The OLS estimator in an LRM with a random sample under exogeneity and heteroscedasticity

Summary

Random sample \(\left( y_i,x_i \right)\) for \(i=1, \ldots ,n\)
Correctly specified linear regression model, \(y_i=x'_iβ+ε_i\) where \(E\left( ε_i \mid x_i \right)=0\) .
Heteroscedasticity so \(Var\left( ε_i \mid x_i \right)=σ^2\) no longer holds
\(b={\left( X'X \right)}^{-1}X'y\) is the OLS estimator of \(β\) .

\(b\) is still an unbiased estimator of \(β\)
\(b\) is still a consistent estimator of \(β\)
\(b\) is no longer BLUE
\(Var\left( b \right|X)=σ^2{\left( X'X \right)}^{-1}\) is no longer valid
All inference based on \(Var\left( b \right|X)=σ^2{\left( X'X \right)}^{-1}\) is invalid