Consider the linear regression model (mathbf{y}=mathbf{X} boldsymbol{beta}+mathbf{u}), where (mathrm{E}[mathbf{u}]=mathbf{0}) and (mathrm{E}left[mathbf{u u}^{prime} ight]=sigma^{2} boldsymbol{Omega}^{*}=boldsymbol{Omega}). By standard results

Consider the linear regression model \(\mathbf{y}=\mathbf{X} \boldsymbol{\beta}+\mathbf{u}\), where \(\mathrm{E}[\mathbf{u}]=\mathbf{0}\) and \(\mathrm{E}\left[\mathbf{u u}^{\prime}\right]=\sigma^{2} \boldsymbol{\Omega}^{*}=\boldsymbol{\Omega}\). By standard results for the OLS estimator \(\widehat{\boldsymbol{\beta}}=\left(\mathbf{X}^{\prime} \mathbf{X}\right)^{-1} \mathbf{X}^{\prime} \mathbf{y}\) (see Section 4.4) we can obtain the correct expression for \(\mathrm{V}[\widehat{\boldsymbol{\beta}}]\) as \(\mathbf{V}_{2}=\left(\mathbf{X}^{\prime} \mathbf{X}\right)^{-1}\left(\mathbf{X}^{\prime} \boldsymbol{\Omega} \mathbf{X}\right)^{-1}\left(\mathbf{X}^{\prime} \mathbf{X}\right)^{-1}\), whereas \(\mathbf{V}_{1}=\widehat{\sigma}^{2}\left(\mathbf{X}^{\prime} \mathbf{X}\right)^{-1}\) with \(\widehat{\sigma}^{2}=\widehat{\mathbf{u}} \widehat{\mathbf{u}} /(N-K)\) is invalid if \(\Omega eq \mathbf{I}\).

(a) Show that the bias of \(\mathbf{V}_{1}\) is given by \(\mathbf{B}=\mathbf{B}_{1}+\mathbf{B}_{2}\), where \(\mathbf{B}_{2}=\left(\mathbf{X}^{\prime} \mathbf{X}\right)^{-1} \mathbf{X}^{\prime}\left(\boldsymbol{\Omega}-\boldsymbol{\sigma}^{2} \mathbf{I}\right) \mathbf{X}\left(\mathbf{X}^{\prime} \mathbf{X}\right)^{-1}\) and \(\mathbf{B}_{1}=(N-K)^{-1} \operatorname{tr}\left\{\mathbf{B}_{2}\left(\mathbf{X}^{\prime} \mathbf{X}\right)\right\}\left(\mathbf{X}^{\prime} \mathbf{X}\right)^{-1}\). (Greenwald refers to \(\mathbf{B}_{2}\) as "direct bias.")

(b) Evaluate the two terms for the special case of \(\mathbf{X}^{\prime} \mathbf{X}=\mathbf{I}_{K}\). Show that \(\mathbf{B} \rightarrow \mathbf{B}_{2}\) as \(N \rightarrow \infty\).