  We define a partition of ${\overline{M}_g^n}$ and show that the cohomology of
${\overline{M}_g^n}$ in a given degree admits a filtration whose respective
quotients are isomorphic to the shifted cohomology groups of the parts if $g$
is sufficiently large. This implies that the map $H^k({\overline{M}_g^n}) \ra
H^k(M_g^n)$ is onto and that the Hodge structure of $H^k(M_g^n)$ is pure of
weight $k$ if $g \geq 2k+1$. Our main ingredient is the stability theorem of
Harer and Ivanov.
