessary condition, that the first partial derivatives with respect to $a$ and $b$, evaluated at $(\hat a, \hat b)$ equal zero. For our result, we need only consider the partial w.r.t. $a$:

But $y_i - \hat a - \hat bx_i = \hat u_i$, i.e. is equal to the residual, so we have that

The above also implies that if the regression specification does not include a constant term, then the sum of residuals will not, in general, be zero.

For the multiple regression,
let $\mathbf X$ be the $n \times k$ matrix containing the regressors, $\hat {\mathbf u}$ the residual vector and $\mathbf y$ the dependent variable vector. Let $\mathbf M = I_n-\mathbf X(\mathbf X'\mathbf X)^{-1}\mathbf X'$ be the “residual-maker” matrix, called thus because we have

It is easily verified that $\mathbf M \mathbf X = \mathbf 0$. Also $\mathbf M$ is idempotent and symmetric.

Now, let $\mathbf i$ be a column vector of ones. Then the sum of residuals is

So we need the regressor matrix to contain a series of ones, so that we get $\mathbf M\mathbf i = \mathbf 0$.