xtabond2 system GMM: questions regarding equation sub option

[Rosa Banlo]

Dear all,

After reading Roodman (2009) and Kripfganz, S. (2019) and this forum extensivly, I still have some questions with respect to sys GMM.

Lets say I have a variable y and want to regress this on its lagged value and several control variables like this:

 y L.y k n i.year

1. When I choose system gmm I can either specify the above like this:

 xi: xtabond2 y L.y k n i.year, gmm(y, lag (2 3)) iv(k n i.year, equation(level))

Where I specify the lagged value of y as endogneous and thus instrument this gmm style wise. I assume k and n to be exogenous.


2. I can also specify the equation differently where the only change is the equation(diff) suboption for the gmm style instrument. Here the lagged levels would be used as instuments for the first difference.

 xi: xtabond2 y L.y k n i.year, gmm(y, lag (2 3) equation(diff)) iv(k n i.year, equation(level))

What would the difference in assumptions be for 1 and 2? I know for the iv style instrument, the instruments for the level equation must be assumed to be uncorrelated with the unobserved effects so the instruments listed under iv (equation(level)) must be just that. Does the same hold for gmm style instruments when equation(diff) is not explicitly specified? Even in tutorials the equation(diff) option is often not used for gmm style.


3. lets say I assume k to be exogenous with respect to the idiosyncratic error but it is not uncorrelated with the unobserved effects. Would the below specification be correct? When would this actually happen?

 xi: xtabond2 y L.y k n i.year, gmm(y, lag (2 3)) iv (n i.year, equation level)) iv (k, equation(diff))

4. Sometimes I see the gmm style instrument included with lag (0 0). I assume this is because k for example cannot be assumed to be stricly exogenous. Why would one however use lag (0 0)?

xi: xtabond2 y L.y k n i.year, gmm(y, lag (2 3)) gmm(k, lag 0 0)) iv (n i.year, equation level))

Thank you in advance,

Rosa

[Sebastian Kripfganz]

1./2.: gmm(y, lag(2 3)) is equivalent to the combination of gmm(y, lag(2 3) equation(diff)) gmm(y, lag(1 1) equation(level)). Because the gmm() option for the level equation creates first differences as instruments, this requires that D.y is uncorrelated with the unobserved effects.

3.:iv(k, equation(diff)) is equivalent to gmm(D.k, lag(0 0) collapse equation(diff)). This would be a valid specification under the mentioned assumptions. These are the same assumptions which are used when you estimate a (static) model with the conventional fixed-effects estimator.

4.: See 3. for an example concerning the use of lag(0 0). This option requires that k is exogenous with respect to the idiosyncratic error term. Additional lags are not needed for the identification of k's coefficient, because the lag 0 is already the strongest possible instrument. However, it might still be useful to add further lags for the identification of other coefficients.