Poisson regression: how do number of observations and offset affect variance of betas?

Question

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.

By dividing a observation into multiple observation I mean the following:

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.

Answer 1

For Poisson likelihoods estimated in log-linear models, the number of observations do not affect the variance of the betas. This is because there is a mean variance relationship. The variance-covariance estimate of a coefficients to a Poisson regression model is given by:

$$ \text{var}(\hat{\beta}) = \left( \mathbf{X}^T \text{diag}(\hat{y}) \mathbf{X} \right) ^T$$

Note there is no use of the offset here. The variance structure is given by the usual $\mathbf{A}$ matrix formulation of exact likelihood based inference, where the variance is the predicted model variance (the variance is the mean for Poisson models).

This all falls apart if you consider quasipoisson models where the mean equals the variance up to a constant value. That constant, the dispersion, must be estimated and leads to inference which does depend on the number and formatting of the observations. Each observation must, in that case, have a tangible meaning and fundamental level of replication which is endemic to the study design. A very illustrative example comes from Agresti's Categorical Data Analysis book and the example of horseshoe crabs mating.