A companion paper presents an algorithm for estimating the actions of orbits in axisymmetric potentials. This algorithm is fast enough for it to be feasible to fit automatically a parametrized distribution function to observational data for the solar neighbourhood. We explore the predictive power of these models and the extent to which global models are constrained by data confined to the solar cylinder. We adopt a gravitational potential that is generated by three discs (gas and both thin and thick stellar discs), a bulge and a dark halo, and fit the thin-disc component of the distribution function to the solar-neighbourhood velocity distribution from the Geneva–Copenhagen survey. We find that the disc's vertical density profile is in good agreement with data at z ≲ 500 pc. The thick-disc component of the distribution function is then used to extend the fit to data from Gilmore & Reid for z ≲ 2.5 kpc. The resulting model predicts excellent fits to the profile of the vertical velocity dispersion σz(z) from the RAVE survey and to the distribution of vφ velocity components at |z| ∼ 1 kpc from the Sloan Digital Sky Survey. The ability of this model to predict successfully data that were not used in the fitting process suggests that the adopted gravitational potential (which is close to a maximum-disc potential) is close to the true one. We show that if another plausible potential is used, the predicted values of σz are too large. The models imply that in contrast to the thin disc, the thick disc has to be hotter vertically than radially, a prediction that it will be possible to test in the near future. When the model parameters are adjusted in an unconstrained manner, there is a tendency to produce models that predict unexpected radial variations in quantities such as scale height. This finding suggests that to constrain these models adequately one needs data that extends significantly beyond the solar cylinder. The models presented in this paper might prove useful to the interpretation of data for external galaxies that have been taken with an integral field unit.