xtabond2 / GMM - Arguments and reasons to choose which equation uses the instruments: equation({diff | level | both}) passthru

[Olaf Hotte]

Dear Statalisters,

I am currently working on static panel data with a predetermined independent variable (x1). Accordingly, a basic model in xtabond2 would look like this:

Diff-GMM

xtabond2 y x1 x2 x3 i.year, gmm(x1, collapse) iv(x2 x3 i.year) noleveleq

Sys-GMM

xtabond2 y x1 x2 x3 i.year, gmm(x1, collapse) iv(x2 x3 i.year)

Now, I would like to go more into details and understand especially the equation() option. The equation() suboption specifies which equation(s) should use the instruments: first-difference only (equation(diff)), levels only (equation(level)), or both (equation(both)), the default. Also by default, the instruments are transformed for use in the transformed equation and entered untransformed for the levels equation. The suboption passthru prevents this transformation.

What are the arguments and reasons to choose which equation uses the instruments? --> equation({diff | level | both}) passthru

Best regards
Olaf

[Sebastian Kripfganz]

First-differencing the regression equation removes the unobserved "fixed" effects. This is the easiest way of dealing with such unobserved heterogeneity. Any instruments then just need to be uncorrelated with the first-differenced idiosyncratic error component. Any time-invariant regressor will be removed as well.

Instruments for the level equation must be assumed to be uncorrelated with the unobserved effects. This is an additional (not always easy to justify) assumption compared to estimating the model in first differences only. Often, researchers are willing to assume that the first differences of those variables are valid instruments for the level equation, which would be the case if these variables (before differencing) have constant correlation over time with the unobserved effects.

Instruments for the level and first-differenced equation can be combined, which requires that the assumptions for both of them hold. This would be more efficient than just estimating the model in first differences (assuming that all assumptions are satisfied) and might mitigate some weak-instruments concerns.

The following presentation might be helpful. While it mostly covers dynamic models, the arguments carry over to static models.

• Kripfganz, S. (2019). Generalized method of moments estimation of linear dynamic panel data models. Proceedings of the 2019 London Stata Conference.