Limit of X’ε/√n

Summary

• Setup:
• a linear regression model \(y=Xβ+ε\) with a random sample
• the Gauss-Markov assumptions, \(E\left( ε \right|X)=0\) and \(Var\left( ε \right|X)=σ^2I\)
• Result:

\[E\left( \frac{1}{\sqrt{n}}X'ε \right)=0\]

• Result:

\[Var\left( \frac{1}{\sqrt{n}}X'ε \right)=σ^2Σ_{xx'}\]

• Result:

\[ \frac{1}{\sqrt{n}}X'ε→N\left( 0,σ^2Σ_{xx'} \right)\]