(a) Consider the classical linear regression model: y = Xβ + u under the Gauss Markov assumptions, such that E uu = σ 2 I . Consider the estimator for σ 2 given by s2∗ = e e/(T − 10) where e is the OLS residual vector: is this an unbiased estimator? If not, how can we correct it to get an unbiased estimator of σ 2 . (b) Consider the generalized linear regression model. State the model in matrix algebra form. Show that there exists a transformation of the model that satisfies all the Gauss Markov assumptions. Derive the GLS estimator and compute its variance-covariance matrix.

(a) Consider the classical linear regression model: y = Xβ + u under the Gauss Markov assumptions, such that E uu = σ 2 I . Consider the estimator for σ 2 given by s2∗ = e e/(T − 10) where e is the OLS residual vector: is this an unbiased estimator? If not, how can we correct it to get an unbiased estimator of σ 2 .

(b) Consider the generalized linear regression model. State the model in matrix algebra form. Show that there exists a transformation of the model that satisfies all the Gauss Markov assumptions. Derive the GLS estimator and compute its variance-covariance matrix.

Consider the simple linear regression model with no intercept yt = xt β + εt . Assume all the Gauss Markov assumptions hold. Furthermore assume that εt ∼ iidN(0, σε^2 ). Derive the ML estimator of β and discuss its asymptotic properties, constructing the asymptotic covariance matrix.

Consider the simplified classical normal linear regression problem written in matrix notation:y = Xβ + u, u ∼ N (0, σ 2 I n ), β ∈ R K , σ > 0

pset 5.2 Prove that $\hat{\boldsymbol{\beta}}_{G I V E}$ can be derived as the OLS estimator of $\boldsymbol{\beta}$ in the linear regression model$$\boldsymbol{y}=X \boldsymbol{\beta}+\hat{U} \boldsymbol{\gamma}+\boldsymbol{\eta}$$where $\hat{U}$ is the matrix of first-stage residuals.