Background: In Poisson regression with an offset, like in this answer, @Hong Ooi writes
Your underlying random variable is still $Y$, but by dividing by $\varepsilon$ we've converted the LHS of the model equation to be a rate of events per unit exposure. But this division also alters the variance of the response, so we have to weight by $\varepsilon$ when fitting the model.
Question: Since the exposure $\varepsilon$ is accounted for when fitting the model, does that mean one can divide a (multiple) observation into multiple observation(s) with new exposure $\varepsilon_i$, where $\sum_i\varepsilon_i=\varepsilon$, without this affecting the variance of the parameter estimates $\beta$?
Attempt: Intuitively I would think that doing this would give me more "observations", which would decrease the variance of each $\beta$ estimate, even though we weigh by $\varepsilon$ when fitting the model.
UPDATE: A while back @Scortchi wrote this in Ten fold:
If you're using the full likelihood formulation (Poisson + offset) nothing will change - 5 counts in 10 hours is the same as 1 count in 6 hours & 4 counts in 4 hours. If you're using the quasi-Poisson formulation you'll get the same point estimates but the standard error will change when your estimate of the dispersion parameter changes.