  Electronic structure of layered van der Waals ferromagnet CrI$_{3}$ is
studied with self consistent diagrammatic approaches beyond GW approximation.
Considerable improvement in the calculated band gap as compared to the
non-self-consistent G0W0 results has been found. Certain spectral features in
the valence bands discovered recently by the angle resolved photoemission
spectroscopy, are reproduced better when we use full frequency dependent self
energy. Density functional theory and quasiparticle self-consistent GW method
which are based on frequency-independent self energy are unable to resolve
these features. Non-locality effects in the diagrams beyond GW approximation
are large for both polarizability and self energy. This finding can potentially
have an impact on the development of methods like GW+DMFT.
