Overdispersion/Excess Zeros and Negative Binomial Panel for Crime data

[Tara McGovern]

Hello,

I have a panel of homicides by municipality over a 12 year period (independent variables include the presence of certain gangs (coded 1/0), Infant Mortality Rate, and Percent males aged 15-29).
I was originally looking at a regular fixed or random effects model using the homicide rate. My professor suggested I use a count model (Poisson/Negative Binomial) and make the dependent variable homicides instead (and consider a zero inflated model).

The problem is 25% of my municipalities/year pairs have 0 homicides and a few have more than 100 (approx. 3%).

My initial understanding was that I should use Poisson, unless the data were overdispersed...which clearly they are:

Mean 18.31811
Std. Dev. 103.1
Variance 10629.61

Reading Hilbe though suggested that Negative Binomial could also be overdispersed...although I couldn't figure out how to check for that.
Reading about the ZINB model, it also concerned me that it suggested a structural reason why there were 0s (e.g., women over 50 who couldn't have children), and there is no structural reason why those municipalities couldn't have a homicide.

Any insight on selecting an appropriate model would be much appreciated.

[Clyde Schechter]

Well, unless your municipalities all have the same population, just using the number of homicides as the response variable will be a mis-specification of the model. You need to put in an -exposure()- option to adjust the analysis for population differences among the cities. Similarly, judging the presence or absence of overdispersion on the distributional statistics of the homicide counts is incorrect. The Poisson model requires that n ~ Poisson(exp(xb)), and that conditional on xb, mean(n) = variance(n). It does not require that mean(n) overall have any particular relationship to variance(n) overall. There may be others on the forum who are better prepared to advise you about ways to test for this. Suffice it to say that in any case overdispersion is usually well handled by using -nbreg-. Since you have panel data, you will probably prefer -xtnbreg- to -nbreg- (or -xtpoisson- to -poisson-).

You are correct that the zero-inflated models are most appropriate when the study population is best understood as a mixture of those who are ineligible to have a non-zero outcome, and those whose outcome follows a Poisson distribution. The -inflate- equation in those models is intended to try to identify the probability of each observation's belonging in each component of the mixture based on observed attributes of the observations. (The -inflate- option is optional, and can be used if a mixture model is credible but component membership is not related to observed attributes.) But if, as you seem to say, a mixture model seems scientifically inappropriate here, then I would not recommend a zero-inflated model.

[Tara McGovern]

Thank you for the quick response. I forgot about the population variable (since it was originally incorporated in the homicide rate) - I do have it in the regression (as the natural log of the total population).

[Joao Santos Silva]

Dear Tara,

Adding to Clyde's excellent points, please note that if you want to use an estimator with fixed effects you are pretty much restricted to Poisson. From what you describe, it looks as if you do not want to compute probabilities of specific events. If that is the case, you can pretty much ignore the overdispersion (but make sure you use robust standard errors).

All the best,

Joao

[gmahofa]

Greetings,

My question is also related on how to implement fixed effects in a Poisson framework. So, is it possible for one to first transform the data (mean difference) and then apply a Poisson estimator to the transformed data. Will this be a fixed effects estimation.

Thanks!

Godfrey

[Joao Santos Silva]

Dear Godfrey,

That won't work. You can either use the -xtpoisson- command or run a Poisson regression including the FE as dummies (do not forget to use clustered standard errors). You should get the same results with the two approaches but -xtpoisson- will be much faster!

Joao

[gmahofa]

Dear Joao,

Many thanks

Godfrey