Homoscedasticity and Gauss-Markov assumptions

Summary

• Setup:
• a linear regression model \(y_i=x'_iβ+ε_i\) with a random sample
• exogeneity, \(E\left( ε_i \right|x_i)=0\)
• Definition: we say that the error terms are homoscedastic if for \(i=1, \ldots ,n\)

\[Var\left( ε_i \right|x_i)=σ^2\]

• In matrix form:

\[Var\left( ε \right|X)=σ^2I\]

• The exogeneity assumption \(E\left( ε_i \right|x_i)=0\) together with the homoscedasticity assumption \(Var\left( ε_i \right|x_i)=σ^2\) are called the Gauss Markov assumptions .