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.

Imagine That You Were Told That the T-Statistic for the Slope

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