Difference gmm vs system gmm, fd vs fod

[daejung yang]

Is there any criteria to decide which estimation of difference gmm vs system gmm, first difference vs forward orthogonal deviation.

[Sebastian Kripfganz]

The comparison of the difference vs. system GMM estimator is typically done using a difference-in-Hansen (also known as incremental Hansen) test which tests the validity of the additional instruments for the system GMM estimator, assuming that the difference GMM estimator is correctly specified. The system GMM estimator requires the stronger assumption that the changes in the variables (which are the instruments) are uncorrelated with the unobserved fixed effects. Joint mean stationarity of all variables is a sufficient condition for this assumption to hold. Sometimes, this assumption can be ruled out already on theoretical grounds.

There is no formal test for deciding between the difference and the FOD GMM estimator. The FOD estimator may have some better properties if the sample is unbalanced with gaps and if the idiosyncratic error term has no serial correlation.

More on GMM estimation for linear dynamic panel data models in Stata:

• Kripfganz, S. (2019). Generalized method of moments estimation of linear dynamic panel data models. Proceedings of the 2019 London Stata Conference.
• XTDPDGMM: new Stata command for GMM estimation of linear (dynamic) panel data models

[daejung yang]

Thank you for your precious comment.
What is the check process about assuming that the difference GMM estimator is correctly specified?
Is it enough to check the Hansen test for all instruments?

[Sebastian Kripfganz]

In addition to the Hansen test, you should typically also check that the AR(2) test does not reject the null hypothesis of no second-order serial correlation in the first-differenced residuals.

[daejung yang]

Professor Sebastian
your comments help me a lot.
Thanks.
bye.

[John Sgr]

Dear Sebastian,

I am looking the dynamic relationship between depression and income with unbalanced data. I use the FOD GMM estimator and I constructed my model by following your 2019 London Stata Conference presentation and Kiviet (2020):

• Kripfganz, S. (2019). Generalized method of moments estimation of linear dynamic panel data models. Proceedings of the 2019 London Stata Conference.
• Kiviet, J. F. (2020). Microeconometric dynamic panel data methods: Model specification and selection issues. Econometrics and Statistics, 13, 16-45.

The FOD GMM estimator brings good results in the coherence tests (m1,m2,J,incJ). Now, I am working on sequential model selection process.

At step 8 in your presentation you say "unless there is no opposing theory add instruments for the level model". I think joint mean stationarity of the dependent variable and the independent variables would not be the case in my model because it is most likely that changes in depression and income are related to time invariant characteristics -- I can control only few of them. Ignoring this threat, I added instruments for the level model and incremental overidentification test rejects very few of the additional moment conditions. So, it seems system GMM should be preferred to FOD. But I am not sure about stationarity assumption.

As Ahn and Schmidt (1995) estimator is robust to this threat, I decided to add the Ahn and Schmidt (1995) moment conditions. I tested the identifying assumption of homoskedasticity by adding 'nl(iid)' option to FOD GMM and the generalized Hausman test do not reject the additional overidentifying restriction from the homoskedasticity assumption (p=0.31). In addition, the difference Hansen test brings p value=0.22 for and the rest of coherence tests performs very well. If I add non-linear condition with 'nl(noserial)' then the Hausman test and difference Hansen test brings p value=0.25.

1. Should I add non-linear moment conditions to my model? If so with 'noserial' or 'iid' option?
2. What are the other things I should consider before doing so?
3. How can I test under identification of model with non-linear conditions? -- or do I need to test it?
4. What is the reference paper to use FOD GMM in case of unbalanced data?

Best regards,
John

[Sebastian Kripfganz]

If you have good theoretical reasons why the mean stationarity assumption does not hold and there is no identification problem with the FOD estimator, then I would stick to the latter and ignore the incremental Hansen tests. These tests are not perfect anyway and decisions should not just be based mechanically on them.

1. Based on your information, adding nonlinear moment conditions seems reasonable. If you are willing to accept the homoskedasticity assumption, the nl(iid) option will give you more efficient estimates. This also depends on your audience. With the nl(noserial) option you are on the safe side, as it is robust to a violation of homoskedasticity. If the results do not differ much, you might choose one of them and simply say that the results are very similar with the other option.

2. The important part before adding nonlinear moment conditions is to make sure that the FOD model is correctly specified, which you seem to have done.

3. Unfortunately, this is not currently possible. You can check for underidentification after the FOD model. If there is no problem in that case, then there will not be a problem either when you add the nonlinear moment conditions.

4. The best reference for FOD is the seminal paper by Arellano and Bover (1995, Journal of Econometrics). With regards to unbalanced panels, I am afraid I cannot tell you a good reference out of the top of my head. If you find one, I would appreciate it if you could post it here.

[John Sgr]

Thanks for your help. I have one further question.

1. How should I interpret failing to reject Hausman test and how does this help me to choose one of these options?

PS: Results are very similar for iid and noserial options.

[Sebastian Kripfganz]

A non-rejection of the Hausman test between nl(noserial) and nl(iid) options indicates that there is no evidence to reject the underlying homoskedasticity assumption of the latter. Thus, nl(iid) would be more efficient and therefore preferred.

(Under the null hypothesis, both nl(noserial) and nl(iid) are consistent. Under the alternative hypothesis, nl(iid) is inconsistent.)

[John Sgr]

Thank you! My next question is about underidentification test.

If overall underidentification test rejects the null hypothesis but sw option shows that some of the endogenous variables do not reject the null, can we say they are poorly predicted by the instruments and we should exclude them? Also I did not see any recommended p value threshold for underidentification test. What is your advise?

[Sebastian Kripfganz]

Unless someone suggests otherwise, the p-values can be interpreted in the conventional way. Excluding an endogenous regressor because it is poorly predicted by the instruments may not solve the problem as it might create a new omitted variables bias for the remaining regressors. A non-rejection by the underidentification test usually indicates that you should try to find stronger instruments, if that is possible. In the present context, adding nonlinear moment conditions might help to solve the problem (although unfortunately we cannot test that). Otherwise, you might want to relax the endogeneity assumption and treat those variables as predetermined, but this of course might lead to concerns about instrument validity. There is no general solution that always works.