What are the alternatives to -nbreg- with fixed effects?

[Igor Martins]

Dear Statalists,


I have two panels, both of which with thousands of individuals (ID) followed through a 35-year period. It has a substantial number of observations in both datasets. Ultimately, I want to estimate the effect of a given exogenous shock on year t=T on the possession of a given asset that is measured through a count variable. I am conducting the estimates separately for each panel.

Since I am working with longitudinal data, it seemed adequate to control for individual's fixed effects using a negative binomial regression. However, after reading about -xtnbreg, fe-, it became clear to me through the works of Allison and Waterman (2002), Greene (2007) and Guimaraes (2008) that this command does not deliver a fixed effect estimation to satisfaction. Apparently, using -xtnbreg, fe- makes a model vulnarable to the "incidental parameters bias".

The most straighforward solution to the problem is to run -nbreg- adding individual dummies to control for fixed effects. Despite computational demanding since I have many individuals, I managed to run the regression in one of my panels for the main sample and also smaller subsamples of interest. Stata did not have any problem converging to optimal results.

My problem starts when I run the exact (the exact!) same model specification for the second panel. Stata does not converge to optimal results. In fact, Stata gets stuck in a non-concave area of likelihood function. I decide to add each variable separately to verify who was the villain. Stata runs the model perfectly until I add i.ID. The moment I add the individual dummies, is the moment Stata fails to maximize the likelihood function. I tried to take a random subset from the main sample hoping that a smaller sample size would make things easier to Stata. No success. I tried to change the iteration mode. I tested them all (nr, bhhh, dfp and bfgs) and none worked. I even tried to relax the maximization assumptions through -tolerate- but my problem is, indeed, lack of concavity.

Additionally, I tried -ppml- to check if my specification had a "maximizable" likelihood function. While -ppml- does not allow me to include factor variables, it did seem that the specification was alright and the villain was, indeed, i.ID. I also read Santos Silva & Tenreyro (2010, 2011) and tried to correct the scaling and magnitude of the covariates - though I do not think this was a problem in my dataset - but, again, no success on convergence.

In closing, it does not seem I will be able to reach results through this method for the second panel. With all that considered, what would be the second-best approach? Could I just use -xtpoisson, fe- and argue that this is a good approximation of -nbreg- with individual dummies? If so, what would be the best references for me to motivate such choice?

Mind you that I do not want to compute probabilities for particular counts. My main interest is to observe direction and significance of the covariates, specially the exogenous shock. So long I can trust the p-values, the direction and the size of the coefficients relative their counterparts in the regression, I am happy.

I would like to thank in advance for any further input.

Best,

\igor

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References:

Allison, P.D. and Waterman, R.P., 2002. 7. Fixed-Effects Negative Binomial Regression Models. Sociological methodology, 32(1), pp.247-265.
Greene, W., 2007. Functional form and heterogeneity in models for count data. Foundations and Trends in Econometrics, 1(2), pp.113-218.
Guimaraes, P., 2008. The fixed effects negative binomial model revisited. Economics Letters, 99(1), pp.63-66.
Silva, J.S. and Tenreyro, S., 2010. On the existence of the maximum likelihood estimates in Poisson regression. Economics Letters, 107(2), pp.310-312.
Santos Silva, J. and Tenreyro, S., 2011. poisson: Some convergence issues.

[Joao Santos Silva]

Dear Igor,

If you have 35 observations per individual, the incidental parameter problem may not be very severe. Anyway, I suggest that you stay away from the NB regression and just use Poisson regression with FE (xtpoisson, fe). This is valid under very general conditions and does not suffer from the IPP. The best reference for it is perhaps:

Wooldridge, J. M., (1999). “Distribution-Free Estimation of Some Nonlinear Panel Data Models,” Journal of Econometrics 90, 77–97.

Best wishes,

Joao

[Igor Martins]

Dear João,

Thanks for your answer (and also for the -ppml- and both papers I mentioned as I assume you are the author!).

Indeed, when I compare -xtpoisson, fe- with the estimates I got for -nbreg- with individual dummies for the one panel it worked, the results are quite similar (Poisson slightly overestimates the parameters, but it does keep the model "together").

My fear of IPP comes from the fact that I have an unbalanced panel and I do not observe every individual for 35 years. On average, I observe them around 5 years until they leave the sample (but in many cases I do seem more than 20 times).

Anyway, I will stick to -xtpoisson, fe-. Thanks again for the input!

Best,

/igor

[Joao Santos Silva]

Dear Igor,

Given your description of the data, the IPP is likely to be severe and needs to be taken seriously. So, Poisson regression is really the only option.

You say that Poisson slightly overestimates the parameters; I would say that NB underestimates them; there is no reason to believe that the NB model with dummies is valid in this case because of the IPP.

Best wishes,

Joao
PS: I am glad you found my contributions useful

[Igor Martins]

João, one last question: is this discussion of IPP still valid in a context of cross-sectional data? Or in such cases the -nbreg- produces consistent estimators?

Best,

/igor

[Joao Santos Silva]

The IPP is not an issue if you do not have fixed effects, but even in the cross-sectional case Poisson is more robust that NB.

Best wishes,

Joao

[Mansi Jain]

Dear Joao Santos Silva,

Greetings! I am having a similar problem. I did xtnbreg, fe for my data and that yielded results, but I now want to cluster standard errors at a particular level. Since xtnbreg does not have any option for clustering standard errors (or does it? perhaps I am wrong about that), I figured the only approach to take would be nbreg with i.id and clustered standard errors. I have 3,617 IDs in an unbalanced panel dataset where each ID has about 9-10 observations for it. I tried running just the simpler version of nbreg Y X i.year i.id without clustering standard errors, but it's been many many hours and it isn't converging--- we're on Iteration 37 now. Should I assume it isn't going to converge at this point or just keep going? And if it isn't going to converge and the Y variable in question is definitely over dispersed, would you still recommend using xtpoisson, fe instead, or would leaving it at xtnbreg, fe without clustering standard errors be better?

Lastly, is there any literature I can refer to that helps me figure out why it is that a negative binomial fixed effects regression doesn't have a standard error clustering option whereas a poisson fixed effects regression does (or perhaps there is no conceptual reason and that's just how the command is built).

Thank you very much! I really appreciate it.

Regards,
Mansi Jain
Stanford University

[Joao Santos Silva]

Dear Mansi Jain,

I would certainly use Poisson regression with fixed effects because it is valid even if you have over-dispersion (unless you want to compute probabilities); notice that you do not know if you have overdispersion after conditioning on the fixed effects and in fact it may be the case that the NB estimator does not converge because you have underdispersion in the model with fixed effects.

Best wishes,

Joao

[Mansi Jain]

Dear Joao Santos Silva,

Thank you so much for your advice! I just saw this, and I was actually looking for similar advice right now so this is perfect timing. How do you suggest checking for over dispersion after including fixed effects in Stata? The usual negative binomial model without fixed effects gives a value for the over dispersion parameter, but I'm not sure how to get that after incorporating the fixed effects.

Secondly, does the fact that the NB estimator did not converge imply anything - that the data is definitely under dispersed, for instance, or can I not interpret such a thing out of the fact of non-convergence? Thank you!

[Joao Santos Silva]

Dear Mansi Jain

Unless you want to compute the probability of some events, I would not worry about the overdispersion. Anyway, I am not aware of any practical way of testing for overdispersion in that context.

Underdispersion is a possible cause, but there are others.

Best wishes,

Joao