Consider the linear regression model y = X + u . (a) Obtain the formula for

Consider the linear regression model $\mathbf{y}=\mathbf{X}\mathit{\beta }+\mathbf{u}$.

(a) Obtain the formula for $\hat{\beta }$ that minimizes $Q(\beta )={\mathbf{u}}^{\mathrm{\prime }}\mathbf{W}\mathbf{u}$, where $\mathbf{W}$ is of full rank. [The chain rule for matrix differentiation for column vectors $\mathbf{x}$ and $\mathbf{z}$ is $\mathrm{\partial }f(\mathbf{x})/\mathrm{\partial }\mathbf{x}=\left(\mathrm{\partial }{\mathbf{z}}^{\mathrm{\prime }}/\mathrm{\partial }\mathbf{x}\right)\times (\mathrm{\partial }f(\mathbf{z})/\mathrm{\partial }\mathbf{z})$, for $f(\mathbf{x})=f(g(\mathbf{x}))=f(\mathbf{z})$ where $\mathbf{z}=g(\mathbf{x})]$.

(b) Show that this simplifies to the OLS estimator if $\mathbf{W}=\mathbf{I}$.

(c) Show that this gives the GLS estimator if $\mathbf{W}={\mathrm{\Omega }}^{-1}$.

(d) Show that this gives the 2SLS estimator if $\mathbf{W}=\mathbf{Z}{\left({\mathbf{Z}}^{\mathrm{\prime }}\mathbf{Z}\right)}^{-1}{\mathbf{Z}}^{\mathrm{\prime }}$.