# Recovering fixed effects after Arrellano-Bond estimation in panel data

I have a data of 74 countries and 15 time periods and want to estimate a dynamic panel model like

$$Y_{i,t} = Y_{i,t-1} + Z_{i,t} + a_i + e_i$$

where $a_i$ is unobserved heterogeneity.

The main aim of my estimation is to compute this fixed effects and then to explain the differences in fixed effects by institutional factors.

So I have read a lot about dynamic panel estimation and the literature suggests that if there is lagged dependent variable and the time periods are small the estimates are biased when using fixed effects estimation, one solution is Arellano-Bond estimation but it uses differenced data!

My question is "Can I recover fixed effects after estimating the equation by Arrellano-Bond technique?"

For simplicity, consider $y_{i,t} = \rho y_{i,t-1} + a_{i} + e_{i,t}$. Let $\hat{\rho}$ be the Arellano-Bond or any consistent estimator of $\rho$. We have $$\hat{e}_{i,t} \equiv y_{i,t} - \hat{\rho}y_{i,t-1} = y_{i,t} - \rho y_{i,t-1} + (\rho-\hat{\rho}) y_{i,t-1} = a_{i} + e_{i,t} + (\rho-\hat{\rho}) y_{i,t-1} = a_{i} + e_{i,t} + o_{p}(1).$$ A natural estimator of $a_{i}$ is to consider $\bar{e}_{i} \equiv T^{-1}\sum_{t=1}^{T}\hat{e}_{i,t} = a_{i} + T^{-1}\sum_{t=1}^{T} e_{i,t} + o_{p}(1)$. This estimator does not converge to $a_{i}$ unless you have long panel (I mean large $T$). However, if your "institutional factors'' for explaining $a_{i}$ are uncorrelated with $e_{i,1}, \dots, e_{i,T}$, it is acceptable to use $\bar{e}_{i}$ as an estimate of $a_{i}$. To be specific, letting $z_{i}$ be a vector your "institutional factors" and considering the linear projection of $\bar{e}_{i}$ on $z_{i}$, the coefficients in the linear projection converges to ${Var(z_{i})}^{-1}Cov(z_{i}, \bar{e}_{i})={Var(z_{i})}^{-1}Cov(z_{i}, a_{i})$.