Heteroscedasticity

Summary

Setup:

• The LRM with random sampling

\[y_i=β_1+β_2x_{i,2}+β_3x_{i,3}+…+β_kx_{i,k}+ε_i, i=1,…,n\]

Heteroscedasticity

• If the error terms are heteroscedastic then the conditional variance of the error terms is not constant and the GM assumptions are violated. We write

\[Var\left( ε_i|x_i \right)=σ_i^2\]

Consequences:

• If the explanatory variables are exogenous then the OLS estimator is still unbiased and consistent (under mild conditions).
• The OLS estimator is no longer efficient .
• The OLS GM variance formula is incorrect . Therefore, the OLS standard errors are inconsistent and all inference will be misleading even when \(n\) is large.