st: xtabond2 with endogenous dummy regressor

["Marc Stuhler" <marc_st@gmx.de>]

Dear all,
I would like to estimate the following equation using system-GMM with xtabond2:  
y_it= y_it-1*lamda + x_it*b_1 + w_it*b_2 + u_it, i=1,...,N; t=1,...,T,
where 
y_it denotes the sales (units) of the analyzed industry in market i (N=38) and time t (T=17),
y_it-1 denotes lagged sales,
x_it are strictly exogenous covariates, and
w_it is an endogenous regressor

All variables are considered in logs and xtabond2 is used to implement system GMM as follows: 
xtabond2 y l.y x w time-dummies, gmm(y w, lag(2 6)) iv(x time-dummies) robust small two pca
This works well.

Then, particular interest is focused on the market entry of one large competitor. Let z_it denote the presence of this competitor in market i at time t, (i.e., a dummy variable = 1 if the competitor is present and 0 otherwise), so that:
y_it= y_it-1*lamda + x_it*b_1 + w_it*b_2 + z_it*b_3 + u_it, i=1,...,N; t=1,...,T

The sales of this competitor are included in y_it and theoretically causality may run in both directions. Therefore, the regressor is expected to be correlated with the error term and thus endogenous. Note that once the market entry occured, the competitor remains in the market (i.e., once the dummy has taken the value 1, it does not change again). Thus, instrumenting current levels with past differrences (as it is the common approach to account for endogeneous regressors under system GMM) does not seem to be an option (?). Any suggestions on how to best specify this endogenous dummy regressor are much appreciated!

Cheers, Marc

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