I have a data set which includes country-years and I am interested in modeling founding and mortality for corporations in each country-year. I am interested in within- as well as between-country differences of predictors such as GDP, education, population, geographical location, etc. What type of models would you recommend that would get at the above? NOTE: Most foundings research uses negative binomial or Poisson regression but the data are usually focused in one geographical area, unlike my case.
Given the additional information you've provided, I think you can use one of three approaches: (1) generalized estimating equations (GEE), (2) a mixed effects generalized regression model, or (3) a fixed-effects regression model. GEE essentially treats the variance as a nuisance parameter and you simply use it to estimate the response for a unit change in the predictor, averaged over the entire population. This is a good approach when you have no real interested in the correlation between to responses (which it doesn't seem to be of any use to you). The mixed model approach is used to estimate the response for a unit change in the predictor for a specific subject (country). I don't want to repeat what others have already described in terms of the differences between the two, so look here for additional details on the differences between GEE and mixed models. To gain a better understanding of the within-country differences only, I would also encourage you to look into something generally called "fixed-effects" models, from the economics literature (not my field). These models essentially allow you to control for all time-invariant characteristics of the subjects, thereby reducing bias. Check out
[Fixed Effects Regression Methods for Longitudinal Data Using SAS] by Paul Allison as a good introduction.
Within your models, I think you will want to use a Poisson or Negative Binomial function. It's difficult to say which without seeing your data. I'd recommend you start with a Poisson model and look at the overdispersion statistics to see if a Poisson model is plausible given them. If not, use a more complex negative binomial. You may even need to use zero-inflated Poisson or regression models, depending on your data. I generally recommend starting off simple and then working your way to more complex models as needed.