Is OLS estimator the only BLUE estimator? 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. 
 A: 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.
A: 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
A: Suppose there were two different Best linear unbiased estimators, $\hat\beta_1$ and $\hat\beta_2$, with (necessarily) the same mean and variance $\sigma^2$. The average of the two would also be a linear unbiased estimator, and it would be Better. Its variance would be  $$(1/2)^2\times (\sigma^2+\sigma^2+2\tau^2)$$
where $\tau^2$ is the covariance of the two.  Since the estimators are different (by assumption), the covariance is less than the variance of each estimator. So the variance of the average is less than either component estimator
A: I slightly disagree with the existing answers, thought it is a matter of how you view the question.
The other answers say that, if you calculate a linear and unbiased estimator that has variance equal to that of the OLS solution, you must get the OLS solution.
That’s fine, but also keep in mind that, when the errors are $iid$ Gaussian (which is an additional assumption on top of the Gauss-Markov assumptions), minimizing square loss in OLS is equivalent to maximum likelihood estimation of the coefficients.
Yes, this gives the same estimator, but that’s the point. There are multiple ways to view getting to the BLUE.
(I’m actually not convinced the other answers prove the uniqueness of the BLUE.)
