The chief difficulty in applying Aitken estimators to large linear systems stems from the dimensionality of the inverse. Here the conjugate gradient algorithm is applied to the problem, leading to substantial savings in storage and some savings in time. Given that the information matrix is not computed, an inference procedure is developed which involves two easily computed statistics which straddle the conventionally estimated standard errors. The paper is intended to open the way for the application of more efficient estimation procedures to large econometric systems.