Testing

\begin{equation}
y_{it} = x_{it}'\beta_1 + y_{i(t-1)}\beta_2 + \alpha_i + \varepsilon_{it}
\end{equation}

As in the conventional linear panel data model the time-invariant unobserved component is related to the regressors but the strategy of taking first differences does not work because of an endogeneity problem. In particular:

\begin{eqnarray}
\Delta y_{it} = \Delta x_{it}'\beta_1 + \Delta y_{i(t-1)} + \Delta \varepsilon_{it}
\end{eqnarray}