The classical assumptions…
- Data is generated by a linear model
- $X$ is non-stochastic and finite
- No multicollinearity. Rank $(X) = k < n$
- Strict Exogeneity. $E(u) = 0$
- Spherical Error Variance (Homoskedasticity) $E(uu’) = \sigma^2I_n\:, \: \sigma^2 \in (0, \infty)$.
Plus… - Normality of the error term. $\mathbf{u \sim N(0,\sigma^2I_n)}$}
Assumptions 4, 5 & 6 lead to the conclusion that
\begin{equation*}
u | X \sim N(0, \sigma^2 I_n)
\end{equation*}
The errors $u$ are independent from the regressors $X$. Maintaining all the assumptions above, the conditional sampling error is distributed
\begin{equation*}
( \tilde{\beta} – \beta) | X \sim N(0, \sigma^2 (X’X)^{-1})
\end{equation*}
Keep track of the numbered assumptions as we will refer to them by their numbers later on.