# Interpretation of non-independent errors in linear regression

I have problems to interpret dependencies in the covariance matrix $$Cov(\epsilon)$$ of errors in the linear Regression model $$Y = X\beta + \epsilon$$​, i.e. when there are non-zero entries apart from the entries in the diagonal. For example let's assume that $$Cov(\epsilon_i,\epsilon_j) > 0$$ for some $$i$$ and $$j$$. If I understand the concept of errors right, $$Cov(\epsilon_i,\epsilon_j) > 0$$ implies that the deviations from the expected $$y_i$$ and $$y_j$$ value (i.e. the expected value given the regressor values in the $$i$$th and $$j$$th row of $$X$$), are somehow positively correlated.

Unfortunately, I cannot attach any meaning to this beyond this mathematical interpretation (if it is correct at all). As I understand it, $$\epsilon_i$$ and $$\epsilon_j$$ belong to two different "populations" that are defined by the respective regressor values in the $$i$$th and $$j$$th row. Why should there be an association between the error terms then? I appreciate any help.

• Can you provide some details on what populations means here? If your populations are aspen groves, then an example about people may not be very illuminating. Nov 19, 2020 at 15:34
• I was referring to the population as a mathematical concept. And I am also trying to reach a conceptual understanding. Only an example might make it easier. Nov 19, 2020 at 15:57