In spatial statistics one often hears the statements like the following:
unaccounted for spatial autocorrelation may lead to spurious significance / understimated uncertainty / too narrow confidence intervals
and so on. The general idea being that ignoring spatial autocorrelation that is unexplained by your covariates is a big no-no. I'm talking here about models in which there are parameters of fixed effects (such as known spatial covariates) on which we seek to make inference.
I am looking for a mathematical description of this principle
I have heard this principle explained by analogy to pseudo-replicates. By ignoring pseudo-replicates we artificially increase our sample size and so our uncertainty is underestimated. Spatial autocorrelation is like the 'continuous' version of this principle.
Obviously I'm unsatisfied with leaving it at this analogy but haven't been able to find a mathematical description of this issue anywhere.