# Estimator for Dynamic Panels with Individual Specific Slopes

I'm working on some economic stuff and the objective is to conduct a panel data analysis. I assumed the following data-generating process: $$$$y_{it} - y_{i,t-1} = \eta z_{i,t-1} + \gamma_i x_{i,t-1} + \beta_i y_{i,t-1} + \delta_t + \alpha_i + u_{it}$$$$ where $$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?

• If $\beta$ and $\gamma$ are meant to vary across $i$, there is barely anything in the model that is still common across $i$. Just the time trend. You could try fitting the model separately for every unit instead of thinking of it as a panel (which is based on the idea that an important part of a relationship between variables is the same for the units, thus making it a good idea to pool the data when estimating it) Dec 12, 2023 at 21:31
• You are right. Let me restate the question by allowing for something more reasonable in a panel data framework. How can I deal with a model like the one above? Dec 12, 2023 at 22:52

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).