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My question is based on this question. Suppose we assume the sample is iid (so time series data is out) and $E[e_i X_i ] = 0$ but we're not sure about $E[e_i \mid X_i]$ = 0. Can you provide a counterexample where the OLS estimator is consistent but not unbiased?

The question I'm referring to has AR(1) as a counterexample, but that violates the iid assumption. I'm struggling to come up with an example.

user1691278
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