# R-squared and variance relation

According to Wiki:

https://en.wikipedia.org/wiki/Fraction_of_variance_unexplained

$$1 - R^2 = VAR_{err}/VAR_{tot}$$

Where $$VAR_{err} = \sum_{i = 1}^N (y_i - \hat{y}_i)^2$$ is the variance of the residuals.

I don't see how this is correct, unless the residuals have mean zero. To see this, define the residuals:

$$e_i = y_i - \hat{y}_i$$

The variance of the residuals is:

$$\sum_{i = 1}^N (e_i - \bar{e}_i)^2 = \sum_{i = 1}^N (y_i - \hat{y}_i - \bar{e}_i)^2$$

Do you disagree or is Wikipedia wrong?

Note that in OLS regression with an intercept term, due to the "first order condition", when you take derivative w.r.t. $$\beta_0$$ you have $$-2\sum_{i=1}^n (y_i - \hat{\beta}_0 - \hat{\beta}_1x_{1i} - \cdots \hat{\beta}_px_{pi}) = -2\sum e_i =0,$$ hence $$\bar{e}_n = 0,$$ thus $$\hat{\sigma}^2 = \frac{1}{n}\sum(e_i - \bar{e})^2 = \frac{\sum e_i ^ 2}{n}.$$