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What are the statistical consequences of heteroskedasticity for regression models where the errors are correlated, e.g., due to spatial or phylogenetic autocorrelation?

For example, consider a phylogenetic regression model of skull length vs. body mass, where errors are correlated due to phylogeny, and heteroskedasticity is induced because measurement error in skull length is positively correlated with body mass.

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  • $\begingroup$ Heteroskedasticity and error correlation have the same effect on OLS estimators in the linear model. The OLS estimator is still unbiased, consistent, and asymptotically normal. However, the usual standard errors are biased and inconsistent, so all confidence intervals and hypothesis tests based on them are wrong. Also, the estimator is no longer efficient (i.e. there is a better unbiased estimator called GLS). Having both problems at once just means that you have two reasons for these facts to be true. Is your real question how you should fix the problem? $\endgroup$ – Bill Sep 22 '17 at 14:37
  • $\begingroup$ No, my question is really just "what are the consequences". I have a book, "Modern Phylogenetic Comparative Methods and Their Application in Evolutionary Biology" in which it is stated that "not much is known about the consequences of... heteroskedasticity... in the framework of phylogenetic generalized least squares" (page 145) , which got me wondering about this question. I also have a situation in which I am collecting data in a way that actually eliminates a common source of heteroskedasticity, which seems like a good thing, but I'm not sure exactly what problems this solves. $\endgroup$ – Slow loris Sep 22 '17 at 18:21
  • $\begingroup$ Your answer does make me think of a follow up question though. If heteroskedasticity does simply lead to biased and inconsistent standard errors for PGLS, am I correct that the problems introduced by heteroskedasticity would be circumvented if I used a permutation test to determine statistical significance (i.e., obtaining a distribution of regression coefficients by repeatedly permuting the response variable across the tips of the tree and refitting the model)? $\endgroup$ – Slow loris Sep 22 '17 at 18:33

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