Hi everyone,

I am conducting research using the dynamic system GMM analysis. My model specification is:
Yi,t=F(Yi,t-1, Xi, t, Fi)
Suppose: X is time-variant variable (access to improved water)
F if time-unvariant variable. (pregnancy care at time t=1)
Y: child malnutrition
t runs from 2 to 5
I aim to estimate the impacts of access to improved water and pregnancy care (initial condition of child health) on child malnutrition when growing up.

Firstly, I would estimate the dynamic panel data analysis with time-invariant regressors. I did follow the paper "Estimation of linear dynamic panel data models with time-invariant regressors" by Sebastian Kripfganz and Claudia Schwarz, published in the Journal of Applied Econometrics (2018). However, the authors assume that X and F are strictly exogenous (Assumption 2, page 528). But in my case, I suppose that X is not strictly exogenous (can be strictly endogenous or weakly endogenous).

Hence, I come back to the system-GMM model which also allows estimating the impact of time-invariant variables (xtabond2). To avoid the over-instrumentation problem, I will use the collapse option. Besides, I also use the orthogonal option to increase the sample size.

However, currently, I face a new problem that F can be a confounder because pregnancy care can affect both the probability of access to improved water after giving birth and the child malnutrition when growing up.
We usually use the different-GMM to overcome the confounding effect by eliminating time-invariant regressors. I don't know whether system-GMM can address this issue?

I hope that anyone can give me some suggestions or papers about using dynamic panel data analysis to estimate my model specification with a time-invariant confounder regressor.

I would appreciate your kind supports!