 O4

Minimizing $\Vert Y-X\theta {\Vert }^2$ gives the normal equations $X^{T}X\hat{\theta }=X^{T}Y$. Since $X$ has full column rank, $X^{T}X$ is positive definite and invertible, and the objective has a unique minimum. Set $A=(X^{T}X{)}^{-1}{X}^T$; then $\hat{\theta }=AY$ and $AX=I_p$. The model gives $\mathbb{E}[Y]=X\theta$ and $\mathrm{Var}(Y)={\sigma }^2{I}_n$, so $\mathbb{E}[\hat{\theta }]=\theta$ and $\mathrm{Var}(\hat{\theta })={\sigma }^{2}A{A}^T={\sigma }^2(X^{T}X{)}^{-1}$. For the alternative estimator the corresponding general formulas are $\mathbb{E}[\tilde{\theta }]=DX\theta$ and $\mathrm{Var}(\tilde{\theta })={\sigma }^{2}D{D}^T$.