  The third moment $d_2$ of the twist-3 part of the nucleon spin structure
function $g_2$ is generalized to arbitrary momentum transfer $Q^2$ and is
evaluated in heavy baryon chiral perturbation theory (HBChPT) up to order
${\mathcal{O}}(p^4)$ and in a unitary isobar model (MAID). We show how to link
$d_2$ as well as higher moments of the nucleon spin structure functions $g_1$
and $g_2$ to nucleon spin polarizabilities. We compare our results with the
most recent experimental data, and find a good description of these available
data within the unitary isobar model. We proceed to extract the twist-4 matrix
element $f_2$ which appears in the $1/Q^2$ suppressed term in the twist
expansion of the spin structure function $g_1$ for proton and neutron.
