  Using precise galaxy stellar mass function measurements in the COSMOS field
we determine the stellar-to-halo mass relationship (SHMR) using a parametric
abundance matching technique. The unique combination of size and highly
complete stellar mass estimates in COSMOS allows us to determine the SHMR over
a wide range of halo masses from $z\sim0.2$ to $z\sim5$. At $z\sim 0.2$ the
ratio of stellar-to-halo mass content peaks at a characteristic halo mass
$M_{\rm h} =10^{12} M_\odot$ and declines at higher and lower halo masses. This
characteristic halo mass increases with redshift reaching $M_{\rm h} =10^{12.5}
M_\odot$ at $z\sim2.3$ and remaining flat up to $z=4$. We considered the
principal sources of uncertainty in our stellar mass measurements and also the
variation in halo mass estimates in the literature. We show that our results
are robust to these sources of uncertainty and explore likely explanation for
differences between our results and those published in the literature. The
steady increase in characteristic halo mass with redshift points to a scenario
where cold gas inflows become progressively more important in driving
star-formation at high redshifts but larger samples of massive galaxies are
needed to rigorously test this hypothesis.
