I have a linear panel data set (murder.dta, standard STATA dataset). First I estimate a first difference model. An assumption from this model is that the first differences of the covariates and the first differences of the individual time-varying error terms are uncorrelated.
My question is, under what circumstances would the first difference of my covariate not be exogenous in the model (first differences) I estimated?
(I hope my question is clearly formulated.)
The first difference of your covariate will not be exogenous in the first-differenced model if, e.g., the original model has a lagged dependent variable as a regressor. Consider e.g.
$$y_{it}=\gamma y_{i,t-1}+x_{it}'\beta+\mu_i+v_{it}, \quad t=1, \ldots, T,$$ where $v_{it}$ is independent of $\mu_i$ and any $x_{is}$. Also assume that $v_{it}$ is independent of $y_{i,t-1}, y_{i,t-2}, \ldots, y_{i0}$.
The model in first differences is $$\Delta y_{it}=\gamma \Delta y_{i,t-1}+\Delta x_{it}'\beta+\Delta v_{it}.$$
Clealry, $$cov(\Delta y_{i,t-1}, \Delta v_{it})=-cov(y_{i,t-1},v_{i,t-1})=-Var[v_{i,t-1}]\neq 0$$ in general.