Probability 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:

\[E\left( \frac{1}{n}X'ε \right)=0\]

• Result:

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

• Result:

\[plim \frac{1}{n}X'ε=0\]