In modeling claim count data in an insurance environment, I began with Poisson but then noticed overdispersion. A Quasi-Poisson better modeled the greater mean-variance relationship than the basic Poisson, but I noticed that the coefficients were identical in both Poisson and Quasi-Poisson models. 

If this isn't an error, why is this happening? What is the benefit of using Quasi-Poisson over Poisson?


**Things to note:**

 - The underlying losses are on an excess basis, which (I believe) prevented the Tweedie from working - but it was the first distribution I tried. I also examined NB, ZIP, ZINB, and Hurdle models, but still found the Quasi-Poisson provided the best fit. 
 - I tested for overdispersion via dispersiontest in the AER
   package. My dispersion parameter was approximately 8.4, with p-value
   at the 10^-16 magnitude.  
 - I am using glm() with family = poisson or quasipoisson and a log link
   for code. 
 - When running the Poisson code, I come out
   with warnings of "In dpois(y, mu, log = TRUE) : non-integer x = ...".


**Helpful SE Threads per Ben's guidance:**

 1. [Basic Math of Offsets in Poisson regression][1]
 2. [Impact of Offsets on Coefficients][2]
 3. [Difference between using Exposure as Covariate vs Offset][3]

  [1]: http://stats.stackexchange.com/questions/11182/when-to-use-an-offset-in-a-poisson-regression
  [2]: http://stats.stackexchange.com/questions/167964/poisson-glm-with-non-count-data-rate-data?lq=1
  [3]: http://stats.stackexchange.com/questions/175349/in-a-poisson-model-what-is-the-difference-between-using-time-as-a-covariate-or?