I'm working on some economic stuff and the objective is to conduct a panel data analysis. I assumed the following data-generating process:
\begin{equation}
y_{it} - y_{i,t-1} = \eta z_{i,t-1} + \gamma_i x_{i,t-1} + \beta_i y_{i,t-1} + \delta_t + \alpha_i + u_{it}
\end{equation}
where y_{it} = log (Y_{it}). Thus, x_{it} and z_{i,t} are two vectors of observable predictors, \delta_t denotes time specific fixed-effects, \alpha_i denotes individual unobserved heterogeneity, with i=1,2,\ldots,N. I must allow for arbitrary correlation between \alpha_i and my predictors.
A general fixed-effects estimator with individual specific slopes is not feasible in this case because having the lagged dependent variable among the predictors would violate the strict exogeneity assumption. I need an instrument for y_{i,t-1}. The GMM should be the natural solution but I cannot find an estimator for dynamic panels allowing for heterogeneity in the slopes, i.e., \beta and \gamma vary across cross-sectional units i. Any suggestion?
\begin{equation}
y_{it} - y_{i,t-1} = \eta z_{i,t-1} + \gamma_i x_{i,t-1} + \beta_i y_{i,t-1} + \delta_t + \alpha_i + u_{it}
\end{equation}
where y_{it} = log (Y_{it}). Thus, x_{it} and z_{i,t} are two vectors of observable predictors, \delta_t denotes time specific fixed-effects, \alpha_i denotes individual unobserved heterogeneity, with i=1,2,\ldots,N. I must allow for arbitrary correlation between \alpha_i and my predictors.
A general fixed-effects estimator with individual specific slopes is not feasible in this case because having the lagged dependent variable among the predictors would violate the strict exogeneity assumption. I need an instrument for y_{i,t-1}. The GMM should be the natural solution but I cannot find an estimator for dynamic panels allowing for heterogeneity in the slopes, i.e., \beta and \gamma vary across cross-sectional units i. Any suggestion?
