A predictive component can contribute to the command signal for smooth pursuit. This is readily demonstrated by the fact that low frequency sinusoidal target motion can be tracked with zero time delay or even with a small lead. The objective of this study was to characterize the predictive contributions to pursuit tracking more precisely by developing analytical models for predictive smooth pursuit. Subjects tracked a small target moving in two dimensions. In the simplest case, the periodic target motion was composed of the sums of two sinusoidal motions (SS), along both the horizontal and the vertical axes. Motions following the same or similar paths, but having a richer spectral composition, were produced by having the target follow the same path but at a constant speed (CS), and by combining the horizontal SS velocity with the vertical CS velocity and vice versa. Several different quantitative models were evaluated. The predictive contribution to the eye tracking command signal could be modeled as a low-pass filtered target acceleration signal with a time delay. This predictive signal, when combined with retinal image velocity at the same time delay, as in classical models for the initiation of pursuit, gave a good fit to the data. The weighting of the predictive acceleration component was different in different experimental conditions, being largest when target motion was simplest, following the SS velocity profiles.