# Poisson regression with panel data and large variation across items

I study the behavior of municipal authorities towards their neighbors. The theory is that some properties of municipalities (like a level of indebtedness) make them more 'aggressive'. I have a database of the statements each municipality made per year, and the number of those categorized as negative (like threat to file a lawsuit). The dataset looks something like that: The problem is that there are huge municipalities (with several millions of inhabitants, like Municipality A in example), and tiny municipalities like B above, with different capacities to produce statements. In addition one important property is that overwhelming majority of statements are neutral or positive (like sign a cooperation agreement), so negative statements usually account for a small fraction, no more than 5% of all statements.

What is the correct way of analyzing such type of data to account for variability across different items?

I could just look at the relative share of 'negative' statements in all statements made, but that would not work for small municipalities because they have a huge variation. Perhaps the correct way of doing it is using count data models such as Poisson regression, but I wonder if these models take into account the different 'sizes' of the items. What I mean is that the change in 10 items for the large municipality generally producing hundreds of statements per year should be treated differently from the similar situation for a small municipality which usually produces something slightly above zero.

The standard Poisson general linear model can be re-written from using the absolute frequency of statements: $$\log S_i = \beta_0 + \beta_1x_{i1} + ...$$ where $S_i$ is the number of statements (either total or negative) for a municipality and the rest is the features and coefficients and intercept of the model, to the following: $$\log \frac{S_i}{s_i} = \beta_0 + \beta_1x_{i1} + ...$$ where $s_i$ is the population size municipality $i$ (which I assuming is known and not random). $\frac{S_i}{s_i}$ is the rate of statements for a municipality given its population size, which could control for the problem you have identified. This can be re-written equivalently as $$\log S_i = \log s_i + \beta_0 + \beta_1x_{i1} + ...$$ which gets you back to a form closer to the normal Poisson regression. The $\log s_i$ term is your offset in the model so that you are modeling the differences in the rate of statements across municipalities instead of the absolute count.