An alternative structure for adaptive linear prediction is proposed in which the adaptive filter is replaced by a cascade of inde- pendently adapting, low-order stages, and the prediction is generated by means of successive refinements. When the adaptation algorithm for the stages is LMS, the associated short filters are less affected by eigenvalue spread and mode coupling problems and display a faster convergence to their steady-state value. Experimental results show that a cascade of second-order LMS filters is capable of successfully modeling most input signals, with a much smaller MSE than LMS or lattice LMS predictors in the early phase of the adaptation. Other adaptation algorithms can be used for the single stages, whereas the overall computational cost remains linear in the number of stages, and very fast tracking is achieved.