Suppose we want a regression model
$y_i = f(x_i^t) + e_i$, where
$x_i$ are collected spatially, and we believe there would be spatial autocorrelation (SAC). I read some popular methods like spatial-lag or spatial-error, and the Moran's I test.
My question is, if we believe that the SAC in
$y_i$ is all caused by SAC
$x_i$, that is, we believe
$e_i$ are independent with each other. Then is there any problem to simply run a normal regression model? It looks to me that this way does not violate any assumptions. I can definitely run a test on the residuals. However several application papers do things in such a way: they first run Moran's I on
$y_i$, if significant, then spatial-lag. O.w., run Moran's I on residuals, if significant, then spatial-error. If looks weird to me.