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Conceptually when do estimated regression coefficients converge with the true values; is it a) when the sample size of observations tends to infinity or b) when the regression with the same predictors is fit repeatedly for an infinite number of samples, and the average of the estimated coefficients in each repetition is considered?

or do they converge in both of these cases?

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a) will happen if the estimator is (asymptotically) consistent. b) will happen if the estimator is unbiased.

In the usual linear regression setup, for the OLS estimator, consistency requires the condition $E(u_i\mathbf x_i) = \mathbf 0$ ("contemporaneous orthogonality"), while unbiasedness requires the stronger condition $E(\mathbf u \mid \mathbf X) = \mathbf 0$ ("strict exogeneity").

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  • $\begingroup$ But $X’u=\sum_{i=1}^nx_iu_i$ so your two conditions are the same. $\endgroup$ – hejseb Mar 16 at 6:25
  • $\begingroup$ @hejseb Typo. Cortected. $\endgroup$ – Alecos Papadopoulos Mar 16 at 7:45
  • $\begingroup$ +1. That said, while unbiased estimators are of course not guaranteed to be consistent, unbiased estimators are typically also consistent. $\endgroup$ – Christoph Hanck Mar 16 at 9:24

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