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.