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

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.

• You might want to see my related question, and clearly the answer seems to be "yes", but only among linear estimators. Apr 22, 2013 at 7:45

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.