Offset variable in Poisson regression makes model worse I'm using a Poisson regression to model count data (number of orders). My observation lengths are different, so I tried to include an offset variable in the model. The problem is that when I estimate length variable coefficient instead of fixing it to 1, model's quality becomes substantially better. But I'm not sure if it is theoretically justified which would be great. Is it a way to explain this?
 A: First make sure that the length really is the exposure.  For example, if you were counting bacteria on a Petri dish the surface area would be a better measure of exposure than the radius of the Petri dish.  In your case, this is very unlikely to be the case since the length of the interval sounds like a good exposure measure for the number of orders.
The other issue is whether the underlying poisson process is heterogeneous or not.  If the intensity of the number of orders vary through out the day, then fitting a single poisson distribution will capture average behaviour.  By itself, this would not cause a good measure of exposure to have a coefficient greater than one.  If, however, you did not select the interval lengths randomly, then the interval length may be correlated with the intensity of the process.  For example, if this data was recorded by hand by employees, they will record less often during busy times, so long times between recordings (a long exposure length) will tend to coincide with high traffic.  
