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:
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?