# Does a linear regression assume that the (unconditional) predictor data is i.i.d?

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:

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

• 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).
– whuber
Jun 21 at 21:09