How Poisson pseudo-maximum Likelihood deals with overdispersion.

[Ray Uddin]

Dear forum,

My question is more theoretical, but I hope I can find an answer here.

I am running a Poisson model in a gravity framework (number of exported products as the dependent variable). My data is over-dispersed and therefore I will use the Poisson pseudo-maximum likelihood (PPML) estimator over the standard Poisson. However, in my data and methodology I would like to explain exactly how the PPML is superior - how it overcomes over dispersion, where the standard Poisson cannot. After reading many resources, I still cannot pinpoint a clear answer.

The general message seems to be that the PPML is consistent even if the distribution is not specified because the first-order condition SUM(yi-exp(xB))xi= 0 as long as yi = exp(xiB). (Where B = beta). Many papers refer this finding of Gourerioux et al. (1984). As I understand it, if overdispersion is present it will not matter for the PPML since there is no equidispersion assumption modelled into it.

But Cameron and Trivedi (1999) say that the standard Poisson is also consistent if the same condition holds. Therefore, what exactly does the PPML change?

I feel that there is large inconsistency and abstractness in the explanations in the non-technical papers that are out there.

I appreciate this is a theoretical matter.

Thank you in advance,

Ray

[Ray Uddin]

Dear Joao Santos Silva, As always with Poisson questions, I guess I am hoping from an answer from yourself. My two options are PPML and Negative Binomial. The Negative Binomial case is quite clear, since it incorporates a dispersion parameter in the variance.

Could you please explain what the PPML does to overcome overdispersion (in the count setting)?

Greatly appreciate your help.

Thank you in advance,

Ray

[Joao Santos Silva]

Dear Ray Uddin,

PPML is just Poisson regression with standard errors that are robust even if there is overdispersion. If you are just estimating the conditional mean, PPML is great because it is very robust and has other interesting properties; this is true with or without overdispersion.

Best wishes,

Joao

[Ray Uddin]

Dear Joao Santos Silva

Is the PPML then the same as running a standard poisson command with the robust option? The difference being, the name changes to 'pseudo' as the standard errors are adjusted?

Thanks in advance.

Kind regards,
Ray

[Joao Santos Silva]

Dear Ray Uddin,

That is correct. The ppml command has an additional feature, but it may not be important in your case.

Best wishes,

Joao

[Ashani Abayasekara]

Hi all,

I'm running difference-in-differences regressions where the outcome variable is log of real income. I initially used OLS fixed-effects models, but I do not think this is the best since there are 0 and negative values which are undefined for logs.

I'm instead interested in estimating fixed-effects Poisson models using PPML. Would this be feasible? Would I then simply use the level form of my real income variable as the outcome variable (without transforming it into log), omitting negative real income values?

Many thanks,
Ashani