Fixed Effects Problem when regressing GDP per capita growth on lagged GDP per capita For my thesis I am studying the impact that economic sanctions can have on the GDP per capita growth rate of targeted countries. I am using panel data for 56 countries spanning a period of 23 years. I have also chosen to use a two-way fixed effects model. My dependant variable is GDP per capita growth rate, but one of my key independent variables is lagged GDP per capita. However I have read that because I am using a fixed effects method, having a 'GDP' term on both the LHS and RHS is innacurate? 
Can anyone please explain why this might be the case, and if it is infact a problem, how it can be circumvented?
 A: Consider a simple version of the model without regressors, which is enough to highlight the problem with fixed effects:
\begin{equation}
y_{im}=\alpha_i+\rho y_{i,m-1}+\eta_{im}\quad(m=1,\ldots,M;\;i=1,\ldots,n)
\end{equation}
Further assume that $E(\alpha_i\eta_{im})=0$ and $E(y_{i0}\eta_{im})=0$ for all $m$, as well as $E(\eta_{i,m}\eta_{i,h})=0$ as well as $E(\eta_{i,m}^2)=\sigma_{\eta}^2$.
Recursive substitution yields
\begin{eqnarray*}
y_{im}&=&\alpha_i+\rho(\alpha_i+\rho y_{i,m-2}+\eta_{i,m-1})+\eta_{i,m}\\
&=&\eta_{i,m}+\rho\eta_{i,m-1}+\rho^2\eta_{i,m-2}+\rho^3 y_{i,m-3}+\alpha_i+\rho\alpha_i+\rho^2\alpha_i\\
&=&\ldots\\
&=&\eta_{i,m}+\rho\eta_{i,m-1}+\rho^2\eta_{i,m-2}+\ldots+\rho^{m-1}\eta_{i,1}+\frac{1-\rho^m}{1-\rho}\alpha_i+\rho^my_{i0}
\end{eqnarray*}
Multiplying through with $\eta_{ih}$ and taking expectations on both sides gives
\begin{eqnarray}
E(y_{im}\eta_{ih})&=&E(\eta_{im}\eta_{ih})+\rho E(\eta_{i,m-1}\eta_{ih})+\rho^2E(\eta_{i,m-2}\eta_{ih})+\ldots\notag\\
&&\quad+\rho^{m-1}E(\eta_{i,1}\eta_{ih})+\frac{1-\rho^m}{1-\rho}E(\alpha_i\eta_{ih})+\rho^mE(y_{i0}\eta_{ih})\notag\\
&=&\begin{cases} 0 &\text{for }h>m\\
\rho^{m-h}\sigma_{\eta}^2&\text{for }h=1,\ldots,m\end{cases}\label{yeta}
\end{eqnarray}
Hence, the regressor $y_{i,m-1}$ will not be uncorrelated with all idiosyncratic errors $\eta_{ih}$, which is the key orthogonality condition for consistency of fixed effects.
Have a look for the Arellano-Bond estimator as an alternative estimation strategy.
