Interpretation of margins after xtpoisson with interaction

[Michael Reuter]

Hi,
this is likely very simple, but I cannot wrap my head around it. I am running an xtpoisson model with two continuous predictors that are both standardized. How do I interpret the margins?
I constructed a little demo below. The dependent variables is a count (obviously), "number of weeks worked" in the demo data. Nevermind that one would likely not standardize any of the independent variables in this dataset, but in my real data I think it would make sense.

webuse nlswork.dta, clear
xtset idcode year

* run regression once to determine sample
xtpoisson wks_work c.birth_yr##c.ttl_exp
keep if e(sample)

* standardize independent variables over full sample
egen z_birth_yr = std(birth_yr)
egen z_ttl_exp = std(ttl_exp)
drop birth_yr ttl_exp
rename z_birth_yr birth_yr
rename z_ttl_exp ttl_exp

* run regression of interest
xtpoisson wks_work c.birth_yr##c.ttl_exp

* obtain margins
margins, at(birth_yr = (-2/1) ttl_exp = (-1 4))

This yields the following result:

(omitted)


. * obtain margins
. margins, at(birth_yr = (-2/1) ttl_exp = (-1 4))

Adjusted predictions                                    Number of obs = 27,831
Model VCE: OIM

Expression: Linear prediction, predict()
1._at: birth_yr = -2
       ttl_exp  = -1
2._at: birth_yr = -2
       ttl_exp  =  4
3._at: birth_yr = -1
       ttl_exp  = -1
4._at: birth_yr = -1
       ttl_exp  =  4
5._at: birth_yr =  0
       ttl_exp  = -1
6._at: birth_yr =  0
       ttl_exp  =  4
7._at: birth_yr =  1
       ttl_exp  = -1
8._at: birth_yr =  1
       ttl_exp  =  4

------------------------------------------------------------------------------
             |            Delta-method
             |     Margin   std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
         _at |
          1  |   3.645507   .0122726   297.04   0.000     3.621453     3.66956
          2  |   4.865549   .0147649   329.53   0.000      4.83661    4.894488
          3  |   3.630847   .0078433   462.92   0.000     3.615475     3.64622
          4  |   4.972935   .0093784   530.25   0.000     4.954554    4.991317
          5  |   3.616188    .005467   661.45   0.000     3.605473    3.626903
          6  |   5.080322   .0066949   758.83   0.000       5.0672    5.093443
          7  |   3.601529   .0074654   482.43   0.000     3.586897    3.616161
          8  |   5.187708    .009413   551.12   0.000     5.169259    5.206157
------------------------------------------------------------------------------

How do I interpret the margins in a way that would allow me to interpret them in a "....and this means x more weeks worked" way?

Thank you so much!

Michael