Hello,
I have a question about the way I should treat unit root in unbalanced panel data analysis and estimating a GMM model

Preliminary:
I have run the fisher test on my key variable of interest x
Code:
xtfisher x, lag(1)
. It appears that unfortunately, it is a Unit Root as I could not reject the H_0 hypothesis.
Hence I transformed my data into d.x. Once transformed variable passes the fisher test and I can reject the H_o hypothesis
Code:
xtfisher D.x, lag(1)
.

The problem
In order to do some theory testing, I am first running a fixed-effects model and then I am using a difference GMM to deal with some endogeneity concerns. Both models should produce comparable results in order to allow conclusion

I have estimated the FE model like this
Code:
xtreg y D.l.x l.c i.Year, fe robust
. where y is my dependent variable, l.c are lagge controls, and D.l.x is the differenced lagged x. I have theoretical reason to enter all right hand side variable lagged.

The Question
my question is the following: how should I enter my variable in the differenced GMM model? should I enter it as it is because then all variables are differenced anyway?

In other words: should I estimate this model

Code:
xtabond2 y l.y l.x l.c yeardum*, gmm(y x c, lag(2 5) collapse) iv(yeardum*) noleveleq small noconstant robust
or this one to account for the UR of my x?

Code:
xtabond2 y l.y D.l.x l.c yeardum*, gmm(y D.x c, lag(2 5) collapse) iv(yeardum*) noleveleq small noconstant robust

thanks a lot in advance for your help


Best regards