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Say I have a linear, cross sectional relationship -

$y_{i}=x_{i}b+e_{i}$.

Where $E(e_{i}|X_{j})=0$ for all relevant $i,j$. Given this, one can prove that the OLS estimator is unbiased.

However, consistency requires us to use the law of large numbers on the samples $x_{i}$ - because we assume that $E(xx')$ is proxied well by its sample counterpart:

Wikipedia proof of consistency

Clearly this only follows if we assume that the predictor data is i.i.d. Am I correct in assuming this?

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    $\begingroup$ It's hard to see in what sense you could make any statement about what happens asymptotically unless you make a specific assumption about the sequence of $x_i.$ Although iid is not the only possibility, the assumption must be something specific (such as second-order stationary). $\endgroup$
    – whuber
    Jun 21 at 21:09

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