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Is it true that Gauss-Markov assumptions (i.e. linearity, full rank, strict exogeneity, and $\sigma^2 I$) can imply "consistency" and "asymptotic normality" of the OLS estimator?

Or do we also need to make assumptions about the distribution of $X$ and $e$?

And can we say that Gauss-Markov assumptions are stronger (or weaker) than asymptotic assumptions?

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G-M implies neither consistency nor asymptotic normality. Here is a counterexample.

Consider the indicator variable model $Y_i = \beta_1 I_i(i = 1) + \beta_2 I_i(i \neq 1) + \epsilon_i$, $i = 1, \dots , n$, where the $\epsilon_i$ are i.i.d. mean zero, not normal, but with common finite variance. Assume $\beta_1 = \beta_2 = 0$. This model satisfies the G-M conditions.

(The model is used in "event studies" in finance, where in the above model, $i=1$ indicates a possible "event" under study.)

Here, $\hat \beta_1 =\epsilon_1$ for all $n$. But $\epsilon_1$ is not zero (for any $n$, regardless of whether $n$ is "large,") and it is not normally distributed.

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