# 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?

• can you give a brief description of other Independent variables including the computation or measurement procedures. – Subhash C. Davar Apr 4 '18 at 9:14
• what is your two way fixed efects model ? – Subhash C. Davar Apr 4 '18 at 9:16
• Y_it= a_i+ β_1 〖Sanction〗_(it )+ β_2 X_it+δ_t+ ε_it - The independent variable would be the sanction variable, and then the other control variables in x, are lagged gdp growth rate, lagged population and a lagged trade openness term which is calculated by the sum of imports + exports /GDP. I also have included controls for the level of democracy in the nation, and a dummy varible coded for 1 if the country was involved in a civil/interstate war at time t. The two way fixed effects model, includes country-fixed effects and year-fixed effects – Sam Jacobs Apr 7 '18 at 16:04
• Do you find any problems with the results for your fixed effects model ? – Subhash C. Davar Apr 7 '18 at 17:14
• You can edit your question - body text to show your model – Subhash C. Davar Apr 7 '18 at 17:16

Consider a simple version of the model without regressors, which is enough to highlight the problem with fixed effects: $$$$y_{im}=\alpha_i+\rho y_{i,m-1}+\eta_{im}\quad(m=1,\ldots,M;\;i=1,\ldots,n)$$$$ Further assume that $$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.