# Hat Matrix off-diagonals, residual covariance in Least Squares Regression

I was looking at the Draper & Smith's 'Applied Regression Analysis', as did the person who asked:this other question on CrossValidated

In short - the variance-covariance matrix of the residuals in regression is given by $(I - H)\sigma^2$, where $H$ is the 'Hat Matrix'. So in general we must assume the residuals are not independent.

Yet whenever I read about the assumptions of regression it says the error terms should be independent (as well as having zero expectation $E[\epsilon_i] = 0$ and equal variance $Var(\epsilon_i) = \sigma^2 \forall i$.

I am missing something, or using a loose definition - my naive, inexperienced reading of this looks contradictory. Thank you, Chris

To understand the dependence of the residuals consider a model with an intercept at which case it is possible to show that the residuals should sum to zero This means that they cannot be uncorrelated as they should always satisfy this condition and deviations from it are not allowed. This correlation will tend to be more pronounced for small samples, for instance consider the case of $n=3$. We then should have
$$e_1 + e_2 + e_3 = 0 \implies e_1 = - e_2 - e_3$$