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  1. I fitted multiple spatial regression models - spatial lag model, spatial error model and spatial durbin model. My question is, how do I check the assumption of normality on errors? In classical linear regression, the diagnostics is done on standardized or studentized residuals, but how do I standardize residuals in these spatial models? When I use function "residuals()" to retrieve residuals from the spatial models and do QQplot, they always have this "S" shape. Does it mean the models are wrong?

  2. The same for weighted spatial error model. Which residuals should I use to check the model assumptions?

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There are two reasons for assuming that the residuals are normal. One is the Central Limit Theorem: if your model captures the major causative influences, but there are also many independent minor ones, then the residuals will be normally distributed.

The other, less convincing, argument is that this assumption justifies some efficient solution algorithms which we can run to make a model.

You say your residuals have an "S" shape. That doesn't sound so bad. Does that mean that you can do a non-linear transformation of the problem that will result in normal residuals? e.g. take logs? (Once I got good results using a cotangent - there was a domain reason why that made sense.)

Sometimes the residuals are just nasty, e.g. a Lorenz distribution on one problem I worked on.

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    $\begingroup$ This reads more like comments to the question rather than an answer to it. Nevertheless, it might be worth noting that the CLT often doesn't apply -- whence the motivation for the question! Indeed, in many applications variation is multiplicative, implying lognormal conditional distributions. Also, an s-shaped QQ plot of residuals will not be cured with a logarithmic transformation of the response (or, indeed, with any power transformation). $\endgroup$
    – whuber
    Apr 28, 2021 at 16:38

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