  Identifying Active Galactic Nuclei (AGNs) through their X-ray emission is
efficient, but necessarily biased against X-ray-faint objects. We aim to
characterize this bias by comparing X-ray-selected AGNs to the ones identified
through optical variability and mid-IR colours. We present a catalogue of AGNs
selected through optical variability using all publicly available z-band Hubble
Space Telescope images in the GOODS-South field. For all objects in the
catalogue, we compute X-ray upper limits or discuss detections in the deepest
available 7 Ms Chandra Deep Field South images and present the Spitzer/IRAC
mid-IR colours. For the variability study, we consider only sources observed
over at least five epochs and over a time baseline of up to ten years. We adopt
the elevated median absolute deviation as a variability indicator robust
against individual outlier measurements and identify 113 variability-selected
AGN candidates. Among these, 26 have an X-ray counterpart and lie within the
conventional AGN area in the FX/Fopt diagram. The candidates with X-ray upper
limits are on average optically fainter, have higher redshifts compared to the
X-ray detected ones and are consistent with low luminosity AGNs. Out of 41
variable optical sources with IR detections, 13 fulfill the IR AGN colour
selection criteria. Our work emphasizes the importance of optical variability
surveys for constructing complete samples of AGNs including the ones that
remain undetected even by the deepest X-ray and IR surveys.
