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st: Re: xtpoisson with endogenous covariate?


From   "Rodrigo Alfaro" <ralfaro76@hotmail.com>
To   <statalist@hsphsun2.harvard.edu>
Subject   st: Re: xtpoisson with endogenous covariate?
Date   Tue, 28 Mar 2006 09:43:03 -0500

Dear Alice,

Note that if Y~Poisson(lambda) then E(Y)=lambda and in MLE framework lambda_hat = mean(Y). When you add exogenous covariates you are adding the following equation to your problem lambda = exp(X*beta). exp() function is because lambda should be positive. On the other hand, IV process assumes that your problem is linear and unbounded, this means that Y is distribute over real line and the relation that you have E(Y) = X*beta. As you can see this is not that you want, which is E(Y) =exp(X*beta).

The "right" solution for your problem is GMM. Basically you have the following moment condition E{[Y-exp(beta*X)]&Z}=0, where & is outer product. In words, the errors of your expected value in the poisson process are orthogonal to some instruments variables (Z). Note GMM in this case is non-linear, but not too difficult to implement by hand using the sample counterpart.

The lazy solution is change the dependent variable and use IV process, using log(Y) instead of Y. Note that this is "statistical" wrong because log[E(Y)] is not equal to E[log(Y)]. Even worse, Y could be zero, in which case you have to use log(Y+a) with a>0.

I hope this helps you
Rodrigo.


----- Original Message ----- From: "ALICE DOBSON" <alice_dobson@hotmail.com>
To: <statalist@hsphsun2.harvard.edu>
Sent: Tuesday, March 28, 2006 8:43 AM
Subject: st: xtpoisson with endogenous covariate?



Hi all,
I use Stata 9.1. What can I do if my count dependent variable is simultaneously determined with a continuous covariate in a panel data setting? In such a case my xtpoisson estimates would be biased. I have good instrumental variables in the data. Should I ignore my count data dependent variable and simply use xtivreg2 instead?
Best,
Alice

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