Homoscedasticity, heteroscedasticity and the Gauss-Markov assumptions

Summary

Setup

• The LRM with random sampling

\[y_i=β_1+β_2x_i+ε_i \quad i=1,…,n\]

• The explanatory variable is exogenous,

\[E\left(\varepsilon_i | x_i \right)=0, \quad i=1,…,n\]

Homoscedasticity / heteroscedasticity

• We say that the error terms are homoscedastic if

\[Var\left(\varepsilon_i | x_i \right)=σ^2, \quad i=1,…,n\]

• That is, the error terms all have the same variance (conditional on \(x_i\) )
• If the error terms are not homoscedastic we say that they are heteroscedastic .

Gauss-Markov assumptions

• We say that the Gauss-Markov assumptions are satisfied for the LRM with random sampling if the explanatory variable is exogenous and the error terms are homoscedastic.