  We calculate reduced moments $\overline \xi_q$ of the matter density
fluctuations, up to order $q=5$, from counts in cells produced by
Particle--Mesh numerical simulations with scale--free Gaussian initial
conditions. We use power--law spectra $P(k) \propto k^n$ with indices
$n=-3,~-2,~-1,~0,~1$. Due to the supposed absence of characteristic times or
scales in our models, all quantities are expected to depend on a single scaling
variable. For each model, the moments at all times can be expressed in terms of
the variance $\overline \xi_2$, alone. We look for agreement with the
hierarchical scaling ansatz, according to which $\overline \xi_q \propto
\overline \xi_2^{~q-1}$. For $n\leq -2$ models we find strong deviations from
the hierarchy, which are mostly due to the presence of boundary problems in the
simulations. A small, residual signal of deviation from the hierarchical
scaling is however also found in $n \geq -1$ models. For the first time, due to
our large dynamic range and careful checks of scaling and shot--noise effects,
we are able to detect evolution away from the perturbation theory result.
