I know it's common to test for the presence of spatial autocorrelation with Moran's I. In our case, running Moran's I on the residuals of our OLS yields a statistic of 0.53 and a z-score of a little over 54, so we certainly need to account for the spatial autocorrelation. In this sense, we've used the Moran I to conclude that OLS is an inappropriate model.
My question: can we use Moran's I test similarly on the residuals of Spatial Autoregressive (SAM) and Spatial Error (SEM) models? If the statistic remains large, does that suggest that we've failed to generate an appropriate spatial weights matrix? If it's small, does that suggest that we've effectively dealt with the spatial autocorrelation? Is this logic valid (formally, or heuristically)?