  The most recent measurement of the fine structure constant leads to a 2.4
$\sigma$ deviation in the electron anomalous magnetic moment $g_{e}$-2, while
the muon anomalous magnetic moment $g_\mu$-2 has a long standing 3.7 $\sigma$
deviation in the opposite direction. We show that these deviations can be
explained in a three-loop neutrino mass model based on an $E_6$ Grand Unified
Theory. We also study the impact such a model can have on the anomalous events
observed by the ANITA experiment and find an insufficient enhancement of the
event rate.
