# 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. – Dimitriy V. Masterov Nov 19 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. – jonaden Nov 19 at 15:57