Estimator for Dynamic Panels with Individual Specific Slopes

Question

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 $y_{it} \equiv log (Y_{it})$. 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?

Answer 1

With the equation you wrote, it seems that you want to study some kind of conditional convergence in economics. You might consider estimating $N$ separate regressions and calculating the coefficient means, leading to the Mean Group (MG) estimator. Alternatively, you could pool the data and assume that the slope coefficients and error variances are identical. For an intermediate procedure, you might rely on the Pooled Mean Group (PMG) estimator, which constrains long-run coefficients to be identical but allows short-run coefficients and error variances to differ across groups (for more details, see Pesaran, Shin, Smith (1999, Journal of the American Statistical Association).