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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?

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  • $\begingroup$ Does anyone have any thoughts on this issue? It doesn't have to be a exact answer, any thoughts will be greatly appreciated! I've been searching around online but cannot find anything related to model checking for the CAR model. $\endgroup$ – Jack Shi Jul 26 '15 at 22:32
  • $\begingroup$ Without the clarifying context you provided, I'd have immediately interpreted "SAR" as "seasonal auto-regressive" not "spatial conditional auto-regressive". I wonder why people who have (ambiguously) called it "SAR" passed up the opportunity to abbreviate it to "SCAR" instead. Besides avoiding ambiguity with an established "AR" related acronym, it's a catchier term. $\endgroup$ – Glen_b Jul 26 '15 at 23:06
  • $\begingroup$ @Glen_b Haha I guess people call it SAR because the SAR(Simultaneous Autoregressive) method does not include any kind of "conditioning." With a appropriate construction of the weight matrix, one can form the log-likelihood without using the Brook's Lemma. But ya, I agree with you in a sense that there are so many different models in the area of statistics, and names,especially abbreviations, of those models can get pretty confusing sometime.. $\endgroup$ – Jack Shi Jul 26 '15 at 23:15

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