Spatial econometrics:heteroskedasticity and autocorrelation consistent(HAC) estimator

Good afternoon,

I have quite basic question however I would like to be sure in this. If I use a HAC estimator of the variance-covariance (VC) matrix for a spatial econometric model, do I still need to test the residuals for spatial autocorrelation and heteroscedasticity? (in particular I am using function stslshac available in statistical software R, package sphet. The estimator is based on Kelejian, H.H. and Prucha, I.R. (2007) HAC estimation in a spatial framework, Journal of Econometrics, 140, pages 131–154).

What if the residuals are spatially autocorrelated and (or) heteroscedastic? Can I still use this estimator with HAC estimate of VC matrix or shall I go for different estimator or specification? Do the estimates have required properties (are they unbiased, consistent, efficient)?

I would be grateful for any reaction.