The distribution of the OLS estimator

Summary

• Setup:
• a linear regression model \(y=Xβ+ε\) with a random sample
• \(b={\left( X'X \right)}^{-1}X'y\) is the OLS estimator of \(β\) (both are \(k×1\) )
• \(e=y-Xb\) are the OLS residuals.
• \(s^2=e'e/(n-k)\)
• Assumption: Normally distributed error terms (conditionally on \(x\) )

\[ε_i|x_i \sim N(0,σ^2)\]

• Note: we are making assumptions stronger than the Gauss Markov assumptions.
• Result:

\[b|X \sim N\left( β,σ^2{\left( X'X \right)}^{-1} \right)\]

• Result:

\[ \frac{\left( n-k \right)s^2}{σ^2} \sim χ_{n-k}^2\]

• Result: \(s^2\) is independent of \(b\) .