We assume our data follow the model:
$$ Y = X\beta +\varepsilon $$
In spatial CAR (SAR) model, we assume that the errors $\varepsilon$ are correlated in a spatial setting. Let's say that we model the autocorrelation among the errors as follows:
$$ \varepsilon_s = \phi \frac{1}{N(s)} \sum_{s'\in N(s)} \varepsilon_{s'} + \delta $$
where $s'$ is an immediate neighbor of $s$, $\phi$ is a normalizing constant, $N(s)$ is the total immediate neighbors that $s$ has, and $\delta \sim \mathcal N_n(0, \tau^2 I_n)$.
Then we use partial log-likelihood (we write $\beta$ and $\tau^2$ in terms of $\phi$) to estimate the parameter $\phi$.
The idea of linear regression should be that our explanatory variables are so good at explaining the response variable, so that the real-world randomness is left to our residuals (I think that's why checking residual plots is very important, since it is an indicator of how comprehensive our trend is).
We model errors as correlated errors in this model. Unless we can find a set of extremely comprehensive explanatory variables to force $\phi$ to be close to zero (pretty unrealistic), the residual plot (usually a map) will look correlated.
So in this case, how would we validate the accuracy and comprehensiveness of our estimate of $\beta$ (besides the $P(>|z|)$ values provided by the spautolm function in R)?
Moreover, is it possible to make the spatial autocorrelation go away with a sufficiently good trend model?
