First time here on CrossValidated!
My question concerns the inclusion of an offset variable in a Poisson regression. I have panel data and my outcome is 'count' distributed. My cross-sectional unit is the precinct and I observe 'shootings' by month in each of those precincts. I am evaluating a government intervention that is 'in effect' for only a few months throughout the year.
I include "precinct" and "month" fixed effects (i.e., a full set of precinct and month dummies enter the model). I have only one independent variable I am assessing. The specification is as follows:
$$ log(y)_{pt} = \alpha + \gamma_{p} + \lambda_{t} + \delta D_{pt} $$
As indicated, the parameters $\gamma_{p}$ and $\lambda_{t}$ are the fixed effects. Most of the covariates I can get a hold of are "time-invariant" and would not be useful to incorporate into an equation looking at "within-precinct" variation. My goal is to include an exposure variable. I want to use population density as an offset but I am having difficulty wrapping my head around it conceptually. Here is how I perceive the offset in a model like this: $$ log(\frac{y_{pt}}{{d_{p}}}) = \alpha + \gamma_{p} + \lambda_{t} + \delta D_{pt} $$
If you algebraically manipulate this, it should become (if I have this right):
$$ log(y)_{pt} = log(d)_{p} + \alpha + \gamma_{p} + \lambda_{t} + \delta D_{pt} $$
It is worth noting that population density only varies across precincts, not so much over the small time dimension I have (24 months). I plan on implementing this in R (in case anyone was curious).
Question:
If I include this as an offset, does this mean that I simply divide the monthly counts by the population density? This might seem weird as population density is already a measure of the population per square mileage.
I thank anyone willing to tackle this question. Also, if you think this exposure variable is not worth my effort then please let me know.
I am open to any and all comments.
Thanks!