Question: Constrained least Squares: Suppose we wish to find the least squares estimator of \(\beta \) gin the model \(Y=X\beta +\varepsilon \) subject to a set of equality constraints on \(\beta \), say \(T\beta =C\). Show that the estimator is

\[\tilde{\beta }=\hat{\beta }+{{\left( X'X \right)}^{-1}}T'{{\left[ T{{\left( X'X \right)}^{-1}}T \right]}^{-1}}\left( C-T\hat{\beta } \right)\]