  We study the connection between the observed star formation rate-stellar mass
(SFR-$M_*$) relation and the evolution of the stellar mass function (SMF) by
means of a subhalo abundance matching technique coupled to merger trees
extracted from a N-body simulation. Our approach consist of forcing the model
to match the observed SMF at redshift $z \sim 2.3$, and let it evolve down to
$z \sim 0.3$ according to a $\tau$ model, an exponentially declining functional
form which describes the star formation rate decay of both satellite and
central galaxies. In this study, we use three different sets of SMFs: ZFOURGE
data from Tomczak et al.; UltraVISTA data from Ilbert et al. and COSMOS data
from Davidzon et al. We also build a mock survey combining UltraVISTA with
ZFOURGE. Our modelling of quenching timescales is consistent with the evolution
of the SMF down to $z \sim 0.3$, with different accuracy depending on the
particular survey used for calibration. We tested our model against the
observed SMFs at low redshift and it predicts residuals (observation versus
model) within $1\sigma$ observed scatter along most of the stellar mass range
investigated, and with mean residuals below 0.1 dex in the range $\sim
[10^{8.7}-10^{11.7}] M_{\odot}$. We then compare the SFR-$M_*$ relation
predicted by the model with the observed one at different redshifts. The
predicted SFR-$M_*$ relation underpredicts the median SFR at fixed stellar mass
relative to observations at all redshifts. Nevertheless, the shapes are
consistent with the observed relations up to intermediate-mass galaxies,
followed by a rapid decline for massive galaxies.
