  Solid inflation is an effective field theory of inflation in which isotropy
and homogeneity are accomplished via a specific combination of anisotropic
sources (three scalar fields that individually break isotropy). This results in
specific observational signatures that are not found in standard models of
inflation: a non-trivial angular dependence for the squeezed bispectrum, and a
possibly long period of anisotropic inflation (to drive inflation, the "solid"
must be very insensitive to any deformation, and thus background anisotropies
are very slowly erased). In this paper we compute the expected level of
statistical anisotropy in the power spectrum of the curvature perturbations of
this model. To do so, we account for the classical background values of the
three scalar fields that are generated on large (superhorizon) scales during
inflation via a random walk sum, as the perturbation modes leave the horizon.
Such an anisotropy is unavoidably generated, even starting from perfectly
isotropic classical initial conditions. The expected level of anisotropy is
related to the duration of inflation and to the amplitude of the squeezed
bispectrum. If this amplitude is close to its current observational limit (so
that one of the most interesting predictions of the model can be observed in
the near future), we find that a level of statistical anisotropy $\gtrsim 3\%$
in the power spectrum is to be expected, if inflation lasted $\gtrsim 20-30$
e-folds more than the final $50-60$ efolds required to generare the CMB modes.
We also comment and point out various similarities between solid inflation and
models of inflation where a suitable coupling of the inflaton to a vector
kinetic term $F^{2}$ gives frozen and scale invariant vector perturbations on
superhorizon scales.
