Count data panel model with fixed effects and endogenous regressor, GMM and Wooldridge (1997) transformation

[Thomas Handke]

I have panel data with a count outcome (highly right-skewed) and want to estimate a model with individual-specific effects. One of the regressors is endogenous. A plausible instrument is available. There are no lagged dependent variables, the model is not dynamic. The time dimension is fairly large, T = 40. I am considering several options, outlined below.

1. A conditional exponential mean model with multiplicative error, estimated by GMM (i.e. the model implemented by ivpoisson gmm or ivpois). However, in the panel case there is the incidental parameters problem if the fixed effects are added as parameters. I could not find any simulation evidence how much of a problem this might be.

2. From my reading of Cameron and Trivedi (2013) and Windmeijer (2000, 2008), a solution to the incidental parameters problem in this scenario is to apply the first difference transformation described by Wooldridge (1997) and then use moment conditions based on the transformation and current (and possibly past) values of the instrument for estimation.

3. An alternative approach is to estimate a control function from a linear first stage, then use the control function in a Poisson fixed effects model and bootstrap the entire procedure.

Option (3) is straightforward. However, I would prefer the GMM approach to (3) due to the weaker assumptions. Is my understanding of the problem correct? Any tips on how to best go about implementing (2)? Thank you for your comments.



References

Cameron, A Colin, and Pravin K Trivedi, 2013, Regression Analysis of Count Data. Vol. 53 (Cambridge University Press).

Windmeijer, Frank, 2000, Moment conditions for fixed effects count data models with endogenous regressors, Economics Letters 68, 21–24.

Windmeijer, Frank, 2008, GMM for Panel Data Count Models, in László Mátyás and Patrick Sevestre (eds.): The Econometrics of Panel Data: Fundamentals and Recent Developments in Theory and Practice, (Springer Berlin, Heidelberg).

Wooldridge, Jeffrey M., 1997, Multiplicative Panel Data Models without the Strict Exogeneity Assumption, Econometric Theory 13, 667–678.

[Nicolas Apfel-Totaro]

Dear Thomas,

I hope I am not too late.
I also am struggling with the use of dynamic FE models for count data at the moment.
Here is what I think I understood:

1. The incidental parameters problem is generally a problem for GLMs when including fixed effects in a short panel (T is small), because in ML estimation inconsistent estimation of the incidental parameters "pushes through" on the parameters of interest.
You shouldn't be concerned with the incidental parameters problem for two reasons:
a. Poisson regression is an exception in that it does not have an IP problem: check out Cameron and Trivedi (2013) 9.4.2 for example.
b. Even if you had an incidental parameters problem: Your T is relatively large and the bias becomes negligible with large T.

2. I think you're confusing Nickell Bias and incidental parameters problem.
The problem with dynamic fixed effects models is the Nickell Bias: the strict exogeneity assumption is violated.
Under weak exogeneity (errors in t are independent from past and current regressors) the Chamberlain and Wooldridge transformations provide transformations (for additive and multiplicative errors respectively) of the data which lead to moment conditions that can be used for GMM.

The Chamberlain transformation would be

y_it * \lambda_it-1 / \lambda_it - y_t-1,

which also equals transformed errors, because the conditional mean cancels out.

We then end up with nonlinear moment conditions.
From reading these slides - http://www.eco.uc3m.es/~ricmora/mei/...estimation.pdf -
I understand that using Stata's gmm() - function, one should also be able to estimate with nonlinear moment conditions.
Instruments used should be past regressors.

However, since I have began reading on the topic today, corrections and comments from more experienced Statalisters would be highly appreaciated.

Best,

Nicolas