Calculation of variance beta hat

Question

I have a regression setting of 1 covariate where $\sigma^2= 4$ and $$(X^TX)^{-1} = \begin{pmatrix} 0.2 & 0.05 \\ 0.05 & 0.14 \\ \end{pmatrix} $$ How can I get $Var(\hat{\beta_1})$ and $Var(\hat{\beta_0} - 0.5 \hat{\beta_1})$

Answer 1

All you need for this problem is that in the standard homoskedastic error setting, the covariance matrix of $\hat\beta = (\hat\beta_0, \hat\beta_1)$ is given by $$\mathsf{Cov}(\hat\beta) = \begin{pmatrix} \mathsf{Var}(\hat\beta_0) & \mathsf{Cov}(\hat\beta_0, \hat\beta_1) \\ \mathsf{Cov}(\hat\beta_0, \hat\beta_1) & \mathsf{Var}(\hat\beta_1) \end{pmatrix} = \sigma^2 (X^\top X)^{-1}.$$ You can directly read off $\mathsf{Var}(\hat\beta_1)$; for the latter, use \begin{align*} \mathsf{Var}(\hat\beta_0 - 0.5\hat\beta_1) &= \mathsf{Var}(\hat\beta_0) + 0.5^2 \mathsf{Var}(\hat\beta_1) - 2\cdot 0.5 \mathsf{Cov}(\hat\beta_0, \hat\beta_1) \\ &= \mathsf{Var}(\hat\beta_0) + 0.25\mathsf{Var}(\hat\beta_1) - \mathsf{Cov}(\hat\beta_0, \hat\beta_1). \end{align*}