This paper reviews the poor performance of standard econometric estimators for autoregressive (and dynamic) panel data applications. It suggests the use of a new estimator. In particular it considers the popular first difference method of moments estimator for highly autoregressive linear panel data models. The situation considered in one of a "typical" household or firm panel data set where there are relatively few time series observations but the cross section dimension is large. The new "system" estimator for such models relies on relatively mild restrictions on the initial conditions process. This estimator is shown to produce a considerable improvement in precision and in finite sample performance for typical economic panel data applications.