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I have seen the Gauss Markov assumption of uncorrelated dependent variables and error term presented in three different ways. I want to make sure that I am correctly interpreting the underlying mathematics.

The three different presentations:

$$cov(x_i, U)=0$$

$$cov(x_i, u_i)=0$$

and

$$cov(X, U)=0$$

As usual x is the explanatory variable, and u the error term. Just to be clear, the "i" on the equations must denote variables and not observations, as it usually denotes (correct?). Covariance between single observations does not make any sense as a concept. So for multiple equation system with multiple explanatory variables we have $cov(x_i, u_i)=0$, which is the same as $cov(X, u_i)=0 = cov(X,U)$. In other words the conditions are exactly the same. Is my understanding here correct?

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    $\begingroup$ Re "covariance between single observations does not make any sense," please see the discussion at stats.stackexchange.com/questions/179346/… . $\endgroup$
    – whuber ♦
    Mar 10, 2016 at 22:19
  • $\begingroup$ @whuber Interesting, but if it indeed is the observation, how come the different definitions? $\endgroup$
    – Dole
    Mar 11, 2016 at 9:30
  • $\begingroup$ I'm unsure what you are referring to: different definitions of what? $\endgroup$
    – whuber ♦
    Mar 11, 2016 at 14:03
  • $\begingroup$ @whuber Apologies, I have observed the three different formulations of Gauss Markov assumptions in different sources. As explained, since there is a difference between $cov()_ii$(cov for each observation) and $cov()$(cov for all observations), is there a mistake on some papers I am reading? (This also relates to a question I asked before): stats.stackexchange.com/questions/198638/… $\endgroup$
    – Dole
    Mar 11, 2016 at 14:36
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    $\begingroup$ I doubt there is a mistake: you need to read each paper on its own terms, using its own definitions and explanations of its notation. Expositions may differ in whether they view the regressors $X$ as being fixed or being random variables. $\endgroup$
    – whuber ♦
    Mar 11, 2016 at 15:21

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