1. (30) Consider a simple regression model:
$y_i = \beta_0 + \beta_1 x_{i1} + \epsilon_i$, $i = 1, \dots, n$.
We assume that the Gauss-Markov conditions are satisfied. Prove that the least
squares estimator $b_0 = \bar{y} - b_1 \bar{x}_1$ is the best linear unbiased estimator (BLUE) of $\beta_0$,
where $\bar{y} = n^{-1} \sum_{i=1}^n y_i$, $\bar{x}_1 = n^{-1} \sum_{i=1}^n x_{i1}$, and
$b_1 = \frac{\sum_{i=1}^n (y_i - \bar{y})(x_{i1} - \bar{x}_1)}{\sum_{i=1}^n (x_{i1} - \bar{x}_1)^2}$.