Model fit statistic for population-average specification

Hello everybody!

For the purpose of my study I am using an exponential Poisson model with population-averaged (PA) effects, as implemented in Stata's -xtpoisson, pa- command. However, neither of the fit statics (e.g., -LogLikelihood, BIC, or AIC) are appropriate after PA estimation.

Are there any other fit statistics that I can use in such case?

Thank you in advance for suggestions.