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I might be misunderstanding something, but are poisson or negative binomial models with an offset term for population size (rate models) and independent variables that are rates as well endogenous? In other words, is the denominator directly correlated with all of the independent variables concerning population? Probably an examples makes this clear.

Example: Imagine the dependent variable is the number of homicides and the offset is the population size so that I am essentially modeling the homicide rate (i.e. homicides divided by population size). Some of the independent variables are the proportion African-American, proportion white, proportion poor etc. Are these variables all endogenous because they are defined as x divided by population size (same denominator as outcome)? What are alternative ways to specify the model?

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These variables may be endogenous but not because "they are divided by the same denominator". For example, the proportion of African-American population will definitely be partially defined by the proportion of white population (you cannot have both of them at 1 for example). On the other hand, the proportion of poor does not have to be defined by any ethnic proportion but it probably is in real cases.

Basically, look at these two cases. Some proportions are mathematically linked like ethnic proportions. Others may be linked by underlying causal phenomena like proportion of poor and proportion of African-American: not for mathematical but for social reasons, they may be linked in one community and not linked in another.

You can check for correlations between variables in your dataset before sticking them into the regression, to get an idea.

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    $\begingroup$ But that is not endogeneity. Endogeneity means that explanatory variables are correlated with the error term. What you are referring to is multicollinearity, which means that independent variables are correlated. Multicollinearity might be an issue in my example but some degree will be in pretty much every regression model. Or am I getting something wrong? $\endgroup$ – greg Oct 6 '16 at 21:36
  • $\begingroup$ No I think you are formally correct, I was just thinking of more related issues that cause problems in real cases. Nevertheless, dividing by a constant does not make your independent variables correlate with the error term if we assume that you are fitting a true model. Perhaps I should try to be clearer: if homicide rate is correlated with population size, it should be in the model, and then the error term is uncorrelated to it. $\endgroup$ – Gena Kukartsev Oct 6 '16 at 21:54

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