# Diagnostics and residual analysis for Poisson regression

Recently, I was asked to check for serial correlation after doing a panel Poisson regression. I haven't seen such a test and in general, researchers (at least in the econom(etr)ics literature) don't report many diagnostics.

There is some discussion going on on this forum about the nature of residuals from a Poisson regression (see for example here, here and here), specifically about the questions 1) whether residuals should normally distributed, and 2) if not, what their theoretical distribution is. My understanding so far is: 1) no, and 2) it's complicated.

Am I correct in thinking that the distribution of residuals for panel Poisson analysis is also complicated, probably more so than for cross-section; and that because of that, there simply are not as many formal diagnostic tests as for linear models?

(My immediate question is about serial correlation in panel Poisson models, but I would be interested in a more general discussion for Poisson models, non-linear models and other diagnostics than serial correlation)

Under the assumption that you can order your residuals (typically in time or space), autocorrelation is just the correlation of the vector $[r_1, \ldots, r_{N-k}]$ with $[r_{1+k}, \ldots, r_{N}]$, for some lag, $k$. Normality doesn't much matter for correlation (cf., Pearson's or Spearman's correlation with non-normal data). It could matter for some tests of the significance of a correlation, but with autocorrelation you are typically more interested in the magnitude than its significance (by analogy, consider: Is normality testing 'essentially useless'?). If, for some reason you really wanted to test the correlation estimate, and didn't want to assume normality, you could bootstrap.