Synthetic Data ( Poisson Regression)

[Oscar Peter]

Hello

I'm trying to get a grip of generating synthetic data but I'm having some difficulties ?

Following Hilbe2010 http://www.stata-journal.com/article...article=st0186
I want to synthetically generate data to analyze using various count data models — starting with poisson.

So I use the code:

clear
set obs 50000
set seed 4744
generate x1 = invnormal(runiform())
generate x2 = invnormal(runiform())
generate xb = 2 + 0.75*x1 - 1.25*x2
generate exb = exp(xb)
generate py = rpoisson(exb)
poisson py x1 x2
% And then I would like to test the overdispersion
%following Cameron1990 http://www.sciencedirect.com/science...0440769090014K
predict muhat,n
generate ystar = ((py-muhat)^2 - py)
regress py muhat, noconstant noheader
%so a alpha =1 indicates overdispersion

So far so good. And now I would like to generate the Data without overdispersion namely Poisson(1).
As far as I understood I have to have a E(exb)=1, and Var(exb)=1 ??
So to get E(exb)=1 E(xb) has to be =0 so any regression with E(xb)=0 would do f.ex. = 1 + b1x1 - b2(=2*b1)x2 when xi N(1,1)
further we have to have Var(exb)=E(exb) here I'm stuck ... ???
Any advice how I could proceed ?

Thank you

[Oscar Peter]

ok I found it....

regress py muhat, noconstant noheader

should of course be :

regress ystar muhat, noconstant noheader

[Joao Santos Silva]

Dear Oscar,

Conditioning on x1 and x2, the data you are generating does not have overdispersion. Anyway, to generate Poisson(1) just do

g y=rpoisson(1)

Best regards,

Joao

[Oscar Peter]

Dear Oscar,

Conditioning on x1 and x2, the data you are generating does not have overdispersion. Anyway, to generate Poisson(1) just do

g y=rpoisson(1)

Best regards,

Joao

Thank you Joao,
my question is how could I formulate everything from the scratch so I get a py ~ Poisson(1). I think I have found a solution:

clear all
set more off                                
set obs 1000000                                
set seed 1234
generate x1=rnormal(1,.1)
generate x2=rnormal(1,.1)
generate xb = 1 + x1 - 2*x2     //E(xb)=0 
generate exb = exp(xb)            //E(exb)=1
generate py = rpoisson(exb)
poisson py x1 x2
predict muhat,n
generate ystar = ((py-muhat)^2 - py)/muhat
regress ystar muhat, noconstant noheader

So far so good.

Going back to my initial Code I have no overdispersion (using

generate ystar = ((py-muhat)^2 - py)
regress py muhat, noconstant noheader

). But when I summarize my py I can see that the variance is way over the mean how is this possible ??

[Joao Santos Silva]

Dear Oscar,

You are mixing up conditional and unconditional equidispersion. You should look at a good textbook covering count data models to see the difference between the two concepts.

In all the cases you are generating you have conditional equidispersion and unconditional overdispersion. The only way to generate data with unconditional equidispersion is to generate the Poisson variable without making it a function of x1 and x2, as I suggested.

Best regards,

Joao