  We examine possible phenomenological constraints for the joint evolution of
supermassive black holes (SMBH) and their host spheroids. We compare all the
available observational data on the redshift evolution of the total stellar
mass and star formation rate density in the Universe with the mass and
accretion rate density evolution of supermassive black holes, estimated from
the hard X-ray selected luminosity function of quasars and active galactic
nuclei (AGN) for a given radiative efficiency, \epsilon. We assume that the
ratio of the stellar mass in spheroids to the black hole mass density evolves
as (1+z)^{-\alpha}, while the ratio of the stellar mass in disks+irregulars to
that in spheroids evolves as (1+z)^{-\beta}, and we derive constraints on
\alpha, \beta and \epsilon. We find that \alpha>0 at the more than 4-sigma
level, implying a larger black hole mass at higher redshift for a given
spheroid stellar mass. The favored values for \beta are typically negative,
suggesting that the fraction of stellar mass in spheroids decreases with
increasing redshift. This is consistent with recent determinations that show
that the mass density at high redshift is dominated by galaxies with irregular
morphology. In agreement with earlier work, we constrain \epsilon to be between
0.04 and 0.11, depending on the exact value of the local SMBH mass density, but
almost independently of \alpha and \beta.
