Difference GMM, building valid instruments with X that is (mostly) time invariant.

[alessandro gst]

Hello,
I have a theoretical question regarding the estimation of Difference GMM (Arellano Bond).
I have a panel with 90 countries and 27 years and am trying to estimate a model similar to the following y=y_t-1+x+z via the Difference GMM estimator. All of the regressors are treated as endogenous and I estimate them via GMM instruments. x is a regular time-varying continuous variable, while z is a dummy that barely changes over time.
In other words, z remains the same (mostly 0) for all the series for many countries. For those countries for which it changes, it changes only once going from 0 to 1 and then it remains 1 for the rest of the series. I was wondering whether with this type of variable is possible to build a valid instrument with these kinds of variables that are barely changing over time.

To be more clear, take three countries i, j and k, the structure of the variable of interest (z) looks something like this:

• For country i, z remains 0 for all the series (77 out of 90 countries are like this)
• For country j, z is 0 at the start of the series, it becomes 1 at time t, and remains 1 for all the subsequent time periods t+n
• For country k, z is 0 at the start of the series, it is 0 at time t, and it becomes 1 at time t+1 remaining 1 for all the subsequent time periods

My question is whether I can build valid instruments using the difference GMM estimator with this kind of variable, or whether I should consider alternatives routes


Thanks in advance for your help

[Sebastian Kripfganz]

Using lags of such break dummies as instruments will not work because the lags are just zero before the break and when they become one, the instrumented first-differenced break dummy is zero. Hence, there is no correlation between the instrument and the dummy.

The main question is why you want to assume that these break dummies are endogenous. If they can be strictly exogenous with respect to the idiosyncratic error term, there will not be any problem as you can just instrument them (in first differences) by themselves. If you really want to treat them as endogenous, I do not think there is much hope to find good internal instruments.

[alessandro gst]

Thank you very much for this reply really clarify the issue.
Unfortunately, I have strong theoretical reasons to believe that these break dummies are endogenous to my dependent variable.

I was wondering on whether I could create internal instruments if i have a variable with limited variation:
Using a more fine-grained version of the variable (z is continuous and ranges from 0 to 18) my variable behaves in the following way
out of 108 countries:
for 33 of them, it remains 0 unchanged.
for 57 of them, it changes one time (breaks)
for 18 it changes values 2 or more times

Would this variation be sufficient to build internal instruments?

thanks a lot again for your help

[Sebastian Kripfganz]

There is still the risk that the instrument is weak. A simple way of getting a quick idea about whether this could work is to check the correlation between your endogenous variable and the proposed instrument.