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Gauss–Markov_theorem states that OLS estimator is a BLUE estimator. My doubt is can there be any other linear estimator, other than OLS, which is also a BLUE estimator?

After going through the proof of why OLS is a BLUE estimator, I feel that only OLS estimator can be the BLUE estimator. Unbiased Linear Estimators from any other techniques should essentially yield the same result as from OLS technique for them to be BLUE.

I hope I am not making any blunders in assuming so.

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    $\begingroup$ The article you link to starts with "the Gauss–Markov theorem, named after Carl Friedrich Gauss and Andrey Markov, states that in a linear regression model in which the errors have expectation zero and are uncorrelated and have equal variances, the best linear unbiased estimator (BLUE) of the coefficients is given by the ordinary least squares (OLS) estimator, provided it exists." $\endgroup$ – Henry Aug 6 '17 at 19:26
  • $\begingroup$ The part Henry quotes gives some immediate clues about what to vary to get something that isn't OLS... $\endgroup$ – Glen_b Aug 7 '17 at 5:28
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When the conditions for linear regression are met, the OLS estimator is the only BLUE estimator. The B in BLUE stands for best, and in this context best means the unbiased estimator with the lowest variance.

If the regression conditions aren't met - for instance, if heteroskedasticity is present - then the OLS estimator is still unbiased but it is no longer best. Instead, a variation called general least squares (GLS) will be BLUE.

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  • $\begingroup$ Why is the OLS estimator the only BLUE estimator? If you look at the statement of the theorem, it's saying that the variance of some other estimator minus the variance of the OLS estimator is positive semi-definite. If the OLS estimator was the only BLUE estimator, then we would expect it to be positive definite. I'm not saying that you're wrong, but it would be nice to have some justification. $\endgroup$ – mlstudent Oct 29 '18 at 1:20
  • $\begingroup$ The OLS estimator does not need to be the only BLUE estimator. For example, the maximum likelihood estimator in a regression setup with normal distributed errors is BLUE too, since the closed form of the estimator is identical to the OLS (but as a method, ML-estimation is clearly different from OLS.). The Gauss–Markov Theorem however tells you that in the class of linear unbiased estimators you don't have too look further than OLS, since every other estimator in this class can not do better under the assumptions. $\endgroup$ – chRrr Jan 25 at 11:01
  • $\begingroup$ do you mean generalized least squares? $\endgroup$ – rep_ho Jan 25 at 11:35
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The Gauss-Markov Theorem states that if a linear regression model fulfils the assumptions of the classical linear regression model the ordinary least squares estimator is the best linear unbiased estimator (BLUE).

You can find a good overview of the Gauss-Markov Theorem here:

https://economictheoryblog.com/2015/02/26/markov_theorem

Here you find the assumptions of the classical linear regression model:

https://economictheoryblog.com/2015/04/01/ols_assumptions

In order for OLS to be BLUE one needs to fulfill assumptions 1 to 4 of the assumptions of the classical linear regression model. The following website provides the mathematical proof of the Gauss-Markov Theorem. That is, it proves that in case one fulfills the Gauss-Markov assumptions, OLS is BLUE.

https://economictheoryblog.com/2016/02/05/proof-gauss-markov-theorem

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