  We present a microscopic theory of the chiral one-dimensional electron gas
system localized on the sidewalls of magnetically-doped Bi$_2$Se$_3$-family
topological insulator nanoribbons in the quantum anomalous Hall effect (QAHE)
regime. Our theory is based on a simple continuum model of sidewall states
whose parameters are extracted from detailed ribbon and film geometry
tight-binding model calculations. In contrast to the familiar case of the
quantum Hall effect in semiconductor quantum wells, the number of microscopic
chiral channels depends simply and systematically on the ribbon thickness and
on the position of the Fermi level within the surface state gap. We use our
theory to interpret recent transport experiments that exhibit non-zero
longitudinal resistance in samples with accurately quantized Hall conductances.
