Estimator of $\sigma^2$: \\
$\hat{\sigma}^2 = \frac{SS_e}{n - p - 1}$, \\
where $p$ is the number of covariates in the model and $SS_e = \sum_{i=1}^n (Y_i - \hat{Y}_i)^2$. \\
(a) Write down the likelihood function of $Y = (Y_1, ..., Y_n)$ given $X = \begin{pmatrix} 1 & X_{1,1} & \dots & X_{1,p} \\ \vdots & \vdots & \vdots \\ 1 & X_{n,1} & \dots & X_{n,p} \end{pmatrix}$ \\
using the matrix notation. Refer to the multivariate normal distribution form for this. \\
(b) Find the maximum likelihood estimator of $\sigma^2$. For this you may write the log-likelihood function $\sum_{i=1}^n \log f(Y_i | X_{i1}, ..., X_{ip}, \beta_0, \beta_1, ..., \beta_p, \sigma^2)$. \\
(c) Use the result that $E[SS_e] = (n - p - 1)\sigma^2$ to show that $\hat{\sigma}_{mle}^2$ is not an unbiased estimator of $\sigma^2$. \\
(d) Show that $E[\hat{\sigma}_{mle}^2]$ is asymptotically $\sigma^2$, that is, tends to $\sigma^2$ as the sample size increases.
