Past exam of the probability, statistics and options I course of the University of Oxford / 2010 / O / 2 / a / Solution

The full column rank makes $X^{\mathsf{T}}X$ invertible. The least-squares normal equations from minimizing squared residuals giveThe errors have covariance matrix ${\sigma }^2{I}_n$ by independence and equal variance. Applying the covariance transformation rule to this linear estimator yieldsThe parameter estimator is unbiased because $\mathbb{E}Y=X\theta$.