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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.

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    $\begingroup$ 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. $\endgroup$
    – dimitriy
    Nov 19, 2020 at 15:34
  • $\begingroup$ 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. $\endgroup$
    – jonaden
    Nov 19, 2020 at 15:57

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The errors or disturbance terms have two common interpretations:

  1. Omission of the influence of innumerable chance events, which constitutes one type of specification error. But this can also include innumerable less systematic influences, such as weather variations, taste changes, earthquakes, epi­demics, or even postal strikes. It's possible for such shocks to be correlated for two observations if they are "close" in some sense (physical, social, trade network). This can either be positive or negative, and decays with distance and time.
  2. Measurement error. Perhaps less of a concern with populations, but, for example, election fraud where ballots are thrown away in some areas that share common characteristics. Another is a survey/surveyor or instrument that have varying fidelity. For example, you might have behavior tracking that works well on your website, but not in the iPhone app, and reasonably well in the Android app, so there will be correlated ME within each platform.
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  • $\begingroup$ Thanks alot! That was very helpful! So the error is, in the cases you described, conceptualized as not actually being drawn from a random distribution but rather as being determined by influences that are present at the time the sample was taken. The assumed random distribution rather describes that these influences might change over time. In this case correlated error terms of course make sense. Did I get your argument right? $\endgroup$
    – jonaden
    Nov 20, 2020 at 10:38
  • $\begingroup$ I think of the disturbances as random in each period, and sometimes persistent across periods. Suppose you want to compare pizzeria sales across US states (regress sales on 50 state indicators). If there is a snow storm in NJ and pizza delivery sales are lower because the roads are bad, pizzerias near NJ would also suffer to a degree, but states a little further away might actually see better sales since customers don't want to go out themselves, but the roads are not so bad that the delivery drivers cannot get through. Stores in CA are not effected. This may not extend into the next week. $\endgroup$
    – dimitriy
    Nov 20, 2020 at 16:49
  • $\begingroup$ In the example above, you have within period correlation that decays as you go farther from NJ, but decays quickly across periods time. There is also some ambiguity in how to model the weather if you had snowfall data, which will also create misspecification error. $\endgroup$
    – dimitriy
    Nov 20, 2020 at 16:52
  • $\begingroup$ I am economist by training, so worrying about the error term is 90% of what we do. $\endgroup$
    – dimitriy
    Nov 21, 2020 at 17:25

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