 Solution

Write $A=(X^{T}X{)}^{-1}{X}^T$. Since $AX=I_p$,The errors have mean zero and covariance matrix ${\sigma }^2{I}_n$. ThereforeThus $\hat{\theta }$ is an unbiased estimator. In addition, the multivariate normal distribution is preserved by linear maps, so $\hat{\theta }$ is multivariate normal with the displayed mean and covariance matrix.