  In rare-earth semiconductors, wide ligand $p$ and rare-earth 5$d$ bands
coexist with localized, partially filled 4$f$ shells. A simultaneous
description for both extended and localized states represents a significant
challenge for first-principles theories. Here, we combine an {\it ab initio}
dynamical mean-field theory approach to strong local correlations with a
perturbative application of the semi-local modified Becke-Johnson exchange
potential to correct the semiconducting gap. We apply this method to calculate
the electronic structure and optical response of the light rare-earth
sesquioxides RE$_2$O$_3$ (RE= La, Ce, Pr and Nd). Our calculations correctly
capture a non-trivial evolution of the optical gap in RE$_2$O$_3$ due to a
progressive lowering of the 4$f$ states along the series and their multiplet
structure. 2$p$ $-$ 4$f$ hybridization is found to induce a substantial upward
shift for the occupied 4$f$ states occurring within the $p-d$ gap, thus
reducing the magnitude of the optical gap. We show that a characteristic
plateau observed in the optical conductivity in the Pr and Nd sequioxides right
above their absorption edge is a fingerprint of 4$f$ states located within the
$p-d$ gap.
