  The available data on $F_L$ suggest the existence of unexpected large higher
twist contributions. We use the $1/N_f$ expansion to analyze the renormalon
contribution to the coefficient function of the longitudinal structure function
$F_L^{p-n}$. The renormalon ambiguity is calculated for all moments of the
structure function thus allowing to estimate the contribution of ``genuine''
twist-4 corrections as a function of Bjorken-$x$. The predictions turn out to
be in surprisingly good agreement with the experimental data.
