  We demonstrate that the processes underlying on-line auction price bids and
many other longitudinal data can be represented by an empirical first order
stochastic ordinary differential equation with time-varying coefficients and a
smooth drift process. This equation may be empirically obtained from
longitudinal observations for a sample of subjects and does not presuppose
specific knowledge of the underlying processes. For the nonparametric
estimation of the components of the differential equation, it suffices to have
available sparsely observed longitudinal measurements which may be noisy and
are generated by underlying smooth random trajectories for each subject or
experimental unit in the sample. The drift process that drives the equation
determines how closely individual process trajectories follow a deterministic
approximation of the differential equation. We provide estimates for
trajectories and especially the variance function of the drift process. At each
fixed time point, the proposed empirical dynamic model implies a decomposition
of the derivative of the process underlying the longitudinal data into a
component explained by a linear component determined by a varying coefficient
function dynamic equation and an orthogonal complement that corresponds to the
drift process. An enhanced perturbation result enables us to obtain improved
asymptotic convergence rates for eigenfunction derivative estimation and
consistency for the varying coefficient function and the components of the
drift process. We illustrate the differential equation with an application to
the dynamics of on-line auction data.
