  We use the sky-average spectrum measured by EDGES High-Band ($90-190$ MHz) to
constrain parameters of early galaxies independent of the absorption feature at
$78$~MHz reported by Bowman et al. (2018). These parameters represent
traditional models of cosmic dawn and the epoch of reionization produced with
the 21cmFAST simulation code (Mesinger & Furlanetto 2007, Mesinger et al.
2011). The parameters considered are: (1) the UV ionizing efficiency ($\zeta$),
(2) minimum halo virial temperature hosting efficient star-forming galaxies
($T^{\rm min}_{\rm vir}$), (3) integrated soft-band X-ray luminosity ($L_{\rm
X\,<\,2\,keV}/{\rm SFR}$), and (4) minimum X-ray energy escaping the first
galaxies ($E_{0}$), corresponding to a typical H${\rm \scriptstyle I}$ column
density for attenuation through the interstellar medium. The High-Band spectrum
disfavors high values of $T^{\rm min}_{\rm vir}$ and $\zeta$, which correspond
to signals with late absorption troughs and sharp reionization transitions. It
also disfavors intermediate values of $L_{\rm X\,<\,2\,keV}/{\rm SFR}$, which
produce relatively deep and narrow troughs within the band. Specifically, we
rule out $39.4<\log_{10}\left(L_{\rm X\,<\,2\,keV}/{\rm SFR}\right)<39.8$
($95\%$ C.L.). We then combine the EDGES High-Band data with constraints on the
electron scattering optical depth from Planck and the hydrogen neutral fraction
from high-$z$ quasars. This produces a lower degeneracy between $\zeta$ and
$T^{\rm min}_{\rm vir}$ than that reported in Greig & Mesinger (2017a) using
the Planck and quasar constraints alone. Our main result in this combined
analysis is the estimate $4.5$~$\leq \log_{10}\left(T^{\rm min}_{\rm vir}/\rm
K\right)\leq$~$5.7$ ($95\%$ C.L.). We leave for future work the evaluation of
$21$~cm models using simultaneously data from EDGES Low- and High-Band.
