Instrumental variable approach in mixed effects model

[Anne-Claire Jo]

Hello,

I am thinking of doing a hybrid model, or Random effects model with within and between estimators to study the innovation (proxy: patent) on the firm-level productivity growth at the downstream-level (non-superstar firms; 95% of the total firms).
Therefore, following Bourlès et al. (2010), my equation (1) would be:
LP_growth_{ist} = alpha1*LP_growth_{fst} + (1-alpha0)*productivity_gap_{is,t-1} + alpha2*[nb_patent_{ist} - mean(nb_patent_{is,t})] + alpha3*[X_ist - mean(X_is)] + alpha4*mean(nb_patent_{is,t}) + alpha5*mean(X_is) + i.sector + i.year + error_term_{ist}
Here, i is non-superstar firm, f is superstar firm, s is sector and t is time.
And alpha2 and alpha3 are the within estimator and alpha 4 and 5 are the between estimators.
In stata, my code would be:

xtreg dlog_lp frontier_growth lagged_gap c_nb_patent c_[FirmControlVars] m_nb_patent m_[FirmControlVars] i.sector i.year, i(firm) re

In more detail, there is coefficient (1-alpha0) since the starting equation was as follows:
log_LP_{ist} = alpha0*log_LP_{ist-1} + alpha1*log_LP_{fst} + alpha2*log_LP_{fst-1} + alpha3*X_{ist} + i.sector + i.year + error_term_{ist}.

In addition, I assume that number of corporate patent (nb_patent) would depend on the number of patent(s) generated by university nearby (nb_univ_patent), therefore nb_univ_patent is correlated with the error term.
In order to incorporate this, I would use instrumental variable in the equation (2) as follows:
nb_patent_{ist} = nb_univ_patent_{ist} + error_term_{ist}

But the problem is that I am not really sure of performing this instrumental variable approach in this setup.
I have tried to find some relevant references and stata FAQ, but I am still unsure.
My question is:
for the equation (2), if I would like to incorporate this in (1), I would like to verify if the equation would be:
nb_patent_{ist} - mean(nb_patent_{is,t}) = beta1*(nb_univ_patent_{ist} - mean(nb_univ_patent_{is})) + error_term_{ist}

 xtivreg c_nb_patent c_nb_univ_patent, vce(cluster firm)

for the within estimator one and
mean(nb_patent_{is,t}) = gamma1*mean(nb_univ_patent_{is}) + error_term_{ist}
for the between estimator one.

 xtivreg m_nb_patent m_nb_univ_patent, vce(cluster firm)

Would it be incorrect ?
(maybe of course, it's wrong..) I am sorry for the stupid question but I thought that since it comprises also the stata code, could someone here advice on the equation/code, please ?

Thank you in advance,
AC

[Anne-Claire Jo]

Or anyone has any advice on using IV in mixed effects model ?
I am wondering whether I could just use IV on fixed effects (c_var) or I should use them on random effects (m_var).