Questions on Testing for Heteroskedasticity/Autocorrelation (statistics)

Question

I just wanted to confirm and see your opinions:

a) True

b) True

c) False

Answer 1

1) False. Let us say that you used some incorrect weights matrix $B$, then
$$ \hat{\beta} = (X'BX)^{-1}X'By, $$ as such $$ E(\hat{\beta}|X) = (X'BX)^{-1}X'BE(y)=(X'BX)^{-1}X'B(X\beta + E\epsilon)=\beta. $$ Thus the weights matrix does not effect the bias of the estimators (however, it surely effects the variance and hence the inference).

2) True. Apply the same logic as at $(1)$. Autocorellation can be viewed as a specification of the covariance matrix, hence as long as $E\epsilon = 0$, it does not effect the expectation of the estimators.

3) False. E.g., Ljung-Box is another valid test for the same task.