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I'm trying to decide whether I should use a difference GMM or a system GMM. I have a panel of 50 states over 19 years.

I know that system GMM has better finite sample properties when the series is persistent and certain stationarity conditions are satisfied. I run an AR(1) model (including year dummies) to test for the series persistence; which is very high (a>0.8), and want to test for series stationarity (I used xtunitroot llc). However, I'm not sure if the series needs to have no unit-roots in levels or first-differences for the system GMM to be preferred.

Also, do I have to test for the stationarity of all variables included in the model or only of the dependent variable?

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