This paper presents a linear analysis on an instability that is driven by the electron temperature gradient and is localized by the density pedestal at the edge. We find that the instability appears only in H-mode plasmas and can survive the observed levels of sheared plasma rotation because of its narrow radial width. For realistic plasma profiles, the mode frequency, growth time, poloidal mode number, and direction of rotation are all comparable to those of the precursors for Type III (small) edge-localized modes (ELMs) observed in DIII-D and ASDEX. We conclude that the instability is a promising candidate for the precursors.