Heteroskedasticity and Regression Coefficient


Can someone explain the apparent anomaly in the following,

  1. Heteroskedasticity does not affect the consistecy of Regression Coefficient, also Heteroskedasticity does not make the estimator unbiased

  2. Heteroskedasticity causes t-tests of each Regression Coefficient to be unreliable as the standard errors of Regression Coefficient become biased

How come the estimators remain BLUE as well as unreliable??? Thanks in advance.