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I'm following a course in econometrics where we have established that for a linear regression model $$y_i=x_i'\beta+\varepsilon_i$$

If $1/N \sum_{i=1}^N x_i x_i'$ converges to a nonsingular matrix and $E[x_i \varepsilon_i]=0$ OLS will be consistent .

What I would like to know if it can happen that OLS is consistent and $E[x_i \varepsilon_i]\neq 0$ and if so this leads to some kind of degeneracy.

I'm not particular proficient in statistics so I hope I have tagged and formulated the question correctly - otherwise feel free to correct me.

htd
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  • May I refer you to an older answer to this question: http://stats.stackexchange.com/questions/61657/understanding-and-interpreting-consistency-of-ols/61679#61679 – Andy Dec 29 '13 at 11:29
  • Apart from looking at various other posts that discuss consistency of OLS, the short answer is: No, if this "orthogonality" condition does not hold, then the OLS will be inconsistent, meaning that it will converge to a constant different than the true value of the parameter, i.e. it will have an asymptotic bias. – Alecos Papadopoulos Dec 29 '13 at 14:20

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