  We report constraints on the global $21$ cm signal due to neutral hydrogen at
redshifts $14.8 \geq z \geq 6.5$. We derive our constraints from low foreground
observations of the average sky brightness spectrum conducted with the EDGES
High-Band instrument between September $7$ and October $26$, $2015$.
Observations were calibrated by accounting for the effects of antenna beam
chromaticity, antenna and ground losses, signal reflections, and receiver
parameters. We evaluate the consistency between the spectrum and
phenomenological models for the global $21$ cm signal. For tanh-based
representations of the ionization history during the epoch of reionization, we
rule out, at $\geq2\sigma$ significance, models with duration of up to $\Delta
z = 1$ at $z\approx8.5$ and higher than $\Delta z = 0.4$ across most of the
observed redshift range under the usual assumption that the $21$ cm spin
temperature is much larger than the temperature of the cosmic microwave
background (CMB) during reionization. We also investigate a `cold' IGM scenario
that assumes perfect Ly$\alpha$ coupling of the $21$ cm spin temperature to the
temperature of the intergalactic medium (IGM), but that the IGM is not heated
by early stars or stellar remants. Under this assumption, we reject tanh-based
reionization models of duration $\Delta z \lesssim 2$ over most of the observed
redshift range. Finally, we explore and reject a broad range of Gaussian models
for the $21$ cm absorption feature expected in the First Light era. As an
example, we reject $100$ mK Gaussians with duration (full width at half
maximum) $\Delta z \leq 4$ over the range $14.2\geq z\geq 6.5$ at $\geq2\sigma$
significance.
