Suppose that we have postulated the model
$Y_i = \beta_1 x_i + \epsilon_i$
$i = 1, 2, \dots, n$,
where the $\epsilon_i$'s are independent and identically distributed random variables with $E(\epsilon_i) = 0$
and $Var(\epsilon_i) = \sigma^2$. Then $\hat{y}_i = \hat{\beta}_1 x_i$ is the predicted value of $y$ when $x = x_i$ and $SSE = \sum_{i=1}^n [y_i - \beta_1 x_i]^2$.
• Derive $\hat{\beta}_1$, the least-squares estimator of the parameter $\beta_1$.
• Show if $\hat{\beta}_1$ is or is not an unbiased estimator of the parameter $\beta_1$.
• Find the variance of $\hat{\beta}_1$.
