I have panel data for counts of new firms in different regions for six years. I am estimating a static poisson regression with multiplicative fixed effects$^*$; I have also tried to estimate a dynamic model by introducing a lagged dependent variable, but could not make that latter model work. Now, I would like to test the residuals from the static model for autocorrelation, so that I have an idea of the importance of the dynamics. However, I cannot find any diagnostic tests for this in a textbook (I've looked at Wooldridge, Cameron & Trivedi, Winkelmann, Greene), and also have not seen such a test in a research paper. Since the individual effects in the model are not identified, I don't know how to compute meaningful residuals in the first place.
Does anyone 1) know how to compute meaningful residuals; and 2) know of any diagnostic tests for these panel fixed effect poisson models?
FYI: I am using Stata (version 12.1) -xtpoisson, fe vce(robust)- command for the static model. Stata's postestimation commands can compute predicted values etc, but only assuming that the individual effects are all zero.
$^*$ The cross-section (or pooled) Poisson regression models the expected number of counts $y$ as $E[y_i|x_i]=\exp(X_i\beta)$, with $\beta$ the coefficients and $X_i$ the variables. A common way to add individual fixed effects with panel data is to let the effects $\alpha_{i}$ enter the model multiplicatively: $E[y_{it}|X_{it},\alpha_i]=\alpha_i\exp(X_{it}\beta)$.
