Under which assumptions does the ordinary least squares method give efficient and unbiased estimators?

Question

Is it true that under the Gauss Markov assumptions the ordinary least squares method gives efficient and unbiased estimators?

So:

$$E(u_t)=0 $$ for all $t$

$$E(u_tu_s)=\sigma^2 $$ for $t=s$

$$E(u_tu_s)=0 $$ for $t\neq s$

where $u$ are the residuals.

Answer 1

The Gauss-Markov Theorem is telling us that in a regression model, where the expected value of our error terms is zero, $ E(\epsilon_{i}) = 0$ and variance of the error terms is constant and finite $\sigma^{2}(\epsilon_{i}) = \sigma^{2} < \infty$ and $\epsilon_{i}$ and $ \epsilon_{j}$ are uncorrelated for all $ i$ and $ j$ the least squares estimator $ b_{0}$ and $ b_{1}$ are unbiased and have minimum variance among all unbiased linear estimators. Note that there might be biased estimator which have a even lower variance.

A proof that actually shows that under the assuptions of the Gauss-Markov Theorem an linear estimator is BLUE can be found under

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