poisson distribution regression exposure proportionality not holding

[Nathan Carter]

Hi all,

I am estimating the intensity of a certain staff groups' rate of hours worked against the total workforce, using an exposure option to contain the total staff hours worked figure. As I understand it, exposure models create a sort of rate, where the proportionality of the rate is fixed over total staff hours:

ln(Staff_A_hours / Total_Hours ) = [explanatory vars], so
ln(Staff_A_hours) = [explanatory vars] + ln(Total_Hours)

Staff_A_hours has many zero values, is in a continuous format bounded by zero, with values like 4.32222, and a normal distribution is unsuitable, so I am using a poisson distribution (even though it is intended for count data I believe is is ok?). My data is an unbalanced panel over 40 months, with 238'000 observations & 10 key variables. Total_hours also counts the hours worked by Staff_A.

I have specified:
xtpoisson staff_A_hours [explanatory vars] , re exposure(Total_Hours) vce(cluster practice_id) irr
And the results look great. But when I instead include the log of total staff hours as another explanatory variable to test the proportionality assumption of exposure variables:

xtpoisson staff_A_hours [explanatory vars] ln(Total_Hours) , re vce(cluster practice_id) irr. :The estimated coefficient for ln(Total_Hours) in this model is 0.01644, very far off of a coefficient of 1, which as I understand violates the underlying assumption of proportionality in an exposure model. The supposed explanation here is that smaller locations with lower levels of total staff are more reliant on Staff_A types. As I understand it, this means the estimator is biased, and I should opt for another approach when estimating this rate dependent variable, as a violation of this proportionality assumption means that a double in total staff hours does not correspond to double the expected staff_a_hours worked.

Apparently the two options I have remaining are :
1 - model the dependent variable as a calculated ratio using a fractional logit model, ie gen ratio = Staff_A_hours / Total_Hours, and use this ratio as the dependent variable. It would be bounded between 0 and 1 as if a percentage, and apparently remain robust despite the value being 0 in many instances.

2 - model the dependent variable as raw Staff_A_hours as an xtpoisson regression, and instead include ln(Total_Hours) as an explanatory variable instead of an offset, so that the proportionality of the relationship is not fixed. However this would change the interpretation of my model slightly, as I would no longer be estimating a staff intensity or use rate.

Which model would be preferable? Am I horribly wrong anywhere above in my modelling assumptions, and is this issue of proportionality a major issue? Thank you!

[Nathan Carter]

Oh my lord I was misinterpreting the IRR. New results of the above regression without IRR give the elasticity as 0.96, with the CI bounds containing 1, so proportionality of my initial exposure model appears to hold.

[Joao Santos Silva]

Dear Nathan Carter,

Glad you sorted it out. I just have a few additional points:

1) If at all possible, use FE rather than RE. The reason is that Jeff Wooldridge has shown that Poisson regression with FE is fully robust to departures from the Poisson assumption, whereas with RE the results are very fragile.
2) Use clustered standard errors; try the user-written ppmlhdfe command for that.
3) It is now standard to use Poisson regression for data that are not counts. John Mullahy was one of the first to do that and I have helped popularising that approach.

Best wishes,

Joao

[Nathan Carter]

Hi Joao Santos Silva , thank you so much for your considered reply, this really addresses my concerns.
It's good to see you here, I actually graduated from Surrey's BSc Econ programme last year and I'm regretting not choosing your applied econometrics course a little right now.

I am also using FE in another model in this study mainly for the characteristics you note. The issue is that the majority of my explanatory variables are either time invariant, or exhibit little volatility over time. I aim to use RE to explain some cross-location variation in staff_A hours, and then critique RE for its assumption that location effects are uncorrelated with the regressors, motivating the addition of location - level averages over time to this RE model. I suppose a panel RE model with Mundlak means would still be prone to this fragility and a pooled Mundlak specification would be more defensible in this case?

Really appreciate your other points too, I've been using some pretty old literature for guidance on Stata and use of the poisson regression.

Many thanks,
Nathan.

[Joao Santos Silva]

Dear Nathan Carter,

I am delighted to see a Surrey graduate using metrics .

Given what you say, I suggest you also try simple pooled Poisson, which is more robust than Poisson with FE.

Best wishes,

Joao

[Nathan Carter]

Thank you Joao Santos Silva ,

I really appreciate the feedback, I'll also be reporting a preliminary pooled model before any panel attempts.

Many thanks,
Nathan