Let $u _ { i }$ Be Distributed N(0 $\sigma _ { u } ^ { 2 }$

Let $u _ { i }$ be distributed N(0, $\sigma _ { u } ^ { 2 }$ ), i.e., the errors are distributed normally with a constant variance (homoskedasticity). This results in $\hat{\beta }_ { 1 }$ being distributed N(?₁, $\sigma _ { \hat { \beta } 1 } ^ { 2 }$ ), where $\sigma _ { p 1 } ^ { 2 } = \frac { \sigma _ { u } ^ { 2 } } { \sum _ { i = 1 } ^ { n } \left( X _ { i } - \bar { X } \right) ^ { 2 } }$ Statistical inference would be straightforward if $\sigma _ { u } ^ { 2 }$ was known. One way to deal with this problem is to replace $\sigma _ { u } ^ { 2 }$ with an estimator $S _ { \hat { u} } ^ { 2 }$ Clearly since this introduces more uncertainty, you cannot expect $\hat{\beta} _ { 1 }$ to be still normally distributed. Indeed, the t-statistic now follows Student's t distribution. Look at the table for the Student t-distribution and focus on the 5% two-sided significance level. List the critical values for 10 degrees of freedom, 30 degrees of freedom, 60 degrees of freedom, and finally ? degrees of freedom. Describe how the notion of uncertainty about $\sigma _ { u } ^ { 2 }$ can be incorporated about the tails of the t-distribution as the degrees of freedom increase.