Consider the following statistical model for $i=1, 2,..., n;\ t=1,2,...,T$:
$y_{i,t}= c_i + \rho y_{i,t-1} + \beta x_{i,t} + \epsilon_{i,t}\ $
where $\epsilon_{i,t}\sim i.i.d\ $, $x_{i,t}\ $ is strictly exogenous and $c_i$ is not observable.
How to obtain consistent and unbiased estimators? With Random Effects assumptions?, with Fixed Effects assumptions? including cross-sectional dummies? Using instruments?
In dynamic panel data models, in which regressors include the lagged dependent, the unobservable $c_i$ is correlated with the regressor $y_ {i, t-1}$, so the Random Effects estimator does not work. One solution is to eliminate $c_i$.
The Fixed Effects (within transformation) estimator which eliminates $c_i$ is biased and inconsistent (see e.g. Verbeek, p. 361). Also, including dummies for cross-sectional observations and running a pooled OLS provide biased and inconsistent estimators, since those are equal to the FE estimators (see e.g. Davidson and MacKinnon (1999), p. 297).
The proposal of Arellano and Bond (1991) is to estimate the model in differences ($\Delta y_{i,t}$) in order to eliminate $c_i$ and then use instrumental variables.
The model in differences still has an endogeneity problem, even though $c_i$ has already been eliminated, because the regressor $\Delta y_{i,t-1} = (y_ {i, t-1} - y_ {i, t-2})$ is correlated with the error $\Delta \epsilon_{i,t} = (\epsilon_ {i, t} - \epsilon_ {i, t-1})$. Nevertheless, with $\epsilon_{i,t}$ that are i.d.d it is true that $E[(\epsilon_{i,t}-\epsilon_{i,t-1})(y_{i,t-2})]=0$ which suggests an instrumental variables approach. Then the AB estimator uses the lags from $t-2$ as the set of potential valid instruments for $\Delta y_{i,t-1}$. If $x_{i,t}$ were not strictly exogenous but only sequentially exogenous, its lags would also serve as potential valid instruments.
The validity of these instruments, nonetheless, do require that the error terms are not serially correlated. The Arellano and Bond's (1991) m-statistic can be used to test for residual serial correlation and a test of over-identifying restrictions can be done with the Sargan statistic.
If $\rho \rightarrow 1$, one can use the orthogonal deviations proposed by Arellano and Bover (1995) to eliminate $c_i$.