According to the Gauss Markov theorem, in a linear regression model, if the errors have expectation zero and are uncorrelated and have equal variances, the best linear unbiased estimator of the regression coefficients is given by the OLS estimator. But it's also true that if we omit an important explanatory variable, the OLS estimator turns out to be biased. As far as the Gauss Markov theorem is concerned, how is this possible? Doesn't this example contradict the theorem?

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    $\begingroup$ Both statements are true at the same time, there is no contradiction. An implicit assumption of the Gauss Markov theorem is that you are estimating the correct model! $\endgroup$ – JohnK May 20 '15 at 21:47
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    $\begingroup$ To elaborate on @JohnK, the errors are not going to have mean zero (conditional on the regressors) if we do not estimate the correct model. $\endgroup$ – Christoph Hanck May 21 '15 at 3:40
  • $\begingroup$ Thank you. @ChristophHanck Shouldn't errors have always mean zero if we include the intercept in the model? I don't understand. $\endgroup$ – John M May 21 '15 at 10:35
  • $\begingroup$ No, the residuals will have, so the estimated errors! Whether the true errors, which are unobservable, have expected value zero is something that cannot be inferred from the residuals. $\endgroup$ – Christoph Hanck May 21 '15 at 10:36
  • $\begingroup$ Great answer. Thank you! Is there a proof somewhere which shows that the errors are not going to have mean zero if we do not estimate the correct model? $\endgroup$ – John M May 21 '15 at 12:51

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