\begin{equation}
y_{it} = y_{i\left(t-1\right)}\beta_1 + x_{it}\beta_2 + \alpha_i + \varepsilon_{it}
\end{equation}
For the model to give you consistent estimates in the Arellano-Bond case you need to satisfy that:
\begin{equation}
E\left[y_{i\left(t-2\right)}\Delta \varepsilon_{it}\right] = 0
\end{equation}
For the previous condition to be satisfied the first differenced errors should have a correlation of order one (or the undifferenced errors should be uncorrelated) and the correlations of higher order should be zero. You can test this using the estat abond postestimation command after xtabond .
In other words:
\begin{eqnarray}
E\left(\Delta \varepsilon_{it} \Delta \varepsilon_{i\left(t-1\right)}\right) &\neq& 0 \\
E\left(\Delta \varepsilon_{it} \Delta \varepsilon_{i\left(t-2\right)}\right) &=& 0
\end{eqnarray}
