  We give a general result on the effective degrees of freedom for nonlinear
least squares estimation, which relates the degrees of freedom to the
divergence of the estimator. We show that in a general framework, the
divergence of the least squares estimator is a well defined but potentially
negatively biased estimate of the degrees of freedom, and we give an exact
representation of the bias. This implies that if we use the divergence as a
plug-in estimate of the degrees of freedom in Stein's unbiased risk estimate
(SURE), we generally underestimate the true risk. Our result applies, for
instance, to model searching problems, yielding a finite sample
characterization of how much the search contributes to the degrees of freedom.
Motivated by the problem of fitting ODE models in systems biology, the general
results are illustrated by the estimation of systems of linear ODEs. In this
example the divergence turns out to be a useful estimate of degrees of freedom
for $\ell_1$-constrained models.
