Exposure/offset variable in Poisson regression with many fixed effects

Question

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!

Answer 1

Model writing has problem. $\varepsilon_{pt}$ does not exist. you can google the Poisson regression to find correct math formula.

Using (log of) population density as offset is not good. My suggestion is use population.

Maybe it is better to treat the effect of precinct $\gamma_{p}$ as random effect.

For your question:

1. right-hand side. maybe not called covariate, because its coeffient is fixed.

2. "being set equal to one" means set the regression coefficient of $log(d)_{p}$ to one, as in your formula.

3. Use population, instead of population density, as offset.

4. Using offset and # of zeros in the output have no direct relation. If you have a lot of zeros, it is possible that response variable does not following Poisson distribution. Then maybe you need to consider zero inflation or other distribution other than Poisson.