Reciprocal Causation in Panel Data I have weekly data on stop and searches for all London Boroughs for ten years (N=32, T=566) and am interested in whether the number of stop and searches has any impact on crime rates. I don't expect it to have had any effect.
Using a simple negative binomial fixed effects model (done in Stata: xtnbreg, fe) with no control variables I can not find an effect (I have experimented with different lags, aggregating by month and quarter, different crime types etc). This simple model however obviously ignores a lot; most pressingly it ignores the possibility of reciprocal effects (i.e. that the number of stop and search might be caused by prior crime rates). However, given that I have failed to find an effect in my more simple models (as hypothesised) would there be any point looking into more complex dynamic and cross-lagged models? It seems possible that if the number of stop and searches is positively correlated with prior crime rates, and prior crime rates predicts current crime rates, then this could be disguising any negative impact that stop and search might have on crime? Bu when I perform the same negative binomial fixed effects regression model with lagged crime rates predicting stop and search my incidence rate ratio is 1.000485 (significant at 99.9%) which looks pretty tiny to me....
If the correlation is so small then would I be warranted in concluding that there is no evidence from my data that the number of stop and searches impacts crime rates? If further analysis is recommended then what sort of models should I be looking into?
 A: If stops are correlated with prior crime, and prior crime affects present crime, then the endogeneity will bias the effect of stops on crime upward.  So it is plausible that your null results so far are picking up a masked effect.  Anyway you cant argue convincingly that there is no effect without clearing up this endogeneity problem.  
As far as I understand your problem (I'm sure that I'm missing nuances), you either need an instrumental variable for the stops, or some other identification strategy (maybe some sort of discontinuity?), or you need to lag your dependent variable.  If you do the latter, you'll need to use panel GMM to avoid Nickell bias, unless there has been some generalization of those techniques to nonlinear (like negative binomial) models that I'm unaware of.  Lagging the dependent variable will only solve the problem if the only "selection" into stops has to do with previous crime rates, which may not be true.
EDIT:  I just noticed that your T is very large.  Nickell bias will be small as T gets big, so yeah, lagging should be more or less fine.
