[Huaxin Wanglu]

If the model without the additional instruments is correctly specified (i.e. the Hansen test excluding this group of instruments does not reject the null hypothesis), then the difference-in-Hansen test could be interpretated as a test for the validity of the additional instruments. In that regard, your understanding is correct.

As to which test results to report, it really depends. You certainly want to report the Hansen test for the full model. On top of that, it makes sense to report difference-in-Hansen tests for particular instruments if their inclusion requires particular justification. For example, if the Arellano-Bond AR(2) test does not reject the null hypothesis of no second-order serial correlation of the first-differenced errors, then you usually need not separately justify the lagged levels of the dependent variable as instruments for the first-differenced model. In contrast, the difference-in-Hansen test for the level instruments is informative because it helps to evaluate whether the Blundell-Bond mean stationarity assumption might be violated.

For example, you could report the Hansen test for the model with the instruments for the first-differenced model only, the Hansen test for the full model, and the respective difference-in-Hansen test. The Hansen test for the first-differenced model tells you something whether your model is dynamically complete (because this implies whether those instruments are valid). The difference-in-Hansen test, as mentioned before, tells you something about the mean stationarity condition needed for the validity of the level instruments. Taking these two test results at face value, the Hansen test for the full model would in principal be redundant but it is still reasonable to provide a complete picture.

Sorry to bring up this post. When reviewing this content again, I am still confused about two things: (1). "if the Arellano-Bond AR(2) test does not reject the null hypothesis of no second-order serial correlation of the first-differenced errors, then you usually need not separately justify the lagged levels of the dependent variable as instruments for the first-differenced model", do you mean if the model passes AR(2) test, it is unnecessary to report "Hansen test excluding group" for "GMM instruments for levels"?; (2). "the difference-in-Hansen test for the level instruments" refers to gmm(beta, eq(level) lag(1 1)) OR gmm(beta, eq(diff) lag(2 3))? If I understand correctly, the level instruments are used in the first differenced model and the differenced instruments are used in the level model, so it should be the latter. But, if we're estimating a two-step system GMM, shouldn't the testing in the level equation be more important? I see your conference slides mention the joint mean stationarity under the system GMM, but I still hope to check with you if I understand which test should I report correctly.

[Sebastian Kripfganz]

(1) No, the GMM instruments for the levels require an additional assumption. Absence of serial correlation is not sufficient for their validity. An additional difference-in-Hansen test can provide useful insights.

(2) The terminology I used in the quoted post might have been misleading. With "level instruments" I meant to refer to the instruments for the level equation.

[Huaxin Wanglu]

(1) No, the GMM instruments for the levels require an additional assumption. Absence of serial correlation is not sufficient for their validity. An additional difference-in-Hansen test can provide useful insights.

(2) The terminology I used in the quoted post might have been misleading. With "level instruments" I meant to refer to the instruments for the level equation.

Thank you so much for the clarification.